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# Polarized currents in Coulomb blockade and Kondo regimes without magnetic fields Anh T. Ngo Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena Institute, Ohio University, Athens, Ohio 45701-2979 Edson Vernek Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena Institute, Ohio University, Athens, Ohio 45701-2979 Instituto de Física - Universidade Federal de Uberlândia, Uberlândia, MG 38400-902, Brazil Sergio E. Ulloa Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena Institute, Ohio University, Athens, Ohio 45701-2979 ###### Abstract We present studies of the Coulomb blockade and Kondo regimes of transport through a quantum dot connected to current leads through spin-polarizing quantum point contacts (QPCs). This structure, arising from the effect of lateral spin-orbit fields defining the QPCs, results in spin-polarized currents even in the absence of external magnetic fields and greatly affects the correlations in the dot. Using equation-of-motion and numerical renormalization group calculations we obtain the conductance and spin polarization for this system under different parameter regimes. We find that the system exhibits spin-polarized conductance in both the Coulomb blockade and Kondo regimes, all in the absence of applied magnetic fields. We analyze the role that the spin-dependent tunneling amplitudes of the QPC play in determining the charge and net magnetic moment in the dot. These effects, controllable by lateral gate voltages, may provide an alternative approach for exploring Kondo correlations, as well as possible spin devices. ###### pacs: 72.15.Qm, 72.25.-b, 72.10.-d, 73.23.Hk ## I Introduction Electronic transport in semiconducting nanostructures is studied both as it has great potential applications in spintronics, 5 and because of its exquisite control of parameters, which allow insightful probes into fundamental physical phenomena. Electrons in such systems experience externally controlled confining environments that result in strong Coulomb interactions with other electrons. As such, quantum dot (QD) structures provide well-characterized and defined systems for studying quantum many-body physics. QDs may also allow the realization of solid state quantum computation devices as well as spintronic semiconductor devices with unprecedented functionalities. QDref Manipulation of spin-polarized current sources is crucial in spintronics. This typically requires efficient spin injection into conventional semiconductors. The difficulties with spin injection from ferromagnetic metal leads has stimulated extensive efforts to produce spin polarized currents out of unpolarized sources. Schmidt In this context, the Rashba spin-orbit (RSO) coupling mechanism provides a basis for possible device applications. Rashba This coupling, arising from interfacial structure asymmetries, depends on the materials used as well as on the confinement geometry of the structures. 6 ; 9 Most interestingly, the Rashba effect allows external tunability, which has been studied experimentally in QDs 7 and quantum point contacts. Bird ; 8 In this paper we study the electronic transport through a quantum dot connected to polarizing quantum point contacts (QPCs) in both the Coulomb blockade (CB) and Kondo regimes. Due to strong spin-orbit interactions, 8 ; 9 QPCs can exhibit spin-dependent hybridization of the QD states with the leads, without applied magnetic fields, opening the possibility for generating spin- polarized transport in an all-electrical setup. These effects are controllable by lateral gate voltages applied on QPCs, resulting in spatially asymmetric structures, as in recent experiments.8 Using the equation-of-motion technique and numerical renormalization group (NRG) calculations we obtain the electronic Green’s function, conductance and spin polarization in different system regimes. Our results show that both the CB and Kondo regimes exhibit non-zero spin-polarized conductance in this system. We analyze how the spin- dependent hybridization of the QPC modifies the charge accumulation in the dot, as well as the density of states (spectral functions) of the system. Interestingly, we find that the polarizing QPCs produce spin polarization and split DOS in the Kondo regime akin to that reported for current injection from ferromagnetic leads, Martinek although here it occurs for unpolarized reservoirs. Our theoretical studies suggest that these effects could be accessible in experiments and result in future spintronic devices. The paper is organized as follows. First, we present the main features of the current polarization in quantum point contact systems with lateral spin-orbit, obtained by scattering matrix methods in section II. The next section describes the model of a quantum dot connected to current leads via polarizing QPCs. In section III.1 we use the equation-of-motion (EOM) technique to obtain the dot Green’s function and discuss numerical results in both the CB and Kondo regimes. In section III.2 we revisit the problem using the numerical renormalization group approach, which provides the most reliable information in the Kondo regime for realistic system parameters. The paper concludes with final remarks and summary in section IV. ## II Polarizing QPC The effective electric field in the $z$ direction that creates a two- dimensional electron gas (2DEG) confined to the $x$-$y$ plane results in the well-known Rashba spin orbit interaction, Rashba ; 6 $\displaystyle H_{SO}^{R}=\frac{\alpha}{\hbar}\left(\sigma_{x}P_{y}-\sigma_{y}P_{x}\right).$ (1) Here, $\sigma_{x}$ and $\sigma_{y}$ are Pauli matrices, and $\alpha$ is Rashba spin orbit coupling constant, which is proportional to the field and is therefore material and structure dependent. Electrons on this 2DEG entering a quantum dot, pass through QPCs defined via a confining potential $V(x,y)$ which can be thought as made of two parts: $U(y)+V_{b}(x,y)$, where $U(y)$ defines a hard-wall potential of width $W$ (related in the experiment to the side-wall etching defining the structure) outlining the overall channel structure, while $V_{b}(x,y)$ is the potential generating the QPC barrier, and effectively modulated by the side gate potentials of the structure. 8 We model such barrier by 12 $\displaystyle V_{b}(x,y)=\frac{1}{2}V_{g}\left(1+\cos\frac{\pi x}{L_{x}}\right)+\frac{1}{2}m\omega^{2}\bar{y}^{2}\Theta(\bar{y})$ (2) with $\bar{y}=y-y_{s}$, and $\displaystyle y_{s}=W_{1}\left(1-\cos\frac{\pi x}{L_{x}}\right)\,,$ (3) where $\Theta(x)$ is the step function, $m$ is the effective mass of the electrons, $L_{x}$ is the unit length of the structure in the $x$ direction (along the current direction) and $\omega$ is the confinement potential frequency. Notice that this potential form is asymmetric in the $y$ direction, to reflect an essential ingredient in the experiments: the QPC potential must lack $y$-reflection symmetry in order to generate the polarizing effect along the $z$-direction. 12 ; 12b ; 12c In fact, the fields forming $V_{b}$ generate a spin-orbit coupling given by 6 $\displaystyle V_{SO}^{\beta}=-\frac{\beta}{\hbar}\nabla V_{b}\cdot(\mathbf{\hat{\sigma}}\times\mathbf{\hat{P}}),$ (4) where $\beta$ is material-specific. Notice that $\nabla V_{b}$ lies in the $x$-$y$ plane, so that the barrier fields induce a lateral spin-orbit coupling. The total Hamiltonian of the QPC will then be given by, $\displaystyle H=\frac{P_{x}^{2}+P_{y}^{2}}{2m}+H_{SO}^{R}+V(x,y)+V_{SO}^{\beta}.$ (5) Figure 1: (color online) Total and spin-dependent conductances for an asymmetric QPC as function of Rashba spin-orbit coupling $\alpha$, obtained from a scattering matrix approach.9 The interplay of vertical and lateral SO effects may result in large asymmetry for the up and down spin components for realistic structure parameters, even in the tunneling regime shown here ($G_{\sigma}\ll 1$). Using a scattering-matrix formalism to study the spin-dependent electron transport in this QPC, 11 ; 9 we find that a net spin-polarized conductance is produced only for $y$-asymmetric potentials. 9 We can calculate the conductance of the structure, assuming that the SO coupling $\alpha$ and $V(x,y)$ are zero at the source and drain 2DEG reservoirs, while both of these terms in the Hamiltonian are turned on in the QPC region. Typical structure parameters in experiments can be cast in terms of characteristic length and energy scales, $L_{0}=32.5$ nm, and $E_{0}=3.12$ meV, with $\alpha_{0}=E_{0}L_{0}=1.0\times 10^{-12}$ eVm, a typical value of spin-orbit coupling. Using $W=L_{x}=2L_{0}$ and $W_{1}=0.6L_{0}$ as width/length of the confining potential, with $\omega=6\times 10^{13}$s-1, and $\beta=0.97\times 10^{-16}$m2, gives results as shown in Fig. 1. This figure shows spin- dependent conductances as functions of Rashba coupling $\alpha$, for a potential barrier which is near its conduction onset (or “pinch-off”, as controlled by the value of $V_{g}$); arrows indicate the results of spin up and down conductances. We see that conductances $G_{\uparrow}$ and $G_{\downarrow}$ can be very different from each other, even in the tunneling regime (where each $G_{\sigma}\ll 1$) in which the QPC would operate to create a quantum dot in the 2DEG. We stress that for these realistic values of structure parameters, one obtains non-zero spin polarization even when there is no external magnetic field and the injection is unpolarized. This interesting result can be understood from anticrossing features in the subband energy structure in the channel region defining the QPC. 8 ; 9 The spin mixings and avoided crossings generate spin rotation as electrons pass through the narrow constriction of the QPC, and can generate large values of the ratio $G_{\uparrow}/G_{\downarrow}$, even in the tunneling regime. Two of these QPCs can then be used to define the QD and result in interesting charging and conductance regimes, as we will see below. ## III Quantum dot with polarizing QPCs In order to address the transport through a quantum dot formed with polarizing QPCs, we consider the single impurity Anderson model given by the following Hamiltonian: $\displaystyle H=\sum_{\ell k\sigma}\varepsilon_{\ell k}c^{\dagger}_{\ell k\sigma}c_{\ell k\sigma}+\sum_{\sigma}\varepsilon_{d}c^{\dagger}_{d\sigma}c_{d\sigma}+Un_{d\uparrow}n_{d\downarrow}$ $\displaystyle+\sum_{\ell k\sigma}t_{\sigma}(c^{\dagger}_{d\sigma}c_{\ell k\sigma}+c^{\dagger}_{\ell k\sigma}c_{d\sigma}),$ (6) where $c^{\dagger}_{d\sigma}(c_{d\sigma})$ is the creation (annihilation) operator of an electron of spin $\sigma$ in the dot. The quantities $\varepsilon_{\ell k}$, $\varepsilon_{d}$ are the energies of the electrons in the $\ell^{\it th}$ conduction band channels ($\ell=L,R$) and the single local energy level in the dot, respectively. $U$ is the Coulomb repulsion between electrons occupying the QD with $n_{d\sigma}=c^{\dagger}_{d\sigma}c_{d\sigma}$, while $t_{\sigma}$ represents the lead-QD hybridization occurring via tunneling through the QPC, and which is assumed to be $k$-independent. The density of states for conduction electrons in each lead is taken to be constant, $\rho_{L}(\varepsilon)=\rho_{R}(\varepsilon)\equiv\rho=(1/2D)\Theta(D-|\varepsilon|)$, where $D$ is the conduction band halfwidth (hereafter taken as our energy unity). The theoretical description of such quantum dot system, especially in the strong correlations regime, has been greatly developed over the years. Mahan Techniques of note include quantum Monte-Carlo, 2 equations of motion for the Green’s functions,3 and the numerical renormalization group approach.costi In what follows, we explore the role that SO interactions play on the Coulomb blockade and Kondo regimes of transport of the QD, utilizing equations of motion and numerical renormalization group formalisms. ### III.1 Equation of motion approach and numerical results To calculate the charge and conductance of the system we calculate Green’s functions (GFs), which allow us to take into account the correlations induced by the Coulomb interaction in the QD. The retarded double-time Green’s functions are defined as ($\hbar=1$)13 $i\langle\langle A;B\rangle\rangle=\int_{-\infty}^{\infty}\langle[A(\tau),B(0)]_{+}\rangle\Theta(\tau)e^{-i\omega\tau}d\tau,$ (7) where $A$ and $B$ are generic fermionic operators, $[A,B]_{+}$ indicates their anticommutator and $\langle\cdots\rangle$ indicates the thermodynamic average for $T>0$, or the ground state expectation value for $T=0$. The GF can be obtained using equation of motion (EOM) techniques, so that $\displaystyle\omega\langle\langle A;B\rangle\rangle=\langle[A,B]_{+}\rangle+\langle\langle[A,H];B\rangle\rangle,$ (8) where $[A,B]$ represents a commutator. Iteration of this formula generates a hierarchy of expressions, starting with the local one-particle GF as $\left(\omega-\varepsilon_{d}-\sum_{k}\frac{\tilde{t}_{\sigma}^{2}}{\omega-\varepsilon_{k}}\right)\langle\langle c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle=1+U\langle\langle c_{d\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle,$ (9) where $\tilde{t}_{\sigma}=\sqrt{2}t_{\sigma}$ and $\bar{\sigma}=-\sigma$. The new (higher order) GF on the right hand side of Eq. (9) can also be determined from (8), giving $\displaystyle(\omega-\varepsilon_{d}-U)\langle\langle c_{d\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle=\langle n_{d\bar{\sigma}}\rangle+\tilde{t}_{\sigma}\sum_{k}\left(\langle\langle c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle-\langle\langle c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\langle\langle c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle\right).$ (10) #### III.1.1 Coulomb blockade regime Although the EOM in Eq. (10) is exact, a solution of the impurity GF requires a procedure to truncate and/or decouple the higher order terms appearing on the right hand side of (10). A solution that captures the Coulomb blockade physics is given by the Hubbard-I approximation:14 $\displaystyle\langle\langle c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\simeq$ $\displaystyle\langle n_{d\bar{\sigma}}\rangle\langle\langle c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\langle\langle c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\simeq$ $\displaystyle\langle c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}\rangle\langle\langle c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\langle\langle c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle\simeq$ $\displaystyle\langle c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}\rangle\langle\langle c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle,$ (11) which allows one to write $\displaystyle G_{d\sigma}(\omega)\equiv\langle\langle c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle_{\omega}=\frac{G^{0}_{d\sigma}(\omega)}{1-G^{0}_{d\sigma}(\omega)\tilde{t}^{2}_{\sigma}\tilde{g}(\omega)},$ (12) where $G^{0}_{d\sigma}(\omega)=\frac{1-\langle n_{d\bar{\sigma}}\rangle}{\omega-\varepsilon_{d}}+\frac{\langle n_{d\bar{\sigma}}\rangle}{\omega-\varepsilon_{d}-U}$ is the local GF in the “atomic” approximation (the exact result for $t_{\sigma}=0$), and $\tilde{g}(\omega)=\sum_{k}(\omega-\epsilon_{k})^{-1}$ is the non-interacting GF of the leads. The DOS of the system (proportional to the imaginary part of $G_{d\sigma}$) contains two Hubbard peaks of width proportional to $\Gamma_{\sigma}=\pi t^{2}_{\sigma}/D$, resulting in the broadening of the poles of $G^{0}_{d\sigma}$. The spectral weights of these peaks are controlled by the dot level occupancy with opposite spin, and caused by the Coulomb interaction in the dot. Notice that the SO-induced polarization of the QPC results in different peak widths for the different spins. The Hubbard-I approximation (11) is known to be valid for a large $U/\Gamma$ ratio, when the Hubbard subbands are well separated in energy scale. Mahan It is the simplest scheme which describes correlated electrons, although, since it ignores the Kondo effect, it is a reasonable description only at temperatures higher than the Kondo scale ($T\gg T_{K}$–see next section). Figure 2: (color online) Occupancies for spin-up $\langle n_{d\uparrow}\rangle$, spin-down $\langle n_{d\downarrow}\rangle$ and total spin $\langle n_{d\uparrow}\rangle+\langle n_{d\downarrow}\rangle$ vs. $\varepsilon_{d}$ at zero temperature. Parameters used are $\Gamma_{\uparrow}=0.06$, $\Gamma_{\downarrow}=0.03$, $U=0.3$, with $D=1$. Notice asymmetry in $\langle n_{d\sigma}\rangle$ on each side of the plateau. Figure 3: (color online) Spin-dependent conductance and polarization as function of $\varepsilon_{d}$ at zero temperature. The other parameters are as in Fig. 2. The occupancies of spin $\uparrow$ and $\downarrow$ are calculated self- consistently from the equation $\displaystyle\langle n_{d\sigma}\rangle=\int f(\omega)\left(-\frac{1}{\pi}\text{Im}\left[G_{d\sigma}(\omega)\right]\right)d\omega\,,$ (13) where $f(\omega)$ is the Fermi function. It is clear that when the QPCs are not polarizing, $t_{\uparrow}=t_{\downarrow}$, the occupancy curves for spin- up and down coincide and a plateau of width $\sim U$ appears when the QD level moves below the Fermi level ($\varepsilon_{d}+U\gtrsim E_{F}\gtrsim\varepsilon_{d}$). This situation changes when the QPCs are polarized (see Fig. 2), as the different $t_{\sigma}$ result in $\Gamma_{\uparrow}\neq\Gamma_{\downarrow}$, which in turn produce different $\langle n_{d\uparrow}\rangle$ and $\langle n_{d\downarrow}\rangle$, especially on both sides of the plateau. The zero-bias conductance is calculated using a Landauer formula generalized for interacting systems 15 $\displaystyle G_{\sigma}=\frac{e^{2}}{h}\Gamma_{\sigma}\int d\epsilon\frac{\partial f(\epsilon)}{\partial\epsilon}\,\text{Im}[G_{d\sigma}(\epsilon)],$ (14) for symmetric coupling of the leads for each spin. One can also calculate the polarization factor $\displaystyle\eta=\frac{G_{\uparrow}-G_{\downarrow}}{G_{\uparrow}+G_{\downarrow}},$ (15) which gives a measure of current polarization in the system. Figure 3 shows the spin-dependent conductances and polarization for the system in the Hubbard-I approximation. As $\langle n_{d\uparrow}\rangle\neq\langle n_{d\downarrow}\rangle$ in general (except at the particle-hole symmetry point, $\varepsilon_{d}=-U/2$), the conductance per spin $G_{\sigma}$ are also different. Notice that the spin-dependent conductance peaks are very asymmetric and non-Lorentzian, due to the peculiar behavior of the occupancies and their different up and down-spin couplings. As we consider here the case $\Gamma_{\uparrow}>\Gamma_{\downarrow}$, one clearly sees that generally $G_{\uparrow}>G_{\downarrow}$ over the entire range of $\varepsilon_{d}$ values. As a consequence, there is a net up-spin polarization ($\simeq 60$%) and conductance around the resonant peaks, with the latter reaching $\simeq 0.3(e^{2}/h)$. #### III.1.2 Kondo regime To study the low-temperature behavior of the system within the EOM we need to consider higher order GFs in Eq. (10). Using Lacroix’s approach, 3 one can obtain relations for the three GFs on the right side of (10) as: $\displaystyle(\omega-\varepsilon_{k\sigma})\langle\langle c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle=$ $\displaystyle\langle[c_{k\sigma}n_{d\bar{\sigma}};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c_{k\sigma}n_{d\bar{\sigma}};H];c^{\dagger}_{d\sigma}\rangle\rangle$ (16) $\displaystyle=$ $\displaystyle\tilde{t}_{\sigma}\langle\langle n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\tilde{t}_{\bar{\sigma}}\sum_{k^{\prime}}[\langle\langle c_{k\sigma}c^{\dagger}_{d\bar{\sigma}}c_{k^{\prime}\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle-\langle\langle c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{d\bar{\sigma}}c_{k\sigma};c^{\dagger}_{d\sigma}\rangle\rangle],$ $\displaystyle(\omega-\varepsilon_{d\sigma}+\varepsilon_{d\bar{\sigma}}-\varepsilon_{k\bar{\sigma}})\langle\langle c_{k\bar{\sigma}}c^{\dagger}_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle=\langle[c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};H];c^{\dagger}_{d\sigma}\rangle\rangle$ (17) $\displaystyle=\langle c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}\rangle+\tilde{t}_{\bar{\sigma}}\langle\langle n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\sum_{k^{\prime}}[-\tilde{t}_{\bar{\sigma}}\langle\langle c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{k\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle+\tilde{t}_{\sigma}\langle\langle c^{\dagger}_{d\bar{\sigma}}c_{k\bar{\sigma}}c_{k^{\prime}\sigma};c^{\dagger}_{d\sigma}\rangle\rangle],$ and $\displaystyle(\omega-\varepsilon_{d\sigma}-\varepsilon_{d\bar{\sigma}}+\varepsilon_{k\bar{\sigma}}-U)\langle\langle c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle=\langle[c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}]_{+}\rangle+\langle\langle[c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}c_{d\sigma};H];c^{\dagger}_{d\sigma}\rangle\rangle$ (18) $\displaystyle=\langle c^{\dagger}_{k\bar{\sigma}}c_{d\bar{\sigma}}\rangle-\tilde{t}_{\bar{\sigma}}\langle\langle n_{d\bar{\sigma}}c_{d\sigma};c^{\dagger}_{d\sigma}\rangle\rangle+\sum_{k^{\prime}}[\tilde{t}_{\sigma}\langle\langle c^{\dagger}_{k^{\prime}\bar{\sigma}}c_{d\bar{\sigma}}c_{k^{\prime}\sigma};c^{\dagger}_{d\sigma}\rangle\rangle$ $\displaystyle-\tilde{t}_{\bar{\sigma}}\langle\langle c^{\dagger}_{k\bar{\sigma}}c_{d\sigma}c_{k^{\prime}\bar{\sigma}};c^{\dagger}_{d\sigma}\rangle\rangle].$ Following the decoupling procedure in Ref. 3, , each GF of the type $\langle\langle A^{*}BC,D^{*}\rangle\rangle$ is replaced by $\displaystyle\approx\langle A^{*}B\rangle\langle\langle C,D^{*}\rangle\rangle-\langle A^{*}C\rangle\langle\langle B,D^{*}\rangle\rangle,$ (19) resulting in an equation for the dot GF given by $\displaystyle G_{d\sigma}(\omega)$ $\displaystyle=$ $\displaystyle\left[U(\omega)-\langle n_{d\bar{\sigma}}\rangle- B_{\bar{\sigma}}(\omega)-B_{\bar{\sigma}}(\omega_{1})\right]$ (20) $\displaystyle\times\left\\{U(\omega)[\omega-\varepsilon_{d\sigma}-\Sigma_{\sigma}(\omega)]+[B_{\bar{\sigma}}(\omega)+B_{\bar{\sigma}}(\omega_{1})\right]\Sigma_{\sigma}(\omega)-A_{\bar{\sigma}}(\omega)+A_{\bar{\sigma}}(\omega_{1})\\}^{-1}\,,$ where $\Sigma_{\sigma}(x)=\sum_{k}|\tilde{t}_{\sigma}|^{2}/(x-\varepsilon_{k\sigma})$, $U(\omega)=[U-\omega+\varepsilon_{d\sigma}-\Sigma_{\sigma}(\omega)+\Sigma_{\bar{\sigma}}(\omega)-\Sigma_{\bar{\sigma}}(\omega_{1})]/U$ and $\omega_{1}=-\omega+\varepsilon_{d\sigma}+U$. The functions $A_{\sigma}(\omega)$ and $B_{\sigma}(\omega)$ are given by $\displaystyle B_{\sigma}(\omega)$ $\displaystyle=$ $\displaystyle\frac{i}{2\pi}\int d\omega^{\prime}f(\omega^{\prime})\left[G_{d\sigma}(\omega^{\prime})\frac{\Sigma_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}-i\delta}-G^{*}_{d\sigma}(\omega^{\prime})\frac{\Sigma^{*}_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}+i\delta}\right]$ $\displaystyle A_{\sigma}(\omega)$ $\displaystyle=$ $\displaystyle\frac{i}{2\pi}\int d\omega^{\prime}f(\omega^{\prime})\left[\left(1+G_{d\sigma}(\omega^{\prime})\Sigma_{\sigma}(\omega^{\prime})\right)\frac{\Sigma_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}-i\delta}-\left(1+G^{*}_{d\sigma}(\omega^{\prime})\Sigma^{*}_{\sigma}(\omega^{\prime})\right)\frac{\Sigma^{*}_{\sigma}(\omega^{\prime})-\Sigma_{\sigma}(\omega)}{\omega-\omega^{\prime}+i\delta}\right],$ (21) and have to be calculated self-consistently. In the limit $U\rightarrow\infty$, the GF (20) acquires a simpler form, $\displaystyle G_{d\sigma}(\omega)=\frac{1-\langle n_{d\bar{\sigma}}\rangle- B_{\bar{\sigma}}(\omega)}{\omega-\varepsilon_{d\sigma}-(1-B_{\bar{\sigma}}(\omega))\Sigma_{\sigma}(\omega)-A_{\bar{\sigma}}(\omega)}.$ (22) Figure 4: (color online) Density of states in QD as a function of $\omega$ in the Kondo regime. Parameters used are $T=10^{-7}$, $\varepsilon_{d}=-0.136$, $\Gamma_{\uparrow}=0.06$ and $\Gamma_{\downarrow}=0.03$ ($D=1$). Sharp feature near the Fermi level ($\omega=0$) is the signature of the Kondo screening, although asymmetric here for the different spin species. In the following, we solve numerically for the spectral function (DOS) in the Kondo regime by the self-consistent iteration of Eqs. (13) and (22). Figure 4 shows the DOS vs. $\omega$ at low temperature ($T=10^{-7}$) for both spin orientations (and total), when the QD electron level $\varepsilon_{d}$ is taken to be at $-0.136$. The three curves exhibit Kondo resonance peaks near the Fermi level ($\omega\simeq 0$), in addition to a much broader peak at $\omega\simeq\varepsilon_{d}$. 3 Since the hybridization between leads and QD are spin dependent in this case, the DOS clearly splits into two different components for spin up and down. The DOS for spin down is shifted upwards in energy with respect to the spin up component, resulting in a lower occupancy for the down spin. Figure 5 shows indeed the occupancy curves for $\langle n_{d\uparrow}\rangle$, $\langle n_{d\downarrow}\rangle$ and total $\langle n\rangle$ vs. $\varepsilon_{d}$ at a given temperature ($T=10^{-7}$). Qualitatively similar to the results in the Coulomb blockade regime, the occupancy curves show that $\langle n_{d\uparrow}\rangle>\langle n_{d\downarrow}\rangle$ for $\omega\gtrsim-0.05$, while the relation is reversed for smaller $\omega$ values, reflecting the asymmetry introduced by the polarizing QPCs, through $\Gamma_{\uparrow}$ and $\Gamma_{\downarrow}$. Notice also that the plateaus at 0.5 are not as well defined here, due to the enhanced spin and charge fluctuations in the Kondo regime. The spin-orbit effect can be understood qualitatively as arising from a shift in the dot level (as well as depending on the occupancy factors): since the effective level position of the electron with spin $\sigma$ is given by $\varepsilon_{d\sigma}\approx\varepsilon_{d}+\text{Re}\Sigma^{{}^{\prime}}_{\sigma}(\omega)$, where $\Sigma^{{}^{\prime}}_{\sigma}(\omega)\propto t_{\sigma}^{2}$ is the self-energy, and thus naturally causes the spin-dependent occupancy seen in the figure. Figure 5: (color online) Occupation in the QD as function of $\epsilon_{d}$. Parameters used as in Fig. 4. Figure 6: (color online) (a) Conductance, and (b) polarization and difference between spin up and down conductance, as functions of $\epsilon_{d}$. Parameters as in Fig. 4. Figure 6(a) shows the spin-dependent conductance curves $G_{\sigma}$ vs. $\varepsilon_{d}$. Several features are noteworthy. As $\varepsilon_{d}$ changes, the spin-dependent conductances exhibit the anticipated peaks at low temperature, with spin up conductance dominating (naturally, as $\Gamma_{\uparrow}>\Gamma_{\downarrow}$). Figure 6(b) shows the difference between spin up and down conductances as well as the spin polarization vs. $\varepsilon_{d}$. In this regime, the difference between the conductance for the two spin orientations reaches $\simeq 0.4(e^{2}/h)$. Correspondingly, the net spin polarization reaches $\eta\simeq 60$%. Let us now analyze in more detail the effect of temperature on the conductance of the system and especially its drop for $\varepsilon_{d}\ll 0$. The conductance curves in Fig. 6(a) are highly asymmetric about the Fermi energy and vanish rapidly away from it. This vanishing for very negative values of $\varepsilon_{d}$ is due to the the Kondo temperature, $T_{K}$, becoming smaller than the temperature of the system. Figure 7 presents the total zero bias conductance as function of $\varepsilon_{d}$ for several values of temperature. As $T$ is lowered, the total conductance increases for a given $\varepsilon_{d}$, and the width of the conductance peak increases in $\varepsilon_{d}$. This explicitly reflects the existence of the Kondo resonant peak in the spectral function, and how the system will reach the unitary limit of conductance for $T=0$ for $\varepsilon_{d}$ well below the Fermi level. Figure 7: (color online) Total conductance as function of $\varepsilon_{d}$ for several temperatures $T$. Notice that $T_{K}$ is strongly suppressed for more negative $\varepsilon_{d}$ values, which lowers the conductance at a given $T$. Parameters as in Fig. 4. Our discussions above are applied to the case of infinite $U$, where the EOM method gives qualitatively accurate results in the Kondo regime. Extension of this approach for finite $U$ is known to be problematic, including its failure to exhibit a Kondo resonance at the particle-hole symmetry point ($\varepsilon_{d}=-U/2$). 18 To carry out our study in the finite $U$ case, we use instead the essentially exact numerical renormalization group approach, as we discuss in the following section. ### III.2 NRG results for finite U For the finite-$U$ case we study the spin polarized conductance using the standard numerical renormalization group approach.Wilson75 ; costi In this case, unlike the previous infinite-$U$ case, the processes involving double occupied states are naturally present in the dynamics of the system, and allow for a reliable description of the low-energy behavior. We set $U=0.5$ and $T=0$. Figure 8 depicts the local density of states calculated with NRG for the same system parameters as before, $\Gamma_{\uparrow}=0.06$, $\Gamma_{\downarrow}=0.03$, and with $\epsilon_{d}=-0.2$. This value of $\epsilon_{d}$ corresponds to a situation where the system is away from the particle-hole symmetric point ($=-U/2$), more suitable to compare to the previous infinite-$U$ calculation (where the system never reaches the p-h symmetry point). Notice that there is a strong spin asymmetry in the DOS; on the negative side of the $\omega$-axis, the DOS for spin up (solid curve) presents a peak near $-0.2$, corresponding to the energy of the local bare orbital $\varepsilon_{d}$, slightly shifted by the real part of the proper self-energy. On the positive side of the $\omega$-axis, the peak near $\epsilon_{d}+U$ would result in the succeeding CB peak, which appears only for the spin-down DOS (dashed curve). In this regime as well, the fact that $\Gamma_{\uparrow}>\Gamma_{\downarrow}$ favors the spin-up occupancy to the detriment of the spin-down occupancy. One also notices the important peaks close to the Fermi level, signature of the Kondo effect, which are slightly split away and suppressed by a seemingly effective magnetic field induced by the spin-asymmetric coupling to the leads. This phenomenon is akin to the suppression discussed by Martinek et al.,Martinek in the context of a QD coupled to ferromagnetic leads. Notice, however, that no external magnetization is present in our system and that the polarization is only arising from the QPCs and due to the lateral SO interaction. Figure 8: (color online) Density of states in the QD as function of energy obtained from NRG calculations. Polarizing QPCs generate effective splitting of the DOS for different spins. System parameters used are $U=0.5$, $\Gamma_{\uparrow}=0.06$, $\Gamma_{\downarrow}=0.03$, $D=1$, and $T=0$. Analogously to the infinite-$U$ case, the spin-asymmetry discussed above induces spin polarized transport in the system. In Fig. 9 we show the conductance as a function of $\varepsilon_{d}$ for the same parameters as Fig. 8. Notice that away from the p-h symmetric point ($\varepsilon_{d}=-0.25$) the conductance for spin up is much larger than for spin down, resulting in a sizeable polarization ($\eta\simeq 70$%), as shown by the (green) dotted curve. At the p-h symmetric point the conductance for both spins reaches the unitary limit and $\eta\rightarrow 0$; this is consistent with the restoration of the Kondo state of the system at the p-h point for ferromagnetic leads. Sindel Figure 9: (color online) Total conductance as function of $\varepsilon_{d}$ obtained from NRG calculations. Same system parameters as in Fig. 8. ## IV Summary In summary, we have investigated the spin-dependent transport properties of quantum dot structures with polarizing quantum point contacts. We have shown that as QPCs can generate finite spin-polarized currents, due to the combination of lateral and perpendicular spin-orbit interactions, they also induce current polarization in quantum dots made with these QPCs. Using equation-of-motion techniques and numerical renormalization group calculations, we obtained the electronic Green’s function, conductance and spin polarization in different parameter regimes. Our results demonstrate that both in the Coulomb blockade and Kondo regimes, the quantum dot exhibits non- zero spin-polarized conductance, even when the injection is unpolarized and there are no applied magnetic fields. The spin-dependent coupling is shown to give rise to nontrivial effects in the density of states of the single QD, resulting in strong modification of the charge distribution in the system. Most importantly, these effects are controllable by lateral gate voltages applied to the QPCs, and together with the ability to create quantum dots, they provide a new approach for exploring spintronic devices, spin polarized sources and spin filters. ## V Acknowledgements We thank helpful discussions with P. Debray and N. Sandler, as well as financial support from CNPq, CAPES, and FAPEMIG in Brazil, and NSF-PIRE, and NSF-MWN/CIAM in the US. ## References * (1) S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y. Chtchelkanova, D. M. Treger, Science 294, 1488 (2001). * (2) R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. Vandersypen, Rev. Mod. Phys. 79, 1217 (2007). * (3) G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip and B. J. van Wees, Phys. Rev. B 62, R4790 (2000). * (4) Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984). * (5) R. Winkler, Spin-Orbit Coupling Effects in Two Dimensional Electron and Hole Systems (Springer, Berlin, 2003). * (6) A. T. Ngo, P. Debray and S. E. Ulloa (arXiv:0908.1080) Phys. Rev. B 81, 115328 (2010). * (7) J. A. Folk, S. R. Patel, K. M. Birnbaum, C. M. Marcus C. I. Duroz and J. S. Harris, Jr. Phys. Rev. Lett. 86, 2101 (2001). * (8) J. P. Bird and Y. Ochiai, Science 303, 1621 (2004). * (9) P. Debray, S. M. S. Rahman, J. Wan, R. S. Newrock, M. Cahay, A. T. Ngo, S. E. Ulloa, S. T. Herbert, M. Muhammad, and M. Johnson, Nature Nanotech. 4, 759 (2009). * (10) J. Martinek, M. Sindel, L. Borda, J. Barnaś, J. König, G. Schön, and J. von Delft, Phys. Rev. Lett. 91, 247202 (2003). * (11) M. Eto, T. Hayashi and Y. Kurotani, J. Phys. Soc. Japan. 74, 1934 (2005). * (12) A. Reynoso, Gonzalo Usaj, C. A. Balseiro, Phys. Rev, B 75, 085321 (2007). * (13) E. N. Bulgakov and A. F. Sadreev, Phys, Rev. B 66, 075331 (2002). * (14) H.Q. Xu, Phys. Rev. B 72, 045347 (2005); H.Q. Xu, Phys. Rev. B 52,5803 (1995). * (15) G. D. Mahan, Many Particle Physics (Plenum, New York, 1981). * (16) J. E. Hirsch and R.M. Fye, Phys. Rev. Lett. 56, 2521 (1986). * (17) C. Lacroix, J. Phys. F: Met. Phys. 11, 2389 (1981). * (18) R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 (2008). * (19) D. N. Zubarev, [Usp. Fiz. Nauk 71, 71 (1960)], Sov. Phys. Usp. 3, 320 (1960). * (20) J. Hubbard, Proc. R. Soc. London, Ser. A 276, 238 (1963). * (21) Y. Meir, N. S. Wingreen and P. A. Lee, Phys. Rev. Lett. 66, 3048 (1991). * (22) V. Kashcheyevs, A. Aharony and O. Entin-Wohlman, Phys, Rev. B 73, 125338 (2005). * (23) K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975); H. R. Krishna-murty, J. W. Wilkins and K. G. Wilson Phys. Rev. B 21, 1003 (1980), and 21, 1044 (1980). * (24) M. Sindel, L. Borda, J. Martinek, R. Bulla, J. König, G. Schön, S. Maekawa, and J. von Delft, Phys. Rev. B 76, 045321 (2007).
arxiv-papers
2010-12-19T15:36:31
2024-09-04T02:49:15.827718
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Anh T. Ngo, Edson Vernek, and Sergio E. Ulloa", "submitter": "Anh Ngo", "url": "https://arxiv.org/abs/1012.4175" }
1012.4290
# Bit recycling for scaling random number generators Andrea C. G. Mennucci Scuola Normale Superiore, Pisa, Italy (a.mennucci@sns.it) ###### Abstract Many Random Number Generators (RNG) are available nowadays; they are divided in two categories, _hardware RNG_ , that provide “true” random numbers, and _algorithmic RNG_ , that generate pseudo random numbers (PRNG). Both types usually generate random numbers $(X_{n})_{n}$ as independent uniform samples in a range $0,\ldots 2^{b}-1$, with $b=8,16,32$ or $b=64$. In applications, it is instead sometimes desirable to draw random numbers as independent uniform samples $(Y_{n})_{n}$ in a range $1,\ldots M$, where moreover $M$ may change between drawings. Transforming the sequence $(X_{n})_{n}$ to $(Y_{n})_{n}$ is sometimes known as _scaling_. We discuss different methods for scaling the RNG, both in term of mathematical efficiency and of computational speed. ###### Contents 1. 1 Introduction 2. 2 Process splitting 1. 2.1 Mathematical formulation 3. 3 Recycling in uniform random number generation 4. 4 Mathematical analysis of the efficiency 5. 5 Speed, simple _vs_ complex algorithms 6. 6 Numerical tests 1. 6.1 Architectures 2. 6.2 Back-end PRNGs 3. 6.3 _Ad hoc_ functions 4. 6.4 Uniform RNGs 5. 6.5 Timing 6. 6.6 Conclusions 7. A Test results 1. A.1 Speed of back-end RNGs and _Ad hoc_ functions 2. A.2 Graphs 1. A.2.1 Uniform random generators 2. A.2.2 Integer arithmetic 8. B Code 1. B.1 Back-end RNGs 2. B.2 _ad hoc_ functions 3. B.3 Uniform random generators ## 1 Introduction We consider the following problem. We want to generate a sequence of random numbers $(Y_{n})_{n}$ with a specified probability distribution, using as input a sequence of random numbers $(X_{n})_{n}$ uniformly distributed in a given range. There are various methods available; these methods involve transforming the input in some way; for this reason, these methods work equally well in transforming both pseudo-random and true random numbers. One such method, called the acceptance-rejection method, involves designing a specific algorithm, that pulls random numbers, transforms them using a specific function, tests whether the result satisfies a condition: if it is, the value is accepted; otherwise, the value is rejected and the algorithm tries again. This kind of method has a defect, though: if not carefully implemented, it throws away many inputs. Let’s see a concrete example. We suppose that we are given a RNG that produces random bits, evenly distributed, and independent111For example, repeatedly tossing a coin with the faces labeled $0,1$.. We want to produce a random number $R$ in the range $\\{1,2,3\\}$, uniformly distributed and independent. Consider the following method. ###### Example 1 We draw two random bits; if the sequence is $11$, we throw it away and draw two bits again; otherwise we return the sequence as $R$, mapping $00,01,10$ to $1,2,3$. This is rather wasteful! The entropy in the returned random $R$ is $\log_{2}(3)=1.585$bits; there is a $1/4$ probability that we throw away the input, so the expected number of (pair of tosses) is $4/3$ and then expected number of input bits is $8/3$; all together we are effectively using only missing$$$\frac{\log_{2}(3)}{8/3}=59\%$$oftheinput.\par ThewastemaybeconsiderasanunnecessaryslowdownoftheRNG:iftheRNGcangenerate1bitin$1μs$,then,aftertheexampleprocedure,theratehasdecreasedto$1.6 μs$perbit.SincealotofeffortwasputindesigningfastRNGinthenearpast,thenslowingdowntherateby$+60%$issimplyunacceptable.\par Anotherproblemintheexamplemethodaboveisthat,althoughitisquiteunlikely,wecanhaveaverylongrunof"00"bits.Thismeansthatwecannotguaranteethattheaboveprocedurewillgeneratethenextnumberinapredeterminedamountoftime.\par Thereareofcoursebettersolutions,asthis\emph{ad hoc method}.\begin{Example}[\cite{DJ}]Wedraweightrandombits,andconsiderthemasanumber$x$intherange$0\ldots 255$;ifthenumberismorethan$3^{5}-1=242$,wethrowitawayanddraweightbitsagain;otherwisewewrite$x$asfivedigitsinbase3andreturnthesedigitsas5randomsamples.\end{Example}Thisismuchmoreefficient!Theentropyinthereturnedfivesamplesis$5 log_2(3)=7.92$bits;thereisa$13/256$probabilitythatwethrowawaytheinput,sotheexpectednumberof8-tuplesofinputsis$256/253$andthenexpectednumberofinputbitsis$2048/253$;alltogetherwearenowusingmissing$ $\frac{5\log_{2}(3)}{2048/253}=97\%$ of the input. In this paper we will provide a mathematical proof (section 2), and discuss some method (section 3), to optimize the scaling of a RNG. Unfortunately after writing this paper it turned out that one of the ideas we are presenting in section 3, was already described in [1]. This paper contains the mathematical proof of the method, the discussion of how best to choose parameters, discussion of its efficiency, and numerical speed tests. ###### Remark 3 A different approach may be to use a decompressing algorithm. Indeed, e.g., the _arithmetic encoder_ decoding algorithm, can be rewritten to decode a stream of bits to an output of symbols with prescribed probability distributions 222If interested, I have the code somewhere in the closet. Unfortunately, it is quite difficult to mathematically prove that such an approach really does transform a stream of independent equidistributed bits into an output of independent random variables. Also, the _arithmetic encoder_ is complex, and this complexity would slow down the RNG, defeating one of the goals. (Moreover, the _arithmetic encoder_ was originally heavily patented.) A note on notations. In all of the paper, ns is a _nanosecond_ , that is $10^{-9}$seconds. When $x$ is a real number, $\lfloor x\rfloor=\mathtt{floor}(x)$ is the largest integer that is less or equal than $x$. ## 2 Process splitting Let ${\mathrm{l\hskip-1.49994ptN}}=\\{0,1,2,3,4,5\ldots\\}$ be the set of natural numbers. Let $(\Omega,{\mathcal{A}},{\mathbb{P}})$ a probability space, let $(E,{\mathcal{E}})$ be a measurable space, and ${\overline{X}}$ a process of i.i.d. random variables $(X_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ defined on $(\Omega,{\mathcal{A}},{\mathbb{P}})$ and each taking values in $(E,{\mathcal{E}})$. We fix an event $S\in{\mathcal{E}}$ such that ${\mathbb{P}}\\{X_{i}\in S\\}\neq 0,1$; we define $p_{S}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}{\mathbb{P}}\\{X_{i}\in S\\}$. We define a formal method of process splitting/unsplitting. The splitting of ${\overline{X}}$ is the operation that generates three processes ${\overline{B}},{\overline{Y}},{\overline{Z}}$, where ${\overline{B}}=(B_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ is an i.i.d. Bernoulli process with parameter $p_{S}$, and ${\overline{Y}}=(Y_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ and ${\overline{Z}}=(Z_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ are processes taking values respectively in $S$ and $E\setminus S$. The unsplitting is the opposite operation. These operations can be algorithmically and intuitively described by the following pseudocode (where processes are thought of as _queues of random variables_). procedure Splitting(${\overline{X}}\mapsto({\overline{B}},{\overline{Y}},{\overline{Z}})$) initialize the three empty queues ${\overline{B}},{\overline{Y}},{\overline{Z}}$ repeat pop X from ${\overline{X}}$ if $X\in S$ then push 1 onto ${\overline{B}}$ push X onto ${\overline{Y}}$ else push 0 onto ${\overline{B}}$ push X onto ${\overline{Z}}$ end if until forever end procedure procedure Unsplitting($({\overline{B}},{\overline{Y}},{\overline{Z}})\mapsto{\overline{X}}$) initialize the empty queue ${\overline{X}}$ repeat pop B from ${\overline{B}}$ if $B=1$ then pop Y from ${\overline{Y}}$ push Y onto ${\overline{X}}$ else pop Z from ${\overline{Z}}$ push Z onto ${\overline{X}}$ end if until forever end procedure The fact that the _splitting_ operation is invertible implies that no entropy is lost when splitting. We will next show a very important property, namely, that the splitting operation preserve probabilistic independence. ### 2.1 Mathematical formulation We now rewrite the above idea in a purely mathematical formulation. We define the Bernoulli process $(B_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ by $B_{n}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}\begin{cases}1&X_{n}\in S\\\ 0&X_{n}\not\in S\end{cases}$ (1) and the times of return to success as $\displaystyle U_{0}$ $\displaystyle=$ $\displaystyle\inf\\{k:k\geq 0,B_{k}=1\\}$ (2) $\displaystyle U_{n}$ $\displaystyle=$ $\displaystyle\inf\\{k:k\geq 1+U_{n-1},B_{k}=1\\},\leavevmode\nobreak\ \leavevmode\nobreak\ n\geq 1$ (3) whereas the times of return to unsuccess are $\displaystyle V_{0}$ $\displaystyle=$ $\displaystyle\inf\\{k:k\geq 0,B_{k}=0\\}$ (4) $\displaystyle V_{n}$ $\displaystyle=$ $\displaystyle\inf\\{k:k\geq 1+U_{n-1},B_{k}=0\\},\leavevmode\nobreak\ \leavevmode\nobreak\ n\geq 1$ (5) it is well known that $(U_{n}),(V_{n})$ are (almost certainly) well defined and finite. We eventually define the processes $(Y_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ and $(Z_{n})_{n\in{\mathrm{l\hskip-1.04996ptN}}}$ by $Y_{n}=X_{U_{n}}\quad Z_{n}=X_{V_{n}}$ (6) ###### Theorem 4 Assume that ${\overline{X}}$ is a process of i.i.d. random variables. Let $\mu$ be the law of $X_{1}$. Then * • the random variables $B_{n},Y_{n},Z_{n}$ are independent; and * • the variables of the same type are identically distributed: the variables $B_{n}$ have parameter ${\mathbb{P}}\\{B_{n}=1\\}=p_{S}$; the variables $Y_{n}$ have law $\mu(\cdot\leavevmode\nobreak\ |\leavevmode\nobreak\ S)$; the variables $Z_{n}$ have law $\mu(\cdot\leavevmode\nobreak\ |\leavevmode\nobreak\ E\setminus S)$. * Proof. It is obvious that ${\overline{B}}$ is a Bernoulli process of independent variables with parameter ${\mathbb{P}}\\{B_{n}=1\\}=p_{S}$. Let $K,M\geq 1$ integers. Let $u_{0}<u_{1}<\ldots u_{K}$ and $v_{0}<v_{1}<\ldots v_{M}$ be integers, and consider the event $A\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}\\{U_{0}=u_{0},\ldots U_{K}=u_{K},V_{0}=v_{0},\ldots V_{M}=v_{M}\\}$ (7) If $A\neq\emptyset$ then $A=\\{B_{0}=b_{0},\ldots,B_{N}=b_{N}\\}$ (8) where $N=\max\\{u_{K},v_{M}\\}$ and $b_{j}\in\\{0,1\\}$ are suitably chosen. Indeed, supposing that $N=u_{K}>v_{M}$, then we use the success times, and set that $b_{j}=1$ iff $j=u_{k}$ for a $k\leq K$; whereas supposing that $N=v_{M}>u_{K}$, then we use the unsuccess times, and set that $b_{j}=0$ iff $j=v_{m}$ for a $m\leq M$. Let ${\mathcal{F}}_{K,M}$ be the family of all above events $A$ defined as per equation (7), for different choices of $(u_{i}),(v_{j})$; let missing$$${\mathcal{F}}=\bigcup_{K,M\geq 1}{\mathcal{F}}_{K,M}\leavevmode\nobreak\ \leavevmode\nobreak\ ;$$let$A^¯B⊂A$bethesigmaalgebrageneratedbytheprocess$¯B$.\par Theaboveequality\eqref{eq:A_as_B}provesthat$F$isa\emph{base}for$A^¯B$:itisstablebyfiniteintersection,anditgeneratesthesigmaalgebra$A^¯B$.\par Consideragain$K,M≥1$integers,andevents$F_i,G_j∈E$for$i=0,…K,j=0,…M$,andtheeventmissing$ $C=\\{Y_{0}\in F_{0},\ldots Y_{K}\in F_{K},Z_{0}\in G_{0},\ldots Z_{M}\in G_{M}\\}\in{\mathcal{A}}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ;$ let $A\in{\mathcal{F}}_{K,M}$ non empty; we want to show that missing$$${\mathbb{P}}(C\leavevmode\nobreak\ |\leavevmode\nobreak\ A)={\mathbb{P}}(C)$$thiswillprovethat$(¯Y,¯Z)$areindependentof$¯B$,byarbitrinessof$(F_i),(G_j),K,M$andsince$F$isabasefor$A^¯B$.\par Wefix$(u_i),(v_j)$anddefine$A$asinequation\eqref{eq:A};welet$N=max{u_K,v_M}$anddefine$(b_n)$asexplainedafterequation\eqref{eq:A_as_B}.Bydefiningmissing$ $S^{1}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}S,S^{0}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}E\setminus S$ for notation convenience, we can write equation (8) as missing$$$A=\\{X_{0}\in S^{b_{0}},\ldots X_{N}\in S^{b_{N}}\\}\leavevmode\nobreak\ \leavevmode\nobreak\ .$$Let$E_0…E_N∈E$bedefinedbymissing$ $E_{n}\stackrel{{\scriptstyle\mbox{\tiny{def}}}}{{=}}\begin{cases}F_{k}&\text{if }n=u_{k}\text{ for a }k\leq K\\\ G_{m}&\text{if }n=v_{m}\text{ for a }m\leq M\\\ E&\text{else}\end{cases}$ then we compute $\displaystyle{\mathbb{P}}(C\leavevmode\nobreak\ |\leavevmode\nobreak\ A)$ $\displaystyle=$ $\displaystyle{\mathbb{P}}(\\{X_{u_{0}}\in F_{0},\ldots X_{u_{K}}\in F_{K},X_{v_{0}}\in G_{0},\ldots X_{v_{M}}\in G_{M}\\}\leavevmode\nobreak\ |\leavevmode\nobreak\ \\{X_{0}\in S^{b_{0}},\ldots X_{N}\in S^{b_{N}}\\})=$ $\displaystyle=$ $\displaystyle\frac{{\mathbb{P}}\\{X_{u_{0}}\in F_{0},\ldots X_{u_{K}}\in F_{K},X_{v_{0}}\in G_{0},\ldots X_{v_{M}}\in G_{M}\leavevmode\nobreak\ ,\leavevmode\nobreak\ X_{0}\in S^{b_{0}},\ldots X_{N}\in S^{b_{N}}\\}}{{\mathbb{P}}\\{X_{0}\in S^{b_{0}},\ldots X_{N}\in S^{b_{N}}\\}}=$ $\displaystyle=$ $\displaystyle\frac{\prod_{n=0}^{N}{\mathbb{P}}\\{X_{n}\in S^{b_{n}}\cap E_{n}\\}}{\prod_{n=0}^{N}{\mathbb{P}}\\{X_{n}\in S^{b_{n}}\\}}=\prod_{n=0}^{N}{\mathbb{P}}(X_{n}\in E_{n}|X_{n}\in S^{b_{n}})=$ $\displaystyle=$ $\displaystyle\prod_{k=0}^{K}\mu(F_{k}\leavevmode\nobreak\ |\leavevmode\nobreak\ S^{1})\prod_{m=0}^{M}\mu(G_{m}\leavevmode\nobreak\ |\leavevmode\nobreak\ S^{0})\leavevmode\nobreak\ \leavevmode\nobreak\ ;$ the last equality is due to the fact that: when $n=u_{k}$ then $b_{n}=1$, when $n=v_{m}$ then $b_{n}=0$, and for all other $n$ we have $E_{n}=E$. Since the last term does not depend on $A$, that is, on $(u_{i}),(v_{j})$, we obtain that $({\overline{Y}},{\overline{Z}})$ are independent of ${\overline{B}}$. The above equality then also shows that missing$$${\mathbb{P}}\\{Y_{0}\in F_{0},\ldots Y_{K}\in F_{K},Z_{0}\in G_{0},\ldots Z_{M}\in G_{M}\\}=\prod_{k=0}^{K}\mu(F_{k}\leavevmode\nobreak\ |\leavevmode\nobreak\ S)\prod_{m=0}^{M}\mu(G_{m}\leavevmode\nobreak\ |\leavevmode\nobreak\ S^{c})$$andthisimpliesthat$¯Y,¯Z$areprocessesofindependentvariables,distributedasinthethesis.Byassociativityoftheindependence,weconcludethattherandomvariables$B_n,Y_n,Z_n$areindependent.\quad\hbox{\leavevmode\hbox to7.77786pt{\hfil\vrule\vbox to6.75003pt{\hrule width=6.00006pt\vfil\hrule}\vrule\hfil}}\par\endtrivlist$ ## 3 Recycling in uniform random number generation We now restrict our attention to the generation of uniformly distributed integer valued random variables. We will say that _$R$ is a random variable of modulus $M$_ when $R$ is uniformly distributed in the range $0,\ldots(M-1)$. We present an algorithm, that we had thought of, and then found (different implementation, almost identical idea) in [1]. We present the latter implementation. The following algorithm Uniform random by bit recycling in figure 1, given $n$, will return a random variable of modulus $n$; note that $n$ can change between different calls to the algorithm. 1:initialize the static integer variables $m=1$ and $r=0$ 2:procedure Uniform random by bit recycling(n) 3: repeat 4: while $m<N$ do$\triangleright$ fill in the state 5: r : = 2*r + NextBit(); 6: m : = 2*m; $\triangleright$ r is a random variable of modulus m 7: end while 8: q := $\lfloor m/n\rfloor$; $\triangleright$ integer division, rounded down 9: if $r<n*q$ then 10: d : = $r\mathop{\operator@font mod}\nolimits n$ $\triangleright$ remainder, is a random variable of modulus n 11: r : = $\lfloor r/n\rfloor$ $\triangleright$ quotient, is random variable of modulus q 12: m : = q 13: return d 14: else 15: r : = r - n*q $\triangleright$ r is still a random variable of modulus m 16: m : = m - n*q $\triangleright$ the procedure loops back to line 3 17: end if 18: until forever 19:end procedure Figure 1: Algorithm Uniform random by bit recycling It uses two internal integer variables, m and r, which are not reset at the beginning of the algorithm (in C, you would declare them as "static"). Initially, $m=1$ and $r=0$. The algorithm has an internal constant parameter $N$, which is a large integer such that $2N$ can still be represented exactly in the computer. We must have $n<N$. 333We will show in next section that it is best to have $n<<N$. The algorithm draws randomness from a function NextBit() that returns a random bit. Here is an informal discussion of the algorithm, in the words of the original author [1]. _At line 10, as r is between 0 and $(n*q-1)$, we can consider r as a random variable of modulus $n*q$. As this is divisible by n, then $d:=(r\mathop{\operator@font mod}\nolimits n)$ will be uniformly distributed, and the quotient $\lfloor r/n\rfloor$ will be uniformly distributed between 0 and $q-1$. _ Note that the theoretical running time is unbounded; we will though show in the next section that an accurate choice of parameters practically cancels this problem. ## 4 Mathematical analysis of the efficiency We recall this simple idea. ###### Lemma 5 Suppose $R$ is a random variable of modulus $MN$; we perform the integer division $R=QN+D$ where $Q\in\\{0,\ldots(M-1)\\}$ is the quotient and $D\in\\{0,\ldots(N-1)\\}$ is the remainder; then $Q$ is a random variable of modulus $M$ and $D$ is a random variable of modulus $N$; and $Q,D$ are independent. ###### Proposition 6 Let us assume that repeated calls of NextBit() return a sequence of independent equidistributed bits. Then the above algorithm Uniform random by bit recycling in figure 1 will return a sequence of independent and uniformly distributed numbers. * Proof. We sketch the proof. We use the lemma above 5 and the theorem 4. Consider the notations in the second section. The bits returned by the call NextBit() builds up the process ${\overline{X}}$. When reaching the if (line 9 in the pseudocode at page 1), the choice $r<n*q$ is the choice of the value of $B_{n}$ in equation (1). This (virtually) builds the process ${\overline{B}}$. At line 10 $r$ is a variable in the process ${\overline{Y}}$; since it is of modulus $nq$, we return (using the lemma) the remainder as $d$, that is a random variable of modulus $n$, and push back the quotient into the state. At line 16 we would be defining a variable in the process ${\overline{Z}}$, that we push back into the state. The “pushing back” of most of the entropy back into the state recycles the bits, and improves greatly the efficiency. The only wasted bits are related to the fact that the algorithm is throwing away the mathematical stream ${\overline{B}}$. Theoretically, if this stream would be feeded back into the state (for example, by employing Shannon-Fano- Elias coding), then efficiency would be exactly $100\%$.444But this would render difficult to prove that the output numbers are independent… Practically, the numbers $N$ and $n$ can be designed so that this is totally unneeded. ###### Remark 7 Indeed, consider the implementation (see the code in the next sections) where the internal state is stored as 64bit unsigned integers, whereas $n$ is restricted to be 32bit unsigned integer; so the internal constant is $N=2^{62}$ while $n\in\\{2\ldots 2^{32}-1\\}$, When reaching the if at line 9, $m$ is in the range $2^{62}\leq m<2^{64}$, and $r$ is uniform of modulus $m$; but $m-n*\lfloor m/n\rfloor$ is less than $n$, that is, less than $2^{32}$; so the probability that $r\geq n*q$ at the if is less than $1/2^{30}$. In particular, this means that each $B_{n}$ in the mathematical stream ${\overline{B}}$ contains $\sim 10^{-8}$bits of entropy, so there is no need to recycle them. Indeed, in the numerical experiments we found out that the following algorithm wastes $\sim 30$ input bits on a total of $\sim 10^{9}$ input bits (!) this is comparable to the entropy of the internal state (and may also be due to numerical error in adding up $\log_{2}()$ values). Also, this choice of parameters ensures that the algorithm will never practically loop twice before returning. When the condition in the if at line 9 is false, we will count it as a failure. In $\sim 10^{10}$ calls to the algorithm, we only experienced 3 failures. 555For this reason, the else block may be omitted with no big impact on the quality of the output – we implement this idea in the algorithm uniform_random_by_bit_recycling_cheating. ## 5 Speed, simple _vs_ complex algorithms We now consider the algorithm Uniform random simple in 2. 1:procedure Uniform random simple(n) 2: repeat 3: r : = GetRandomBits(b); $\triangleright$ fill the state with b bits 4: q = $\lfloor N/n\rfloor$; $\triangleright$ integer division, rounded down 5: if $r<n*q$ then 6: return $r\mathop{\operator@font mod}\nolimits n$ $\triangleright$ remainder, is random variable of modulus n 7: end if$\triangleright$ otherwise, start all over again 8: until forever 9:end procedure Figure 2: Uniform random simple ; in our tests $N=2^{b}$ or $N=2^{b}-1$, whereas $b=32,40,48,64$ Again, when the condition in the if at line 5 is false, we will count it as a failure. This algorithm will always call the original RNG to obtain $b$ bits, regardless of the value of $n$. When the algorithm fails, it starts again and again draws $b$ bits. This is inefficient in terms of entropy: for small values of $n$ it will produce far less entropy than it consume. But, will it be slower or faster than our previous algorithm? It turns out that the answer pretty much depends on the speed of the back-end RNG (and this is unsurprising); but also on how much time it takes to compute the basic operations “integer multiplications” $q*n$ and “integer division” $\lfloor N/n\rfloor$: we will see that, in some cases, these operations are so slow that they defeat the efficiency of the algorithm Uniform random by bit recycling. ## 6 Numerical tests ### 6.1 Architectures The tests were performed in six different architectures, (HW1) _Intel® Core ™ 2 Duo CPU E7500 2.93GHz_ , in i686 mode, (HW2) _Intel® Core ™ 2 Duo CPU P7350 2.00GHz_ , in i686 mode, (HW3) _AMD Athlon™ 64 X2 Dual Core Processor 4200+_ , in x86_64 mode, (HW4) _AMD Athlon ™ 64 X2 Dual Core Processor 4800+_ , in x86_64 mode, (HW5) _Intel® Core ™ 2 Duo CPU P7350 2.00GHz_ , in x86_64 mode, (HW6) _Intel ® Xeon ® CPU 5160 3.00GHz_ , in x86_64 mode. In the first five cases, the host was running a _Debian GNU/Linux_ or _Ubuntu_ O.S. , and the code was compiled using _gcc 4.4_ , with the optimization flags ` -march=native -O3 -finline-functions -fno-strict-aliasing -fomit-frame- pointer -DNDEBUG ` . In the last case, the O.S. was _Gentoo_ and the code was compiled using _gcc 4.0_ with flags ` -march=nocona -O3 -finline-functions -fno-strict-aliasing -fomit-frame- pointer -DNDEBUG `. ### 6.2 Back-end PRNGs To test the speed of the following algorithms, we used four different back-end PRNGs. (sfmt_sse) The _SIMD oriented Fast Mersenne Twister(SFMT)_ ver. 1.3.3 by Mutsuo Saito and Makoto Matsumoto [2] (compiled with SSE support) (xorshift) The _xorshift_ generator by G. Marsaglia [4] (sfmt_sse_md5) as sfmt_sse above, but moreover the output is cryptographically protected using the MD5 algorithm (bbs260) The Blum-Blum-Shub algorithm [5], with two primes of size $\sim 130$bit. The last two were home-made, as examples of slower but (possibly) cryptographically strong RNG 666The author makes no guarantees, though, that the implemented versions are really good and cryptographically strong RNGs — we are interested only in their speeds.. All of the above were uniformized to implement two functions, my_gen_rand32() and my_gen_rand64(), that return (respectively) a 32bit or a 64bit unsigned integer, uniformly distributed. The C code for all the above is in the appendix B.1. The speeds of the different RNGs are listed in the tables in sec. A.1. We also prepared a simple _counter_ “RNG” algorithm, that returns numbers that are in arithmetic progression; since it is very simple, it is useful to assess the overhead complexity in the testing code itself; this overhead is on the order of 2 to to 4 ns, depending on the CPUs. ### 6.3 _Ad hoc_ functions We implemented some _ad hoc_ functions, that are then used by the uniform RNGs (that are described in the next section). NextBit returns a bit Next2Bit returns two bits NextByte returns 8 bits NextWord returns 16 bits For any of the above, we prepared many variants, that internally call either the my_gen_rand32() or my_gen_rand64() calls (see the C code in sec. B.2) and then we benchmarked them in all architecture, to choose the faster one (that is then used by the uniform RNGs). 777We had to make an exception for when the back-end RNG is based on SFMT, since SFMT cannot mix 64bit and 32bit random number generations: in that case, we forcibly used the 32bit versions (that in most of our benchmarks are anyway slightly faster). We also prepared a specific method (that is not used for the uniform RNGs): NextCard returns a number uniformly distributed in the range $0\ldots 51$ (it may be thought of as a card randomly drawn from a deck of cards). The detailed timings are in the tables in sec. A.1. ### 6.4 Uniform RNGs We implemented nine different versions of uniform random generators. The C code is in B.3; we here briefly describe the ideas. Four versions are based on the “simple” generator in fig. 2: uniform_random_simple32 uses 32bit variables internally, $N=2^{32}-1$, and consumes a 32bit random number, (a call to my_gen_rand32()) each time uniform_random_simple40 uses 64bit variables internally, $N=2^{40}$, and calls my_gen_rand32() and NextByte() each time uniform_random_simple48 uses 64bit variables internally, $N=2^{48}$, and calls my_gen_rand32() and NextWord() each time uniform_random_simple64 uses 64bit variables internally, $N=2^{64}-1$, and calls my_gen_rand64() each time. Then there are three versions based on the “bit recycling” generator in fig. 1 (all use 64bit variables internally): uniform_random_by_bit_recycling is the code in fig. 1 (but it refills the state by popping two bits at a time) uniform_random_by_bit_recycling_faster it refills the state by popping words, bytes and pairs of bits, for improved efficiency uniform_random_by_bit_recycling_cheating as the “faster” one, but the _if/else_ block is not implemented, and the modulus $r\mathop{\operator@font mod}\nolimits n$ is always returned; this is not mathematically exact, but the probability that it is inexact is $\sim 2^{-30}$. Moreover there are “mixed” methods uniform_random_simple_recycler uses 32bit variables internally, keeps an internal state that is sometimes initialized but not refilled each time (so, it is useful only for small $n$), uniform_random_by_bit_recycling_32 when $n<2^{29}$, it implements the “bit recycling” code using 32bit variables; when $2^{29}\leq n<2^{32}$, it implements a “simple”–like method, using only bit shifting. We tested them in all of the architectures, for different values of the modulus $n$, and graphed the results (see appendix A.2.1). ### 6.5 Timing To benchmark the algorithms, we computed the process time using both the Posix call clock() (that returns an approximation of processor time used by the program) and the CPU TSC (that counts the number of CPU ticks). When benchmarking one of the above back-end RNGs or the _ad hoc_ functions, we called it in repeated loops of $2^{24}$ iterations each, repeating them for at least 1 second of processor time; and then compared the data provided by TSC and clock(). We also prepared a statistics of the values missing$$$\mathtt{cycles\\_per\\_clock}:=\frac{\Delta\texttt{TSC}}{\Delta\texttt{clock()}}$$sothatwecouldconvertCPUcyclestonanoseconds;weverifiedthatthestandarddeviationofthelogarithmoftheabovequantitywasusuallylessthan$1%$.\par Toavoidover- optimizationofthecompiler,theresultsofanybenchmarkedfunctionwas\emph{xor}-edina\emph{bucket}variable,thatwasthenprintedonscreen.\par Duringbenchmarking,wedisabledtheCPUpower- savingfeatures,forcingtheCPUtobeatmaximumperformance(usingthe\texttt{cpufreq- set}command)andalsowetiedtheprocesstoonecore(usingthe\texttt{taskset}command).\par Unfortunatelythe\texttt{clock()}call,inGNU/Linuxsystems,hasatimeresolutionof$0.01sec$,soitwastoocoarsetobeusedforthegraphsinsection\ref{sec:graph_speeds}:forthosegraphs,onlythe\texttt{TSC}wasused(andthencycleswereconvertedtonanoseconds,usingtheaveragevalueof\texttt{cycles\\_per\\_clock}).\par$ ### 6.6 Conclusions While efficiency is exactly mathematically assessed, computational speed is a more complex topic, and sometimes quite surprising. We report some considerations. 1. 1. In our Intel™ CPUs running in 32bit mode, integers divisions and remainder computation using 64bit variables are quite slow: in HW1, each such operations cost $\sim 15$ns. 2. 2. Any bit recycling method that we could think of needs at least four arithmetic operations for each result it produces; moreover there is some code to refill the internal state. 3. 3. In the same CPUs, the cost of _bit shifting_ or _xor_ operations are on the order of 3 ns, even on 64bit variables; moreover the back-ends sfmt_sse and xorshift can produce a 32bit random number in $\sim 5$ ns. 4. 4. So, unsurprisingly, when the back end is sfmt_sse and xorshift, and the Intel™ CPU runs in 32bit mode, the fastest methods are the “simple32” and “simple_recycler” methods, that run in $\sim 10$ns; and the bit recycling methods are at least 5 times slower than those. 5. 5. When the back-end is sfmt_sse and xorshift, but the the Intel™ CPU runs in 64bit mode, the fastest method are still the “simple32” and “simple_recycler” methods; the bit recycling methods are twice slower. 6. 6. When the back-ends RNGs are sfmt_sse or xorshift, in the AMD™ CPUs, the “simple32” and “simple_recycler” take $\sim 40$ns; this is related to the fact that 32bit division and remainder computation need $\sim 20$ns (as is shown in sec. A.2.2). So these methods are much slower than the back-ends RNGs, that return a 32bit random number in $\sim 6$ns. One consequence is that, since the uniform_random_by_bit_recycling_32 for $n>2^{29}$ uses a “simple”–like method with only bit shifting, then it is much faster than the “simple32” and “simple_recycler”. What we cannot explain is that, in the same architectures, the NextCard32 function, that implements the same type of operations, runs in $\sim 8$ns (!) (We also tried to test the above with different optimizations. Using the xorshift back-end, setting optimization flags to be just -O0, NextCard32 function takes $\sim 21$ns; setting it to -O, it takes $\sim 12$ns.) 7. 7. The back-end RNGs sfmt_sse_md5 or bbs260, are instead much slower, that is, sfmt_sse_md5 needs $\sim 300$ ns to produce a 32bit number, and bbs260 needs $\sim 500$ ns when the CPU runs in 64bit mode and more than a microsecond (!) in 32bit mode. In this case, the bit recycling methods are usually faster. Their speed is dominated by how many times the back-end RNGs is called, so it can be estimated in terms of _entropy bitrate_ , and indeed the graphs are (almost) linear (since the abscissa is in logarithmic scale). 8. 8. One of the biggest surprises comes from the NextCard functions: there are four implementations, * – using 64bit or 32bit variables; * – computing a result for each call, or precomputing them and storing in an array (the “prefilled” versions). The speed benchmarks give discordant results. When using the faster back-ends sfmt_sse and xorshift, the “prefilled” versions are slower. When using the slower back-ends sfmt_sse_md5 or bbs260, the 64bit “prefilled” version is the fastest in Intel™ CPUs; but it is instead much slower than the “non prefilled” version in AMD™ CPUs. It is possible that the cache misses are playing a rôle in this, but we cannot provide a good explanation. 9. 9. Curiously, in some Intel™ CPUs, the time needed for an integer arithmetic operation depends also on the _values_ of the operands (and not only on the bit sizes of the variable)! See the graph in sec. A.2.2. So, the speed of the functions depend on the value of the modulus $n$. This is the reason why some the graphs are all oscillating in nature. In particular, when we looked at the graphs for _Core 2_ architectures in 32bit mode, by looking at the graphs of the functions simple_40, simple_48 (where $N=2^{40},2^{48}$ constant) we noted that the operations $q:=N/n,qn:=n*q$ are $\sim 10$ ns slower when $n<N2^{-32}$ than when $n>N2^{-32}$. This is similar to what is seen in the graphs in sec. A.2.2. Instead the speed graphs in AMD™ CPUs are almost linear, and this is well explained by the average number of needed operations. Summarizing, the speeds are quite difficult to predict; if a uniform random generator is to be used for $n$ in a certain range, and the back-end RNG takes approximatively as much time as 4 integer operations in 64bits, then the only sure way to decide which algorithm is the fastest one is by benchmarking. If a a uniform random generator is to be used for a constant and specific $n$ (such as in the case of the NextCard function), there may be different strategies to implement it, and again the only sure way to decide which algorithm is the fastest one is by benchmarking. ## Appendix A Test results ### A.1 Speed of back-end RNGs and _Ad hoc_ functions These tables list the average time (in nanoseconds) of the back-end RNGs and the _ad hoc_ functions (see the C code in sec. B.2); these same data are plotted as red crosses in the plots of the next section. For each family, the fastest function is marked blue; competitors that differ less than $10\%$ are italic and blue; competitors that are slower more than $50\%$ are red. | sfmt_sse | xorshift ---|---|--- | HW1 | HW2 | HW3 | HW4 | HW5 | HW6 | HW1 | HW2 | HW3 | HW4 | HW5 | HW6 Next2Bit32 | 2.6 | 3.9 | 4.9 | _4.8_ | _3.9_ | _2.6_ | 2.5 | 3.7 | _4.7_ | _4.2_ | 3.7 | 2.8 Next2Bit64 | 4.2 | 4.7 | _5.0_ | 4.4 | 3.7 | 2.5 | 3.3 | 6.3 | 4.5 | 4.0 | 3.7 | 2.4 NextBit32 | 2.5 | 3.7 | 4.6 | 4.2 | _3.7_ | _2.8_ | 2.5 | 3.6 | _4.4_ | _3.9_ | 3.6 | 2.4 NextBit32_by_mask | 2.5 | 3.7 | _4.9_ | _4.4_ | _3.7_ | _2.8_ | 2.5 | 3.6 | _4.4_ | _3.9_ | 3.6 | 2.7 NextBit64 | 3.2 | 4.6 | _4.7_ | 4.2 | 3.6 | 2.8 | 3.2 | 4.7 | 4.3 | 3.8 | 3.6 | 2.4 NextByte32 | 3.5 | 5.1 | 6.1 | 5.4 | 5.1 | 3.8 | _3.0_ | _4.4_ | 5.1 | 4.5 | _4.5_ | 3.3 NextByte64 | _3.3_ | 5.6 | _6.5_ | _5.8_ | 4.4 | 3.1 | _3.0_ | _4.4_ | 6.2 | 5.2 | 4.3 | 2.8 NextByte64_prefilled | 3.1 | 4.6 | _6.3_ | _5.6_ | _4.5_ | 3.4 | 2.9 | 4.3 | 5.8 | 6.0 | _4.6_ | _3.0_ NextCard32 | 3.5 | 5.1 | 8.0 | 7.1 | 5.5 | 7.0 | 3.4 | 4.9 | 5.6 | 5.0 | 5.2 | 6.9 NextCard32_prefilled | 6.1 | 9.0 | 24.4 | 21.7 | 11.4 | 7.1 | 6.0 | 15.6 | 22.2 | 19.7 | 11.1 | 7.1 NextCard64 | 22.0 | 32.2 | 6.5 | 5.8 | 7.0 | 4.9 | 22.4 | 32.8 | 6.6 | 5.8 | 7.0 | 4.9 NextCard64_prefilled | 22.0 | 32.2 | 37.9 | 33.5 | 20.5 | _5.4_ | 24.0 | 37.0 | 37.4 | 33.1 | 20.4 | _5.3_ NextWord32 | 4.2 | 6.2 | 7.5 | 6.7 | 6.5 | 4.5 | 3.4 | 5.0 | 5.7 | 5.0 | _5.0_ | 3.3 NextWord64 | 4.2 | _6.5_ | 6.5 | 5.8 | 5.2 | 3.7 | 4.2 | 5.9 | 5.7 | 5.0 | 4.9 | _3.4_ my_gen_rand32 | 3.7 | 5.4 | 6.5 | 5.7 | 5.4 | 3.9 | 3.5 | 5.1 | 5.0 | 4.4 | 5.4 | 3.5 my_gen_rand64 | 4.2 | 6.3 | 8.4 | 7.4 | 6.3 | 5.2 | 4.5 | 6.9 | 6.8 | 6.0 | 7.4 | 4.8 | sfmt_sse_md5 | bbs260 ---|---|--- | HW1 | HW2 | HW3 | HW4 | HW5 | HW6 | HW1 | HW2 | HW3 | HW4 | HW5 | HW6 Next2Bit32 | 18.7 | 27.5 | 30.2 | 26.7 | 32.0 | 23.1 | 55.2 | 81.0 | 32.9 | 31.4 | 36.3 | 28.6 Next2Bit64 | 11.6 | 17.0 | 17.6 | 15.6 | 18.1 | 12.9 | 35.6 | 52.2 | 18.9 | 17.5 | 20.0 | 15.2 NextBit32 | 10.5 | 15.5 | 17.2 | 15.2 | 17.8 | 12.6 | 28.7 | _42.2_ | 19.4 | 16.4 | 30.4 | 15.3 NextBit32_by_mask | 10.3 | 15.2 | 17.6 | 15.6 | 18.2 | 12.9 | 28.7 | _42.4_ | 18.9 | 16.8 | 25.0 | 15.5 NextBit64 | 7.5 | 10.9 | 11.1 | 9.8 | 11.1 | 7.8 | 19.7 | 41.5 | 11.7 | 10.4 | 11.8 | 9.8 NextByte32 | 67.6 | 99.5 | 108.1 | 95.7 | 117.9 | 84.8 | 212.5 | 315.5 | 118.2 | 113.7 | 136.3 | 102.7 NextByte64 | _36.2_ | _53.5_ | 57.1 | 50.5 | 60.5 | 43.8 | 130.6 | 192.9 | 62.4 | _59.7_ | 69.2 | _56.2_ NextByte64_prefilled | 35.7 | 52.5 | 56.9 | 50.3 | 60.6 | 43.7 | 131.0 | 192.7 | 62.6 | 56.4 | 69.7 | 54.1 NextCard32 | 56.2 | 83.3 | 88.1 | 77.9 | 96.5 | 71.7 | 173.9 | 255.6 | 97.9 | 85.9 | 110.4 | 86.4 NextCard32_prefilled | 60.7 | 89.7 | 104.3 | 92.2 | 102.9 | _37.6_ | 176.3 | 259.6 | 115.6 | 106.3 | 117.5 | 45.3 NextCard64 | 46.4 | 68.1 | 43.5 | 38.4 | 49.1 | 34.8 | 119.0 | 193.4 | 48.5 | 43.1 | 55.3 | 40.7 NextCard64_prefilled | 46.0 | 67.5 | 74.7 | 66.0 | 62.0 | _35.2_ | 118.7 | 173.6 | 79.2 | 72.9 | 68.1 | _43.8_ NextWord32 | 132.7 | 195.7 | 209.2 | 185.4 | 228.3 | 166.7 | 429.4 | 633.0 | 236.0 | 204.2 | 265.3 | 202.9 NextWord64 | 69.3 | 102.8 | 108.1 | 95.6 | 117.8 | 84.5 | 264.2 | 387.4 | 117.9 | 105.6 | 136.2 | 108.7 my_gen_rand32 | 262.5 | 391.5 | 413.3 | 366.7 | 457.3 | 328.7 | 859.8 | 1263.3 | _459.2_ | 403.1 | 525.4 | 399.5 my_gen_rand64 | 263.0 | 390.4 | 413.1 | 368.4 | 457.7 | 328.4 | 1018.1 | 1513.1 | 454.3 | 402.9 | 523.9 | 398.5 | counter | | | | | | ---|---|---|---|---|---|---|--- | HW1 | HW2 | HW3 | HW4 | HW5 | HW6 | | | | | | Next2Bit32 | 2.5 | 3.6 | _4.6_ | _4.1_ | _3.6_ | _2.4_ | | | | | | Next2Bit64 | 3.4 | 4.9 | 4.4 | 3.9 | 3.6 | 2.4 | | | | | | NextBit32 | 2.4 | 3.6 | _4.3_ | _3.8_ | 3.6 | 2.4 | | | | | | NextBit32_by_mask | 2.4 | 3.6 | _4.4_ | _3.9_ | 3.6 | 2.4 | | | | | | NextBit64 | 3.1 | 4.6 | 4.2 | 3.7 | 3.6 | 2.7 | | | | | | NextByte32 | _2.9_ | 4.3 | 4.8 | 4.2 | 4.3 | 2.8 | | | | | | NextByte64 | _2.9_ | 4.3 | 5.3 | 4.7 | _3.8_ | _2.9_ | | | | | | NextByte64_prefilled | 2.8 | 3.8 | _5.2_ | _4.6_ | 3.6 | _3.0_ | | | | | | NextCard32 | 2.8 | 4.2 | 5.1 | 4.5 | 4.5 | 6.4 | | | | | | NextCard32_prefilled | 5.2 | 7.7 | 21.9 | 19.4 | 8.9 | 6.9 | | | | | | NextCard64 | 22.0 | 32.7 | 6.0 | 5.4 | 6.0 | 4.2 | | | | | | NextCard64_prefilled | 21.7 | 31.9 | 36.9 | 32.7 | 19.7 | 4.9 | | | | | | NextWord32 | 2.9 | 4.3 | _5.0_ | _4.4_ | 4.8 | 3.2 | | | | | | NextWord64 | 3.3 | 4.9 | 4.6 | 4.1 | 4.0 | 2.7 | | | | | | my_gen_rand32 | 2.0 | 3.0 | 3.6 | 3.2 | 3.5 | 2.0 | | | | | | my_gen_rand64 | 2.4 | 3.5 | 3.6 | 3.2 | 3.5 | 2.3 | | | | | | ### A.2 Graphs In all of the following graphs, the abscissa is $n$, (that is the modulus of the uniform RNGs); the abscissa is in log-scale (precisely, it contains all $n$ from 2 to 32, and then $n$ is incremented by $\lfloor n/32\rfloor$ up to $2^{32}$, for a total of 733 samples). The number in parentheses near the graph labels are the average time for call (in nanoseconds; averaged in the aforementioned log scale). #### A.2.1 Uniform random generators To reduce the size of the labels, we abbreviated uniform_random_by_bit_recycling as bbr, and uniform_random_simple as simple. #### SFMT #### xorshift #### SFMT + MD5 #### bbs260 #### A.2.2 Integer arithmetic typedef uint64_t st; st div32(uint32_t n) { return my_gen_rand32() / n ; } st div32_24(uint32_t n) { return (my_gen_rand32() & 0xFF000000) / n ; } st div48(uint32_t n) { uint64_t r = my_gen_rand32(), m = r << 16; return m / n ; } st div64(uint32_t n) { uint64_t m = my_gen_rand64() / n ; return m; } st mod32(uint32_t n) { return my_gen_rand32() % n ; } st mod32_24(uint32_t n) { return (my_gen_rand32() & 0xFF000000) % n ; } st mod48(uint32_t n) { uint64_t r = my_gen_rand32(), m = r << 16; return m % n ; } st mod64(uint32_t n) { uint64_t m =my_gen_rand64() % n ; return m; } st prod32(uint32_t n) { return my_gen_rand32() * n ; } st prod32_24(uint32_t n) { return (my_gen_rand32() & 0xFF) * n ; } st prod48(uint32_t n) { uint64_t r = my_gen_rand32(), m = r << 16; return m * n ; } st prod64(uint32_t n) { uint64_t m = my_gen_rand64() * n ; return m; } ## Appendix B Code The complete C code is available on request. The code that we wrote is licensed according to the _Gnu Public License v2.0_. (The _SFMT_ code is licensed according to a modified BSD license — they are considered to be compatible). In the following code, the macro COUNTBITS was used in testing efficiency; and was disabled while testing speeds. ### B.1 Back-end RNGs // //to compile this code, define RNG, by using ’gcc .... -DRNG=n’, where n is // 1 -> SFMT , by M. Saito and M. Matsumoto // 2 -> SFMT + md5 // 3 -> xorshift , by Marsaglia // 4 -> Blum Blum Shub with ~128bit (product of two ~31bit primes) (only on amd64, using gcc 128 int types) // 5 -> Blum Blum Shub with ~260bit modulus (product of two ~130bit primes) //disclaimer: methods 2,4,5 are not guaranteed to generate high quality pseudonumbers; they were used // only to test the code speed /*************** SFMT ***/ #if 1==RNG #ifdef HAVE_SSE2 char *RNGNAME="SFMT (sse)"; char *RNGNICK="sfmt_sse"; #else char *RNGNAME="SFMT"; char *RNGNICK="sfmt"; #endif #include "SFMT.h" uint32_t my_gen_rand32() { uint32_t r=gen_rand32(); COUNTBITS(32); return r; } uint64_t my_gen_rand64() { uint64_t r=gen_rand64(); COUNTBITS(64); return r; } void my_init_gen_rand(uint32_t seed) { init_gen_rand(seed); } /*************** SFMT + md5 ***/ #elif 2==RNG #ifdef HAVE_SSE2 char *RNGNAME="SFMT (sse) + md5"; char *RNGNICK="sfmt_sse_md5"; #else char *RNGNAME="SFMT + md5"; char *RNGNICK="sfmt_md5"; #endif #include "SFMT.h" #include "md5.h" uint32_t my_gen_rand32() { uint32_t I[8]; for(int i=0;i<8;i++) I[i]=gen_rand32(); unsigned char *UCp, output[16]; UCp=(unsigned char*)I; md5(UCp, 32, output); uint32_t *U32p=(uint32_t *)output, r=*U32p; COUNTBITS(32); return r; } uint64_t my_gen_rand64() { uint32_t I[8]; for(int i=0;i<8;i++) I[i]=gen_rand32(); unsigned char *UCp, output[16]; UCp=(unsigned char*)I; md5(UCp, 32, output); uint64_t *U64p=(uint64_t *)output, r=*U64p; COUNTBITS(64); return r; } void my_init_gen_rand(uint32_t seed) { if(md5_self_test(0)) exit(4); init_gen_rand(seed); } /*************** xorshift , by Marsaglia***/ #elif 3==RNG char *RNGNAME="xorshift"; char *RNGNICK="xorshift"; static uint32_t __xorshift__x = 123456789; static uint32_t __xorshift__y = 362436069; static uint32_t __xorshift__z = 521288629; static uint32_t __xorshift__w = 88675123; uint32_t my_gen_rand32(void) { uint32_t t; t = __xorshift__x ^ (__xorshift__x << 11); __xorshift__x = __xorshift__y; __xorshift__y = __xorshift__z; __xorshift__z = __xorshift__w; COUNTBITS(32); return __xorshift__w = __xorshift__w ^ (__xorshift__w >> 19) ^ (t ^ (t >> 8)); } uint64_t my_gen_rand64() { uint64_t a=my_gen_rand32(), r = (a << 32) | my_gen_rand32(); return r; } void my_init_gen_rand(uint32_t seed) { __xorshift__x = 123456789 ^ seed; __xorshift__y = 362436069; __xorshift__z = 521288629; __xorshift__w = 88675123; } /******************** Blum Blum Shub (needs amd64 arch) *******************/ #elif 4==RNG char *RNGNAME="Blum Blum Shub (64bit)"; char *RNGNICK="bbs64"; uint64_t my_seed=0x987fed5; typedef __uint128_t uint128_t; uint32_t my_gen_rand32(void) { const uint64_t p = 4222234259UL; //~ 2^31.9 const uint64_t q = 4222231271UL; const uint64_t M = p * q ; const uint64_t mask = 0xFFFFFFFFFFFFFFFFUL; uint128_t a = my_seed, b = a * a , my_seed = b % M; my_seed = c & mask ; COUNTBITS(32); return my_seed & 0xFFFFFFFFF; } uint64_t my_gen_rand64() { uint64_t a=my_gen_rand32(), r = (a << 32) | my_gen_rand32(); return r; } void my_init_gen_rand(uint32_t seed) { my_seed=seed ^ 0x987fed5 ; } /***********************************************************/ /******************** Blum Blum Shub *******************/ #elif RNG==5 char *RNGNAME="Blum Blum Shub (260bit)"; char *RNGNICK="bbs260"; #include "gmp.h" mpz_t bbs__pq__ , bbs__n__, bbs__64__, bbs ; int initialized = 0; void my_init_gen_rand(uint32_t seed) { if(!initialized) { //http://primes.utm.edu/lists/small/small.html mpz_t p; mpz_init_set_str (p, "3615415881585117908550243505309785526231", 10); assert(mpz_probab_prime_p(p,12)); mpz_t q; mpz_init_set_str (q, "5570373270183181665098052481109678989411", 10); assert(mpz_probab_prime_p(q,12)); mpz_mul(bbs__pq__ ,p ,q); mpz_clear(p); mpz_clear(q); mpz_init(bbs__n__); initialized=1; } unsigned long s=seed+0x100000000; mpz_set_ui(bbs__n__, s); } void bbs_step() { mpz_t sqr; mpz_init(sqr); mpz_mul(sqr, bbs__n__, bbs__n__); mpz_tdiv_r(bbs__n__, sqr, bbs__pq__); mpz_clear(sqr); } uint32_t my_gen_rand32() { bbs_step(); unsigned long int r=mpz_get_ui(bbs__n__); COUNTBITS(32); if( sizeof(unsigned long int) == 4) { return r; } else { uint32_t rr=r & 0xFFFFFFFF; return rr; } } uint64_t my_gen_rand64() { bbs_step(); unsigned long int r=mpz_get_ui(bbs__n__); COUNTBITS(64); if ( sizeof(unsigned long int) == 8 ) { return r; } else { mpz_t q; mpz_init(q); mpz_tdiv_q_2exp (q, bbs__n__, 32); uint64_t r2=mpz_get_ui(q); mpz_clear(q); uint64_t rr=r | (r2 << 32); return rr; } } /*************** a counter , to test speeds ***/ #elif 11==RNG char *RNGNAME="counter"; char *RNGNICK="counter"; static uint32_t my_seed32=0; static uint64_t my_seed64=0; uint32_t my_gen_rand32(void) { COUNTBITS(32); return my_seed32 += 0x4c1; } uint64_t my_gen_rand64() { COUNTBITS(64); return my_seed64 += 0x4c7000004c1; } void my_init_gen_rand(uint32_t seed) { my_seed32=seed; my_seed64=seed; } #else #error "please define RNG" #endif // ### B.2 _ad hoc_ functions // unsigned int NextByte32() { static int l=0; static uint32_t R=0; if(unlikely(l<=0)) { R=my_gen_rand32(); l=4; } unsigned int byte=R&255; l--; if(l) R>>=8; return byte; } unsigned int NextByte64() { static int l=0; static uint64_t R=0; if(unlikely(l<=0)) { R=my_gen_rand64(); l=8; } unsigned int byte=R & 255; l--; if(l) R>>=8; return byte; } unsigned int __saved_bytes[8]; unsigned int NextByte64_prefilled() { static int l=0; if(unlikely(l<=0)) { uint64_t R=my_gen_rand64(); __saved_bytes[0] = R & 255; R >>= 8; __saved_bytes[1] = R & 255; R >>= 8; __saved_bytes[2] = R & 255; R >>= 8; __saved_bytes[3] = R & 255; R >>= 8; __saved_bytes[4] = R & 255; R >>= 8; __saved_bytes[5] = R & 255; R >>= 8; __saved_bytes[6] = R & 255; R >>= 8; __saved_bytes[7] = R & 255; l=8; } l--; return __saved_bytes[l]; } unsigned int NextWord32() { static int l=0; static uint32_t R=0; if(l<=0) { R=my_gen_rand32(); l=2; } unsigned int bytes=R & 0xFFFF; R>>=16; l--; return bytes; } unsigned int NextWord64() { static int l=0; static uint64_t R=0; if(unlikely(l<=0)) { R=my_gen_rand64(); l=4; } unsigned int bytes=R & 0xFFFF; R>>=16; l--; return bytes; } unsigned int Next2Bit32() { static int l=0; static uint32_t R=0; if(unlikely(l<=0)) { R=my_gen_rand32(); l=16; } unsigned int bit=R&3; R>>=2; l--; return bit; } unsigned int Next2Bit64() { static int l=0; static uint64_t R=0; if(unlikely(l<=0)) { R=my_gen_rand64(); l=32; } unsigned int bit=R&3; R>>=2; l--; return bit; } const static uint32_t bitmasks[32] = { 0x1, 0x2, 0x4, 0x8 , 0x10, 0x20, 0x40, 0x80, 0x100, 0x200, 0x400, 0x800, 0x1000, 0x2000, 0x4000, 0x8000, 0x10000, 0x20000, 0x40000, 0x80000, 0x100000, 0x200000, 0x400000, 0x800000, 0x1000000, 0x2000000, 0x4000000, 0x8000000, 0x10000000, 0x20000000, 0x40000000, 0x80000000 }; unsigned int NextBit32_by_mask() { static int l=0; static uint32_t R=0; if(unlikely(l<=0)) { R=my_gen_rand32(); l=32; } l--; return (R & bitmasks[l]) ? 1 : 0; } unsigned int NextBit32() { static int l=0; static uint32_t R=0; if(unlikely(l<=0)) { R=my_gen_rand32(); l=32; } unsigned int bit=R&1; R>>=1; l--; return bit; } unsigned int NextBit64() { static int l=0; static uint64_t R=0; if(unlikely(l<=0)) { R=my_gen_rand64(); l=64; } unsigned int bit=R&1; R>>=1; l--; return bit; } //draws one card from a deck of 52 cards, using 32bit variables unsigned int NextCard32() { static int l=0; static uint32_t R=0; if(unlikely(l<=0)) { R=my_gen_rand32(); l=5; } l--; unsigned int c = R % 52; R /= 52; return c; } //draws one card from a deck of 52 cards, using 64 bit variables unsigned int NextCard64() { static int l=0; static uint64_t R=0; if(unlikely(l<=0)) { R=my_gen_rand64(); l=11; } l--; unsigned int c = R % 52; R /= 52; return c; } //draws one card from a deck of 52 cards, using 32 bit variables //and prefilling an array in memory unsigned char __32saved_cards[5]; unsigned int NextCard32_prefilled() { static int l=0; if(unlikely(l<=0)) { uint32_t R=my_gen_rand32(); //52**5 < 2**32 __32saved_cards[0] = R % 52; R /= 52; __32saved_cards[1] = R % 52; R /= 52; __32saved_cards[2] = R % 52; R /= 52; __32saved_cards[3] = R % 52; R /= 52; __32saved_cards[4] = R % 52; l=5; } l--; return __32saved_cards[l]; } //draws one card from a deck of 52 cards, using 64 bit variables //and prefilling an array in memory unsigned char __64saved_cards[11]; unsigned int NextCard64_prefilled() { static int l=0; if(unlikely(l<=0)) { uint64_t R=my_gen_rand64(); //52**11 < 2**64 __64saved_cards[0] = R % 52; R /= 52; __64saved_cards[1] = R % 52; R /= 52; __64saved_cards[2] = R % 52; R /= 52; __64saved_cards[3] = R % 52; R /= 52; __64saved_cards[4] = R % 52; R /= 52; __64saved_cards[5] = R % 52; R /= 52; __64saved_cards[6] = R % 52; R /= 52; __64saved_cards[7] = R % 52; R /= 52; __64saved_cards[8] = R % 52; R /= 52; __64saved_cards[9] = R % 52; R /= 52; __64saved_cards[10] = R % 52; l=11; } l--; return __64saved_cards[l]; } // ### B.3 Uniform random generators In the following code, the macro COUNTFAILURES was used in testing efficiency; and was disabled while testing speeds. // /********** uniform_random_by_bit_recycling this implements the pseudocode in page 5 (but pops 2 bits at a time) *******************/ uint32_t uniform_random_by_bit_recycling(uint32_t n) { static uint64_t m = 1, r = 0; const uint64_t N62=((uint64_t)1)<<62; while(1) { while(m<N62) { //fill the state r = (r<<2) | Next2Bit(); m = m<<2; } const uint64_t q=m/n, nq = n * q; if( likely(r < nq) ) { uint32_t d = r % n; //remainder, is a random variable of modulus n r = r/n; //quotient, is random variable of modulus q m = q; return d; } else { COUNTFAILURES(); m = m - nq; r = r - nq; // r is still a random variable of modulus m } } } /********** uniform_random_by_bit_recycling this implements the pseudocode in page 5 , but pops words,bytes,and pairs of bits *******************/ uint32_t uniform_random_by_bit_recycling_faster(uint32_t n) { static uint64_t m = 1, r = 0; const uint64_t N62=((uint64_t)1)<<62, N56=((uint64_t)1)<<56, N48=((uint64_t)1)<<48; while(1) { //fill the state if(m<N48) { r = (r<<16) | NextWord(); m = m<<16; } while(m<N56) { r = (r<<8) | NextByte(); m = m<<8; } while(m<N62) { r = (r<<2) | Next2Bit(); m = m<<2; } const uint64_t q=m/n, nq=n*q; if( likely(r < nq) ) { uint32_t d = r % n; //remainder, is a random variable of modulus n r = r/n; //quotient, is random variable of modulus q m = q; return d; } else { COUNTFAILURES(); m = m - nq; r = r - nq; // r is still a random variable of modulus m } } } /********** uniform_random_by_bit_recycling_cheating this implements the pseudocode in page 6 , but does not implement the "else" block, so it is not mathematically perfect; at the same time, since the probability of the "else" block would be less than 1/2^24 the random numbers generated by this function are good enough for most purposes *******************/ uint32_t uniform_random_by_bit_recycling_cheating(uint32_t n) { static uint64_t m = 1, r = 0; const uint64_t N48=((uint64_t)1)<<48, N56=((uint64_t)1)<<56; if(m<N48) { //fill the state r = (r<<16) | NextWord(); m = m<<16; } while(m<N56) { r = (r<<8) | NextByte(); m = m<<8; } uint32_t d = r % n; r = r/n; m = m/n; return d; } /********** uniform_random_by_bit_recycling32 this implements the pseudocode in page5 but only using 32bit variables, so it has some special methods when n>=2^29 *******************/ uint32_t uniform_random_by_bit_recycling32(uint32_t n) { const uint32_t N29=((uint32_t)1)<<29, N30=((uint32_t)1)<<30, N31=((uint32_t)1)<<31, N24=((uint32_t)1)<<24; //special methods if(n>=N31) { while(1) { uint32_t r = my_gen_rand32(); if( likely(r < n) ) { return r; } else { COUNTFAILURES(); } }} if(n>=N30) { while(1) { uint32_t r = my_gen_rand32() >> 1; if( likely(r < n) ) { return r; } else { COUNTFAILURES(); } }} if(n>=N29) { while(1) { uint32_t r = my_gen_rand32() >> 2; if( likely(r < n) ) { return r; } else { COUNTFAILURES(); } }} //usual bit recycling static uint32_t m = 1, r = 0; while(1) { while(m<N24) { //fill the state r = (r<<8) | NextByte(); m = m<<8; } while(m<N30) { //fill the state r = (r<<2) | Next2Bit(); m = m<<2; } uint32_t q=m/n, nq=q*n; if(likely( r < nq) ) { uint32_t d = r % n; //remainder, is a random variable of modulus n r = r/n; //quotient, is random variable of modulus q m = q; return d; } else { COUNTFAILURES(); m = m - nq; r = r - nq; // r is still a random variable of modulus m } } } /********** uniform_random_simple_64 this is a simple implementation, found in many random number libraries this version uses 64 bit variables *******************/ uint32_t uniform_random_simple_64(uint32_t n) { const uint64_t N=0xFFFFFFFFFFFFFFFFU; //2^64-1; uint64_t q = N/n, nq=((uint64_t)n) * q; while(1) { uint64_t r = my_gen_rand64(); if( likely(r < nq) ) { uint32_t d = r % n; //remainder, is a random variable of modulus n return d; } else { COUNTFAILURES(); } } } /********** uniform_random_simple this is a simple implementation, found in many random number libraries this version uses 32 bit variables *******************/ uint32_t uniform_random_simple_32(uint32_t n) { const uint32_t N=0xFFFFFFFFU; // 2^32-1; uint32_t q = N/n; while(1) { uint32_t r = my_gen_rand32(); if( likely(r < n * q)) { uint32_t d = r % n; //remainder, is a random variable of modulus n return d; } else { COUNTFAILURES(); } } } /********** uniform_random_simple_recycler this is a simple implementation, with recycling for small n this version uses 32 bit variables *******************/ uint32_t uniform_random_simple_recycler(uint32_t n) { static uint32_t _r=0, _m=0; const uint32_t N=0xFFFFFFFFU; //2^32-1 if (_m>n) { uint32_t d = _r % n; _m/=n; _r/=n; return d; } const uint32_t q = N/n, nq=n*q; while(1) { uint32_t newr = my_gen_rand32(); if(newr < nq) { uint32_t d = newr % n; //newr is a random variable of modulus n if(_m<q) { //there is more entropy in newr than in _r _m=q; _r=newr/n; } return d; } else { COUNTFAILURES(); } } } /********** uniform_random_simple some alternative versions, using 64 bit variables *******************/ uint32_t uniform_random_simple_40(uint32_t n) { const uint64_t N=((uint64_t)1)<<40; uint64_t q = N/n, nq=((uint64_t)n) * q; while(1) { uint64_t r = my_gen_rand32(); r=(r<<8) | NextByte(); //create 40 bits random numbers if( likely(r < nq) ) { uint32_t d = r % n; //remainder, is a random variable of modulus n return d; } else { COUNTFAILURES(); } } } uint32_t uniform_random_simple_48(uint32_t n) { const uint64_t N=((uint64_t)1)<<48; uint64_t q = N/n, nq=((uint64_t)n) * q; while(1) { uint64_t r = my_gen_rand32(); r=(r<<16) | NextWord(); //create 48 bits random numbers if( likely(r < nq) ) { uint32_t d = r % n; //remainder, is a random variable of modulus n return d; } else { COUNTFAILURES(); } } } // ## Acknowledgments The author thanks Professors S. Marmi and A. Profeti for allowing access to hardware. ## Bibliography ## References * [1] “Doctor Jacques” at the Math forum http://mathforum.org/library/drmath/view/65653.html (2004) * [2] Mutsuo Saito and Makoto Matsumoto, _SIMD oriented Fast Mersenne Twister(SFMT): a 128-bit Pseudorandom Number Generator_ Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, 2008, pp. 607 – 622. DOI 10.1007/978-3-540-74496-2_36. Code (ver. 1.3.3 ) available from http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/SFMT/index.html (See Mutsuo Saito’s Master’s Thesis for more detailed information). * [3] Mutsuo Saito and Makoto Matsumoto, _"A PRNG Specialized in Double Precision Floating Point Number Using an Affine Transition"_ , Monte Carlo and Quasi-Monte Carlo Methods 2008, Springer, 2009, pp. 589 – 602. DOI 10.1007/978-3-642-04107-5_38 * [4] George Marsaglia _Xorshift RNGs_ Journal of Statistical Software, 8, 1-9 (2003). http://www.jstatsoft.org/v08/i14/. * [5] L. Blum, M. Blum, and M. Shub, _A Simple Unpredictable Pseudo-Random Number Generator_ SIAM J. Comput. 15, 364 (1986), DOI 10.1137/0215025
arxiv-papers
2010-12-20T11:27:57
2024-09-04T02:49:15.837560
{ "license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/", "authors": "Andrea C. G. Mennucci", "submitter": "Andrea Carlo Giuseppe Mennucci", "url": "https://arxiv.org/abs/1012.4290" }
1012.4292
# Infall and outflow detections in a massive core JCMT 18354-0649S Tie Liu11affiliation: Department of Astronomy, Peking University, 100871, Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn , Yuefang Wu11affiliation: Department of Astronomy, Peking University, 100871, Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn , Qizhou Zhang22affiliation: Harvard- Smithsonian Center for Astronomy, 60 Garden St., Cambridge, MA 02138, USA , Zhiyuan Ren11affiliation: Department of Astronomy, Peking University, 100871, Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn , Xin Guan11affiliation: Department of Astronomy, Peking University, 100871, Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn 33affiliation: Current affiliation: I. Physikal. Institut, Universität zu Köln, Zülpicher St.77,D-50937 Köln, Germany and Ming Zhu44affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012 ###### Abstract We present a high-resolution study of a massive dense core JCMT 18354-0649S with the Submillimeter Array. The core is mapped with continuum emission at 1.3 mm, and molecular lines including CH3OH ($5_{23}$–$4_{13}$) and HCN (3–2). The dust core detected in the compact configuration has a mass of $47~{}M_{\odot}$ and a diameter of $2\arcsec$ (0.06 pc), which is further resolved into three condensations with a total mass of $42~{}M_{\odot}$ under higher spatial resolution. The HCN (3–2) line exhibits asymmetric profile consistent with infall signature. The infall rate is estimated to be $2.0\times 10^{-3}~{}M_{\odot}\cdot$yr-1. The high velocity HCN (3-2) line wings present an outflow with three lobes. Their total mass is $12~{}M_{\odot}$ and total momentum is $121~{}M_{\odot}\cdot$km s-1, respectively. Analysis shows that the N-bearing molecules especially HCN can trace both inflow and outflow. Massive core:pre-main sequence-ISM: molecular-ISM: kinematics and dynamics- ISM: jets and outflows-stars: formation ††slugcomment: Accepted by APJ ## 1 Introduction Studies of high-mass star formation have received much attention during recent years. One of the main questions is whether massive stars form through an accretion-disk-outflow process, similar to low-mass counterparts (Shu, Adams, & Lizano 1987), or via collision-coalescence (Wolfire & Cassinelli 1987; Bonnell, Bate, & Zinnecker. 1998). Studying the characteristics of massive cores at the early stages is critical for understanding their formation process. High-Mass Protostellar Objects (HMPOs) are precursors of UC Hii regions, and represent an essential phase in high-mass star formation (Churchwell 2002). HMPOs often have strong dust emission and high bolometric luminosity. But their radio emission is weak or non detectable at a level of approximately 1 mJy (Molinari et al. 1996, 2000; Sridharan et al. 2002; Beuther, Schilke, & Menten. 2002; Wu et al. 2006).. Their natal clouds have not been affected significantly by the star forming process. Thus, they present the information about the early kinematic processes of high mass star formation. The dense core JCMT 18354-0649S was first detected in an ammonia survey of high-mass star forming regions with Max-Planck-Institut für Radioastronomie (MPIfR) 100 m telescope at Effelsberg (Wu et al. 2006), and was later confirmed by the observation with the Submillimeter Common-User Bolometric Array (SCUBA) of James Clerk Maxwell telescope (JCMT) (Wu et al. 2005). Another SCUBA core which harbors a UC H ii region G35.4NW is located about $1\arcmin$ north of JCMT 18354-0649S. The kinetic distance of the two SCUBA cores is 5.7 or 9.6 kpc (Wu et al. 2005), and 5.7 kpc was adopted in this paper. Core JCMT 18354-0649S has no counterpart in radio continuum. Multiple lines towards this core including HCN (3–2), H13CO+ (3–2), and C17O (2–1) reveal typical ”blue profile” (Wu et al. 2005), indicating that the core is undergoing gravitational collapse (Keto, Ho & Haschick. 1988; Zhou et al. 1993; Zhang, Ho, & Ohashi. 1998; Wu & Evans. 2003; Wu et al. 2005; Fuller, Williams, & Sridharan. 2005; Wyrowski 2006; Wu et al. 2007; Birkmann et al. 2007; Klaassen & Wilson 2007; Sun & Gao 2008; Velusamy et al. 2008). The core is also associated with a near-infrared point source, corresponding to a star of 6-11 $M_{\odot}$ (Zhu et al.2010, submitted). Carolan et al. (2009) observed sixteen different molecular line transitions including CO, HCN, HCO+ and their isotopes in this region, and modeled the source with a chemically depleted rotating envelope collapsing onto a central protostellar source which has evolved sufficiently to generate a molecular outflow. All the evidence suggests that JCMT 18354-0649S is forming high mass protostellar object(s)(HMPO). However, single-dish observations with a resolution of $15\arcsec-40\arcsec$ can not reveal detailed kinematics in the core at a distance as large as 5.7 kpc. In this paper we report the results of a high resolution study with the Submillimeter Array (SMA111Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.) in order to probe the details of the core and its kinematic features. The observation and initial results are presented in sections 2 and 3. Properties of infall and outflow motions are discussed further in section 4, and a brief summary is given in section 5. ## 2 Observations The observations of JCMT 18354-0649S was carried out with the SMA in July 2005 with seven antennas in its compact configuration and in September 2005 with six antennas in its extended configuration. The 345 GHz receivers were tuned to 265 GHz for the lower sideband (LSB) and 275 GHz for the upper sideband (USB). The frequency spacing across the spectral band is 0.8125 MHz or $\sim$1 km s-1 for both configurations. The phase reference center of both observations was R.A.(J2000) = 18h38m08.10s and DEC.(J2000) = -$6\arcdeg 46\arcmin 52.17\arcsec$. In the observations with the compact configuration, Jupiter, Uranus and QSO 3c454.3 were observed for antenna-based bandpass correction. An amplitude offset was found on some baselines and the baseline-based errors in bandpass were further corrected using the point source QSO 3c454.3. QSOs 1741-038 and 1908-201 were employed for antenna-based gain correction. Uranus was observed for flux-density calibration. The synthesized beam size is $3.76\arcsec\times 2.72\arcsec$ (PA=-54$\arcdeg$). For the extended configuration, QSO 3c454.3 was used as bandpass calibrator, QSOs 1741-038 and 1908-201 as gain calibrators, and Uranus as a flux calibrator, respectively. The synthesized beam size is about $1\arcsec$. Miriad222http://carma.astro.umd.edu/miriad was employed for calibration and imaging. The 1.3 mm continuum data were acquired by averaging all the line- free channels over both the 2 GHz of upper and lower spectral bands. MIRIAD task ”selfcal” was employed to perform self-calibration on the continuum data. Since the dust emission is weak, self-calibration with phase only was performed. The gain solutions from the self-calibration were applied to the line data. The continuum data combined from both configurations yield a synthesized beam of $1.63\arcsec\times 1.28\arcsec$ (PA=-81.4$\arcdeg$), and 1 $\sigma$ rms of 2.5 mJy in the naturally weighted maps. HCN (3-2) and CH3OH ($5_{23}$–$4_{13}$) were detected in the compact configuration. The shortest baseline in compact configuration observations is 16.5 m, corresponding to a spatial scale of $20\arcsec$. Spatial structures more extended than this limit, such as HCN maps close to the cloud velocity, would be filtered out. HCN (3–2) data in JCMT archive (Wu et al. 2005; Carolan et al. 2009) were used to recover the missing flux. The JCMT archive data were reduced using the KAPPA and GAIA packages in the STARLINK suite. The JCMT beam size for HCN (3–2) was $18.3\arcsec$, and the main beam efficiency was 0.69. The combination of the SMA compact and JCMT HCN (3–2) data was done using the task ”immerge” in MIRIAD. ## 3 Results ### 3.1 Dust core The 1.3 mm continuum images are shown in Fig.1. The left panel is obtained in the compact array and the right panel from the combined data of the compact and extended configurations. An elongated core is revealed with the 1.3 mm continuum emission observed with the compact array, and is further resolved into three condensations by the continuum emission using the combined data from both configurations. The three condensations are named as MM1, MM2 and MM3. The peak position of MM1 is R.A.(J2000)=$18^{\rm h}38^{\rm m}08.1^{\rm s}$, DEC.(J2000)=$-6\arcdeg 46\arcmin 52.98\arcsec$. MM2 peaks at R.A.(J2000)=$18^{\rm h}38^{\rm m}07.9^{\rm s}$, DEC.(J2000)=$-6\arcdeg 46\arcmin 51.36\arcsec$, and MM3 peaks at R.A.(J2000)=$18^{\rm h}38^{\rm m}08.05^{\rm s}$, DEC.(J2000)=$-6\arcdeg 46\arcmin 51.36\arcsec$. From the fit of an elliptical Gaussian, the core revealed by the compact array is found to be elongated from south-east to north-west. It has an average FWHM diameter of 0.06 pc ($\sim 2\arcsec$) at a distance of 5.7 kpc, smaller than the beam size of the compact configuration. The total integrated flux is 0.47 Jy. The total dust and gas mass can be obtained with the formula $M=S_{\nu}D^{2}/\kappa_{\nu}B_{\nu}(T_{d})$, where $S_{\nu}$ is the flux at 1.3 mm, D is the distance, and $B_{\nu}(T_{d})$ is the Planck function. We adopt a dust opacity $k_{1330}$=$1.4\times 10^{-2}$ cm2g-1 at 1.3 mm calculated from Ossenkopf & Henning (1994) with a dust opacity index $\beta=2$. Here the ratio of gas to dust is taken as 100. Molecular line CH3OH ($5_{23}$–$4_{13}$) is detected, and its emission peak coincides with the dust core very well (see Sec.3.2). The upper energy level of the CH3OH ($5_{23}$–$4_{13}$) line is 57 K above the ground, indicating a relatively warm conditions. Assuming $T_{d}$=57 K, a total dust and gas mass of $47~{}M_{\odot}$ is derived. A beam-average gas/dust density amounts to $2.0\times 10^{6}$ cm-3, which is larger than $1.1\times 10^{6}$ cm-3 obtained from single-dish telescope (Wu et al. 2005). The three condensations (MM1, MM2 and MM3) have total integrated flux of 0.42 Jy, leading to a total mass of $42~{}M_{\odot}$ (assuming Td = 57 K as above). MM1 has a diameter of 0.02 pc, and a mass of $30~{}M_{\odot}$. MM1 is centrally concentrated and compact, while MM2 and MM3 are much more diffuse and extended. With UKIRT (United Kingdom Infrared Telescope) Zhu et al. (2010, in preparation) detected three near-infrared sources IRS1a, IRS1b, IRS1c in H, K and L bands. Their positions are marked with crosses in Fig.1. IRS1a ($18^{\rm h}38^{\rm m}08.135^{\rm s}$, $-6\arcdeg 46\arcmin 51.57\arcsec$) lies about $1.6\arcsec$ north-east of MM1. IRS1b ($18^{\rm h}38^{\rm m}08.026^{\rm s}$, $-6\arcdeg 46\arcmin 56.24\arcsec$) and IRS1c ($18^{\rm h}38^{\rm m}07.929^{\rm s}$, $-6\arcdeg 46\arcmin 55.34\arcsec$) are about $3\arcsec$ southwest from MM1 and are much fainter. ### 3.2 Gas core The molecular line CH3OH ($5_{23}$–$4_{13}$) is detected in the compact configuration. Fig.2 presents its spectrum at three positions and the integrated emission overlaid on the 1.3 mm continuum image. The central velocity of the CH3OH ($5_{23}$–$4_{13}$) spectra is 96.7 km s-1, which is taken as the systemic velocity of the core. The central velocity of CH3OH ($5_{23}$–$4_{13}$) does not shift at different positions (see Fig.2), which should exclude rotation at the core. The P-V diagram of CH3OH ($5_{23}$–$4_{13}$) is shown in Fig.3, indicating a compact gas core without rotation. The emission center of CH3OH ($5_{23}$–$4_{13}$) (R.A.(J2000)=$18^{\rm h}38^{\rm m}08.092^{\rm s}$, DEC.(J2000)=$-6\arcdeg 46\arcmin 52.318\arcsec$) coincides with MM1 very well. While there are no CH3OH components corresponding with MM2 and MM3. The deconvolved size of the gas core revealed by CH3OH ($5_{23}$–$4_{13}$) is $3.78\arcsec\times 2.76\arcsec$ (PA=-29$\arcdeg$), comparable to the synthesized beam size of the compact array. The HCN (3–2) (265.886GHz) spectra obtained from the SMA compact configuration and from the data combined from both compact and extended configurations are presented in the left panel of Fig.4. Both of the two spectra are averaged over a region of $5\arcsec\times 5\arcsec$, which show a redshifted absorption dip and broad wings. The line profiles observed with the SMA and JCMT, as well as a combination of the two are presented in the right panel of Fig.4. All the spectra in the right panel of Fig.4 are convolved with the JCMT beam ($18.3\arcsec$) for comparing. One can see that the SMA compact array observations recover less than 10$\%$ of JCMT flux around the systematic velocity, but recover more than 30$\%$ flux at the wings. The combination of the SMA and JCMT data recovers more than 70$\%$ of the JCMT flux at all the velocity channels. ### 3.3 Kinematic signatures of lines #### 3.3.1 Infall motion The left panel of Fig.4 shows the most prominent feature (”blue profile”) of the HCN (3–2) line at the core. The absorption gap is more than 8 km s-1 wide, ranging from 93 to 101 km s-1. Fig.5 presents the channel maps of the HCN (3–2) emission from 80 km s-1 to 109 km s-1 constructed from the combined data, which is convolved with the beam of the SMA compact configuration. The absorption is obvious in the velocity range (95,99) km s-1. The absorption dip is also clearly seen in the P-V diagrams (see Fig.6), which is much deeper than that revealed by the single-dish observation (Wu et al. 2005). From the left panel of Fig.4, it is clearly to see that the centeral velocity of the absorption dip (98 km s-1) is redshifted from the systematic velocity (96.7 km s-1) by 1.3 km s-1. Such a blue asymmetric line profile where the blue emission peak is at a higher intensity than the red one is a collapse signature of molecular cores (Zhou et al. 1993). The spectra constructed from JCMT data and the combined data (the right panel of Fig.4) also show significant ”blue profile”, confirming the existence of infall motions. #### 3.3.2 Molecular outflow Besides the absorption dip, the HCN (3–2) line exhibits remarkable broad wings extending more than 40 km s-1. High-velocity gas also can be easily identified in P-V diagrams of the HCN (3–2) emission along the direction of P.A.=15$\arcdeg$ and P.A.=90$\arcdeg$ as shown in Fig.6. The HCN (3–2) emission obtained from SMA compact configuration is integrated from 80 to 87 km s-1 for the blue lobe and from 103 to 109 km s-1 for the red lobe, respectively. The contour map of the integrated flux are shown in Fig.7. As in the channel maps (Fig.5), we can see several clumps in each lobe in the integrated map. The integrated HCN (3–2) emission seems to comprise an S-shaped structure from north-east to south. Another jet-like structure extended more than $10\arcsec$ is also seen at the west of the continuum emission center. IRS1a seems to be the driving source of the outflow. The southern redshifted lobe (S-lobe) comprises two clumps named ”Clump1” and ”Clump2”. In the north-east blueshifted lobe (NE-lobe), two clumps are also found and named ”Clump3” and ”Clump4”. These clumps are distributed along the direction of the outflow and likely to be outward gas knots. They are probably not physically related with other stellar sources except the driving source though Clump2 is close to IRS1b and IRS1c. ## 4 Discussion ### 4.1 Infall motion Although the HCN emission is extended over a region larger than the compact configuration beam, the infall region is still difficult to confine due to the contamination of the outflow. Since the size of the gas core traced by CH3OH ($5_{23}$–$4_{13}$) is comparable to the compact configuration beam size, the beam size of the compact configuration was taken as the radius ($R_{in}$) of the infall region (Wu et al. 2009). The kinematic mass infall rate can be calculated using dM/dt=$4{\pi}n\mu_{G}m_{H_{2}}R_{in}^{2}V_{in}$, where $V_{in}$, $\mu_{G}=1.36$, $m_{H_{2}}$ and n=$2.0\times 10^{6}$ cm-3 are the infall velocity, the mean molecular weight, the H2 mass, and the beam-average gas/dust density, respectively. The infall velocity $V_{in}$ is 1.3 km s-1 by comparing the systemic velocity (96.7 km s-1) and the velocity of the redshifted absorbing dip (98 km s-1) in the HCN (3-2) spectrum (Welch et al. 1987), leading to a kinematic mass infall rate of 2.0$\times 10^{-3}~{}M_{\odot}\cdot$yr-1. In core G10.6-0.4, the redshifted NH3 indicates large infall velocity $5.0\pm 1.7$ km s-1 at about 0.05 pc, and a mass infall rate as high as 5$\times 10^{-3}~{}M_{\odot}\cdot$yr-1 (Keto, Ho & Haschick. 1987). Also with NH3 inverse lines, Zhang & Ho (1997) obtained a high infall velocity $\sim 3.5$ km s-1 within a region smaller than 0.02 pc towards core W51e2. Large infall velocities ($>1.5$ km s-1) and mass infall rates ($>1\times 10^{-3}~{}M_{\odot}\cdot$yr-1) were also detected towards G10.47 and G34.26 with HCO+ (4–3) line (Klaassen & Wilson 2007). In core G19.61+0.23, an infall velocity of 2.5 km s-1 and a mass infall rate as high as $6.1\times 10^{-3}~{}M_{\odot}\cdot$yr-1 were derived (Wu et al. 2009). It seems high mass infall rate is required by high-mass star formation. The results of the core JCMT 18354-0649S are comparable with those of the above sources. For comparison, the $V_{in}$ from pure free-infall assumption is also derived with the formula $V_{in}^{2}~{}=~{}2GM/R_{in}$. The pure free-infall velocity inferred is 2.9 km s-1, larger than the infall velocity obtained from the spectrum. Wu et al. (2005) obtained a small infall velocity ($\sim 0.3~{}km~{}s^{-1}$) at a radius of 4$\arcsec$. The absorption dip of HCN (3–2) line seen by the SMA is much deeper and broader than that observed by JCMT. However, the kinematic mass infall rate $\dot{M}_{in}$ ($2.0\times 10^{-3}~{}M_{\odot}\cdot$ yr-1) obtained here is well coincident with that obtained with JCMT (Wu et al. 2005), $3.4\times 10^{-3}~{}M_{\odot}\cdot$ yr-1. ### 4.2 Properties of HCN (3-2) outflow The column density of HCN at each velocity channel in each outflow lobe can be obtained through (Garden et al. 1991): $N_{HCN}(v)=\frac{3k}{8\pi^{3}B\mu^{2}}\frac{exp[hBJ(J+1)/kT_{ex}]}{(J+1)}\frac{(T_{ex}+hB/3k)}{1-exp(-h\nu/kT_{ex})}\int\tau_{v}dv$ (1) Where $v$ is the central velocity of the channel relative to the systemic velocity, the rotational constant B=44.315976 GHz and permanent dipole moment $\mu=3$ debye for HCN, the velocity channel width is smoothed to be $1~{}km~{}s^{-1}$. Assuming HCN emission in the line wings to be optically thin and excitation temperature of Tex=30 K (Wu et al. 2004), the optical depth $\tau_{v}$ can be derived with the equation: $\tau_{v}=\frac{kT_{r}(v)}{h\nu}(\frac{1}{exp(h\nu/kT_{ex})-1}-\frac{1}{exp(h\nu/kT_{bg})-1})^{-1}$ (2) where $T_{r}(v)$ is the excess brightness temperature of HCN(3–2) emission at $v$. Adopting $X_{HCN}=[HCN]/[H_{2}]=1\times 10^{-10}$ (Carolan et al. 2009), the mass of each lobe at $v$ can be calculated with: $M(v)=X_{HCN}^{-1}\mu_{G}m_{H_{2}}D^{2}{\int}N_{HCN}(v)d{\Omega}$ (3) where D, $\Omega$ are the cloud distance and the solid angle. Thus the total mass of each lobe is given by $M=\sum$$M(v)$, the total momentum by $P=\sum$$M(v)v$, and the energy by $E={\frac{1}{2}}\sum$$M(v)v^{2}$. The dynamical timescale $t_{dyn}$ is estimated as $R/V_{max}$, where R is the outflow extent, and $V_{max}$ is the maximum velocity of the outflow lobe. The mechanical luminosity L, and the mass-loss rate $\dot{M}$ are calculated as L=E/t, $\dot{M}=P/(tV_{w}$), where the wind velocity $V_{w}$ is assumed to be 500 km s-1 (Lamers et al. 1995). The derived parameters are listed in Table.1. The total mass, momentum, energy of the three lobes are 12 $M_{\odot}$, 121 $M_{\odot}\cdot$ km s-1 and $1.3\times 10^{46}$ erg, respectively. The average dynamical timescale is about $1.6\times 10^{4}$ yr, and the total mass-loss rate $1.6\times 10^{-5}~{}M_{\odot}\cdot$ yr-1. The outflow is massive with parameters similar to that of IRAS 05274+3345E and the other outflows detected towards five massive star formation regions (Zhang et al. 2007b; Klaassen & Wilson 2008). The Position-Velocity diagram at the left panel of Fig.6 shows that at low velocities, the NE-lobe and S-lobe both have compact morphology near the core center. At higher velocities the southern lobe becomes further away from the center. As shown in Fig.7, the outflow axis traced by Clump1 and Clump2 differs from that traced by Clump3 and Clump4. The different outflow orientations in the large and small scales may be attributed to the precession of the outflow axis (Su et al. 2007). From the right panel of the P-V diagram, a high-velocity component (V $<$ 85 km s-1) with velocities decreasing with distance from the protostar, and a second component tracing the low-velocity material (V $>$ 85 km s-1) extending about $15\arcsec$ along the axis of the W-lobe are clearly seen. Such convex spur PV structure was also revealed in a simulation of a pulsed jet driven outflow (Lee et al. 2001). ### 4.3 HCN — tracer of both inflow and outflow motions HCN is among the most abundant molecular species with a high critical density larger than $10^{6}$ cm-3 (for HCN (1–0)) (Carolan et al. 2009), and is believed to trace dense molecular cores. HCN is detected in both low mass class 0 and I sources (Park, Kim, & Ming. 1999; Yun et al. 1999), and high- mass hot cores (Boonman et al. 2001). HCN is thought as a good tracer of inflow motions (Wu & Evans. 2003). The infall asymmetry in the HCN spectra is found to be more prevalent, and more prominent than in any other previously used infall tracers such as CS (2–1), DCO+ (2–1), and N2H+(1–0) during a survey toward 85 starless cores (Sohn et al. 2007). Among the small group of pre- and protostellar objects in L1251B, infall signature was also detected in the HCN emission (Lee et al. 2007). HCN also traces inflow motions very well in massive star-formation regions. Wu and Evans. (2003) found 12 sources showing ”blue profile” in the HCN lines during a spectroscopic survey of 28 massive cores with water maser. Besides HCN, other nitrogen bearing molecules such as N2H+ are also tracers of inflow motions (Tsamis et al. 2008; Schnee et al. 2007; Crapsi et al. 2005). Recently inverse P Cygni profile of CN line in hot cores was found (Zapata et al. 2008; Wu et al. 2009). These results suggest nitrogen bearing molecular species be good tracers of inflowing motions in star-formation regions. Outflows traced by HCN are often detected not only in low-mass star-formation regions but also in massive star-formation regions (Bachiller, Gutiérrez, & Pérez. 1997; Choi 2001; Su et al. 2007; Zhang et al. 2007a). HCN outflow of the core JCMT 18454-0649S is another good sample. Additionally, Zhu et al. (2010 in preparation) and Cyganowski (2008) found excess emission at 4.5 $\micron$ at the position of source IRS1a, which is close to the center of the NE-lobe and S-lobe. Such excess emission at the 4.5 $\micron$ band could be shock-excited. In IRAS 20126+4104, HCN emission is also found to be closely related to the shock-excited near-IR H2 knots and was identified to be associated with shock wings (Su et al. 2007). The inner clumps (Clump1 in the S-lobe and Clump3 in the NE-lobe) of the core JCMT 18354-0649S should also be coincident with shocks. In fact, models have already demonstrated a dramatic increase of HCN molecules, during the intense interaction between outflow and ambient gas, or slow shock front (Mitchell 1984; Nejad, Williams, & Charnley. 1990). In this process sulfur and nitrogen react with hydrocarbons to produce various compounds, wherein HCN abundance gets higher than the rest of the products (Nejad, Williams, & Charnley. 1990). Thus HCN may trace the outflow even better than sulfur containing molecules. ## 5 Summary Both dust continuum at 1.3 mm and CH3OH emission detected with SMA reveal a compact core in JCMT 18354-0649S. The core observed with the compact configuration has a mass of $47~{}M_{\odot}$ and an average density of $2.0\times 10^{6}$ cm-3. With the combination of the compact and extended configurations, the core is resolved to three condensations with a total mass of $42~{}M_{\odot}$. HCN (3-2) spectra exhibit an infall signature in this region. The red shifted absorption seen in the SMA observation is deeper and broader than that in the JCMT observation. The infall rate is $2.0\times 10^{-3}~{}M_{\odot}\cdot$ yr-1. High velocity gas is detected in HCN (3-2) emission. The outflow has three lobes and their total mass is 12 $M_{\odot}$ and momentum of 121 $M_{\odot}\cdot$ km s-1. The average dynamical timescale and the total mass- loss rate are about $1.6\times 10^{4}$ yr and $1.6\times 10^{-5}~{}M_{\odot}\cdot$ yr-1, respectively. All the findings indicate a high-mass protostar is forming via rapid accretion. Our results suggest that nitrogen bearing molecules especially HCN are good for probing both infall and outflows. ## Acknowledgment We are grateful to the SMA staff. 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M. 1993, ApJ, 404, 232 Table 1: Outflow parameters of each lobe. outflow | Vmax | $t_{dyn}$ | Mass | momentum | Energy | $L$ | $\dot{M}_{out}$ ---|---|---|---|---|---|---|--- | km s-1 | (10${}^{4}~{}yr$) | ($M_{\odot}$) | ($M_{\odot}$ km s-1) | (10${}^{45}~{}erg$) | ($L_{\odot}$) | ($10^{-6}M_{\odot}$ yr-1) southern lobe | 11.7 | 1.4 | 3.3 | 29 | 3.0 | 1.8 | 4.0 north-eastern lobe | 15.3 | 1.0 | 4.2 | 42 | 4.5 | 3.6 | 8.0 western lobe | 15.3 | 2.3 | 4.6 | 50 | 5.5 | 2.1 | 4.0 Figure 1: The 1.3 mm continuum emission towards JCMT 18354-0649S. The left one is obtained with the compact configuration. The rms level is 3 mJy beam-1 (1 $\sigma$). The contours are at -6, 3, 6, 12, 21, 33, 48, 66, 87$\sigma$. The right panel gives the contours of the continuum emission combined from both configurations. The rms level is 2.5 mJy beam-1 (1 $\sigma$) and the contours are at -6, 3, 6, 12, 21, 33, 48$\sigma$. The three near-infrared sources are marked with crosses. Figure 2: The lower-left panel is the contours of the CH3OH integrated intensity overlaid on the 1.3 mm continuum image (grey scale). The contours start from 30$\%$ in steps of 10$\%$ of the peak emission (15 Jy beam${}^{-1}\cdot$km s-1). The three near-infrared sources are marked with crosses. The beam-averaged spectrum of CH3OH at three positions are presented in the other panels. The gaussian fit towards each spectrum is shown with solid lines. Figure 3: Position-velocity diagrams of CH3OH along a P.A. of 0$\arcdeg$. The contour levels are from 15% to 90% in steps of 15% of the peak intensity in both panels. The intensity at the peak is 3.73 Jy beam-1. Figure 4: The HCN (3-2) spectra. Left: the solid black line exhibits the spectrum constructed from SMA compact array and the dashed line shows the spectrum obtained from combining the compact and extended data together. Both of the two spectra are integrated over a region of $5\arcsec\times 5\arcsec$. Right: the solid black line shows the spectrum constructed from the combined SMA and JCMT data, which is convolved with the JCMT beam ($18.3\arcsec$); the dashed line shows the spectrum from JCMT only; the dash-dotted gray line shows the spectrum obtained with the SMA compact array and convolved with the JCMT beam. The vertical dashed lines in both panels mark the position of the systematic velocity (96.7 km s-1). Figure 5: Combined JCMT and SMA HCN (3-2) channel maps from 80 km s-1 to 109 km s-1, which is convolved with the beam of the SMA compact configuration ($3.76\arcsec\times 2.72\arcsec$, PA=-54$\arcdeg$). The contours are in steps of 0.5 Jy beam-1 (3$~{}\sigma$) from 0.5 Jy beam-1 (3$~{}\sigma$). The velocity of each channel is plotted at the upper-left of each panel, and the beam size at the lower-right. The positions of MM1, MM2 and MM3 are marked with crosses. Figure 6: Position-velocity diagrams of HCN (3-2) outflow observed by the SMA along a P.A. of 15$\arcdeg$ (left panel) and 90$\arcdeg$ (right panel). The image is smoothed to 2 km s-1 velocity resolution. The contour levels are from 15% to 90% in steps of 15% of the peak intensity in both panels. The peak is 1.84 Jy beam-1 in the left panel and 1.92 Jy beam-1 in the right panel. The four clumps are labeled by solid lines with arrows. The clumps are distinguish by the thick dashed lines. Figure 7: The high-velocity HCN (3-2) intensity contours overlaid on the 1.3 mm continuum image, integrated from 80 to 87 km s-1 for the blueshifted lobes (solid contours) and from 103 to 109 km s-1 for the redshifted lobe (dashed contours), with contours from 30$\%$ in steps of 10$\%$ of the peak emission. The peak is 8.55 Jy beam-1$\cdot$km s-1 for the blueshifted lobes and 7.16 Jy beam-1$\cdot$km s-1 for the redshifted lobe. The empty and solid ellipses in the lower-right corner represent the synthesized beams of HCN (3-2) emission and 1.3 mm continuum emission combined from both configurations, respectively.
arxiv-papers
2010-12-20T11:43:11
2024-09-04T02:49:15.846574
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Tie Liu, Yuefang Wu, Qizhou Zhang, Zhiyuan Ren, Xin Guan and Ming Zhu", "submitter": "Tie Liu", "url": "https://arxiv.org/abs/1012.4292" }
1012.4308
11institutetext: 1 Euratom-VR Association, Department of Radio and Space Science, Chalmers University of Technology, SE-412 96 Göteborg, Sweden. 2 Max-Planck-Institut für Plasmaphysik EURATOM-IPP, D-85748 Garching, Germany. # Impurity transport in ITG and TE mode dominated turbulence A. Skyman1 H. Nordman1 P. Strand1 F. Jenko2 F. Merz2 ## 1 Introduction Figure 1: Illustration of $PF_{0}$ and the linearity of $\Gamma_{Z}\left(\nabla n_{Z}\right)$; ITG dominated quasilinear GENE result for $Ne$ with $k\rho=0.3$ The transport properties of impurities is of high relevance for the performance and optimisation of magnetic fusion devices. For instance, if impurities from the plasma-facing surfaces accumulate in the core, wall- impurities of relatively low density suffice to dilute the plasma and lead to unacceptable energy losses in the form of radiation. In the present study, turbulent impurity transport in Deuterium tokamak plasmas, driven by Ion Temperature Gradient (ITG) and Trapped Electron (TE) modes, has been investigated using fluid and gyrokinetic models. The impurity diffusivity ($D_{Z}$) and convective velocity ($V_{Z}$) are calculated, and from these the zero-flux peaking factor ($PF_{0}$) is derived. This quantity expresses the impurity density gradient at which the convective and diffusive transport of impurities are exactly balanced. The sign of $PF_{0}$ is of special interest, as it determines whether the impurities are subject to an inward ($PF_{0}>0$) or outward ($PF_{0}<0$) pinch. Quasilinear results obtained from the GENE code [1, 2] are compared with two- fluid results [3] for both ITG and TE mode dominated turbulence. Scalings of $PF_{0}$ with impurity charge ($Z$) and various plasma parameters, such as magnetic shear ($\hat{s}$), are studied. Of particular interest are conditions favouring an outward convective impurity flux. ## 2 Theoretical background The transport of a trace impurity species can locally be described by a _diffusive_ and a _convective_ part. The former is characterized by the diffusion coefficient $D_{Z}$, the latter by a convective velocity or “pinch” $V_{Z}$, see equation (1) [4]. From these, the _zero flux peaking factor_ is defined as $PF_{0}=\frac{-R\,V_{Z}}{D_{Z}}|_{\Gamma=0}$, see figure 1. $PF_{0}$ is important in reactor design, as it quantifies the balance of convective and diffusive transport. This can be seen from equation (1), where $\Gamma_{Z}$ is the impurity flux, $n_{Z}$ the density of the impurity species and $R$ the major radius of the tokamak. For the domain studied – a narrow flux tube – the gradient of the impurity density is constant: $\nabla n_{Z}/n_{Z}=1/L_{n_{Z}}$. Setting $\Gamma_{Z}=0$ in equation (1) yields the interpretation of $PF_{0}$ as the gradient of zero impurity flux. $\Gamma_{Z}=-D_{Z}\nabla n_{Z}+n_{Z}V_{Z}\Leftrightarrow\frac{R\Gamma}{n_{Z}}=-D_{Z}\frac{R}{L_{n_{Z}}}+RV_{Z}$ (1) ## 3 Fluid model Though the main results presented in this study have been obtained using quasilinear gyrokinetic simulations, their physical meaning is interpreted by comparing with the Weiland multi-fluid model [3]. The fluid equations for each included species ($j=i,\,te,\,Z$, representing Deuterium ions, trapped electrons, and trace impurities) are: $\displaystyle\frac{\partial n_{j}}{\partial t}+\nabla\cdot\left(n_{j}\boldsymbol{v}_{j}\right)=0$ (2) $\displaystyle m_{i,Z}n_{i,Z}\frac{\partial v_{||i,Z}}{\partial t}+\nabla_{||}\left(n_{i,Z}T_{i,Z}\right)+n_{i,Z}e\nabla_{||}\varphi=0$ (3) $\displaystyle\frac{3}{2}n_{j}\frac{\mathrm{d}T_{j}}{\mathrm{d}t}+n_{j}T_{j}\nabla\cdot\boldsymbol{v}_{j}+\nabla\cdot\boldsymbol{q}_{j}=0$ (4) Here $\boldsymbol{q}_{j}$ is the diamagnetic heat flux, and $\boldsymbol{v_{j}}$ is the sum of the $\boldsymbol{E}\times\boldsymbol{B}$, diamagnetic drift, polarization drift, and stress-tensor drift velocities. To solve the equations, it is assumed that $\boldsymbol{q}_{j}$ is the only heat flux for all species, that passing electrons are adiabatic, and that quasineutraility (equation (5)) holds. Going to the trace limit for the impurities, i.e. letting $Zf_{Z}\rightarrow 0$ in equation (5), an eigenvalue equation for ITG and TE modes is obtained. The impurity particle flux in equation (1) is then obtained from $\Gamma_{nj}=\langle\delta n_{j}\boldsymbol{v}_{\boldsymbol{E}\times\boldsymbol{B}}\rangle$, where the averaging is performed over all unstable modes for a fixed length scale $k\rho$ of the turbulence. $\frac{\delta n_{e}}{n_{e}}=\left(1-Zf_{Z}\right)\frac{\delta n_{i}}{n_{i}}+Zf_{Z}\frac{\delta n_{Z}}{n_{Z}},\enspace f_{Z}=\frac{n_{Z}}{n_{e}}$ (5) ## 4 Quasilinear gyrokinetic simulations GENE is a parallel gyrokinetic code employing a fixed grid in five dimensional phase space and a flux-tube geometry [1]. The simulations were performed on the _HPC-FF_ cluster111HPC-FF (_High Performance Computing For Fusion_) is an EFDA funded computer situated at Forschungszentrum Jülich. Germany, dedicated to fusion research with GENE running in eigenvalue mode. Growth rates and impurity fluxes were thus computed for ITG and TE mode dominated cases, for which a number of parameters were varied and trends observed. The main parameters used are presented in table 1. Table 1: Parameters used in all simulations | ITG: | TEM: ---|---|--- $T_{D}/T_{e}$: | $1.0$ | $1.0$ $\hat{s}$: | $0.8$ | $0.8$ $q_{0}$: | $1.4$ | $1.4$ $\varepsilon$: | $0.14$ | $0.14$ $R/L_{T_{D}},R/L_{T_{Z}}$: | $7.0$ | $3.0$ $R/L_{T_{e}}$: | $3.0$ | $7.0$ $N_{x}\times N_{ky}\times N_{z}$: | $5\times 1\times 24$ | $4\times 1\times 24$ $N_{v_{||}}\times N_{\mu}$: | $64\times 12$ | $64\times 12$ ## 5 Results ### Impurity charge $Z$: The main results obtained are the scalings of the peaking factor with the charge of the impurity species. These are presented in figures 2(a) and 2(b), showing ITG and TE mode dominated turbulence respectively. (a) ITG mode dominated case (b) TEM dominated case Figure 2: Scalings of $PF_{0}$ with impurity charge $Z$; quasilinear GENE and fluid results The difference between figure 2(a) and 2(b) can be understood from the properties of the convective velocity in (1). $V_{Z}$ contains a thermodiffusive term $V_{T_{Z}}\sim\frac{1}{Z}\frac{R}{L_{T_{Z}}}$ and a parallel impurity compression term $V_{p_{Z}}\sim\frac{Z}{A_{Z}}k_{||}^{2}\sim\frac{Z}{A_{Z}q^{2}}$. The former is generally outward ($V_{T_{Z}}>0$) for ITG and inward ($V_{T_{Z}}<0$) for TE mode dominated transport, whereas for the latter the opposite is generally the case. ### Magnetic shear $\hat{s}$: The effect of magnetic shear on the peaking factor is shown in figures 3(a) and 3(b). It is worth noting that a flux reversal, i.e. a change of sign in $PF_{0}$, owing to a change in sign of $V_{Z}$, occurs for negative $\hat{s}$ for $Z\gtrsim 6$ in the TE mode dominated case, indicating a net outward transport of the heavier elements. Similar trends are not seen in fluid simulations, and this warrants further investigation. (a) ITG mode dominated case (b) TEM dominated case Figure 3: Scalings of $PF_{0}$ with magnetic shearing $\hat{s}$; quasilinear GENE results with $k\rho=0.3$ ### Other parameters: Scans of the dependence $PF_{0}$ on other parameters, such as $k\rho$ and $L_{T}$, have also been carried out. The results are similar to those reported in [5], [6] and [7] respectively. In most cases, only a weak dependence of $PF_{0}$ is observed. ## 6 Conclusions and outlook Quasilinear GENE simulations and fluid results show that peaking factor increases with impurity charge $Z$ for ITG mode dominated transport, whereas the opposite holds for TE mode dominated transport. In both cases $PF_{0}$ saturates for high $Z$. For magnetic shear, a flux reversal is observed for negative magnetic shear in the TEM dominated case. This is not seen in fluid simulations, and will be a focus of future studies. For other parameters investigated, weak scalings for $PF_{0}$ are observed, in agreement with previous work. ## References * [1] F. Jenko, W. Dorland, M. Kotschenreuther, and B. N. Rogers. Electron temperature gradient driven turbulence. Physics of Plasmas, 7(5):1904–10, May 2000. * [2] F. Merz. Gyrokinetic Simulation of Multimode Plasma Turbulence. Monography, Westfälischen Wilhelms-Universität Münster, 2008. * [3] J. Weiland. Collective Modes in Inhomogeneous Plasma. Institute of Physics Publishing, London, UK, 2000. * [4] C. Angioni and A. G. Peeters. Direction of impurity pinch and auxiliary heating in tokamak plasmas. PRL, 96:095003–1–4, 2006. * [5] T. Dannert. Gyrokinetische Simulation von Plasmaturbulenz mit gefangenen Teilchen und elektromagnetischen Effekten. Monography, Technischen Universität München, January 2005. * [6] H. Nordman, R. Singh, and T. Fülöp et al. Influence of the radio frequency ponderomotive force on anomalous impurity transport in tokamaks. PoP, 15:042316–1–5, 2007. * [7] T. Fülöp and H. Nordman. Turbulent and neoclassical impurity transport in tokamak plasmas. PoP, 16:032306–1–8, 2009. This work benefited from an allocation on the EFDA HPC-FF computer.
arxiv-papers
2010-12-20T12:55:46
2024-09-04T02:49:15.853512
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Andreas Skyman, Hans Nordman, P\\\"ar Strand, Frank Jenko, Florian Merz", "submitter": "Andreas Skyman", "url": "https://arxiv.org/abs/1012.4308" }
1012.4309
Core transport studies in fusion devices Pär Strand, Andreas Skyman and Hans Nordman Department of radio and space science, Chalmers university of technology, EURATOM-VR association SE-412 96 Göteborg, Sweden & EFDA Task Force on Integrated Tokamak Modelling elfps@chalmers.se http://www.chalmers.se/rss/EN/research/research- groups/transport-theory/ ## 1 Introduction Comprehensive first principles modelling of fusion plasmas is a numerically challenging: the complicated magnetic geometry and long range electromagnetic interactions between multiple species introduce complex collective behaviour in the plasma. In addition, steep density and temperature gradients combined with an inhomogeneous magnetic field drives instabilities, resulting in non- linear dynamics and turbulence. The turbulence in magnetically confined fusion plasmas has important and non- trivial effects on the quality of the energy confinement. These effects are hard to make a quantitative assessment of analytically. The problem investigated in this article is the transport of energy and particles, in particular impurities, in a Tokamak plasma. Impurities from the walls of the plasma vessel cause energy losses if they reach the plasma core. It is therefore important to understand the transport mechanisms to prevent impurity accumulation and minimize losses. This is an area of research where turbulence plays a major role and is intimately associated with the performance of future fusion reactors, such as ITER. With the rapid growth and increased accessibility of high performance computing (HPC) over the last few decades, plasma modelling has matured towards an increased predictive capability. Particular emphasis has been put on simulation of drift wave physics, widely accepted as the main source of transport in the plasma core. Theory, reduced physics as well as first principles modelling, and software are developed in a coordinated European effort to produce a virtual Tokamak, a tool that will become indispensable, both when it comes to developing and running ITER, and in the planning of future reactors aimed at energy production ParFusEngDes. ## 2 Physical background To arrive at a set of equations that are both meaningful and solvable, some approximation is necessary. The advances in high performance computing have allowed fusion modellers to move from fluid descriptions of the plasma to kinetic descriptions as the basis for turbulence modelling. In _kinetic theory_ the plasma is described through distribution functions of velocity and position for the plasma species. Hence, kinetic equations are inherently six- dimensional, however, magnetically confined particles are constrained to tight orbits along field lines. This motivates averaging over the gyration, reducing the problem to five-dimensional _gyrokinetic_ equations ## References * [1] * [2] This is a considerable gain, and the foundation of most current plasma codes. In this project, GENE, a European code developed by IPP-Garching * [3] , has been used. GENE employs a second order accurate explicit finite difference scheme, and has demonstrated excellent parallel performance using in excess of $10000$ cores * [4] . * [5] * [6] ## 3 Modelling plasmas As mentioned above, gradients drive turbulence. Here, plasma core turbulence induced by the so called _ion temperature gradient_ (ITG) mode * [7] , has been studied. Parameters were taken from discharge #67730 of the _Joint European Torus_ (JET). A slice of the simulation domain, illustrating the turbulence, is shown in figure 1(a). * [8] The GENE code employs a fixed grid in five dimensional phase space and a flux-tube geometry. For a typical simulation for main ions and one trace species, with electrons considered adiabatic, a resolution of $n_{x}\times n_{ky}\times n_{z}=48\times 48\times 32$ grid points in real space and of $n_{v}\times n_{\mu}=64\times 12$ in velocity space is necessary. A normal run with these parameters uses a minimum of $8000$ CPU hours and $384$ cores. Incorporating kinetic electrons increases the demand for high resolution in all of phase space, but most notably in velocity space, and also requires a shorter time step. Typically, such simulations require $40000$ CPU hours, occupying $1024$ cores. * [9] The computations produce tens of gigabyte of data to be analysed. The non-linear data presented in figure 1(b) is the result of approximately twenty runs on the HPC cluster _Akka_ , and from the discussion above, it is readily understood that HPC is vital for this kind of study. * [10] * [11] ## 4 Transport in ITER-like plasmas In general, transport of a species with atomic number $Z$ can locally be described by a diffusive and a convective contribution. The former is characterized by the diffusion coefficient $D_{Z}$, the latter by a convective velocity or “pinch” $V_{Z}$. The _zero flux peaking factor_ , defined as $PF_{0}=-R\,V_{Z}/D_{Z}$, is important in reactor design because it quantifies the balance of convective and diffusive transport. This can be understood from equation (1), where $\Gamma_{Z}$ is the impurity flux, $n_{Z}$ the density of the impurity species and $R$ the major radius of the tokamak HansArtikel. For the regime studied $\nabla n_{Z}$ is regarded as a constant, such that $-\nabla n_{Z}/n_{Z}=1/L_{n_{Z}}$. Setting $\Gamma_{Z}=0$ in equation (1) yields the interpretation of $PF_{0}$ as the gradient at which the impurity flux vanishes. * [12] $\Gamma_{Z}=-D_{Z}\nabla n_{Z}+n_{Z}V_{Z}\Leftrightarrow\frac{R\Gamma}{n_{Z}}=D_{Z}\frac{R}{L_{n_{Z}}}+RV_{Z}$ (1) * [13] A positive sign of $PF_{0}$ indicates a net inward transport. This might lead to an accumulation of wall impurities in the plasma core, which can seriously hamper the efficiency of the fusion device. Understanding under what circumstances a negative peaking factor can be achieved is therefore an important issue for ITER and future fusion reactors. * [14] Time series were generated by GENE on the _Akka_ HPC cluster for multiple values of $L_{n_{Z}}$, and from these $\Gamma_{Z}$ was extracted. The parameters $D_{Z}$ and $RV_{Z}$ were estimated and the peaking factor calculated as their quotient. This was repeated for several different nuclear charges $Z$. Results have been reported in HansArtikel, HansEPS and SkymanEPS. * [15] (a) crossection of the Deuterium density profile showing turbulent features (b) zero-flux peaking factor for different $Z$ according to three different models with adiabatic and one with kinetic electrons Figure 1: Results from the non-linear simulations on Akka * [16] Investigating where different models are in agreement is one of the aims of this kind of study: it is the first step towards an understanding of the physics that underlie their differences. In figure 1(b) the non-linear results from _Akka_ are compared with quasi-linear kinetic results and results from a fluid model developed at Chalmers * [17] . As can be seen, the three models are in good qualitative agreement with one another for $Z>4$ (Be). This owes to the domination of the single strong ITG mode for these particular parameters, and it cannot be guaranteed to hold for other cases. Also in figure 1(b), the quasi-linear result with kinetic electrons is shown for comparison. From that one expects kinetic effects to be most pronounced for low $Z$. At the time of writing, the fully kinetic non-linear case is still being simulated. * [18] * [19] 00 ParFusEngDes P. Strand et al.: Simulation & high performance computing – building a predictive capability for Fusion, Fusion Engineering and Design (accepted), 2010 HansArtikel H. Nordman, A. Skyman, P Strand et al: Fluid and gyrokinetic simulations of impurity transport in JET, (manuscript), 2010 HansEPS H. Nordman, A. Skyman, P. Strand et al.: Modelling of impurity transport experiments at the Joint European Torus, Proceedings of EPS 2010 (accepted), 2010 SkymanEPS A. Skyman, H. Nordman, P. Strand et al.: Impurity transport in ITG and TE mode dominated turbulence, Proceedings of EPS 2010 (accepted), 2010 * [20] 00 Merz2008 F. Merz: Gyrokinetic Simulation of Multimode Plasma Turbulence, PhD thesis, Westfälischen Wilhelms-Universität Münster, 2008 Jenko2000 F. Jenko et al.: Electron temperature gradient driven turbulence, Physics of Plasmas, 7, pp. 1904–10, 2000 Weiland2000 J. Weiland: Collective Modes in Inhomogeneous Plasma, Institute of Physics Publishing, London (UK), 2000 [‡]GENE http://www.ipp.mpg.de/~fsj/gene/ * [21]
arxiv-papers
2010-12-20T12:56:13
2024-09-04T02:49:15.858012
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "P\\\"ar Strand, Andreas Skyman, Hans Nordman", "submitter": "Andreas Skyman", "url": "https://arxiv.org/abs/1012.4309" }
1012.4507
020008 2010 A. Vindigni A. A. Fedorenko, CNRS-Lab. de Physique, ENS de Lyon, France. 020008 We numerically study the geometry of a driven elastic string at its sample- dependent depinning threshold in random-periodic media. We find that the anisotropic finite-size scaling of the average square width $\overline{w^{2}}$ and of its associated probability distribution are both controlled by the ratio $k=M/L^{\zeta_{\mathrm{dep}}}$, where $\zeta_{\mathrm{dep}}$ is the random-manifold depinning roughness exponent, $L$ is the longitudinal size of the string and $M$ the transverse periodicity of the random medium. The rescaled average square width $\overline{w^{2}}/L^{2\zeta_{\mathrm{dep}}}$ displays a non-trivial single minimum for a finite value of $k$. We show that the initial decrease for small $k$ reflects the crossover at $k\sim 1$ from the random-periodic to the random-manifold roughness. The increase for very large $k$ implies that the increasingly rare critical configurations, accompanying the crossover to Gumbel critical-force statistics, display anomalous roughness properties: a transverse-periodicity scaling in spite that $\overline{w^{2}}\ll M$, and subleading corrections to the standard random- manifold longitudinal-size scaling. Our results are relevant to understanding the dimensional crossover from interface to particle depinning. # Anisotropic finite-size scaling of an elastic string at the depinning threshold in a random-periodic medium S. Bustingorry [inst1] A. B. Kolton[inst1] E-mail: sbusting@cab.cnea.gov.arE- mail: koltona@cab.cnea.gov.ar (20 October 2010; 1 December 2010) ††volume: 2 99 inst1 CONICET, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Río Negro, Argentina. ## 1 Introduction The study of the static and dynamic properties of $d$-dimensional elastic interfaces in $d+1$-dimensional random media is of interest in a wide range of physical systems. Some concrete experimental examples are magnetic [1, 2, 3, 4] or ferroelectric [5, 6] domain walls, contact lines of liquids [7], fluid invasion in porous media [8, 9], and fractures [10, 11]. In all these systems, the basic physics is controlled by the competition between quenched disorder (induced by the presence of impurities in the host materials) which promotes the wandering of the elastic object, against the elastic forces which tend to make the elastic object flat. One of the most dramatic and worth understanding manifestations of this competition is the response of these systems to an external drive. The mean square width or roughness of the interface is one of the most basic quantities in the study of pinned interfaces. In the absence of an external drive, the ground state of the system is disordered but well characterized by a self-affine rough geometry with a diverging typical width $w\sim L^{\zeta_{\mathrm{eq}}}$, where $L$ is the linear size of the elastic object and $\zeta_{\mathrm{eq}}$ is the equilibrium roughness exponent. When the external force is increased from zero, the ground state becomes unstable and the interface is locked in metastable states. To overcome the barriers separating them and reach a finite steady-state velocity $v$ it is necessary to exceed a finite critical force, above which barriers disappear and no metastable states exist. For directed $d$-dimensional elastic interfaces with convex elastic energies in a $D=d+1$ dimensional space with disorder, the critical point is unique, characterized by the critical force $F=F_{c}$ and its associated critical configuration [12]. This critical configuration is also rough and self-affine such that $w\sim L^{\zeta_{\mathrm{dep}}}$ with $\zeta_{\mathrm{dep}}$ the depinning roughness exponent. When approaching the threshold from above, the steady-state average velocity vanishes like $v\sim(F-F_{c})^{\beta}$ and the correlation length characterizing the cooperative avalanche-like motion diverges as $\xi\sim(F-F_{c})^{-\nu}$ for $F>F_{c}$, where $\beta$ is the velocity exponent and $\nu$ is the depinning correlation length exponent [13, 14, 15, 16]. At finite temperature and for $F\ll F_{c}$, the system presents an ultra-slow steady-state creep motion with universal features [17, 18] directly correlated with its multi-affine geometry [19, 20]. At very small temperatures the absence of a divergent correlation length below $F_{c}$ shows that depinning must be regarded as a non-standard phase transition [20, 21] while exactly at $F=F_{c}$, the transition is smeared-out with the velocity vanishing as $v\sim T^{\psi}$, with $\psi$, the so-called thermal rounding exponent [22, 23, 24, 25, 26, 27]. During the last years, numerical simulations have played an important role to understand the physics behind the depinning transition thanks to the development of powerful exact algorithms. In particular, the development of an exact algorithm able to target efficiently the critical configuration and critical force for a given sample [28, 29] has allowed to study, precisely, the self-affine rough geometry at depinning [7, 28, 29, 30, 31], the sample- to-sample critical force distribution [32], the critical exponents of the depinning transition [26, 27, 33], the renormalized disorder correlator [34], and the avalanche-size distribution in quasistatic motion [35]. Moreover, the same algorithm has allowed to study, precisely, the transient universal dynamics at depinning [36, 37], and an extension of it has allowed to study low-temperature creep dynamics [20, 21]. In practice, the algorithm for targeting the critical configuration [28, 29] has been numerically applied to directed interfaces of linear size $L$ displacing in a disordered potential of transverse dimension $M$, applying periodic boundary conditions in both directions in order to avoid border effects. This is thus equivalent to an elastic string displacing in a disordered cylinder. The aspect ratio between longitudinal $L$ and transverse $M$ periodicities must be carefully chosen, in order to have the desired thermodynamic limit corresponding to a given experimental realization. In Ref. [32] it was indeed shown that the critical force distribution $P(F_{c})$ displays three regimes associated with $M$: (i) At very small $M$ compared with the typical width $L^{\zeta_{\mathrm{dep}}}$ of the interface, the interface wraps the computational box several times in the transverse direction, as shown schematically in Fig. 1(b), and therefore the periodicity of the random medium is relevant and $P(F_{c})$ is Gaussian; (ii) At very large $M$ compared with $L^{\zeta_{\mathrm{dep}}}$, as shown schematically in Fig. 1(c), periodicity effects are absent but then the critical force, being the maximum among many independent sub-critical forces, obeys extreme value statistics and $P(F_{c})$ becomes a Gumbel distribution; (iii) In the intermediate regime, where $M\approx L^{\zeta_{\mathrm{dep}}}$ and periodicity effects are still irrelevant, as shown schematically in Fig. 1(a), the distribution function is in between the Gaussian and the Gumbel distribution. It has been argued that only the last case, where $M\approx L^{\zeta_{\mathrm{dep}}}$, corresponds to the random-manifold depinning universality class (periodicity effects absent) with a finite critical force in the thermodynamic limit $L,M\to\infty$. This criterion does not give, however, the optimal value of the proportionality factor between $M$ and $L^{\zeta_{\mathrm{dep}}}$, and must be modified at finite velocity since the crossover to the random-periodic universality class at large length-scales depends also on the velocity [38]. To avoid this problem, it has been therefore proposed to define the critical scaling in the fixed center of mass ensemble [39]. The crossover from the random-manifold to the random-periodic universality class is, however, physically interesting, as it can occur in periodic elastic systems such as elastic chains. Remarkably, although the mapping from a periodic elastic system (with given lattice parameter) in a random potential to a non-periodic elastic system (such as an interface) in a random potential with periodic boundary conditions is not exact, it was recently shown that the lattice parameter does play the role of $M$ for elastic interfaces with regard to the geometrical or roughness properties [38]. Since the periodicity can often be experimentally tuned in such periodic systems it is thus worth studying in detail the geometry of critical interfaces of size $L$ as a function of $M$ with periodic boundary conditions, and thus complement the study of the critical force in such systems [32]. In this paper, we study in detail, using numerical simulations, the geometrical properties of the one-dimensional interface or elastic string critical configuration in a random-periodic pinning potential as a function of the aspect ratio parameter $k$, conveniently defined as $k=M/L^{\zeta_{\mathrm{dep}}}$. We show that $k$ is indeed the only parameter controlling the finite-size scaling (i.e. the dependence of observables with the dimensions $L$ and $M$) of the average square width and its sample-to- sample probability distribution. The scaled average square width $\overline{w^{2}}L^{-2{\zeta_{\mathrm{dep}}}}$ is described by a universal function of $k$ displaying a non-trivial single minimum at a finite value of $k$. We show that while for small $k$ this reflects the crossover at $k\sim 1$ from the random-periodic to the random-manifold depinning universality class, for large $k$ it implies that in the regime where the depinning threshold is controlled by extreme value (Gumbel) statistics, critical configurations also become rougher, and display an anomalous roughness scaling. Figure 1: (a) Elastic string driven by a force $F$ in a random-periodic medium with periodic boundary conditions. It is described by a displacement field $u(z)$ and has a mean width $w$. The anisotropic finite-size scaling of width fluctuations are controlled by the aspect-ratio parameter $k=M/L^{\zeta_{\mathrm{dep}}}$, with ${\zeta_{\mathrm{dep}}}$ the random- manifold roughness exponent at depinning. In the case $k\ll 1$ (b) periodicity effects are important, while when $k\gg 1$ (c) they are not important but the roughness scaling of the critical configuration is anomalous. ## 2 Method The model we consider here is an elastic string in $(1+1)$ dimensions described by a single valued function $u(z,t)$, which gives the transverse displacement $u$ as a function of the longitudinal direction $z$ and the time $t$ [see Fig. 1(a)]. The zero-temperature dynamics of the model is given by $\gamma\,\partial_{t}u(z,t)=c\,\partial^{2}_{z}u(z,t)+F_{p}(u,z)+F,$ (1) where $\gamma$ is the friction coefficient and $c$ the elastic constant. The first term in the right hand side derives from an harmonic elastic energy. The effects of a random-bond type disorder is given by the pinning force $F_{p}(u,z)=-\partial_{u}U(u,z)$. The disorder potential $U(u,z)$ has zero average and sample-to-sample fluctuations given by $\overline{\left[U(u,z)-U(u^{\prime},z^{\prime})\right]^{2}}=\delta(z-z^{\prime})\,R^{2}(u-u^{\prime}),$ (2) where the overline indicates average over disorder realizations and $R(u)$ stands for a correlator of finite range $r_{f}$ [18]. Finally, $F$ represents the uniform external drive acting on the string. Physically, this model can phenomenologically describe, for instance, a magnetic domain wall in a thin film ferromagnetic material with weak and randomly located imperfections [1], being $F$ proportional to an applied external magnetic field pushing the wall in the energetically favorable direction. In order to numerically solve Eq. (1), the system is discretized in the $z$-direction in $L$ segments of size $\delta z=1$, i.e. $z\to j=0,...,L-1$, while keeping $u_{j}(t)$ as a continuous variable. To model the continuous random potential, a cubic spline is used, which passes through $M$ regularly spaced uncorrelated Gaussian number points [30]. For the numerical simulations performed here we have used, without loss of generality, $\gamma=1$, $c=1$ and $r_{f}=1$ and a disorder intensity $R(0)=1$. In both spatial dimensions we have used periodic boundary conditions, thus defining a $L\times M$ system. The critical configuration $u_{c}(z)$ and force $F_{c}$ are defined from the pinned (zero-velocity) configuration with the largest driving force $F$ in the long time limit dynamics. They are thus the real solutions of $c\,\partial^{2}_{z}u(z)+F_{p}(u,z)+F=0,$ (3) such that for $F>F_{c}$ there are no further real solutions (pinned configurations). Middleton theorems [12] assure that for Eqs. (3) the solution exists and it is unique for both $u_{c}(z)$ and $F_{c}$, and that above $F_{c}$ the string trajectory in an $L$ dimensional phase-space is trapped into a periodic attractor (for a system with periodic boundary conditions as the one we consider). In other words, the critical configuration is the marginal fixed point solution or critical state of the dynamics, being $F_{c}$ the critical point control parameter of a Hopf bifurcation. Solving the $L$-dimensional system of Eqs. (3) for large $L$ directly is a formidable task, due to the non-linearity of the pinning force $F_{p}$. On the other hand, solving the long-time dynamics at different driving forces $F$ to localize $F_{c}$ and $u_{c}$ is very inefficient due to the critical slowing down. Fortunately, Middleton theorems, and in particular the “non-passing rule”, can be used again to devise a precise and very efficient algorithm which allows to obtain the critical force $F_{c}$ and the critical configuration $u^{c}_{j}$ for each independent disorder realization iteratively without solving the actual dynamics nor directly inverting the system of Eqs. (3) [30]. Once the critical force and the critical configuration are determined with this algorithm, we can compute the different observables. In particular, the square width or roughness of the string at the critical point for a given disorder realization is defined as $w^{2}=\frac{1}{L}\sum_{j=0}^{L-1}\left[u^{c}_{j}-\frac{1}{L}\sum_{k=0}^{L-1}u^{c}_{k}\right]^{2}.$ (4) Computing $w^{2}$ for different disorder realizations allows us to compute its disorder average $\overline{w^{2}}$ and the sample-to-sample probability distribution $P(w^{2})$. In addition, the average structure factor associated to the critical configuration is $S_{q}=\frac{1}{L}\overline{\left|\sum_{j=0}^{L-1}u^{c}_{j}\,e^{-iqj}\right|^{2}},$ (5) where $q=2\pi n/L$, with $n=1,...,L-1$. One can show, using a simple dimensional analysis, that given a roughness exponent $\zeta$, such that $\overline{w^{2}}\sim L^{2\zeta}$, the structure factor behaves as $S(q)\sim q^{-(1+2\zeta)}$ for small $q$, thus yielding an estimate to $\zeta$ without changing $L$. To compute averages over disorder and sample-to-sample fluctuations, we consider between $10^{3}$ and $10^{4}$ independent disorder realizations depending on the size of the system. ## 3 Results ### 3.1 Roughness at the critical point Figure 2: The scaling of $\overline{w^{2}}$ for the critical configuration at different $M$ values as indicated. The curves for $M=64$ and $16384$ are shifted upwards for clarity. The dashed and dotted lines are guides to the eye showing the expected slopes corresponding to the different roughness exponents. Figure 2 shows the scaling of the square width of the critical configuration $\overline{w^{2}}$ with the longitudinal size of the system $L$ for $L=32,64,128,256,512$ and different values of $M$. When $M$ is small, $M=8$, for all the $L$ values shown we observe $\overline{w^{2}}\sim L^{2\zeta_{\mathrm{L}}}$ with $\zeta_{\mathrm{L}}=1.5$, corresponding to the Larkin exponent in $(1+1)$ dimensions. This value is different from the value $\zeta_{\mathrm{dep}}=1.25$ [33, 40] expected for the random-manifold universality class, and is thus indicating that the periodicity effects are important for this joint values of $M$ and $L$. This situation is schematically represented in Fig. 1(b). This result is a numerical confirmation of the two-loop functional renormalization group result of Ref. [16] which shows that the $\zeta=0$ fixed point, leading to a universal logarithmic growth of displacements at equilibrium is unstable. The fluctuations are governed, instead, by a coarse-grained generated random-force as in the Larkin model, yielding a roughness exponent $\zeta_{\mathrm{L}}=(4-d)/2$ in $d$ dimensions [16], which agrees with our result for $d=1$. We can thus say that for small enough $M$ (compared to $L$) the system belongs to the same random-periodic depinning universality class as charge density wave systems [14, 41], which strictly correspond to $M=1$. When $M$ is large, on the other hand, $M=16384$ in Fig. 2, for all the $L$ values considered the exponent is consistent with $\zeta_{\mathrm{dep}}$, of the random-manifold universality class. This situation is schematically represented in Fig. 1(c), and we will show later that, for this elongated samples, the effects of extreme value statistics are already visible. For intermediate values of $M$, such as $M=64$ in Fig. 2, we can observe the crossover in the scale-dependent roughness exponent $\zeta(L)\sim\frac{1}{2}\frac{d\log w^{2}}{d\log L}$ changing from $\zeta_{\mathrm{dep}}$ to $\zeta_{\mathrm{L}}$ as $L$ increases, as indicated by the dashed and dotted lines. This crossover, from the random-manifold to the random-periodic depinning geometry, occurs at a characteristic distance $l^{*}\sim M^{1/\zeta_{\mathrm{dep}}}$, when the width in the random-manifold regime reaches the transverse dimension or periodicity $M$. At finite velocity, this crossover length remains constant up to a non-trivial characteristic velocity and then decreases with increasing velocity [38]. Figure 3: Structure factor of the critical configuration for $L=256$ and different $M$ values, as indicated. The curves for $M=64$ and $16384$ are shifted upwards for clarity. The dashed and dotted lines are guides to the eye showing the expected slopes corresponding to the different roughness exponents. The above mentioned geometrical crossover can be studied in more details through the analysis of the structure factor $S(q)$, for a line of fixed size $L$. In Fig. 3 we show $S(q)$ for $L=256$ and $M=8,64,16384$. For the intermediate value $M=64$ a crossover between the two regimes is visible, and can be described by $S_{q}\sim\left\\{\begin{array}[]{ll}q^{-(1+2\zeta_{\mathrm{L}})}&q\ll q^{*},\\\ q^{-(1+2\zeta_{\mathrm{dep}})}&q\gg q^{*}.\\\ \end{array}\right.$ (6) with $q^{*}$ expected to scale as $q^{*}\sim l^{*-1}\sim M^{-1/\zeta_{\mathrm{dep}}}$. Therefore, the structure factor should scale as $S_{q}M^{-(2+1/\zeta_{\mathrm{dep}})}=H(x)$, where the scaled variable is $x=q\,M^{1/\zeta_{\mathrm{dep}}}\sim q/q^{*}$ and the scaling function behaves as $H(x)\sim\left\\{\begin{array}[]{ll}x^{-(1+2\zeta_{\mathrm{L}})}&x\ll 1,\\\ x^{-(1+2\zeta_{\mathrm{dep}})}&x\gg 1.\\\ \end{array}\right.$ (7) The collapse of Fig. 4 for $L=256$ and different values of $M=2^{p}$ with $p=3,4,...,14$ shows that this scaling form is a very good approximation. However, as we show below, small corrections can be expected fully in the random-manifold regime in the large $ML^{-\zeta_{\mathrm{dep}}}$ limit of very elongated samples. Figure 4: Scaling of the structure factor of the critical configuration for $L=256$ and different values of the transverse size $M=2^{p}$ with $p=3,4,...,14$ $M$. Although the values of the two exponents are very close, the change in the slope of the scaling function against the scaling variable $x=q\,M^{1/\zeta_{\mathrm{dep}}}$ is clearly observed. Figure 5: (a) Squared width of the critical configuration as a function of $M$ for different system sizes $L$ as indicated. (b) Scaling of the width in (a), showing that the relevant control parameter is $M/L^{\zeta_{\mathrm{dep}}}$. The dashed line in (a) and (b) corresponds to $\overline{w^{2}}=M^{2}$, which is always to the left of the minimum of $\overline{w^{2}}$ occurring at $k^{*}=m^{*}L^{-\zeta_{\mathrm{dep}}}$. The solid line indicates $k^{2(1-\zeta_{\mathrm{L}}/\zeta_{\mathrm{dep}})}$ which is the behavior expected purely from the random-periodic to random-manifold crossover at the characteristic distance $l^{*}\sim M^{1/\zeta_{\mathrm{dep}}}$. In Fig. 5(a), we show $\overline{w^{2}}$ as a function of the transverse periodicity $M$ for different values of the longitudinal periodicity $L$. Remarkably, $\overline{w^{2}}$ is a non-monotonic function of $M$. For small $M$ it decreases towards an $L$ dependent minimum $m^{*}$, and then increases with increasing $M$, in the regime where the extreme value statistics starts to affect the distribution of the critical force [32]. Since the only typical transverse scale in Fig. 5(a) is set by the minimum $m^{*}$, we can expect $\overline{w^{2}}\sim m^{*2}G(M/m^{*})$ with $G(x)$ some universal function. On the other hand, since the only relevant characteristic length-scale of the problem is set by the crossover between the random-periodic regime and the random-manifold regime, we can simply write $m^{*}\sim L^{\zeta_{\mathrm{dep}}}$ and therefore $\overline{w^{2}}\,L^{-2\zeta_{\mathrm{dep}}}\sim G(M\,L^{-\zeta_{\mathrm{dep}}}).$ (8) This scaling form is confirmed in Fig. 5(b) and shows that the aspect-ratio parameter $k=ML^{-\zeta_{\mathrm{dep}}}$ fully controls the anisotropic finite-size scaling of the problem. It is worth, however, noting some interesting consequences of the result of Fig. 5(b), as we describe below. Since at very small $k$ the interface is in the random-periodic regime, Eq. (8) should led to $\overline{w^{2}}\sim L^{2\zeta_{\mathrm{L}}}$ and therefore one deduces that, $G(k)\sim k^{2(1-{\zeta_{\mathrm{L}}}/{\zeta_{\mathrm{dep}}})},\;\;\;k\ll k^{*},$ (9) where $k^{*}=m^{*}L^{-\zeta_{\mathrm{dep}}}$. The fact that the random- periodic roughness exponent ${\zeta_{\mathrm{L}}}=3/2$ is larger than the random-manifold one ${\zeta_{\mathrm{dep}}}\approx 5/4$ consequently implies an initial decrease of $G(k)$ as $G(k)\sim k^{-2/5}$, as shown in Fig. 5(b) by the solid line. Periodicity effects, or the crossover from random-periodic to random-manifold, thus explain the initial decrease of $G(k)$ seen in Fig. 5(b), or the initial decrease of $\overline{w^{2}}$ against $M$ for fixed $L$, seen in Fig. 5(a). At this respect, it is then worth noting that the line $\overline{w^{2}}=M^{2}$, shown by a dashed line, lies completely in the regime $k<k^{*}$ implying that the naive criterion $\overline{w^{2}}<M^{2}$ is not enough to avoid periodicity effects, and to have the system fully in the random-manifold regime. As we show later, this is related with the shape of the probability distribution of $P(w^{2})$ which displays sample-to-sample fluctuations of the order of the average $\overline{w^{2}}$. The presence of a minimum at $k^{*}$ in the function $G(k)$ and in particular its slower than power-law increase for $k>k^{*}$ is non-trivial and constitutes one of the main results of the present work. This result shows that corrections to the standard scaling $\overline{w^{2}}\sim L^{\zeta_{\mathrm{dep}}}$ may arise from the aspect-ratio dependence of the prefactor $G(k)$. On the one hand, $\overline{w^{2}}$ now grows with $M$ for $L$ fixed, in spite that $\overline{w^{2}}\ll M^{2}$, i.e. transverse- size/periodicity scaling is present. On the other hand, the scaling of $\overline{w^{2}}$ with $L$ is slower in this regime, due to subleading scaling corrections coming from $G(k)$. The precise origin of these interesting leading and subleading corrections in the finite-size anisotropic scaling are highly non-trivial. Since the critical configurations in this regime have the constant roughness exponent $\zeta_{\mathrm{dep}}$ of the random-manifold universality class, the slow increase of $G(k)$ cannot be attributed to a geometrical crossover effect, as for the case $k<k^{*}$. However, we might relate this effect to the crossover in the critical force statistics, from Gaussian to Gumbel, in the $k\gg k^{*}$ limit [32]. In the Gumbel regime, the average critical force is expected to increase as $F_{c}\sim\log(M/L^{\zeta}_{\mathrm{dep}})\equiv\log k$ [39], since the sample critical force can be roughly regarded as the maximum among $M/L^{\zeta}_{\mathrm{dep}}$ independent sub-critical forces and configurations [32]. The increase in the critical force might be therefore correlated with the slow increase of roughness. The physical connection between the two is subtle though, since a large critical force in a very elongated sample could be achieved both by profiting very rare correlated pinning forces such as accidental columnar defects, or by profiting very rare non-correlated strong pinning forces. Since in the first case the critical configuration would be more correlated and in general less rough than for less elongated samples (smaller $k$), contrary to our numerical data of Fig. 5(b), we think that the second cause is more plausible. We can thus think that in the $k\gg k^{*}$ limit of extreme value statistics of $F_{c}$, the effective disorder strength on the critical configuration increases with $k$. This might be translated into the universal function $G(k)$, such that $\overline{w^{2}}\approx L^{2\zeta_{\mathrm{dep}}}G(k)$ can increase for increasing values of $k$ at fixed $L$ in such regime. A quantitative description of these scaling corrections remains an open challenge. ### 3.2 Distribution function Figure 6: Scaling function $\Phi(x)$ for $L=256$ and different values of $M=8,128,2048,16384$, which shows the change with the transverse size $M$. We now analyze sample-to-sample fluctuations of the square width $w^{2}$ by computing its probability distribution $P(w^{2})$. This property is relevant as $w^{2}$ fluctuates even in the thermodynamic limit for critical interfaces with a positive roughness exponent [42]. It has been computed for models with dynamical disorder such as random-walk [43] or Edwards–Wilkinson interfaces [44, 45], the Mullins Herrings model [46] and for non-Markovian Gaussian signals in general [47, 48]. It has also been calculated for non-linear models such as the one-dimensional Kardar–Parisi–Zhang model [49, 50] and for the quenched Edwards–Wilkinson model at equilibrium [51]. In particular, the probability distribution $P(w^{2})$ of critical interfaces at the depinning transition was studied analytically [52], numerically [31] and also experimentally for contact lines in partial wetting [7]. Remarkably, non-Gaussian effects in depinning models are found to be smaller than $0.1\%$ [31, 52], thus showing that $P(w^{2})$ is strongly determined by the self- affine (critical) geometry itself, rather than by the particular mechanism producing it. As in all the above mentioned systems the width distribution $P(w^{2})$ at different universality classes of the depinning transition was found to scale as $\overline{w^{2}}P(w^{2})\approx\Phi_{\zeta}\left(\frac{w^{2}}{\overline{w^{2}}}\right).$ (10) with $\Phi_{\zeta}$ an universal function, which only depends on the roughness exponent $\zeta$ and on boundary conditions when the global width is considered [47, 48]. In this way, $\overline{w^{2}}$ is the only characteristic length-scale of the system, absorbing the system longitudinal size $L$, and all the non-universal parameters of the model such as the elastic constant of the interface, the strength of the disorder and/or the temperature. Since $\Phi_{\zeta}$ can be easily generated using non-Markovian Gaussian signals [53], the quantity $\overline{w^{2}}P(w^{2})$ is a good observable to extract the roughness exponent of a critical interface from experimental data. Figure 7: Scaling function $\Phi(x)$ for different values of $L=32,64,128,256$ while keeping (a) $k=M/L^{\zeta_{\mathrm{dep}}}\approx 1$ and (b) $k=M/L^{\zeta_{\mathrm{dep}}}\approx 0.025$. The dotted line corresponds to the scaling function of the non-disordered Edwards–Wilkinson equation [43], while the continuous and dashed lines correspond to the scaling functions of Gaussian signals with $\zeta=1.25$ and $\zeta=1.5$, respectively [31, 53]. In Fig. 6, we show how the scaled distribution function $\Phi(x)\equiv\overline{w^{2}}\,P(x\;\overline{w^{2}})$ looks like for the depinning transition in a random-periodic medium for a fixed value $L=256$ and different values of $M$. We see that $\Phi(x)$ depends on $M$ for small $M$ but converges to a fixed shape for large $M$. We also note that for all $M$ $\Phi(x)$ extends appreciably beyond $x=1$ explaining why the criterion $\overline{w^{2}}\lesssim M^{2}$ is not enough to be fully in the random- manifold regime, as noted in Fig. 5. In Fig. 7, we show the scaling function $\Phi(x)$ for different values of $L$ and $M$ but fixing the aspect-ratio parameter $k=M/L^{\zeta_{\mathrm{dep}}}$, $k\approx 1>k^{*}$ in Fig. 7(a) and $k\approx 0.025\ll k^{*}$ in Fig. 7(b), with $k^{*}$ the minimum of $\overline{w^{2}}$. Since data for the same $k$ practically collapses into the same curve, we can write for our case: $\overline{w^{2}}P(w^{2})=\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right).$ (11) Therefore, the anisotropic scaling of the probability distribution is fully controlled by $k$, as it was found for $\overline{w^{2}}$. In Figs. 7(a) and (b), we also show the universal functions $\Phi_{\zeta_{L}}$ and $\Phi_{\zeta_{\mathrm{dep}}}$ generated using non-Markovian Gaussian signals [31, 53], and for comparison we also show $\Phi_{1/2}$ corresponding to the Markovian periodic Gaussian signal or the Edwards–Wilkinson equation [43]. Comparing this with the collapsed data for depinning, we see that the function $\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right)$ respects the limits $\displaystyle\Phi\left(x,k\to 0\right)$ $\displaystyle=$ $\displaystyle\Phi_{\zeta_{\mathrm{L}}}(x),$ $\displaystyle\Phi\left(x,k\gtrsim k^{*}\right)$ $\displaystyle\approx$ $\displaystyle\Phi_{\zeta_{\mathrm{dep}}}(x),$ (12) as expected from the existence of the geometric crossover between the roughness exponents ${\zeta_{\mathrm{L}}}$ for $k\to 0$ and ${\zeta_{\mathrm{dep}}}$ for $k>k^{*}$. For intermediate values $k<k^{*}$, however, $\Phi\left(\frac{w^{2}}{\overline{w^{2}}},k\right)$ does not necessarily coincide with the one of a Gaussian signal function $\Phi_{\zeta}$ for a given $\zeta$, since the critical configuration includes a crossover length $l^{*}\lesssim L$. Whether multi-affine or effective exponent self- affine non-Markovian Gaussian signals can be used to describe satisfactorily these intermediate cases is an interesting open issue. ## 4 Conclusions We have numerically studied the anisotropic finite-size scaling of the roughness of a driven elastic string at its sample-dependent depinning threshold in a random medium with periodic boundary conditions in both the longitudinal and transverse directions. The average square width $\overline{w^{2}}$ and its probability distribution are both controlled by the parameter $k=M/L^{\zeta_{\mathrm{dep}}}$. A non-trivial single minimum for a finite value of $k$ was found in $\overline{w^{2}}/L^{2\zeta_{\mathrm{dep}}}$. For small $k$, the initial decrease of $\overline{w^{2}}$ reflects the crossover from the random-periodic to the random-manifold roughness. For very large $k$, the growth with $k$ implies that the crossover to Gumbel statistics in the critical forces induces corrections to $G(k)$, that grow with $k$, to the string roughness scaling $\overline{w^{2}}\approx G(k)L^{2\zeta_{\mathrm{dep}}}$. These increasingly rare critical configurations thus have an anomalous roughness scaling: they have a transverse-size/periodicity scaling in spite that its width is $\overline{w^{2}}\ll M^{2}$, and subleading (negative) corrections to the standard random-manifold longitudinal-size scaling. Our results could be useful for understanding roughness fluctuations and scaling in finite experimental systems. The crossover from random-periodic to random-manifold roughness could be studied in periodic elastic systems with variable periodicity, such as confined vortex rows [54] and single-files of macroscopically charged particles [55] or colloids [56], with additional quenched disorder. 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arxiv-papers
2010-12-20T23:42:44
2024-09-04T02:49:15.869618
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sebastian Bustingorry, Alejandro B. Kolton", "submitter": "Luis Ariel Pugnaloni", "url": "https://arxiv.org/abs/1012.4507" }
1012.4542
# Impact of Mistiming on the Achievable Information Rate of Rake Receivers in DS-UWB Systems Chunhua Geng, Yukui Pei, Jiaqi Zhang and Ning Ge This work is supported by National Nature Science Foundation of China No. 60928001 and 60972019, National Basic Research Program of China under grant No. 2007CB310608, and the National Science & Technology Major Project under grant No. 2009ZX03006-007-02 and 2009ZX03006-009. State Key Laboratory on Microwave and Digital Communications Tsinghua National Laboratory for Information Science and Technology Department of Electronic Engineering, Tsinghua University, Beijing 100084, China Email: {gengch07,zhangjq06}@mails.tsinghua.edu.cn, {peiyk,gening}@tsinghua.edu.cn ###### Abstract In this paper, we investigate the impact of mistiming on the performance of Rake receivers in direct-sequence ultra-wideband (DS-UWB) systems from the perspective of the achievable information rate. A generalized expression for the performance degradation due to mistiming is derived. Monte Carlo simulations based on this expression are then conducted, which demonstrate that the performance loss has little relationship with the target achievable information rate, but varies significantly with the system bandwidth and the multipath diversity order, which reflects design trade-offs among the system timing requirement, the bandwidth and the implementation complexity. In addition, the performance degradations of Rake receivers with different multipath component selection schemes and combining techniques are compared. Among these receivers, the widely used maximal ratio combining (MRC) selective-Rake (S-Rake) suffers the largest performance loss in the presence of mistiming. ## I Introduction Ultra-wideband (UWB) is promising for wireless high rate and short range communications [1]. Direct-sequence UWB (DS-UWB) [2] has received considerable interest due to its fine properties of coherent processing of the occupied bandwidth and the widest contiguous bandwidth [3]. To exploit the ample multipath diversity, the Rake reception is widely employed in DS-UWB systems [4]. Various types of Rake receivers, like selective Rake (S-Rake) and partial Rake (P-Rake), are proposed recently [5]. However, Rake receivers have stringent requirements for timing accuracy [6]. In practical DS-UWB systems, mistiming due to acquisition and tracking errors is inevitable, thus its effects on the performance degradation is worthy of investigation. Several studies have explored this issue in UWB systems [7]-[9]. In [7], it is shown that the system throughput degrades significantly with relatively modest increase in timing errors over additive white Gaussian noise (AWGN) channels. In [8] and [9], the authors analyze the bit error rate (BER) degradation induced by mistiming for both fixed and random channels in UWB systems based on Rake reception. Compared with throughput and BER, the achievable information rate, which identifies the maximum mutual information between the input and output of one communication system, is a more fundamental measurement for system performance, and is also a subject of continuing research in UWB systems [10]. To the best of the authors’ knowledge, the effect of mistiming on the achievable information rate of Rake receivers in DS-UWB systems has not been investigated yet to date. In this paper, a systematic approach is presented to evaluate the impact of imperfect timing on Rake receivers in DS-UWB systems from the perspective of the achievable information rate. The influence of key system parameters on the performance of various types of Rake receivers is also investigated. In our analysis, a two-step procedure is adopted. First, a generalized expression of the system performance degradation due to timing mismatch is derived. Then based on this expression, the numerical results are obtained by averaging over a sufficiently large number of channel realizations. The major contributions of this paper lie in the following: (1) As for the widely used maximal ratio combining (MRC) S-Rake receiver, we observe that the performance degradation has little relationship with the target information rate, but varies significantly with the occupied bandwidth and the diversity order, which reflects design trade-offs among the system timing requirement, the bandwidth and the implementation complexity. (2) The performance degradation of various Rake receivers, including MRC S-Rake, MRC P-Rake and equal gain combining (EGC) P-Rake, are compared. Such comparisons shed light on the robustness of various multipath component selection schemes and combining techniques to the variation of system parameters in the presence of mistiming. This paper is organized as follows: Section II describes the DS-UWB system model. In Section III, from the perspective of the achievable information rate, we derive a generalized expression for the system performance degradation induced by timing mismatch. In Section IV, Monte Carlo simulations based on the analytic derivation are conducted to investigate the influence of some key parameters on the system performance and compare the performance degradation of various Rake receivers under mistiming. Section V draws conclusions. ## II System Models Motivated by current DS-UWB system implementations, we confine our discussions to binary phase-shift keying (BPSK) modulation. The equivalent complex-valued system baseband model considered throughout this paper is shown in Fig.1. Figure 1: Block diagrams for the Rake receiver with pulse shaping filters in DS-UWB systems ### II-A Transmitter Model In DS-UWB systems, the random source symbol is spreaded and then modulated with chip pulse $p_{T}(t)$. For each symbol, the transmitted waveform is defined as $S(t)=\sum\limits_{n=0}^{N-1}c[n]p_{T}(t-nT_{c})$ (1) where $c[n]$ denotes the $n$-th chip of the spreading code of length $N$, and $T_{c}$ is the chip duration. ### II-B UWB Channel Model In this model, the IEEE802.15.3a UWB indoor channel for wireless personal area networks (WPAN) is considered [11]. It states that the magnitude of channel amplitude better agrees with the lognormal distribution, corresponding to the shadowing phenomenon which arises from a more serious fluctuation than ordinary fading in the impulse response [12]. In addition, multipath arrivals are grouped into two categories: cluster arrivals, and ray arrivals within each cluster. The channel impulse response is defined as: $H(t)=X\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\alpha_{k,l}\delta(t-T_{l}-\tau_{k,l})$ (2) where $\delta(t)$ represents the impulse function, $X$ stands for the log- normal shadowing, $\alpha_{k,l}$ denotes the multipath gain coefficient, $T_{l}$ is the delay of $l$-th cluster and $\tau_{k,l}$ is the delay of the $k$-th multipath component relative to the $l$-th cluster arrival time ($T_{l}$). By definition, we have $\tau_{0,l}=0$ for $l\in\\{0,1,...,L-1\\}$. ### II-C Reception Model At the receiver, $p_{R}(t)$ is matched to the impulse response of the transmit filter $p_{T}(t)$. In current DS-UWB systems, the raised cosine filter is commonly employed as the pulse shaping filter, which is always achieved by implementing root raised cosine filters as the transmit and receive filters [13]. Therefore, throughout the rest of this paper, we will consider that the overall impulse response $p(t)=p_{T}(t)*p_{R}(t)$ corresponds to a raised cosine filter, which means that $p(t)$ can be written as $p(t)=p_{T}(t)*p_{R}(t)=\frac{\sin(\pi t/T)}{\pi t/T}\frac{\cos(\alpha\pi t/T)}{1-4\alpha^{2}t^{2}/T^{2}}$ (3) where $\alpha$ is the roll-off factor, which represents the excess bandwidth of the filter and is a real number ranging from 0 to 1. In this reception model, we assume that the perfect channel information of the UWB channel is available at the Rake receiver. The impulse response of the Rake receiver can be written as $k(t)=\sum\limits_{j=1}^{J}w_{j}\sigma(t-t_{j})$ (4) where $J$ stands for the Rake diversity orders i.e. finger numbers, $w_{j}$ is the path weights, and $t_{j}$ denotes the path delay satisfying $t_{j}<t_{j+1}$. In P-Rake receiver, the first arrival $J$ multipath components are combined, while the S-Rake receiver selects out the most $J$ strongest multipath components and then combines them together. The Rake weights $w_{j}$ are selected according to different linear combining techniques. For MRC, $w_{j}=a^{*}_{j}$, while for EGC, $w_{j}=sign(\alpha_{j})$, where $a_{j}$ denotes the actual path amplitude, $(.)^{*}$ represents complex conjugation, and $sign(.)$ is the signum function. Finally, de-spreading is performed to get the symbol-level estimation of transmitted data $y(n)$. The whole DS-UWB receiver, including the matched filter $p_{R}(t)$, the Rake receiver $k(t)$ and the de-spreading operation, can be expressed as $\displaystyle R(t)=$ $\displaystyle\sum\limits_{j=1}^{J}\sum\limits_{n=0}^{N-1}c[n]w_{j}p_{T}(t-nT_{c}-t_{j})$ (5) $\displaystyle=$ $\displaystyle\sum\limits_{j=1}^{J}w_{j}S(t-t_{j})$ Finally, let the impulse response given by $S(t)*H(t)*R(t)$ be denoted by $g(t)$, and its symbol-sampled version be $g(n)$. Then we can write $y(n)$ as $y(n)=\sum\limits_{k}x(n-k)g(k)+w(n)$ (6) where $w(n)$ represents the noise component at the Rake output. In (6), $w(n)$ is the symbol-sampled version of $w(t)$, and $w(t)=Z(t)*R(t)$ (7) where $Z(t)$ represents the channel noise which is modeled as AWGN. ## III Performance Analysis under Mistiming In this section, the performance degradation induced by timing mismatch for Rake receivers in DS-UWB systems is derived in terms of the achievable information rate. In the DS-UWB system model, when the length of spreading code $N$ is sufficiently large, the autocorrelation property of spreding code is ideal. Hence the equivalent channel response between the source symbols $x(n)$ and the symbol-level received data $y(n)$ can be simplified to $h(t)=p_{T}(t)*H(t)*p_{R}(t)*k(t)$ (8) Its symbol-sampled version is denoted as $h(k)$. When mistiming is caused by acquisition or tracking errors in the DS-UWB receiver, the branch delays in the Rake receiver get inaccurate. Denote this timing mismatch in all branches as $\Delta t:=t^{\prime}_{j}-t_{j}(\forall j\in\\{1,2...J\\})$ (9) where $t_{j}$ is the actual path delay for path $j$, and $t^{\prime}_{j}$ is the estimated path delay. In this case, the impulse response of the Rake receiver is given by $\displaystyle k^{\prime}(t)=$ $\displaystyle\sum\limits_{j=1}^{J}w_{j}\sigma(t-t^{\prime}_{j})$ (10) $\displaystyle=$ $\displaystyle\sum\limits_{j=1}^{J}w_{j}\sigma(t-t_{j}-\Delta t)$ $\displaystyle=$ $\displaystyle k(t-\Delta t)$ The corresponding equivalent channel with timing errors is then expressed as $\displaystyle f(t)=$ $\displaystyle p_{T}(t)*H(t)*p_{R}(t)*k(t-\Delta t)$ (11) $\displaystyle=$ $\displaystyle h(t-\Delta t)$ Its symbol-sampled version is written as $f(k)$. This timing mismatch will result in performance loss in DS-UWB systems. It is also worthwhile noting that, when the spreading code attains ideal autocorrelation property and the amplitude modulation schemes, especially those with bipolar modulation, e.g. BPSK, are employed, the inter-chip interference (ICI) can be reduced to a negligible order [14]. Furthermore, to simplify the analysis, we also assume the excess multipath delay is smaller than several symobl periods, therefore the effect of inter-symbol interference (ISI) on the DS-UWB system is also limited. In this case, where the ICI and ISI are ignorable, the noise component at the Rake output $w(n)$ can be regarded as AWGN [14]. In the system model described in section II, the source data are constrained to be independent and identically distributed (i.i.d). The system achievable information rate, which corresponds to the maximum mutual information between the input $x(n)$ and the output $y(n)$, is given by [15] 111When the white Gaussian noise at the Rake output is not valid, i.e. in the case of colored Gaussian noise, the calculation of achievable information rate can be performed by using the method of water pouring [16]. $C=\frac{1}{4\pi}\int\limits_{-\pi}^{\pi}\log_{2}[1+2\frac{E_{s}}{N_{0}}|H(e^{j\theta})|^{2}]d\theta$ (12) where $E_{s}$ is the symbol energy, and $H(e^{j\theta})$ is the Fourier transform of the equivalent channel impulse response. In the scenario of perfect synchronization, $H(e^{j\theta})$ stands for the Fourier transform of $h(k)$; in the scenario with timing errors, $H(e^{j\theta})$ represents the Fourier transform of $f(k)$. From (12), it is obvious that the achievable information rate $C$ is derived from $E_{s}/N_{0}$ considerations, and a certain $E_{s}/N_{0}$ is required to achieve a specified $C$. Let $R$ be the target achievable information rate. In the perfect synchronization scenario, we have $C|_{H(e^{j\theta})=F[h(k)]}=R$ (13) where $F\\{\\}$ represents the Fourier transform operator. Assume the needed $E_{s}/N_{0}$ in dBs at this point is $SNR_{h}$. AS for the scenario with timing errors, let $SNR_{f}$ be the $E_{s}/N_{0}$ in dBs at which $C|_{H(e^{j\theta})=F[f(k)]}=R$ (14) Finally, the performance degradation $L$ induced by timing mismatch is obtained as $L=SNR_{f}-SNR_{h}$ (15) ## IV Numerical Results and Discussions In this section, Monte Carlo simulations based on the generalized expressions (12) and (15) are conducted to evaluate the effect of various system configurations on the performance degradation $L$ induced by timing mismatch in DS-UWB systems. In the following simulations, in order to keep the simulation complexity on a reasonable level, the timing mismatch $\Delta t$ on Rake diversity branches is set as $(0,0.1,0.2,...0.9)\times T_{s}$, where $T_{s}$ denotes the duration of one symbol. In addition, the numerical results are averaged over the best 900 out of 1000 IEEE 802.15.3a CM1 channel realizations, following the recommended instructions in [11] that the worst 10% channels are ignored in the simulation. The rest of this section consists of two parts: In the first part, the influence of timing mismatch on the performance of the widely used MRC S-Rake receiver is investigated in details under different system parameters; In the second part, we compare the performance loss of various Rake receivers, including MRC S-Rake, MRC P-Rake and EGC P-Rake, in the presence of mistiming. The Influence of Timing Mismatch on MRC S-Rake Receivers: The needed $E_{s}/N_{0}$ curves for MRC S-Rake receivers in DS-UWB systems with different roll-off factors $\alpha$ are plotted in Fig.2. It is observed that if no timing mismatch exists, as $\alpha$ increases, the needed $E_{s}/N_{0}$ to achieve the target information Rate $R$ gets smaller. However, the receiver with larger $\alpha$ is more sensitive to the timing mismatch $\Delta t$. As seen in this figure, when mistiming is small, the receiver with larger $\alpha$ needs less $E_{s}/N_{0}$ to obtain the target $R$; however, they requires more $E_{s}/N_{0}$ as mistiming aggravates. Figure 2: The needed $E_{s}/N_{0}$ (dB) for MRC S-Rake receivers in DS-UWB systems with different roll-off factors $\alpha$ when $J=8$ and $R=0.3$ Fig.3 depicts the $E_{s}/N_{0}$ needed for MRC S-Rake receivers with different diversity orders to achieve $R=0.3$ when $\alpha$ equals to 1.0. As one can expect, when no timing errors exhibit, the MRC S-Rake receiver with more diversity branches needs less $E_{s}/N_{0}$ to achieve the target $R$. It is further seen that the sensitivity to timing mismatch increases with increasing $J$ in the MRC S-Rake receiver. Hence there also exists a design trade-off between the robustness to mistiming and the implementation complexity. Figure 3: The needed $E_{s}/N_{0}$ (dB) for MRC S-Rake receivers with different diversity orders $J$ when $\alpha=1.0$ and $R=0.3$ The needed $E_{s}/N_{0}$ for MRC S-Rake receivers to achieve various target information rates $R$ is illustrated in Fig.4 when $J$ is 8 and $\alpha$ equals to 0.3. It shows that as $R$ increases, the needed $E_{s}/N_{0}$ increases obviously. However, the performance loss induced by timing mismatch rarely varies with the increase of the target achievable information rate in MRC S-Rake receivers. Figure 4: The needed $E_{s}/N_{0}$ (dB) for MRC S-Rake receivers with different target achievable information rates $R$ when $\alpha=0.3$ and $J=8$ The Comparison of Various Rake Receivers under Mistiming: In this part, the mistiming is assumed to follow uniform distribution [17]. We consider two cases: one is the worst case with the maximum performance degradation; the other is the average case, which represents the average degradation over the set of timing mismatch $\Delta t$. Fig.5 \- Fig.7 demonstrate the performance degradation of three types of Rake receivers, including MRC S-Rake, MRC P-Rake and EGC P-Rake, with respect to the variation of the excess bandwidth, the diversity order and the target achievable information rate under timing mismatch respectively. Fig.5 shows that with roll-off factor increasing, all types of Rake receivers obtain worse performance due to timing errors. Among all the receivers, MRC S-Rake is most sensitive to the increase of excess bandwidth and the EGC P-Rake is least sensitive. From Fig.6, it is observed that as the diversity order increases, the performance degradation of all the three kinds of Rake receivers gets larger. Fig.6 also shows that when MRC is employed, P-Rake is more sensitive to the change of Rake finger numbers than S-Rake, and in P-Rake receiver the EGC technique is more robust compared with MRC. Fig.7 demonstrates that the target information rate rarely impacts the performance loss of all the Rake receivers under timing mismatch, and the MRC S-Rake suffers the largest performance loss compared with other Rake receivers. ## V Conclusion The effect of imperfect timing has been evaluated for the Rake reception in DS-UWB systems from the perspective of the achievable information rate. A generalized expression for the system performance degradation is derived, then corresponding simulations are conducted to investigate the effect of timing mismatch on the widely used MRC S-Rake receiver with respect to the excess bandwidth induced by the roll-off factor in RRC filters, the multipath diversity order and the target information rate. Simulation results illustrate that the performance loss has little relationship with the target information rate, but varies significantly with the system bandwidth and the diversity order, which further demonstrates that there exist fundamental trade-offs among the system timing requirement, the occupied bandwidth and the implementation complexity of DS-UWB systems. In addition, the performance degradation of various types of Rake receivers, including MRC S-Rake, MRC P-Rake and EGC P-Rake, is compared, and the sensitivity of different multipath component selection schemes and diversity combining techniques to the variation of system parameters are obtained. The numerical results also show that among the three types of Rake receivers, the MRC S-Rake suffers the most performance degradation in the presence of mistiming. Figure 5: Performance degradation $L$ (dB) as a function of the roll-off factor $\alpha$ for various Rake receivers when $J=8$ and $R=0.3$ Figure 6: Performance degradation $L$ (dB) as a function of the diversity order $J$ in various Rake receivers when $\alpha=1.0$ and $R=0.3$ Figure 7: Performance degradation $L$ (dB) as a function of the target achievable information rate $R$ for various Rake receivers when $\alpha=0.3$ and $J=8$ ## References * [1] S. Roy, J. R. Foerster, V. S. Somayazulu, and D. G. 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Alagoz, ”Achievable information rates of PPM impulse radio for UWB channels and rake reception” , IEEE Transactions on Communications, vol.58, no.5, pp.1524-1535, May 2010\. * [11] Channel-Modeling-Subcommittee-Report-Final, IEEE P802.15. Dec. 2002. * [12] R. C. French, ”The effect of fading and shadowing on channel reuse in mobile radio”, IEEE Transactions on Vehicular Technology, vol.28, no.3, pp.171-181, Aug. 1979. * [13] A. Parihar, L. Lampe, R. Schober, and C. Leung, ”Equalization for DS-UWB Systems Part I: BPSK Modulation,” IEEE Transactions on Communications, vol.55, no.6, pp.1164-1173, Jun. 2007. * [14] J. Zhang, R. A. Kennedy, and T. D. Abhayapala, ”Performance of Rake reception for ultra-wideband signals in a lognormal-fading channel”, Proc. IWUWBS, pp. 5-9, 2003. * [15] W. Hirt, and J. L. Massey, ”Capacity of the discrete-time Gaussian channel with intersymbol interference,” IEEE Transactions on Information Theory, vol.34, no.3, pp.380-388, May 1988. * [16] W. Xiang, and S. S. Pietrobon. ”On the capactity abd normalization of ISI channels”, IEEE Transactions on Information Theory, vol.49, no.9, pp.2263-2268, Sep. 2003. * [17] M. O. Sunay, and P. J. Mclane, ”Diversity combining for DS CDMA systems with synchronization errors”, IEEE International Conference on Communications (ICC), pp.83-89, 1996.
arxiv-papers
2010-12-21T04:28:18
2024-09-04T02:49:15.877654
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Chunhua Geng, Yukui Pei, Jiaqi Zhang and Ning Ge", "submitter": "Chunhua Geng", "url": "https://arxiv.org/abs/1012.4542" }
1012.4556
# Selective Multipath Interference Canceller with Linear Equalization for DS- UWB Systems with Low Spreading Factor Chunhua Geng, Yukui Pei, and Ning Ge This paper was presented in part at the IEEE International Conference on Ultra-Wideband (ICUWB), Vancouver, Canada, 2009. The authors are with the State Key Laboratory on Microwave and Digital Communications, Tsinghua National Laboratory for Information Science and Technology, Tsinghua University. (email: gengch07@mails.tsinghua.edu.cn, peiyk@tsinghua.edu.cn, gening@tsinghua.edu.cn) ###### Abstract In high rate DS-UWB systems with low spreading factor, the selective multipath interference canceller with linear equalization (SMPIC-LE) is developed to alleviate severe multipath interferences induced by the poor orthogonality of spreading codes. The SMPIC iteratively mitigates the strongest inter-path interference, inter-chip interference and inter-symbol interference, while the former two are unresolvable in conventional RAKE-decision feedback equalizer (DFE) receivers. The numerical results and complexity analysis demonstrate that SMPIC-LE with proper parameters provides an attractive overall advantage in performance and computational complexity compared with RAKE-DFE. In addition, it approaches the matched filter bound well as the RAKE finger in SMPIC increases. ###### Index Terms: _Equalization, iterative receiver, multipath interference, RAKE, ultra- wideband_ ## I Introduction Ultra-wideband (UWB) is a promising technology for wireless high rate and short range communications [2]. Direct-sequence spreading based UWB (DS-UWB) and multiband orthogonal frequency-division multiplexing UWB (MB-OFDM UWB) are two main physical layer standards for high data rate wireless personal area networks (WPAN) [3], [4]. Because of the fine properties of coherent processing of the occupied bandwidth and the widest contiguous bandwidth, DS- UWB has received considerable attention from both academia and industry [5], [6]. For high data rate DS-UWB systems supporting transmission rate ranging from several Megabit per second to more than one Gigabit per second, most recent research on the receiver design focuses on the RAKE reception with symbol level decision feedback equalizer (DFE) [7]-[9]. In practical high rate DS-UWB systems, limited by state-of-the-art ADC technology, the spreading factor (SF) cannot be large enough to maintain the ideal orthogonality between spreading codes [10]. Therefore, the conventional RAKE-DFE receiver would suffer significant performance loss due to severe inter-path interference (IPI), inter-chip interference (ICI) and inter-symbol interference (ISI) [11]-[14]. The former two kinds of interference can not be mitigated by the RAKE-DFE receiver effectively. Furthermore, in order to combat the severe ISI induced by long channel delay spread, the DFE tap number has to be quite large. The demanding computational complexity of DFE always exceeds that of the RAKE receiver significantly by far and becomes a heavy burden for system design. To resolve the above problems, the selective multipath interference canceller (SMPIC) with symbol level linear equalization (LE) is proposed in this paper for practical high rate DS-UWB systems with low SF. The SMPIC is capable of mitigating the IPI, ICI and ISI by reconstructing and subtracting the selected strongest multipath interferences from the received signal in an iterative way. Then the symbol level LE is concatenated to alleviate the remaining ISI. In addition, to validate the effectiveness of the SMPIC-LE receiver, we derive the matched filter bound (MFB), which takes into account such practical constrains as the sampling rate and the RAKE diversity order, i.e. the finger number. Simulation results and complexity analysis show that the proposed SMPIC-LE can achieve similar or even better performance with much lower computational complexity compared with the conventional RAKE-DFE receiver in various realistic UWB channels. Moreover, as the RAKE diversity order increases, the performance of SMPIC-LE receivers can approach the derived MFB well. The remainder of this paper is organized as follows. The DS-UWB system model with the proposed SMPIC-LE receiver is introduced in Section II. In Section III, the computational complexity and performance of the SMPIC-LE receiver are analyzed, and the MFB is also derived. In Section IV, the corroborating simulation results are presented. Section V summarizes the whole paper. ## II System Model In this paper, the IEEE802.15.3a UWB indoor channel model is employed [15]. The equivalent complex-valued baseband model with the proposed SMPIC-LE receiver is shown in Fig.1. Figure 1: Diagram of the DS-UWB System model with the SMPIC-LE receiver ### II-A Transmitter In this paper, we only focus on binary phase-shift keying (BPSK) modulation, which is the mandatory transmission mode for DS-UWB systems. At the transmitter, the random source symbol is spread and modulated with chip pulse $g_{T}(t)$. For each symbol, the pulse shape is defined as $g(t)=\sum\limits_{n=0}^{N-1}c[n]g_{T}(t-nT_{c})$ (1) where $c[n]$ denotes the $n$-th chip of the spreading code of length $N$, and $T_{c}$ is the chip duration. Assume $M$ symbols are contained in each frame, and each transmitted frame can be written as $s(t)=\sum\limits_{m=0}^{M-1}b[m]g(t-mT_{b})$ (2) where $b[m]\in\\{-1,+1\\}$ represents the $m$-th symbol of each frame, and $T_{b}=NT_{c}$ is the symbol interval. ### II-B UWB channels In order to compare standardization proposals for high data rate WPANs, IEEE802.15.3a task group developed a standard channel model for UWB systems [15]. This model is based on the Saleh-Valenzuela model [16] with some modification to account for the properties of realistic UWB channels. In this model, multipath arrivals are grouped into two categories: cluster arrivals and ray arrivals within each cluster. The channel impulse response is defined as: $h(t)=X\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\alpha_{k,l}\delta(t-T_{l}-\tau_{k,l})$ (3) where $X$ represents the log-normal shadowing, $\alpha_{k,l}$ is the multipath gain coefficient, $T_{l}$ is the delay of $l$-th cluster and $\tau_{k,l}$ is the delay of the $k$-th multipath component relative to the $l$-th cluster arrival time ($T_{l}$). By definition, we have $\tau_{0,l}=0$ for $l\in\\{0,1,...,L-1\\}$. ### II-C SMPIC-LE Receiver In the DS-UWB system, the SMPIC-LE receiver is developed to alleviate the severe multipath interferences induced by the poor orthogonality of spreading codes. In the first stage, the SMPIC is employed to specifically mitigate the strongest multipath interference components, including IPI, ICI and ISI. Then, a conventional symbol level LE with small tap number is concatenated to combat the residual ISI. In the sequel of this section, we mainly focus on the proposed SMPIC scheme. The structure of SMPIC is presented in Fig.2. The SMPIC works in an iterative manner. Its purpose is to eliminate the interference induced by multipath delay at each RAKE finger. Similar with selective-RAKE (SRAKE) [17], the SMPIC selects the instantaneously strongest $J$ multipath components and combines them together at first. Then the interference is estimated and subtracted from the received data at each RAKE finger to get more precise input signals for the RAKE reception in the next iteration. In order to reduce the complexity, the interference is reconstituted by using the hard decision of the SRAKE output. Through this iterative process, the precision of the correlation result in each RAKE finger is improved, so is the output of the receiver. In DS-UWB systems, the received signal of each frame is given by $\displaystyle r(t)=$ $\displaystyle\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{N-1}a_{k,j}b[m]c[n]$ (4) $\displaystyle g_{T}(t-T_{l}-\tau_{k,j}-nT_{c}-mT_{b})+z(t)$ where $z(t)$ is additive white Gaussian noise (AWGN) with mean being zero and power spectral density being $N_{0}/2$ W/Hz. We assume the receiver can get the perfect channel knowledge. Received data $r(t)$ is first fed into maximal ratio combining (MRC) SRAKE in SMPIC. After conventional RAKE processing, the output is sent to hard-decision module. The estimation of $m$-th bit of each frame at the output of hard-decision module is denoted as $\widetilde{b}^{(0)}[m]$. This estimated sequence is then spread, modulated and processed by a very simple multipath interference regenerator (MIR). In this sub-module, the modulated sequence is multiplied by the amplitude of selected $J$ paths and delayed by corresponding time. So the reconstituted $j$-th path signal can be expressed as $\displaystyle\widetilde{r}_{j}^{(0)}(t)=$ $\displaystyle\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{N-1}a_{k_{j},l_{j}}\widetilde{b}^{(0)}[m]c[n]$ (5) $\displaystyle g_{T}(t-T_{l_{j}}-\tau_{k_{j},l_{j}}-nT_{c}-mT_{b})$ where $j\in\\{1,2,...,J\\}$, and $\alpha_{k_{j},l_{j}}$ is the multipath gain coefficient corresponding to the $j$-th RAKE finger. $T_{l_{j}}$ and $\tau_{k_{j},l_{j}}$ denote the delays. In the next iteration, the input signal to the $j$-th finger is represented as $\displaystyle r_{j}^{(1)}(t)=r(t)-w\sum\limits_{j^{\prime}=1,j^{\prime}\neq j}^{J}\widetilde{r}_{j^{\prime}}^{(0)}(t)$ (6) where $w\in[0,1]$ is a constant named as interference rejection weight, which allows to reduce the impact of possible errors presented in the estimated multipath interference replicas. Then $r_{j}^{(1)}(t)$ is delivered to the MRC SRAKE receiver in the next iteration. The SRAKE output can be used as either the input of MIR for the following iterations, or the output of the SMPIC receiver if the pre-defined iteration time $p$ is achieved. Finally, the output of the SMPIC is sent to the LE to reduce the remaining ISI. Figure 2: Block diagram of SMPIC (the iteration times $p$ are 2 in this diagram) ## III Performance and Computational Complexity Analysis ### III-A Performance Analysis and the Matched Filter Bound In this subsection, the effect of multipath components (MPCs) on conventional SRAKE receivers and the validity of SMPIC are analyzed first. The energy of $g_{T}(t)$ is defined as $E_{g}$, $E_{g}=\int_{-\infty}^{+\infty}g_{T}^{2}(t)dt$ (7) The normalized autocorrelation function of $g_{T}(t)$ expresses as $R_{g}(\triangle t)=\frac{1}{E_{g}}\int_{-\infty}^{+\infty}g(t)g(t+\triangle t)dt$ (8) The channel is assumed perfectly known at the receiver. The local template of the $\widetilde{m}$-th bit in the $j$-th finger of SRAKE is given by $v_{j}(t)=\sum\limits_{\widetilde{n}=0}^{N-1}\alpha_{k_{j},l_{j}}^{*}c[\widetilde{n}]g_{T}(t-T_{l_{j}}-\tau_{k_{j},l_{j}}-\widetilde{m}T_{b}-\widetilde{n}T_{c})$ (9) where $(.)^{*}$ denotes complex conjugation. The correlation output of the $j$-th finger is $R_{j}(t)=\int\limits_{\widetilde{m}T_{b}}^{(\widetilde{m}+1)T_{b}}r(t)v_{j}(t)dt=b[\widetilde{m}](S+I_{1}+I_{2})+I_{3}+Z$ (10) where $Z$ represents the effect of noise, and $S=NE_{g}|\alpha_{k_{j},l_{j}}|^{2}$ (11) is the signal component. $I_{1}$, $I_{2}$ and $I_{3}$ are the IPI, ICI and ISI respectively, $I_{1}=NE_{g}\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}a_{k,j}a_{k_{j},l_{j}}^{*}R_{g}(T_{l}-T_{l_{j}}+\tau_{k,j}-\tau_{k_{j},l_{j}})$ (12) where $l\neq l_{j}$ or $k\neq k_{j}$, $\displaystyle I_{2}=$ $\displaystyle E_{g}\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\sum\limits_{n=0}^{N-1}\sum\limits_{\widetilde{n}=0}^{N-1}a_{k,j}a_{k_{j},l_{j}}^{*}c[n]c[\widetilde{n}]$ (13) $\displaystyle R_{g}(T_{l}-T_{l_{j}}+\tau_{k,j}-\tau_{k_{j},l_{j}}+(n-\widetilde{n})T_{c})$ where $n\neq\widetilde{n}$, $\displaystyle I_{3}=$ $\displaystyle E_{g}\sum\limits_{m=0}^{M-1}\sum\limits_{l=0}^{L-1}\sum\limits_{k=0}^{K-1}\sum\limits_{n=0}^{N-1}\sum\limits_{\widetilde{n}=0}^{N-1}a_{k,j}a_{k_{j},l_{j}}^{*}b[m]c[n]c[\widetilde{n}]$ (14) $\displaystyle R_{g}(T_{l}-T_{l_{j}}+\tau_{k,j}-\tau_{k_{j},l_{j}}+(n-\widetilde{n})T_{c}$ $\displaystyle+(m-\widetilde{m})T_{b})$ where $m\neq\widetilde{m}$. The accuracy of the RAKE output is closely related to the statistical properties of IPI, ICI, and ISI, which follow an impulsive distribution [11]. The conventional symbol level equalizer can only combat long ISI at the cost of high computational complexity, but fails to mitigate IPI and ICI effectively. When the SF is small, which means that the autocorrelation property of the spreading code is poor, the multipath interferences degrade the performance of the RAKE-DFE receiver dramatically. From (5) and (6), we can see that the proposed SMPIC can subtract the $J$-1 strongest interference components in every finger before the correlation and combining at each iteration, hence the strongest interferences, including $I_{1}$, $I_{2}$, and $I_{3}$, in (10) can be mitigated effectively. To validate the effectiveness of the SMPIC-LE receiver, in the following simulation part, the performance of SMPIC-LE is compared with the MFB of DS- UWB systems, which yields the absolute performance limit for equalization schemes. In order to obtain expressions for the bit error rate (BER) of the MFB, we define the signal-to-noise ratio (SNR) as $\gamma_{r}=\frac{E_{b}(r)}{N_{0}}$ (15) where $E_{b}(r)$ is the received energy per bit for the $r$th UWB channel realization. The corresponding BER($\gamma_{r}$) for BPSK in one particular channel realization can be written as $BER(\gamma_{r})=Q(\sqrt{2\gamma_{r}})$ (16) where $Q(*)$ stands for $Q$ function. The average BER is obtained semi- analytically by averaging over $R$ channel realizations $BER_{MFB}=\frac{1}{R}\sum_{r=1}^{R}BER(\gamma_{r})$ (17) As for the DS-UWB systems employing $J$-finger RAKE receiver, where $J$ is much smaller than the total number of resolvable multipath components for complexity reasons, we obtain $E_{b}(r)=\sum_{j=1}^{J}|h(t_{j})|^{2}$ (18) where $t_{j}$ ($j\in{1,2,...J}$) denotes the positions of the strongest $J$ multipath components in one channel realization. The resolution of $t_{j}$ equals to the sampling rate at the receiver. In this paper, the derived MFB takes into account the sampling rate at the receiver and the effect of selective RAKE combining with a limited number of RAKE fingers. Therefore, it demonstrates an accurate performance bound for practical receivers in DS-UWB systems. ### III-B Computational Complexity Analysis In this paper, the computational complexity of SMPIC-LE and SRAKE-DFE receiver is calculated in terms of multiplications and divisions per output symbol (MADPOS) [18]. The proposed SMPIC is comprised of two kinds of basic sub-modules: one is the MRC SRAKE, and the other is the MIR. When the SF is small, correlators, the main part of SRAKE, are quite simple. The MIR can be seen as the inverse procedure of RAKE processing from Section II. Therefore, the computational complexity of MIR is comparable with that of RAKE combining. Moreover, the computational complexity of SMPIC is independent of the magnitude of path delays, hence it can be kept under a relatively low level in various transmission scenarios. The computational complexity of SRAKE and SMPIC is given by $\displaystyle C_{SRAKE}=2\times J$ (19) $\displaystyle C_{SMPIC}=2\times(p+1)\times J+p\times 3J$ (20) where $J$ stands for the RAKE finger number and $p$ denotes the iteration time in SMPIC. For equalization, the widely used adaptive Kalman recursive least-square (K-RLS) algorithm is employed for adjusting tap coefficients to ensure fast convergence and lower stead-state mean square error (MSE), and hence a favorable detection performance in UWB system [19]. The adaptive equalizer works in two stages: in training stage, a training sequence is employed to initially adjust the tap weights; in decision directed stage, the decisions at the output of the equalizer are used to continue the coefficients adaption process. The computational complexity of equalizers based on K-RLS is approximately given by [18] $\displaystyle C_{K-RLS}=2.5\times N^{2}+4.5\times N$ (21) where $N$ represents the total tap number in equalizers. ## IV Numerical Results and Discussion ### IV-A System Parameters Monte Carlo simulations have been run to access the performance of the proposed receiver in high rate DS-UWB systems with low SF. The spreading code is set as [-1 +1] with SF being an extreme of 2. The sampling rate is $T_{c}/4$. At the receiver, the value of interference rejection weight $w$ in SMPIC is chosen as 0.9 by investigating the effect of different weights on the system BER performance. The iteration time $p$ is set as 2, which can guarantee the performance convergence in most cases through our simulations. The forgetting factor in K-RLS algorithm is 0.99999. As for equalization, without notable instructions, the lengths of LE tap $L$, DFE feedforward tap $FF$ and feedback tap $FB$ are set as 15, 25 and 20, respectively. The IEEE 802.15.3a CM1 line-of-sight (LOS) and CM4 extreme non-LOS (NLOS) UWB indoor channel models are considered here. According to the recommended instructions in [15], the numerical results are averaged over the best 90 out of 100 channel realizations. ### IV-B Bit Error Rate Performance As a function of $E_{b}/N_{0}$ at the input of receivers, the BER performance of SRAKE-DFE and SMPIC-LE is evaluated and compared with MFB. Figure 3: BER performance of SRAKE-DFE and SMPIC-LE receivers in CM1 channel Figure 4: BER performance of SRAKE-DFE and SMPIC-LE receivers in CM4 channel First, we present BER curves of SMPIC-LE and SRAKE-DFE receivers with different RAKE finger numbers and transmission data rates. Fig.3 shows the system performance in the CM1 channel model. As seen in this figure, the SMPIC-LE outperforms the conventional SRAKE-DFE receiver. When the transmission data rate equals to 250Mbps, the $J$=32 SMPIC-LE gets a performance gain about 0.2dB over $J=32$ SRAKE-DFE at a BER of $10^{-4}$, and the loss in power efficiency compared with the derived MFB is within 1dB. As the data rate increases to 1.5Gbps, the advantage of SMPIC-LE over SRAKE-DFE gets more significant. It is shown when $J$ equals to 32, the performance gain is up to about 1dB, and the performance of SMPIC-LE approaches the MFB well. From Fig.4, it is observed that in the case of CM4 channels, when data rate is 250Mbps, the proposed SMPIC-LE receiver only suffers negligible performance loss compare with SRAKE-DFE with the same RAKE fingers. As the data rate increases to 1.5Gbps, the SMPIC-LE receiver can lower the error floor. From the above two figures, we can conclude that as the data rate increases, the performance gain of SMPIC-LE over SRAKE-DFE improves. This can be attributed to the fact that with the data rate increasing, more resolvable strong interferences, which degrade the system performance dramatically, occur at the receiver, and the proposed SMPIC-LE receiver can alleviate these interferences in a more effective iterative way compared with the conventional SRAKE-DFE receiver. Figure 5: BER performance of SMPIC-LE receivers with different equalizer tap lengths in CM4 channel when the transmission data rate is 250Mbps Then the effect of the LE tap number on the BER performance of the SMPIC-LE receiver is also investigated. From Fig.5, it shows that as LE tap number $L$ increases, the system performance improves as well. In addition, as the RAKE finger $J$ increases, the SMPIC-LE receiver yields a close-to-optimum performance and the performance gain by increasing LE taps become unobvious. For instance, when the finger number $J$ is 16, with $L$ increasing from 15 to 25, the SMPIC-LE receiver can obtain a performance gain of more than 1dB. When $J$ increases to 32, the performance improvement is only about 0.4dB. This is due to the fact that as the RAKE finger number gets larger, the strong ISIs are mitigated by SMPIC effectively, hence increasing LE taps cannot get additional significant performance gain. This fact provides a useful pointer for system designers when specifying system parameters. Our findings also suggest that the SMPIC-LE receiver with more RAKE fingers outperforms the receiver with less RAKE fingers but more equalizer taps, which demonstrates the key role of the proposed SMPIC scheme to mitigate severe multipath interferences in UWB channels. ### IV-C Computational Complexity Comparison Finally, the computational complexity of SMPIC-LE and SRAKE-SFE receivers adopted in the simulations are compared. The MADPOS of both SMPIC-LE and SRAKE-DFE is shown in Table I. As seen in this table, when $J$ equals to 16, the MADPOS in the SMPIC-LE receiver with $L$ = 15 is 822, which is only 15.5% of that in the SRAKE-DFE with $FF$ = 25 and $FB$ = 20. When $J$ increases to 32, the SMPIC-LE can still save 81% MADPOS than SRAKE-DFE. These results demonstrate that the computational complexity of SMPIC-LE scheme is much less than that of conventional SRAKE-DFE receivers. TABLE I: Computational Complexity Comparison (MADPOS) | | | SRAKE-DFE --- ($FF$=25, $FB$=20) | SMPIC-LE --- ($L$=15, $p$=2) Saving | $J$ = 16 | 5297 | 822 | 84.5% | $J$ = 32 | 5329 | 1014 | 81.0% ## V Conclusions The scheme presented in this paper offers a low computational complexity alternative to the conventional SRAKE-DFE receiver, which provides a more efficient way for UWB signal detection by mitigating significant multipath interference components specifically. In this proposed SMPIC-LE scheme, the receiver can alleviate the strongest IPI, ICI and ISI, while the former two interferences are unresolvable in conventional RAKE-DFE receivers. The MFB, which takes into account the effects of sampling rate and the number of RAKE fingers at the receiver, is also derived. Numerical results and complexity analysis show that compared with SRAKE-DFE, the SMPIC-LE receiver, with much lower computational complexity, can achieve similar or even better performance in high rate DS-UWB systems with low SF for various UWB propagation scenarios. In addition, as the RAKE finger number increases, the low-complexity SMPIC-LE receiver approaches the derived MFB limit well. ## Acknowledgment This work is supported by National Nature Science Foundation of China No. 60928001 and 60972019, National Basic Research Program of China under grant No. 2007CB310608, and the National Science & Technology Major Project under grant No. 2009ZX03006-007-02 and 2009ZX03006-009. ## References * [1] * [2] L. Q. Yang, and G. B. Giannakis, “Ultra-wideband communications: an idea whose time has come,” IEEE Signal Process. Mag., vol.21, no.6, pp.26-54, Nov. 2004. * [3] R. Fisher, R. Kohno, M. McLaughlin, and M. Welbourn, ”DS-UWB physical layer submission to 802.15 task group 3a,” IEEE P802.15-04/0137r4, Jan. 2005. * [4] IEEE P802.15. Multi-band OFDM physical layer proposal for IEEE 802.15 task group 3a (Doc. Number P802.15-03/268r3). 2004. * [5] P. Runkle, J. McCorkle, T. Miller, and M. 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Valenzuela, “A statistical model for indoor multipath propagation,” IEEE J. Sel. Areas Commun., vol.5, no.2, pp.128-137, Feb. 1987. * [17] D. Cossioli, M. Z. Win, F. Vatalaro, and A. F. Molish,“Low complexity Rake receivers in ultra-wideband channels,” IEEE Trans. Wireless Commun., vol.6, no.4, pp.1265-1275, Apr. 2007. * [18] J. G. Proakis, ed., Digital Communications. New York, NY, McGraw-Hill, Inc., 4th Ed., 2001. * [19] C. C. Hu, and J. F. Chang, “DS-UWB forward link adaptive chip-equalizer using subband decomposition technique,” Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), pp.2802-2806, 2009.
arxiv-papers
2010-12-21T07:11:26
2024-09-04T02:49:15.884109
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Chunhua Geng, Yukui Pei, and Ning Ge", "submitter": "Chunhua Geng", "url": "https://arxiv.org/abs/1012.4556" }
1012.4643
¡html¿ ¡head¿ ¡title¿CERN-2010-002¡/title¿ ¡/head¿ ¡body¿ ¡h1¿¡a href=”http://physicschool.web.cern.ch/PhysicSchool/2009/Info/Bautzen_Info.html”¿2009 European School of High-Energy Physics¡/a¿¡/h1¿ ¡h2¿Bautzen, Germany, 14 - 27 June 2009¡/h2¿ ¡h2¿Proceedings - CERN Yellow Report ¡a href=”http://cdsweb.cern.ch/record/1119304?ln=en”¿CERN-2010-002¡/a¿¡/h2¿ ¡h3¿editors: C. Grojean and M. Spiropulu ¡/h3¿ The European School of High-Energy Physics is intended to give young physicists an introduction to the theoretical aspects of recent advances in elementary particle physics. These proceedings contain lecture notes on quantum field theory, quantum chromodynamics, physics beyond the Standard Model, flavour physics, effective field theory, cosmology, as well as statistical data analysis. ¡h2¿Lectures¡/h2¿ ¡!– Quantum field theory and the standard model –¿ LIST:1012.3883 ¡br¿ ¡!– Elements of QCD for hadron colliders –¿ LIST:1011.5131 ¡br¿ ¡!– Beyond the Standard Model –¿ LIST:arXiv:1005.1676 ¡br¿ ¡!– Introduction to flavor physics –¿ LIST:arXiv:1006.3534 ¡br¿ Title: ¡a href=”http://cdsweb.cern.ch/record/1281952”¿Effective field theory - concepts and applications¡/a¿ ¡br¿ Author: M. Beneke ¡br¿ Journal-ref: CERN Yellow Report CERN-2010-002, pp. 145-148 ¡br¿ ¡!– The violent universe: the Big Bang –¿ LIST:arXiv:1005.3955 ¡br¿ ¡!– Topics in statistical data analysis for high-energy physics –¿ LIST:arXiv:1012.3589 ¡br¿ ¡/body¿ ¡/html¿
arxiv-papers
2010-12-21T13:09:41
2024-09-04T02:49:15.891319
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "C. Grojean and M. Spiropulu", "submitter": "Scientific Information Service Cern", "url": "https://arxiv.org/abs/1012.4643" }
1012.4726
# Pathways of Distinction Analysis: A new technique for multi-SNP analysis of GWAS data Rosemary Braun and Kenneth Buetow National Cancer Institute, NIH, Bethesda, MD. (March 4, 2011) ###### Abstract Genome-wide association studies have become increasingly common due to advances in technology and have permitted the identification of differences in single nucleotide polymorphism (SNP) alleles that are associated with diseases. However, while typical GWAS analysis techniques treat markers individually, complex diseases (cancers, diabetes, and Alzheimers, amongst others) are unlikely to have a single causative gene. There is thus a pressing need for multi-SNP analysis methods that can reveal system-level differences in cases and controls. Here, we present a novel multi-SNP GWAS analysis method called Pathways of Distinction Analysis (PoDA). The method uses GWAS data and known pathway-gene and gene-SNP associations to identify pathways that permit, ideally, the distinction of cases from controls. The technique is based upon the hypothesis that if a pathway is related to disease risk, cases will appear more similar to other cases than to controls (or vice versa) for the SNPs associated with that pathway. By systematically applying the method to all pathways of potential interest, we can identify those for which the hypothesis holds true, i.e., pathways containing SNPs for which the samples exhibit greater within- class similarity than across classes. Importantly, PoDA improves on existing single-SNP and SNP-set enrichment analyses in that it does not require the SNPs in a pathway to exhibit independent main effects. This permits PoDA to reveal pathways in which epistatic interactions drive risk. In this paper, we detail the PoDA method and apply it to two GWA studies: one of breast cancer, and the other of liver cancer. The results obtained strongly suggest that there exist pathway-wide genomic differences that contribute to disease susceptibility. PoDA thus provides an analytical tool that is complementary to existing techniques and has the power to enrich our understanding of disease genomics at the systems-level. ## Author Summary We present a novel method for multi-SNP analysis of genome-wide association studies. The method is motivated by the intuition that if a set of SNPs is associated with disease, cases and controls will exhibit more within-group similarity than across-group similarity for the SNPs in the set of interest. Our method, Pathways of Distinction Analysis (PoDA), uses GWAS data and known pathway-gene and gene-SNP associations to identify pathways that permit the distinction of cases from controls. By systematically applying the method to all pathways of potential interest, we can identify pathways containing SNPs for which the cases and controls are distinguished and infer those pathway’s role in disease. We detail the PoDA method and describe its results in breast and liver cancer GWAS data, demonstrating its utility as a method for systems- level analysis of GWAS data. ## Introduction Genome-wide association studies (GWAS) have become a powerful and increasingly affordable tool to study the genetic variants associated with disease. Modern GWAS yield information on millions of single nucleotide polymorphism (SNPs) loci distributed across the human genome, and have already yielded insights into the genetic basis of complex diseases [1, 2], including diabetes, inflammatory bowel disease, and several cancers [3, 4, 5, 6, 7]; a complete list of published GWAS can be found at the National Cancer Institute–National Human Genome Research Institute (NCI-NHGRI) catalog of published genome-wide association studies [8]. Typically, the data produced in GWAS are analyzed by considering each SNP independently, testing the alleles at each locus for association with case status; significant association is indicative of a nearby genetic variation which may play a role in disease susceptibility. Genomic regions of interest may also be subject to haplotype analysis, in which a handful of alleles transmitted together on the same chromosome are tested for association with disease; in this case, the loci which are jointly considered are located within a small genomic region, often confined to the neighborhood of a single gene. Recently, however, there has been increasing interest in multilocus, systems- based analyses. This interest is motivated by a variety of factors. First, few loci identified in GWAS have large effect sizes (the problem of “missing heritability”) and it is likely that the common–disease, common–variant hypothesis [9, 10] does not hold in the case of complex diseases. Second, single marker associations identified in GWAS often fail to replicate. This phenomenon has been attributed to underlying epistasis [11], and a similar problem in gene expression profiling has been mitigated through the use of gene-set statistics. Most importantly, it is now well understood that because biological systems are driven by complex biomolecular interactions, multi-gene effects will play an important role in mapping genotypes to phenotypes; recent reviews by Moore and coworkers describe this issue well [10, 12]. Additionally, the finding that epistasis and pleiotropy appear to be inherent properties of biomolecular networks [13] rather than isolated occurences motivates the need for systems-level understanding of human genetics. The impact that biological interaction networks have on our ability to identify genomic causes of complex disease is readily apparent. Consider a biologically crucial mechanism with several potential points of failure, such that an alteration to any will confer disease risk. Because no single alteration is predominant amongst cases, none attain a significant association; indeed, it has long been observed that even in histologically identical tumors, only a fraction may share the same set of mutations (see references in [14] for examples). Additionally, in a robust system, multiple alterations may be necessary to alter the activity of an interaction network; here, healthy individuals may share a subset of the deleterious alleles found in cases, and again these loci will not be detected. This complexity, noted by [10, 12, 13, 14] and others, has generated considerable interest in multi- locus analysis techniques that take advantage of known interaction information. Several multi-SNP GWAS analysis approaches have been described in the literature. Thorough reviews are provided in [15, 16], and we briefly describe several here. Building on the well-established Gene Set Enrichment Analysis [17] method initially developed for gene expression data, two articles have proposed an extension of GSEA for SNP data [18, 19]. In these techniques, each SNP is assigned a statistic based on a $\chi^{2}$ test of association with the phenotype; a running sum is then used to assess whether large statistics occur more frequently amongst a SNP set of interest than could be expected by chance. While GSEA-type approaches have proven quite useful, their reliance on single-marker statistics means that systematic yet subtle changes in a gene set will be missed if the individual genes do not have a strong marginal association. In the case of a purely epistatic interaction between two SNPs in a set, the set may fail to reach significance altogether. To address this issue, Yang and colleagues proposed SNPHarvester [20], designed to detect multi-SNP associations even when the marginal effects are weak. To reduce the search space of possible multi-SNP effects, SNPHarvester [20] first removes any SNPs with univarite significance. Using a novel searching algorithm, they identify groups of $l$ SNPs that show association with status in a $\chi^{2}$ test with $3^{l}-1$ degrees of freedom. While this approach can reveal epistatic effects that would be missed by the GSEA-type schemes [18, 19], it has other drawbacks. First, the combinatorial explosion of SNP groups puts a limit on the number of SNPs $l$ that may simultaneously be examined. Second, the the arbitrary groupings of SNPs and the exclusion of SNPs with marginal effects can make the biological interpretation of the analysis results difficult. The notion that cases will more closely resemble other cases than they will controls has motivated a number of interesting distance-based approaches for detecting epistasis. Multi-dimensionality reduction (MDR) has been proposed and applied to SNP data [21, 22, 23]. In this technique, sets of $l$ SNPs are exhaustively searched for combinations that will best partition the samples by examining the $3^{l}$ cells in that space (corresponding to homozygous minor, heterozygous, or homozygous major alleles for each locus) for overrepresentation of cases. While this method both finds epistatic interactions without requiring marginal effects and can be structured to incorporate expert knowledge, it is limited by the fact the the total number of loci to be combinatorially explored must be restricted to limit computational cost. To address this, an “interleaving” approach in which models are constructed hierarchically has been suggested [22] to reduce the combinatorial search space. A recent and powerful MDR implementation [24] taking advantange of the CUDA parallel computing architecture for graphics processors has made feasible a genome-wide analysis of pairwise SNP interactions. Still, MDR remains computationally challenging, such that expanding the search to other SNP set sizes (rather than restricting to pairwise interactions) can be impeded by combinatorial complexity if an exhaustive search is to be performed. In order to narrow down the combinatorial complexity of discovering SNP sets using techniques such as MDR, feature selection may be employed. Of particular importance here is the distance-based approach of the Relief family of algorithms [25, 26, 27, 28]. These are designed to identify features of interest by weighting each feature through a nearest-neighbor approach. The weights are constructed in the following way: for each attribute, one selects samples at random and asks whether the nearest neighbor (across all attributes) from the same class and the nearest neighbor from the other class have the same or different values from the randomly chosen sample. Attributes for which in-class nearest neighbors tend to have the same value are weighted more strongly. Because the distances are computed across all attributes, Relief-type algorithms can identify SNPs that form part of an epistatic group and they provide a means of filtration that does not have the drawbacks of other significance filters. While these methods have so far been applied to finding small groups of interacting SNPs, one may instead be interested in whether cases and controls exhibit differential distance when considering a large number of genes. A multi-SNP statistic has been proposed in the literature [29, 30, 31] for determining whether an individual of interest is on average (across a large number of SNPs) “closer” to one population sample than to another. The method, originally proposed by Homer [29], is motivated by the idea that a subtle but systematic variation across a large number of SNPs can produce a discernible difference in the closeness of an individual to one population sample relative to another. While this statistic was originally designed to identify the proband as a member of one of the population samples, it was shown in [30] that out-of-pool cases from a case-control breast cancer study were in general closer (as defined by the statistic presented in [29]) to in-pool cases than they were to in-pool controls, suggesting that the combination of multiple alleles has the potential to distinguish cases from controls. Building on these ideas, we present a new technique that finds pathway-based SNP-sets that differentiate cases from controls; we call this method Pathways of Distinction Analysis (PoDA). In PoDA, SNP sets are defined based on known relationships (e.g., SNPs in genes sharing a common pathway), and thus incorporate expert knowledge to reduce the search space and provide biological interpretability. Motivated by the differential “closeness” of cases and controls as discussed about and as observed in [30], we hypothesize that if the SNPs come from a pathway that plays a role in disease, there will be greater in-class similarity than across-class similarity in the genotypes for those SNPs; i.e., a case will show greater genetic similarity to other cases than to controls for the SNPs on a disease-related pathway, but will be equidistant for the SNPs on a non-disease-related pathway. Based on this notion, PoDA seeks to identify pathways for which differential heterogeneity is exhibited in cases and controls. In each pathway, PoDA returns a statistic $S$ for each sample that quantifies that sample’s distance to the remaining cases relative to its distance to the remaining controls for a given pathway’s SNPs. PoDA then examines whether the distributions of $S$ for the controls differ from those of the cases by computing and testing for significance a Pathway Distinction Score $DS$ that quantifies the differences in case and control $S$ distributions. In this manuscript, we detail the PoDA method and report the results of its application to two data sets. As we will describe, PoDA improves and complements existing approaches in a number of respects. First, it permits the investigation of arbitrarily large pathways, circumventing the dimensionality issues that are encountered with MDR and SNP-Harvester. Second, it is able to detect pathways that contain an over-abundance of highly-significant markers as well as pathways whose markers have a small but consistent association that would be missed by GSEA-type approaches. Finally, it uses a leave-one-out technique to return for each sample an unsupervised relative distance statistic that can then be used to model disease risk via logistic regression. In addition to providing an effect size for the pathway, this allows the odds of disease for new samples to be obtained by computing its relative distance statistic with respect to the known samples and applying the model. ## Methods Following our conjecture that SNPs associated with the genes in a pathway involved in disease will exhibit more within-group similarity than across- group similarity, we propose Pathways of Distinction Analysis (PoDA), a method designed to address the following questions: * • Given some set of SNPs, do we find that, on average, cases are “closer” to other cases than to controls (or that controls are “closer” to other controls than to cases)? * • If we look for these distinctions systematically over all SNP-sets of potential interest, can we use it to single out SNP-sets which may be associated with disease? In PoDA, a set of SNPs are selected, and for each sample we compute whether it is closer to the pool of remaining cases or controls across that SNP set, using the relative distance statistic described below. Once this is done for every sample, the distribution of the relative distance statistic is compared in the cases and controls using a nonparametric statistic, addressing the first question above. This may be carried out amongst all SNP sets of interest, adjusting the $p$-value for the multiple hypotheses, to find SNP sets for which cases more strongly resemble the population of remaining cases while controls more strongly resemble the population of remaining controls. We begin with a discussion of how we measure the relative distance of an individual to the other cases vs. other controls. A simple but computationally intensive approach is to represent each sample by a vector in an $l$-dimensional space, where $l$ is the number of SNPs in the group of interest. One can then compute, between each sample pair, their distance in this $l$-dimensional space using a Euclidean, Manhattan, or Hamming metric. For each sample, we would define its relative distance statistic as the mean of its distance to other controls minus the mean of its distance to other cases; a sample that is more similar to cases will exhibit a positive statistic, whereas one that is more similar to other controls will exhibit a negative statistic. For the given SNP set, we would then have for each sample a value quantifying its relative distance that was computed without knowledge of that sample’s class (i.e., using a leave-one-out scheme) and could then be used in further tests. By doing this for all pathways of interest, one derives a relative distance value for each sample in each pathway. This brute-force approach, while conceptually clear, has two significant drawbacks. The first is that the distance computation is $\mathcal{O}(l\cdot n^{2})$ where $n$ is the total number of samples in the study—a considerable undertaking, particularly if many SNP sets are to be analyzed, and one that becomes exceedingly troublesome in the context of permutation tests. The second drawback is that because we are taking the mean of the distances, a sample that is situated squarely within a cluster of cases may have a large case-distance value due to the dispersion of cases around it. Both of these issues are circumvented by instead considering the relative distance to the centroids of the cases and controls in the $l$-dimensional space, a computation that can be performed in $\mathcal{O}(l\cdot n)$ for all $n$ samples. It is this approach that PoDA employs, as follows: In [30, 29], the authors consider a measure of individual $Y$’s distance to two population samples $F$ and $G$ at locus $i$, $D_{Y,i}=\left|{y_{i}-f_{i}}\right|-\left|{y_{i}-g_{i}}\right|\,.$ (1) where $f_{i}$ and $g_{i}$ are the minor allele frequencies (MAFs) of SNP $i$ in samples $F$ and $G$, and $y_{i}\in\\{0,0.5,1\\}$ is $Y$’s genotype at $i$ corresponding to homozygous major, heterozygous, and homozygous minor alleles, respectively (i.e., the frequency of minor allele in that individual. The first term quantifies how different $Y$’s MAF is from $F$’s for a given locus $i$; the second term quantifies how different $Y$’s MAF is from $G$’s at locus $i$; and so $D_{Y,i}$ gives the distance of $Y$ relative to $F$ and $G$ at locus $i$. Since the minor allele frequencies $f_{i}$ and $g_{i}$ are computed by averaging the genotypes (again, written as $\\{0,0.5,1\\}$) in samples $F$ and $G$ respectively, it is clear that $\left|{y_{i}-f_{i}}\right|$ is the distance from $Y$ to the centroid of $F$ along the coordinate $i$ (and likewise for the $g_{i}$ term). It can be seen from Eq. 1 that positive $D_{Y,i}$ implies that $y_{i}$ is closer to $g_{i}$ than to $f_{i}$, and that negative $D_{Y,i}$ implies that $y_{i}$ is closer to $f_{i}$ than to $g_{i}$. By computing $D_{Y,i}$ at each locus $i$ and taking the standardized mean across the $l$ loci, [29] obtain a distance score $S$ which quantifies how close $Y$ is relative to $F$ and $G$ across all $l$ loci under consideration, $S_{Y}=\frac{\mathsf{E}({D_{Y,i}})}{\sqrt{\mathsf{Var}({D_{Y,i}})/l}}\,,$ (2) where $\mathsf{E}({D_{Y,i}})$ denotes the mean of $D_{Y,i}$ across all loci $i$. That is, $S$ provides a means to quantify whether $Y$’s MAFs are closer to $G$’s MAFs or $F$’s MAFs on average for the loci under consideration. It is instructive to consider the geometrical interpretation of Eq. 2. Is clear upon inspection that the numerator in Eq. 2 is equal, up to a factor of $l$, to the difference in Manhattan distances between $Y$ and the (nonstandardized) $G$ centroid and $Y$ and the (nonstandardized) $F$ centroid; in this sense, Eq. 2 resembles a nearest-centroid classifier. However, the denominator scales the relative distances by their variance across the $l$ SNPs; that is, a sample $Y$ who is consistently closer to $G$ than to $F$ for each of the $l$ SNPs will obtain a higher $S$ than an individual who is variously closer to either across the $l$ SNPs under consideration. By assigning the (non-$Y$) controls as $F$ and the (non-$Y$) cases as $G$, we can compute a statistic $S_{Y}$ quantifying $Y$’s distance to other cases relative to $Y$’s distance to other controls. If we then apply this systematically to all individuals in the study population (removing that individual, computing the MAF’s $f_{i}$ and $g_{i}$ for the remaining individuals who comprise $F$ and $G$, and then computing $S_{Y}$ in a leave- one-out manner), we can obtain distributions of $S_{Y}$ statistics in cases and controls that may be compared. Here, the null hypothesis is that case and control $S_{Y}$ distributions do not differ, with the alternative hypothesis that the cases have higher $S$ values than do controls, i.e., that they are closer (via Eqs. 1-2) to other cases than are controls. We can use $S$ in the following manner to answer the questions posed above by applying it in a leave-one-out manner in each pathway: 1. 1. For a given pathway $P$, select the $l_{P}$ SNPs associated with that pathway; 2. 2. For every sample $Y$, remove $Y$ from the case or control group as needed, and compute $S_{Y,P}$ with respect to the remaining cases and controls using the SNPs chosen in step 1. 3. 3. Quantify the differences in distribution of $S_{Y,P}$’s for the case samples versus that of the controls and test for significance. By systematically carrying out the above steps on all pathways of interest, we can identify pathways for which there appears to be differential homogeneity in cases and controls, indicating that the pathway may play a disease-related role. The details of the algorithm are explained below, and summarized in Table 1. In [30], we examined Eqs. 1-2 and found that the magnitude of $S$ is influenced both by the MAF differences $f_{i}-g_{i}$ (that is, how distant the centroids of $F$ and $G$ are) and by correlations between the SNPs under consideration (due to the penalization for variance in $D_{i}$ provided by the denominator of Eq. 2. These properties are extremely well-suited to the application we propose: pathways with few highly-significant SNPs will yield large $S$ differences (due to the influence of $f_{i}-g_{i}$), as will pathways with SNPs that are highly correlated yet have subtle individual MAF differences, corresponding to the concerted action of multiple SNPs. At the same time, however, we wish to ensure that the pathways we select as having differential $S$ are not being influenced large LD blocks covered by the SNPs in the genes on the pathway. That is, we wish to ensure that the SNP correlations which drive $S$ are reflective of epistatic effects between different genes rather than recombination events within a gene. To this end, we select a single SNP to represent each gene, based on the desire to detect multi-gene rather than multi-SNP effects. In practice, SNPs are selected as follows: for each pathway represented in the Pathway Interaction Database [32] (PID, http://www.pid.nci.gov, containing annotations from BioCarta, Reactome, and the NCI/Nature database [32]) and KEGG [33], we select the associated genes. Using dbSNP [34], we retrieve the SNPs associated with the pathway genes that are present in the data, excluding those with $>20\%$ missing data or with minor allele frequency $<0.05$ in either case of control group. We necessarily exclude pathways for which only one gene is probed by the remaining SNPs. Because we are interested in $S$ values that are driven by correlations across genes (and not in individual genes covered by many SNPs with high LD), we select for each gene its most significant SNP in a univariate test of association (Fisher’s exact test). It should be noted here that while the SNP chosen for each gene is the most significant of that gene’s SNPs, it is not necessarily significantly associated with disease. Our goal here is not to filter based on SNP significance, but rather to select, for each gene, a single marker that is as informative as possible. Having selected the SNPs of interest, we compute $D_{Y,i}$ at each locus for every sample by selectively removing it and comparing it to the remaining cases and controls, as described above. For each pathway $P$, we compute $S_{Y,P}$ for $l_{P}$ the SNPs $i$ that comprise it, yielding a distribution of $S_{Y,P}$ for cases and another distribution for controls. The difference in the location of the case and control $S_{Y,P}$ distributions is then quantified nonparametrically by computing the Wilcoxon rank sum statistic, defined as $W_{P}=\sum_{Y\in\mathrm{case}}R_{Y,P}-\frac{n_{\mathrm{case}}(n_{\mathrm{case}}+1)}{2}\,$ (3) where $R_{Y,P}$ is the rank of $S_{Y,P}$ amongst all samples $Y$ for a given pathway $P$. Eq. 3 thus quantifies, non-parametrically, the degree to which cases are “closer” to other cases and controls “closer” to other controls across a set of SNPs for all individuals in the GWAS. To illustrate the above, we consider a simulated GWAS of $250$ cases and $250$ controls and $50$ SNPs, shown in Figure 1, and ask whether we are able to detect a 12-SNP pathway in which a subset of SNPs appear to have an epistatic interaction. Alleles were simulated as binomial samples from a source population with MAFs ranging from $0.1$ to $0.4$ across the $50$ SNPs, and case labels were assigned such that a combintion of homozygous minor alleles at SNPs 1 and 2 or 3 (i.e., $(y_{1}=1)\land((y_{2}=1)\lor(y_{2}=1))$) conferred a three-fold relative risk, mimicking an epistatic interaction between SNPs 1 and 2 and SNPs 1 and 3 (Figure 1(a)). Alone, none of the $50$ SNPs showed any association with case status, nor was any SNP significantly out of HWE in either cases or controls. However, the “shared pattern” in SNPs 1–3 is such that a 12 SNP pathway comprising SNPs 1–12 yields greater $S$ in cases than in controls as can been seen in Figure 1(b), while a random 12 SNP pathway selected from the 50 SNPs (that includes SNP 3, but neither SNP 1 or 2) shows no difference in $S$ values as seen in Figure 1(c). While the Wilcoxon statistic $W$ is normal in the large-sample limit and can be directly compared to a Gaussian, to truly evaluate the significance of $W_{P}$ for a given pathway $P$, we must address two sources of bias: the number of SNPs per gene, and the size of the pathway. To address these issues, we introduce a normalized Pathway Distinction Score $DS_{P}$ that we test for significance using a resampling procedure. First, we expect that because we have selected for each gene the single most informative SNP, we are pre-disposed to seeing higher $W_{P}$ for pathways that contain large genes. Because large genes will be more likely to contain highly-significant SNPs by chance, the concern has been raised that [35, 18] selecting the single most significant SNP as a proxy for the gene (as is done here) will lead to a bias toward pathways that contain an abundance of large genes. To account for this, we follow the approach in [18] and normalize the score via a permutation-based procedure. First, we permute the phenotype labels and in each permutation recalculate $W_{P}$ as described above, but using the permuted case and control labels. The permuted labels are used both to select the most informative SNP per gene and to compute $f_{i}$, $g_{i}$, and $W_{P}$ in Eqns. 1–3). This yields a distribution of $W^{*}_{P}$ under the null hypothesis that the magnitude of $W$ is independent of the true case/control classifications. We then normalize the true $W_{P}$ by the distribution from the permutation procedure, yielding a Distinction Score $DS_{P}$ for pathway $P$ that effectively adjusts for different sizes of genes and preserves correlations of SNPs in the same gene: $DS_{P}=\frac{W_{P}-\mathsf{E}({W^{*}_{P}})}{\mathsf{SD}({W^{*}_{P}})}\,,$ (4) where $W^{*}_{P}$ are the set of $W_{P}$ obtained for pathway $P$ across the permutations. (In practice, 100 permutations are used.) Because the permuted labels are used in the SNP selection, this normalization adjusts for the bias introduced by the fact that large genes have more opportunity to contain significant SNPs by chance. The Pathway Distinction Score $DS_{P}$ thus provides a model-free, gene-size adjusted metric for quantifying the degree to which cases are “closer” to other cases (higher $S_{P}$) than controls. To test whether $DS_{P}$ is significant, we note that larger pathways may yield high $DS_{P}$ values simply due to the fact that they sample the case anc control differences more thoroughly. Indeed, the question of significance that we wish to address is not simply whether a pathway permits the distinction of cases and controls, but whether it does so better than a random collection of as many SNPs, wherein the SNPs are still selected to be the most informative by gene. To account for the fact that the pathways are of differing sizes, significance of the Distinction Score for a given pathway is assessed through resampling by choosing, at random, the same number of SNPs that are present in that pathway ($l_{P}$) from the total set of most- informative-SNP-per-gene and recomputing $DS_{P}$ for the random pathway. The $p$ value is obtained by counting the fraction of random $l_{P}$-SNP sets which give a larger $DS_{P}$ than the true pathway SNPs in $10^{4}$ resamplings. In this way, we are able to detect pathways that yield large differences of case and control $S$ distributions due to their particular SNPs, rather than simply being the result of choosing many SNPs. The $p$ value obtained addresses the question of whether the pathway under consideration permits greater separation of cases and controls than would a random collection of most-informative-SNP-per-gene, i.e., whether there exists a more extreme aggregated effect in that pathway than expected by chance. ## Results We applied PoDA to 2287 genotypes obtained from the Cancer Genomic Markers of Susceptibility (CGEMS) breast cancer study. The samples were sourced as described in [4]. Briefly, the samples comprised 1145 breast cancer cases and a comparable number (1142) of matched controls from the participants of the Nurses Health Study. All the participants were American women of European descent. The samples were genotyped against the Illumina 550K arrays, which assays over 550,000 SNPs across the genome. We also applied it to a smaller liver cancer GWAS [36] comprising 386 hepatocellular carcinoma (HCC) patients and 587 healthy controls from a Korean population. Samples were genotyped against Affymetrix SNP6.0 arrays, which provides SNP information at approximately one million loci. ### Breast cancer GWAS results We begin by applying PoDA to the CGEMS breast cancer GWAS data. Having observed (Figure 1) that PoDA performs as expected for the simulated data, we first turn our attention to a simple test in which we select a SNP set comprising the four SNPs in intron 2 of $\mathit{FGFR2}$ that were reported to show significant association with case status in [4] (rs11200014, rs2981579, rs1219648, rs2420946). We expect to see a strong difference in the test case and test control distributions, and indeed we do: the cases more frequently have positive $S$ than do controls in Fig. 2. (The discrete peaks in the distribution are a result of the fact that with four SNPs there exist fewer available values of $S$.) Using a nonparametric Wilcoxon rank sum test with the alternative hypothesis that cases have greater $S$ than controls, $p=1.016\cdot 10^{-6}$ is obtained, confirming our intuition. We next applied PoDA systematically to the pathways represented in PID [32] using CGEMS data. Associations between genes and SNPs were made using dbSNP build 129 [34]. 1081 pathways were non-trivially covered in the data set; 69453 SNPs in the data could be associated with at least one of the pathways. Because these 69453 SNPs were associated with 4446 unique genes, 4446 were kept in the analysis (the most significant SNP for each gene of interest). The 1081 pathways ranged from 2 to 229 genes, with a mean of 19. $S_{Y,P}$ was computed in each pathway $P$ for each of the 2287 samples $Y$ via Eq. 2, and the distinction score $DS_{P}$ (Eq. 4) quantifying differential $S$ distributions in cases and controls was computed for each pathway. Significance was assessed as described above, by resampling “dummy” pathways of the same length and computing the fraction of greater $DS_{P}$ scores. Because PoDA provides for each sample a measure $S$ (Eq. 2 of that sample’s relative distance from the remaining ones that is obtained without regard to that sample’s true class membership, we can use the $S$ value as a metric by which to predict the odds of disease. Here, we construct a logistic regression model of case status as a function of $S$ to obtain the odds ratio. $p$-values were adjusted for the multiplicity of pathways using FDR adjustment [37, 38]. Pathways with significant $DS_{P}$ and odds ratios are reported in Table 2 and plots of $S$ for four of them are illustrated in Figure 3. Although the cases and controls are not crisply separable, a unit increase in $S$ over its range from approximately -3 to 3 yields between a 1.5 and 2.0-fold increase in odds. Importantly, given known minor allele frequencies for cases and controls for this set of SNPs, we can model the increase in odds for an unknown individual based on her “closeness” to other cases. In order to ensure that the pathways listed were not interrogating the same set of genes, we carried out two checks. First, we computed the SNP overlap between all pairs of significant pathways, sequentially removing pathways that shared in excess of 60% of their genes with another pathway. Because this is done using a greedy algorithm that depends on the order of the pathways input, the culling algorithm was run with different starting orders, and the most frequent output was kept. No pathway remaining in Table 2 shares more than 60% of its SNPs with another pathway. (An un-culled list may be found in Supplementary Table S-1.) Second, we computed the correlation of $S$ values between each pair of pathways to assess whether any pathway’s $S$ statistic was reflecting the same genetic variation as another (i.e., whether samples that had high $S$ values for one pathway consistently did so in another). The maximum correlation of $S$ values observed between any two pathways in Table 2 was 0.58, suggesting that a different subset of samples is affected in each pathway. Many of the pathways listed in Table 2 fulfill biological functions that are well known to be cancer-associated, playing a strong role in cell proliferation and migration, processes which are perturbed in malignancies. Purine metabolism—the most significantly associated pathway—has been observed to be altered in cancer cells [39, 40], and the majority of the other significant pathways are directly related to cell migration (e.g., ErbB signaling and gap junction pathways) and cellular signalling (e.g., calcium signaling, PKC-catalyzed phosphorylation of myosin phosphatase, attenuation of GPCR signaling, and activation of PKC through GPCRs) processes that have been implicated in a variety of cancers. In addition, eicosanoids and unsaturated fatty acid metabolism have been associated with breast cancer specifically [41]. In general, the findings in Table 2 suggest that there exist germline genetic differences in these mechanisms that confer a predisposition to disease. Interestingly, the GnRH (gonadotropin releasing hormone) signaling pathway appears to be significant. GnRH has been linked with HR-positive breast cancer and the use of GnRH analogues in breast cancer treatment has already been proposed [42, 43]. However, a recent large sequencing study found no association of GnRH1 or GnRH receptor gene polymorphisms with breast cancer risk [44], contrary to the author’s hypothesis that common, functional polymorphisms of GnRH1 and GnRHR could influence breast cancer risk by modifying hormone production. In contrast to their null findings, our result suggests that there are system-wide variations in GnRH signalling that contribute to risk that are not evident when considering the GnRH1 and GnRHR SNPs independently. Of the 1081 pathways considered, four—FGF signaling, MAPK signaling, regulation of actin cytoskeleton, and prostate cancer—contained $\mathit{FGFR2}$, the gene found to be significantly associated in the initial CGEMS analysis [4]. However, only one—prostate cancer—was significant in comparison to randomly generated pathways of the same length. It may reasonably be asked, then, whether the high significance of the prostate cancer pathway in Table 2 is a result of $\mathit{FGFR2}$. To address this, we eliminated the $\mathit{FGFR2}$ SNP from the prostate cancer pathway; the resampling-based test remained significant ($p(DS_{P})=0.044,OR=0.3,q(OR)=$ 8.2e-09), suggesting that the association of the prostate cancer pathway is not driven solely by differences in $\mathit{FGFR2}$. ### Liver cancer GWAS results We carried out the same procedure in using data from the liver cancer GWAS described above. Here, 1049 pathways were non-trivially covered in the data set; 53079 SNPs in the data could be associated with at least one of the pathways. Because these 53079 SNPs were associated with 3718 unique genes, 3718 were kept in the analysis (the most significant SNP for each gene of interest). The 1081 pathways ranged from 2 to 193 genes, with a mean of 16. As above, $DS_{P}$ scores for differential $S$ distributions in cases and controls were computed for each pathway, resampled $p$ values obtained for each pathway size, odds ratios for $S$ were obtained, and the multiple hypotheses were corrected using FDR adjustment [37, 38]. Significant pathways are listed in Table 3, and plots of the top three pathways are given in Figures 4a-d. As in the breast cancer data above, we removed pathways which had over 60% their SNPs covered by another pathway (a complete list, with overlapping pathways, is give in Supplementary Table S-2) and examined the correlation in $S$ for all remaining pathways (maximum $\rho=0.42$). The results here are interesting. First, we observe that a couple pathways are significant in both the CGEMS breast and liver GWAS with similar effect sizes, namely ErbB signaling and biosynthesis of unsaturated fatty acids. ErbB has a well–established association with cancer; unsaturated fatty acid biosynthesis may link diet to cancer risk, and its appearance may suggest a gene- environment interaction. The commonality of these known cancer-associated pathways across the two studies suggest that there may exist genetic patterns that confer carcinogenesis risk irrespective of the disease site. Along with those shared in the breast cancer data, many of the other significant pathways in the liver cancer data well known to be tumorassociated, including cell adhesion molecules, Wnt signaling, c-Kit receptor, and angiogenesis pathways, further supporting the notion that germline genetic differences in these mechanisms contribute to cancer risk. The appearance of many neuronal pathways is also supported by our understanding of carcinogenesis: thes contain well- known signal transduction molecules including Ras and PKA that may both be driving their conferring increased cancer risk and driving the significance of the pathway [45]. Additionally, six of the 25 significant liver cancer pathways are immune– and inflammation–related, namely, antigen processing and presentation (two, with $<$60% overlap), classical complement pathway, corticosteroids, IL12 signaling mediated by STAT4, and NO2-dependent IL-12 pathway in NK cells. This is a particularly interesting finding in light of the fact that the original analysis of the liver data [36] suggested that altered T-cell activation plays a direct role in the onset of liver cancer. The involvement of the immune system in liver cancer development has been established in clinical studies and research involving model organisms. Increased activity of helper T-cells, which promote inflammation, is associated with hepatocellular carcinogenesis [46] while activation and proliferation of cytotoxic T-lymphocytes is suppressed in liver cancers [47, 48]. The inflammatory immune response, mediated by interleukins, has also been closely connected to liver cancers in mice [49] and humans [50, 51, 52]. These findings, coupled with the observation of several significant immune-related pathways in our data, are suggestive of germline polymorphisms in immune response that lead to hepatocellular carcinoma risk. ### Combining pathways In both the breast and liver cancer results, we see observe that even though significant pathways yield between a 1.5 and 2.0-fold increase in odds for each unit increase in $S$ (over its typical range of approximately $-3$ to $3$), the cases and controls are not crisply separable based on $S$ values. These findings suggest that it may be possible to combine pathways to yield a model that is more predictive than a single pathway alone. However, the $S$ values must not simply be put into the regression model because the overlap in pathways will result in some SNPs being double-counted. Rather, we combine pathways by taking the union of their SNPs, and recomputing the statistics. Doing this sequentially for the top pathways in the order as listed in Tables 2 and 3 yields the values given in Tables 4 and 5, respectively. Considerably higher ORs are obtained when combining the significant pathways. An illustration of the case and control distributions when using a “superpathway” comprised of the top three pathways in the breast and liver data, respectively, is given in Figure 5. These findings support the notion that the genomic variation contributing to risk is spread over several mechanisms, rather than being concentrated in a single gene. ## Discussion We have introduced the Pathways of Distinction analysis method (PoDA) for identifying pathways which can be used to distinguish between phenotype groups. PoDA identifies sets of SNPs in GWAS studies for which cases and controls exhibit differential “closeness” to other cases and controls; that is, it permits one to infer whether cases are more similar to other cases than are controls across a given set of SNPs. Because PoDA is designed to detect the joint effects of multiple SNPs, it presents an approach to GWAS analysis that augments single-SNP or single-gene tests. We applied PoDA to two GWAS data sets, with highly promising results. In the breast cancer data, we found a number of pathways which are known to play a role in cancers generally and breast cancer specifically, suggesting that differences in these mechanisms which confer disease risk may exist at the germline DNA level. In the liver cancer data, we found an extreme abundance of immune-related pathways, further corroborating the known link between inflammation and hepatocellular carcinoma, and bolstering the observation in [36] that germ-line differences in immune function may play a role in liver carcinogenesis. PoDA may be used as a complement to other multi-SNP analysis techniques [18, 19, 20, 21]. Unlike gene-set enrichment type approaches [17, 18, 19], which search for an overabundance of significant markers in a gene set of interest, PoDA finds both sets containing highly significant markers as well as sets that have a subtle but consistent pattern across all the markers in the set. This permits the detection of pathways in which the joint action of several alterations produce a phenotype and those for which any of several possible alterations, none of them the dominant one, confer predisposition to disease. Indeed, many of the pathways indicated in our analysis of the breast cancer data (Table 2) were not detected using SNP-set enrichment [17, 18, 19] (data not shown), including the highly significant purine metabolism and GnRH signaling pathways, both of which are biologically relevant (purine metabolism has been implicated in cancers generally due to its role in DNA and RNA synthesis [40], and GnRH has been shown to be clinically important in breast and gynecological cancers [53]). These pathways, along with others that were indicated using PoDA but not enrichment analysis (data not shown), have a statistically significant difference in case and control $S$ distributions and remain significant in comparison with randomly-generated pathways of the same length. Because PoDA effectively measures the closeness of each individual to remaining cases and controls, it bears a conceptual relationship to nearest- neighbor and nearest-centroid classifiers [54, 55], as well as to the distance-based feature selection algorithms like Relief-F and its derivatives [25, 26, 27, 28]. However, it must be remembered that the goal of PoDA is to indicate mechanisms that may be deleteriously hit at the genomic level even when those hits are heterogeneous, whereas the goal of nearest-centroid classifiers and Relief-F–type feature selection is to derive a minimal set of markers that best classify cases and controls (and thus are the most homogeneously hit). These approaches are complementary, and one can easily envision an application in which (e.g.) Relief-F is run within pathways that are highly significant in the PoDA analysis in order to single out the SNPs driving the effect. In fact, this approach may improve ReliefF’s ability to find those genes, since the nearest neighbors from which the Relief SNP weights are calculated would be the nearest-neighbors for that specific pathway, thus discounting heterogeneity introduced by mechanistically unrelated genes. For instance, in the provided example (Fig 1), ReliefF fails to identify the significance of SNPs 1–3 when run using the complete 50-SNP data, but places at least two of SNPs 1, 2 or 3 in the top third of selected features when restricted to SNPs 1–12. While PoDA has many benefits, it should be noted that when epistasis drives a phenotype with no differences in the minor allele frequencies for the epistatically-interacting genes (as opposed to a slight yet consistent one shown in the example), PoDA as computed via Eqs. 1,2 will miss the pathway. Geometrically, such a situation would mean that the case and control groups have the same centroids while having a different distribution of samples about those centroids. A famous example of this is provided through the non-linearly separable XOR (exclusive or): consider two epistatic loci $(X,Y)$ such that all controls have genotypes in the set $\\{(0,0),(1,1)\\}$ and all cases have genotypes in the set $\\{(0,1),(1,0)\\}$ (i.e., that a genotype of 1 at either locus can be compensated by a genotype of 1 at the other, but having just one alone—1 at exclusively $X$ or $Y$—is deleterious). If the loci $X$ and $Y$ each have the same MAF in cases and controls, it is plain to see that the centroids will be in the same location for both groups, and Eq. 1 will yield zero for both cases and controls. If instead of using Eq. 1, we compute pairwise sample-sample distances, we can circumvent this limitation and find such epistatic situations (it is this pairwise approach that permits Relief-F to also uncover nonlinearly interacting SNPs). While we provide the facility for this in the PoDA package, the cost of carrying out the pairwise computation is a considerable increase in computational complexity. A number of potential avenues exist to extend the application of PoDA further. One possible application is in improving the reproducibility of GWAS results. We note that several of the pathways identified in the breast cancer GWAS data were also implicated in the liver cancer data, which suggests that there may be common features which distinguish individuals to cancer generally. Because different GWA studies—even those of the same phenotypes—often yield different results at the SNP level, it may be possible to find common alterations at the pathway level across disparate GWAS using PoDA. Extending PoDA further, the $DS_{P}$ scores obtained for each pathway may be examined for over-representation of extreme values in pathways that comprise a particular biological subsystem—one may think of this as a “pathway-set” enrichment analysis (which would be conducted using the a running-sum statistic analogous GSEA [17]), and could use it to answer whether (e.g.) immune-related pathways are hit in liver cancer more often than expected by chance. Alternatively, boosting [56, 57] could be used to find sets of pathways which are more predictive of case status than individual pathways. Either of these approaches would yield a richer, systems-wide view of the connection between genotype and phenotype. Finally, because PID contains topological information regarding pathway connectivity, one may consider sub- networks of pathways, permitting one to find potential chemopreventive and therapeutic targets. Alternatively, Relief-F can be used, as mentioned above, in a pathway–specific manner to yield the subset of SNPs that drive the distinction of cases and controls in high-$DS_{P}$ pathways. PoDA provides an advantage over existing GWAS analysis methods. Because it does not rely on the significance of individual markers, it has the power to aid in identifying the genomic causes of complex diseases that would not be detected in single-gene tests or enrichment analyses. The size of the SNP set is not limited in PoDA, and since PoDA leverages known biological relationships to find multi-SNP effects, the results are readily interpretable. 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Statistical Science 22: 477–505. * [57] Meir R, Ratsch G (2003) An introduction to boosting and leveraging. Lecture Notes in Computer Science 2600: 118–183. Figure 1: PoDA applied to simulated data. Alleles at 50 loci for 250 cases and 250 controls were simulated such that each SNP was in HWE and not associated with case status, but homozygous minor (red) at both loci 1 and 2 or 1 and 3 yielded a three-fold relative risk (a). A 12-SNP pathway comprising SNPs 1–12 shows differential $S$ distributions (b); a random 12-SNP pathway does not (c). Boxplots are overlayed on the scatterplots of $S$ for clarity. Figure 2: PoDA applied to four highly-significant SNPs. Shown is the distribution of $S$ values in CGEMS cases (red) and controls (black) for a SNP-set comprised of four highly-significant SNPs located in the $\mathit{FGFR2}$ gene [4]. As expected, there is a substantial difference in case and control $S$ values, with the cases having higher $S$ (i.e., closer to other cases) than controls. The discreteness of the distributions are due to the fact that with four SNPs, a finite number of $S$ values are possible. Figure 3: Four significant pathways in breast cancer data. Scatter plots of $S_{Y,P}$ for each pathway are overlayed with boxplots are given in the left panel; higher values of $S$ indicate that the sample is closer to other cases than it is to other controls. Distributions of $S$ for cases (red) and controls (black) are given to the right. A significant shift toward higher $S$ values is seen in the cases. Odds ratios and FDR-adjusted OR $p$ values are given. Figure 4: Four significant pathways in liver cancer data. Scatter plots of $S_{Y,P}$ for each pathway are overlayed with boxplots are given in the left panel; higher values of $S$ indicate that the sample is closer to other cases than it is to other controls. Distributions of $S$ for cases (red) and controls (black) are given to the right. A significant shift toward higher $S$ values is seen in the cases. Odds ratios and FDR-adjusted OR $p$ values are given. Figure 5: Union of top three pathways. SNPs from the top three pathways are combined to compute $S$ for the breast cancer data (a) and the liver cancer data (b). Distributions of $S$ for cases (red) and controls (black) are given to the right. A significant shift toward higher $S$ values is seen in the cases. | Procedure for Pathways of Distinction Analysis ---|--- 1. | For a each pathway $P$, select all associated genes from pathway database such as PID [32]; 2. | For each gene on the pathway, select associated SNPs (e.g., using dbSNP) and choose the one | with the strongest association with case status, determined using Fisher’s exact test; 3. | For each sample $Y$ in the GWAS, select the controls $F$ and cases $G$ which do not include it, | compute MAFs $f_{i}$ and $g_{i}$ for the SNPs $i$ selected in step 2, and compute $S_{Y,P}$ for each sample $Y$; 4. | Compare the distribution of $S_{Y,P}$ obtained in step 2 for cases to that of controls by computing | the Wilcoxon statistic $W_{P}$ based on the $S_{Y,P}$ for that pathway; 5. | Repeat steps 2–5 using permuted case/control labels, and normalize $W_{P}$ by the distribution | of $W^{*}_{P}$ obtained with permuted labels, yielding the distinction score $DS_{P}$; 6. | Compare the distinction score $DS_{P}$ obtained in step 5 to that obtained for random sets of $l_{P}$ genes, | where $l_{P}$ is the number of genes in the pathway of interest. Table 1: Procedure for Pathways of Distinction Analysis Pathway | Source | Length | $DS_{P}$ | $p(DS_{P})$ | O.R. | $q$(O.R.) ---|---|---|---|---|---|--- Purine metabolism | Kegg | 136 | 1.86 | 6.36e-03 | 1.59 | 4.15e-21 Calcium signaling pathway | Kegg | 100 | 1.38 | 1.82e-03 | 1.55 | 6.99e-20 Melanogenesis | Kegg | 84 | 2.36 | 4.55e-03 | 1.53 | 1.47e-18 Gap junction | Kegg | 80 | 1.54 | 5.45e-03 | 1.49 | 1.49e-16 ErbB signaling pathway | Kegg | 81 | 1.36 | 1.45e-02 | 1.46 | 4.68e-15 Long-term potentiation | Kegg | 60 | 1.71 | 9.09e-04 | 1.45 | 4.34e-15 GnRH signaling pathway | Kegg | 79 | 1.36 | 1.18e-02 | 1.44 | 1.32e-14 TCR signaling in naive CD4+ T cells | NCI-Nature | 60 | 2.11 | 5.45e-03 | 1.42 | 7.80e-13 Prostate cancer | Kegg | 75 | 1.45 | 4.09e-02 | 1.38 | 4.37e-11 PKC-catalyzed phosphorylation …myosin phosphatase | BioCarta | 20 | 1.97 | $<$1e-04 | 1.30 | 5.82e-09 CCR3 signaling in eosinophils | BioCarta | 21 | 1.59 | 1.09e-02 | 1.29 | 8.86e-08 Biosynthesis of unsaturated fatty acids | Kegg | 18 | 1.69 | 2.45e-02 | 1.26 | 1.38e-06 Attenuation of GPCR signaling | BioCarta | 11 | 1.75 | 1.09e-02 | 1.25 | 2.41e-06 Stathmin and breast cancer resistance to antimicrotubule agents | BioCarta | 18 | 1.84 | 4.82e-02 | 1.24 | 4.96e-06 Visual signal transduction: Cones | NCI-Nature | 20 | 1.56 | 4.73e-02 | 1.24 | 2.24e-06 Dentatorubropallidoluysian atrophy (DRPLA) | Kegg | 11 | 1.84 | 2.73e-03 | 1.24 | 2.24e-06 Intrinsic prothrombin activation pathway | BioCarta | 22 | 1.35 | 3.18e-02 | 1.23 | 4.61e-06 Eicosanoid metabolism | BioCarta | 19 | 1.69 | 1.91e-02 | 1.23 | 3.44e-06 Effects of botulinum toxin | NCI-Nature | 7 | 1.44 | 2.27e-02 | 1.20 | 3.50e-05 Activation of PKC through G-protein coupled receptors | BioCarta | 10 | 1.50 | 9.09e-03 | 1.20 | 8.42e-06 Streptomycin biosynthesis | Kegg | 9 | 1.36 | 3.55e-02 | 1.17 | 1.89e-04 PECAM1 interactions | Reactome | 6 | 2.70 | 5.45e-03 | 1.17 | 7.28e-05 HDL-mediated lipid transport | Reactome | 8 | 1.47 | 2.00e-02 | 1.14 | 1.59e-03 Granzyme A mediated apoptosis pathway | BioCarta | 8 | 1.97 | 1.73e-02 | 1.12 | 6.60e-04 Table 2: PID pathways with significant $DS_{P}$ in the CGEMS breast cancer GWAS. (Pathways with over 60% SNPs covered by another pathway have been removed; for the complete list, see Supplemental Table S-1). Pathway-length based resampled $p$-values, denoted $p(DS_{P})$, are given for significant pathways, along with the odds ratios and associated FDRs for a logistic regression model. Pathway | Source | Length | $DS_{P}$ | $p(DS_{P})$ | O.R. | $q$(O.R.) ---|---|---|---|---|---|--- Cell adhesion molecules (CAMs) | Kegg | 86 | 1.57 | 9.09e-03 | 1.66 | 3.56e-13 ErbB signaling pathway | Kegg | 76 | 1.45 | 3.45e-02 | 1.61 | 2.59e-10 Signaling events mediated by Stem cell factor receptor (c-Kit) | NCI-Nature | 40 | 2.35 | 5.45e-03 | 1.58 | 7.31e-10 Neurotrophic factor-mediated Trk receptor signaling | NCI-Nature | 50 | 1.60 | 2.36e-02 | 1.55 | 2.49e-08 Lissencephaly gene (LIS1) in neuronal migration and development | NCI-Nature | 21 | 2.02 | 7.27e-03 | 1.52 | 1.44e-07 Angiopoietin receptor Tie2-mediated signaling | NCI-Nature | 40 | 2.36 | 1.36e-02 | 1.51 | 5.77e-08 Reelin signaling pathway | NCI-Nature | 28 | 1.62 | 5.45e-03 | 1.46 | 7.35e-08 Syndecan-4-mediated signaling events | NCI-Nature | 27 | 1.74 | 1.64e-02 | 1.46 | 1.19e-06 Galactose metabolism | Kegg | 19 | 1.65 | 2.27e-02 | 1.44 | 5.01e-06 Vibrio cholerae infection | Kegg | 35 | 1.84 | 2.64e-02 | 1.43 | 6.67e-07 Paxillin-independent events mediated by a4b1 and a4b7 | NCI-Nature | 19 | 2.14 | 1.00e-02 | 1.40 | 6.67e-07 Antigen processing and presentation | Kegg | 34 | 3.26 | 1.36e-02 | 1.40 | 3.71e-08 Corticosteroids and Cardioprotection | BioCarta | 21 | 1.98 | 3.55e-02 | 1.39 | 1.24e-05 Lissencephaly gene (Lis1) in neuronal migration and development | BioCarta | 15 | 1.60 | 1.36e-02 | 1.37 | 2.52e-05 IL12 signaling mediated by STAT4 | NCI-Nature | 25 | 1.93 | 4.55e-02 | 1.37 | 1.58e-05 Biosynthesis of unsaturated fatty acids | Kegg | 13 | 1.76 | 1.64e-02 | 1.36 | 6.44e-05 Growth hormone signaling pathway | BioCarta | 18 | 1.75 | 3.18e-02 | 1.36 | 7.46e-05 Canonical Wnt signaling pathway | NCI-Nature | 28 | 1.92 | 4.73e-02 | 1.35 | 9.36e-06 NO2-dependent IL-12 pathway in NK cells | BioCarta | 8 | 1.82 | 2.73e-03 | 1.32 | 5.83e-05 Signaling events mediated by HDAC Class III | NCI-Nature | 19 | 2.12 | 3.91e-02 | 1.32 | 4.19e-05 Removal of aminoterminal propeptides from $\gamma$-carboxylated proteins | Reactome | 7 | 3.12 | 5.45e-03 | 1.29 | 8.46e-05 Aminophosphonate metabolism | Kegg | 13 | 1.91 | 3.36e-02 | 1.26 | 8.17e-04 Antigen processing and presentation | BioCarta | 6 | 2.61 | 1.82e-03 | 1.22 | 3.36e-05 Classical complement pathway | BioCarta | 12 | 2.27 | 1.55e-02 | 1.19 | 1.67e-04 Chylomicron-mediated lipid transport | Reactome | 7 | 1.94 | 3.27e-02 | 1.16 | 1.49e-02 Table 3: PID pathways with significant $DS_{P}$ in the liver cancer GWAS. (Pathways with over 60% SNPs covered by another pathway have been removed; for the complete list, see Supplemental Table S-2). Pathway-length based resampled $p$-values, denoted $p(DS_{P})$, are given for significant pathways, along with the odds ratios and associated FDRs for a logistic regression model. Pathway | Length | $p(DS_{P})$ | O.R. | $q$(O.R.) ---|---|---|---|--- Top-2 | 318 | $<$1e-04 | 2.02 | 1.63e-46 Top-3 | 397 | 1.00e-04 | 2.19 | 2.07e-54 Top-4 | 474 | $<$1e-04 | 2.33 | 3.65e-62 Top-5 | 522 | $<$1e-04 | 2.45 | 6.83e-66 Top-6 | 544 | $<$1e-04 | 2.44 | 8.51e-66 Top-7 | 558 | 2.00e-04 | 2.47 | 1.22e-67 Top-8 | 626 | $<$1e-04 | 2.59 | 1.01e-73 Top-9 | 658 | $<$1e-04 | 2.64 | 9.84e-75 Top-10 | 700 | $<$1e-04 | 2.77 | 9.72e-79 Top-11 | 710 | $<$1e-04 | 2.80 | 1.42e-79 Top-12 | 723 | $<$1e-04 | 2.82 | 2.06e-80 Top-13 | 739 | $<$1e-04 | 2.89 | 3.31e-82 Top-14 | 744 | $<$1e-04 | 2.93 | 2.86e-83 Top-15 | 770 | $<$1e-04 | 2.96 | 6.41e-85 Top-16 | 774 | $<$1e-04 | 2.97 | 5.10e-85 Top-17 | 791 | $<$1e-04 | 2.95 | 2.43e-85 Top-18 | 800 | $<$1e-04 | 3.06 | 1.15e-87 Top-19 | 814 | $<$1e-04 | 3.14 | 1.19e-89 Top-20 | 832 | $<$1e-04 | 3.26 | 4.51e-92 Top-21 | 837 | $<$1e-04 | 3.28 | 2.92e-92 Top-22 | 839 | $<$1e-04 | 3.29 | 2.41e-92 Top-23 | 845 | $<$1e-04 | 3.34 | 1.45e-93 Top-24 | 854 | $<$1e-04 | 3.38 | 4.62e-95 Table 4: PoDA results for sucessive unions of significant pathways in the CGEMS breast cancer data. Pathway-length based resampled $p$ values, denoted$p(DS_{P})$, are given along with the odds ratios and associated FDRs for a logistic regression model. Pathway | Length | $p(DS_{P})$ | O.R. | $q$(O.R.) ---|---|---|---|--- Top-2 | 321 | 5.38e-02 | 2.37 | 1.20e-27 Top-3 | 402 | 2.80e-03 | 2.63 | 1.40e-34 Top-4 | 474 | 1.10e-03 | 2.86 | 6.50e-38 Top-5 | 539 | 9.00e-04 | 3.22 | 4.03e-42 Top-6 | 560 | 1.00e-04 | 3.39 | 1.19e-43 Top-7 | 580 | $<$1e-04 | 3.50 | 1.39e-44 Top-8 | 589 | 6.00e-04 | 3.50 | 1.35e-44 Top-9 | 603 | 4.00e-04 | 3.52 | 1.23e-44 Top-10 | 624 | $<$1e-04 | 3.60 | 1.33e-45 Top-11 | 640 | $<$1e-04 | 3.73 | 3.69e-47 Top-12 | 646 | $<$1e-04 | 3.78 | 1.68e-47 Top-13 | 667 | $<$1e-04 | 3.81 | 9.29e-48 Top-14 | 709 | 3.00e-04 | 3.88 | 1.90e-48 Top-15 | 751 | $<$1e-04 | 4.09 | 2.11e-49 Top-16 | 761 | $<$1e-04 | 4.09 | 1.76e-49 Top-17 | 797 | $<$1e-04 | 4.45 | 1.29e-50 Top-18 | 805 | $<$1e-04 | 4.46 | 5.24e-51 Top-19 | 823 | $<$1e-04 | 4.56 | 2.20e-51 Top-20 | 838 | $<$1e-04 | 4.56 | 1.73e-51 Table 5: PoDA results for sucessive unions of significant pathways in the liver cancer data. Pathway-length based resampled $p$ values, denoted $p(DS_{P})$, are given along with the odds ratios and associated FDRs for a logistic regression model. Pathway | Source | Length | $DS_{P}$ | $p(DS_{P})$ | O.R. | $q$(O.R.) ---|---|---|---|---|---|--- Purine metabolism | Kegg | 136 | 1.86 | 6.36e-03 | 1.59 | 4.15e-21 Calcium signaling pathway | Kegg | 100 | 1.38 | 1.82e-03 | 1.55 | 6.99e-20 Melanogenesis | Kegg | 84 | 2.36 | 4.55e-03 | 1.53 | 1.47e-18 Gap junction | Kegg | 80 | 1.54 | 5.45e-03 | 1.49 | 1.49e-16 ErbB signaling pathway | Kegg | 81 | 1.36 | 1.45e-02 | 1.46 | 4.68e-15 Long-term potentiation | Kegg | 60 | 1.71 | 9.09e-04 | 1.45 | 4.34e-15 GnRH signaling pathway | Kegg | 79 | 1.36 | 1.18e-02 | 1.44 | 1.32e-14 TCR signaling in naive CD4+ T cells | NCI-Nature | 60 | 2.11 | 5.45e-03 | 1.42 | 7.80e-13 TCR signaling in naive CD8+ T cells | NCI-Nature | 48 | 2.03 | 7.27e-03 | 1.38 | 1.11e-11 Prostate cancer | Kegg | 75 | 1.45 | 4.09e-02 | 1.38 | 4.37e-11 PKC-catalyzed phosphorylation …myosin phosphatase | BioCarta | 20 | 1.97 | $<$1e-04 | 1.30 | 5.82e-09 CCR3 signaling in eosinophils | BioCarta | 21 | 1.59 | 1.09e-02 | 1.29 | 8.86e-08 Biosynthesis of unsaturated fatty acids | Kegg | 18 | 1.69 | 2.45e-02 | 1.26 | 1.38e-06 Attenuation of GPCR signaling | BioCarta | 11 | 1.75 | 1.09e-02 | 1.25 | 2.41e-06 Stathmin and breast cancer resistance to antimicrotubule agents | BioCarta | 18 | 1.84 | 4.82e-02 | 1.24 | 4.96e-06 Visual signal transduction: Cones | NCI-Nature | 20 | 1.56 | 4.73e-02 | 1.24 | 2.24e-06 Dentatorubropallidoluysian atrophy (DRPLA) | Kegg | 11 | 1.84 | 2.73e-03 | 1.24 | 2.24e-06 Intrinsic prothrombin activation pathway | BioCarta | 22 | 1.35 | 3.18e-02 | 1.23 | 4.61e-06 Eicosanoid metabolism | BioCarta | 19 | 1.69 | 1.91e-02 | 1.23 | 3.44e-06 Effects of botulinum toxin | NCI-Nature | 7 | 1.44 | 2.27e-02 | 1.20 | 3.50e-05 Activation of PKC through G-protein coupled receptors | BioCarta | 10 | 1.50 | 9.09e-03 | 1.20 | 8.42e-06 Ca-calmodulin-dependent protein kinase activation | BioCarta | 8 | 1.70 | 1.00e-02 | 1.19 | 5.67e-05 Streptomycin biosynthesis | Kegg | 9 | 1.36 | 3.55e-02 | 1.17 | 1.89e-04 PECAM1 interactions | Reactome | 6 | 2.70 | 5.45e-03 | 1.17 | 7.28e-05 HDL-mediated lipid transport | Reactome | 8 | 1.47 | 2.00e-02 | 1.14 | 1.59e-03 Granzyme A mediated apoptosis pathway | BioCarta | 8 | 1.97 | 1.73e-02 | 1.12 | 6.60e-04 Table S-1: Full list PID pathways with significant $DS_{P}$ in the breast cancer GWAS, including highly “overlapping” pathways. Pathway-length based resampled $p$-values, denoted $p(DS_{P})$, are given for significant pathways, along with the odds ratios and associated FDRs for a logistic regression model. Pathway | Source | Length | $DS_{P}$ | $p(DS_{P})$ | O.R. | $q$(O.R.) ---|---|---|---|---|---|--- Cell adhesion molecules (CAMs) | Kegg | 86 | 1.57 | 9.09e-03 | 1.66 | 3.56e-13 ErbB signaling pathway | Kegg | 76 | 1.45 | 3.45e-02 | 1.61 | 2.59e-10 Signaling events mediated by Stem cell factor receptor (c-Kit) | NCI-Nature | 40 | 2.35 | 5.45e-03 | 1.58 | 7.31e-10 Neurotrophic factor-mediated Trk receptor signaling | NCI-Nature | 50 | 1.60 | 2.36e-02 | 1.55 | 2.49e-08 Lissencephaly gene (LIS1) in neuronal migration and development | NCI-Nature | 21 | 2.02 | 7.27e-03 | 1.52 | 1.44e-07 Angiopoietin receptor Tie2-mediated signaling | NCI-Nature | 40 | 2.36 | 1.36e-02 | 1.51 | 5.77e-08 Reelin signaling pathway | NCI-Nature | 28 | 1.62 | 5.45e-03 | 1.46 | 7.35e-08 Syndecan-4-mediated signaling events | NCI-Nature | 27 | 1.74 | 1.64e-02 | 1.46 | 1.19e-06 Galactose metabolism | Kegg | 19 | 1.65 | 2.27e-02 | 1.44 | 5.01e-06 TPO signaling pathway | BioCarta | 17 | 2.61 | 6.36e-03 | 1.44 | 3.80e-06 Vibrio cholerae infection | Kegg | 35 | 1.84 | 2.64e-02 | 1.43 | 6.67e-07 Paxillin-independent events mediated by a4b1 and a4b7 | NCI-Nature | 19 | 2.14 | 1.00e-02 | 1.40 | 6.67e-07 Antigen processing and presentation | Kegg | 34 | 3.26 | 1.36e-02 | 1.40 | 3.71e-08 Corticosteroids and cardioprotection | BioCarta | 21 | 1.98 | 3.55e-02 | 1.39 | 1.24e-05 Lissencephaly gene (Lis1) in neuronal migration and development | BioCarta | 15 | 1.60 | 1.36e-02 | 1.37 | 2.52e-05 IL12 signaling mediated by STAT4 | NCI-Nature | 25 | 1.93 | 4.55e-02 | 1.37 | 1.58e-05 Biosynthesis of unsaturated fatty acids | Kegg | 13 | 1.76 | 1.64e-02 | 1.36 | 6.44e-05 Growth hormone signaling pathway | BioCarta | 18 | 1.75 | 3.18e-02 | 1.36 | 7.46e-05 Canonical Wnt signaling pathway | NCI-Nature | 28 | 1.92 | 4.73e-02 | 1.35 | 9.36e-06 NO2-dependent IL-12 pathway in nk cells | BioCarta | 8 | 1.82 | 2.73e-03 | 1.32 | 5.83e-05 Signaling events mediated by HDAC Class III | NCI-Nature | 19 | 2.12 | 3.91e-02 | 1.32 | 4.19e-05 Removal of aminoterminal propeptides from gamma-carboxylated proteins | Reactome | 7 | 3.12 | 5.45e-03 | 1.29 | 8.46e-05 Gamma-carboxylation, transport, and amino-terminal cleavage of proteins | Reactome | 6 | 3.25 | 1.82e-03 | 1.28 | 6.64e-05 Transport of $\gamma$-carboxylated protein precursors … | Reactome | 6 | 3.25 | 1.82e-03 | 1.28 | 6.64e-05 Paxillin-dependent events mediated by a4b1 | NCI-Nature | 17 | 1.84 | 2.00e-02 | 1.28 | 3.41e-05 Gamma-carboxylation of protein precursors | Reactome | 7 | 2.86 | 3.64e-03 | 1.28 | 1.38e-04 Aminophosphonate metabolism | Kegg | 13 | 1.91 | 3.36e-02 | 1.26 | 8.17e-04 Antigen processing and presentation | BioCarta | 6 | 2.61 | 1.82e-03 | 1.22 | 3.36e-05 Lectin induced complement pathway | BioCarta | 11 | 1.91 | 2.18e-02 | 1.20 | 1.55e-04 Classical complement pathway | BioCarta | 12 | 2.27 | 1.55e-02 | 1.19 | 1.67e-04 Chylomicron-mediated lipid transport | Reactome | 7 | 1.94 | 3.27e-02 | 1.16 | 1.49e-02 Table S-2: Full list PID pathways with significant $DS_{P}$ in the liver cancer GWAS, including highly “overlapping” pathways. Pathway-length based resampled $p$-values, denoted $p(DS_{P})$, are given for significant pathways, along with the odds ratios and associated FDRs for a logistic regression model.
arxiv-papers
2010-12-21T16:50:59
2024-09-04T02:49:15.897586
{ "license": "Public Domain", "authors": "Rosemary Braun and Kenneth Buetow", "submitter": "Rosemary Braun", "url": "https://arxiv.org/abs/1012.4726" }
1012.5022
# Numeric and symbolic evaluation of the pfaffian of general skew-symmetric matrices C. González-Ballestero L.M. Robledo G. F. Bertsch Departamento de Física Teórica, Universidad Autónoma de Madrid, E-28049 Madrid, Spain Department of Physics and Institute for Nuclear Theory, University of Washington, Seattle, WA 98195–1560 USA ###### Abstract Evaluation of pfaffians arises in a number of physics applications, and for some of them a direct method is preferable to using the determinantal formula. We discuss two methods for the numerical evaluation of pfaffians. The first is tridiagonalization based on Householder transformations. The main advantage of this method is its numerical stability that makes unnecessary the implementation of a pivoting strategy. The second method considered is based on Aitken’s block diagonalization formula. It yields to a kind of LU (similar to Cholesky’s factorization) decomposition (under congruence) of arbitrary skew-symmetric matrices that is well suited both for the numeric and symbolic evaluations of the pfaffian. Fortran subroutines (FORTRAN 77 and 90) implementing both methods are given. We also provide simple implementations in Python and Mathematica for purpose of testing, or for exploratory studies of methods that make use of pfaffians. ###### keywords: Skew symmetric matrices , Pfaffian ††journal: Computer Physics Communications PROGRAM SUMMARY Manuscript Title: Numeric and symbolic evaluation of the pfaffian of general skew-symmetric matrices Authors: C. Gonzalez-Ballestero, L.M.Robledo and G.F. Bertsch Program Title: Pfaffian Journal Reference: Catalogue identifier: Licensing provisions: Programming language: Fortran 77 and 90 Computer: Operating system: RAM: bytes Number of processors used: Supplementary material: Keywords: Skew symmetric matrices, Pfaffian Classification: 4.8 Linear Equations and Matrices External routines/libraries: BLAS Subprograms used: Catalogue identifier of previous version:* Journal reference of previous version:* Does the new version supersede the previous version?:* Nature of problem: Evaluation of the Pfaffian of a skew symmetric matrix. Evaluation of pfaffians arises in a number of physics applications involving fermionic mean field wave functions and their overlaps. Solution method: Householder tridiagonalization. Aitken’s block diagonalization formula. Reasons for the new version:* Summary of revisions:* Restrictions: Unusual features: Additional comments: Python and Mathematica implementations are provided in the main body of the paper Running time: Depends on the size of the matrices. For matrices with 100 rows and columns a few miliseconds are required. ## 1 Introduction In a number of fields in physics, the formal equations derived from the theory make use of the pfaffian of some skew-symmetric matrix appearing in the theory. For example, the pfaffian arises in the treatment of electronic structure with quantum Monte Carlo methods [1], the description of two- dimensional Ising spin glasses [2], and the evaluation of entropy and its relation to entanglement [3]. Pfaffians occur naturally in field theory and nuclear physics in formalisms based on fermionic coherent states [4, 5, 6, 7]. A recent application is to the overlap of two Hartree Fock Bogoliubov (HFB) product wave functions [8], needed for nuclear structure theory. While there is a simple formula for the pfaffian of a skew-symmetric matrix $M$ in terms of the determinant, $\textrm{pf}(A)=\sqrt{{\rm det}(A)}$ (1) the so-called “sign problem of the overlap” [9] associated with the square root motivates the use of numerical algorithms that evaluate it directly. The most straightforward method, the rule of “expanding in minors” [10], has bad scaling with the size of the matrix and is prohibitive for large matrices. In this paper we discuss two alternative methods that have the same scaling property as the normal $N^{3}$ algorithms for the determinant. The methods are implemented in the FORTRAN 77 and 90 subroutines provided in the accompanying program library. We also comment on the practical implementation of the two methods in Mathematica and in the Python programming language. ## 2 Evaluation of the Pfaffian The Pfaffian $\textrm{pf}(A)$ is reduced to a simple form that is easily evaluated by making repeated use of transformation formula given in A, $\textrm{pf}(B^{t}AB)={\rm det(B)}\textrm{pf}(A).$ (2) In order to perform the numerical evaluation of the Pfaffian of a complex skew-symmetric matrix $A$ we reduce the skew-symmetric matrix to a tridiagonal form $A_{TR}$ by using unitary matrices $U$. Once it is in this form, the evaluation of the pfaffian is straightforward (see below). ### 2.1 Reduction to tridiagonal form by mean of Householder transformations In this method, we will use the well-known Householder transformations [11] to reduce $A$ to tridiagonal form. We present it in some detail because the generalization to the complex number field is not entirely trivial. Complex Householder transformations have the form $P=\mathbb{I}-2\frac{u\otimes u^{+}}{\left|u\right|^{2}}$ (3) where $u$ is an arbitrary complex complex row vector $u=(u_{1},u_{2},\ldots,u_{N})$ and $\left(u\otimes u^{+}\right)_{ij}=u_{i}u_{j}^{*}$. The vector $u$ must be chosen to zero all the elements of a vector $x$ except a given one. If we take $u=x\mp e^{i\arg(x_{j})}|x|e_{j}$, with $(e_{j})_{k}=\delta_{jk}$, it can be easily proved that $P_{u}x=\pm e^{i\arg(x_{j})}|x|e_{j}$ as required. The freedom on the sign in the expression defining the vector $u$ can be used to make sure that the vector $u$ is non zero. The rest of the Householder tridiagonalization procedure follows exactly as in the real case. Consider a skew-symmetric matrix of dimension N (even) $A=\left(\begin{array}[]{c|ccc}0&a_{12}&a_{13}&\ldots\\\ \hline\cr-a_{12}\\\ -a_{13}&&{}^{(N-1)}A\\\ \vdots\end{array}\right)$ (4) The Householder transformation matrix is $P_{1}=\left(\begin{array}[]{c|ccc}1&0&0&\ldots\\\ \hline\cr 0\\\ 0&&{}^{(N-1)}P_{1}\\\ \vdots\end{array}\right)$ where ${}^{(N-1)}P_{1}$ is built by using Eq. (3) and taking the vector $x$ (of dimension N-1) as $(a_{12},a_{13},\ldots)^{T}$. The resulting transformed matrix is given by $P_{1}AP_{1}^{T}=\left(\begin{array}[]{c|ccc}0&k_{1}&0&\ldots\\\ \hline\cr- k_{1}\\\ 0&&{}^{(N-1)}\tilde{A}\\\ \vdots\end{array}\right)$ where $k_{1}=\pm e^{i\arg(a_{12})}|x|$ and the matrix ${}^{(N-1)}\tilde{A}$ is skew-symmetric and given by ${}^{(N-1)}\tilde{A}={}^{(N-1)}P_{1}{}^{(N-1)}A{}^{(N-1)}P_{1}^{T}$. Performing this procedure a total of N-2 times we end up with a tridiagonal and skew-symmetric matrix $P_{N-2}\ldots P_{2}P_{1}AP_{1}^{T}P_{2}^{T}\ldots P_{N-2}^{T}=\left(\begin{array}[]{cccccc}0&k_{1}&0&0&\ldots&0\\\ -k_{1}&0&k_{2}&0&\ldots&0\\\ 0&-k_{2}&0&\ddots&\ldots&\vdots\\\ &&\ddots&0&\ddots\\\ 0&&&\ddots&0&k_{N-1}\\\ \vdots&&\vdots&&-k_{N-1}&0\end{array}\right)$ (5) Using now a known property the Pfaffian (see A) we can deduce from the above identity that $\det(P_{1})\ldots\det(P_{N-2})\textrm{pf}(A)=\textrm{pf}(A_{TR})$ where $A_{TR}$ is the triagonal and skew-symmetric matrix of the right hand side of Eq. (5). Taking into account that the determinant of any Householder matrix is -1 and that $N$ is even, we can express the Pfaffian of $A$ in terms of the pfaffian of the tridiagonal $A_{TR}$ $\textrm{pf}(A)=\textrm{pf}(A_{TR})$ As will be shown below the Pfaffian of a tridiagonal skew-symmetric matrix is simply given by $k_{1}k_{3}\ldots k_{N-1}=\prod_{i=1}^{N/2}k_{2i-1}$ (this result can also be obtained using the “minor expansion” formula [10] ) and finally we obtain $\textrm{pf}(A)=\prod_{i=1}^{N/2}k_{2i-1}.$ (6) In terms of numerical stability, the Householder transformation is very robust and there is no need to consider any “pivoting” strategy common to other methods. However, the presence of the square root of $x$ and the argument $\arg(x_{j})$ of complex quantities prevents an easy implementation of the Householder tridiagonalization procedure for symbolic computation. For this purpose the second method described in the next section is far easier to implement. ### 2.2 Aitken’s block diagonalization formula There is an alternative method for the calculation of the pfaffian, which is also well suited for a symbolic implementation and that relies on an expression for the pfaffian of a bipartite skew-symmetric matrix. Let us start with a general skew-symmetric matrix $A$ (dimension N, even) given by $A=\left(\begin{array}[]{cc}R&Q\\\ -Q^{T}&S\end{array}\right)$ (7) where $R$ and $S$ are square skew-symmetric matrices and $Q$ is a general rectangular matrix (to account for the case where $R$ and $S$ have different dimensions). Using Aitken’s block diagonalization formula (see [12] for an early use of the formula and [13] for a recent and thorough reference) for a bipartite matrix we obtain $\displaystyle\left(\begin{array}[]{cc}\mathbb{I}&0\\\ Q^{T}R^{-1}&\mathbb{I}\end{array}\right)\left(\begin{array}[]{cc}R&Q\\\ -Q^{T}&S\end{array}\right)\left(\begin{array}[]{cc}\mathbb{I}&-R^{-1}Q\\\ 0&\mathbb{I}\end{array}\right)$ $\displaystyle=$ (14) $\displaystyle\left(\begin{array}[]{cc}R&0\\\ 0&S+Q^{T}R^{-1}Q\end{array}\right)$ (17) where the matrix $S+Q^{T}R^{-1}Q$ is referred to in the literature as the Schur complement of the matrix $A$ (see, for instance, [13]). For the special case of a skew-symmetric matrix $A$, the matrices $R$ and $S$ are also skew- symmetric and the transformation of the matrix $A$ is a congruence (i.e. the matrix acting on the left hand side of $A$ is the transpose of the one acting on the right hand side). Denoting $P_{1}=\left(\begin{array}[]{cc}\mathbb{I}&0\\\ Q^{T}R^{-1}&\mathbb{I}\end{array}\right)$ (18) Eq. (17) becomes $P_{1}AP_{1}^{T}=\left(\begin{array}[]{cc}R&0\\\ 0&S+Q^{T}R^{-1}Q\end{array}\right)$ An equivalent expression involving $S^{-1}$ instead of $R^{-1}$ is easily obtained $P_{2}AP_{2}^{T}=\left(\begin{array}[]{cc}R+QS^{-1}Q^{T}&0\\\ 0&S\end{array}\right)$ with $P_{2}=\left(\begin{array}[]{cc}\mathbb{I}&-QS^{-1}\\\ 0&\mathbb{I}\end{array}\right)$ (19) By using the property $\textrm{pf}(P^{T}AP)=\textrm{det}(P)\textrm{pf}(A)$ (see A) and taking into account that $\det P_{1}=\det P_{2}$=1, we come to $\displaystyle\textrm{pf}(A)$ $\displaystyle=$ $\displaystyle\textrm{pf}(R)\textrm{pf}(S+Q^{T}R^{-1}Q)$ (20) $\displaystyle=$ $\displaystyle\textrm{pf}(R+QS^{-1}Q^{T})\textrm{pf}(S)$ (21) Another nice property of the matrices $P_{1}$ and $P_{2}$ is that their inverses can be obtained very easily $P_{1}^{-1}=\left(\begin{array}[]{cc}\mathbb{I}&0\\\ -Q^{T}R^{-1}&\mathbb{I}\end{array}\right)$ (22) and $P_{2}^{-1}=\left(\begin{array}[]{cc}\mathbb{I}&QS^{-1}\\\ 0&\mathbb{I}\end{array}\right)$ (23) These expressions of the inverses explicitly show that both $P_{1}$ and $P_{2}$ are not orthogonal matrices. Let us now apply the above result to an arbitrary skew-symmetric matrix of dimension $N=2M$ which is written in block form as $A=\left(\begin{array}[]{ccc}A^{(1)}&A_{N-1}&A_{N}\\\ -A_{N-1}^{T}&0&a_{N-1,N}\\\ -A_{N}^{T}&-a_{N-1,N}&0\end{array}\right)$ (24) where $A^{(1)}$ is a skew-symmetric square matrix of dimension $N-2=2(M-1)$ and $A_{N-1}$ and $A_{N}$ are column vectors $A_{N-1}=\\{A_{i,N-1},\,i=1,N-2\\}$ and $A_{N}=\\{A_{i,N},\,i=1,N-2\\}$ both of dimension $(N-2)\times 1$. In the language of Eq (7) the matrix $R$ is the matrix $A^{(1)}$, the matrix $Q$ is a rectangular matrix of dimension $2\times(N-2)$ made of the two column vectors, $A_{N-1}$ and $A_{N}$ and finally the matrix $S$ is the $2\times 2$ skew-symmetric matrix with matrix element $S_{12}=a_{N-1,N}$. Using the ideas of Aitken’s block diagonalization formula, it is easy to shows that the matrix $\tilde{A}=D_{1}^{T}AD_{1}$ is in block diagonal form $\tilde{A}=\left(\begin{array}[]{ccc}\mbox{$\tilde{A}$}^{(1)}&0&0\\\ 0&0&a_{N-1,N}\\\ 0&-a_{N-1,N}&0\end{array}\right)$ (25) with a matrix $D_{1}$ of the form $D_{1}=\left(\begin{array}[]{ccc}\mathbb{I}_{N-2}&0&0\\\ X&1&0\\\ Y&0&1\end{array}\right)$ (26) where $\mathbb{I}_{N-2}$ stands for the identity matrix of dimension $N-2$ and both $X$ and $Y$ are row vectors of dimension $1\times(N-2)$ and given by $X=-a_{N-1,N}^{-1}A_{N}^{T}$ and $Y=a_{N-1,N}^{-1}A_{N-1}^{T}$. In the above equation 25, the skew-symmetric matrix $\tilde{A}^{(1)}$ is given by $\tilde{A}^{(1)}=A^{(1)}+A_{N}(a_{N-1,N})^{-1}A_{N-1}^{T}-A_{N-1}a_{N-1,N}^{-1}A_{N}^{T}$ (27) Taking into account that $\det D_{1}=1$ then $\textrm{pf}(A)=\textrm{pf}(\tilde{A})=a_{N-1.N}\textrm{pf}(\tilde{A}^{(1)})$. The algorithm can be applied recursively to $\tilde{A}^{(1)}$ to obtain $\textrm{pf}(A)=a_{N-1,N}\tilde{a}_{N-3,N-2}^{(1)}\textrm{pf}(\tilde{A}^{(2)})$ so as to reduce, after $M-1$ iterations, the computation of the pfaffian to the product of the corresponding elements. This procedure can be easily implemented for a skew-symmetric tridiagonal matrix, as the transformed matrices in Eq (27) coincide with the original ones; for instance, $\tilde{A}^{(1)}=A^{(1)}$. As a consequence, the pfaffian of a tridiagonal matrix is given by $\textrm{pf }\left(\begin{array}[]{cccccc}0&d_{1}&0&0&\ldots&0\\\ -d_{1}&0&d_{2}&0&\ldots&0\\\ 0&-d_{2}&0&\ddots&\ldots&\vdots\\\ 0&0&\ddots&0&\ddots&0\\\ \vdots&\vdots&\vdots&\ddots&0&d_{2N-1}\\\ 0&0&\cdots&0&-d_{2N-1}&0\end{array}\right)=d_{1}d_{3}\ldots d_{2N-1}=\prod_{i=1}^{N}d_{2i-1}$ #### 2.2.1 Pivoting As a consequence of the division by matrix elements like $a_{N-1,N}$ in the first iteration, the numerical stability of the algorithm requires the use of pivoting strategy in the implementation of the method. Full pivoting amounts to search the whole matrix for the matrix element with the largest modulus and exchange it with the required matrix element. For instance, in the first iteration of the procedure, the matrix element $a_{p,q}$ ($p<q)$ with the largest modulus is searched for and exchanged with the matrix element $a_{N-1,N}$. In this way we avoid dangerous divisions by small (or even zero) matrix elements. We have to take into account that in the present case, the exchange of both columns and rows is required to preserve the skew-symmetric nature of the matrices involved. To carry out the exchange of rows and columns we will use the exchange matrix $P(ij)$ that, when applied to the right of an arbitrary matrix, exchanges columns $i$ and $j$. The exchange matrix is given by the matrix elements $\displaystyle P(ij)_{kl}$ $\displaystyle=$ $\displaystyle\delta_{kl}-\delta_{i,l}\delta_{i,k}-\delta_{j,l}\delta_{j,k}+\delta_{i,l}\delta_{j,k}+\delta_{j,l}\delta_{i,k}.$ (28) To exchange the corresponding rows we have to apply $P(ij)^{T}$ to the left of the matrix (notice that $P(ij)$ is symmetric). With the help of these matrices we can write the matrix after pivoting $a_{p,q}$ with $a_{N-1,N}$ (and $a_{q,p}$ with $a_{N,N-1})$ as $A_{P}=P^{T}(N-1,p)P^{T}(N,q)\,A\,P(N-1,p)P(N,q)$ As a consequence of such exchange and taking into account that $\det P(ij)=-1$ we can conclude that the pfaffian of $A$ does not change by the pivoting procedure. Finally we obtain $A=P(N,q)P(N-1,p)\,A_{P}\,P^{T}(N-1,p)P^{T}(N,q)$ $=P(N,q)P(N-1,p)D_{1}^{T\,-1}\tilde{A}_{P}D_{1}^{-1}P^{T}(N-1,p)P^{T}(N,q)$ where $\tilde{A}_{P}$ has the same structure as $\tilde{S}$ in Eq. (25). As before, $\textrm{pf}(A)=\textrm{pf}(\tilde{A}_{P})=\left(A_{P}\right)_{N-1,N}\textrm{pf}(\tilde{A}_{P}^{(1)})$ and repeating recursively the whole procedure $M-1$ times we obtain the pfaffian as the product of the corresponding matrix elements. #### 2.2.2 Cholesky like decomposition of a skew-symmetric matrix Although it is not necessary in order to compute the pfaffian, it can be useful to show that even with pivoting we can write the matrix $A$ as $A=PL^{T}\tilde{A}LP$ (29) where $P$ is the product of exchange matrices as in Eq (28), $L$ is the product of matrices of the $D^{-1}$ type, Eq (26), and therefore is a lower triangular matrix with ones in the main diagonal and finally, $\tilde{A}$ is a skew-symmetric matrix in canonical form, i.e. a block diagonal matrix with skew-symmetric, $2\times 2$ blocks in the diagonal. This decomposition of a general skew-symmetric matrix $A$ resembles the Cholesky decomposition of a general matrix and can be useful in formal manipulations like, for instance, the inversion of the matrix $A$. In order to show that Eq (29) holds the only required property is that, when applying the pivoting procedure to $\tilde{A}_{P}^{(1)}$ the exchange matrices required $P(N-2,s)P(N-3,r)$ have the property of preserving the structure of the matrix $D_{1}$(and its inverse). For instance, $D_{1}^{T\,-1}P(N-2,s)P(N-3,r)=P(N-2,s)P(N-3,r)\tilde{D}_{1}^{T\,-1}$ with $\tilde{D}_{1}^{T\,-1}$ a matrix that is obtained from $D_{1}^{T\,-1}$ by exchanging rows $N-2$ and $s$ and rows $N-3$ and $r$ and therefore has the same upper triangular structure with ones in the diagonal as the original matrix $D_{1}^{T\,-1}$ . Using this property we can move all the exchange matrices to the right (or to the left) and the remaining matrix will be the product of triangular matrices (lower for products involving $D^{-1}$) with ones in the diagonal. As mentioned earlier, Aitken’s method is better adapted to symbolic evaluations. However, one must take care that in each step of the process some specific matrix elements are non-zero. ## 3 Fortran implementation The implementation of the algorithms considered in this paper in a high level computer language is straightforward. However, specific code in FORTRAN (both 77 and 90, real and complex arithmetic) is provided along with this paper. The algorithms are easy to follow and the comments included in the code are useful guides. Just a few comments are in order: to implement the tridiagonalization procedure in Fortran, it is advantageous to use the BLAS package [14] to perform the required matrix by vector multiplication and rank two update. Unfortunately there are no equivalent in the skew-symmetric case of the routines SYM (to multiply a symmetric matrix by a vector) or SYR2 (to perform a symmetric rank two update) but the general procedures GEMV and GERU can be used instead. On output, both the pfaffian of the matrix and the set of vectors required to bring it to tridiagonal form are returned. For the implementation of the method based on Aitken’s block diagonalization formula a pivoting strategy is required. We have used full pivoting in our implementation due to its robustness. The routines provided only require the upper part of the skew-symmetric matrix. The lower part is destroyed and replaced with the tridiagonal transformation matrix that brings the skew- symmetric matrix to canonical form upon congruence. An integer vector is also returned to reconstruct the required exchange of rows and columns. Perhaps the best test to check the validity of the two implementations is to compute the pfaffian of a skew-symmetric matrix using both procedures in order to compare the output. If it is the same up to a given accuracy then it is very likely that the two implementations are correct. We have writen a test program (also included in the distribution) that generates skew-symmetric matrices of given dimension with random entries and compute the pfaffian using both techniques. In our tests the pfaffians computed both ways coincide up to one part in $10^{10}$ with dimensions of the matrices of one thousand. This result also supports the adequacy of the implementation in terms of numerical stability. Another possibility to test the numerical implementation is to use the analytical formula given in B for a specific kind of $8\times 8$ matrices. A test program implementing this approach has also been included in the distribution. To finish this section we will briefly comment on the timing of the FORTRAN numerical implementations mentioned. In a modern personal computer under Linux the computation of the pfaffian of a $100\times 100$ matrix takes a few milliseconds in both implementations and the timing scales roughly as the cube of the dimension of the matrix in such a way that for matrices of a $1000\times 1000$ dimension the time is of the order of a few seconds. ## 4 A simple Python implementation We provide here a simple implementation of the tridiagonal reduction method (see [12] and [15]) in Python, which may be useful for testing purposes. It is similar to the Householder, but it only use simple row and column operations that have determinants of unity. The code is: from numpy import * def pfaff_py(m) : mat=copy(m) ndim = shape(mat)[0] t1=1.0 for j in range(ndim/2) : t1 *= mat[0,1] print ’t1’, t1 if j <ndim/2-1 : ndimr=shape(mat)[0] for i in range(2,ndimr) : if mat[0,1] != 0.0 : tv=mat[1,:]*mat[i,0]/mat[1,0] mat[i,: ] -= tv tv=mat[:,1]*mat[0,i]/mat[0,1] mat[:,i ] -= tv else : print ’need to pivot’ raise Exception mat=mat[2:,2:] return t1 The user should be cautioned that the algorithm is not guaranteed to be stable without an additional pivot step. Also, the matrix is assumed to have been constructed with the array function in the Numpy library. ## 5 A simple Mathematica implementation We also provide a simple Mathematica implementation of the method based on Aitken’s block diagonalization formula. As mentioned above, this method requires pivoting to avoid divisions by small (or zero) numbers. In the symbolic implementation, this issue is solved by replacing the denominator by a variable (OO on the implementation below) in case it is zero and an additional limit when the variable tends to zero is performed at the end. The two Mathematica modules required are: Aitken[M_,n_,OO_]:= Module[{MM=M,i,j,p}, If[MM[[n-1,n]]==0,MM[[n-1,n]]=OO;MM[[n,n-1]]=-OO]; p=MM[[n-1,n]]; For[i=1,i<=n-2,i++, For[j=1,j<=n-2,j++, MM[[i,j]]=M[[i,j]]+(MM[[i,n]]*MM[[j,n-1]]-MM[[j,n]]*MM[[i,n-1]])/p ] ]; MM]; pfaffian[S_]:= Module[{T=S,n,p}, n=Length[T]/2; If[T[[2*n-1,2*n]]==0,T[[2*n-1,2*n]]=OO;T[[2*n,2*n-1]]=-OO]; For[n=Length[T]/2;p=T[[2*n-1,2*n]],n>1,n--, T=Aitken[T,2*n,OO]; p=p*T[[2*(n-1)-1,2*(n-1)]] ]; Limit[p,OO->0]]; ## 6 Conclusions The issue of how to compute both numerically and symbolically the pfaffian of a skew-symmetric matrix has been addressed using two different approaches. Numerical stability issues are discussed and methods to assure the desired accuracy are fully incorporated. A collection of subroutines and test programs in FORTRAN (both 77 and 90, double precision and complex) are provided. A few comments on the implementation of the algorithms in Mathematica and Python are also given. ## 7 Acknowledgements We acknowledge K. Roche for a careful reading of the manuscript and several suggestions. This work was supported by MICINN (Spain) under research grants FPA2009-08958, and FIS2009-07277, as well as by Consolider-Ingenio 2010 Programs CPAN CSD2007-00042 and MULTIDARK CSD2009-00064. ## Appendix A Definition and basic properties of the pfaffian The pfaffian of a skew-symmetric matrix $R$ of dimension $2N$ and with matrix elements $r_{ij}$ is defined as $\textrm{pf}(R)=\frac{1}{2^{n}}\frac{1}{n!}\sum_{\textrm{Perm}}\epsilon(P)r_{i_{1}i_{2}}r_{i_{3}i_{4}}r_{i_{5}i_{6}}\ldots r_{2n-1,2n}$ where the sum extends to all possible permutations of $i_{1},\ldots,i_{2n}$ and $\epsilon(P)$ is the parity of the permutation. For matrices of odd dimension the pfaffian is by definition equal to zero. As an example, the pfaffian of a $2\times 2$ matrix $R$ is $\textrm{pf}(R)=r_{12}$ and for a $4\times 4$ one $\textrm{pf}(R)=r_{12}r_{34}-r_{13}r_{24}+r_{14}r_{23}$. Useful properties of the pfaffian are $\textrm{pf}(P^{T}RP)=\textrm{det}(P)\textrm{pf}(R),$ (30) $\text{{pf}}\left(\begin{array}[]{cc}0&R\\\ -R^{T}&0\end{array}\right)=(-1)^{N(N-1)/2}\det(R)$ where the matrix $R$ is $N\times N$ and $\text{{pf}}\left(\begin{array}[]{cc}R_{1}&0\\\ 0&R_{2}\end{array}\right)=\text{{pf}}(R_{1})\text{{pf}}(R_{2})$ where $R_{1}$ and $R_{2}$ are skew-symmetric matrices. The matrices may be defined on the real or on the complex number fields. ## Appendix B Pfaffian of a test matrix In this appendix we give the expression of the pfaffian of a test matrix which is big enough as not to be trivial but on the other hand is small enough as to render the explicit expression of the pfaffian manageable. The expression given below can be used to check both numerical and symbolic implementations of the pfaffian. Consider the two general skew-symmetric matrices of dimension 4 $M=\left(\begin{array}[]{cccc}0&f_{1}&m_{11}&m_{12}\\\ -f_{1}&0&m_{21}&m_{22}\\\ -m_{11}&-m_{21}&0&f_{2}\\\ -m_{12}&-m_{22}&-f_{2}&0\end{array}\right)$ and $N=\left(\begin{array}[]{cccc}0&g_{1}&n_{11}&n_{12}\\\ -g_{1}&0&n_{21}&n_{22}\\\ -n_{11}&-n_{21}&0&g_{2}\\\ -n_{12}&-n_{22}&-g_{2}&0\end{array}\right)$ where the matrix elements can be complex numbers. With these two matrices and the identity $4\times 4$ matrix we build the skew-symmetric matrix $S=\left(\begin{array}[]{cc}N&-\mathbb{I}\\\ \mathbb{I}&-M^{*}\end{array}\right)$ of dimension $8\times 8$ (see Ref [8] for the physical context of this matrix). It is relatively easy to compute its pfaffian $\textrm{pf}[S]=1+f_{1}^{*}g_{1}+f_{2}^{*}g_{2}+m_{11}^{*}n_{11}+m_{22}^{*}n_{22}+m_{12}^{*}n_{12}+m_{21}^{*}n_{21}+$ $+(f_{1}^{*}f_{2}^{*}-m_{11}^{*}m_{22}^{*}+m_{12}^{*}m_{21}^{*})(g_{1}g_{2}-n_{11}n_{22}+n_{12}n_{21})$ ## References * [1] M. Bajdich, L. Mitas and L.K. Wagner, Phys. Rev B77, 115112 (2008) * [2] C.K. Thomas and A. A. Middleton, Phys. Rev E80, 046708 (2009) * [3] J.-M. Stéphan, S. Furukawa, G. Misguich, and V. Pasquier, Phys. Rev B80, 184421 (2009) * [4] F.A. Berezin, _The Method of Second Quantization_ (Academic Press, New York, 1966) * [5] Y. Ohnuki, and T. Kashiwa, Prog. Theor. Phys. 60, 548 (1978). * [6] John R. Klauder, Bo-Sture Skagerstam, Coherent states: applications in physics and mathematical physics (World Scientific, Singapore, 1985) * [7] G.H. Lang, C.W. Johnson, S.E. Koonin, and W.E. Ormand, Phys. Rev. C48, 1518 (1993) * [8] L.M. Robledo, Phys. Rev. C79, 021302 (2009) * [9] K. Neergard, and E. Wüst, Nucl. Phys. A402, 311 (1983) * [10] E.R. Caianiello, _Combinatorics and renormalization in Quantum Field Theory_ (W.A. Benjamin, Massachusetts, 1973) * [11] G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, Baltimore, 1996). * [12] J. R. Bunch, Math of Comp. 38, 475 (1982) * [13] F. Zhang Ed., The Schur Complement and Its Applications (Numerical Methods and Algorithms) (Springer, Berlin, 2005) * [14] J. J. Dongarra, J. Du Croz, S. Hammarling, and R. J. Hanson, ACM Trans. Math. Soft. 14, 1 (1988), * [15] J. O. Aasen, BIT 11, 233 (1971)
arxiv-papers
2010-12-22T16:05:07
2024-09-04T02:49:15.916797
{ "license": "Public Domain", "authors": "C. Gonz\\'alez-Ballestero, L.M. Robledo and G. F. Bertsch", "submitter": "Luis Robledo", "url": "https://arxiv.org/abs/1012.5022" }
1012.5043
# Near-Optimal and Explicit Bell Inequality Violations Harry Buhrman CWI and University of Amsterdam, buhrman@cwi.nl. Supported by a Vici grant from NWO, and EU-grant QCS. Oded Regev Blavatnik School of Computer Science, Tel Aviv University, and CNRS, ENS Paris. Supported by the Israel Science Foundation, by the Wolfson Family Charitable Trust, and by a European Research Council (ERC) Starting Grant. Part of the work done while a DIGITEO visitor in LRI, Orsay. Giannicola Scarpa CWI Amsterdam, g.scarpa@cwi.nl. Supported by a Vidi grant from NWO, and EU-grant QCS. Ronald de Wolf CWI and University of Amsterdam, rdewolf@cwi.nl. Supported by a Vidi grant from NWO, and EU-grant QCS. ###### Abstract Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension $n$ (e.g., $\log n$ EPR-pairs), while we show that the winning probability of any classical strategy differs from $\frac{1}{2}$ by at most $O(\log n/\sqrt{n})$. The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here $n$-dimensional entanglement allows to win the game with probability $1/(\log n)^{2}$, while the best winning probability without entanglement is $1/n$. This near-linear ratio (“Bell inequality violation”) is near-optimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players. ## 1 Introduction One of the most striking features of quantum mechanics is the fact that _entangled_ particles can exhibit correlations that cannot be reproduced or explained by classical physics, i.e., by “local hidden-variable theories.” This was first noted by Bell [Bel64] in response to Einstein-Podolsky-Rosen’s challenge to the completeness of quantum mechanics [EPR35]. Experimental realization of such correlations is the strongest proof we have that nature does not behave according to classical physics: nature cannot simultaneously be “local” (meaning that information doesn’t travel faster than the speed of light) and “realistic” (meaning that properties of particles such as its spin always have a definite—if possibly unknown—value). Many such experiments have been done. All behave in accordance with quantum predictions, though so far none has closed all “loopholes” that would allow some (usually very contrived) classical explanation of the observations based on imperfect behavior of, for instance, the photon detectors used. Here we study quantitatively how much such “quantum correlations” can deviate from what is achievable classically. The setup for a game $G$ is as follows. Two space-like separated parties, called Alice and Bob, receive inputs $x$ and $y$ according to some fixed and known probability distribution $\pi$, and are required to produce outputs $a$ and $b$, respectively. There is a predicate specifying which outputs $a,b$ “win” the game on inputs $x,y$. The definition of the game $G$ consists of this predicate and the distribution $\pi$. The goal is to design games where entanglement-based strategies have much higher winning probability than the best classical strategy. While this setting is used to study non-locality in physics, the same set-up is also used extensively to study the power of entanglement in computer science contexts like multi-prover interactive proofs [KKM+08, KKMV08], parallel repetition [CSUU07, KRT08], and cryptography. Quantum strategies start out with an arbitrary fixed entangled state. No communication takes place between Alice and Bob. For each input $x$ Alice has a measurement, and for each input $y$ Bob has a measurement. They apply the measurements corresponding to $x$ and $y$ to their halves of the entangled state, producing classical outputs $a$ and $b$, respectively. Their goal is to maximize the winning probability. The _entangled value_ $\omega^{*}(G)$ of the game is the supremum of the expected winning probability, taken over all entangled strategies. When restricting to strategies that use entanglement of local dimension $n$, the value is denoted $\omega^{*}_{n}(G)$. This should be contrasted with the _classical value_ $\omega(G)=\omega^{*}_{0}(G)$ of the game, which is the maximum among all classical, non-entangled strategies. Shared randomness between the two parties is allowed, but is easily seen not to be beneficial. The remarkable fact, alluded to above, that some “quantum correlations” cannot be simulated classically, corresponds to the fact that there are games $G$ where the entangled value $\omega^{*}(G)$ is strictly larger than the classical value $\omega(G)$. For reasons explained in Section 2, such examples are known in the physics literature as “Bell inequality violations.” The CHSH game is one particularly famous example [CHSH69]. Here the inputs $x\in\\{0,1\\}$ and $y\in\\{0,1\\}$ are uniformly distributed, and Alice and Bob win the game if their respective outputs $a\in\\{0,1\\}$ and $b\in\\{0,1\\}$ satisfy $a\oplus b=x\wedge y$; in other words, $a$ should equal $b$ unless $x=y=1$. The classical value of this game is $\omega(G)=3/4$, while the entangled value is $\omega^{*}(G)=\cos(\pi/8)^{2}\approx 0.85$. The entangled value is achieved already with 2-dimensional entanglement (i.e., one EPR-pair), so $\omega^{*}(G)=\omega^{*}_{2}(G)$ for this game [Tsi87]. In the physics literature it is common to quantify the violation demonstrated by a given game $G$ by the ratio of entangled and classical values. The larger this ratio the better, both for philosophical reasons (to show the divergence between classical and quantum worlds) and for practical reasons (a larger violation is typically more noise-resistant and easier to realize in the noisy circumstances of an actual laboratory). To be precise, Bell violations are defined for a slight generalization of the notion of a game, which we call a _Bell functional_ 111There is unfortunately no good name for this notion in the literature. It is often called “Bell inequality,” but this is a misnomer since (1) it is not an inequality, and (2) the term Bell inequality is used to describe upper bounds on the classical value., where roughly speaking, some outputs might lead to a loss (or a negative gain) to the players. Since this discussion is not too relevant for our main results, we postpone it to Section 2. For now, suffice it to say that when considering the violation exhibited by a game, instead of comparing the maximum winning probabilities as we did above, one can also compare the maximum achievable deviation of the winning probability from (say) $1/2$. This is exactly what we will do in our first game in Section 3. In two recent papers, Junge et al. [JPP+10, JP11] studied how large a Bell inequality violation one can obtain. In terms of upper bounds, [JPP+10] proved that the maximum Bell inequality violation $\omega^{*}_{n}(G)/\omega(G)$ obtainable with entangled strategies of local dimension $n$, is at most $O(n)$, and [JP11, Theorem 6.8] proved that if Alice and Bob have at most $k$ possible outputs each, then the violation $\omega^{*}(G)/\omega(G)$ is at most $O(k)$, irrespective of the amount of entanglement they can use. (This improved an earlier $O(k^{2})$ upper bound due to Degorre et al. [DKLR09], and was also obtained for the special case of games by Dukaric [Duk10, Theorem 4].) These upper bounds hold for all Bell functionals, and not just for games. In terms of lower bounds, [JPP+10] showed the existence of a Bell inequality violation of order $\sqrt{n}/(\log n)^{2}$, where $n$ is both the entanglement-dimension and the number of outputs of Alice and Bob. This was recently improved to $\sqrt{n}/\log n$ in [JP11]. Both constructions are probabilistic, and the proofs show that with high probability the constructed Bell functionals exhibit a large violation. Their proofs are heavily based on connections to the mathematically beautiful areas of Banach spaces and operator spaces, but as a result are arguably somewhat inaccessible to those unfamiliar with these areas, and it is difficult to get a good intuition for them. (It is actually possible to analyze their game and reprove many of their results—often with improved parameters—using elementary probabilistic techniques [Reg11].) Our main result in this paper is to exhibit two stronger and fully explicit Bell inequality violations. Interestingly, both of our games address a question in theoretical physics but are inspired by theoretical computer science (communication complexity and unique games, respectively), and the tools used to analyze them are very much the tools from theoretical computer science. In fact, one aim of this paper is to export our techniques to mathematical physics. In the remainder of this introduction we provide an overview of our two non- local games, followed by some discussion and comparison. ### 1.1 The Hidden Matching game The “Hidden Matching” problem was introduced in quantum communication complexity by Bar-Yossef et al. [BJK08], and many variants of it were subsequently studied [GKRW09, GKK+08, Gav09]. The original version is as follows. Let $n$ be a power of 2. Alice is given input $x\in\\{0,1\\}^{n}$ and Bob is given a perfect matching $M$ (i.e., a partition of $[n]$ into $n/2$ disjoint pairs $(i,j)$). Both inputs are uniformly distributed.222All our results also hold with minor modifications for the case that Bob’s matching is chosen uniformly from the set $\\{M_{k}~{}|~{}k\in\\{0,\ldots,n/2-1\\}\\}$, where the matching $M_{k}$ consists of the pairs $(i,j)$ where $i\leq n/2$ and $j=n/2+1+(i+k-1\mbox{ mod }n/2)$. This has the advantage of lowering the number of possible inputs to Bob to $n/2$. We allow one-way communication from Alice to Bob, and Bob is required to output a pair $(i,j)\in M$ and a bit $v\in\\{0,1\\}$. They win if $v=x_{i}\oplus x_{j}$. In Section 3.1 we show that if Alice sends Bob a $c$-bit message, then their optimal winning probability is $\frac{1}{2}+\Theta(\frac{c}{\sqrt{n}})$. Bar- Yossef et al. [BJK08] earlier proved this for $c=\Theta(\sqrt{n})$, using information theory. However, their tools seem unable to give good bounds on the success probability for much smaller $c$. Instead, the main mathematical tool we use in our analysis is the so-called “KKL inequality” [KKL88] from Fourier analysis of Boolean functions (see [O’D08, Wol08] for surveys of this area). Roughly speaking, this inequality implies that if the message that Alice sends about $x$ is short, then Bob will not be able to predict the parity $x_{i}\oplus x_{j}$ well for many $(i,j)$ pairs. His matching $M$ is uniformly distributed, independent of $x$, and contains only $n/2$ of all ${n\choose 2}$ possible $(i,j)$ pairs. Hence it is unlikely that he can predict any one of those $n/2$ parities well. The KKL inequality was used before to analyze another variant of Hidden Matching in [GKK+08], though their analysis is different and more complicated because their variant is a promise problem with a non-product input distribution. The non-local game based on Hidden Matching is as follows: the inputs $x$ and $M$ are the same as before, but now Alice and Bob don’t communicate. Instead, Alice outputs an $a\in\\{0,1\\}^{\log n}$, Bob outputs $d\in\\{0,1\\}$ and $(i,j)\in M$, and they win the game if the outputs satisfy the relation $(a\cdot(i\oplus j))\oplus d=x_{i}\oplus x_{j}$, where the dot indicates inner product (modulo 2) of two $\log n$-bit strings. Observe that Alice has $n$ possible outputs and Bob also has $2\cdot n/2=n$ possible outputs. A classical strategy that wins this game induces a protocol for the original Hidden Matching problem with communication $c=\log n$ bits and the same winning probability $p$: Alice sends Bob the $\log n$-bit output $a$ from the non-local strategy, allowing Bob to compute $v=(a\cdot(i\oplus j))\oplus d$. Since $v=x_{i}\oplus x_{j}$ with probability $p$, Bob can now output $(i,j),v$. Hence our bound for the original communication problem implies that no classical strategy can win with probability that differs from $1/2$ by more than $O(\frac{\log n}{\sqrt{n}})$. In contrast, there is a strategy that wins with probability 1 using $\log n$ EPR-pairs, which shows $\omega^{*}_{n}(G)=1$.333The reader might be a bit confused by the seeming overloading of the meaning of ‘$n$’. Formally, ‘$n$’ is a parameter in the specification of the game. As it happens, for both of our games it’s also the number of possible outputs for each player, _and_ the local dimension of the entangled state that our quantum protocol uses (though we don’t claim that this entanglement-dimension $n$ is needed to achieve the best-possible entangled value). This game therefore exhibits a Bell violation of $\Omega(\sqrt{n}/\log n)$ (by measuring the maximal deviation of the winning probability from $1/2$). This order is the same as that obtained by Junge et al. [JPP+10, JP11], but our game is fully explicit and arguably simpler (which would help any future experimental realization). One might feel that our reduction to a communication complexity lower bound is responsible for losing the $\log n$ factor; however in Theorem 6 we exhibit a classical strategy with winning probability $1/2+\Omega(\sqrt{\log(n)/n})$. This shows that at least the square root of the log-factor is really necessary. ### 1.2 The Khot-Vishnoi game Our second non-local game derives from the work of Khot and Vishnoi [KV05] on the famous _Unique Games Conjecture_ (UGC), which was introduced by Khot [Kho02]. The UGC is a hardness-of-approximation assumption for a specific graph labeling problem, the details of which need not concern us here. The conjecture implies many other hardness-of-approximation results that do not seem obtainable using the more standard techniques based on the PCP theorem. Khot and Vishnoi exhibited a large integrality gap for the standard semidefinite programming (SDP) relaxation of this labeling problem, showing that at least SDP-solvers will not be able to efficiently approximate the value of the optimal labeling. We use essentially the same set-up for our game, though for our purposes we will not have to worry about SDPs or the UGC. Kempe, Regev, and Toner [KRT08] already observed that they could combine their “quantum rounding” technique with the game of [KV05] to get a Bell inequality violation of $n^{\varepsilon}$ for some small constant $\varepsilon>0$, where $n$ is the entanglement dimension and the number of possible outputs. Our main contribution in the second part of this paper is a refined (and at the same time simpler) analysis of both the Khot-Vishnoi game and of the quantum rounding technique, showing that, somewhat surprisingly, nearly optimal violations can be obtained using this method. The game is parameterized by an integer $n$, which we assume to be a power of 2, and a “noise-parameter” $\eta\in[0,1/2]$. Consider the group $\\{0,1\\}^{n}$ of all $n$-bit strings with ‘$\oplus$’ denoting bitwise addition mod 2, and let $H$ be the subgroup containing the $n$ Hadamard codewords. This subgroup partitions $\\{0,1\\}^{n}$ into $2^{n}/n$ cosets of $n$ elements each. Alice receives a uniformly random coset $x$ as input, which we can think of as $u\oplus H$ for uniformly random $u\in\\{0,1\\}^{n}$. Bob receives a coset $y$ obtained from Alice’s by adding a string of low Hamming weight, namely $y=x\oplus z=u\oplus z\oplus H$, where each bit of $z\in\\{0,1\\}^{n}$ is set to 1 with probability $\eta$, independently of the other bits. Notice that addition of $z$ gives a natural bijection between the two cosets, mapping each element of the first coset to a relatively nearby element of the second coset; namely, the distance between the two elements is the Hamming weight of $z$, which is typically around $\eta n$. Each player is supposed to output one element from its coset, and their goal is for their elements to match under the bijection. In other words, Alice outputs an element $a\in x$, Bob outputs $b\in y$, and they win the game iff $a\oplus b=z$.444Note that the winning condition for this game is a “randomized predicate”, as there are $n$ possible predicates (one for each $z$ in $x\oplus y$) corresponding to each pair of inputs $x,y$. However, it is easy to see that with very high probability exactly one of these $n$ constraints dominates (namely, the one corresponding to a $z$ of Hamming weight around $\eta n$). This allows one to modify the game in a straightforward manner, making it a game with a deterministic predicate (although there is usually no reason to do so). Notice that the number of possible inputs to each player is $2^{n}/n$ and the number of possible outputs for each player is $n$. Based on the integrality gap analysis of Khot and Vishnoi, we show that no classical strategy can win this game with probability greater than $1/n^{\eta/(1-\eta)}$. We also sketch a classical strategy that achieves roughly this winning probability. In contrast, using a simplified version of the “quantum rounding” technique of [KRT08], we exhibit a quantum strategy that uses the $n$-dimensional maximally entangled state and wins with probability at least $(1-2\eta)^{2}$. This strategy follows from the observation that each coset of $H$ defines an orthonormal basis of $\mathbb{R}^{n}$ in which we can do a measurement. Summarizing, we have entangled value $\omega^{*}_{n}(G)\geq(1-2\eta)^{2}$ and classical value $\omega(G)\leq 1/n^{\eta/(1-\eta)}$ for this game. Setting the noise-rate to $\eta=1/2-1/\log n$, the entangled value is roughly $1/(\log n)^{2}$ while the classical value is roughly $1/n$, leading to a Bell inequality violation $\omega^{*}_{n}(G)/\omega(G)=\Omega(n/(\log n)^{2})$. Up to the polylogarithmic factor, this is optimal both in terms of the local dimension, and in terms of the number of possible outputs. ### 1.3 Discussion and open problems The main advantage of the Khot-Vishnoi game is its strong, near-linear Bell inequality violation of about $n/(\log n)^{2}$. This is quadratically stronger than the violation of $\sqrt{n}/\log n$ given by Hidden Matching and by [JP11] and almost matches the $O(n)$ bound proved in [JPP+10] for arbitrary Bell inequality violations with entangled states of local dimension $n$. One open question is to tweak the KV game (or define another game) to get rid of the polylogarithmic term in the ratio, making it optimal up to a constant factor. A second advantage of KV over HM is that the value is just the winning probability rather than the bias from $1/2$. This might make the KV game more relevant for physical non-locality experiments, as it is the probability of winning that matters most there. This also means that the corresponding Bell functional (see Section 2) is nonnegative, a case that has been investigated in [JP11]. One advantage of HM over KV is that the entangled strategy wins the game with probability 1. In contrast, the entangled strategy for KV wins only with probability about $1/(\log n)^{2}$, which means any quantum experiment needs to be repeated about $(\log n)^{2}$ times before we expect to see the first win. Another advantage is that HM’s description is a bit simpler than KV’s. Throughout this paper we considered the Bell violation as a function of the number of outputs of the players and/or of the dimension of entanglement. One can also analyze the violation in terms of the number of possible _inputs_. We recall that in the KV game both players have inputs taken from an exponentially large set, and that in the HM game (when modified as in Footnote 2) Bob has only $n/2$ possible inputs, but Alice still has an exponentially large set of inputs. The Bell inequality violation of $\sqrt{n}/\log n$ presented by Junge and Palazuelos [JP11] has the advantage that the number of inputs is only $O(n)$. Accordingly, another open question presents itself: can we find a game with a (near-)linear Bell inequality violation, and linear number of inputs and outputs for both Alice and Bob? Finally, while this paper focuses on the two-party setting, obtaining stronger Bell inequality violations for settings with three or more parties is also a worthy goal. Pérez-García et al. [PWP+08] gave a randomized construction of a three-party _XOR game_ (in such a game each party outputs a bit, and winning or losing depends only on the XOR of those three bits) that gives a Bell inequality violation of roughly $\sqrt{d}$ using an entangled state in dimensions $d\times D\times D$ (with $D\gg d$).555They also showed that using GHZ states cannot give a superconstant Bell inequality violation for XOR games (see also [BBLV09]). In contrast, it is a known consequence of Grothendieck’s inequality that such non-constant separations do not exist for _two_ -party XOR games. We do not know how large Bell inequality violations can be for arbitrary three-party games. Note that it is easy to make a three-party version of Hidden Matching: Alice gets input $x\in\\{0,1\\}^{n}$, Bob gets input $y\in\\{0,1\\}^{n}$, and Charlie gets a matching $M$ as input (all uniformly distributed). The goal is that Alice outputs $a\in\\{0,1\\}^{\log n}$, Bob outputs $b\in\\{0,1\\}^{\log n}$, Charlie outputs $d\in\\{0,1\\}$ and $(i,j)\in M$, such that $((a\oplus b)\cdot(i\oplus j))\oplus d=x_{i}\oplus x_{j}\oplus y_{i}\oplus y_{j}$. By modifying the two-party proofs in this paper, it is not hard to show that the winning probability using an $n$-dimensional GHZ state is 1, while the best classical winning probability deviates from $1/2$ by at most $(\log n)^{2}/n$. So going from two to three parties roughly squares the Bell inequality violation for Hidden Matching. This improvement unfortunately does not scale up with more than three parties, because one can show the classical winning probability is always at least $1/2+\Omega(1/n)$. ## 2 A more formal look at Bell violations Before we precisely analyze the two games mentioned above, let us first say something more about the mathetical treatment of general Bell inequalities. Readers who are happy with the above (more concrete) approach in terms of winning probabilities of games, may safely skip this section. Consider a game with $n$ possible inputs to each player and $k$ possible outputs. The observed behavior of the players (whether they use a classical or an entangled strategy) can be summarized in terms of $n^{2}$ probability distributions, each on the set $[k]\times[k]$. We denote by $P(ab|xy)$ the probability of producing outputs $a$ and $b$ when given inputs $x$ and $y$. As described in the introduction, a game is defined by a probability distribution $\pi$ on the input set $[n]\times[n]$, as well as a (possibly randomized) predicate on $[k]\times[k]$ for each input pair $(x,y)$.The winning probability of the players can be written as $\langle{M},{P}\rangle=\sum_{abxy}M^{ab}_{xy}P(ab|xy).$ where $M_{xy}^{ab}$ is defined as the probability of the input pair $(x,y)$ multiplied by the probability that the output pair $(a,b)$ is accepted on this input pair. We call $M=(M_{xy}^{ab})$ the _Bell functional corresponding to the game_. More generally, a _Bell functional_ is an arbitrary tensor $M=(M_{xy}^{ab})$ containing $n^{2}k^{2}$ real numbers. We define the _classical value_ of a Bell functional $M$ as $\omega(M)=\sup_{P}|\langle{M},{P}\rangle|,$ where the supremum is over all classical strategies. Similarly, the _entangled value_ of $M$ is defined as $\omega^{*}(M)=\sup_{P}|\langle{M},{P}\rangle|,$ where the supremum now is over all quantum strategies (using an entangled state of arbitrary dimension). If the entangled state is restricted to local dimension $n$, the value is denoted $\omega^{*}_{n}(M)$. We note that if $M$ is the Bell functional corresponding to a game, then these definitions coincide with our definitions from the introduction (and in this case the absolute value is unnecessary since $M$ is non-negative). A _Bell inequality_ is an upper bound on $\omega(M)$ for some Bell functional $M$; it shows a limitation of _classical_ strategies.666An upper bound on $\omega^{*}(M)$ is known as a _Tsirelson inequality_ , and shows a limitation of entangled strategies. The _Bell inequality violation_ demonstrated by a Bell functional $M$ is defined as the ratio between the entangled and the classical value $\frac{\omega^{*}(M)}{\omega(M)}.$ This provides a convenient quantitative way to measure the extra power provided by entangled strategies. This definition of Bell violation enjoys a rich mathematical structure (as witnessed by the numerous connections found to Banach space and operator space theory [JPP+10, JP11, Duk10]), and also has a beautiful geometrical interpretation as the “distance” between the set of all classical strategies and the set of all quantum strategies (see Section 6.1 in [JP11]). Clearly, any game $G$ for which $\omega^{*}(G)\geq K\omega(G)$ gives a Bell violation of $K$ by just taking the functional corresponding to $G$. A more interesting case is when we consider the largest deviation of the winning probability from (say) $1/2$. We claim that if $G$ is a game for which the winning probability of any classical strategy cannot deviate from $1/2$ by more than $\delta_{1}$ and, moreover, there is a quantum strategy obtaining winning probability at least $1/2+\delta_{2}$, then we obtain a Bell violation of $\delta_{2}/\delta_{1}$. To see why, let $M$ be the functional corresponding to the game, and let $M^{\prime}$ be the functional obtained by subtracting from each $M_{xy}^{ab}$ half the probability of input pair $(x,y)$. Then it is easy to see that for each strategy $P$, $\langle{M^{\prime}},{P}\rangle=\langle{M},{P}\rangle-1/2$. Hence, $\omega(M^{\prime})$ and $\omega^{*}(M^{\prime})$ measure the largest possible deviation of the winning probability from $1/2$ of classical and entangled strategies, respectively. The claim follows. The converse to this statement is also true: any Bell functional can be converted to a game (by simply scaling and adding a constant) in such a way that the Bell violation demonstrated by the functional is equal to the ratio between the largest possible deviation of winning probability from $1/2$ obtainable by classical and entangled strategies. ## 3 Hidden Matching problem In this section we define and analyze the Hidden Matching game. ### 3.1 The Hidden Matching problem in communication complexity While our focus is non-locality, it will actually be useful to first study the original version of the Hidden Matching problem in the context of protocols where communication from Alice to Bob is allowed. Both the problem and the efficient quantum protocol below come from [BJK08]. ###### Definition 1 (Hidden Matching (HM)). Let $n$ be a power of 2 and ${\cal M}_{n}$ be the set of all perfect matchings on the set $[n]=\\{1,\ldots,n\\}$ (a perfect matching is a partition of $[n]$ into $n/2$ disjoint pairs $(i,j)$). Alice is given $x\in\\{0,1\\}^{n}$ and Bob is given $M\in{\cal M}_{n}$, distributed according to the uniform distribution. We allow one-way communication from Alice to Bob, and Bob outputs an $(i,j)\in M$ and $v\in\\{0,1\\}$. They win if $v=x_{i}\oplus x_{j}$. ###### Theorem 1. There is a protocol for HM with $\log n$ qubits of one-way communication that wins with probability 1 (i.e., $v=x_{i}\oplus x_{j}$ always holds). ###### Proof. The protocol is the following: 1. 1. Alice sends Bob the state $|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}(-1)^{x_{i}}|i\rangle$. 2. 2. Bob measures $|\psi\rangle$ in the $n$-element basis $B=\\{\frac{1}{\sqrt{2}}(|i\rangle\pm|j\rangle)\mid(i,j)\in M\\}$. If the outcome of the measurement is a state $\frac{1}{\sqrt{2}}(|i\rangle+|j\rangle)$ then Bob outputs $(i,j)$ and $v=0$. If the outcome of the measurement is a state $\frac{1}{\sqrt{2}}(|i\rangle-|j\rangle)$, Bob outputs $(i,j)$ and $v=1$. For each $(i,j)\in M$, the probability to get $\frac{1}{\sqrt{2}}(|i\rangle+|j\rangle)$ is $2/n$ if $x_{i}\oplus x_{j}=0$ and 0 otherwise, and similarly for $\frac{1}{\sqrt{2}}(|i\rangle-|j\rangle)$. Hence Bob’s output is always correct. ∎ #### 3.1.1 Limits of classical protocols for HM Here we show that classical protocols with little communication cannot have good success probability. To start, note that a protocol that uses shared randomness is just a probability distribution over deterministic protocols, hence the maximal winning probability is achieved by a deterministic protocol. ###### Theorem 2. Every classical deterministic protocol for HM with $c$ bits of one-way communication, where Bob outputs $(i,j),v$, has $\Pr[v=x_{i}\oplus x_{j}]\leq\frac{1}{2}+O\left(\frac{c}{\sqrt{n}}\right).$ The intuition behind the proof is the following. If the communication $c$ is small, the set $X_{m}$ of inputs $x$ for which Alice sends message $m$, will typically be large (of size about $2^{n-c}$), meaning Bob has little knowledge of most of the bits of $x$. By the KKL inequality, this implies that for most of the ${n\choose 2}$ $(i,j)$-pairs, Bob cannot guess the parity $x_{i}\oplus x_{j}$ well. Of course, Bob has some freedom in which $(i,j)$ he outputs, but that freedom is limited to the $n/2$ $(i,j)$-pairs in his matching $M$, and it turns out that on average he will not be able to guess any of those parities well. ###### Proof. Fix a classical deterministic protocol. For each $m\in\\{0,1\\}^{c}$, let $X_{m}\subseteq\\{0,1\\}^{n}$ be the set of Alice’s inputs for which she sends message $m$. These sets $X_{m}$ together partition Alice’s input space $\\{0,1\\}^{n}$. Define $p_{m}=\frac{|X_{m}|}{2^{n}}$. Note that $\sum_{m}p_{m}=1$, so $\\{p_{m}\\}$ is a probability distribution over the $2^{c}$ messages $m$. Define $\varepsilon$ such that $\Pr_{\mathcal{U}}[v=x_{i}\oplus x_{j}]=\frac{1}{2}+\varepsilon$, and $\varepsilon_{m}$ such that $\Pr_{\mathcal{U}}[v=x_{i}\oplus x_{j}\mid\mbox{Bob received }m]=\frac{1}{2}+\varepsilon_{m}$. Then $\varepsilon=\sum_{m}p_{m}\varepsilon_{m}$. For each $m$, define the following probability distribution over all $(i,j)\in[n]^{2}$: $q_{m}(i,j)=\Pr_{M\in{\cal M}_{n}}[\mbox{Bob outputs }(i,j)\mid\mbox{Bob received }m].$ We have $q_{m}(i,j)\leq\frac{1}{n-1}$, because we assume Bob always outputs an element from $M$ and for fixed $i\neq j$ we have $\Pr_{M}[(i,j)\in M]=1/(n-1)$ (each $j$ is equally likely to be paired up with $i$). The best Bob can do when guessing $x_{i}\oplus x_{j}$ given message $m$, is to output the value of $x_{i}\oplus x_{j}$ that occurs most often among the $x\in X_{m}$. Define $\beta^{m}_{ij}=\mathop{\mathbb{E}}_{x\in X_{m}}[(-1)^{x_{i}}\cdot(-1)^{x_{j}}]$. The fraction of $x\in X_{m}$ where $x_{i}\oplus x_{j}=0$ is $1/2+\beta^{m}_{ij}/2$, hence Bob’s optimal success probability when guessing $x_{i}\oplus x_{j}$ is $1/2+|\beta^{m}_{ij}|/2$. This implies, for fixed $m$, $\mathop{\mathbb{E}}_{(i,j)\sim q_{m}}\left[\frac{1}{2}+\frac{|\beta^{m}_{ij}|}{2}\right]\geq\Pr_{\stackrel{{\scriptstyle x\in X_{m}}}{{M\in{\cal M}_{n}}}}[v=x_{i}\oplus x_{j}]=\frac{1}{2}+\varepsilon_{m},$ where the notation $x\sim Q$ stands for “$x$ chosen according to probability distribution $Q$.” As explained in [Wol08, Section 4.1], it follows from the KKL inequality [KKL88] that $\sum_{i,j:i\neq j}(\beta^{m}_{ij})^{2}\leq O\left(\log\frac{1}{p_{m}}\right)^{2}.$ (1) This allows us to upper bound $\varepsilon_{m}$: $2\varepsilon_{m}\leq\mathop{\mathbb{E}}_{(i,j)\sim q_{m}}[|\beta^{m}_{ij}|]=\sum_{i,j}q_{m}(i,j)|\beta^{m}_{ij}|\stackrel{{\scriptstyle(*)}}{{\leq}}\sqrt{\sum_{i,j}q_{m}(i,j)^{2}}\cdot\sqrt{\sum_{i,j}(\beta^{m}_{ij})^{2}}\stackrel{{\scriptstyle(**)}}{{\leq}}\frac{1}{\sqrt{n-1}}\cdot O\left(\log\frac{1}{p_{m}}\right),$ where $(*)$ is Cauchy-Schwarz and $(**)$ follows from $\sum_{i,j}q_{m}(i,j)^{2}\leq\max_{i,j}q_{m}(i,j)\cdot\sum_{i,j}q_{m}(i,j)\leq\max_{i,j}q_{m}(i,j)\leq\frac{1}{n-1}$ and Eq. (1). Now we can bound $\varepsilon$: $\varepsilon=\sum_{m}p_{m}\varepsilon_{m}\leq\sum_{m}p_{m}\frac{O(\log(1/p_{m}))}{\sqrt{n-1}}=\frac{1}{\sqrt{n-1}}\sum_{m}p_{m}O(\log(1/p_{m}))=\frac{1}{\sqrt{n-1}}O(H(p))=O\left(\frac{c}{\sqrt{n}}\right)$ where $H$ denotes the binary entropy function, and $H(p)\leq c$ since the distribution $\\{p_{m}\\}$ is on $2^{c}$ elements. ∎ ### 3.2 Classical protocol for HM Here we design a classical protocol that achieves the above upper bound on the success probability. This protocol has no bearing on the large Bell inequality violations that are our main goal in this paper, but it is nice to know the previous upper bound on the maximal success probability is essentially tight. ###### Theorem 3. For every positive integer $c\leq\sqrt{n}$, there exists a classical protocol for HM with $c$ bits of one-way communication, such that for all inputs $x,M$, $\Pr[v=x_{i}\oplus x_{j}]=\frac{1}{2}+\Omega\left(\frac{c}{\sqrt{n}}\right).$ ###### Proof. Assume for simplicity that $c$ is even and sufficiently large. Alice and Bob use shared randomness to choose two disjoint subsets $S_{1},S_{2}$ of $[n]$ of size $\sqrt{n}$ each. Let $y$ denote the bits of $x$ located in the indices given by the first subset, and $z$ the bits located in the indices given by the second subset. Alice and Bob use shared randomness to produce $2^{c/2}$ random $\sqrt{n}$-bit strings $y^{(1)},\ldots,y^{(2^{c/2})}$. For each $\ell$, the distance $d(y,y^{(\ell)})$ is distributed binomially, as the sum of $\sqrt{n}$ fair coin flips. There exists a constant $\delta>0$ such that $\Pr[d(y,y^{(\ell)})\leq\sqrt{n}/2-\beta n^{1/4}]\geq 2^{-\delta\beta^{2}}$ (this can be seen for instance by estimating ${k\choose k/2-\beta\sqrt{k}}$ using Stirling’s approximation). Hence with probability close to 1, there will be an $\ell$ such that $y$ and $y^{(\ell)}$ are at relative distance $\leq 1/2-\Omega(c^{1/2}/n^{1/4})$. If so, Alice sends Bob the first such $\ell$, and otherwise she tells him there is no such $\ell$. This costs $c/2$ bits of communication. Similarly, at the expense of another $c/2$ bits of communication, Bob obtains an approximation of $z$ with relative distance at most $\leq 1/2-\Omega(c^{1/2}/n^{1/4})$. It is easy to see that with probability at least 1/2, Bob’s matching $M$ contains an $(i,j)$ with $i$ in $S_{1}$ and $j$ in $S_{2}$. Bob can predict $x_{i}$ with success probability $1/2+\Omega(c^{1/2}/n^{1/4})$ from his approximation of $y$, and can predict $x_{j}$ with success probability $1/2+\Omega(c^{1/2}/n^{1/4})$ from his approximation of $z$. These success probabilities are independent, hence he can predict $x_{i}\oplus x_{j}$ with success probability $1/2+\Omega(c/\sqrt{n})$. If there is no such $(i,j)\in M$, or if he didn’t get good approximations to $y$ or $z$, then Bob just outputs any $(i,j)\in M$ and a random bit for $v$, giving success probability $1/2$. Putting everything together, we have a protocol that wins with probability $1/2+\Omega(c/\sqrt{n})$. ∎ ### 3.3 Non-local version of Hidden Matching, and a quantum protocol We now port our results to the non-local setting. The following non-local version of HM and the subsequent protocol for it were originally due to Buhrman, and related problems were studied in [GKRW09, Gav09]. ###### Definition 2 (Non-Local Hidden Matching ($\mbox{\rm HM}_{nl}$)). Let $n$ be a power of 2 and ${\cal M}_{n}$ be the set of all perfect matchings on the set $[n]$. Alice is given $x\in\\{0,1\\}^{n}$ and Bob is given $M\in{\cal M}_{n}$, distributed according to the uniform distribution. Alice’s output is a string $a\in\\{0,1\\}^{\log n}$ and Bob’s output is an $(i,j)\in M$ and $d\in\\{0,1\\}$. They win the game if and only if $(a\cdot(i\oplus j))\oplus d=x_{i}\oplus x_{j}.$ (2) ###### Theorem 4. There exists a quantum protocol for $\mbox{\rm HM}_{nl}$ using a maximally entangled state with local dimension $n$, such that condition (2) is always satisfied. ###### Proof. The protocol is as follows. Alice and Bob share $|\psi\rangle=\frac{1}{\sqrt{n}}\sum_{i\in\\{0,1\\}^{\log n}}|i\rangle|i\rangle.$ 1. 1. Alice performs a phase-flip according to her input $x$. The state becomes $\frac{1}{\sqrt{n}}\sum_{i\in\\{0,1\\}^{\log n}}(-1)^{x_{i}}|i\rangle|i\rangle$. 2. 2. Bob performs a projective measurement with projectors $P_{ij}=|i\rangle\langle i|+|j\rangle\langle j|$, with $(i,j)\in M$. The state collapses to $\frac{1}{\sqrt{2}}[(-1)^{x_{i}}|i\rangle|i\rangle+(-1)^{x_{j}}|j\rangle|j\rangle]$ for some $(i,j)\in M$ known to Bob. 3. 3. Both players apply Hadamard transforms $H^{\otimes\log{n}}$, and the state becomes $\frac{1}{\sqrt{2}n}\sum_{a,b\in\\{0,1\\}^{\log n}}\left((-1)^{x_{i}+a\cdot i+b\cdot i}+(-1)^{x_{j}+a\cdot j+b\cdot j}\right)|a\rangle|b\rangle.$ Notice that in the latter state, any pair $a,b$ with nonzero amplitude must satisfy that $(a\cdot(i\oplus j))\oplus(b\cdot(i\oplus j))=x_{i}\oplus x_{j}.$ Hence, if the players measure the state, Alice outputs $a$, and Bob outputs $(i,j)$ and the bit $d=b\cdot(i\oplus j)$, they win the game with certainty. ∎ ###### Theorem 5. The winning probability of any classical protocol for $\mbox{\rm HM}_{nl}$ differs from $\frac{1}{2}$ by at most $O\left(\log{n}/{\sqrt{n}}\right)$. ###### Proof. A protocol that wins $\mbox{\rm HM}_{nl}$ with success probability $1/2+\varepsilon$ can be turned into a protocol for HM with $\log{n}$ bits of communication and the same probability to win: the players play $\mbox{\rm HM}_{nl}$, with Alice producing $a$ and Bob producing $i,j,d$; Alice then sends $a$ to Bob, who outputs $i,j,(a\cdot(i\oplus j))\oplus d$. The latter bit equals $x_{i}\oplus x_{j}$ with probability $1/2+\varepsilon$. This requires $c=\log{n}$ bits of communication, so Theorem 2 gives the upper bound on the winning probability. The lower bound follows similarly. ∎ ### 3.4 Classical protocols for $\mbox{\rm HM}_{nl}$ Next we show that our upper bound on the success probability of classical strategies for $\mbox{\rm HM}_{nl}$ is nearly optimal: we can achieve advantage at least $\Omega(\sqrt{\log(n)/n})$. (In Appendix A we also give an alternative protocol with a slightly weaker advantage $\Omega(1/\sqrt{n})$.) ###### Theorem 6. There exists a classical deterministic protocol for $\mbox{\rm HM}_{nl}$ with winning probability $\frac{1}{2}+\Omega\left(\sqrt{\frac{\log n}{n}}\right)$ (under the uniform input distribution). ###### Proof. The protocol is as follows. Given $x$, Alice finds an $a\in\\{0,1\\}^{\log n}$ that maximizes $J_{ax}:=|\\{j\neq 1\mid a\cdot j=x_{1}\oplus x_{j}\\}|$. Bob outputs $(1,j)$, where $j$ is the unique element matched to 1 by $M$, and $d=0$. With these choices, and letting the number 1 correspond to the string $0^{\log n}$, the winning condition $(a\cdot(i\oplus j))\oplus d=x_{i}\oplus x_{j}$ is equivalent to $a\cdot j=x_{1}\oplus x_{j}$. Accordingly, for fixed $x$ and uniformly distributed $M$ (and hence uniformly distributed $j\in[n]\backslash\\{0^{\log n}\\}$), the winning probability equals $p_{x}:=\max_{a}J_{ax}/(n-1)$. Below we use the second moment method to show $\mathop{\mathbb{E}}_{x}[p_{x}]\geq 1/2+\Omega(\sqrt{\log(n)/n})$. For a fixed $a$ and uniformly random $x\in\\{0,1\\}^{n}$, $J_{ax}$ behaves like the sum of $n-1$ fair 0/1-valued coin flips. Let $Z_{a}$ be the indicator random variable for the event that $J_{ax}\geq(n-1)/2+\beta\sqrt{n}$. Choosing $\beta$ a sufficiently small constant multiple of $\sqrt{\log n}$, we have $\mathop{\mathbb{E}}_{x}[Z_{a}]\geq 1/\sqrt{n}$ for each $a$ (this follows from the $2^{-\delta\beta^{2}}$ probability lower bound mentioned in the proof of Theorem 3). Let $Z=\sum_{a}Z_{a}$, then $\mathop{\mathbb{E}}_{x}[Z]\geq\sqrt{n}$ by linearity of expectation. For $a\neq b$, the covariance $\mbox{Cov}[Z_{a},Z_{b}]$ is non-positive, informally because if $x$ is “well-aligned” with the string $(a\cdot j)_{j}$ then it’s less likely to be well-aligned with the orthogonal string $(b\cdot j)_{j}$. Hence we can bound the variance of $Z$: $\mbox{Var}[Z]=\sum_{a}\mbox{Var}[Z_{a}]+\sum_{a\neq b}\mbox{Cov}[Z_{a},Z_{b}]\leq\sum_{a}\mbox{Var}[Z_{a}]\leq\sum_{a}\mathop{\mathbb{E}}_{x}[Z_{a}^{2}]=\sum_{a}\mathop{\mathbb{E}}_{x}[Z_{a}]=\mathop{\mathbb{E}}_{x}[Z].$ By Chebyshev’s inequality, we have $\Pr[Z=0]\leq\Pr\left[|Z-\mathop{\mathbb{E}}_{x}[Z]|\geq\mathop{\mathbb{E}}_{x}[Z]\right]\leq\Pr\left[|Z-\mathop{\mathbb{E}}_{x}[Z]|\geq\sqrt{\mathop{\mathbb{E}}_{x}[Z]}\sqrt{\mbox{Var}[Z]}\right]\leq\frac{1}{\mathop{\mathbb{E}}_{x}[Z]}\leq\frac{1}{\sqrt{n}}.$ Hence with probability at least $1-1/\sqrt{n}$ over the choice of $x$, we have $Z>0$, meaning there is at least one $a$ with $J_{ax}\geq(n-1)/2+\beta\sqrt{n}$, and hence $p_{x}\geq 1/2+\Omega(\sqrt{\log(n)/n})$. This implies $\mathop{\mathbb{E}}_{x}[p_{x}]\geq\left(1-\frac{1}{\sqrt{n}}\right)\left(1/2+\Omega(\sqrt{\log(n)/n})\right)=1/2+\Omega(\sqrt{\log(n)/n}).$ ∎ ## 4 The Khot-Vishnoi game ### 4.1 The classical value In this section we analyze the classical value of the Khot-Vishnoi game. Our main result is an upper bound on the classical value of $1/n^{\eta/(1-\eta)}$, based on the analysis from [KV05]. Before we give that upper bound, let us first argue that it is essentially tight, i.e., there exists a strategy whose winning probability is roughly $1/n^{\eta/(1-\eta)}$. To get some intuition for this game, first think of $\eta$ as some small constant (even though we will eventually choose it close to $1/2$), and consider the following natural classical strategy: > Alice and Bob each output the element of their coset that has highest > Hamming weight. The idea is that if $a$ is the element of highest Hamming weight in Alice’s coset $x$, we expect $a\oplus z$ to also be of high Hamming weight (because it is close to $a$ in Hamming distance), and so Bob is somewhat likely to pick it. We now give a brief back-of-the-envelope calculation suggesting that the winning probability of this strategy is of order $1/n^{\eta/(1-\eta)}$; since it is not required for our main result, we will not attempt to make this argument rigorous. Let $t\geq 0$ be such that the probability that a binomial $B(n,1/2)$ variable is greater than $(n+t)/2$ is $1/n$. Recalling that a binomial distribution $B(n,p)$ can be approximated by the normal distribution $N(np,np(1-p))$, and that the probability that a normal variable is greater than its mean by $s$ standard deviations is approximately $e^{-s^{2}/2}$, we can essentially choose $t$ to be the solution to $e^{-t^{2}/(2n)}=1/n$ (so $t=\sqrt{2n\ln n}$). Then we expect Alice’s $n$-element coset to contain exactly one element of Hamming weight greater than $(n+t)/2$. Since the element $a$ that Alice picks is the one of highest Hamming weight, we assume for simplicity that its Hamming weight is $(n+t)/2$. The players win the game if and only if $a\oplus z$ has the highest weight among Bob’s $n$ elements, which we heuristically approximate by the event that $a\oplus z$ has Hamming weight at least $(n+t)/2$. The Hamming weight of $a\oplus z$ is distributed as the sum of $B((n+t)/2,1-\eta)$ and $B((n-t)/2,\eta)$, which can be approximated as above by the normal distribution $N((n+t)/2-\eta t,n\eta(1-\eta))$. Hence for the Hamming weight of $a\oplus z$ to be at least $(n+t)/2$, the normal variable needs to be greater than its mean by $\eta t/\sqrt{n\eta(1-\eta)}$ standard deviations, and the probability of this happening is approximately $e^{-\eta^{2}t^{2}/(2n\eta(1-\eta))}=1/n^{\eta/(1-\eta)}$, as claimed. Now we show that no classical strategy can be substantially better. The main technical tool used in the proof is the so-called Bonami-Beckner hypercontractive inequality, which is applicable to our setting because we choose $u$ uniform and $u\oplus z$ may be viewed as a “noisy version” of $u$. ###### Theorem 7. For any $n$ which is a power of 2, and any $\eta\in[0,1/2]$, every classical strategy for the Khot-Vishnoi game (as defined in Section 1.2) has winning probability at most $1/n^{\eta/(1-\eta)}$. ###### Proof. Recall that the inputs are generated as follows: we choose a uniformly random $u\in\\{0,1\\}^{n}$ and an $\eta$-biased $z\in\\{0,1\\}^{n}$, and define the respective inputs to be the cosets $u\oplus H$ and $u\oplus z\oplus H$. We can assume without loss of generality that Alice’s and Bob’s behavior is deterministic. Define functions $A,B:\\{0,1\\}^{n}\to\\{0,1\\}$ by $A(u)=1$ if and only if Alice’s output on $u\oplus H$ is $u$, and similarly for Bob. Notice that by definition, these functions attain the value $1$ on exactly one element of each coset. Recall that the players win if and only if the sum of Alice’s output and Bob’s output equals $z$. Hence for all $u,z$, $\sum_{h\in H}A(u\oplus h)B(u\oplus z\oplus h)$ is $1$ if the players win on input pair $u\oplus H,u\oplus z\oplus H$ and $0$ otherwise. Therefore, the winning probability is given by $\displaystyle\mathop{\mathbb{E}}_{u,z}\left[\sum_{h\in H}A(u\oplus h)B(u\oplus z\oplus h)\right]$ $\displaystyle=\sum_{h\in H}\mathop{\mathbb{E}}_{u,z}\left[A(u\oplus h)B(u\oplus z\oplus h)\right]$ $\displaystyle=n\mathop{\mathbb{E}}_{u,z}[A(u)B(u\oplus z)],$ where the second equality uses the fact that for all $h$, $u\oplus h$ is uniformly distributed. We use the framework of hypercontractivity (see e.g. [O’D08, Wol08]), which we briefly explain now. Specifically, for a function $F:\\{0,1\\}^{n}\rightarrow\mathbb{R}$, define its $p$-norm by ${\left\|{F}\right\|}_{p}=(\mathop{\mathbb{E}}_{x}[|F(u)|^{p}])^{1/p}$, where the expectation is uniform over all $u\in\\{0,1\\}^{n}$. The _noise-operator_ $T_{1-2\eta}$ adds “$\eta$-noise” to each of $F$’s input bits; more precisely, $(T_{1-2\eta}F)(u)=\mathop{\mathbb{E}}_{z}[F(u\oplus z)]$, where $z$ is an $\eta$-biased “noise string.” The linear operator $T_{\rho}$ is diagonal in the Fourier basis: it just multiplies each character function $\chi_{S}$ ($S\subseteq[n]$) by the factor $\rho^{|S|}$. It is easy to see that $\mathop{\mathbb{E}}_{u}[F(u)\cdot(T_{\rho}G)(u)]=\mathop{\mathbb{E}}_{u}[(T_{\sqrt{\rho}}F)(u)\cdot(T_{\sqrt{\rho}}G)(u)]$. The Bonami-Beckner inequality implies ${\left\|{T_{\rho}F}\right\|}_{2}\leq{\left\|{F}\right\|}_{1+\rho^{2}}$ for all $\rho\in[0,1]$. We now have, $\displaystyle\mathop{\mathbb{E}}_{u,z}[A(u)B(u\oplus z)]$ $\displaystyle=\mathop{\mathbb{E}}_{u}[A(u)(T_{1-2\eta}B)(u)]$ $\displaystyle=\mathop{\mathbb{E}}_{u}[(T_{\sqrt{1-2\eta}}A)(u)\cdot(T_{\sqrt{1-2\eta}}B)(u)]$ $\displaystyle\leq{\left\|{T_{\sqrt{1-2\eta}}A}\right\|}_{2}\cdot{\left\|{T_{\sqrt{1-2\eta}}B}\right\|}_{2}$ $\displaystyle\leq{\left\|{A}\right\|}_{2-2\eta}\cdot{\left\|{B}\right\|}_{2-2\eta}$ $\displaystyle=\left(\mathop{\mathbb{E}}_{u}[A(u)]\right)^{1/(2-2\eta)}\cdot\left(\mathop{\mathbb{E}}_{u}[B(u)]\right)^{1/(2-2\eta)}$ $\displaystyle=\frac{1}{n^{1/(1-\eta)}}.$ Here the first inequality is Cauchy-Schwarz, and the second is the hypercontractive inequality. We complete the proof by noting that $n/n^{1/(1-\eta)}=1/n^{\eta/(1-\eta)}$. ∎ ### 4.2 Lower bound on the entangled value In this section we describe a good quantum strategy for the Khot-Vishnoi game, following the ideas of Kempe, Regev, and Toner [KRT08] and the SDP-solution of [KV05]. ###### Theorem 8. For any $n$ which is a power of 2, and any $\eta\in[0,1/2]$, there exists a quantum strategy that wins the Khot-Vishnoi game with probability at least $(1-2\eta)^{2}$, using a maximally entangled state with local dimension $n$. ###### Proof. For $a\in\\{0,1\\}^{n}$, let $v^{a}\in\mathbb{R}^{n}$ denote the unit vector $((-1)^{a_{i}}/\sqrt{n})_{i\in[n]}$. Notice that for all $a,b$, we have $\langle{v^{a}},{v^{b}}\rangle=1-2d(a,b)/n$, where $d(a,b)$ denotes the Hamming distance between $a$ and $b$. In particular, the $n$ vectors $v^{a}$, as $a$ ranges over a coset of $H$, form an orthonormal basis of $\mathbb{R}^{n}$. The quantum strategy is as follows. Alice and Bob start with the $n$-dimensional maximally entangled state. Alice, given coset $x=u\oplus H$ as input, performs a projective measurement in the orthonormal basis given by $\\{v^{a}\mid a\in x\\}$ and outputs the value $a$ given by the measurement. Bob proceeds similarly with the basis $\\{v^{b}\mid b\in y\\}$ induced by his coset $y=x\oplus z\oplus H$. A standard calculation now shows that the probability to obtain the pair of outputs $a,b$ is $\langle{v^{a}},{v^{b}}\rangle^{2}/n$. Since the players win iff $b=a\oplus z$, the winning probability on inputs $x,y$ is given by $\frac{1}{n}\sum_{a\in x}\langle{v^{a}},{v^{a\oplus z}}\rangle^{2}=\frac{1}{n}\sum_{a\in x}(1-2d(a,a\oplus z)/n)^{2}=(1-2|z|/n)^{2},$ where $|z|$ denotes the Hamming weight (number of 1s) of the $\eta$-biased string $z$. Taking expectation and using convexity, the overall winning probability is $\mathop{\mathbb{E}}_{z}[(1-2|z|/n)^{2}]\geq\left(\mathop{\mathbb{E}}_{z}[1-2|z|/n]\right)^{2}=(1-2\eta)^{2}.$ ∎ ##### Acknowledgements. We thank Jop Briët, Dejan Dukaric, Carlos Palazuelos, and Thomas Vidick for useful discussions and comments. ## References * [BBLV09] J. Briët, H. Buhrman, T. Lee, and T. Vidick. Multiplayer XOR games and quantum communication complexity with clique-wise entanglement. arXiv:0911.4007, 2009. * [Bel64] J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195–200, 1964. * [BJK08] Z. Bar-Yossef, T. S. Jayram, and I. Kerenidis. Exponential separation of quantum and classical one-way communication complexity. SIAM Journal on Computing, 38(1):366–384, 2008. Earlier version in STOC’04. * [CHSH69] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. Proposed experiment to test local hidden-variable theories. Physical Review Letters, 23(15):880–884, 1969. * [CSUU07] R. Cleve, W. Slofstra, F. Unger, and S. Upadhyay. Strong parallel repetition theorem for quantum XOR proof systems. In Proceedings of 22nd IEEE Conference on Computational Complexity, pages 282–299, 2007. quant-ph/0608146. * [DKLR09] J. Degorre, M. Kaplan, S. Laplante, and J. Roland. The communication complexity of non-signaling distributions. In Proceedings of 34th MFCS, pages 270–281, 2009. arXiv:0804.4859. * [Duk10] D. Dukaric. The Hilbertian tensor norm and its connection to quantum information theory. arXiv:1008.1948v2, 2010. * [EPR35] A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical Review, 47:777–780, 1935. * [Gav09] D. Gavinsky. Classical interaction cannot replace quantum nonlocality. arXiv:0901.0956, 2009. * [GKK+08] D. Gavinsky, J. Kempe, I. Kerenidis, R. Raz, and R. de Wolf. Exponential separation for one-way quantum communication complexity, with applications to cryptography. SIAM Journal on Computing, 38(5):1695–1708, 2008. Earlier version in STOC’07. quant-ph/0611209. * [GKRW09] D. Gavinsky, J. Kempe, O. Regev, and R. de Wolf. Bounded-error quantum state identification and exponential separations in communication complexity. SIAM Journal on Computing, 39(1):1–24, 2009. Special issue on STOC’06. quant-ph/0511013. * [JP11] M. Junge and C. Palazuelos. Large violation of Bell inequalities with low entanglement. Communications in Mathematical Physics, 2011. To appear. Preprint at arXiv:1007.3043v2. * [JPP+10] M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, and M. Wolf. Unbounded violations of bipartite Bell inequalities via Operator Space theory. Communications in Mathematical Physics, 300(3):715–739, 2010. arXiv:0910.4228. Shorter version appeared in PRL 104:170405, arXiv:0912.1941. * [Kho02] S. Khot. On the power of unique 2-prover 1-round games. In Proceedings of 34th ACM STOC, pages 767––775, 2002. * [KKL88] J. Kahn, G. Kalai, and N. Linial. The influence of variables on Boolean functions. In Proceedings of 29th IEEE FOCS, pages 68–80, 1988. * [KKM+08] J. Kempe, H. Kobayashi, K. Matsumoto, B. Toner, and T. Vidick. Entangled games are hard to approximate. In Proceedings of 49th IEEE FOCS, pages 447–456, 2008. * [KKMV08] J. Kempe, H. Kobayashi, K. Matsumoto, and T. Vidick. Using entanglement in quantum multi-prover interactive proofs. In Proceedings of 23th IEEE Complexity, pages 211–222, 2008. * [KRT08] J. Kempe, O. Regev, and B. Toner. Unique games with entangled provers are easy. In Proceedings of 49th IEEE FOCS, pages 457–466, 2008. arXiv:0710.0655. * [KV05] S. Khot and N. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into $\ell_{1}$. In Proceedings of 46th IEEE FOCS, pages 53–62, 2005. * [O’D08] R. O’Donnell. Some topics in analysis of boolean functions. Technical report, ECCC Report TR08–055, 2008. Paper for an invited talk at STOC’08. * [PWP+08] D. Pérez-García, M.M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge. Unbounded violations of tripartite Bell inequalities. Communications of Mathematical Physics, 279:455, 2008. quant-ph/0702189. * [Reg11] O. Regev. Bell violations through independent bases games. arXiv:1101.0576, 3 Jan 2011. * [Tsi87] B. S. Tsirelson. Quantum analogues of the bell inequalities. the case of two spatially separated domains. Journal of Soviet Mathematics, 36:557–570, 1987. * [Wol08] R. de Wolf. A brief introduction to Fourier analysis on the Boolean cube. Theory of Computing, 2008. ToC Library, Graduate Surveys 1. ## Appendix A An alternative strategy for $\mbox{\rm HM}_{nl}$ Here we give an alternative and slightly weaker version of Theorem 6, with advantage $\Omega(1/\sqrt{n})$ instead of $\Omega(\sqrt{\log(n)/n})$. ###### Proof. Fix arbitrary inputs $x,M$. Bob always outputs $i=1$ and $j$ is whatever is matched to $i$ by $M$. Consider the following two unit vectors in $\mathbb{R}^{n}$, $u=\left((-1)^{x_{1}\oplus x_{k}}/\sqrt{n}\right)_{k=1}^{n}\qquad\qquad v=e_{j}$ where $e_{j}$ is the vector with $1$ in the $j$th coordinate and zero elsewhere. Notice that Alice knows $u$, Bob knows $v$, and that $\langle{u},{v}\rangle=(-1)^{x_{1}\oplus x_{j}}/\sqrt{n}$. The players use shared randomness to choose a random unit vector $w\in\mathbb{R}^{n}$. Bob outputs $d=0$ if $\langle{w},{v}\rangle>0$, and $d=1$ otherwise. Alice outputs $a=0^{\log n}$ if $\langle{w},{u}\rangle>0$, and a uniform $a\in\\{0,1\\}^{\log n}$ otherwise. We now analyze the success probability. Assume that $x_{1}\oplus x_{j}=0$ (the other case being similar). It is easy to see that the probability of both $\langle{w},{u}\rangle$ and $\langle{w},{v}\rangle$ being positive is $\frac{1}{2}-\frac{1}{2\pi}\arccos\langle{u},{v}\rangle$, as this is essentially a two-dimensional question. They have the same probability of both being negative, and probability $\frac{1}{2\pi}\arccos\langle{u},{v}\rangle$ to be in each of the two remaining cases. In the two cases that $\langle{w},{u}\rangle\leq 0$ (an event that happens with probability $1/2$), $a\cdot(i\oplus j)$ is a uniform bit (since $i\neq j$) and the players win with probability exactly $1/2$. Otherwise (i.e., if $\langle{w},{u}\rangle>0$), the players win if and only if $d=0$ (i.e., if also $\langle{w},{v}\rangle>0$). Hence, using that $\arccos(z)=\pi/2-\Theta(z)$ for small $z$, the overall winning probability is $\frac{1}{2}\cdot\frac{1}{2}+\frac{1}{2}-\frac{1}{2\pi}\arccos\langle{u},{v}\rangle=\frac{1}{2}+\Theta\left(\frac{1}{\sqrt{n}}\right).$ ∎
arxiv-papers
2010-12-22T17:20:09
2024-09-04T02:49:15.923719
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Harry Buhrman, Oded Regev, Giannicola Scarpa, Ronald de Wolf", "submitter": "Oded Regev", "url": "https://arxiv.org/abs/1012.5043" }
1012.5086
Current address: ]Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA # Magnetic Structure in Fe/Sm-Co Exchange Spring Bilayers with Intermixed Interfaces Yaohua Liu yhliu@anl.gov S. G. E. te Velthuis tevelthuis@anl.gov J. S. Jiang Y. Choi [ S. D. Bader Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA A. A. Parizzi H. Ambaye V. Lauter Spallation Neutron Source Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA ###### Abstract The depth profile of the intrinsic magnetic properties in an Fe/Sm-Co bilayer fabricated under nearly optimal spring-magnet conditions was determined by complementary studies of polarized neutron reflectometry and micromagnetic simulations. We found that at the Fe/Sm-Co interface the magnetic properties change gradually at the length scale of 8 nm. In this intermixed interfacial region, the saturation magnetization and magnetic anisotropy are lower and the exchange stiffness is higher than values estimated from the model based on a mixture of Fe and Sm-Co phases. Therefore, the intermixed interface yields superior exchange coupling between the Fe and Sm-Co layers, but at the cost of average magnetization. ###### pacs: 75.70.Cn, 61.05.fj ## I Introduction Exchange-coupled, high-magnetization (soft) and high-anisotropy (hard) magnetic phases have potential applications as both ultra-strong permanent magnet KnellerHawigIEEE1991 and ultra-high-density recording media. VictoraIEEE2005 While the intrinsic properties of the two phases play the most important role, optimization of the interface properties are also important to achieve the best performance. For example, the interface morphology need to be optimized for good exchange coupling between the soft and hard phases. Micromagnetic simulations suggested that magnetically graded interfaces, whose magnetic properties are gradually changed over a distance $\sim 10$ nm, is more effective than sharp interfaces, in order to increase the nucleation field in the soft phase and decrease the switching barrier of the hard phase. JiangAPL2004 ; SuessAPL2006 Magnetically graded interfaces can be fabricated from chemically intermixing phases via nanotechnology. ChoiAPL2007 ; GollJAP2008 ; KirbyPRB2010 Nanoscale spatial resolution is needed to experimentally determine the intrinsic magnetic properties of hard- soft heterostructures. Interestingly, for Fe/Sm-Co spring magnets under optimized fabrication conditions for the maximum energy product, the Fe layer and the Sm-Co layer have an intermixed interface over a length scale $\sim 5-10$ nm. ChoiPRB2007 ; LiuAPL2008 Although the intrinsic magnetic properties of individual Fe films and Sm-Co films have been well studied, FullertonPRB1998 there lacks quantitative knowledge about the intrinsic magnetic properties of the intermixed Fe/Sm-Co interface due to its complex composition. ChoiPRB2007 ; LiuAPL2008 Polarized neutron reflectometry (PNR) is a sensitive tool to study the depth profile of magnetic structures within multilayers with sub-nanometer depth resolution, Fitz2005 ; Chatterji2006 and has been used to study exchange coupling and anisotropy. DonovanPRL2002 ; FitzPRB2006 For example, in a NiFe/FePt spring magnet, O’Donovan et al. have determined that the twisted magnetic structure of the spring magnet is not confined to the magnetically soft layer, but also penetrates into the hard magnetic phase. DonovanPRL2002 Recently, Kirby et al. showed qualitatively that the structural gradation yields a graded anisotropy in Co/Pd multilayers. KirbyPRB2010 In the present work, we report PNR studies on a Fe/Sm-Co bilayer, which was fabricated under nearly optimized spring magnet conditions. We first confirmed that there is a structurally intermixed interface of $\sim 8$-nm wide between the Fe and Sm-Co layers via combined X-ray reflectometry (XRR) and PNR studies. We also determined the depth profile of the saturation magnetization $M_{S}$. Furthermore, the profiles of the magnetic anisotropy $K$ and the exchange stiffness $A$ were obtained by analyzing the PNR data with the aid of micromagnetic simulations. The magnetic properties in the intermixed region were compared to a model based on a mixture of Fe and Sm-Co phases. The saturation magnetization is slightly lower than the value estimated from the model, suggesting new compounds formed in the intermixed region. The intrinsic anisotropy is also lower than the value from the model. However, the exchange stiffness is higher, so that the interface efficiently couples the Fe and Sm- Co layers. ## II Experimental Methods The Fe/Sm-Co thin film was fabricated via _dc_ magnetron sputtering onto an MgO (110) single-crystal substrate with a nominal structure of (10 nm Cr)/(10 nm Fe)/(20 nm Sm-Co)/(20 nm Cr)/MgO. The sample size is $10\times 10$ mm2. The Cr (211) buffer layer and the Sm-Co layer were grown at 400 ∘C, and the Fe layer was grown at 100 ∘C. ChoiAPL2007 The Sm-Co layer, nominally Sm2Co7 composition, has an uniaxial, in-plane magnetic easy axis. BenaissaIEEE1998 Magnetic hysteresis loops were obtained by means of vibrating sample magnetometer (VSM). XRR studies were performed with a X-ray diffractometer using Cu $K_{\alpha}$ radiation. PNR experiments were conducted at the SNS at Oak Ridge National Laboratory, at the Magnetism Reflectometer. LauterPhysicaB2009 This is a time-of-flight (TOF) instrument with a wavelength band 2 - 5 Å and the polarization efficiency of $\sim 98\%$. All experiments were conducted at room temperature. Figure 1: (Color online) Easy-axis magnetic hysteresis loop. The film is saturated at 1 T. The two circles label the magnetization states measured by PNR, with applied fields of +1.11 and +0.39 T, respectively. The external field was applied along the magnetic easy-axis for PNR experiments. For convenience, we define the field direction as $+\hat{x}$ and the sample surface normal as $-\hat{z}$. Figure 1 shows the easy-axis magnetic hysteresis loop. The film saturates at fields above $+1$ T. There are two major reversal processes along the increasing field branch, centered at $+0.35$ and $+0.75$ T, respectively. A minor soft-phase was also observed, which reverses below $+0.05$ T. PNR measurements were performed at $+1.11$ T and in a demagnetized state at $+0.39$ T, after saturation with a $-1.11$ T field, in order to determine the depth profiles of the saturation magnetization, the exchange stiffness and the magnetic anisotropy. Reflectometry is a non-destructive method to determine scattering length density (SLD) profiles. In the specular condition, XRR yields the depth profile of the electron density, which can be used to reconstruct the chemical structures. Neutrons interact with both nuclei and the internal magnetic field. Spin-up and spin-down neutrons feel the same nuclear scattering potential, but an opposite magnetic scattering potential. From subsequent measurement with oppositely polarized neutron beams, the two contributions can be separated in order to reconstruct both depth profiles of the chemical structure and the magnetization vector. Fitz2005 ; Chatterji2006 The contrast of the neutron SLDs between Fe and Sm-Co is high, so that the PNR experiments are sensitive to the interfacial structure of interest. There are four reflectivities in PNR, including two non-spin-flip (NSF) reflectivities $R^{++}$ and $R^{--}$, and two spin-flip (SF) reflectivities $R^{+-}$ and $R^{-+}$. SF scattering occurs when the sample’s magnetization vector has a non-zero component perpendicular to both the neutron’s polarization direction and the momentum transfer direction. During magnetization reversal, the SF scattering occurred in a sufficiently high magnetic field so that it was necessary to take into account the Zeeman effect when analyzing the data. FelcherPB1996 (See Appendix A.) For PNR, Fredrikze’s formalism FredrikzePB2001 was used to determine the optics’ polarization efficiency and the direct beam (DB) spectra from the measured four reflectivities of the DB. The polarization corrections with the error propagation were made following Wilder’s formalism. WildesNN2006 Simulations of the reflectivities were based on the Parratt formalism. Reflpak A rough interface was modeled as a sequence of very thin slices whose SLDs vary, following an error function so as to interpolate between adjacent layers. Fitz2005 The instrumental resolution was handled by Gaussian convolution. The GenCurvefit program GenCurvefit using the genetic algorithm was employed for the model optimization. ## III Experimental Results and Analysis ### III.1 Depth profile of Saturated Magnetization Figure 2: (Color online) (a) XRR data and (b) PNR data taken at saturation field $+1.11$ T. (c) Depth profiles of neutron nuclear SLD (thin black line, left), neutron magnetic SLD (thick red line, left) and X-ray SLD (dashed green line, right). Only the real parts of the X-ray and the nuclear SLDs are shown. The XRR data and the specular PNR data taken at a saturation field of $+1.11$ T are displayed in Figs. 2a and 2b, respectively. The reflectivities are shown as functions of the momentum transfer perpendicular to the film plane $Q_{z}=4\pi sin\theta/\lambda$, where $\theta$ is the incident angle and $\lambda$ is the wavelength of the radiation source. Since there is no spin flip scattering exists at saturation, we measured only $R^{+}$ (=$R^{++}$) and $R^{-}$ (=$R^{--}$). Because of the sample’s high saturation magnetization, $R^{+}\gg R^{-}$ for $Q_{z}$ above the critical edges ($Q_{z}>0.018$ Å-1). There is an interesting feature in the PNR data below the critical edges: both $R^{+}$ and $R^{-}$ showed frustrated total reflection. This is due to the enhanced absorption at resonant conditions. MaazaPLA1996 Simulations show that the $Q$ separation between the two dips strongly depends on the magnetization along the field direction, especially that of the Sm-Co layer, which is the locus of absorption. Therefore, this separation, as well as the splitting between $R^{+}$ and $R^{-}$ above the critical edge, are direct indications of the sample’s magnetization. The chemical and the magnetic SLD profiles were determined by fitting the XRR and the PNR data simultaneously with the same structural model.Model Since there are well tabulated values for the neutron and the X-ray scattering length and absorption length for each element, both the real and imaginary parts of the X-ray and the neutron’s nuclear SLDs were calculated from the chemical composition and the film density. Therefore a single parameter was used to determine both the X-ray and the neutron nuclear SLDs for each layer that has a well defined composition. This parameter is the mass density for that particular layer. However, the intermixed interface has a complex chemical composition that extends $>~{}5$ nm in depth. ChoiPRB2007 ; LiuAPL2008 The X-ray and the neutron’s nuclear SLDs in the intermixed region need to be determined separately, which was done by introducing an intermixed layer between the Fe and Sm-Co layers. Therefore, both the nuclear and the magnetic SLD profiles can vary in more sophisticated ways and they are less correlated to each other over the intermixed region. Actually, the peak and dip positions in $R^{-}$ between 0.025 and 0.045 Å can not be reproduced without introducing this intermixed layer, which indicates that a single error function is not sufficient to model the nuclear and/or the magnetic SLD profiles at the Fe/Sm-Co interface. Table 1: The layer thickness, _rms_ interface roughness, the nuclear and magnetic SLDs, saturated magnetization $M_{S}$, exchange stiffness $A$ and uniaxial anisotropy $K$ in the layers of interest from the best fits. Nuclear and magnetic SLDs and $M_{S}$ are from fitting the $+1.11$-T data (Sec. III.1). $A$ and $K$ are from fitting the $+0.39$-T data (Sec. III.2). Typical literature values are $M_{S}=1700~{}(550)$ emu/cc, $A=2.8~{}(1.2)$ ergs/cm, and $K=10^{3}~{}(5\times 10^{7}$) ergs/cc for the Fe (Sm-Co) layer. FullertonPRB1998 (∗Literature value of $K_{Fe}$ was used during the optimization.) | Fe | | mixed | | Sm-Co ---|---|---|---|---|--- Thickness (nm) | $7.4\pm 0.3$ | | $4.2\pm 0.3$ | | $20.1\pm 0.2$ Roughness (nm) | | $4.3\pm 0.3$ | | $2.6\pm 0.2$ | $\rho_{nuc}$ ($10^{-6}$ Å-2) | $7.9\pm 0.2$ | | $5.2\pm 0.2$ | | $1.64\pm 0.04$ $\rho_{mag}$ ($10^{-6}$ Å-2) | $4.9\pm 0.1$ | | $2.5\pm 0.1$ | | $1.62\pm 0.03$ $M_{s}$ (emu/cc) | $1700\pm 50$ | | $890\pm 50$ | | $570\pm 10$ $M_{+0.39~{}T}$ (emu/cc) | $1540\pm 10$ | | $730\pm 30$ | | $420\pm 10$ $A$ (10-6 ergs/cm) | $2.6\pm 0.1$ | | $2.6\pm 0.1$ | | $1.2\pm 0.1$ $K$ (106 ergs/cc) | 0.001∗ | | $3.0\pm 0.2$ | | $27\pm 1$ The best-fit curves for the reflectivity data are overlaid on the data, and the SLD depth profiles are shown in Fig. 2c. The SLD profiles show sharp transitions at the Cr/MgO and the Sm-Co/Cr interfaces with _rms_ roughnesses of $\sim$ 0.4 nm for both interfaces. However, the SLD profiles gradually change between the Fe and Sm-Co layers over a distance of $\sim 8$ nm. Such a large intermixed region is consistent with previous energy-dispersive- spectroscopy (EDS) results from samples fabricated under the same conditions. ChoiPRB2007 In the intermixed region, there appears a shoulder in the nuclear SLD profile (Fig. 2c), showing that it is chemically rich in Fe. However, this feature is not present in the magnetic SLD profile. This will be discussed below. The parameters of interest are listed in Table 1, which shows that both the Fe and the Sm-Co layers have their saturation magnetizations close to the literature values FullertonPRB1998 despite the large intermixing at the interface. ### III.2 Depth Profiles of Exchange Stiffness and Uniaxial Anisotropy Figure 3: (Color online) (a) PNR data taken at $+0.39$ T after saturation in $-1.11$ T. $R^{+-}$ and $R^{-+}$ are offset by a factor of 0.1 for clarity. (b) The depth profiles of the magnetization vector $\vec{M}$ (obtained from micromagnetic simulations) that yields the best fit to the PNR data. The field direction is along $+\hat{x}$. In order to get insight into the intrinsic magnetic properties, besides $M_{S}$, in the intermixed region, we also studied the magnetization structure in a demagnetized state. The PNR data were collected at $+0.39$ T after the negative saturation in $-1.11$ T. The field is sufficiently high so that the specular SF reflection showed at off-specular positions on the detector due to the Zeeman effect, as shown in Fig. 6. Therefore, in the analysis the momentum transfer $Q_{z}$ was modified according to Eq. 1. FelcherNature1995 ; FelcherPB1996 It is worth noting that the off-specular scattering in the SF reflectivity is dominated by the Zeeman effect in our data and possible off- specular scattering due to in-plane magnetic domains is not distinguishable from the background, and is therefore not considered in the data. The PNR data are shown in Fig. 3a. In contrast to the saturation case, $R^{--}$ is higher than $R^{++}$ for most $Q_{z}$ values above the critical edge, which indicates that the average $M_{x}$ is still along the negative applied field direction. Both the splitting between $R^{--}$ and $R^{++}$ and the separation between the dips below the critical edge are smaller because the average $M_{x}$ is much lower than $M_{s}$. At the same time, there is significant SF scattering of the same amplitude as the NSF scattering, indicating a large in-plane magnetization component perpendicular to the field direction ($M_{y}$). This is caused by the magnetization spiral structure observed in exchange-coupled bilayers during demagnetization. DonovanPRL2002 Figure 3b plots the depth profile of the magnetization vector $\vec{M}$ that yields the best fit, which is obtained from micromagnetic simulations. Our PNR studies do not reveal the chirality of $\vec{M}$ so that a positive $M_{y}$ is used for convenience. As expected, the magnetic moment of the Sm-Co layer is along the negative field direction and the magnetization vector rotates into the intermixed region and the Fe layer. Figure 4: (Color online) Top: The depth profile of the nuclear SLD in gray scale in the unit of $10^{-6}$ Å-2. Bottom: The depth profile of the exchange stiffness $A$ (left, red solid line) and the uniaxial anisotropy $K$ (right, blue dashed line) which gives the best fit to the PNR data. Rohlsberger _et al._ have shown that the depth profile of the magnetization vector in exchange coupled bilayers agrees with the micromagnetic simulation from the 1D spin-chain model. RohlsbergerPRL2002 Fitzsimmons _et al._ further determined the micromagnetic parameters for individual layers in the exchange coupled DyFe${}_{2}2$/YFe2 superlattice using combined studies of PNR and micromagnetic simulations. FitzPRB2006 For the Fe/Sm-Co spring magnet studied here, the Fe and the Sm-Co layers are largely intermixed for $\sim 8$ nm at the interface and it is naturally to expect that the intrinsic magnetic properties, i.e. uniaxial anisotropy $K$ and exchange stiffness $A$, vary gradually due to the gradation of the chemical composition. Hence, we extended Fitzsimmons’ approach and constructed depth profiles of the intrinsic magnetic properties. Given a profile of the micromagnetic parameters, the equilibrium magnetic structure can be simulated by energy minimization via the 1D spin- chain model, FullertonPRB1998 from which the neutron’s magnetic scattering potential can be directly calculated followed by the spin-dependent neutron reflectivities. Therefore, the intrinsic magnetic properties can be optimized to yield the best fit to the experimental PNR data. These profiles of $K$ and $A$ are built up of layers identical to those of the chemical structure as determined from the fits to the XRR and the PNR data in saturations. The gradual changes between the layers are computed using the interface roughness in the same way as is done for the SLDs. When fitting, the chemical structure and the roughnesses are fixed to those already determined. $M$ is allowed to be lower than $M_{s}$ to account for the effect from the minor soft phase mentioned above. The model optimization is not sensitive to the $K_{Fe}$ because the associated magnetic energy is negligible, and therefore the literature value of $K_{Fe}$ is used. The error bars of the parameters only reflect the statistical error and the accuracy of the values depends on the model. Therefore it is important here to have the additional layer between the Fe and the Sm-Co layers in the model because the exact mapping function from the chemical structure to the micromagnetic properties is unknown. The layer has its own free micromagnetic parameters so that $M$, $A$ and $K$ in the intermixed region are able to vary more independently from those in the Fe layer and the Sm-Co layer. We checked the robustness of the model by allowing different interface roughnesses for $M$, $A$, and $K$ and found that the optimized micromagnetic parameters have similar depth dependencies. The depth profiles of $A$ and $K$ from the best fit are shown in Fig. 4 and the values of $M$, $A$ and $K$ for layers of interest are listed in Table 1. Both the parameters $A$ and $K$ show monotonic depth dependence as expected. $M_{Sm-Co}$ is $\sim 20\%$ lower than the saturation value, from which we estimated that $\sim 10\%$ of the Sm-Co was reversed due to the minor phase. Therefore, the 1D spin-chain model is approximately valid and yields fairly good fits, which were overlaid on the data in Fig. 3a. As also listed in Table 1, in the intermixed region, $A$ is close to that of the Fe layer, but $K$ is much lower than the average value between the Fe and the Sm-Co layers. ## IV Discussion and Summary Figure 5: (Color online) Depth profiles of (a) the saturation magnetization, and (b) the exchange stiffness (left) and the uniaxial anisotropy (right) calculated by assuming that the intermixed region is a mixture of the Fe phase and the Sm-Co phase (dashed lines). The experimentally determined profiles are plotted as solid lines. Also shown in the top panel is the depth profile of the nuclear SLD in gray scale. The unit is $10^{-6}$ Å-2. By assuming that the intermixed region is composed of a mixture of the Fe and Sm-Co phases, the saturation magnetization can be computed from the relative volume of both phases via $M=f_{s}M_{s}+f_{h}M_{h}$, where $f_{s}$ ($M_{s}$) and $f_{h}$ ($M_{h}$) are the relative volume (the saturation magnetization) of the Fe phase and the Sm-Co phase, respectively. From the depth profile of the nuclear SLD, the relative volume of the two phases can be determined. Following this approach, the calculated saturation magnetization profile is shown in Fig. 5a. Also shown is the saturation magnetization profile calculated from the magnetic SLD profile. Clearly, the mixture model predicted slightly larger saturation magnetization in the intermixed region. This agrees with the suggestion that new compound(s) is (are) formed in the intermixed region, JiangAPL2004 possibly due to interdiffusion between the Fe and Sm-Co layers. The estimation shows that the average magnetization of the whole sample is reduced by $\sim 4\%$ due to the intermixing of the two components. The exchange stiffness and the anisotropy of a two phase mixture can also be approximately estimated by the relative volume: $K=f_{s}K_{s}+f_{h}K_{h}$ and $A=f_{s}A_{s}+f_{h}A_{h}$. SkomskiPRB1993 The results are shown in Fig. 5b. In comparison to those determined from the PNR experiments, the model predicts higher anisotropy but lower exchange stiffness in the intermixed region. Therefore, we conclude that the optimized Fe/Sm-Co interface for a spring magnet reduces the average magnetization. Interfaces with graded magnetic properties are desirable for superior performances in exchange coupling systems, such as spring magnets and magnetic media. JiangAPL2004 ; VictoraIEEE2005 A straightforward approach to achieve magnetically graded interfaces is to make structurally graded interfaces. ChoiAPL2007 ; GollJAP2008 ; KirbyPRB2010 The micromagnetic properties could be predicted from the composition. However, it is challenging if multiple phases coexist even when the micromagnetic properties of each individual phase are known, due to lacking of prior knowledge of the interphase coupling. As shown above, the intrinsic magnetic properties in the intermixed Fe/Sm-Co interface deviates from the results predicted by the mixture model. Therefore, a complex correlation between the magnetic properties and the nominal composition may exist at the chemically graded interfaces. In summary, we determined the intrinsic magnetic properties of a Fe/Sm-Co bilayer under the nearly optimized fabrication condition for spring magnets. We determined that the intermixed interface between the Fe and Sm-Co layers extends $\sim 8$ nm, where the magnetic properties changes gradually, as expected. We compared the magnetic properties with the prediction of a mixture model and found, in the intermixed region, the saturation magnetization is slightly lower than that estimated from the model, but the exchange stiffness is higher. This observation indicates that the intermixed interface is efficient for magnetically coupling the Fe and Sm-Co layers but at the cost of the average magnetization. The intrinsic anisotropy is also lower than the value from the model. Overall, the intrinsic magnetic properties in the structurally intermixed region may not be predicted correctly by the mixture model of nominal compositions, which is worth keeping in mind when designing the magnetically graded interfaces. ###### Acknowledgements. We thank Gian P. Felcher for helpful discussions. Research at Argonne was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No.DE- AC02-06CH11357. Research at Oak Ridge National Laboratory’s Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U. S. Department of Energy. ## Appendix A Zeeman effect and off-specular scattering The Zeeman effect in PNR was firstly clarified by Felcher _et. al_. FelcherNature1995 ; FelcherPB1996 The effect shows up in the SF reflectivities when the difference of the Zeeman energy for spin-up and spin- down neutrons is not negligible in a sufficiently high magnetic field. Let $k_{\perp}$ and $k_{//}$ label the perpendicular and the parallel components of the wavevector with respect to the sample surface in the _vacuum_ region, _i.e._ , before neutrons enter the magnetic-field region of interest. $k_{\perp}=2\pi sin\theta/\lambda\ll k_{//}$ at glancing angles. There is negligible energy exchange between the sample and the neutrons for elastic scattering, therefore the Zeeman energy change after SF scattering accompanies a change of the kinetic energy, which is associated with $\vec{k}$. The energy change after spin-flip is twice the Zeeman energy. If the sample is magnetically homogenous in the film plane as seen by the neutrons, the energy change is totally associated with $k_{\perp}$. Since $k_{\perp}$’s are small quantities, the outgoing angles of the spin-flipped neutrons are different from the incoming angles. Therefore the SF reflections appear at off-specular positions, which have a characteristic field dependence. FelcherNature1995 ; FelcherPB1996 The Zeeman effect on the SF reflections becomes clear in a magnetic field on $\sim 0.1$ T or higher. Figure 6 shows an example. It is clear that the locus of the SF reflections deviates away from the specular position, but are described by the prediction after considering the Zeeman effect. Beside the reflection due to the Zeeman effect, there is no noticeable off-specular scattering in the SF channels; therefore, the scattering from the in-plane magnetic domains are not considered during the data analysis. Consequently, the momentum transfer $Q_{z}$ are no longer equal to $2k_{\perp}$ for spin-flipped neutrons, but follows $\begin{array}[]{rcl}Q_{z}^{+-}&=&k_{\perp}+\sqrt{k_{\perp}^{2}+2CB}\\\ Q_{z}^{-+}&=&k_{\perp}+\sqrt{k_{\perp}^{2}-2CB},\end{array}$ (1) where $C=|2m_{n}\mu_{n}/\hbar^{2}|=2.906\times 10^{-5}$ Å-2/T and $B=\mu_{0}H$ is the applied magnetic field. $R^{-+}$ is forbidden when $k_{\perp}<\sqrt{2CB}$ because there is no enough kinetic energy to compensate the Zeeman energy change. vandeKruijsPB2000 At the same time, the minimum $Q_{z}^{+-}$ is also $\sqrt{2CB}$ for finite reflectivity. Therefore, there is a cutoff $Q_{z}$ for non-zero SF scattering, $Q_{SFcutoff}=\sqrt{2CB}$. Figure 6: (Color online) The coutour map of the neutron intensities measured with the polarization analysis, which are presented as functions of the neutron wavelengths $\lambda$. The data were collected from a Fe/Sm-Co spring magnet sample at an incident angle $\theta_{i}=0.24^{o}$ in an external field of +0.39 T after a negative saturation. The dashed lines indicate the specular reflection positions without considering the Zeeman effect, i.e. $\theta_{f}=\theta_{i}$. The solid lines in the SF channels indicate the off- specular scattering following the the Zeeman effect, i.e. $\sin^{2}\theta_{f}=\sin^{2}\theta_{i}\pm\frac{cB\lambda^{2}}{2\pi^{2}}$. Actually, the kinetic energy also changes when neutrons enter and leave the magnetic-field region of interest, but the change of $\vec{k}$ is essentially in $k_{//}$ in these cases, Chatterji2006 which changes the time-of-flight. However, the relative change of $k_{//}$ is so small in laboratory fields, and TOF only changes $\sim 10$ ns, while the width of the neutron pulse is $\sim 0.2$ ms. Therefore, it does not result in any observable affect. SF reflection is typically weak when the Zeeman effect matters, therefore the Zeeman effect is usually not considered during data reduction and analysis. Reflpak ; Fitz2005 However, we observed significant SF scattering at $+0.39$ T, which is sufficiently high so that the Zeeman effect needs to be considered. We adopted the generalized algorithm of the GEPORE Chatterji2006 for the PNR simulations. $\vec{B}$ rather than $\vec{M}$ is used to calculate the magnetic scattering potential since $H$ is now comparable to $M$ and the spin eigenstates are aligned along $\vec{B}$ rather than $\vec{M}$. Considering the problem along the normal direction of the sample surface (let it be $\hat{z}$), it is reduced to a pair of coupled 1D differential wave equations, Chatterji2006 $\begin{array}[]{rcl}\left[-\frac{\hbar^{2}}{2m}k_{0}^{2}+V_{++}(z)-E\right]\Psi_{+}(z)+V_{+-}\Psi_{-}(z)&=&0\\\ \left[-\frac{\hbar^{2}}{2m}k_{0}^{2}+V_{--}(z)-E\right]\Psi_{-}(z)+V_{-+}\Psi_{+}(z)&=&0.\end{array}$ (2) $V$ is the potential operator. $\Psi_{\pm}(z)$’s are the wave functions for spin-up and spin down states. $k_{0}$ is the wavevector in vacuum, which can be either a real number or a pure imaginary number. States with ${k_{0}}^{2}\pm CB<0$ correspond to evanescent waves outside the sample. 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arxiv-papers
2010-12-22T20:18:14
2024-09-04T02:49:15.933981
{ "license": "Public Domain", "authors": "Yaohua Liu, S. G. E. te Velthuis, J. S. Jiang, Y. Choi, S. D. Bader,\n A. A. Parizzi, H. Ambaye, and V. Lauter", "submitter": "Yaohua Liu", "url": "https://arxiv.org/abs/1012.5086" }
1012.5293
# Phase Estimation with Non-Unitary Interferometers: Information as a Metric Thomas B. Bahder Aviation and Missile Research, Development, and Engineering Center, US Army RDECOM, Redstone Arsenal, AL 35898, U.S.A. ###### Abstract Determining the phase in one arm of a quantum interferometer is discussed taking into account the three non-ideal aspects in real experiments: non- deterministic state preparation, non-unitary state evolution due to losses during state propagation, and imperfect state detection. A general expression is written for the probability of a measurement outcome taking into account these three non-ideal aspects. As an example of applying the formalism, the classical Fisher information and fidelity (Shannon mutual information between phase and measurements) are computed for few-photon Fock and N00N states input into a lossy Mach-Zehnder interferometer. These three non-ideal aspects lead to qualitative differences in phase estimation, such as a decrease in fidelity and Fisher information that depends on the true value of the phase. ###### pacs: PACS number 07.60.Ly, 03.75.Dg, 06.20.Dk, 07.07.Df ## I Introduction Optical interferometers Hariharan (2003) and matter wave interferometers Cronin et al. (2009) have been of great interest because of their practical applications in metrology. Interferometers have been used to measure such diverse quantities as electric, magnetic, and gravitational fields, gravitational waves Cronin et al. (2009); Thorne (1980); Caves (1981), and there are plans to use them to test the theory of general relativity Dimopoulos et al. (2008). Classical optical interferometers Lefevre (1993) have been routinely used for sensing rotation in gyroscopic applications based on the Sagnac effect Sagnac (1913a, b, 1914); Post (1967); Chen et al. (2008) and experiments with Sagnac interferometers have been done with single-photons Bertocchi et al. (2006), with Bose-Einstein condensates(BEC) Gupta et al. (2005); Wang et al. (2005); Tolstikhin et al. (2005), and schemes using entangled particles have been proposed that are capable of Heisenberg limited precision measurements that scale as $1/N$, where $N$ is the number of particles Cooper et al. (2010). On a more fundamental level, there is interest in interferometers because they are a vehicle to study the limits of precision of quantum measurements Godun et al. (2001); Giovannetti et al. (2006); Berry et al. (2009). Perhaps the simplest generic measurement problem consists of determining the relative phase shift between two arms of an interferometer from measurements made at the output ports of the interferometer Combes and Wiseman (2005); Nagata et al. (2007); Durkin and Dowling (2007); Pezze and Smerzi (2008); Dorner et al. (2009); Cable and Durkin (2010). This phase shift may be related to a classical external field incident on a phase shifter in one arm of the interferometer, in which case the interferometer can be used as a sensor of the field Bahder and Lopata (2006a). The determination of the phase shift is a specific example of the more general problem of parameter estimation, whose goal is to determine one or more parameters from measurements Cramér (1958); Helstrom (1967, 1976); Holevo (1982); Braunstein and Caves (1994); Braunstein et al. (1996); Barndorff-Nielsen and Gill (2000); Barndorff-Nielsen et al. (2003). Recently, there have been experimental demonstrations using entangled states to estimate the phase shift in one arm of a Mach-Zehnder interferometer Walther et al. (2004); Mitchell et al. (2004); Nagata et al. (2007); Okamoto et al. (2008). Even more recently, the effect of losses on phase determination was studied experimentally Kacprowicz et al. (2010). In real experiments, there are three non-ideal elements of the interferometer system: state preparation Mitchell et al. (2004); Thomas-Peter et al. (2009), photon losses in the interferometer Kacprowicz et al. (2010) and non-ideal photon-number detection Nagata et al. (2007); Okamoto et al. (2008). Phase estimation has been theoretically investigated by separately taking into account non- deterministic state preparation Helstrom (1967, 1976); Holevo (1982); Braunstein and Caves (1994); Braunstein et al. (1996); Barndorff-Nielsen and Gill (2000); Barndorff-Nielsen et al. (2003), photon losses in the interferometer itself Kim et al. (1998); Durkin et al. (2004); Rubin and Kaushik (2007); Gilbert et al. (2008); Dorner et al. (2009); Demkowicz- Dobrzanski et al. (2009); Cable and Durkin (2010); Ono and Hofmann (20010) and photon-number counting efficiency Okamoto et al. (2008); D Ariano et al. (2000); Cable and Durkin (2010). In this work, I write down a formalism that simultaneously takes into account, in a unified way, all three of the non-ideal elements in experiments: non- deterministic state preparation, propagation through a lossy interferometer, and imperfect state detection. Non-deterministic state preparation must be described by a density matrix, rather than by a pure state, thereby allowing for the finite probability of creating states other than intended. When the optical state is created, it enters the interferometer, where propagation may be non-ideal because photon absorption and scattering can occur. Finally, when the optical state leaves the interferometer, it enters the detection system, which may also be non-ideal: the state registered by the detection system may not be the true state that entered the detection system. Much of the previous work was focused on determining the optimum measurements for determining phase and hence the quantum Fisher information was of primary interest, because it gives a bound on the variance of the phase associated with the optimum measurement Helstrom (1967, 1976); Holevo (1982); Braunstein and Caves (1994); Braunstein et al. (1996); Barndorff-Nielsen and Gill (2000); D Ariano et al. (2000); Monras (2006); Olivares and Paris (2009); Gaiba and Paris (2009). In contrast, in this work I look at the information gain from specific, simple, photon-number counting measurements that can be easily implemented in the laboratory, and hence the classical Fisher information is the quantity of interest because it depends on the particular measurement that is performed. In Section II, I briefly review the theory of phase determination based on parameter estimation (Fisher information) and on fidelity (Shannon mutual information between measurements and phase). In Section III, I introduce an example of a non-ideal interferometer system, where state evolution is non- unitary. I write a statistical expression for the probability of measurement outcomes that takes into account the three non-ideal components of the interferometer system described above. I use this probability in the classical Fisher information in Eq. (2) and in the fidelity in Eq. (9) to analyze the determination of phase in a non-ideal interferometer system. As simple examples of the formalism, in sub-sections of Section III, I look at few photon examples of non-deterministic state preparation, propagation through a non-unitary (lossy) interferometer, and imperfect state detection. Finally, in Section IV, I make some concluding remarks. My goal is to look at examples of few-photon states that can be implemented experimentally, with the hope that the examples and method described here can be helpful for analyzing real experiments. Furthermore, in this work, I restrict myself to the simple case of non-adaptive measurements Higgins et al. (2009), where the measurement is fixed before phase estimation. ## II Theoretical background The accuracy of estimating a (single) one-dimensional parameter, $\phi$, is described in terms of the classical Cramer-Rao bound Cover and Thomas (2006), which gives a lower bound on the variance $(\delta\phi)^{2}$ of an unbiased estimator of the parameter $\phi$: $\left({\delta\phi}\right)^{2}\geq\frac{1}{F_{cl}(\phi;M)}$ (1) where $F_{cl}(\phi;M)$ is the classical Fisher information given by Cramér (1958); Cover and Thomas (2006) $F_{cl}(\phi;M)=\sum\limits_{\xi}{\frac{1}{{P(\xi|\phi,\rho)}}\,\left[{\frac{{\partial P(\xi|\phi,\rho)}}{{\partial\phi}}}\right]^{2}}$ (2) The classical Fisher information is described in terms of the conditional probability distribution, $P(\xi|\phi,\rho)$, for measurement outcome, $\xi$, which can take one or more continuous values, or, one or more discreet values. If $\xi$ takes continuous values, the sum over $\xi$ in Eq. (2) is an integral. For the case of quantum measurements, these probabilities are given by $P(\xi|\phi,\rho)={\rm{tr}}\left({\hat{\rho}_{\phi}\,\hat{\Pi}_{o}\left(\xi\right)}\right)={\rm{tr}}\left({\hat{\rho}_{o}\,\hat{\Pi}_{\phi}\left(\xi\right)}\right)$ (3) where the state is specified by the Schrödinger picture density matrix, $\hat{\rho}_{\phi}$, and the measurements by the positive-operator valued measure (POVM) in the Schrödinger picture, $\hat{\Pi}_{o}(\xi)$. The POVM are set a of non-negative Hermitian operators, $M=\\{\hat{\Pi}(\xi)\\}$, representing a given physical measurement and so the expectation value of each operator $\hat{\Pi}_{o}(\xi)$ is non-negative, and satisfies $\sum\limits_{\xi}{\hat{\Pi}_{o}(\xi)}=\hat{I}$, where $\hat{I}$ is the identity operator. Alternatively, in the Heisenberg picture, the state is given by the density matrix, $\hat{\rho}_{o}$, and probabilities of measurements by POVM, $\hat{\Pi}_{\phi}(\xi)$. Operators in the Schrodinger and Heisenberg pictures, $\hat{O}_{S}$ and $\hat{O}_{H}(\phi)$, respectively are related by $\hat{O}_{H}(\phi)=\hat{U}^{\dagger}(\phi)\,\hat{O}_{S}\,\hat{U}(\phi)$, and $\hat{U}(\phi)=\exp(-i\phi\hat{h})$, where $\hat{h}$ is the infinitesimal displacement operator for the parameter $\phi$, satisfying $i\frac{\partial}{\partial\phi}\left|\psi(\phi)\right\rangle=\hat{h}\,\left|\psi(\phi)\right\rangle$ (4) The quantum Fisher information, $F_{Q}(\phi)$ is obtained by maximizing the classical Fisher information, $F_{cl}(\phi;M)$, over all possible measurements $M$, at a given value of $\phi$. Braunstein and Caves Braunstein and Caves (1994); Braunstein et al. (1996) have shown that an improved lower bound is possible for the variance, $\left({\delta\phi}\right)^{2}$, in terms of the quantum Fisher information: $\left({\delta\phi}\right)^{2}\geq\frac{1}{{F_{cl}(\phi;M)}}\geq\frac{1}{{F_{Q}(\phi)}}$ (5) where $F_{Q}(\phi)$ is independent of the measurement $M$. The quantum Fisher information is defined by $F_{Q}\left(\phi\right)=\rm{tr}\left[\hat{\rho}_{\phi}\hat{\Lambda}_{\phi}^{2}\right]$ (6) where the Hermitian operator, $\Lambda_{\phi}$, is the symmetric logarithmic derivative (S.L.D.), defined implicitly by $\frac{\partial\hat{\rho}_{\phi}}{\partial\phi}=\frac{1}{2}\left[\hat{\Lambda}_{\phi}\,\hat{\rho}_{\phi}+\hat{\rho}_{\phi}\hat{\Lambda}_{\phi}\right]$ (7) The right side and left side of the inequality in Eq. (5) is sometimes called the quantum Cramer-Rao bound, see also the work by Helstrom Helstrom (1967, 1976) and Holevo Holevo (1982), and discussion by Barndorff-Nielsen et al. Barndorff-Nielsen and Gill (2000); Barndorff-Nielsen et al. (2003). The expression in Eq. (5) provides a bound on the variance of an unbiased estimator for an optimum measurement. However, the theory does not give a procedure for determining the optimum measurement. For one-dimensional parameter estimation and for simple (non-adaptive) measurements, Barndorff- Nielsen and Gill have shown that in general the optimum measurement $M$ will depend on the parameter $\phi$, which is unknown prior to estimation Barndorff-Nielsen and Gill (2000). Consequently, Barndorff-Nielsen and Gill have proposed a two-stage adaptive measurement procedure that will give $F_{cl}(\phi;M)=F_{Q}\left(\phi\right)$ (8) for optimum measurement $M$ for all $\phi$. For the case of a pure state, $\left|{\psi_{o}}\right\rangle$, where the density matrix is $\rho_{o}=\left|{\psi_{o}}\right\rangle\,\left\langle{\psi_{o}}\right|$, and where the path is generated by a unitary transformation, $\hat{U}(\phi)$, the quantum Fisher information, $F_{Q}(\phi)$, does not depend on $\phi$ Braunstein et al. (1996); Olivares and Paris (2009); Gaiba and Paris (2009), and is given by the fluctuations of the generator $\hat{h}$ by $F_{Q}\left(\phi\right)=4(\Delta h)^{2}$. Furthermore, Hofman has shown that for pure states having a path symmetry Hofmann (2009), the quantum Cramer-Rao bound in Eq.(5) can be achieved at any value of phase $\phi$. The condition for optimal measurements for the case of pure states has also been investigated Durkin (2010). In Section III below, I consider the case of a lossy Mach-Zehnder interferometer, where the state evolution is effectively non-unitary, thereby leading to a classical Fisher information that depends on the true value of the phase $\phi$. The above discussion of parameter estimation is based on classical and quantum Fisher informations, which are local descriptions of phase estimation, because they depend on the true value of $\phi$. Complementary to the above local descriptions, is a global description given by the fidelity Bahder and Lopata (2006a): $\displaystyle H(M)$ $\displaystyle=$ $\displaystyle\sum_{\xi}\int_{-\pi}^{+\pi}d\,\phi\,\,P(\xi|\phi,\rho)\,p(\phi)\,\,\times\,$ (9) $\displaystyle\log_{2}\left[\frac{P(\xi|\phi,\rho)\,}{\int_{-\pi}^{+\pi}\,\,\,P(\xi|\phi^{\prime},\rho)\,p(\phi^{\prime})\,\,d\,\phi^{\prime}}\right].$ where $P(\xi|\phi,\rho)$ is given in Eq. (3). The fidelity, $H(M)$, is the Shannon mutual information Shannon (1948); Cover and Thomas (2006) between the measurement $M$ and the unknown parameter $\phi$. The fidelity, $H(M)$, gives the average amount of information (in bits) about the parameter $\phi$ that can be obtained from the measurement $M$ for one use (one measurement cycle) of the interferometer. The fidelity does not depend on $\phi$ because it is an average over all possible phases $\phi$ and over all probabilities of measurement outcomes for a given POVM $M$. (For an alternative discussion of local versus global phase estimation, see Ref.Durkin and Dowling (2007).) The fidelity also depends on the prior information about the parameter $\phi$ through the prior probability distribution $p(\phi)$. Consequently, the fidelity characterizes the quality of the interferometer system as a whole, in terms of mutual information between the measurement $M$ and the parameter $\phi$. Note that the fidelity depends on the input state density matrix, $\hat{\rho}$, and the measurement $M$, and therefore can be used to optimize the system with respect to the input state and measurement. The fidelity is a measure of the information that flows from the phase $\phi$ to the measurements, which is analogous to a communication problem where Alice sends messages to Bob. In the case of the measurement problem, quantum fluctuations in the initial state, in the channel (interferometer), and the type of measurement, determine the amount of information that is obtained about the parameter $\phi$ from the measurements. The fidelity has been applied to compare the use of Fock states and N00N states when no prior information is present about the phase Bahder and Lopata (2006a) and when there is significant prior information about the phase Bahder and Lopata (2006b). The complimentary measures of fidelity and Fisher information may be contrasted as follows. Assume that I want to shop to purchase the best measurement device to determine the unknown parameter $\phi$. If I do not know the true value of the parameter $\phi$, I would compare the overall performance specifications of several devices and I would purchase the device with the best over-all specifications for measuring $\phi$. The fidelity, $H(M)$, is the over-all specification for the quality of the device, so I would purchase the device with the largest fidelity. After I have purchased the device, I want to use it to determine a specific value of the parameter $\phi$ based on several measurements (data). This involves parameter estimation, which requires the use of Fisher information, and depends on the true value of the parameter $\phi$. Historically, the variance, $(\delta\phi)^{2}$, of the estimated parameter $\phi$ has been discussed in terms of the standard quantum limit, $\delta\phi_{SL}=$ $1/\sqrt{N}$, and the Heisenberg limitCaves (1981); Ou (1997); Giovannetti et al. (2004, 2006), $\delta\phi_{HL}=$ $1/N$, where $N$ is the number of particles or quanta that enter the interferometer during each measurement cycle. The value $\delta\phi$ is presumably the width of some probability distribution, $p(\phi|\xi,\rho)$, such as the distribution calculated from Bayes’ rule, see Eq. (25). Detailed calculation of $p(\phi|\xi,\rho)$ for a number of input states shows that these distributions have multiple peaks Bahder and Lopata (2006a). Consequently, rather than using the widths of these distributions as a metric for determining $\phi$, I use the information measures, Fisher information and fidelity, which naturally handle distributions with multiple peaks. Figure 1: (Color) Interferometer system shown with three components: state preparation, interferometer, and detection system. ## III Non-Ideal Optical System As described in the introduction, an interferometer system can be divided into three parts: state creation, state evolution through the optical interferometer, and detection of the output state. In a real experiment, each of these three parts can be non-ideal, see Fig. 1. For example, I may want to create a quantum state $|\psi^{in}\rangle$ as input into the interferometer. However, instead, the resulting state may be a mixture of states, each with some probability, $P_{S}(\psi_{k}^{in})$, for $k=1,2,\cdots$. Such a quantum state is described by the density matrix $\hat{\rho}$: $\hat{\rho}=\sum\limits_{k}{\,\;P_{S}(\psi_{k}^{in})}\;\left|{\psi_{k}^{in}}\right\rangle\left\langle{\psi_{k}^{in}}\right|$ (10) The state $\hat{\rho}$ is then input into the interferometer, where there may be absorption and scattering of photons. For example, a two-photon state may enter the interferometer and a one-photon state may exit the interferometer, because one photon was absorbed inside the interferometer. Alternatively, a two-photon state may enter the interferometer and a three-photon state may exit the interferometer, due to light scattering into the interferometer from the environment. I can describe these processes generally by a transfer matrix, ${P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)}$, which gives the conditional probability for state $\left|{\psi_{j}^{out}}\right\rangle$ to exit the interferometer given that state $\left|{\psi_{k}^{in}}\right\rangle$ entered the interferometer. The transfer matrix, ${P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)}$, is general enough to describe non-unitary propagation of the quantum state through the interferometer, and so can take into account losses and scattering. Note that the transfer matrix may depend on the state of the interferometer, which I specify here by single parameter $\phi$. Finally, the detection of the quantum state that leaves the interferometer can be non-ideal. For example, the detection system may register a measurement $\xi$, when state $\left|{\psi_{i}^{out}}\right\rangle$ enters the detection system, whereas the true state that entered the detection system was $\left|{\psi_{j}^{out}}\right\rangle$. I can represent such an imperfect detection system by the conditional probability $P_{D}(\xi|\psi_{j}^{out},\phi)$, which gives the probability for making a measurement $\xi$ when state $\psi_{j}^{out}$ entered the detection system. Note that in general this probability may or may not depend on $\phi$, a parameter describing the state of the interferometer. For a non-ideal interferometer system, the probability of obtaining a measurement $\xi$ is given by Jaynes (2009) $\small P(\xi|\phi)=\sum\limits_{j}{P_{D}}(\xi|\psi_{j}^{out},\phi)\;\sum\limits_{k}{\,P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)\;P_{S}(\psi_{k}^{in})}$ (11) where we must have each of the three probabilities sum to unity: $\sum\limits_{k}{\,P_{S}(\psi_{k}^{in})}=1$ (12) $\sum\limits_{j}{\,P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)}=1$ (13) $\sum\limits_{\xi}{P_{D}(\xi|\psi_{j}^{out},\phi)}=1$ (14) Equation (11) is a general statistical relation for the probability of obtaining a measurement outcome $\xi$ for given phase shift $\phi$, taking into account the three non-ideal aspects of interferometer systems. Note that Eq. (11) is sufficiently general that it can be applied to the case where states are represented by density matrices. In this case, in Eq. (11) we can make the replacements $\psi_{k}^{in}\rightarrow\rho_{k}^{in}$ and $\psi_{j}^{out}\rightarrow\rho_{j}^{out}$, where $\rho_{k}^{in}$ and $\rho_{j}^{out}$ are a set of input and output density matrices labeled by integers $j,k=1,2,\cdots$. In order to compute the probabilities of measurement outcomes, $P(\xi|\phi)$, Eq. (11) must be augmented by a detailed model of input and output states. I give several examples of applying Eq. (11) in the sections that follow. The probability of measurement outcome, given by Eq. (11), enters into the Fisher information and into the Shannon mutual information, in Eq. (2) and Eq. (9), respectively. In the next three subsections, A, B, and C, I give examples of the effects of non-deterministic state preparation, state evolution through an interferometer when absorption is present, and imperfect output state detection, respectively, using Fisher and Shannon mutual informations as metrics of performance of the interferometer. ### III.1 Non-Deterministic State Preparation Consider an optical interferometer system that has non-deterministic state preparation, but has no losses in the interferometer and has perfect state detection. When I try to prepare a certain quantum state for input into the interferometer, there is always a non-zero probability that another state than intended will be prepared. This non-deterministic state preparation is expressed by a density matrix for the input state, which assigns probabilities for creating various quantum states, see Eq. (10). Since state detection is assumed perfect, $P_{D}(\xi|\psi^{out},\phi)=1$ when the measurement $\xi$ corresponds to the true state that entered the detection system, $\psi^{out}$, and otherwise $P_{D}(\xi|\psi^{out},\phi)=0$. A general interferometer with no losses is characterized by a unitary scattering matrix, $S_{ij}(\phi)$, that connects the $N_{p}$ input-mode field operators, $\hat{\alpha}_{i}$, to the $N_{p}$ output field operators, $\hat{\beta}_{i}$: $\hat{\beta}_{i}=\sum_{j=1}^{N_{p}}S_{ij}(\phi)\,\hat{\alpha}_{j}=\hat{U}^{\dagger}(\phi)\hat{\alpha}_{i}\hat{U}(\phi)$ (15) where $\hat{U}(\phi)$ is a unitary evolution operator, $i,j=1,2,\cdots,N_{p}$, and $\phi$ is one or more parameters (e.g., phase shift) that describe the state of the interferometer. For simplicity, I consider a Mach-Zehnder interferometer, with no losses, with input ports labeled, “a” and “b”, and output ports, “c” and “d”, having a scattering matrix $S_{ij}(\phi)=\frac{1}{2}\left(\begin{array}[]{cc}-i\left(1+e^{i\phi}\right)&\left(-1+e^{i\phi}\right)\\\ \left(-1+e^{i\phi}\right)&i\left(1+e^{i\phi}\right)\end{array}\right)$ (16) where $\hat{\alpha}_{i}=\\{\hat{a},\hat{b}\\}$ and $\hat{\beta}_{i}=\\{\hat{c},\hat{d}\\}$. The probabilities, $P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)$, that relate the input state $\psi_{k}^{in}$ to the output state $\psi_{j}^{out}$ of the interferometer are given in terms of the projection operators $\hat{\Pi}_{\phi}\left({n_{c},n_{d}}\right)$: $P_{I}(\psi_{j}^{out}|\psi_{k}^{in},\phi)=\left\langle{\psi_{k}^{in}}\right|\,\hat{\Pi}_{\phi}\left({n_{c},n_{d}}\right)\,\left|{\psi_{k}^{in}}\right\rangle$ (17) where the output state $\psi_{j}^{out}$ is specified by two integers, $\left\\{n_{c},n_{d}\right\\}$, giving the photon numbers output in ports “c” and “d”. In terms of the unitary evolution operator, $\hat{U}(\phi)$, the output state in the Schrödinger picture is $\left|\psi^{out}(\phi)\right\rangle=\hat{U}(\phi)\left|\psi^{in}\right\rangle$, where $\left|\psi^{in}\right\rangle$ is the Heisenberg picture input state. In Eq. (11), the sum over $k$ is now a double sum over all non-negative values of the two integers $n_{c}$ and $n_{d}$. For a generic Mach-Zehnder interferometer, with input ports “a” and “b”, and output ports “c” and “d”, the projective operators are Bahder and Lopata (2006a) $\hat{\Pi}_{\phi}\left({n_{c},n_{d}}\right)=\frac{1}{{n_{c}!\,n_{d}!}}\,\left({\hat{c}^{\dagger}}\right)^{n_{c}}\,({\hat{d}^{\dagger}})^{n_{d}}\,\,\left|0\right\rangle\,\left\langle 0\right|\,\,\left({\hat{c}}\right)^{n_{c}}\,({\hat{d}})^{n_{d}}$ (18) where the vacuum state $\left|0\right\rangle=\left|0\right\rangle_{a}\,\otimes\,\left|0\right\rangle_{b}$. As an example, consider the simplest case of input given by a mixed state represented by the density matrix $\hat{\rho}=P_{0}\,\left|0\right\rangle\left\langle 0\right|+P_{1}\,\left|{10}\right\rangle\left\langle{10}\right|$ (19) where $P_{1}$ is the probability for 1-photon input into port “a” and vacuum input into port “b”, and $P_{0}$ is the probability of vacuum input into both ports “a” and “b”, and $P_{0}+P_{1}=1$. In Eqs. (10) and (11), $P_{S}(\psi^{in})=P_{o}$ when $\left|\psi^{in}\right\rangle=\left|0\right\rangle$ and $P_{S}(\psi^{in})=P_{1}$ when $\left|\psi^{in}\right\rangle=\left|10\right\rangle$. Figure 2: (Color) Lossy Mach-Zehnder interferometer is shown, with input modes $a_{1}$, $a_{2}$, $v_{1}$, $v_{2}$, and output modes $b_{1}$, $b_{2}$, $d_{1}$ and $d_{2}$. Here modes $v_{1}$ and $v_{2}$ have vacuum input and the output modes $d_{1}$ and $d_{2}$ are loss channels in each arm. Equation (17) for the interferometer transfer matrix can then be written as $P(n_{c},n_{d}|\phi,\rho)={\rm{tr}}\left({\hat{\rho}\,\hat{\Pi}_{\phi}\left({n_{c},n_{d}}\right)}\right)$ (20) where the input state is represented by the density matrix $\hat{\rho}$. It seems that there can be only two possible measurement outcomes, $\xi=\\{n_{c},n_{d}\\}=\\{1,0\\}$ and $\xi=\\{n_{c},n_{d}\\}=\\{0,1\\}$. However, the probabilities for the two measurement outcomes do not sum to unity because, $P(10|\phi,\rho)+P(01|\phi,\rho)=P_{1}$. Therefore, there is a non-zero probability of an inconclusive measurement outcome associated with the probability $P_{0}$ of vacuum injected into both input ports “a” and “b”. I introduce an inconclusive measurement operator, $\hat{\Pi}_{\phi}\left(i\right)$, so that the sum of the three operators is equal to the identity operator $\hat{I}$: $\hat{\Pi}_{\phi}\left({10}\right)+\hat{\Pi}_{\phi}\left({01}\right)+\hat{\Pi}_{\phi}\left(i\right)=\hat{I}$ (21) Using Eqs.(18)–(21), the probabilities given by Eq. (11), $P(\xi|\phi)$, for measurement outcomes $\xi$ are given by $\displaystyle P(10|\phi)$ $\displaystyle=$ $\displaystyle P_{1}\cos^{2}\left(\frac{\phi}{2}\right)$ (22) $\displaystyle P(01|\phi)$ $\displaystyle=$ $\displaystyle P_{1}\sin^{2}\left(\frac{\phi}{2}\right)$ (23) $\displaystyle P(i|\phi)$ $\displaystyle=$ $\displaystyle 1-P_{1}$ (24) where $P(i|\phi)$ is the probability for an inconclusive measurement outcome. Using Bayes’ rule and Eq. (11), we can write the conditional probability distributions for the phase, $p(\phi|\xi)$, given measurement outcome, $\xi$, as $p(\phi|\xi)=\frac{{P(\xi|\phi)\,p(\phi)}}{{\int\limits_{-\pi}^{+\pi}{P(\xi|\phi^{\prime})\,p(\phi^{\prime})\,d\phi^{\prime}}}}$ (25) where $p(\phi)$ is the prior probability distribution specifying our prior information about the phase $\phi$. Assuming no prior information about the phase, $p(\phi)=1/(2\pi)$, and using Bayes’ rule in Eq. (25), the conditional probability distributions for the phase for a given measurement outcome are given by $\displaystyle p(\phi|10)$ $\displaystyle=$ $\displaystyle\frac{1}{\pi}\cos^{2}\left(\frac{\phi}{2}\right)$ (26) $\displaystyle p(\phi|01)$ $\displaystyle=$ $\displaystyle\frac{1}{\pi}\sin^{2}\left(\frac{\phi}{2}\right)$ (27) $\displaystyle p(\phi|i)$ $\displaystyle=$ $\displaystyle\frac{1}{{2\pi}}$ (28) where $\xi=(n_{c},n_{d})$. As we would expect, for an inconclusive measurement outcome the phase probability distribution, $p(\phi|i)$, is flat since an inconclusive measurement result cannot be used to distinguish different values of the phase. Note that the probabilities for the three measurement outcomes in Eq. (26)–(28) sum to unity. When $P_{0}=0$, or equivalently $P_{1}=1$, state preparation is deterministic, and the probability for an inconclusive measurement outcome is zero. In this case, the probabilities for measurement outcomes $(10)$ and $(01)$ reduce to the values for the case of a single- photon input state created with probability unity. Note that this single photon input case corresponds to the case of a classical interferometer fed by a laser in one input port. Larger photon-number input states have measurement outcome probabilities that differ from the probabilities for a classical interferometer. For the input state in Eq. (19), the classical Fisher information, defined in Eq. (2), is given by $F(\phi)=P_{1}$ (29) where I dropped the subscript ${cl}$ on the classical Fisher information, a convention that I follow in rest of this work. According to the Cramer-Rao bound in Eq. (1), when the probability of creating a single photon approaches zero, $P_{1}\rightarrow 0$, the variance $(\delta\phi)^{2}$ becomes arbitrarily large, because the probability for inconclusive measurement outcomes approaches unity. Assuming no prior information about the phase, therefore taking $p(\phi)=1/(2\pi)$, the fidelity (Shannon mutual information) of the system defined in Eq. (9) is given by $H(M)=P_{1}\,\left({\frac{1}{{\ln 2}}-1}\right)$ (30) Similar to the Fisher information, the fidelity $H(M)$ also approaches zero when the probability $P_{1}$ of having one photon in the input of each shot approaches zero. The fidelity is the amount of information (in bits) that is gained on average about the phase from a single use of the interferometer, averaged over all possible phase values $\phi$. ### III.2 Lossy Mach-Zehnder Interferometer Next, I consider an interferometer with absorption losses—so state evolution is non-unitary. I assume that state preparation is deterministic (ideal) and that state detection is perfect (no errors). Equation (11) is general enough to describe processes other than losses in the interferometer, such as photons scattering into the interferometer from the environment, in which case there are more photons leaving the output ports than entering the input ports. However, in what follows, I restrict myself to simple absorption in the interferometer. I model losses in each arm of a Mach-Zehnder interferometer by inserting two beam splitters, $S_{3}$ and $S_{4}$, one in each path, see Fig. 2. While a lossless Mach-Zehnder interferometer has two input and two output ports, a general lossy Mach-Zehnder interferometer can be represented by four input and four output ports, see Fig. 2. I label the input modes as $a_{1}$, $a_{2}$, $v_{1}$, and $v_{2}$, where $v_{1}$, and $v_{2}$ have vacuum input and I label the output modes as $b_{1}$, $b_{2}$, $d_{1}$, and $d_{2}$, where $d_{1}$, and $d_{2}$ are the modes where probability amplitude is “dissipated”. I take the phase shifts at the two mirrors, $M_{1}$ and $M_{2}$ to be equal to $\pi$. Furthermore, I assume that the interferometer is balanced, so that path lengths satisfy, $\displaystyle L$ $\displaystyle=$ $\displaystyle l_{1}+l_{3}+l_{5}=l_{2}+l_{4}+l_{6}$ $\displaystyle l$ $\displaystyle=$ $\displaystyle l_{1}+l_{3}=l_{2}+l_{4}$ (31) see Fig. 2. A calculation gives the input and output modes related by the 4$\times$4 unitary scattering matrix $S_{ij}(\phi)$ $\left({\begin{array}[]{*{20}c}{b_{1}}\\\ {b_{2}}\\\ {d_{1}}\\\ {d_{2}}\\\ \end{array}}\right)=\left[{\begin{array}[]{*{20}c}{}\hfil&{}\hfil&{}\hfil\\\ {}\hfil&{S_{ij}\left(\phi\right)}&{}\hfil\\\ {}\hfil&{}\hfil&{}\hfil\\\ \end{array}}\right]\cdot\left({\begin{array}[]{*{20}c}{a_{1}}\\\ {a_{2}}\\\ {v_{1}}\\\ {v_{1}}\\\ \end{array}}\right)$ (32) where the phase-dependent scattering matrix is given by $S_{ij}(\phi)=\left[{\begin{array}[]{*{20}c}{\frac{i}{2}e^{i\frac{{L\omega}}{c}}\left({\sqrt{1-r_{y}^{2}}-e^{i\phi}\sqrt{1-r_{x}^{2}}}\right)}&{-\frac{1}{2}e^{i\frac{{L\omega}}{c}}\left({\sqrt{1-r_{y}^{2}}+e^{i\phi}\sqrt{1-r_{x}^{2}}}\right)}&{\frac{i}{{\sqrt{2}}}r_{x}\,e^{i\left({L-l}\right)\frac{\omega}{c}}}&{\frac{1}{{\sqrt{2}}}r_{y}\,e^{i\left({L-l}\right)\frac{\omega}{c}}}\\\ {-\frac{1}{2}e^{i\frac{{L\omega}}{c}}\left({\sqrt{1-r_{y}^{2}}+e^{i\phi}\sqrt{1-r_{x}^{2}}}\right)}&{-\frac{i}{2}e^{i\frac{{L\omega}}{c}}\left({\sqrt{1-r_{y}^{2}}-e^{i\phi}\sqrt{1-r_{x}^{2}}}\right)}&{\frac{1}{{\sqrt{2}}}r_{x}\,e^{i\left({L-l}\right)\frac{\omega}{c}}}&{\frac{i}{{\sqrt{2}}}r_{y}\,e^{i\left({L-l}\right)\frac{\omega}{c}}}\\\ {-\frac{i}{{\sqrt{2}}}r_{x}\,e^{i\left({\phi+\frac{{l\omega}}{c}}\right)}}&{-\frac{1}{{\sqrt{2}}}r_{x}\,e^{i\left({\phi+\frac{{l\omega}}{c}}\right)}}&{-i\sqrt{1-r_{x}^{2}}}&0\\\ {-\frac{1}{{\sqrt{2}}}r_{y}\,e^{i\frac{{l\omega}}{c}}}&{-\frac{i}{{\sqrt{2}}}r_{y}\,e^{i\frac{{l\omega}}{c}}}&0&{-i\sqrt{1-r_{y}^{2}}}\\\ \end{array}}\right]$ (33) It is easy to check that the scattering matrix is unitary, $S^{\dagger}\,S=I$ where $I$ is the 4$\times$4 unit matrix. The parameters, $r_{x}$ and $r_{y}$, are the reflection amplitudes for beams splitters $S_{3}$ and $S_{4}$, respectively, and they represent the strength of the loss or dissipation, see Fig. 2. When the system is considered in terms of two input modes, $a_{1}$ and $a_{2}$, and two output modes, $b_{1}$ and $b_{2}$, the evolution of the input state in not unitary. The 4$\times$4 unitary scattering matrix $S_{ij}(\phi)$ has some simple properties. The case when $r_{x}=r_{y}=0$ corresponds to no dissipation. In this case, the 4$\times$4 S-matrix reduces to two diagonal 2$\times$2 blocks. The upper left 2$\times$2 block couple modes $a_{1}$ and $a_{2}$ to modes $b_{1}$ and $b_{2}$, and this 2$\times$2 block (up to a phase) is given by Eq. (16), which is the scattering matrix for the Mach-Zehnder interferometer with no losses. For this case of no loss, in Eq. (33) the lower right 2$\times$2 block couples the dissipative modes, $d_{1}$, and $d_{2}$, to the vacuum modes, $v_{1}$, and $v_{2}$. The case $r_{x}=r_{y}=1$ corresponds to maximum dissipation, and the 4$\times$4 S-matrix again decouples, into two off-diagonal 2$\times$2 blocks. The upper right 2$\times$2 block couples the two vacuum modes, $v_{1}$ and $v_{2}$, to the two output modes, $b_{1}$ and $b_{2}$. The lower left 2$\times$2 block of this S-matrix couples the loss modes, $d_{1}$, and $d_{2}$, to the input modes $a_{1}$ and $a_{2}$. For this case of maximum dissipation, the input modes, $a_{1}$ and $a_{2}$, are decoupled from the output modes, $b_{1}$ and $b_{2}$. The probabilities for various measurement outcomes $\xi=(n,m)$ are given by the analog of Eq. (3): $P(n,m|\phi,\rho)={\rm{tr}}\left({\hat{\rho}_{o}\,\hat{\Pi}_{\phi}(n,m)}\right)$ (34) where $n$ and $m$ are the number of photons leaving ports $b_{1}$ and $b_{2}$, respectively. The trace is over the complete space of four direct product Fock basis states, ${\left|n_{1}\right\rangle_{a_{1}}\otimes\left|n_{2}\right\rangle_{a_{2}}\otimes\left|n_{3}\right\rangle_{v_{1}}\otimes\left|n_{4}\right\rangle_{v_{2}}}$. The input state density matrix, $\hat{\rho}_{o}$, is defined in terms of a sum of products of creation operators $\hat{a}_{1}^{\dagger}$, $\hat{a}_{2}^{\dagger}$, $\hat{v}_{1}^{\dagger}$ and $\hat{v}_{2}^{\dagger}$ acting on the vacuum $\left|0\right\rangle=\left|0\right\rangle_{a_{1}}\otimes\left|0\right\rangle_{a_{2}}\otimes\left|0\right\rangle_{v_{1}}\otimes\left|0\right\rangle_{v_{2}}$ and has the form $\hat{\rho}_{0}=\sum\limits_{n,m}{c_{nm}\,\left|{nm00}\right\rangle}\;\left\langle{nm00}\right|$ (35) where I use the short-hand notation ${\left|{nm00}\right\rangle\equiv\left|n\right\rangle_{a_{1}}\otimes\left|m\right\rangle_{a_{2}}\otimes\left|0\right\rangle_{v_{1}}\otimes\left|0\right\rangle_{v_{2}}}$. I am using the Heisenberg picture, so the input density matrix $\hat{\rho}_{o}$ is independent of time (phase), while the operators $\hat{\Pi}_{\phi}\left({n,m}\right)$ evolve in time (phase) and so they depend on $\phi$. The projective measurement operators are given by $\hat{\Pi}_{\phi}\left({n,m}\right)=\frac{1}{{n!\,m!}}\;\sum\limits_{k,l=0}^{\infty}{\frac{1}{{k!\,l!}}\left({\hat{b}_{1}^{\dagger}}\right)^{n}\left({\hat{b}_{2}^{\dagger}}\right)^{m}\left({\hat{d}_{1}^{\dagger}}\right)^{k}\left({\hat{d}_{2}^{\dagger}}\right)^{l}\left|0\right\rangle\left\langle 0\right|\left({\hat{b}_{1}}\right)^{n}\left({\hat{b}_{2}}\right)^{m}\left({\hat{d}_{1}}\right)^{k}\left({\hat{d}_{2}}\right)^{l}}$ (36) where $\hat{b}_{1}^{\dagger}$, $\hat{b}_{2}^{\dagger}$, $\hat{d}_{1}^{\dagger}$, and $\hat{d}_{2}^{\dagger}$ are creation operators for the output modes $b_{1}$, $b_{2}$, $d_{1}$, and $d_{2}$, respectively. The sums over $k$ and $l$ take into account the probabilities for losing photons into ports $d_{1}$ and $d_{2}$. I use the short-hand notation for the input modes $\\{\alpha_{i}\\}=\\{\hat{a}_{1},\hat{a}_{2},\hat{v}_{1},\hat{v}_{2}\\}$ and for the output modes $\\{\beta_{i}\\}=\\{\hat{b}_{1},\hat{b}_{2},\hat{d}_{1},\hat{d}_{2}\\}$, see Eq. (15). Note that, even though dissipation is being modeled, it is easy to check that there is global photon number conservation, $\sum\limits_{i=0}^{4}{\hat{\alpha}_{i}^{\dagger}\hat{\alpha}_{i}=}\sum\limits_{i=0}^{4}{\hat{\beta}_{i}^{\dagger}\hat{\beta}_{i}}$ (37) since the S-matrix in Eq. (33) is unitary. The measurement outcomes, $\xi=(n,m)$, are specified by two integers, which label the number of photons that are output at ports $b_{1}$ and $b_{2}$, see Fig. 2. So the probability of output state, $\psi_{j}^{out}$, as given in Eq.(34), is expressed by the two integers $n$ and $m$, see also Eq. (17). Note that in general the number of photons $n+m$ in the output state, $\psi_{j}^{out}$, is not equal to the number of photons in the input state, $\psi_{in}$, because $\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}\neq\hat{b}_{1}^{\dagger}\hat{b}_{1}+\hat{b}_{2}^{\dagger}\hat{b}_{2}$ (38) However, the sum of probabilities for all possible measurement outcomes is unity: $\sum\limits_{m,n=0}^{\infty}{P(n,m|\phi,\rho)}=1$ (39) where $P(n,m|\phi,\rho)$ is given by Eq. (34), and is a result of $\sum\limits_{n,m}{\hat{\Pi}_{\phi}\left({n,m}\right)}=\hat{I}$ (40) For the case of a pure input state, $\left|{\psi_{in}}\right\rangle$, it is convenient to define the operators $\hat{N}\left({n,m,k,l}\right)=\frac{1}{{n!\,m!\,k!\,l!}}\left({\hat{b}_{1}}\right)^{n}\left({\hat{b}_{2}}\right)^{m}\left({\hat{d}_{1}}\right)^{k}\left({\hat{d}_{2}}\right)^{l}$ (41) and the probabilities in Eq. (34) are then given by $P(n,m|\phi,\psi_{in})=\sum\limits_{k,l=1}^{\infty}{\left|{\left\langle 0\right|\hat{N}\left({n,m,k,l}\right)\left|{\psi_{in}}\right\rangle}\right|}^{2}$ (42) Equation (34), or Eq. (42) for the case of pure states, defines a unitary mapping, $|nm00\rangle\rightarrow|n^{\prime}m^{\prime}kl\rangle$, between interferometer input states, $|nm00\rangle$, and output states, $|n^{\prime}m^{\prime}kl\rangle$, because the photon number is conserved: ${n+m=n^{\prime}+m^{\prime}+k+l}$, see Eq. (37). If we restrict our attention to measurement outcomes projected onto the Hilbert subspace with basis ${\left|n^{\prime}\right\rangle_{b_{1}}\otimes\left|m^{\prime}\right\rangle_{b_{2}}}$, Eq. (34) or Eq. (42) defines the non-unitary mapping ${\cal E}$: ${\cal E}\left[\left|n\right\rangle_{a_{1}}\otimes\left|m\right\rangle_{a_{2}}\right]\rightarrow\left|n^{\prime}\right\rangle_{b_{1}}\otimes\left|m^{\prime}\right\rangle_{b_{2}}$ (43) since photon number is not conserved: $n+m\neq n^{\prime}+m^{\prime}$, see Eq. (38), which represents losses in the Mach-Zehnder interferometer. Here the Fock states ${|n^{\prime}\rangle_{b_{i}}}$, for $i=1,2$, are created from the vacuum state, $|0\rangle_{b_{i}}$, by application of creation operators $\hat{b}_{i}^{\dagger}$ in the usual way. The mapping ${\cal E}$ depends on two parameters, $r_{x}$ and $r_{y}$, which specify the strength of the dissipation or losses in each arm of the interferometer. In the limit of no dissipation, when $r_{x}=0$ and $r_{y}=0$, the mapping ${\cal E}$ becomes a unitary transformation and $n+m=n^{\prime}+m^{\prime}$. In what follows, I use the short-hand notation $|nm\rangle$ for the input state $|nm00\rangle\equiv|n\rangle_{a_{1}}\otimes|m\rangle_{a_{2}}\otimes|0\rangle_{v_{1}}\otimes|0\rangle_{v_{2}}$. In the next two subsections, III.3 and III.4, I discuss the Fisher information and the fidelity (Shannon mutual information) for specific cases of few-photon Fock state and N00N state input into the lossy Mach-Zehnder (MZ) interferometer. ### III.3 Fock State Input into Lossy MZ Interferometer Consider the $N$-photon Fock state $\left|{\psi_{N}}\right\rangle=\frac{1}{{\sqrt{N!}}}\left({\hat{a}_{1}^{\dagger}}\right)^{N}\,\left|0\right\rangle=\left|{N000}\right\rangle\equiv\left|{N0}\right\rangle$ (44) input into a lossy Mach-Zehnder interferometer given by the scattering matrix in Eq. (33). The probabilities for measurement outcomes $\xi=(n,m)$ are given by: $\displaystyle P(n,m|\phi,\psi_{N})=\frac{N!}{n!m!}\sum\limits_{k=0}^{N}\sum\limits_{l=0}^{N}\frac{1}{k!l!}\times$ $\displaystyle\left|S_{11}^{n}S_{21}^{m}S_{31}^{k}S_{41}^{l}\right|^{2}\,\delta_{n+m+k+l,N}$ (45) where $S_{ij}$ are the matrix elements of Eq. (33) and $\delta_{m,n}$ is the Kronecker delta function. A direct calculation of the classical Fisher information for the $N$-photon Fock state input, $F_{N}(\phi)$, gives $F_{N}(\phi)=NF_{1}(\phi)$ (46) where $F_{1}(\phi)$ is the classical Fisher information for one-photon input, given in Eq. (53). This shows that for the lossy MZ interferometer with $N$-photon Fock state input, the standard deviation $(\delta\phi)$ scales as $1/\sqrt{N}$. From another point of view, since the Fisher information is additive for independent events, the $N$-photon Fock state acts like $N$ independent 1-photon states. When $r_{x}=r_{y}$, then $F_{1}(\phi)=1$, and the $N$-photon Fisher information becomes $F_{N}(\phi)=N$. This means that dissipation in the (non-unitary) interferometer has the effect of introducing a phase dependence into the Fisher information, see the discussion below. #### III.3.1 1-Photon Fock State Input into Lossy MZ Interferometer As the simplest example of the effect of dissipation, I consider the 1-photon Fock state input into the lossy Mach-Zehnder interferometer with scattering matrix given by Eq. (33) $\left|\psi_{in}\right\rangle=\hat{a}_{1}^{{\dagger}}\left|0\right\rangle=\left|1000\right\rangle\equiv\left|10\right\rangle$ (47) where I use the short-hand notation $\left|10\right\rangle$ for the input state $\left|1000\right\rangle$. The probabilities for measurement outcomes are given by $P(n,m|\phi,\psi_{in})$, where $n,m$ specify the photon numbers output in port $b_{1}$ and $b_{2}$, respectively, see Fig. 2. The probabilities $P(n,m|\phi,\psi_{in})$ for the three measurement outcomes are: $\small\begin{array}[]{lllll}P(10|\phi,10)&=&\frac{1}{4}\left({2-r_{x}^{2}-r_{y}^{2}-2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\phi}\right)\\\ P(01|\phi,10)&=&\frac{1}{4}\left({2-r_{x}^{2}-r_{y}^{2}+2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\phi}\right)\\\ P(00|\phi,10)&=&\frac{1}{2}\left({r_{x}^{2}+r_{y}^{2}}\right)\\\ \end{array}$ (48) The probability $P(00|\phi,10)$ is associated with an inconclusive measurement outcome, since for this case zero photons leave the output ports, i.e., the photon that entered in port “a” was absorbed in the interferometer, or more precisely the photon was output in either port $d_{1}$ or $d_{2}$. From Bayes’ rule in Eq. (25), the phase probability distributions, $p(\phi|m\,n,\psi_{in})$, for input state $\left|\psi_{in}\right\rangle$ given in Eq. (47), are given by $\begin{array}[]{lllll}p(\phi|10,10)&=&\frac{1}{{2\pi}}\frac{{2-r_{x}^{2}-r_{y}^{2}-2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\phi}}{{2-r_{x}^{2}-r_{y}^{2}}}\\\ p(\phi|01,10)&=&\frac{1}{{2\pi}}\frac{{2-r_{x}^{2}-r_{y}^{2}+2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\phi}}{{2-r_{x}^{2}-r_{y}^{2}}}\\\ p(\phi|00,10)&=&\frac{1}{{2\pi}}\\\ \end{array}$ (49) When $r_{x}\neq r_{y}$, there is a loss of contrast in the phase probability distributions $p(\phi|m\,n,\psi_{in})$, see Fig. 3. When the absorption probabilities are the same in both arms, in the limit $r_{x}=r_{y}$, the phase probability distributions in Eq. (49) reduce to the case of a single photon input without losses in the interferometer, which are given in Eq. (26)-(28), with trivial phase change given by the replacements $\sin\rightarrow\cos$. The phase probability density, $p(\phi|00,10)$, is associated with the inconclusive outcome, $p(\phi|i)$. It is a remarkable feature that for equal loss in both arms, $r_{x}=r_{y}$, the phase probability densities, $p(\phi|m\,n,\psi_{in})$, in Eq.(49) do not depend on the size of the loss, $r_{x}$. However, there is loss of information with increasing absorption, $r_{x}$ and $r_{y}$, which is reflected in the information measures, see below. The effect of equal dissipation in both arms of the interferometer is the same as the effect of non-deterministic state preparation, specified by input state characterized by a density matrix in Eq. (19). However, when the dissipation in both arms is not equal, say for $r_{y}=0$ and $r_{x}$ is finite, the phase probability distributions show a loss of contrast, see Fig. 3. This feature may be useful in applications to null-type measurements. Figure 3: (Color) For the 1-photon input state $\left|10\right\rangle$, given by Eq. (47), the probability distribution for the phase, $p(\phi|10,10)$, in Eq. (49) is plotted for absorption $r_{y}=0$ and $r_{x}=$0.0, 0.90, 0.95, 0.98, and 1.0. As $r_{x}\rightarrow$1.0, the probability distribution becomes flat and does not distinguish between different phase values. Figure 4: (Color) For 1-photon Fock input state, $\left|10\right\rangle$, given by Eq. (47), the fidelity (Shannon mutual information) $H(M)$ seems linear in $r_{x}^{2}$ for $r_{x}=r_{y}$, however, this is not true for general values of $r_{x}$ and $r_{y}$, see also Fig. 5. Figure 5: (Color) For 1-photon Fock input state $\left|1000\right\rangle=\left|10\right\rangle$, given by Eq. (47), the fidelity (Shannon mutual information) $H(M)$ is plotted as a function of loss parameters $r_{x}^{2}$ and $r_{y}^{2}$. Figure 6: (Color) The Fisher information $F(\phi)$ for 1-photon input state $\left|1000\right\rangle$ (short-hand notation $\left|10\right\rangle$), is plotted as a function of $r_{x}$ and $r_{y}$, for different values of $\phi=0,0.125,\pi/2,3.0,3.1,\pi$, left to right in top row and bottom row. In the discussion that follows, I assume no prior information on the phase, so I take $p(\phi)=1/(2\pi)$. When there is no loss in the interferometer, $r_{x}=r_{y}=0$, the Shannon mutual information (fidelity) as defined in Eq. (9) is a constant: $H(M)=\frac{1}{{\ln 2}}-1$ (50) When the losses in both arms are equal, $r_{x}=r_{y}$, we have the exact result $H(M)=\left({\frac{1}{{\ln 2}}-1}\right)\left({1-r_{x}^{2}}\right)$ (51) For general values of $r_{x}$ and $r_{y}$, the expression for $H(M)$ is large and complicated, but for small ${r_{x}\ll 1}$ and ${r_{y}\ll 1}$, I can expand it in a power series, $\small H(M)=\left({\frac{1}{{\ln 2}}-1}\right)\left[{1-\frac{1}{2}\left({r_{x}^{2}+r_{y}^{2}}\right)}\right]+O(r_{x}^{4})+O(r_{y}^{4})$ (52) where I dropped fourth order terms in $r_{x}$ and $r_{y}$. When $r_{x}=r_{y}$, from Eq. (52), we may expect the fidelity (Shannon mutual information) $H(M)$ to be quadratic in $r_{x}$, however, this is not true for general values $r_{x}$ and $r_{y}$, see Fig. 4, which shows $H(M)$ vs. $r_{x}^{2}$ for the case $r_{x}=r_{y}$ and for $r_{y}=0$. In Figure 5 the fidelity (Shannon mutual information) $H(M)$ is plotted as a function of the dissipation parameters, $r_{x}$ and $r_{y}$. When the dissipation in either arm is a maximum, $r_{x}=1$ or $r_{y}=1$, the fidelity $H(M)=0$, indicating that we obtain zero information from each photon. For the 1-photon Fock state (given in Eq. (47)) input into the lossy Mach- Zehnder interferometer given in Eq. (33), the classical Fisher information (defined by Eq. (2)) is given by: $F_{1}(\phi)=\frac{2\left(1-r_{x}^{2}\right)\left(1-r_{y}^{2}\right)\left(2-r_{x}^{2}-r_{y}^{2}\right)\sin^{2}(\phi)}{\left(2-r_{x}^{2}-r_{y}^{2}\right)^{2}-4\left(1-r_{x}^{2}\right)\left(1-r_{y}^{2}\right)\cos^{2}(\phi)}$ (53) see plots in Fig. 6. From these plots, it is clear that for a lossy interferometer, the Fisher information depends strongly on the true value of $\phi$. Through the Cramer-Rao bound in Eq. (1), this translates to a dependence of the variance $(\delta\phi)^{2}$ on the true value of $\phi$. Therefore, Fisher information $F_{1}(\phi)$ for 1-photon Fock state input into a lossy interferometer is qualitatively different than for an ideal interferometer without loss, see Eq. (53) for the case $r_{x}=r_{y}=0$. As a consequence of Eq.(46), the Fisher information for $N$-photon Fock state input into a lossy interferometer is qualitatively different than for a lossless interferometer, where it is a constant given by $F_{N}(\phi)=N$. Specifically, for a lossy interferometer the Fisher information depends on the value of the true phase, see plots in Fig. 6. For values of the phase given by $\phi=0$ and $\phi=\pm\pi$, the Fisher information vanishes for Fock state input, independent of the value of the dissipation parameters, $r_{x}$ and $r_{y}$, see comment 111This statement applies to a balanced interferometer, whose path lengths satisfy Eq. (31).. According to the Cramer-Rao bound, the variance $(\delta\phi)^{2}$, is large for values of true phase near $\phi=0$ and $\phi=\pm\pi$. However, when there is no dissipation, $r_{x}=r_{y}=0$, the Fisher information is independent of $\phi$ and so is the bound on the variance $(\delta\phi)^{2}$. The presence of dissipation in the interferometer introduces a dependence of the variance, $(\delta\phi)^{2}$, on true phase $\phi$. The exception to this is when $r_{x}=r_{y}$, where $F_{1}(\phi)$ reduces to $F_{1}(\phi)=1-r_{x}^{2}$, and is independent of $\phi$. #### III.3.2 2-Photon Fock State Input into Lossy MZ Interferometer Consider now the 2-photon Fock state input into the lossy Mach-Zehnder interferometer with S-matrix given by Eq. (33): $\left|\psi_{in}\right\rangle=\frac{1}{{\sqrt{2}}}\left({a_{1}^{\dagger}}\right)^{2}\left|0\right\rangle=\left|2000\right\rangle\equiv\left|20\right\rangle$ (54) where I use the short-hand notation $\left|20\right\rangle$ for the state $\left|2000\right\rangle$. The probabilities $P(m,n|\phi,\psi_{in})$ for the six measurement outcomes are given by $\displaystyle P(20|\phi,20)$ $\displaystyle=$ $\displaystyle\frac{1}{{16}}\left[{6-6r_{x}^{2}-6r_{y}^{2}+4\sqrt{\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)}\left({r_{x}^{2}+r_{y}^{2}-2}\right)\cos(\phi)+2\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)\cos(2\phi)+4r_{x}^{2}r_{y}^{2}+r_{x}^{4}+r_{y}^{4}}\right]$ $\displaystyle P(02|\phi,20)$ $\displaystyle=$ $\displaystyle\frac{1}{{16}}\left[{6-6r_{x}^{2}-6r_{y}^{2}-4\sqrt{\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)}\left({r_{x}^{2}+r_{y}^{2}-2}\right)\cos(\phi)+2\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)\cos(2\phi)+4r_{x}^{2}r_{y}^{2}+r_{x}^{4}+r_{y}^{4}}\right]$ $\displaystyle P(11|\phi,20)$ $\displaystyle=$ $\displaystyle\frac{1}{8}\left[{2-2r_{x}^{2}-2r_{y}^{2}+r_{x}^{4}+r_{y}^{4}-2\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)\cos\left({2\phi}\right)}\right]$ $\displaystyle P(10|\phi,20)$ $\displaystyle=$ $\displaystyle\frac{1}{4}\left({r_{x}^{2}+r_{y}^{2}}\right)\left[{2-r_{x}^{2}-r_{y}^{2}-2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\left(\phi\right)}\right]$ $\displaystyle P(01|\phi,20)$ $\displaystyle=$ $\displaystyle\frac{1}{4}\left({r_{x}^{2}+r_{y}^{2}}\right)\left[{2-r_{x}^{2}-r_{y}^{2}+2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\,\cos\left(\phi\right)}\right]$ $\displaystyle P(00|\phi,20)$ $\displaystyle=$ $\displaystyle\frac{1}{4}\left(r_{x}^{2}+r_{y}^{2}\right)^{2}$ (55) The sum of the probabilities for the six possible measurement outcomes in Eq. (55) is unity. Since a two-photon Fock state has been used as input, the probability that no photon was absorbed is equal to the the sum $P(20|\phi)+P(02|\phi)+P(11|\phi)=\frac{1}{4}\left({2-r_{x}^{2}-r_{y}^{2}}\right)^{2}$, while the probability that exactly one photon was absorbed is $P(10|\phi)+P(01|\phi)=\frac{1}{2}\left({2r_{x}^{2}+2r_{y}^{2}-2r_{x}^{2}r_{y}^{2}-r_{x}^{4}-r_{y}^{4}}\right)$. The conditional probability distributions for the phase, $p(\phi|m\,n,\psi_{in})$ , with input state $\left|\psi_{in}\right\rangle$ in Eq. (54) and measurement outcome $\xi=(m,n)$ are given by $\displaystyle p(\phi|20,20)$ $\displaystyle=$ $\displaystyle\frac{1}{{2\pi}}\left[{1+\frac{{4\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\left({r_{x}^{2}+r_{y}^{2}-2}\right)\cos(\phi)+2\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)\cos(2\phi)}}{{6-6(r_{x}^{2}+r_{y}^{2})+r_{x}^{4}+r_{y}^{4}+4r_{x}^{2}r_{y}^{2}}}}\right]$ $\displaystyle p(\phi|02,20)$ $\displaystyle=$ $\displaystyle\frac{1}{{2\pi}}\left[{1-\frac{{4\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\left({r_{x}^{2}+r_{y}^{2}-2}\right)\cos(\phi)+2\left({r_{x}^{2}-1}\right)\left({r_{y}^{2}-1}\right)\cos(2\phi)}}{{6-6(r_{x}^{2}+r_{y}^{2})+r_{x}^{4}+r_{y}^{4}+4r_{x}^{2}r_{y}^{2}}}}\right]$ $\displaystyle p(\phi|11,20)$ $\displaystyle=$ $\displaystyle\frac{1}{{2\pi}}\left[{1-\frac{{2\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)\cos(2\phi)}}{{2-2(r_{x}^{2}+r_{y}^{2})+r_{x}^{4}+r_{y}^{4}}}}\right]$ $\displaystyle p(\phi|10,20)$ $\displaystyle=$ $\displaystyle\frac{1}{{2\pi}}\left[{1-\frac{{2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\cos(\phi)}}{{2-r_{x}^{2}-r_{y}^{2}}}}\right]$ $\displaystyle p(\phi|01,20)$ $\displaystyle=$ $\displaystyle\frac{1}{{2\pi}}\left[{1+\frac{{2\sqrt{\left({1-r_{x}^{2}}\right)\left({1-r_{y}^{2}}\right)}\cos(\phi)}}{{2-r_{x}^{2}-r_{y}^{2}}}}\right]$ $\displaystyle p(\phi|00,20)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi}$ In the limit of high photon absorption, $r_{x}=1$ and $r_{y}=1$, the phase probability densities $p(\phi|m\,n,\,20)=1/(2\pi)$ for all measurement outcomes, so that different phase values are not distinguishable because of photon losses. A remarkable property of the phase probability distributions in Eq. (LABEL:2-PhotonPhaseProbs) (similar to that of Fock state input in Eq. (49)) is that for equal losses in both arms of the interferometer, $r_{x}=r_{y}$, the phase probability distributions are independent of the magnitude of the loss $r_{x}$, having values $\tiny\frac{1}{\pi}\left\\{{\frac{4}{3}\sin^{4}\left({\frac{\phi}{2}}\right),\frac{4}{3}\cos^{4}\left({\frac{\phi}{2}}\right),\sin^{2}(\phi),\sin^{2}\left({\frac{\phi}{2}}\right),\cos^{2}\left({\frac{\phi}{2}}\right),\,\frac{1}{2}}\right\\}$ (57) where I have written the values of the functions (from top to bottom) in Eq. (LABEL:2-PhotonPhaseProbs) as components of a vector left to right in Eq. (57). One may think that for $r_{x}=r_{y}$ the contrast in the photon number measurements is lost, however, setting $r_{x}=r_{y}$ in Eq. (55) and expanding for small $r_{x}\ll 1$ leads to $\left({\begin{array}[]{*{20}c}{P(20|\phi,20)}\\\ {P(02|\phi,20)}\\\ {P(11|\phi,20)}\\\ {P(10|\phi,20)}\\\ {P(01|\phi,20)}\\\ {P(00|\phi,20)}\\\ \end{array}}\right)=\left({\begin{array}[]{*{20}c}{\sin^{4}\left({\frac{\phi}{2}}\right)-2r_{x}^{2}\sin^{4}\left({\frac{\phi}{2}}\right)+O\left({r_{x}^{4}}\right)}\\\ {\cos^{4}\left({\frac{\phi}{2}}\right)-2r_{x}^{2}\cos^{4}\left({\frac{\phi}{2}}\right)+O\left({r_{x}^{4}}\right)}\\\ {\frac{{\sin^{2}(\phi)}}{2}-r_{x}^{2}\sin^{2}(\phi)+O\left({r_{x}^{4}}\right)}\\\ {r_{x}^{2}(1-\cos(\phi))+O\left({r_{x}^{4}}\right)}\\\ {r_{x}^{2}(\cos(\phi)+1)+O\left({r_{x}^{4}}\right)}\\\ {O\left({r_{x}^{4}}\right)}\\\ \end{array}}\right)$ (58) which shows that the probabilities become vanishingly small for the measurement outcomes $P(10|\phi,20)$, $P(01|\phi,20)$, and $P(00|\phi,20)$, which correspond to one or both photons being absorbed, but contrast for different measurement outcomes is not lost. For the case of equal loss in both arms, $r_{x}=r_{y}$, with no prior information on the phase, $p(\phi)=1/(2\pi)$, the fidelity is given by $\small H(M)=\frac{1}{{4\ln 2}}\left({1-r_{x}^{2}}\right)\left[{8-4\ln 2-3\ln 3+2r_{x}^{2}{\rm{arctanh}}\left({\frac{{11}}{{43}}}\right)}\right]$ (59) For the case of small (but not equal) losses in both arms, $r_{x}\ll 1$ and $r_{y}\ll 1$, the fidelity is given by $\displaystyle H(M)=$ $\displaystyle\frac{{8-4\ln 2-3\ln 3}}{{4\ln 2}}+\left({r_{x}^{2}+r_{y}^{2}}\right)\left({\frac{{3\ln 3}}{{4\ln 2}}-\frac{1}{{\ln 2}}}\right)+$ (60) $\displaystyle r_{x}^{2}r_{y}^{2}\left({\frac{{1+\ln 2-\ln 3}}{{2\ln 2}}}\right)+O\left({r_{x}}\right)^{4}+O\left({r_{y}}\right)^{4}$ where I have dropped terms of fourth order in $r_{x}$ and $r_{y}$. Equation (60) gives the dependence on the dissipation parameters, $r_{x}$ and $r_{y}$, of the information gain about $\phi$, for single use of the interferometer, when there is no prior information about $\phi$. ### III.4 N00N State Input into Lossy MZ Interferometer Next, I consider the $N$-photon N00N state input into the lossy Mach-Zehnder interferometer with scattering matrix given by Eq. (33): $\displaystyle\left|{\psi_{N00N}}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{{\sqrt{2N!}}}\left[{\left({\hat{a}_{1}^{\dagger}}\right)^{N}+\left({\hat{a}_{2}^{\dagger}}\right)^{N}}\right]\left|0\right\rangle$ (61) $\displaystyle=$ $\displaystyle\frac{1}{{\sqrt{2}}}\left[{\left|{N000}\right\rangle+\left|{0N00}\right\rangle}\right]$ (62) For the input state in Eq. (62), the probability for measurement outcome $\xi=(n,m)$ is given by $\displaystyle P(n,m|\phi,\psi_{N00N})=\frac{N!}{2n!m!}\sum\limits_{k=0}^{N}\sum\limits_{l=0}^{N}\frac{1}{k!l!}\times$ $\displaystyle\left|S_{11}^{n}S_{21}^{m}S_{31}^{k}S_{41}^{l}+S_{12}^{n}S_{22}^{m}S_{32}^{k}S_{42}^{l}\right|^{2}\,\delta_{n+m+k+l,N}$ (63) where $n$ and $m$ are the number of photons output in ports $b_{1}$ and $b_{2}$, respectively. Using the Fisher information, I compare how well Fock states and N00N states perform in the presence of absorption losses. In Fig. 7, I plot the classical Fisher information for $N=$3, 4, and 5 photon Fock states and N00N states, plotted vs. $r_{x}$ for the special case where $r_{x}=r_{y}$. The plots show that, for equal dissipation in both arms, and for equal photon number, Fock states perform better for phase estimation than N00N states, for the same amount of dissipation $r_{x}$, see Eq. (1). For N00N states, the Fisher information vanishes at the phase values: $\phi=0,\pm\pi/2,\pm\pi$. While for Fock states, the Fisher information vanished only at $\phi=0,\pm\pi$. For $\phi$ close to these values, phase estimation may have large standard deviation, see Eq.(1). Figure 8 shows the classical Fisher information for the case where Fock and N00N states are input into a Mach-Zehnder interferometer with small dissipation (losses) and when the losses are not equal in both arms. Generally, Fock states perform better (have larger Fisher information) for all values of dissipation $r_{x}$ except at the very highest values of $r_{x}\sim 0.95$. The comparison is made at a true value of $\phi=\pi/4$, where the Fisher informations do not vanish. Figure 9 shows a plot of the classical Fisher information for Fock and for N00N states for $N=$ 3, 4 and 5 photons for the case of large dissipation in one arm of the Mach-Zehnder interferometer, $r_{y}=0.9$. The comparison is complicated, since for the $N=$3 photon case, Fock states perform better than N00N states for large dissipation $r_{x}\sim 0.8$, whereas the situation is reversed for small dissipation $r_{x}\sim 0.05$. In Figures 8 and 9, the comparisons are made at a true value of phase $\phi=\pi/4$, where the classical Fisher informations (for Fock and N00N states) do not vanish. When dissipation is present, the classical Fisher information has a complicated behavior as a function of the true phase $\phi$, see Fig. 10. This shows that phase estimation using simple photon counting is a sensitive procedure whose accuracy depends on the true value of phase. The classical Fisher information for a lossy Mach-Zehnder interferometer depends on the true value of the phase $\phi$. The fidelity (Shannon mutual information) is an information measure that averages over all phases, for prior information given by $p(\phi)$, see Eq. (9). Figure 11 shows a comparison of the fidelity versus dissipation $r_{x}$ for Fock and N00N states for equal dissipation in both arms, $r_{x}=r_{y}$. The fidelity of 1-photon Fock and N00N states is equal, see the discussion below. For a given amount of dissipation, $r_{x}$, for Fock states the fidelity increases with input photon number $N$. The fidelity for 2-photon N00N state input is exactly zero for all values of dissipation $r_{x}$ because this state carries no information about the phase in a Mach-Zehnder interferometer, see the discussion below. #### III.4.1 1-Photon N00N State Input into Lossy MZ Interferometer Consider now the 1-photon entangled N00N state: $\left|{\psi_{1}^{N00N}}\right\rangle=\frac{1}{{\sqrt{2}}}\left({a_{1}^{\dagger}+a_{2}^{\dagger}}\right)\left|0\right\rangle=\frac{1}{\sqrt{2}}\left[\left|10\right\rangle+\left|01\right\rangle\right]$ (64) where again, I use the short-hand notation $\left|10\right\rangle$ for $\left|1000\right\rangle$ and $\left|01\right\rangle$ for $\left|0100\right\rangle$. The probabilities for the measurement outcomes for this input state are given by Eq. (48) with the replacement $\cos\phi\rightarrow\sin\phi$. Similarly, the phase probability distributions, assuming no prior information, $p(\phi)=1/(2\pi)$, are given by Eq. (49) with the replacement $\cos\phi\rightarrow\sin\phi$. The fidelity for this input state is the same as for the 1-photon Fock state, given by Eq. (50)–(52). Therefore, according to Shannon mutual information (fidelity), the presence of entanglement in the 1-photon N00N state has not improved the information on the phase. The Fisher information for this entangled state is given by the 1-photon Fock state Fisher information in Eq. (53) with the replacements $\phi\rightarrow\frac{\pi}{2}-\phi$. Therefore, the entanglement simply has the effect of changing the phase of the classical Fisher information. This phase change changes the places where $F(\phi)=0$, which, for this entangled state, is now $\phi=\pm\pi/2$. Comparison of the 1-photon Fock state to the the 1-photon entangled N00N state shows that the introduction of entanglement does not remove the $\phi$ dependence of the Fisher information when arbitrary losses $r_{x}$ and $r_{y}$ are present. However, when $r_{x}=r_{y}$”, the Fisher information, $F_{1}(\pi/2-\phi)$, is independent of $\phi$. This is in agreement with the result of Chen and Jiang, who derived the Fisher information for N00N state input using a master equation for a quantum continuous variable system for the case of symmetrical losses Chen and Jiang (2007). When losses are absent, $r_{x}=r_{y}=0$, the Fisher information for input state given by Eq. (64) reduces to $F(\phi)=1$, independent of $\phi$, as in the 1-photon Fock state without losses. Figure 7: (Color) The Fisher information is plotted vs. $r_{x}$ for $r_{x}=r_{y}$, for Fock states (red) and N00N states (blue), for $\phi=\pi/4$, where the Fisher information is non-zero for both Fock states and N00N states. Figure 8: (Color) The Fisher information is plotted vs. $r_{x}$ for small dissipation in one arm, $r_{y}=0.10$, for Fock states (red) and N00N states (blue), for $\phi=\pi/4$. Figure 9: (Color) The Fisher information is plotted vs. $r_{x}$ for large dissipation in one arm, $r_{y}=0.90$, for Fock states (red) and N00N states (blue), for $\phi=\pi/4$. This example of high dissipation shows that the situation is complicated at high values of the dissipation in one arm and small values of dissipation in the other arm. Figure 10: (Color) The Fisher information is plotted vs. $\phi$ for loss parameters $r_{x}=0.3$ and $r_{y}=0.4$, for N00N state input into a Mach- Zehnder interferometer for $N=$1, 2, 3, 4, and 5 photons. Figure 11: (Color) The fidelity (Shannon mutual information) is plotted as a function of dissipation $r_{x}$, for $r_{x}=r_{y}$, for one-, two- and three-photon Fock state input, and for one- and two-photon N00N state input. #### III.4.2 2-Photon N00N State Input into Lossy MZ Interferometer The 2-photon N00N state, $\left|{\psi_{2}^{N00N}}\right\rangle=\frac{1}{2}\left[{\left({\hat{a}_{1}^{\dagger}}\right)^{2}+\left({\hat{a}_{2}^{\dagger}}\right)^{2}}\right]\,\left|0\right\rangle=\frac{1}{\sqrt{2}}\left[\left|20\ \right\rangle+\left|02\ \right\rangle\right]$ (65) has a peculiar behavior when input into a Mach-Zehnder interferometer with losses. The probabilities distributions, given by Eqs. (42) and (63), for the six measurement outcomes are independent of $\phi$ and are given by $\displaystyle P(20|\phi,20)$ $\displaystyle=$ $\displaystyle P(02|\phi,20)=\frac{1}{2}\left(1-r_{x}^{2}-r_{y}^{2}+r_{x}^{2}r_{y}^{2}\right)$ $\displaystyle P(11|\phi,20)$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle P(10|\phi,20)$ $\displaystyle=$ $\displaystyle P(01|\phi,20)=\frac{1}{2}\left(r_{x}^{2}+r_{y}^{2}-2r_{x}^{2}r_{y}^{2}\right)$ $\displaystyle P(00|\phi,20)$ $\displaystyle=$ $\displaystyle r_{x}^{2}r_{y}^{2}$ The measurement outcomes in Eq.(66) are independent of $\phi$ because of the Hilbert space geometry of the measurement operators, $\hat{\Pi}_{\phi}(n,m)$, and input state vector in Eq.(65), see Eq.(34). For no prior information on the phase, $p(\phi)=1/(2\pi)$, using Bayes’ rule in Eq. (25), the phase probability densities are independent of $\phi$, and are given by $p(\phi|m\,n,20)=1/(2\pi)$, for all measurement outcomes $\xi=(m,n)$. Therefore, the 2-photon N00N state cannot be used in a Mach-Zehnder interferometer for determining the phase $\phi$. However, such an arrangement can be useful in applications that require phase in-sensitive interferometry to be performed. In the limit of no loss, $r_{x}=r_{y}=0$, the interferometer acts as a beam splitter and both photons come out the same port. Figure 12: (Color) The conditional phase probability density, $p(\phi|10)$, is plotted as a function of $\phi$ for increasing detector error probabilities $p_{x}=0,\,0.2,\,0.4,\,0.5$, showing a loss of phase distinguishability. At $p_{x}=0.5$, all phases $\phi$ are equally probable. Figure 13: (Color) The fidelity (Shannon mutual information) between the measurements and the phase is plotted vs. the probability of incorrect detection, $p_{x}$. Consistent with the phase probability density plotted in Fig. 12, the fidelity decreases to zero at $p_{x}=0.5$ because there is no information on the phase in the measurements, so there is no discrimination between different phases. ### III.5 Imperfect Photon Number Detection Next, I consider the simplest example of imperfect photon-number detection. I assume that state preparation is deterministic and that the interferometer is ideal, so there are no losses. I also assume that the input state is a pure state, in Eq. (11) taking $P_{S}(\psi^{in})=\left\\{\begin{array}[]{l}1,\quad{\rm{if}}\;\left|{\psi^{in}}\right\rangle=\left|{10}\right\rangle\\\ 0,\quad{\rm{otherwise}}\\\ \end{array}\right.$ (67) where again I use the short-hand notation $\left|{10}\right\rangle$ for $\left|1000\right\rangle$. For this 1-photon input state, $\left|{\psi^{in}}\right\rangle=\left|{10}\right\rangle$, the no-loss Mach- Zehnder interferometer transfer matrix is given by $\small P_{I}(\psi^{out}|\psi^{in},\phi)=\left\\{\begin{array}[]{l}\sin^{2}\phi,\;{\rm{for}}\;\left|{\psi^{in}}\right\rangle=\left|{10}\right\rangle{\rm{and}}\;\left|{\psi^{out}}\right\rangle=\left|{10}\right\rangle\\\ \cos^{2}\phi,\;{\rm{for}}\;\left|{\psi^{in}}\right\rangle=\left|{10}\right\rangle{\rm{and}}\;\left|{\psi^{out}}\right\rangle=\left|{01}\right\rangle\\\ 0,\quad{\rm{otherwise}}\\\ \end{array}\right.$ (68) For the detection system, I assume that there is a probability $p_{d}$ to detect the state correctly and a probability $p_{x}$ to detect the state incorrectly, where $p_{d}+p_{x}=1$. I am neglecting the possibility of an inconclusive measurement outcome. The matrix, $P_{D}(\xi|\psi^{out},\phi)$, in Eq. (11) describing the state detection is then given by $P_{D}(\xi|\psi^{out},\phi)=\left\\{{\begin{array}[]{*{20}c}{p_{d},\quad\xi=\psi^{out}}\\\ {p_{x},\quad\xi\neq\psi^{out}}\\\ \end{array}}\right.$ (69) From Eq. (11), the probabilities $P(\xi|\phi)$ for measurement outcomes are $\begin{array}[]{l}P(10|\phi)=p_{d}\sin^{2}\phi+p_{x}\cos^{2}\phi\\\ P(01|\phi)=p_{x}\sin^{2}\phi+p_{d}\cos^{2}\phi\\\ \end{array}$ (70) where $\xi=(m,n)$ specifies that $m$ and $n$ photons are detected in output ports “c” and “d”, respectively, see Eq. (15)–(20). From Bayes’ rule in Eq. (25), I find the conditional probability density, $p(\phi|\xi)$, for the phase shift $\phi$ for a given measurement outcome $\xi$ to be $\begin{array}[]{l}p(\phi|10)=\frac{1}{\pi}\left[{\left({1-p_{x}}\right)\sin^{2}\phi+p_{x}\cos^{2}\phi}\right]\\\ p(\phi|01)=\frac{1}{\pi}\left[{p_{x}\sin^{2}\phi+\left({1-p_{x}}\right)\cos^{2}\phi}\right]\\\ \end{array}$ (71) Figure 12 shows a plot of the phase probability density, $p(\phi|10)$ vs. $\phi$, for different values of detector error probability $p_{x}$. With increasing probability $p_{x}$ of detecting the state incorrectly, the constrast in the phase probability decreases. Note that this contrast is not a “visibility” because $p(\phi|10)$ is a probability, and not an optical intensity. The fidelity (Shannon mutual information) defined in Eq. (9) is plotted in Fig. 13. As expected for this simple model, the fidelity $H(M)$ decreases with increasing probability of incorrect detection, $p_{x}$, reaching zero at $p_{x}=0.5$. Note that the fidelity is symmetric about $p_{x}=0.5$. ## IV Conclusion I considered the experimentally relevant problem of determining the phase shift in one arm of a quantum interferometer when state creation is not perfectly deterministic, state propagation through the interferometer is non- unitary due to absorption losses in the interferometer, and state detection is not ideal. In Section II, I have argued that two types of information are useful for evaluating the quality of a parameter estimation device, such as a quantum optical system used to determine phase shifts. First, fidelity (Shannon mutual information between measurements and parameter) is useful for deciding the overall quality of the optical system. The fidelity represents an average over probabilities of all possible measurements and parameter values (phases). The fidelity is the metric to use when choosing or designing a system and the prior parameter (phase) distribution is unknown. Once a system is chosen, it is to be used in estimating the parameter based on measurements (data), which is an estimation problem. At this point, the (classical or quantum) Fisher information can be exploited, using the classical or quantum Cramer-Rao theorem, to estimate the variance of the parameter associated with its unbiased estimator. In Eq. (11), I have written down a general statistical expression for the probability of a measurement outcome that simultaneously takes into account the three non-ideal aspects of real experiments: non-deterministic state preparation, losses in the interferometer, and non-ideal quantum state detection. This expression requires detailed models for each of the three non- ideal elements. In Section III, using simple, few-photon Fock states and N00N states, I give examples of applying Eq. (11). In subsection, A, of Section III, I look at a simple example of the effect of non-deterministic state creation, where there is a probability of creating one photon and a probability of creating vacuum, as input into a Mach-Zehnder interferometer. As expected, the non-zero probability of creating a vacuum input leads to a probability for an inconclusive measurement outcome, which in turn reduces the information on phase, as measured by Fisher information and fidelity (Shannon mutual information). In subsection B, of Section III, I have constructed a scattering matrix for a lossy (non-unitary) Mach-Zehnder interferometer. I find that for simple photon counting measurements, losses introduce a strong phase dependence in the classical Fisher information, making accuracy of phase estimation dependent on the unknown true phase. 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arxiv-papers
2010-12-23T20:40:57
2024-09-04T02:49:15.947040
{ "license": "Public Domain", "authors": "Thomas B. Bahder", "submitter": "Thomas B. Bahder", "url": "https://arxiv.org/abs/1012.5293" }
1012.5494
¡html¿ ¡head¿ ¡title¿COMP6411 Summer 2010¡/title¿ ¡/head¿ ¡body¿ ¡a href=”http://users.encs.concordia.ca/ mokhov”¿Serguei A. Mokhov¡/a¿¡br /¿ mokhov@cse.concordia.ca ¡h1¿TOC¡/h1¿ ¡ul¿ ¡li¿¡a href=”#abstract”¿Abstract¡/a¿¡/li¿ ¡li¿¡a href=”#lecture- notes”¿Lecture Notes¡/a¿¡/li¿ ¡!–li¿¡a href=”#project-brief”¿Brief Project Overview¡/a¿¡/li–¿ ¡li¿¡a href=”#reports”¿Reports¡/a¿¡/li¿ ¡/ul¿ ¡h1¿Abstract¡/h1¿ ¡a id=”abstract” name=”abstract” /¿ ¡p¿This index covers the lecture notes and the final course project reports for COMP6411 Summer 2010 at Concordia University, Montreal, Canada, Comparative Study of Programming Languages by 4 teams trying compare a set of common criteria and their applicability to about 10 distinct programming languages, where 5 language choices were provided by the instructor and five were picked by each team and each student individually compared two of the 10 and then the team did a summary synthesis across all 10 languages. Their findings are posted here for further reference, comparative studies, and analysis. ¡/p¿ ¡hr /¿ ¡h1¿Lecture Notes¡/h1¿ ¡a id=”lecture-notes” name=”lecture-notes” /¿ ¡ul¿ ¡li¿Current COMP6411 Lecture Notes ¡ul¿ ¡li¿ LIST:arXiv:1007.2123 ¡/li¿ ¡/ul¿ ¡/li¿ ¡/ul¿ ¡hr /¿ ¡!– ¡h1¿Brief Project Overview¡/h1¿ ¡a id=”project-brief” name=”project-brief” /¿ ¡hr /¿ –¿ ¡h1¿COMP6411 Summer 2010 Select Final Project Reports¡/h1¿ ¡a id=”reports” name=”reports” /¿ ¡ul¿ ¡li¿Team 5’s Approach ¡ul¿ ¡li¿ LIST:arXiv:1008.3434 ¡/li¿ ¡/ul¿ ¡/li¿ ¡li¿Team 7’s Approach ¡ul¿ ¡li¿ LIST:arXiv:1009.0305 ¡/li¿ ¡/ul¿ ¡/li¿ ¡li¿Team 10’s Approach ¡ul¿ ¡li¿ LIST:arXiv:1008.3561 ¡/li¿ ¡/ul¿ ¡/li¿ ¡li¿Team 11’s Approach ¡ul¿ ¡li¿ LIST:arXiv:1008.3431 ¡/li¿ ¡/ul¿ ¡/li¿ ¡/ul¿ ¡hr /¿ ¡/body¿ ¡/html¿
arxiv-papers
2010-12-26T03:33:09
2024-09-04T02:49:15.961285
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Serguei A. Mokhov", "submitter": "Serguei Mokhov", "url": "https://arxiv.org/abs/1012.5494" }
1012.5508
# A new approach in modeling the behavior of RPC detectors L. Benussi S. Bianco S.Colafranceschi F.L. Fabbri M. Giardoni L. Passamonti D. Piccolo D. Pierluigi A. Russo G. Saviano S. Buontempo A. Cimmino M. de Gruttola F. Fabozzi A.O.M. Iorio L. Lista P. Paolucci P. Baesso G. Belli D. Pagano S.P. Ratti A. Vicini P. Vitulo C. Viviani A. Sharma A. K. Bhattacharyya INFN Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044 Frascati, Italy Sapienza Università degli Studi di Roma “La Sapienza”, Piazzale A. Moro, Roma, Italy CERN CH-1211 Genéve 23 F-01631 Switzerland INFN Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, edificio 6, 80126 Napoli, Italy Università di Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, edificio 6, 80126 Napoli, Italy INFN Sezione di Pavia and Università degli studi di Pavia, Via Bassi 6, 27100 Pavia, Italy ###### Abstract The behavior of RPC detectors is highly sensitive to environmental variables. A novel approach is presented to model the behavior of RPC detectors in a variety of experimental conditions. The algorithm, based on Artificial Neural Networks, has been developed and tested on the CMS RPC gas gain monitoring system during commissioning. ###### keywords: RPC , CMS , Neural Network , muon detectors HEP , , , , , , , , , , , , , , , , , , , , , , , , , ††thanks: Corresponding author: Stefano Colafranceschi E-mail address: stefano.colafranceschi@cern.ch ## 1 Introduction Resistive Plate Chamber (RPC) detectors [1] are widely used in HEP experiments for muon detection and triggering at high-energy, high-luminosity hadron colliders [2, 3], in astroparticle physics experiments for the detection of extended air showers [4], as well as in medical and imaging applications [5]. At the LHC, the muon system of the CMS experiment[6] relies on drift tubes, cathode strip chambers and RPCs[7]. In this paper a new approach is proposed to model the behavior of an RPC detector via a multivariate strategy. Full details on the developed algorithm and results can be found in Ref.[8]. The algorithm, based on Artificial Neural Networks (ANN), allows one to predict the behavior of RPCs as a function of a set of variables, once enough data is available to provide a training to the ANN. At the present stage only environmental variables (temperature $T$, atmospheric pressure $p$ and relative humidity $H$) have been considered. Further studies including radiation dose are underway and will be the subject of a forthcoming paper. In a preliminary phase we trained a neural network with just one variable and we found out, as expected, that the predictions are improved after adding more variables into the network. The agreement found between data and prediction has to be considered a pessimistic evaluation of the validity of the algorithm, since it also depends on the presence of unknown variables not considered for training. The data for this study have been collected utilizing the gas gain monitoring (GGM) system [9][10][11] of the CMS RPC muon detector during the commissioning with cosmic rays in the ISR test area at CERN. The GGM system is composed by the same type of RPC used in the CMS detector (2 mm-thick Bakelite gaps) but of smaller size (50$\times$50 cm2). Twelve gaps are arranged in a stack. The trigger is provided by four out of twelve gaps of the stack, while the remaining eight gaps are used to monitor the working point by means of a cosmic ray telescope based on RPC detectors. In this study, the GGM was operated in open loop mode with a Freon 95.5%, Isobutane 4.2%, SF6 0.3% gas mixture. Six out of eight monitoring gaps were used, two out of eight monitoring gaps failed during the study and were therefore excluded from the analysis. The monitoring is performed by measuring the charge distributions of each chamber. The six gaps are operated at different high voltages, fixed for each chamber, in order to monitor the total range of operating modes of the gaps (Table 1). The operation mode of the RPC changes as a function of the voltage applied, in particular the chamber will change from avalanche mode to streamer mode when increasing HV. Table 1: Applied high voltage for power supplies for GGM RPC detectors used in this study | CH1 | CH2 | CH3 | CH6 | CH7 | CH8 ---|---|---|---|---|---|--- Applied high voltage (kV) | 10.2 | 9.8 | 10.0 | 10.4 | 10.2 | 10.4 ## 2 The Artificial Neural Network simulation code An Artificial Neural Network (ANN) is an information processing paradigm that is inspired by the way biological nervous systems, such as the brain, process information[12]. The most common type of artificial neural network (Fig. 1) consists of three groups, or layers, of units: a layer of input units is connected to a layer of hidden units, which is connected to a layer of output unit. The activity of the input units represents the raw information that is fed into the network. Figure 1: Example of a simple Neural Network configuration. The activity of each hidden unit is determined by the activities of the input units and the weights on the connections between the inputs and the hidden units. The behavior of the output units depends on the activity of the hidden units and the weights between the hidden and output units. For this study temperature, humidity and pressure have been selected as inputs and anodic charge as output variable. It was demonstrated[13] that the number of layers is not critical for the network performance, so we decided to go with 3 layers and give to the neural network a sufficient number of hidden units automatically optimized by a genetic algorithm that can take into account several configurations. For each configuration a genetic algorithm performs the training process with an estimation of the global error; then the configuration is stored and the genetic algorithm continues to evaluate a slightly different configuration. Once the algorithm has taken into account all the possible configurations the best one in terms of global error is chosen. During the training phase the network is taught with environmental data as input, the output depends on the neuronal weights, that at the very beginning are initialized with random numbers. The network output is compared to the experimental data we want to model, then the network estimates the error and modifies the neurons weights in order to minimize the estimated error. The training phase consists of determining both weights and configuration (nunber of neurons and number of layers) by minimizing the error, i.e., the difference between data and output. ## 3 Environmental variables and datasets The environmental variables are monitored by an Oregon Scientific weather station WMR100. The DAQ has been modified in order to acquire via USB the environmental informations and merge environmental variables with output variables. The accuracy of the temperature sensor is $\pm 1^{o}$C in the range $0-40^{o}$C and the resolution is $0.1^{o}$C. The relative humidity sensor has an operating range from 2% to 98% with a 1% resolution, $\pm 7\%$ absolute accuracy from $25\%$ to $40\%$, and $\pm 5\%$ from $40\%$ to $80\%$. The barometer operational range is between 700 mbar and 1050 mbar with a 1 mbar resolution and a $\pm 10$ mbar accuracy. The online monitoring system records the ambient temperature, pressure and humidity of the GGM box that contains the RPC stack. Pressure and temperature are mainly responsible of different detector behavior as well as the humidity for the bakelite and gas properties. The used dataset is composed of four periods, each period composed of runs (about 270 each). Each run contains $10^{4}$ cosmic ray events where environmental variables and GGM anodic output charges (Q) are collected. The acquisition rate is typically 9.5 Hz. ## 4 Results Typical ANN outputs show generally good agreement between data and prediction during training phase. (Fig. 2 $(a)$). In periods where the prediction is not accurate, the discrepancy is typically concentrated in narrow regions (“spikes”). Fig 2 $(b)$ shows the prediction on period 3 using the period 1 as training, the discrepancy around run 137 and run 256 are due to a set of environmental variables not available in the training period as shown in Fig. 2 $(c)$ and Fig. 3. Figure 2: $(a)$ Gap 7 trained on the period 3 - prediction on period 3; the prediction is performed on the same period used as training with very good agreement between experimental data and prediction. $(b)$ Gap 7 trained on the period 1 - prediction on period 3, the prediction is performed on a period different from the training one, the agreement depends on dispersion of environmental variables. $(c)$ Environmental variables during the period 3. Figure 3: Environmental variables during the period 1 The comparison between data and prediction is shown in Fig. 4 where the quantity $\frac{\Delta Q}{Q}\equiv\frac{Q_{EXP}-Q_{ANN}}{Q_{EXP}}$ (1) is plotted for all four periods both for training (top) and predictions (bottom), divided for training and prediction respectively. The error distribution for the predictions is much wider than for the training, as expected. Figure 4: Error for training (top) and prediction (bottom) for all runs. Gaussian fit superimposed. The quantity $\hat{\sigma}$ is the width of the gaussian fit to the data in a reduced range which excludes the nongaussian tails. The gaussian fit superimposed (Fig. 4) is not able to fit the data properly due to the presence of large nongaussian tails, which are caused by runs with very large discrepancy between data and prediction. To evaluate the width $\hat{\sigma}$ of the error distribution we perform a gaussian fit in a reduced range which does not take into account the nongaussian tails. The distribution of the error for the predictions shows a $\hat{\sigma}$ = $6.7\%$. In the Table 3 there is a summary with error for training and predictions. The cases with very large discrepancy were studied in detail, and found to be characterized by a $(p,T,H)$ value at the edges of the variables space. To determine the measure of the dispersion of the environmental variables considering all the runs ($N$) we computed the: $\frac{\Delta X}{X}\equiv\sqrt{\sum_{j=1,3}\biggr{[}\frac{(x_{j}-X_{j})}{X_{j}}\biggr{]}^{2}}$ (2) $X_{j}\equiv\sum_{i=1,N}(x_{j})_{i}\quad;\quad{\bf x}\equiv(p,T,H)\quad$ (3) The distribution of the $\frac{\Delta Q}{Q}$ error as a function of the dispersion of environmental variables $\frac{\Delta X}{X}$ (Fig. 5) shows three distinct structures. The satellite bands with very large error were studied in detail. All data point in such bands belong to period four and gap six for which problems were detected. Period four and gap six therefore were excluded in the analysis. The distribution of the error as a function of dispersion of environmental variables after this selection has a $\hat{\sigma}\sim 4\%$ width and nongaussian tails extending up to $\frac{\Delta Q}{Q}=200\%$. Figure 5: Distribution of $\frac{\Delta Q}{Q}$ as a function of the dispersion of environmental variables $\frac{\Delta X}{X}$ for all periods, six gaps and both training and prediction. Each training period is included once, each prediction is included 4 times, due to different training period chosen. A selection on the fiducial volume in the x variables space (Table 2) was applied in order to exclude from the analysis data with $(p,T,H)$ close to the edges of the variable space. After the selection cuts, prediction on two periods based on training on the third period were performed. The nongaussian (NG) tails were defined as the fractional area outside the region $\pm 4\hat{\sigma}$. The selection cuts slightly reduce the width ($\hat{\sigma}<3.7\%$), while drastically reducing the nongaussian tails (Table 3). Table 2: Synopsis of the selection cuts for fiducial volume applied to predicted data. $(958<p<968){\rm mbar}$ | $(19.4<T<20.4)^{o}$C | $(34<H<44)\%$ ---|---|--- Table 3: Summary of errors $\hat{\sigma}$ and nongaussian (NG) tails for various selection cuts and samples. Data sets | $\hat{\sigma}$ | NG tail ---|---|--- | % | % All six chambers, all four periods training | $2.7$ | $2.26$ All six chambers, all four periods prediction | $6.7$ | $6.60$ Chamber six and period four excluded prediction | $3.0$ | $4.63$ Predict. on per. 2 and 3, train. on per. 1 | $4.0$ | $3.52$ Predict. on per. 3 and 1, train. on per. 2 | $3.4$ | $2.95$ Predict. on per. 1 and 2, train. on per. 3 | $3.8$ | $1.63$ Predict. on per. 2 and 3, train. on per. 1, fiducial cuts | $3.7$ | $0.49$ Predict. on per. 3 and 1, train. on per. 2, fiducial cuts | $2.9$ | $0.98$ Predict. on per. 1 and 2, train. on per. 3, fiducial cuts | $3.3$ | $0.29$ ## 5 Discussion In this study the GGM is the system used to train the neural network with anode charge as output variable and $(p,T,H)$ as input variable. The addition of the dark current as a output variable and dose as input variable is expected to improve predictions and will be implemented. The main advantage of this approach is that several variables can be used together in order to predict chamber behavior without the needs of studying the surface corrosion, environmental/radiation dependence and bakelite aging due to chemical reactions and deposits; also in the ANN analysis, given enough data, it is possible to decouple the effect of the chosen variables used as output. This approach, once properly trained, could spot immediately and online pathological chambers whose behavior is shifting from the normal one. Further studies are in progress to determine and cure the residual nongaussian tails of the $\frac{\Delta Q}{Q}$ errors distributions, to deal with training and prediction on detectors with different high voltage supply, to widen the sample of environmental conditions, and in adding new dimensions to the variables space such as radiation levels. ## 6 Conclusions A new approach for modeling the RPC behavior, based on ANN, has been introduced and preliminary results obtained using data from the CMS RPC GGM system. The ANN was trained for predicting the behavior of the anode charge Q (output variables) as function of the environmental variables $(p,T,H)$ (input variables), resulting in a prediction error $\frac{\Delta Q}{Q}=4\%$. In a forthcoming work we plan to include the dose as input variable and the dark current as output variable, aiming at a further improvement on the predictions. Acknowledgements The skills of M. Giardoni, L. Passamonti, D. Pierluigi, B. Ponzio and A. Russo (Frascati) in setting up the experimental setup are gratefully acknowledged. The technical support of the CERN gas group is gratefully acknowledged. Thanks are due to R. Guida (CERN Gas Group), Nadeesha M. Wickramage, Yasser Assran for discussions and help. This research was supported in part by the Italian Istituto Nazionale di Fisica Nucleare and Ministero dell’ Istruzione, Università e Ricerca. ## References * [1] R. Santonico and R. Cardarelli, Nucl. Instrum. Meth. 187 (1981) 377. * [2] CMS Collaboration, JINST 0803 (2008) S08004. doi:10.1088/1748-0221/3/08/S08004. * [3] The ATLAS Collaboration, G. Aad et al., CERN Large Hadron Collider , JINST 3 (2008) S08003. * [4] G. D’Ali Staiti [ARGO-YBJ Collaboration], Nucl. Instrum. Meth. A 588 (2008) 7. * [5] P. Fonte, IEEE Transactions on Nuclear Science, vol. 49, no. 3, June 2002. * [6] CMS Collaboration, JINST 3 (2008) S08004. * [7] CMS Collaboration,CERN-LHCC-97-032 ; CMS-TDR-003. Geneva, CERN, 1997. * [8] L. Benussi et al., CMS NOTE 2010/076. * [9] M. Abbrescia et al., LNF-06-34-P, LNF-04-25-P, Jan 2007. 9pp. Presented by S. Bianco on behalf of the CMS RPC Collaboration at the 2006 IEEE Nuclear Science Symposium (NSS), Medical Imaging Conference (MIC) and 15th International Room Temperature Semiconductor Detector Workshop, San Diego, California, 29 Oct - 4 Nov 2006. arXiv:physics/0701014. * [10] L. Benussi et al., Nucl. Instrum. Meth. A 602 (2009) 805 [arXiv:0812.1108 [physics.ins-det]]. * [11] L. Benussi et al., JINST 4 (2009) P08006 [arXiv:0812.1710 [physics.ins-det]]. * [12] W. S. Mc Culloch, W. Pitts, Bulletin of Mathematical Biophysics 5 (1943) 115\. * [13] K. Hornik, M. Stinchcombe and H. White, Neural Networks, vol. 2, pp. 359, 1989.
arxiv-papers
2010-12-26T11:55:58
2024-09-04T02:49:15.965545
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "L. Benussi (1), S. Bianco (1), S.Colafranceschi (1 and 2 and 3), F.L.\n Fabbri (1), M. Giardoni (1), L. Passamonti (1), D. Piccolo (1), D. Pierluigi\n (1), A. Russo (1), G. Saviano (1 and 2), S. Buontempo (4), A. Cimmino (4 and\n 5), M. de Gruttola (4 and 5), F Fabozzi (4), A.O.M. Iorio (4 and 5), L. Lista\n (4), P. Paolucci (4), P. Baesso (6), G. Belli (6), D. Pagano (6), S.P. Ratti\n (6), A. Vicini (6), P. Vitulo (6), C. Viviani (6), A. Sharma (3), A. K.\n Bhattacharyya (3) ((1) INFN Laboratori Nazionali di Frascati Via E. Fermi,\n Italy, (2) Sapienza Universit`a degli Studi di Roma \"La Sapienza\" Piazzale A.\n Moro Roma Italy, (3) CERN CH-1211 Gen\\'eve 23 F-01631 Switzerland, (4) INFN\n Sezione di Napoli Complesso Universitario di Monte Sant'Angelo, Italy, (5)\n Universit`a di Napoli Federico II Complesso Universitario di Monte\n Sant'Angelo, Italy, (6) INFN Sezione di Pavia and Universit`degli studi di\n Pavia Via Bassi, Italy)", "submitter": "Stefano Colafranceschi", "url": "https://arxiv.org/abs/1012.5508" }
1012.5511
# STUDY OF GAS PURIFIERS FOR THE CMS RPC DETECTOR L. Benussi S. Bianco S.Colafranceschi F.L. Fabbri F. Felli M. Ferrini M. Giardoni T. Greci A. Paolozzi L. Passamonti D. Piccolo D. Pierluigi A. Russo G. Saviano S. Buontempo A. Cimmino M. de Gruttola F Fabozzi A.O.M. Iorio L. Lista P. Paolucci P. Baesso G. Belli D. Pagano S.P. Ratti A. Vicini P. Vitulo C. Viviani R. Guida A. Sharma INFN Laboratori Nazionali di Frascati, Via E. Fermi 40, I-00044 Fr ascati, Italy Sapienza Università degli Studi di Roma ”La Sapienza”, Piazzale A. Moro, Roma, Italy CERN CH-1211 Genéve 23 F-01631 Switzerland INFN Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, edificio 6, 80126 Napoli, Italy Università di Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, edificio 6, 80126 Napoli, Italy INFN Sezione di Pavia and Università degli studi di Pavia, Via Bassi 6, 27100 Pavia, Italy ###### Abstract The CMS RPC muon detector utilizes a gas recirculation system called closed loop (CL) to cope with large gas mixture volumes and costs. A systematic study of CL gas purifiers has been carried out over 400 days between July 2008 and August 2009 at CERN in a low-radiation test area, with the use of RPC chambers with currents monitoring, and gas analysis sampling points. The study aimed to fully clarify the presence of pollutants, the chemistry of purifiers used in the CL, and the regeneration procedure. Preliminary results on contaminants release and purifier characterization are reported. ###### keywords: RPC , CMS , gas , purifier detectors HEP muon ††thanks: Corresponding author: Giovanna Saviano E-mail address: giovanna.saviano@uniroma1.it ## 1 Introduction The Resistive Plate Chamber (RPC) [1] muon detector of the Compact Muon Solenoid (CMS) experiment[2] utilizes a gas recirculation system called closed loop (CL) [3], [4] to cope with large gas mixture volumes and costs. A systematic study of Closed Loop gas purifiers has been carried out in 2008 and 2009 at the ISR experimental area of CERN with the use of RPC chambers exposed to cosmic rays with currents monitoring and gas analysis sampling points. Goals of the study [5] were to observe the release of contaminants in correlation with the dark current increase in RPC detectors, to measure the purifier lifetime, to observe the presence of pollutants and to study the regeneration procedure. Previous results had shown the presence of metallic contaminants, and an incomplete regeneration of purifiers [6],[7]. The basic function of the CMS CL system is to mix and purify the gas components in the appropriate proportions and to distribute the mixture to the individual chambers. The gas mixture used is 95.2% of C2H2F4 in its environmental-friendly version R137a, 4.5% of $i$C4H10, and 0.3% SF6 to suppress streamers and operate in saturated avalanche mode. Gas mixture is humidified at the 45% RH (Relative Humidity) level typically to balance ambient humidity, which affects the resistivity of highly hygroscopic bakelite, and to improve efficiency at lower operating voltage. The CL is operated with a fraction of fresh mixture continuously injected into the system. Baseline design fresh mixture fraction for CMS is 2%, the test CL system was operated at 10% fresh mixture. The fresh mixture fraction is the fraction of the total gas content continuously replaced in the CL system with fresh mixture. The filter configuration is identical to the CMS experiment. ## 2 Experimental setup and data sample In the CL system gas purity is guaranteed by a multistage purifier system: * • The purifier-1 consisting of a cartridge filled with 5Å (10%) and 3Å (90%) Type LINDE[12] molecular sieve[11] based on Zeolite manufactured by ZEOCHEM[8]; * • The purifier-2 consisting of a cartridge filled with 50% Cu-Zn filter type R12 manufactured by BASF[9] and 50% Cu filter type R3-11G manufactured by BASF; * • The purifier-3 consisting of a cartridge filled with Ni AlO3 filter type 6525 manufactured by LEUNA[10]. The experimental setup (Fig. 1) is composed of a CL system and an open mode gas system. A detailed description of the CL, the experimental setup, and the filters studied can be found in [7]. The CL is composed of mixer, purifiers (in the subunit called “filters“ in the Fig. 1) , recirculation pump and distribution to the RPC detectors. Eleven double-gap RPC detectors are installed, nine in CL and two in open mode. Each RPC detector has two gaps (upstream and downstream) whose gas lines are serially connected. The the gas flows first in the upstream gap and then in the downstream gap. The detectors are operated at a 9.2 kV power supply. The anode dark current drawn because of the high bakelite resistivity is approximately 1-2 $\mu$A. Gas sampling points before and after each filter in the closed loop allow gas sampling for chemical and gaschromatograph analysis. The system is located in a temperature and humidity controlled hut, with online monitoring of environmental parameters. Figure 1: Test setup with CL and open loop. Chemical analyses have been performed in order to study the dynamical behaviour of dark currents increase in the double-gap experimental setup and correlate to the presence of contaminants, measure lifetime of unused purifiers, and identify contaminant(s) in correlation with the increase of currents. In the chemical analysis set-up (Fig. 2) the gas is sampled before and after each CL purifier, and bubbled into a set of PVC flasks. The first flask is empty and acts as a buffer, the second and third flasks contain 250 ml solution of LiOH (0.001 mole/l corresponding to 0.024 g/l, optimized to keep the pH of the solution at 11). The bubbling of gas mixture into the two flasks allows one to capture a wide range of elements that are likely to be released by the system, such as Ca, Na, K, Cu, Zn, Cu, Ni, F. At the end of each sampling line the flow is measured in order to have the total gas amount for the whole period of sampling. Tygon filters ($0.45\mu$m) have been installed upstream the flasks. Figure 2: Chemical setup The sampling points (Fig. 3) are located before the whole filters unit at position HV61, after purifier-1 (Zeolite) at HV62, after purifier-2 (Cu/Zn filter) at HV64 and after the Ni filter at position HV66. RPC are very sensitive to environmental parameters (atmospheric pressure, humidity, temperature), this study has been performed in environmentally controlled hut with pressure, temperature and relative humidity online monitoring. The comparison of temperature and humidity inside and outside the hut is displayed in Fig. 4 and Fig. 5, respectively, over the whole time range of the test. The inside temperature shows a variation of less than $\pm 0.5\,^{\circ}{\rm C}$; the inside humidity still reveals seasonal structures between 35% and 50%, it is, however, much smaller than the variation outside. Gas mixture composition was monitored twice a day by gaschromatography, which also provided the amount of air contamination, stable over the entire data taking run and below 300 (100) ppm in closed (open) loop. Purifiers were operated with unused filter material. Figure 3: Chemical setup sampling points Figure 4: Temperature distribution inside and outside the experimental hut. Figure 5: Relative humidity inside and outside the experimental hut. ## 3 Results and discussion The data-taking run was divided into cycles where different phenomena were expected. We have four cycles (Fig. 6), i.e., initial stable currents (cycles 1 and 2), at the onset of the raise of currents (cycle 3), in the full increase of currents (cycle 4). Cycle 4 was terminated in order not to damage permanently the RPC detectors. The currents of all RPC detectors in open loop were found stable over the four cycles. Fig. 6 shows the typical behaviour of one RPC detector in CL. While the current of the downstream gap is stable throughout the run, the current of the upstream gap starts increasing after about seven months. Such behaviour is suggestive of the formation of contaminants in the CL which are retained in the upstream gap, thus causing its current to increase, and leaving the downstream gap undisturbed. While the production of F- is constant during the run period, significant excess of K and Ca is found in the gas mixture in cycles 3 and 4. The production of F- is efficiently depressed by the zeolite purifier (Fig. 7). The observed excess of K and Ca could be explained by a damaging effect of HF (continuously produced by the system) on the zeolite filter whose structure contains such elements. Further studies are in progress to confirm this model. Figure 6: Dark currents increase in the upstream gap and not in the downstream gap, correlated to the detection of contaminants in gas. Figure 7: Concentration of F- in gas as measured by the chemical setup. Sampling point HV61 is before the zeolite purifier, the others after each purifier. ## 4 Conclusions Preliminary results show that the lifetime of purifiers using unused material is approximately seven months. Contaminants (K, Ca) are released in the gas in correlation with the dark currents increase. The currents increase is observed only in the upstream gap. The study suggests that contaminants produced in the system stop in the upstream gap and affect its noise behaviour, leaving the downstream gap undisturbed. The presence of an excess production of K and Ca in coincidence with the currents increase suggests a damaging effect of HF produced in the system on the framework of zeolites which is based on K and Ca. Further studies are in progress to fully characterize the system over the four cycles from the physical and the chemical point of view. The main goal is to better schedule the operation and maintenance of filters for the CMS experiment, where for a safe and reliable operation the filter regeneration is presently performed several times per week. A second run is being started with regenerated filter materials to measure their lifetime and confirm the observation of contaminants. Finally, studies in high-radiation environment at the CERN gamma irradiation facility are being planned. Acknowledgements The technical support of the CERN gas group is gratefully acknowledged. Thanks are due to F. Hahn for discussions, and to Nadeesha M. Wickramage, Yasser Assran for help in data taking shifts. This research was supported in part by the Italian Istituto Nazionale di Fisica Nucleare and Ministero dell’ Istruzione, Università e Ricerca. ## References * [1] R. Santonico and R. Cardarelli, “Development Of Resistive Plate Counters,” Nucl. Instrum. Meth. 187 (1981) 377. * [2] CMS Collaboration, “The CMS experiment at the CERN LHC”, JINST 3 (2008) S08004. * [3] M. Bosteels et al., “CMS Gas System Proposal”, CMS Note 1999/018. * [4] L. Besset et al., “Experimental Tests with a Standard Closed Loop Gas Circulation System”, CMS Note 2000/040. * [5] M. Abbrescia et al., “Proposal for a Systematic Study of the CERN Closed Loop Gas System Used by the RPC Muon Detectors in CMS”, Frascati preprint LNF-06/27(IR), available at http://www.lnf.infn.it/sis/preprint/ . * [6] G. Saviano et al., “Materials studies for the RPC detector in CMS ”, presented at the RPC07 Conference, Mumbai (India), January 2008. * [7] S.Bianco et al., “Chemical analyses of materials used in the CMS RPC muon detector”, CMS NOTE 2010/006. * [8] Manufactured by ZEOCHEM, 8708 Uetikon (Switzerland). * [9] BASF Technical Bulletin. * [10] LEUNA Data Sheet September 9, 2003, Catalyst KL6526-T. * [11] GRACE Davison Molecular Sieves data sheet. * [12] LINDE Technical Bullettin. * [13] L. Benussi et al., “Sensitivity and environmental response of the CMS RPC gas gain monitoring system,” JINST 4 (2009) DOI:10.1088 1748-0221 4 08 P08006 [arXiv:0812.1710 [physics.ins-det]]. * [14] M. Abbrescia et al., “HF Production In Cms-Resistive Plate Chambers,” Nucl. Phys. Proc. Suppl. 158 (2006) 30. NUPHZ,158,30; * [15] G. Aielli et al., “Fluoride production in RPCs operated with F-compound gases”, 8th Workshop on Resistive Plate Chambers and Related Detectors, Seoul, Korea, 10-12 Oct 2005. Published in Nucl.Phys.Proc.Suppl. 158 (2006) 143.
arxiv-papers
2010-12-26T12:20:14
2024-09-04T02:49:15.970286
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "L. Benussi (1), S. Bianco (1), S.Colafranceschi (1 and 2 and3), F.L.\n Fabbri (1), F. Felli (1 and 2), M. Ferrini (1 and 2), M. Giardoni (1), T.\n Greci (1 and 2), A. Paolozzi (1 and 2), L. Passamonti (1), D. Piccolo (1), D.\n Pierluigi (1), A. Russo (1), G. Saviano (1 and 2), S. Buontempo (4), A.\n Cimmino (4 and 5), M. de Gruttola (4 and 5), F Fabozzi d A.O.M. Iorio (4 and\n 5), L. Lista (4), P. Paolucci (4), P. Baesso (6), G. Belli (6), D. Pagano\n (6), S.P. Ratti (6), A. Vicini (6), P. Vitulo (6), C. Viviani (6), R. Guida\n (3), A. Sharma (3) ((1) INFN Laboratori Nazionali di Frascati Via E. Fermi,\n Italy, (2) Sapienza Universit\\'a degli Studi di Roma \"La Sapienza\" Piazzale\n A. Moro Roma Italy, (3) CERN CH-1211 Gen\\'eve 23 F-01631 Switzerland, (4)\n INFN Sezione di Napol Complesso Universitario di Monte Sant'Angelo edificio 6\n 80126 Napoli Italy, (5) Universit\\'a di Napoli Federico II Complesso\n Universitario di Monte Sant'Angelo edificio 6 80126 Napoli Italy, (6) INFN\n Sezione di Pavia and Universit\\'a degli studi di Pavia Via Bassi, Italy)", "submitter": "Stefano Colafranceschi", "url": "https://arxiv.org/abs/1012.5511" }
1012.5538
Generating functions for the Bernstein polynomials: A unified approach to deriving identities for the Bernstein basis functions Yilmaz Simsek Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya, Turkey E-mail: ysimsek@akdeniz.edu.tr Abstract > The main aim of this paper is to provide a unified approach to deriving > identities for the Bernstein polynomials using a novel generating function. > We derive various functional equations and differential equations using this > generating function. Using these equations, we give new proofs both for a > recursive definition of the Bernstein basis functions and for derivatives of > the $n$th degree Bernstein polynomials. We also find some new identities and > properties for the Bernstein basis functions. Furthermore, we discuss > analytic representations for the generalized Bernstein polynomials through > the binomial or Newton distribution and Poisson distribution with mean and > variance. Using this novel generating function, we also derive an identity > which represents a pointwise orthogonality relation for the Bernstein basis > functions. Finally, by using the mean and the variance, we generalize Szasz- > Mirakjan type basis functions. 2010 Mathematics Subject Classification. 14F10, 12D10, 26C05, 26C10, 30B40, 30C15. Key Words and Phrases. Bernstein polynomials; generating function; Szasz- Mirakjan basis functions; Bezier curves; Binomial distribution; Poisson distribution. ## 1\. Introduction and main definition The Bernstein polynomials have many applications in approximations of functions, in statistics, in numerical analysis, in $p$-adic analysis and in the solution of differential equations. It is also well-known that in Computer Aided Geometric Design polynomials are often expressed in terms of the Bernstein basis functions. Many of the known identities for the Bernstein basis functions are currently derived in an ad hoc fashion, using either the binomial theorem, the binomial distribution, tricky algebraic manipulations or blossoming. The main purpose of this work is to construct novel generating functions for the Bernstein polynomials. Using these novel generating functions, we develop a unify approach both to standard and to new identities for the Bernstein polynomials. The following definition gives us generating functions for the Bernstein basis functions: ###### Definition 1. Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ a be positive integer and let $x\in\left[a,b\right]$. Then the Bernstein basis functions $\mathbb{Y}_{k}^{n}(x;a,b,m)$ are defined by means of the following generating function: $\displaystyle f_{\mathbb{Y},k}(x,t;a,b,m)$ $\displaystyle=$ $\displaystyle\sum_{j=0}^{\infty}\sum_{l=0}^{k}\left(\begin{array}[]{c}j+m-1\\\ j\end{array}\right)(-1)^{k-l}\frac{t^{k}x^{l}a^{j+k-l}b^{-m-j}e^{(b-x)t}}{l!(k-l)!}$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!},$ where $t\in\mathbb{C}$ and $0^{j}=\left\\{\begin{array}[]{cc}0&\text{if }j\neq 0,\\\ 1&\text{if }j=0.\end{array}\right.$ The remainder of this study is organized as follows: Section 2: We find many functional equations and differential equations of this novel generating function. Using these equations, many properties of the Bernstein basis functions can be determined. For instance, we give new proofs of the recursive definition of the Bernstein basis functions as well as a novel derivation for the two term formula for the derivatives of the $n$th degree Bernstein basis functions. We also prove many other properties of the Bernstein basis functions via functional equations. Jetter and Stöckler [9] proved an identity for multivariate Bernstein polynomials on a simplex, which is considered a pointwise orthogonality relation. The integral version of this identity provides a new representation for the polynomial basis dual to the Bernstein basis. An identity for the reproducing kernel is used to define quasi-interpolants of arbitrary order. As an application of the identity of Jetter and Stöckler, Abel and Li [1] gave Proposition 1, in Section 3. Their method is based on generating functions, which reveals the general structure of the identity. As an applications of Proposition 1 they derive generating functions for the Baskakov basis functions and the Szasz-Mirakjan basis functions. Using Eq-(2.7) in Section 2, they exhibit a special case of the identity of Jetter and Stöckler for the Bernstein basis functions. In Section 3; we give relations between the Bernstein basis functions, the binomial distribution and the Poisson distribution. Using the Poisson distribution, we give generating functions for the Szasz-Mirakjan type basis functions. By using Abel and Li’s [1] method, and applying our generating functions to Proposition 1, we derive identities which give pointwise orthogonality relations for the Bernstein polynomials and the Szasz-Mirakjan type basis functions. ## 2\. Unified approach to deriving new proofs of the identities and properties for the Bernstein polynomials The Bernstein polynomials and related polynomials have been studied and defined in many different ways, for examples by $q$-series, complex functions, $p$-adic Volkenborn integrals and many algorithms. In this section, we provide fundamental properties of the Bernstein basis functions and their generating functions. We introduce some functional equations and differential equations of the novel generating functions for the Bernstein basis functions. We also give new proofs of some well known properties of the Bernstein basis functions via functional equations and differential equations. ### 2.1. Generating Functions We now modify (1) as follows: By the negative binomial theorem, we have $\frac{1}{b^{m}(1-\frac{a}{b})^{m}}=\frac{1}{b^{m}}\sum_{j=0}^{\infty}\left(\begin{array}[]{c}j+m-1\\\ j\end{array}\right)a^{j}b^{-m-j}.$ (2.1) Substituting (2.1) into (1), we get $\displaystyle f_{\mathbb{Y},k}(x,t;a,b,m)$ $\displaystyle=$ $\displaystyle\frac{t^{k}e^{(b-x)t}}{(b-a)^{m}k!}\sum_{l=0}^{k}\left(\begin{array}[]{c}k\\\ l\end{array}\right)(-1)^{k-l}x^{l}a^{k-l}$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}.$ Thus we obtain the following novel generating function, which is a modification of (1): $\displaystyle f_{\mathbb{Y},k}(x,t;a,b,m)$ $\displaystyle=$ $\displaystyle\frac{t^{k}\left(x-a\right)^{k}e^{(b-x)t}}{(b-a)^{m}k!}$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}.$ ###### Remark 1. If we set $a=0$ and $b=1$ in (2.1), we obtain a result given by Simsek and Acikgoz [13] and Acikgoz and Arici [2]: $\frac{(xt)^{k}}{k!}e^{(1-x)t}=\sum_{n=0}^{\infty}B_{k}^{n}(x)\frac{t^{n}}{n!},$ so that, obviously; $\mathbb{Y}_{k}^{n}(x;0,1,m)=B_{k}^{n}(x),$ where $B_{k}^{n}(x)$ denote the Bernstein polynomials. By using the Taylor series for $e^{(b-x)t}$ in (2.1), we get $\frac{(x-a)^{k}}{(b-a)^{m}k!}\sum_{n=0}^{\infty}\mathbb{(}b-x\mathbb{)}^{n}\frac{t^{n+k}}{n!}=\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}.$ Comparing the coefficients of $t^{k}$ on the both sides of the above equation, we arrive at the following theorem: ###### Theorem 1. Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a positive integer and let $x\in\left[a,b\right]$. Let $k$ and $n$ be non- negative integers with $n\geq k$. Then $\mathbb{Y}_{k}^{n}(x;a,b,m)=\left(\begin{array}[]{c}n\\\ k\end{array}\right)\frac{\left(x-a\right)^{k}(b-x)^{n-k}}{(b-a)^{m}},$ (2.4) where $k=0$, $1$,$\cdots$, $n$, and $\left(\begin{array}[]{c}n\\\ k\end{array}\right)=\frac{n!}{k!(n-k)!}$. ###### Remark 2. For $m=n$, the Bernstein basis functions of degree $n$ are defined by (2.4). ###### Remark 3. In the special case when $m=n$, Theorem 1 immediately yields the corresponding well known results concerning the Bernstein basis functions $B_{k}^{n}(x)$ that appears for example in Goldman [5, p. 384, Eq.(24.6)] and cf. [3]: $\mathbb{Y}_{k}^{n}(x;a,b,n)=B_{k}^{n}(x;a,b)=\left(\begin{array}[]{c}n\\\ k\end{array}\right)\frac{\left(x-a\right)^{k}(b-x)^{n-k}}{(b-a)^{n}},$ where $k=0$, $1$,$\cdots$, $n$ and $x\in[a,b]$. One can easily see that $B_{k}^{n}(x)=\left(\begin{array}[]{c}n\\\ k\end{array}\right)x^{k}(1-x)^{n-k},$ (2.5) where $k=0,1,\cdots,n$ and $x\in[0,1]$ cf. [1]-[13]. In [5], Goldman gives many properties of the Bernstein polynomials $B_{k}^{n}(x,a,b)$. The functions $B_{0}^{n}(x,a,b),\cdots,B_{n}^{n}(x,a,b)$ are called the Bernstein basis functions. Goldman [5], in Chapter 26, shows that the Bernstein basis functions form a basis for the polynomials of degree $n$. The Bezier curve $B(t)$ with control points $P_{0}$,$\cdots$, $P_{n}$ is defined as follows: $B(t)=\sum_{k=0}^{n}P_{k}B_{k}^{n}(x,a,b)\text{ cf. \cite[cite]{[\@@bibref{}{GoldmanBOOK}{}{}]}.}$ ###### Remark 4. By using (2.4), we have $\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\begin{array}[]{c}n\\\ k\end{array}\right)\frac{\left(x-a\right)^{k}(b-x)^{n-k}}{(b-a)^{m}}\frac{t^{n}}{n!}.$ From this equation, we obtain $\sum_{n=0}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n}}{n!}=\frac{\left(x-a\right)^{k}t^{k}}{k!(b-a)^{m}}\sum_{n=k}^{\infty}(b-x)^{n-k}\frac{t^{n-k}}{\left(n-k\right)!}.$ The series on the right hand side is the Taylor series for $e^{(b-x)t}$; thus we arrive at (2.1). Substituting $m=n$ in (2.4), we now give another well-known generating function for the Bernstein basis functions: $\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)t^{k}\right)\frac{z^{n}}{n!}=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\left(\begin{array}[]{c}n\\\ k\end{array}\right)t^{k}\left(\frac{x-a}{b-a}\right)^{k}\left(\frac{b-x}{b-a}\right)^{n-k}\right)\frac{z^{n}}{n!}.$ By using the Cauchy product in the above equation, we have $\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)t^{k}\right)\frac{z^{n}}{n!}=\sum_{n=0}^{\infty}\left(t\frac{x-a}{b-a}\right)\frac{z^{n}}{n!}\sum_{n=0}^{\infty}\left(\frac{b-x}{b-a}\right)\frac{z^{n}}{n!}.$ From this equation, we find that $\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)t^{k}\right)\frac{z^{n}}{n!}=e^{z\left(\frac{b-x}{b-a}+t\frac{x-a}{b-a}\right)}.$ After some elementary calculations in the above relation, we arrive at the following generating function for the Bernstein basis functions: $\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)t^{k}=\left(\frac{b-x}{b-a}+t\frac{x-a}{b-a}\right)^{n}.$ (2.6) ###### Remark 5. If we set $a=0$, $b=1$ and $m=n$ in (2.6), then we have $\sum_{k=0}^{n}B_{k}^{n}(x)t^{k}=\left(\left(1-x\right)+tx\right)^{n}.$ (2.7) This generating functions is given by Goldman [7]-[6, Chapter 5, pages 299-306]. Goldman [7]-[6, Chapter 5, pages 299-306] also constructs the following generating functions the univariate and bivariate Bernstein basis functions: $\sum_{k=0}^{n}B_{k}^{n}(x)e^{ky}=\left(\left(1-x\right)+te^{y}\right)^{n},$ $\sum_{i+j+k=n}B_{i,j,k}^{n}(s,t)x^{i}y^{j}=\left(\left(1-s-t\right)+sx+ty\right)^{n},$ where $B_{i,j,k}^{n}(s,t)=\left(\begin{array}[]{c}n\\\ ijk\end{array}\right)s^{i}t^{j}\left(1-s-t\right)^{k}\text{ and }\left(\begin{array}[]{c}n\\\ ijk\end{array}\right)=\frac{n!}{i!j!k!}$ and $\sum_{i+j+k=n}B_{i,j,k}^{n}(s,t)e^{ix}e^{jy}=\left(\left(1-s-t\right)+se^{x}+te^{y}\right)^{n}.$ Below are some well-known properties of the Bernstein basis functions: Non-negative property: $\mathbb{Y}_{k}^{n}(x;a,b,m)\geq 0\text{, for }0\leq a\leq x\leq b.$ (2.8) Symmetry property: $\mathbb{Y}_{k}^{n}(x;a,b,m)=\mathbb{Y}_{n-k}^{n}(b+a-x;a,b,m).$ (2.9) Corner values: $\mathbb{Y}_{k}^{n}(a;a,b,m)=\left\\{\begin{array}[]{cc}0&\text{if }k\neq 0,\\\ 1&\text{if }k=0,\end{array}\right.$ (2.10) and $\mathbb{Y}_{k}^{n}(b;a,b,m)=\left\\{\begin{array}[]{cc}0&\text{if }k\neq n,\\\ 1&\text{if }k=n.\end{array}\right.$ (2.11) Alternating sum: Substituting $m=n$ in (2.4), we get $\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}(-1)^{k}\mathbb{Y}_{k}^{n}(x;a,b,n)\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\frac{\left(\frac{a-x}{b-a}\right)^{k}\left(\frac{b-x}{b-a}\right)^{n-k}}{k!(n-k)!}\right)t^{n}.$ By using the Cauchy product in the above equation, we have $\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}(-1)^{k}\mathbb{Y}_{k}^{n}(x;a,b,n)\right)\frac{t^{n}}{n!}=e^{\left(\frac{a+b-2x}{b-a}\right)t}.$ From this relation, we arrive at the following formula for the alternating sum. $\sum_{k=0}^{n}(-1)^{k}\mathbb{Y}_{k}^{n}(x;a,b,n)=\left(\frac{a+b-2x}{b-a}\right)^{n}.$ (2.12) ###### Remark 6. If we set $a=0$, $b=1$ and $m=n$, then Eq-(2.8)-Eq-(2.12) reduce to Goldman’s results [7]-[6, Chapter 5, pages 299-306]. In [7] and [6, Chapter 5, pages 299-306], Goldman also gives many identities and properties for the univariate and bivariate Bernstein basis functions, for example boundary values, maximum values, partitions of unity, representation of monomials, representation in terms of monomials, conversion to monomial form, linear independence, Descartes’ law of sign, discrete convolution, unimodality, subdivision, directional derivatives, integrals, Marsden identities, De Boor-Fix formulas, and the other properties. A Bernstein polynomial $\mathcal{P}(x,a,b,m)$ is a polynomial represented in the Bernstein basis functions: $\mathcal{P}(x,a,b,m)=\sum_{k=0}^{n}c_{k}^{n}\mathbb{Y}_{k}^{n}(x;a,b,m).$ (2.13) ###### Remark 7. If we set $a=0$, $b=1$ and $m=n$ (2.13), then we have $P(x)=\sum_{k=0}^{n}c_{k}^{n}B_{k}^{n}(x)$ cf. [4]. By using (2.1), we obtain the following functional equation: $f_{\mathbb{Y},k_{1}}(x,t;a,b,m_{1})f_{\mathbb{Y},k_{2}}(x,t;a,b,m_{2})=\frac{\left(\begin{array}[]{c}k_{1}+k_{2}\\\ k_{1}\end{array}\right)}{2^{k_{1}+k_{2}}}f_{\mathbb{Y},k_{1}+k_{2}}(x,2t;a,b,m_{1}+m_{2}),$ where $\left(\begin{array}[]{c}k_{1}+k_{2}\\\ k_{1}\end{array}\right)=\left(\begin{array}[]{c}k_{1}+k_{2}\\\ k_{2}\end{array}\right)=\frac{\left(k_{1}+k_{2}\right)!}{k_{1}!k_{2}!}.$ By using the definition of the novel generating function $f_{\mathbb{Y},k}(x,t;a,b,m)$ in the preceding equation, we get $\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k_{1}}^{n}(x;a,b,m_{1})\frac{t^{n}}{n!}\sum_{n=0}^{\infty}\mathbb{Y}_{k_{2}}^{n}(x;a,b,m_{2})\frac{t^{n}}{n!}$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k_{1}+k_{2}}^{n}(x;a,b,m_{1}+m_{2})\frac{2^{n-k_{1}-k_{2}}\left(k_{1}+k_{2}\right)!t^{n}}{n!k_{1}!k_{2}!}.$ And using the Cauchy product in this equation, we have $\displaystyle\sum_{n=0}^{\infty}\left(\mathop{\displaystyle\sum}\limits_{j=0}^{n}\left(\begin{array}[]{c}n\\\ j\end{array}\right)\mathbb{Y}_{k_{1}}^{j}(x;a,b,m_{1})\mathbb{Y}_{k_{2}}^{n-j}(x;a,b,m_{2})\right)\frac{t^{n}}{n!}$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\mathbb{Y}_{k_{1}+k_{2}}^{n}(x;a,b,m_{1}+m_{2})\frac{2^{n-k_{1}-k_{2}}\left(k_{1}+k_{2}\right)!t^{n}}{n!k_{1}!k_{2}!}.$ Comparing the coefficients of $\frac{t^{n}}{n!}$ on the both sides of the above equation, we arrive at the following theorem: ###### Theorem 2. Let $m_{1}$ and $m_{2}$ be integers. Then the following identity holds: $\mathbb{Y}_{k_{1}+k_{2}}^{n}(x;a,b,m_{1}+m_{2})=\frac{2^{k_{1}+k_{2}-n}k_{1}!k_{2}!}{\left(k_{1}+k_{2}\right)!}\mathop{\displaystyle\sum}\limits_{j=0}^{n}\left(\begin{array}[]{c}n\\\ j\end{array}\right)\mathbb{Y}_{k_{1}}^{j}(x;a,b,m_{1})\mathbb{Y}_{k_{2}}^{n-j}(x;a,b,m_{2}).$ Observe that if we set $a=0$ and $b=1$, then we have $B_{k_{1}+k_{2}}^{n}(x)=\frac{2^{k_{1}+k_{2}-n}k_{1}!k_{2}!}{\left(k_{1}+k_{2}\right)!}\mathop{\displaystyle\sum}\limits_{j=0}^{n}\left(\begin{array}[]{c}n\\\ j\end{array}\right)B_{k_{1}}^{j}(x)B_{k_{2}}^{n-j}(x).$ Note that many new identities can be found via functional equations for the novel generating functions of the Bernstein basis functions. We derive some functional equations and identities related to the generating functions and the Bernstein basis functions in the remainder of this section. ### 2.2. Subdivision property The following functional equation of the novel generating functions is fundamental to driving the subdivision property for the Bernstein basis functions. Let we us define $f_{\mathbb{Y},j}(xy,t;a,b,n)=f_{\mathbb{Y},j}\left(x,t\left(\frac{y-a}{b-a}\right);a,b,n\right)e^{t\left(\frac{b-y}{b-a}\right)}.$ (2.15) From this generating function, we have the following theorem: ###### Theorem 3. Let $a\leq yx\leq b$. Then the following identity holds: $\mathbb{Y}_{j}^{n}(xy;a,b,n)=\mathop{\displaystyle\sum}\limits_{k=j}^{n}\mathbb{Y}_{j}^{k}(x;a,b,k)\mathbb{Y}_{k}^{n}(y;a,b,n-k).$ ###### Proof. By equations (2.1) and (2.15), we obtain $\displaystyle\sum_{n=j}^{\infty}\mathbb{Y}_{j}^{n}(xy;a,b,n)\frac{t^{n}}{n!}$ $\displaystyle=$ $\displaystyle\left(\sum_{n=0}^{\infty}\mathbb{Y}_{j}^{n}(x;a,b,n)\left(\frac{y-a}{b-a}\right)^{n}\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}\frac{\left(\frac{b-y}{b-a}\right)^{n}t^{n}}{n!}\right).$ Using the Cauchy product in this equation, we get $\sum_{n=j}^{\infty}\mathbb{Y}_{j}^{n}(xy;a,b,m)\frac{t^{n}}{n!}=\sum_{n=j}^{\infty}\left(\mathop{\displaystyle\sum}\limits_{k=j}^{n}\mathbb{Y}_{j}^{n}(x;a,b,k)\frac{\left(\frac{y-a}{b-a}\right)^{k}\left(\frac{b-y}{b-a}\right)^{n-k}}{k!\left(n-k\right)!}\right)t^{n}.$ Substituting (2.4) into the above equation then after some elementary manipulations, we arrive at the desired result. ###### Remark 8. Substituting $a=0$, $b=1$ and $m=n$ into Theorem 3, we have $B_{j}^{n}(xy)=\mathop{\displaystyle\sum}\limits_{k=j}^{n}B_{j}^{k}(x)B_{k}^{n}(y).$ (2.16) The above identity is essentially the subdivision property for the Bernstein basis functions. This identity is a bit tricky to prove with algebraic manipulations. ###### Remark 9. Goldman [7]-[6, Chapter 5, pages 299-306] proves equation (2.16) with algebraic manipulations. He also proves the following subdivision properties: $B_{j}^{n}(\left(1-y\right)x+y)=\mathop{\displaystyle\sum}\limits_{k=0}^{j}B_{j-k}^{n-k}(x)B_{k}^{n}(y),$ and $B_{j}^{n}(\left(1-y\right)x+yz)=\mathop{\displaystyle\sum}\limits_{k=0}^{n}\left(\mathop{\displaystyle\sum}\limits_{p+q=j}B_{p}^{n-k}(x)B_{q}^{k}(z)\right)B_{k}^{n}(y)$ for the others see cf. [7]-[6, Chapter 5, pages 299-306]. ### 2.3. Differentiating the generating function In this section we give higher order derivatives of the Bernstein basis functions by differentiating the generating function in (2.1) with respect to $x$. Using Leibnitz’s formula for the $l$th derivative, with respect to $x$, of the product $f_{\mathbb{Y},k}(x,t;a,b,m)$ of two functions $g(t,x;a,b)=\frac{t^{k}\left(x-a\right)^{k}}{(b-a)^{m}k!}$ with $a\neq b$ and $h(t,x;b)=e^{(b-x)t}$, we obtain the following higher order partial derivative equation: $\frac{\partial^{l}f_{\mathbb{Y},k}(x,t;a,b,m)}{\partial x^{l}}=\mathop{\displaystyle\sum}\limits_{j=0}^{l}\left(\begin{array}[]{c}l\\\ j\end{array}\right)\left(\frac{\partial^{j}g(t,x;a,b)}{\partial x^{j}}\right)\left(\frac{\partial^{l-j}h(t,x;b)}{\partial x^{l-j}}\right).$ From this equation, we arrive at the following theorem: ###### Theorem 4. Let $l$ be a non-negative integer. Then $\frac{\partial^{l}f_{\mathbb{Y},k}(x,t;a,b,m)}{\partial x^{l}}=\mathop{\displaystyle\sum}\limits_{j=0}^{l}\left(\begin{array}[]{c}l\\\ j\end{array}\right)(-1)^{l-j}\frac{t^{l}}{(b-a)^{j}}f_{\mathbb{Y},k-j}(x,t;a,b,m-j).$ By using Theorem 4, we obtain higher order derivatives of the Bernstein basis functions by the following theorem: ###### Theorem 5. Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a positive integer and let $x\in\left[a,b\right]$. Let $k$, $l$ and $n$ be nonnegative integers with $n\geq k$. Then $\frac{d^{l}\mathbb{Y}_{k}^{n}(x;a,b,m)}{dx^{l}}=\mathop{\displaystyle\sum}\limits_{j=0}^{l}(-1)^{l-j}\left(\begin{array}[]{c}n\\\ n-l,l-j,j\end{array}\right)\frac{l!}{(b-a)^{j}}\mathbb{Y}_{k-j}^{n-l}(x;a,b,m-j),$ where $\left(\begin{array}[]{c}n\\\ x,y,z\end{array}\right)=\frac{n!}{x!y!z!}\text{, with }n=x+y+z.$ ###### Remark 10. Substituting $a=0$, $b=1$ and $m=n$ into Theorem 5, we have $\frac{d^{l}B_{k}^{n}(x)}{dx^{l}}=\mathop{\displaystyle\sum}\limits_{j=0}^{l}(-1)^{l-j}\left(\begin{array}[]{c}n\\\ n-l,l-j,j\end{array}\right)l!B_{k-j}^{n-l}(x),$ or $\frac{d^{l}B_{k}^{n}(x)}{dx^{l}}=\frac{n!}{(n-l)!}\mathop{\displaystyle\sum}\limits_{j=0}^{l}(-1)^{l-j}\left(\begin{array}[]{c}n\\\ j\end{array}\right)B_{k-j}^{n-l}(x),$ cf. ([7], [6, Chapter 5, pages 299-306]). Substituting $l=1$ into Theorem 5, we arrive at the following corollary: ###### Corollary 1. Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a positive integer and let $x\in\left[a,b\right]$. Let $k$ and $n$ be nonnegative integers with $n\geq k$. Then $\frac{d}{dx}\mathbb{Y}_{k}^{n}(x;a,b,m)=n\left(\frac{\mathbb{Y}_{k-1}^{n-1}(x;a,b,m-1)-\mathbb{Y}_{k}^{n-1}(x;a,b,m-1)}{b-a}\right).$ ###### Remark 11. By setting $m=n$ in Corollary 1, we arrive at the known known result recorded by Goldman [5]: $\frac{d}{dx}B_{k}^{n}(x;a,b)=n\left(\frac{B_{k-1}^{n-1}(x;a,b)-B_{k}^{n-1}(x;a,b)}{b-a}\right).$ ###### Remark 12. One can also see the following special case of Theorem 1 when $a=0$ and $b=1$: $\frac{d}{dx}B_{k}^{n}(x)=n\left(B_{k-1}^{n-1}(x)-B_{k}^{n-1}(x)\right)$ cf. [1]-[13]. ### 2.4. Recurrence Relation In this section by using higher order derivatives of the novel generating function with respect to $t$, we derive a partial differential equation. Using this equation, we shall give a new proof of the recurrence relation for the Bernstein basis functions. Differentiating Eq-(1) with respect to $t$, we prove a recurrence relation for the polynomials $\mathbb{Y}_{k}^{n}(x;a,b,m)$. This recurrence relation can also be obtained from Eq-(2.4). By using Leibnitz’s formula for the $v$th derivative, with respect to $t$, of the product $f_{\mathbb{Y},k}(x,t;a,b,m)$ of two function $g(t,x;a,b)=\frac{t^{k}\left(x-a\right)^{k}}{(b-a)^{m}k!}$ with $a\neq b$ and $h(t,x;b)=e^{(b-x)t}$, we obtain another higher order partial differential equation as follows: $\frac{\partial^{v}f_{\mathbb{Y},k}(x,t;a,b,m)}{\partial t^{v}}=\mathop{\displaystyle\sum}\limits_{j=0}^{v}\left(\begin{array}[]{c}v\\\ j\end{array}\right)\left(\frac{\partial^{j}g(t,x;a,b)}{\partial t^{j}}\right)\left(\frac{\partial^{v-j}h(t,x;b)}{\partial t^{v-j}}\right).$ From the above equation, we have the following theorem: ###### Theorem 6. Let $v$ be an integer number. Then $\frac{\partial^{v}f_{\mathbb{Y},k}(x,t;a,b,m)}{\partial t^{v}}=\mathop{\displaystyle\sum}\limits_{j=0}^{v}(b-a)^{v-j}\mathbb{Y}_{j}^{v}(x;a,b,v)f_{\mathbb{Y},k-j}(x,t;a,b,m-j),$ where $f_{\mathbb{Y},k}(x,t;a,b,m)$ and $\mathbb{Y}_{j}^{v}(x;a,b,v)$ are defined in (2.1) and (2.4), respectively. Using definition (2.1) and (2.4) in Theorem 6, we obtain a recurrence relation for the Bernstein basis functions by the following theorem: ###### Theorem 7. Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a positive integer and let $x\in\left[a,b\right]$. Let $k$, $v$ and $n$ be nonnegative integers with $n\geq k$. Then $\mathbb{Y}_{k}^{n}(x;a,b,m)=\mathop{\displaystyle\sum}\limits_{j=0}^{v}(b-a)^{v-j}\mathbb{Y}_{j}^{v}(x;a,b,v)\mathbb{Y}_{k-j}^{n-v}(x;a,b,m-j).$ ###### Remark 13. Substituting $a=0$ and $b=1$ into Theorem 7, we obtain the following result: $B_{k}^{n}(x)=\mathop{\displaystyle\sum}\limits_{j=0}^{v}B_{j}^{v}(x)B_{k-j}^{n-v}(x).$ Substituting $v=1$ into Theorem 7, we arrive at the following corollary: ###### Corollary 2. (Recurrence Relation) Let $a$ and $b$ be nonnegative real parameters with $a\neq b$. Let $m$ be a positive integer and let $x\in\left[a,b\right]$. Let $k$ and $n$ be nonnegative integers with $n\geq k$. Then $\displaystyle\mathbb{Y}_{k}^{n}(x;a,b,m)$ $\displaystyle=$ $\displaystyle\frac{x-a}{b-a}\mathbb{Y}_{k-1}^{n-1}(x;a,b,m-1)$ $\displaystyle+\frac{b-x}{b-a}\mathbb{Y}_{k}^{n-1}(x;a,b,m-1).$ ###### Remark 14. Differentiating equation (1) with respect to $t$, we also get $\displaystyle\frac{x-a}{b-a}f_{\mathbb{Y},k-1}(x,t;a,b,m-1)+\frac{b-x}{b-a}f_{\mathbb{Y},k}(x,t;a,b,m-1)$ $\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}\mathbb{Y}_{k}^{n}(x;a,b,m)\frac{t^{n-1}}{\left(n-1\right)!}.$ From this equation, one can also obtain Corollary 2. ###### Remark 15. By setting $a=0$ and $b=1$ in (2), one obtains the following relation: $B_{k}^{n}(x)=(1-x)B_{k}^{n-1}(x)+xB_{k-1}^{n-1}(x).$ ### 2.5. Multiplication and division by powers of $(\frac{x-a}{b-a})^{d}$ and $(\frac{b-x}{b-a})^{d}$ In [4], Buse and Goldman present much background material on computations with Bernstein polynomials. They provide formulas for multiplication and division of Bernstein polynomials by powers of $x$ and $1-x$ and for degree elevation of Bernstein polynomials. Our method is similar to that of Buse and Goldman’s [4]. In this section we find two functional equations. Using these equations, we also give new proofs of both the multiplication and division properties for the Bernstein polynomials. By using the generating function in (1), we provide formulas for multiplying Bernstein polynomials by powers of $(\frac{x-a}{b-a})^{d}$ and $(\frac{b-x}{b-a})^{d}$ and for degree elevation of the Bernstein polynomials. Using (2.1), we obtain the following functional equation: $(\frac{x-a}{b-a})^{d}f_{\mathbb{Y},k}(x,t;a,b,n)=\frac{(k+d)!}{k!t^{d}}f_{\mathbb{Y},k}(x,t;a,b,n).$ After elementary manipulations in this equation, we get $(\frac{x-a}{b-a})^{d}\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{n!(k+d)!}{k!(n+d)!}\mathbb{Y}_{k+d}^{n+d}(x;a,b,n+d).$ (2.18) Substituting $d=1$, we have $(\frac{x-a}{b-a})\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{k+1}{n+1}\mathbb{Y}_{k+1}^{n+1}(x;a,b,n+1).$ (2.19) ###### Remark 16. Substituting $a=0$ and $b=1$ into (2.19), we have $xB_{k}^{n}(x)=\frac{k+1}{n+1}B_{k+1}^{n+1}(x).$ The above relation can also be proved by (2.5) cf. [4]. Similarly, using (2.4), we obtain $(\frac{b-x}{b-a})^{d}\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{n!(n+d-k)!}{\left(n+d\right)!(n-k)!}\mathbb{Y}_{k}^{n+d}(x;a,b,n+d).$ Substituting $d=1$ into the above equation, we have $(\frac{b-x}{b-a})\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{n+1-k}{n+1}\mathbb{Y}_{k}^{n+1}(x;a,b,n+1).$ (2.20) Consequently, by the same method as in [4], if we have (2.13), then $(\frac{x-a}{b-a})^{d}\mathcal{P}(x,a,b)=\sum_{k=0}^{n}c_{k}^{n}\frac{n!(k+d)!}{k!(n+d)!}\mathbb{Y}_{k+d}^{n+d}(x;a,b,n+d),$ (2.21) and $(\frac{b-x}{b-a})^{d}\mathcal{P}(x,a,b)=\sum_{k=0}^{n}c_{k}^{n}\frac{n!(n+d-k)!}{\left(n+d\right)!(n-k)!}\mathbb{Y}_{k}^{n+d}(x;a,b,n+d).$ (2.22) We now consider division properties. We assume that (2.13) holds and that we are given an integer $j>0$. Since $(\frac{x-a}{b-a})^{j}$ divides $\mathbb{Y}_{k}^{n}(x;a,b,n)$ for all $k\geq j$, it follows that $(\frac{x-a}{b-a})^{j}$ divides $\mathcal{P}(x,a,b)$. Similarly, using (2.1), we obtain the following functional equation: $\frac{f_{\mathbb{Y},k}(x,t;a,b,n)}{(\frac{x-a}{b-a})^{j}}=\frac{(k-f)!t^{j}}{k!}f_{\mathbb{Y},k-j}(x,t;a,b,n-j).$ For $k\geq j$, from the above equation, we have $\frac{\mathbb{Y}_{k}^{n}(x;a,b,n)}{(\frac{x-a}{b-a})^{j}}=\frac{n!(k-j)!}{k!(n-j)!}\mathbb{Y}_{k-j}^{n-j}(x;a,b,n-j).$ By a calculation similar to the calculation in [4], for $j\leq n-k$, we have $\frac{\mathbb{Y}_{k}^{n}(x;a,b,n)}{(\frac{b-x}{b-a})^{j}}=\frac{n!(n-j-k)!}{\left(n-k\right)!(n-j)!}\mathbb{Y}_{k}^{n-j}(x;a,b,n-j).$ Therefore $\frac{\mathcal{P}(x,a,b)}{(\frac{x-a}{b-a})^{j}}=\sum_{k=j}^{n}c_{k}^{n}\frac{n!(k-j)!}{k!(n-j)!}\mathbb{Y}_{k-j}^{n-j}(x;a,b,n-j),$ (2.23) and $\frac{\mathcal{P}(x,a,b)}{(\frac{b-x}{b-a})^{j}}=\sum_{k=0}^{n-j}c_{k}^{n}\frac{n!(n-j-k)!}{\left(n-k\right)!(n-j)!}\mathbb{Y}_{k}^{n-j}(x;a,b,n-j).$ (2.24) ### 2.6. Degree elevation According to Buse and Goldman [4], given a polynomial represented in the univariate Bernstein basis of degree $n$, degree elevation computes representations of the same polynomial in the univariate Bernstein bases of degree greater than $n$. Degree elevation allows us to add two or more Bernstein polynomials which are not represented in the same degree Bernstein basis functions. Adding (2.19) and (2.20), we obtain the degree elevation formula for the Bernstein basis functions: $\mathbb{Y}_{k}^{n}(x;a,b,n)=\frac{k+1}{n+1}\mathbb{Y}_{k+1}^{n+1}(x;a,b,n+1)+\frac{n+1-k}{n+1}\mathbb{Y}_{k}^{n+1}(x;a,b,n+1).$ Substituting $d=1$ into (2.22), and adding these two equations gives the following degree elevation formula for the Bernstein polynomials: $\mathcal{P}(x,a,b)=\sum_{k=0}^{n}\left(\frac{k}{n+1}c_{k-1}^{n}+\frac{n+1-k}{\left(n+1\right)}c_{k}^{n}\right)\mathbb{Y}_{k}^{n+1}(x;a,b,n+1),$ (2.25) where $c_{k}^{n+1}=\frac{k}{n+1}c_{k-1}^{n}+\frac{n+1-k}{\left(n+1\right)}c_{k}^{n}.$ ###### Remark 17. If we set $a=0$ and $b=1$, then (2.25) reduces to Eq-(2.5) in [4, p. 853]. ## 3\. Relation between the generating functions $f_{\mathbb{Y},k}(x,t;a,b,m)$, Poisson distribution and Szasz-Mirakjan type basis functions The identity of Jetter and Stöckler represents a pointwise orthogonality relation for the multivariate Bernstein polynomials on a simplex. This identity give us a new representation for the dual basis which can be used to construct general quasi-interpolant operators cf. (See, for details, [9], [1]). As an application of the generating functions for the basis functions to the identity of Jetter and Stöckler, Abel and Li [1] proved Proposition 1, which is given in this section. Applying our generating functions to Proposition 1, we give pointwise orthogonality relations for the Bernstein polynomials and the Szasz-Mirakjan basis functions. In this section, we give relations between the Bernstein basis functions, the binomial distribution and the Poisson distribution. First we we consider the generalized binomial or Newton distribution (probability function). Suppose that $0\leq\frac{x-a}{b-a}\leq 1$ and $0\leq\frac{b-x}{b-a}\leq 1$. Set $\mathbb{Y}_{k}^{n}(x;a,b,n)=\left(\begin{array}[]{c}n\\\ k\end{array}\right)\left(\frac{x-a}{b-a}\right)^{k}\left(\frac{b-x}{b-a}\right)^{n-k}.$ (3.1) From the above definition, one can see that $\sum_{k=0}^{n}\mathbb{Y}_{k}^{n}(x;a,b,n)=1.$ ###### Remark 18. If we set $a=0$ and $b=1$, then (3.1) reduces to $\mathbb{Y}_{k}^{n}(x;0,1,n)=\left(\begin{array}[]{c}n\\\ k\end{array}\right)x^{k}(1-x)^{n-k}$ which is the binomial or Newton distribution (probabilities) function. If $0\leq x\leq 1$ is the probability of an event $E$, then $\mathbb{Y}_{k}^{n}(x;0,1,n)$ is the probability that $E$ will occur exactly $k$ times in $n$ independent trials cf. [11]. Expected value or mean and variance of $\mathbb{Y}_{k}^{n}(x;a,b,n)$ are given by $\mu=\sum_{k=0}^{n}k\mathbb{Y}_{k}^{n}(x;a,b,n)=n\left(\frac{x-a}{b-a}\right),$ and $\sigma^{2}=\sum_{k=0}^{n}k^{2}\mathbb{Y}_{k}^{n}(x;a,b,n)-\mu^{2}=\frac{n\left(x-a\right)\left(b-x\right)}{\left(b-a\right)^{2}}.$ If we let $n\rightarrow\infty$ in (3.1), then we arrive at the well-known Poisson distribution function: $\mathbb{Y}_{k}^{n}(\frac{b-a}{n}\mu+a;a,b,n)\rightarrow\frac{\mu^{k}e^{-\mu}}{k!}.$ (3.2) The following proposition is proved by Abel and Li [1, p. 300, Proposition 3]: ###### Proposition 1. Let the system $\\{f_{n}(x)\\}$ of functions be defined by the generating function $A_{t}(x)=\mathop{\displaystyle\sum}\limits_{n=0}^{\infty}f_{n}(x)t^{n}.$ If there exists a sequence $w_{k}=w_{k}(x)$ such that $\mathop{\displaystyle\sum}\limits_{k=0}^{\infty}w_{k}\mathcal{D}^{k}A_{t}(x)\mathcal{D}^{k}A_{z}(x)=A_{tz}(x)$ with $\mathcal{D}=\frac{d}{dx}$, then for $i,j=0,1,\cdots$, $\mathop{\displaystyle\sum}\limits_{k=0}^{\infty}w_{k}\mathcal{D}^{k}f_{i}(x)\mathcal{D}^{k}f_{j}(x)=\delta_{i,j}f_{i}(x).$ As an application of Proposition 1, Abel and Li [1] use the generating function in Eq-(2.7) for the Bernstein basis functions. They also use generating functions for the Szasz-Mirakjan basis functions and Baskakov basis functions. In this section, we apply our novel generating functions to Proposition 1, which give pointwise orthogonality relations for the Bernstein polynomials and the Szasz-Mirakjan type basis functions, respectively. As applications of Proposition 1, we give the following examples: ###### Example 1. For given $n$ and $k$, the Bernstein basis functions $f_{i}(x,n;a,b)=\mathbb{Y}_{i}^{n}(x;a,b,n)=\left(\begin{array}[]{c}n\\\ i\end{array}\right)\left(\frac{x-a}{b-a}\right)^{k}(\frac{b-x}{b-a})^{n-k}$ are generated by the function in (2.1), that is $A_{t}(x)=\frac{t^{k}\left(x-a\right)^{k}e^{(b-x)t}}{(b-a)^{n}k!}=\mathop{\displaystyle\sum}\limits_{i=0}^{\infty}\frac{f_{i}(x,n;a,b)}{i!}t^{i}.$ It is easy to check that Proposition 1 holds with $w_{k}=w_{k}(x)=\mathbb{Y}_{k}^{n}(x;a,b,n)$. ###### Example 2. Using (3.2), for $i\geq 0$, we generalize the Szasz-Mirakjan type basis functions as follows $f_{i}(x,n;a,b)=\frac{(n\frac{x-a}{b-a})^{i}e^{-n\frac{x-a}{b-a}}}{i!},$ where $a$ and $b$ are nonnegative real parameters with $a\neq b$, $n$ is a positive integer and $x\in\left[a,b\right]$. The functions $f_{i}(x,n;a,b)$ are generated by $A_{t}(x)=\exp\left((t-1)n\left(\frac{x-a}{b-a}\right)\right)=\mathop{\displaystyle\sum}\limits_{i=0}^{\infty}f_{i}(x,n;a,b)t^{i},$ where $\exp(x)=e^{x}$. In this case, Proposition 1 holds with $w_{k}=w_{k}(x)=\frac{\left(\frac{x-a}{b-a}\right)^{k}}{n^{k}k!}$. Therefore, we have $\mathop{\displaystyle\sum}\limits_{k=0}^{\infty}\frac{\left(\frac{x-a}{b-a}\right)^{k}}{n^{k}k!}\mathcal{D}^{k}f_{i}(x,n;a,b)\mathcal{D}^{k}f_{i}(x,n;a,b)=\delta_{i,j}f_{i}(x,n;a,b).$ ###### Remark 19. If $a=0$ and $b=1$ in Example 2, then we arrive at the Szasz-Mirakjan basis functions which are given in [1, p. 300, Example 2]. ###### Acknowledgement 1. The author would like to thank Professor Ronald Goldman (Rice University, Houston, USA) for his very valuable comments, criticisms and for his very useful suggestions on this present paper. The present investigation was supported by the Scientific Research Project Administration of Akdeniz University. ## References * [1] U. Abel and Z. Li, A new proof of an identity of Jetter and Stöckler for multivariate Bernstein polynomials, Computer Aided Geometric Design, 23(3) (2006), 297-301. * [2] M. Acikgoz and S. Araci, On generating function of the Bernstein polynomials, Numerical Analysis and Applied Mathematics, Amer. Inst. Phys. Conf. Proc. CP1281, (2010) 1141-1143. * [3] S. N. Bernstein, Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Comm. Soc. Math. Charkow Sér. 2 t. 13, 1-2 (1912-1913). * [4] L. Busé and R. Goldman, Division algorithms for Bernstein polynomials, Computer Aided Geometric Design, 25(9) (2008), 850-865. * [5] R. Goldman, An Integrated Introduction to Computer Graphics and Geometric Modeling, CRC Press, Taylor and Francis, New York, 2009. * [6] R. Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, Academic Press, San Diego, 2002. * [7] R. Goldman, Identities for the Univariate and Bivariate Bernstein Basis Functions, Graphics Gems V, edited by Alan Paeth, Academic Press, (1995), 149-162. * [8] L. C. Jang, W.-J. Kim and Y. Simsek, A study on the $p$-adic integral representation on $\mathbb{Z}p$ associated with Bernstein and Bernoulli polynomials, Advances in Difference Equations 2010 (2010), Article ID 163217, 6pp. * [9] K. Jetter and J. Stöckler, An identity for multivariate Bernstein poynomials, Computer Aided Geometric Design, 20 (2003), 563-577. * [10] M. S. Kim, D. Kim, and T. Kim, On the $q$-Euler numbers related to modified $q$-Bernstein polynomials, Abstr. Appl. Anal. 2010, Art. ID 952384, 15 pages, doi:10.1155/2010/952384, arXiv:1007.3317v1. * [11] G. G. Lorentz, Bernstein Polynomials, Chelsea Pub. Comp. New York, N. Y. 1986. * [12] G. M. Phillips, Interpolation and approximation by polynomials, CMS Books in Mathematics/ Ouvrages de Mathématiques de la SMC, 14. Springer-Verlag, New York, (2003). * [13] Y. Simsek and M. Acikgoz, A new generating function of ($q$-) Bernstein-type polynomials and their interpolation function, Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010. doi:10.1155/2010/769095.
arxiv-papers
2010-12-21T10:01:40
2024-09-04T02:49:15.975820
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yilmaz Simsek", "submitter": "Yilmaz Simsek", "url": "https://arxiv.org/abs/1012.5538" }
1012.5636
Alexandrov meets Kirszbraun S. Alexander, V. Kapovitch, A. Petrunin We give a simplified proof of the generalized Kirszbraun theorem for Alexandrov spaces, which is due to Lang and Schroeder. We also discuss related questions, both solved and open. § INTRODUCTION Kirszbraun's theorem states that any short map (i.e. 1-Lipschitz map) from a subset of Euclidean space to another in Euclidean space can be extended as a short map to the whole space. This theorem was proved first by Kirszbraun in <cit.>. Later it was reproved by Valentine in <cit.> and <cit.>, where he also generalized it to pairs of Hilbert spaces of arbitrary dimension as well as pairs of spheres of the same dimension and pairs of hyperbolic spaces with the same curvature. J. Isbel in <cit.> studied target spaces that satisfy the above condition for any source space. Valentine was also interested in pairs of metric spaces, say $\spc{U}$ and $\spc{L}$, which satisfy the above property, namely, given a subset $Q\subset\spc{U}$, any short map $Q\to\spc{L}$ can be extended to a short map $\spc{U}\to \spc{L}$. It turns out that this property has a lot in common with the definition of Alexandrov spaces (see theorems <ref>, <ref> and <ref>). Surprisingly, this relationship was first discovered only in the 1990's; it was first published by Lang and Schroeder in <cit.>. (The third author of this paper came to similar conclusions a couple of years earlier, and told it to the first author, but did not publish the result.) We slightly improve the results of Lang and Schroeder. Our proof is based on the barycentric maps introduced by Kleiner in <cit.>. The material of this paper will be included in the book on Alexandrov geometry that we are currently writing, but it seems useful to publish it now. Structure of the paper. We introduce notations in Section <ref>. In section <ref> we give altternative definitions of Alexandrov spaces based on the Kirszbraun property for 4-point sets. The generalized Kirszbraun theorem is proved in Section <ref>. In the sections <ref> and <ref> we describe some comparison properties of finite subsets of Alexandrov spaces. In Section <ref> we discuss related open problems. Appendices <ref> and <ref> describe Kleiner's barycentric map and an analog of Helly's theorem for Alexandrov spaces. Historical remark. Not much is known about the author of this remarkable theorem. The theorem appears in Kirszbraun's master's thesis which he defended in Warsaw University in 1930. His name is Mojżesz and his second name is likely to be Dawid but is uncertain. He was born either in 1903 or 1904 and died in a ghetto in 1942. After university, he worked as an actuary in an insurance company; <cit.> seems to be his only publication in mathematics. We want to thank S. Ivanov, N. Lebedeva and A. Lytchak for useful comments and pointing out misprints. Also we want to thank L. Grabowski for bringing to our attention the entry about Kirszbraun in the Polish Biographical Dictionary. § PRELIMINARIES In this section we mainly introduce our notations. Metric spaces. Let $\spc{X}$ be a metric space. The distance between two points $x,y\in\spc{X}$ will be denoted as $\dist{x}{y}{}$ or $\dist{x}{y}{\spc{X}}$. Given $R\in[0,\infty]$ and $x\in \spc{X}$, the sets \begin{align*} \oBall(x,R)&=\{y\in \spc{X}\mid \dist{x}{y}{}<R\}, \\ \cBall[x,R]&=\{y\in \spc{X}\mid \dist{x}{y}{}\le R\}. \end{align*} are called respectively the open and closed ball of radius $R$ with center at $x$. A metric space $\spc{X}$ is called if for any $\eps>0$ and any two points $x,y\in \spc{X}$ with $\dist{x}{y}{}<\infty$ there is an $\eps$-midpoint for $x$ and $y$; i.e. there is a point $z\in \spc{X}$ such that $\dist{x}{z}{},\dist{z}{y}{}<\tfrac{1}{2}\cdot \dist[{{}}]{x}{y}{}+\eps$. Model space. $\Lob{m}{\kappa}$ denotes $m$-dimensional model space with curvature $\kappa$; i.e. the simply connected $m$-dimensional Riemannian manifold with constant sectional curvature $\kappa$. Set $\varpi\kappa=\diam\Lob2\kappa$$\varpi\kappa$, so $\varpi\kappa=\infty$ if $\kappa\le0$ and $\varpi\kappa=\pi/\sqrt{\kappa}$ if $\kappa>0$. (The letter $\varpi{}$ is a glyph variant of lower case $\pi$, but is usually pronounced as pomega.) Ghost of Euclid. Let $\spc{X}$ be a metric space and $\II$ be a real interval. A globally isometric map $\gamma\:\II\to \spc{X}$ will be called a unitspeed geodesic. A unitspeed geodesic between $p$ and $q$ will be denoted by $\geod_{[p q]}$. We consider $\geod_{[p q]}$ with parametrization starting at $p$; i.e. $\geod_{[p q]}(0)=p$ and $\geod_{[p q]}(\dist{p}{q}{})=q$. The image of $\geod_{[p q]}$ will be denoted by $[p q]$ and called a geodesicgeodesic. Also we will use the following short-cut notation: \begin{align*} \l] p q \r[&=[p q]\backslash\{p,q\}, \l] p q \r]&=[p q]\backslash\{p\}, \l[ p q \r[&=[p q]\backslash\{q\}. \end{align*} A metric space $\spc{X}$ is called if for any two points $x,y\in \spc{X}$ there is a geodesic $[x y]$ in $\spc{X}$. Given a geodesic $[p q]$, we denote by $\dir{p}{q}$ its direction at $p$. We may think of $\dir{p}{q}$ as belonging to the space of directions $\Sigma_p$ at $p$, which in turn can be identified with the unit sphere in the tangent space $\T_p$ at $p$. Further we set $\ddir{p}{q}=\dist[{{}}]{p}{q}{}\cdot\dir{p}{q}$; it is a tangent vector at $p$, that is, an element of $\T_p$. For a triple of points $p,q,r\in \spc{X}$, a choice of triple of geodesics $([q r], [r p], [p q])$ will be called a triangle and we will use the notation $\trig p q r=([q r], [r p], [p q])$. If $p$ is distinct from $x$ and $y$, a pair of geodesics $([p x],[p y])$ will be called a hingehinge, and denoted by $\hinge p x y=([p x],[p y])$. A locally Lipschitz function $f$ on a metric space $\spc{X}$ is called $\lambda$-convex ($\lambda$-concave) if for any geodesic $\geod_{[p q]}$ in $\spc{X}$ the real-to-real function $$t\mapsto f\circ\geod_{[p q]}(t)-\tfrac\lambda2\cdot t^2$$ is convex (respectively concave). In this case we write $f''\ge \lambda$ (respectively $f''\le \lambda$). A function $f$ is called strongly convex (strongly concave) if $f''\ge \delta$ (respectively $f''\le -\delta$) for some $\delta>0$. Model angles and triangles. Let $\spc{X}$ be a metric space, $p,q,r\in \spc{X}$ and $\kappa\in\RR$. Let us define a model triangle $\trig{\~p}{\~q}{\~r}$ $\trig{\~p}{\~q}{\~r}=\modtrig\kappa(p q r)$) to be a triangle in the model plane $\Lob2\kappa$ such that \ \ \dist{\~q}{\~r}{}=\dist{q}{r}{}, \ \ \dist{\~r}{\~p}{}=\dist{r}{p}{}.$$ If $\kappa\le 0$, the model triangle is said to be defined, since such a triangle always exists and is unique up to an isometry of $\Lob2\kappa$. If $\kappa>0$, the model triangle is said to be defined if in addition $$\dist{p}{q}{}+\dist{q}{r}{}+\dist{r}{p}{}< 2\cdot\varpi\kappa.$$ In this case the triangle also exists and is unique up to an isometry of $\Lob2\kappa$. If for $p,q,r\in \spc{X}$, the model triangle $\trig{\~p}{\~q}{\~r}=\modtrig\kappa(p q r)$ is defined and $\dist{p}{q}{},\dist{p}{r}{}>0$, then the angle measure of $\trig{\~p}{\~q}{\~r}$ at $\~p$ will be called the model angle of the triple $p$, $q$, $r$, and will be denoted by $\angk\kappa p q r$. Curvature bounded below. We will denote by $\CBB{}{\kappa}$, complete intrinsic spaces $\spc{L}$ with curvature $\ge\kappa$ in the sense of Alexandrov. Specifically, $\spc{L}\in \CBB{}{\kappa}$ if for any quadruple of points $p,x^1,x^2,x^3\in \spc{U}$ , we have $$\angk\kappa p{x^1}{x^2} +\angk\kappa p{x^2}{x^3} +\angk\kappa p{x^3}{x^1}\le 2\cdot\pi.\eqlbl{Yup-kappa}$$ or at least one of the model angles $\angk\kappa p{x^i}{x^j}$ is not defined. Condition <ref> will be called (1+3)-point comparison. According to Plaut's theorem <cit.>, any space $\spc{L}\in \CBB{}{}$ is $G_\delta$-geodesic; that is, for any point $p\in \spc{L}$ there is a dense $G_\delta$-set $W_p\subset\spc{L}$ such that for any $q\in W_p$ there is a geodesic $[p q]$. We will use two more equivalent definitions of $\CBB{}{}$ spaces (see <cit.>). Namely, a complete $G_\delta$-geodesic space is in $\CBB{}{}$ if and only if it satisfies either of following conditions: * (point-on-side comparison) For any geodesic $[x y]$ and $z\in \l]x y\r[$, we have $$\angk\kappa x p y\le\angk\kappa x p z; \eqlbl{POS-CBB}$$ or, equivalently, $$\dist{\~p}{\~z}{}\le \dist{p}{z}{},$$ where $\trig{\~p}{\~x}{\~y}=\modtrig\kappa(p x y)$, $\~z\in\l] \~x\~y\r[$, $\dist{\~x}{\~z}{}=\dist{x}{z}{}$. * (hinge comparison) For any hinge $\hinge x p y$, the angle $\mangle\hinge x p y$ is defined and $$\mangle\hinge x p y\ge\angk\kappa x p y.$$ Moreover, if $z\in\l]x y\r[$, $z\not=p$ then for any two hinges $\hinge z p y$ and $\hinge z p x$ with common side $[z p]$ $$\mangle\hinge z p y + \mangle\hinge z p x\le\pi.$$ We also use the following standard result in Alexandrov geometry, which follows from the discussion in the survey of Plaut <cit.>. Let $\spc{L}\in \CBB{}{}$. Given an array of points $(x^1,x^2\dots,x^n)$ in $\spc{L}$, there is a dense $G_\delta$-set $W\subset\spc{L}$ such that for any $p\in W$, all the directions $\dir{p}{x^i}$ lie in an isometric copy of a unit sphere in $\Sigma_p$. (Or, equivaletntly, all the vectors $\ddir{p}{x^i}$ lie in a subcone of the tangent space $\T_p$ which is isometric to Euclidean space.) Curvature bounded above. We will denote by $\Cat{}{\kappa}$ the class of metric spaces $\spc{U}$ in which any two points at distance $<\varpi\kappa$ are joined by a geodesic, and which have curvature $\le\kappa$ in the following global sense of Alexandrov: namely, for any quadruple of points $p^1,p^2,x^1,x^2\in \spc{U}$, we have \angk{\kappa}{p^1}{x^1}{x^2} \le \angk{\kappa}{p^1}{p^2}{x^1}+\angk{\kappa}{p^1}{p^2}{x^2}, \ \t{or}\ \angk{\kappa} {p^2}{x^1}{x^2}\le \angk{\kappa} {p^2}{p^1}{x^1} + \angk{\kappa} {p^2}{p^1}{x^2}, \eqlbl{gokova:eq:2+2}$$ one of the six model angles above is undefined. The condition <ref> will be called (2+2)-point comparison (or (2+2)-point $\kappa$-comparison if a confusion may arise). We denote the complete $\Cat{}{\kappa}$ spaces by $\cCat{}{\kappa}$. The following lemma is a direct consequence of the definition: Any complete intrinsic space $\spc{U}$ in which every quadruple $p^1,p^2,x^1,x^2$ satisfies the (2+2)-point $\kappa$- is a $\cCat{}{\kappa}$ space (that is, any two points at distance $<\varpi\kappa$ are joined by a geodesic). In particular, the completion of a $\Cat{}{\kappa}$ space again lies in $\Cat{}{\kappa}$. We have the following basic facts (see [1]): In a $\Cat{}{\kappa}$ space, geodesics of length $<\varpi\kappa$ are uniquely determined by, and continuously dependent on, their endpoint pairs. In a $\Cat{}{\kappa}$ space, any open ball $\oBall(x,R)$ of radius $R\le\varpi\kappa/2$ is convex, that is, $\oBall(x,R)$ contains every geodesic whose endpoints it contains. We also use an equivalent definition of $\Cat{}{\kappa}$ spaces (see <cit.>). Namely, a metric space $\spc{U}$ in which any two points at distance $<\varpi\kappa$ are joined by a geodesic is a $\Cat{}{\kappa}$ space if and only if it satisfies the following condition: * (point-on-side comparison) for any geodesic $[x y]$ and $z\in \l]x y\r[$, we have $$\angk\kappa x p y\ge\angk\kappa x p z,$$ or equivalently, $$\dist{\~p}{\~z}{}\ge \dist{p}{z}{}, \eqlbl{POS-CAT}$$ where $\trig{\~p}{\~x}{\~y}=\modtrig\kappa(p x y)$, $\~z\in\l] \~x\~y\r[$, $\dist{\~x}{\~z}{}=\dist{x}{z}{}$. We also use Reshetnyak's majorization theorem <cit.>. Suppose $\~\alpha$ is a simple closed curve of finite length in $\Lob2{\kappa}$, and $D\subset\Lob2{\kappa}$ is a closed region bounded by $\~\alpha$. If $\spc{X}$ is a metric space, a length-nonincreasing map $F\:D\to\spc{X}$ is called majorizing if it is length-preserving on $\~\alpha$. In this case, we say that $D$ majorizes the curve $\alpha=F\circ\~\alpha$ under the map $F$. Reshetnyak's majorization theorem Any closed curve $\alpha$ of length $<2\cdot \varpi\kappa$ in $\spc{U}\in\Cat{}{\kappa}$ is majorized by a convex region in $\Lob2\kappa$. Ultralimit of metric spaces. Given a metric space $\spc{X}$, its ultrapower (i.e. ultralimit of constant sequence $\spc{X}_n=\spc{X}$) will be denoted as $\spc{X}^\o$; here $\o$ denotes a fixed nonprinciple ultrafilter. For definitions and properties of ultrapowers, we refer to a paper of Kleiner and Leeb <cit.>. We use the following facts about ultrapowers which easily follow from the definitions (see <cit.> for details): * $\spc{X}\in\cCat{}{\kappa}\ \Longleftrightarrow\ \spc{X}^\o\in\cCat{}{\kappa}$. * $\spc{X}\in\CBB{}{\kappa}\ \Longleftrightarrow\ \spc{X}^\o\in\CBB{}{\kappa}$. * $\spc{X}$ is intrinsic if and only if $\spc{X}^\o$ is geodesic. Note that if $\spc{X}$ is proper (namely, bounded closed sets are compact), then $\spc{X}$ and $\spc{X}^\o$ coincide. Thus a reader interested only in proper spaces may ignore everything related to ultrapower in this article. section.thm. #1.
arxiv-papers
2010-12-27T16:54:34
2024-09-04T02:49:15.984552
{ "license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/", "authors": "Stephanie Alexander, Vitali Kapovitch and Anton Petrunin", "submitter": "Anton Petrunin", "url": "https://arxiv.org/abs/1012.5636" }
1012.5709
Hawking temperature for constant curvature black bole and its analogue in de Sitter Space Rong-Gen Cai1,***Email address: cairg@itp.ac.cn and Yun Soo Myung2,†††Email address: ysmyung@inje.ac.kr 1Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China 2Institute of Basic Science and School of Computer Aided Science, Inje University, Gimhae 621-749, Korea Abstract The constant curvature (CC) black holes are higher dimensional generalizations of BTZ black holes. It is known that these black holes have the unusual topology of ${\cal M}_{D-1}\times S^{1}$, where $D$ is the spacetime dimension and ${\cal M}_{D-1}$ stands for a conformal Minkowski spacetime in $D-1$ dimensions. The unusual topology and time-dependence for the exterior of these black holes cause some difficulties to derive their thermodynamic quantities. In this work, by using globally embedding approach, we obtain the Hawking temperature of the CC black holes. We find that the Hawking temperature takes the same form when using both the static and global coordinates. Also it is identical to the Gibbons-Hawking temperature of the boundary de Sitter spaces of these CC black holes. Employing the same approach, we obtain the Hawking temperature for the counterparts of CC black holes in de Sitter spaces. ## 1 Introduction Over the past years, the so-called BTZ (Banados-Teitelboim-Zanelli) [1] black hole solutions have played the important role in understanding microscopic degrees of freedom of black hole. The BTZ black hole is an exact solution of Einstein field equations with a negative cosmological constant in three dimensions. It is well known the BTZ black hole can be constructed by identifying points along the orbit of a Killing vector in a three dimensional anti-de Sitter (AdS) space. The BTZ black hole has a topology of ${\cal M}_{2}\times S^{1}$, where $M_{2}$ denotes a conformal Minkowski space in two dimensions. Following the same way as done in three dimensions, one can construct analogues of the BTZ solution, the so-called constant curvature (CC) black holes in higher $(D\geq 4$) dimensional AdS spaces [2, 3, 4]. However, such black holes have topology of ${\cal M}_{D-1}\times S^{1}$ in $D$ dimensions, which is quite different from the known topology of ${\cal M}_{2}\times S^{D-2}$ for the usual black holes in $D$ dimensions. In addition, the exterior region of these CC black holes is time-dependent and thus, there is no global time-like Killing vector [2]. Because of this, it is difficult to discuss Hawking radiation and thermodynamics associated with these black holes. For example, see [5, 6, 7] and references therein. On the other hand, these spacetimes are interesting examples of smooth time- dependent solutions. Particularly, they are consistent background spacetimes for string theory at least to leading order since they are vacuum solutions to Einstein field equations with a negative cosmological constant too. Further we note that these spacetimes are time-dependent, the boundary metric is also time-dependent, and it is asymptotically AdS. Therefore, it might open a window to investigate dual strong coupling field theory in the time-dependent backgrounds through the AdS/CFT correspondence [8]. Especially, the $D$-dimensional CC black holes have the boundary topology of $dS_{D-2}\times S^{1}$, where $dS_{D-2}$ denotes a $(D-2)$-dimensional de Sitter (dS) space. Resorting the AdS/CFT correspondence, these CC black holes are gravity duals to strong coupling conformal field theories living on $dS_{D-2}\times S^{1}$. Finally it is observed in [9] that these CC black holes have a close connection to the so-called “bubbles of nothing” in AdS space [10, 11]. The bubbles of nothing were constructed by analytically continuing (Schwarzschild, Reissner-Nordström, and Kerr) black holes in AdS spaces. The stress-energy tensor for dual conformal field theories to these CC black holes was calculated in [9, 11]. It is well known that there is the Gibbons-Hawking temperature $T_{\rm GH}$ for a comoving observer in a dS space [12]. This temperature may be viewed as the Hawking temperature $T_{\rm HK}$ associated with cosmological horizon of dS space. A $D$-dimensional dS space can be embedded as a hypersurface into a $(D+1)$-dimensional Minkowski space. Then, the comoving observer in dS space is identical to an observer with a constant acceleration in Minkowski space. According to Davies [13] and Unruh [14], an observer with a constant acceleration in Minkowski space will see a hot bath with the Davies-Unruh temperature $T_{\rm DU}=a/2\pi$ where $a$ is the acceleration of the observer. It turns out that the Gibbons-Hawking temperature of dS space is equivalent to the Davies-Unruh temperature of the corresponding observer in Minkowski space. One decade ago, it was shown that an observer with a constant acceleration $a$ in dS space will detect a temperature given by $\sqrt{a^{2}+1/l^{2}}/2\pi$, where $l$ is the radius of the dS space [15]. This was soon generalized to the cases of dS/AdS space by Deser and Levin [16] with temperatures of $\sqrt{a^{2}\pm 1/l^{2}}/2\pi$. Further, Deser and Levin have shown that the temperature is equivalent to the Davies-Unruh temperature for the corresponding observer in Minkowski space. Further examples for the equivalence have been shown by globally embedding curved spaces including BTZ, Schwarzschild, Schwarzschild-AdS (dS), and Reissner-Nordström solutions into higher dimensional Minkowski spaces in Ref.[17]. For more examples on the equivalence, see [18] and references therein. In this work, the “globally embedding approach” will be employed to determine the Hawking temperature of CC black holes and positive CC spaces. This approach shows a clear way to compute the Hawking temperature, in comparison to other methods with ambiguity to calculate it. In the next section, we show that the Hawking temperature of the constant curvature black holes is given by $T_{\rm HK}=r_{+}/(2\pi l)$ using both the static and global coordinates in AdS spaces. Further it is shown that the Hawking temperature is identical to the Gibbons-Hawking temperature of the boundary dS space. In section 3 we consider the counterparts of the CC black holes in dS spaces. These are constant curvature (CC) spaces with the cosmological horizon. We find that the Hawking temperature for these spaces are given by $T_{\rm HK}=r_{+}/(2\pi l)$, where $r_{+}$ and $l$ are the cosmological horizon radius and the radius of dS spaces, respectively. We give our conclusions and discussions in section 4\. In this paper, we confine ourselves to the five dimensional space. The generalization to other dimensions is straightforward. ## 2 Hawking temperature of CC black holes A five dimensional AdS space is defined as the universal covering space of a surface obeying $-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}-x_{5}^{2}=-l^{2},$ (2.1) where $l$ is the AdS radius. This surface has fifteen Killing vectors of seven rotations and eight boosts. We consider the boost $\xi=(r_{+}/l)(x_{4}\partial_{5}+x_{5}\partial_{4})$ with its norm $\xi^{2}=r_{+}^{2}(-x_{4}^{2}+x_{5}^{2})/l^{2}$ where $r_{+}$ is an arbitrary real constant. The so-called CC black hole is constructed by identifying points along the orbit of the Killing vector $\xi$. Since the starting point is the AdS, the resulting black hole has a constant curvature as the AdS does show. The topology of the black holes is changed to be ${\cal M}_{4}\times S^{1}$, which is quite different from the usual topology of ${\cal M}_{2}\times S^{3}$ for five dimensional black holes. Here ${\cal M}_{n}$ denotes a conformal Minkowski space in $n$ dimensions. For more details for the construction of the black hole, see [3, 4]. The CC black holes can be nicely described by using Kruskal coordinates. For this purpose, a set of coordinates on the AdS for the region of $\xi^{2}>0$ has been introduced in Ref.[3]. The six dimensionless local coordinates $(y_{i},\varphi)$ are given by $\displaystyle x_{i}=\frac{2ly_{i}}{1-y^{2}},\ \ \ i=0,1,2,3$ $\displaystyle x_{4}=\frac{lr}{r_{+}}\sinh\left(\frac{r_{+}\varphi}{l}\right),$ $\displaystyle x_{5}=\frac{lr}{r_{+}}\cosh\left(\frac{r_{+}\varphi}{l}\right),$ (2.2) with $r=r_{+}\frac{1+y^{2}}{1-y^{2}},\ \ \ y^{2}=-y_{0}^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}.$ (2.3) Here the allowed regions are $-\infty<y_{i}<\infty$ and $-\infty<\varphi<\infty$ with the restriction $-1<y^{2}<1$. In these coordinates, the boundary at $r\to\infty$ corresponds to the hyperbolic “ball” which satisfies $y^{2}=1$. The induced line element can be written down $ds^{2}=\frac{l^{2}(r+r_{+})^{2}}{r_{+}^{2}}(-dy_{0}^{2}+dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2})+r^{2}d\varphi^{2}.$ (2.4) Obviously, the Killing vector is given by $\xi=\partial_{\varphi}$ with its norm $\xi^{2}=r^{2}$. The black hole spacetime could be obtained by identifying $\varphi\sim\varphi+2\pi n$, and the topology of the black hole takes the form of ${\cal M}_{4}\times S^{1}$ clearly. On the other hand, the CC black holes can also be described by introducing Schwarzschild coordinates. The local “spherical” coordinates ($t,r,\theta,\chi$) in the hyperplane $y_{i}$ are $\displaystyle y_{0}=f\sin\theta\sinh(r_{+}t/l),\ \ \ y_{1}=f\sin\theta\cosh(r_{+}t/l),$ $\displaystyle y_{2}=f\cos\theta\sin\chi,\ \ \ \ \ \ y_{3}=f\cos\theta\cos\chi,$ (2.5) where $f=[(r-r_{+})/(r+r_{+})]^{1/2}$, $0\leq\theta\leq\pi/2$, $0\leq\chi\leq 2\pi$, and $r_{+}\leq r<\infty$. One finds that the solution (2.4) can be rewritten as $ds^{2}=l^{2}N^{2}d\Omega_{3}^{2}+N^{-2}dr^{2}+r^{2}d\varphi^{2},$ (2.6) where $N^{2}=\frac{r^{2}-r_{+}^{2}}{l^{2}},\ \ \ d\Omega_{3}^{2}=-\sin^{2}\theta dt^{2}+\frac{l^{2}}{r_{+}^{2}}(d\theta^{2}+\cos^{2}\theta d\chi^{2}).$ (2.7) This is the black hole solution expressed in terms of Schwarzschild coordinates. Here $r=r_{+}$ is the location of black hole horizon. In these coordinates the solution seems static. However, we observe from (2.6) that the form (2.7) does not cover the whole exterior region of black hole since the difference of $y^{2}_{1}-y_{0}^{2}$ is required to be positive in the region covered by these coordinates. Indeed, it has been proved that there is no globally timelike Killing vector in this geometry [2]. Now we consider a static observer with constant $(r>r_{+},\theta,\chi,\varphi)$ in the black hole background (2.6). To this observer, we find that an associated acceleration $a_{5}$ is given by $a_{5}^{2}=\frac{1}{l^{2}(r^{2}-r_{+}^{2})}\frac{1}{\sin^{2}\theta}\left(r^{2}\sin^{2}\theta+r_{+}^{2}\cos^{2}\theta\right).$ (2.8) On the other hand, the acceleration of $a_{6}$ for the corresponding observer in six dimensional embedding Minkowski space is given by $a_{6}^{-2}=x_{1}^{2}-x_{0}^{2}=\frac{l^{2}(r^{2}-r_{+}^{2})}{r_{+}^{2}}\sin^{2}\theta.$ (2.9) It is easy to check that these two accelerations obey the relation $a_{6}^{2}=-\frac{1}{l^{2}}+a^{2}_{5}.$ (2.10) This shows that the Davies-Unruh temperature for the local observer in six dimensional Minkowski space is $T_{\rm DU}=\frac{a_{6}}{2\pi}=\frac{r_{+}}{2\pi l\sqrt{r^{2}-r_{+}^{2}}}\frac{1}{\sin\theta}.$ (2.11) We note that the redshift factor of $\sqrt{-g_{00}}=lN\sin\theta$ for the black hole (2.6) is necessary to define the Hawking temperature. Hence we conclude that the Hawking temperature of the CC black hole is $T_{\rm HK}=\sqrt{-g_{00}}~{}T_{\rm DU}=\frac{r_{+}}{2\pi l}.$ (2.12) We notice that the Hawking temperature $T_{\rm HK}$ is consistent with the inverse period of the Euclidean time derived from the solution (2.7). As the case in four dimensions [4], there is another set of coordinates covering the whole exterior of the Minkowskian black hole geometry as [9] $\displaystyle y_{0}=f\sinh(r_{+}t/l),\ \ \ y_{1}=f\cos\theta\cosh(r_{+}t/l),$ $\displaystyle y_{2}=f\sin\theta\cos\chi\cosh(r_{+}t/l),\ \ \ y_{3}=f\sin\theta\sin\chi\cosh(r_{+}t/l),$ (2.13) where $f$ is given by (2) and the allowed regions are $0\leq\theta\leq\pi$, $r_{+}\leq r<\infty$, and $0\leq\chi\leq 2\pi$. In these coordinates, the solution can be expressed as $ds^{2}=N^{2}l^{2}d\Omega_{3}^{2}+N^{-2}dr^{2}+r^{2}d\varphi^{2},$ (2.14) where $N^{2}=(r^{2}-r_{+}^{2})/l^{2}$ and $d\Omega_{3}^{2}=-dt^{2}+\frac{l^{2}}{r_{+}^{2}}\cosh^{2}(r_{+}t/l)(d\theta^{2}+\sin^{2}\theta d\chi^{2}).$ (2.15) The time-dependence of the solution is manifest in this coordinate system. We introduce a static observer located at constant $r>r_{+}$, $\varphi$ and $\chi$, but $\theta=0$ due to the spherical symmetry of the solution [16]. Here, we find that the acceleration $a_{5}$ associated with the observer is $a_{5}^{2}=\frac{r^{2}}{l^{2}(r^{2}-r_{+}^{2})},$ (2.16) while the acceleration $a_{6}$ of the corresponding observer in six dimensional Minkowski space is given by $a^{-2}_{6}=x_{1}^{2}-x_{0}^{2}=\frac{r_{+}^{2}}{l^{2}(r^{2}-r_{+}^{2})}.$ (2.17) They satisfy the relation (2.10) too. In this case, the Davies-Unruh temperature is given by $T_{\rm DU}=\frac{a_{6}}{2\pi}=\frac{r_{+}}{2\pi l\sqrt{r^{2}-r_{+}^{2}}}.$ (2.18) Considering the redshift factor of $\sqrt{-g_{00}}$, we get the Hawking temperature of the black hole in the line element of (2.14) as $T_{\rm HK}=\frac{r_{+}}{2\pi l}.$ (2.19) Thus we have obtained the Hawking temperature of the CC black hole by employing globally embedding approach combined with the Davies-Unruh temperature in six dimensional Minkowski space. The Hawking temperatures (2.12) and (2.19) are our main results. Here some remarks are in order. First, in general, Hawking temperature of black hole depends on coordinates used to calculate it. That is, the Hawking temperature may be different when using different coordinates, even for the same black hole. In our case, we obtained the same Hawking temperature for the CC black hole even when used the different coordinate systems (2.6) and (2.14). Second, we mention that the Hawking temperature (2.12) is the same as the inverse period of the Euclidean time for the Euclidean sector of the solution (2.6). However, when used the coordinates (2.15), the Hawking temperature is no longer the same as the inverse period of the Euclidean time. In order to see this, let us consider carefully the Euclidean sector of the black hole solution which can be obtained by replacing the time $t$ by $-i(\tau+\pi l/(2r_{+}))$ in (2.15). In this case, $d\Omega_{3}^{2}$ becomes $d\Omega_{3}^{2}=d\tau^{2}+\frac{l^{2}}{r_{+}^{2}}\sin^{2}(r_{+}\tau/l)(d\theta^{2}+\sin^{2}\theta d\chi^{2}).$ (2.20) In order that $d\Omega_{3}^{2}$ be a regular three-sphere, $\tau$ must have the period of $\tau\sim\tau+\tilde{\beta}$ with $\tilde{\beta}=\frac{\pi l}{r_{+}}.$ (2.21) Clearly this is not the inverse of Hawking temperature. This shows that the Euclidean method does not always provide a correct Hawking temperature of CC black holes. However, using both coordinates (2) and (2), the Euclidean time $\tau=it$ obtained by Wick rotation leads to the fact that it has a periodicity with period $2\pi l/r_{+}$, which gives a correct Hawking temperature of the black hole. This may be related to the issue of the factor 2 in [19]. Finally, we observe from (2.14) that the black hole solution has a boundary topology $dS_{3}\times S^{1}$ at $r=\infty$. The three dimensional de Sitter space $dS_{3}$ has a Hubble constant $H=r_{+}/l$. It is well known that for a de Sitter space with a Hubble constant $H$, there is the Gibbons-Hawking temperature $T=H/2\pi$ for a comving observer. We find that the Gibbons- Hawking temperature in our case is identical to the Hawking temperature of the CC black hole $T_{\rm GH}=\frac{H}{2\pi}=\frac{r_{+}}{2\pi l}=T_{\rm HK}.$ (2.22) ## 3 Hawking temperature of a positive CC space In this section we consider the analogue of the CC black hole in dS space. This space is constructed by identifying points along the orbit of a Killing vector in dS space. In fact, this space is a generalization of the three- dimensional Schwarzschild-de Sitter solution in higher dimensions. This space has a cosmological event horizon, and its topology is ${\cal M}_{D-1}\times S^{1}$ where ${\cal M}_{D-1}$ denotes a $(D-1)$-dimensional conformal Minkowski spacetime. Such space was constructed in Ref.[20]. As the case with a negative cosmological constant, we consider a five dimensional de Sitter space, which can be viewed as a hypersurface embedded into a six dimensional Minkowski space, satisfying $-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}=l^{2}$ (3.1) with $l$ the radius of the dS space. This dS space has fifteen Killing vectors of five boosts and ten rotations. We consider a rotational Killing vector $\xi=(r_{+}/l)(x_{4}\partial_{5}-x_{5}\partial_{4})$ with its norm $\xi^{2}=r_{+}^{2}/l^{2}(x_{4}^{2}+x_{5}^{2})$ where $r_{+}$ is an arbitrary real constant. Identifying points along the orbit of a Killing vector $\xi$, another one-dimensional manifold becomes compact and it is isomorphic to $S^{1}$. Thus, we obtain a spacetime of topology ${\cal M}^{4}\times S^{1}$ with cosmological horizon. For details of the construction of the space, see [20]. We can describe the spacetime in the region with $0\leq\xi^{2}\leq r_{+}^{2}$ by introducing six dimensionless local coordinates $(y_{i},\phi)$, $\displaystyle x_{i}=\frac{2ly_{i}}{1+y^{2}},\ \ \ i=0,1,2,3$ $\displaystyle x_{4}=\frac{lr}{r_{+}}\sin\left(\frac{r_{+}\phi}{l}\right),$ $\displaystyle x_{5}=\frac{lr}{r_{+}}\cos\left(\frac{r_{+}\phi}{l}\right),$ (3.2) where $r=r_{+}\frac{1-y^{2}}{1+y^{2}},\ \ \ y^{2}=-y_{0}^{2}+y_{1}^{2}+y_{2}^{2}+y_{3}^{2}.$ (3.3) Here the allowed regions are $-\infty<y_{i}<+\infty$ and $-\infty<\phi<+\infty$ with the restriction $-1<y^{2}<1$ to have a positive $r$. In the coordinates (3), the induced line element is $ds^{2}=\frac{l^{2}(r+r_{+})^{2}}{r_{+}^{2}}(-dy_{0}^{2}+dy_{1}^{2}+dy_{2}^{2}+dy_{3}^{2})+r^{2}d\phi^{2},$ (3.4) which is the same form as the case of a negative constant curvature [3]. However, it is noted that the coordinates (3) and the definition of $r$ differ from those in the CC black holes. In this coordinate system, it is evident that the Killing vector is $\xi=\partial_{\phi}$ with norm $\xi^{2}=r^{2}$. Imposing the identification $\phi\sim\phi+2\pi n$, the solution has the topology ${\cal M}_{4}\times S^{1}$. We now introduce the Schwarzschild coordinates to describe the solution. Using local “spherical” coordinates $(t,r,\theta,\chi)$ defined as $\displaystyle y_{0}=f\sin\theta\sinh(r_{+}t/l),\ ~{}~{}~{}~{}~{}y_{1}=f\sin\theta\cosh(r_{+}t/l),$ (3.5) $\displaystyle y_{2}=f\cos\theta\sin\chi,\ ~{}~{}~{}~{}~{}y_{3}=f\cos\theta\cos\chi,$ where $f=[(r_{+}-r)/(r+r_{+})]^{1/2}$, and the allowed coordinate ranges are $0<\theta<\pi/2$, $0<\chi<2\pi$, and $0<r<r_{+}$. The line element can be expressed as $ds^{2}=l^{2}N^{2}d\Omega_{3}^{2}+N^{-2}dr^{2}+r^{2}d\phi^{2}.$ (3.6) Here $N^{2}=(r_{+}^{2}-r^{2})/l^{2}$ and $d\Omega_{3}^{2}=-\sin^{2}\theta dt^{2}+\frac{l^{2}}{r_{+}^{2}}(d\theta^{2}+\cos^{2}\theta d\chi^{2}).$ (3.7) Clearly the location of $r=r_{+}$ represents a cosmological horizon. This solution is the counterpart of a five dimensional CC black hole described in the previous section. The only difference is that $N^{2}=(r^{2}-r_{+}^{2})/l^{2}$ is replaced by $N^{2}=(r_{+}^{2}-r^{2})/l^{2}$ here. Further, in three dimensions, the corresponding induced line element takes the form $ds^{2}=-(r_{+}^{2}-r^{2})dt^{2}+\frac{l^{2}}{r_{+}^{2}-r^{2}}dr^{2}+r^{2}d\phi^{2},$ (3.8) After a suitable rescaling of coordinates, it can be transformed to three dimensional Schwarzschild-de Sitter solution [21]. In this sense, the solution (3.6) can be viewed as an analogue of the three dimensional Schwarzschild-de Sitter solution in five dimensions. The solution (3.6) seems to be static, but it does not cover the whole region within the cosmological horizon. It can be seen from the definition of coordinates (3.5) because they must obey the constraint: $y_{1}^{2}-y_{0}^{2}=f^{2}\cos^{2}\theta\geq 0$. Considering a static observer located at constant $(r<r_{+},\theta,\chi$) in the background (3.6), we find that the static observer has a constant acceleration $a_{5}$ as $a_{5}^{2}=\frac{1}{l^{2}(r_{+}^{2}-r^{2})\sin^{2}\theta}\left(r^{2}\sin^{2}\theta+r_{+}^{2}\cos^{2}\theta\right),$ (3.9) while the observer in six dimensional Minkowski space has a constant acceleration $a_{6}$ as $a_{6}^{-2}=x_{1}^{2}-x_{0}^{2}=\frac{l^{2}(r_{+}^{2}-r^{2})}{r_{+}^{2}}\sin^{2}\theta.$ (3.10) These two accelerations are related to each other as $a_{6}^{2}=1/l^{2}+a_{5}^{2}.$ (3.11) According to Davies and Unruh, the observer has a temperature as $T_{\rm DU}=\frac{a_{6}}{2\pi}=\frac{r_{+}}{2\pi l\sqrt{r_{+}^{2}-r^{2}}\sin\theta}.$ (3.12) Taking into account the redshift factor $\sqrt{-g_{00}}$ of the observer, one has the Hawking temperature as $T_{\rm HK}=\frac{r_{+}}{2\pi l}.$ (3.13) On the other hand, one has another set of coordinates which covers the whole region within the cosmological horizon, $\displaystyle y_{0}=f\sinh(r_{+}t/l),\ ~{}~{}~{}~{}~{}~{}y_{1}=f\cos\theta\cosh(r_{+}t/l),$ $\displaystyle y_{2}=f\sin\theta\cos\chi\cosh(r_{+}t/l),\ ~{}~{}~{}~{}~{}~{}y_{3}=f\sin\theta\sin\chi\cosh(r_{+}t/l).$ (3.14) In this case, the line element is described by $ds^{2}=l^{2}N^{2}\tilde{d\Omega_{3}^{2}}+N^{-2}dr^{2}+r^{2}d\phi^{2},$ (3.15) where $N^{2}=(r_{+}^{2}-r^{2})/l^{2}$ and $\tilde{d\Omega_{3}^{2}}=-dt^{2}+\frac{l^{2}}{r_{+}^{2}}\cosh^{2}(r_{+}t/l)(d\theta^{2}+\sin^{2}\theta d\chi^{2}).$ (3.16) We consider a static observer located at constant position of $(r<r_{+},\chi,\varphi)$ and $\theta=0$. For such an observer, we have a constant acceleration $a_{5}$ as $a_{5}^{2}=\frac{r^{2}}{l^{2}(r^{2}_{+}-r^{2})}.$ (3.17) In six dimensional Minkowski space, the acceleration $a_{6}$ associated with the corresponding observer takes the form $a_{6}^{-2}=x_{1}^{2}-x_{0}^{2}=\frac{l^{2}(r_{+}^{2}-r^{2})}{r_{+}^{2}}.$ (3.18) We check that they obey the relation $a_{6}^{2}=a_{5}^{2}+1/l^{2}$. We conclude that the observer has the Davies-Unruh temperature $T_{\rm DU}=\frac{a_{6}}{2\pi}=\frac{r_{+}}{2\pi l\sqrt{r_{+}^{2}-r^{2}}}.$ (3.19) Considering the redshift factor for the observer in the background (3.15), we have the Hawking temperature observed as $T_{\rm HK}=\frac{r_{+}}{2\pi l}.$ (3.20) Consequently, we find the same Hawking temperature as (3.13) obtained when using the coordinates (3.6). ## 4 Conclusions and Discussions A $D$-dimensional CC black hole has unusual topological structure ${\cal M}_{D-1}\times S^{1}$ and there is no globally timelike Killing vector in the geometry of the black hole. Hence it was quite difficult to discuss thermodynamic properties and Hawking temperature associated with this black hole. For example, Banados has considered a five dimensional rotating CC black hole and embedded it into a Chern-Simons supergravity theory [3]. By computing related conserved charges, it was shown that the black hole mass is proportional to the product of outer horizon $r_{+}$ and inner horizon $r_{-}$, while the angular momentum is proportional to the sum of two horizons. In this case, the entropy of black hole is found to be proportional not to the outer horizon $r_{+}$ but the inner horizon $r_{-}$. This approach has two drawbacks. One is that the result cannot be degenerated to the non- rotating case. The other is that it cannot be generalized to other dimensions. Creighton and Mann have considered the quasilocal thermodynamics of a four dimensional CC black hole in general relativity by computing thermodynamic quantities at a finite boundary which encloses the black hole [5]. They have shown that the entropy is not associated with the event horizon, but the Killing horizon of a static observer which is tangent to the event horizon of the black hole. The quasilocal energy density [see (11) of [5]] is negative. In this work, we have derived Hawking temperature of CC black holes by employing the globally embedding approach since these black holes can be embedded into higher dimensional Minkowski space. We found that the Hawking temperature of CC black holes is given by $r_{+}/2\pi l$ when using both static and global coordinates. Here $r_{+}$ and $l$ are black hole horizon and the radius of AdS space. Furthermore we found that the Hawking temperature is also identical to the Gibbons-Hawking temperature of the boundary dS space of the CC black holes. Importantly, we mention that the Hawking temperature obtained in this work is the same as that obtained from semi-classical tunneling method [19]. It turns out that the globally embedding technique is powerful to determine the Hawking temperature of CC black hole without any ambiguity. Using the same approach, we also obtained Hawking temperature of a positive CC space which is counterpart of CC black hole in dS space. Finally, we comment that those solutions including CC black holes and positive CC spaces depend on an arbitrary real constant $r_{+}$. The $r_{+}$-dependence can be made disappear by rescaling coordinates. In this case, the Hawking temperature is given by $1/2\pi l$. ## Acknowledgments This work was initiated during the APCTP joint focus program: frontiers of black hole physics, Dec. 6-17, Pohang, Korea and Inje workshop on gravity and numerical relativity during Dec. 16-18, Busan, Korea, the warm hospitality extended to the authors in both places are grateful. RGC thanks S.P. Kim for useful discussions during APCTP focus program. This work was supported in part by a grant from Chinese Academy of Sciences and in part by the National Natural Science Foundation of China under Grant Nos. 10821504, 10975168 and 11035008, and by the Ministry of Science and Technology of China under Grant No. 2010CB833004. YSM was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No.2009-0062869). ## References * [1] M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992) [arXiv:hep-th/9204099]; M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48, 1506 (1993) [arXiv:gr-qc/9302012]. * [2] S. Holst and P. Peldan, Class. Quant. Grav. 14, 3433 (1997) [arXiv:gr-qc/9705067]. * [3] M. Banados, Phys. Rev. D 57, 1068 (1998) [arXiv:gr-qc/9703040]. * [4] M. Banados, A. Gomberoff and C. Martinez, Class. Quant. Grav. 15, 3575 (1998) [arXiv:hep-th/9805087]. * [5] J. D. Creighton and R. B. Mann, Phys. Rev. D 58, 024013 (1998) [arXiv:gr-qc/9710042]. * [6] S. F. Ross and G. Titchener, JHEP 0502, 021 (2005) [arXiv:hep-th/0411128]. * [7] J. A. Hutasoit, S. P. Kumar and J. Rafferty, JHEP 0904, 063 (2009) [arXiv:0902.1658 [hep-th]]. * [8] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]; E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998) [arXiv:hep-th/9802150]. * [9] R. G. Cai, Phys. Lett. B 544, 176 (2002) [arXiv:hep-th/0206223]. * [10] D. Birmingham and M. Rinaldi, Phys. Lett. B 544, 316 (2002) [arXiv:hep-th/0205246]. * [11] V. Balasubramanian and S. F. Ross, Phys. Rev. D 66, 086002 (2002) [arXiv:hep-th/0205290]. * [12] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977). * [13] P. C. W. Davies, J. Phys. A 8, 609 (1975). * [14] W. G. Unruh, Phys. Rev. D 14, 870 (1976). * [15] H. Narnhofer, I. Peter and W. E. Thirring, Int. J. Mod. Phys. B 10, 1507 (1996). * [16] S. Deser and O. Levin, Class. Quant. Grav. 14, L163 (1997) [arXiv:gr-qc/9706018]; S. Deser and O. Levin, Class. Quant. Grav. 15, L85 (1998) [arXiv:hep-th/9806223]. * [17] S. Deser and O. Levin, Phys. Rev. D 59, 064004 (1999) [arXiv:hep-th/9809159]. * [18] Y. W. Kim, J. Choi and Y. J. Park, Int. J. Mod. Phys. A 25, 3107 (2010) [arXiv:0909.3176 [gr-qc]]. * [19] A. Yale, arXiv:1012.2114 [gr-qc]. * [20] R. G. Cai, Phys. Lett. B 552, 66 (2003) [arXiv:hep-th/0207053]. * [21] S. Deser and R. Jackiw, Annals Phys. 153, 405 (1984); M. I. Park, Phys. Lett. B 440, 275 (1998) [arXiv:hep-th/9806119].
arxiv-papers
2010-12-28T07:44:51
2024-09-04T02:49:15.991087
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Rong-Gen Cai and Yun Soo Myung", "submitter": "Rong-Gen Cai", "url": "https://arxiv.org/abs/1012.5709" }
1012.5832
# On the Existence of Bertrand-Nash Equilibrium Prices Under Logit Demand W. Ross Morrow and Steven J. Skerlos Iowa State University wrmorrow@iastate.edu The University of Michigan, Department of Mechanical Engineering skerlos@umich.edu ###### Abstract. This article proves the existence of equilibrium prices in Bertrand competition with multi-product firms using the Logit model of demand. The most general proof, an application of the Poincare-Hopf Theorem, does not rely on restrictive assumptions such as single-product firms, firm homogeneity or symmetry, homogeneous product costs, or even concavity of the utility function with respect to prices. This proof relies on new conditions for the indirect utility function, along with fixed-point equations derived from the first- order conditions and a direct analysis of the second-order conditions that proves the uniqueness of profit-maximizing prices. The degree to which our conditions are as weak as possible is discussed. Models with finite purchasing power and convex total costs are also addressed. Analysis of equilibrium prices for multi-product firms with constant unit costs suggests that Bertrand-Nash equilibrium cannot adequately describe multi-product pricing in differentiated product markets. Significant portions of this research were undertaken while W. Ross Morrow was a Ph.D. student in Mechanical Engineering at the University of Michigan. The National Science Foundation, the University of Michigan Transportation Research Institute’s Doctoral Studies Program, and Iowa State University provided support for this research. The authors wish to thank Fred Feinberg, Erin MacDonald, Jong-Shi Pang, Che-Lin Su, and Norman Shiau for helpful suggestions. ## 1\. Introduction Bertrand competiton has been a prominent paradigm for the empirical study of differentiated product markets for over twenty years [14, 10, 19, 51, 24, 20, 49, 43, 11, 48, 1]. Most of these empirical applications have been undertaken without theoretical assurances of the existence, uniqueness, and even “plausibility” of Bertrand-Nash equilibrium prices. This article proves the existence of equilibrium prices for differentiated product market models based on the Logit model under weak conditions on the (indirect) utility function and convex total costs (i.e., unit costs that increase with volume). Further analysis reveals some counter-intuitive properties of equilibrium prices that suggest more complex models than Logit are required to adequately model differentiated product markets in price competition. Most existing theoretical analyses of Bertrand competition are based on assumptions too restrictive to suit empirical applications of Bertrand competition and that obscure potentially counterintuitive properties of equilibrium. For example, there are few theoretical studies that consider multi-product firms (see, e.g., [46, 4, 6]), but real firms in differentiated product markets almost always offer more than one product. Theoretical analyses of Bertrand-Nash equilibrium prices have also typically relied on homogeneity or “symmetry” between firms with respect to the costs and “values”111Authors in the theoretical literature use the term “quality” to describe the non-price utility of a product [35]. This use of the term is confounded with the way it would be interpreted by many engineers, marketers, operations researchers, or laypeople as a measure of reliability. of the products offered. Real markets are heterogeneous with respect to the number of products offered, the values consumers derive from these products, and the costs with which these products are produced. In one analysis, Anderson & dePalma [6] state that > “empirical application[s] would have to relax the symmetry assumptions and > allow firms to produce products of different qualities, allow for > heterogeneity across firms, and differing costs to introducing products.” > [6, pg. 98] Thus, existing theoretical analyses currently offer limited support to empirical applications of differentiated product market models or other models in which Logit models might be useful, such as those described by Gallego et. al [18]. A theoretical understanding of Bertrand-Nash equilibrium prices begins with the conditions under which equilibrium prices exist. Perloff [42] provided an early existence proof for Bertrand-Nash equilibrium under a general Random Utility Maximization (RUM) model. Firms in Perloff’s model are ‘systematically homogeneous’ in that product differentiation exists only through random brand preference, rather than differentiated product characteristics and unit costs. Anderson & dePalma [3] undertake an analysis of equilibrium with single- product firms focusing on the linear-in-price utility Logit model. They characterize equilibrium prices with a closed-form expression when there is no outside good, and as solutions to a fixed-point equation when an outside good exists. Milgrom & Roberts [28]m Caplin & Nalebuff [15], and Gallego et. al [18] have also provided equilibrium existence proofs for Bertrand competition between single-product firms that apply to the Logit RUM, assuming utility is linear in price. Such results have been used to ensure that empirical single- product firm models based on Bertrand competition are well-posed [40, 7, 21, 5]. More recently, Sandor [45] and Konovalov & Sandor [27] have proven the existence and uniqueness of equilibrium prices with multi-product firms and the linear-in-price utility Logit model. Beyond models with single-product firms and linear utility functions, the literature lacks general conditions under which equilibrium exists. Without this understanding, it is not known if empirical examples cannot have equilibrium prices. This article provides one example where this has already occurred. Unfortunately the mathematical methods employed in these works cannot be extended to establish the existence of equilibrium prices for models with multi-product firms and general non-linear utility functions. Perloff’s [42] and Anderson & dePalma’s [3] analyses are specific to symmetric equilibrium between homogeneous single-product firms. While Milgrom & Roberts [28] apply a general property $-$ “supermodularity” $-$ to prove the existence (and uniqueness) of single-product firm equilibrium prices under Logit, Sandor [45] has shown that multi-product firm profit functions under linear-in-price utility Logit fail to be supermodular arbitrarily near equilibrium prices, ruling out the application of this property for multi-product firms; Appendix C extends Sandor’s proof to any Logit model within the class studied here. Similarly, the proofs from Caplin & Nalebuff [15] and Gallego et. al [18] rely on quasi-concavity of the firms’ profit functions. Hanson & Martin [22], however, have observed that multi-product firm profits are not quasi-concave under the Logit model. Thus new mathematical tools are needed to prove the existence of equilibrium prices for Bertrand competition between multi-product firms. The remainder of this article proceeds as follows. Section 2 presents a framework for Bertrand competition under an arbitrary RUM model assuming unit costs are constant (equivalently, total costs depend linearly on quantity sold). Special attention is paid to the interpretation of an RUM model as the generator of a stochastic choice process leading to random demand, highlighting some implicit assumptions that may not be commonly acknowledged. Section 3 specializes this framework to the Logit RUM model, equivalent to the “attraction demand model” being used by some researchers in revenue management (e.g., [18]). Specifically, Section 3 presents a new set of utility specifications, defines the Logit choice probabilities, and identifies when firm profits under the Logit model are bounded. Section 4 proves the existence of Bertrand-Nash equilibrium prices using fixed-point equations derived from the first-order or “Simultaneous Stationarity” condition when unit costs are constant. Most existing analyses of equilibrium prices also rely on the first-order condition. Moreover, fixed- point expressions have already been utilized to characterize equilibrium under linear-in-price utility Logit models [3, 10, 12, 45, 27] and even for more complex Mixed Logit models [10, 31, 33, 32]. Here three fixed-point equations for Logit models are derived, two of which generalize to Mixed Logit models; see [31, 33, 32]. As is common in analyses of equilibrium, the existence proof has two parts: proving that (i) there exist simultaneously stationary prices and (ii) simultaneously stationary prices are in fact equilibria. Existence proofs using both Brouwer’s fixed-point theorem and the Poincare-Hopf Theorem [29, Chapter 6] are given. The Poincare-Hopf approach to this problem was first taken in [31], and has also been used by Konovalov & Sandor [27], restricting the utility function to be linear-in-price. Simsek et. al [47] provide several references to applications of the Poincare-Hopf Theorem in general equilibrium models. Identifying simultaneously stationary prices with equilibria requires a direct analysis of the second-order conditions, combined with a second application of the Poincare-Hopf Theorem to prove the uniqueness of profit-maximizing prices, circumventing the lack of quasi-concavity in Logit profits for multi-product firms. These results are based on new, general conditions on the utility function that are weaker than most assumptions currently applied in theoretical economics, econometrics, and marketing. Sections 5 and 6 generalize the analysis in Section 4 to address non-constant unit costs and populations with finite purchasing power, respectively. The treatment of non-constant unit costs is a fairly straightforward extension of the analysis for the constant unit cost case, at least when total costs are convex (that is, unit costs are increasing in volume). Limits on purchasing power qualitatively change the behavior that may occur in equilibrium: While every product has a non-zero (if small) probability of being purchased in equilibrium with no such limit, some products can be profit-optimally “priced out of the market” when there is a finite limit on purchasing power. The fixed-point approach from the traditional, no-limit case is extended to characterize equilibrium prices when there is a limit, and existence is proved with essentially the same methods. Finally, Section 7 identifies several structural properties of equilibrium prices under the Logit model. First, if the consumer population systematically values some product’s characteristics more than the same firm’s other offerings, that product must be given lower profit-optimal markup by the firm in equilibrium when unit costs are constant. This counterintuitive result cannot be observed in analyses with single-product firms, and relies only on the common assumptions that (i) utility is concave in price and separable in price and characteristics and (ii) unit costs, though constant, increase with the value of product characteristics [35, 14]. As a consequence, Bertand competition under Logit with conventional utility specifications and constant unit costs cannot have fixed percentage markups as an equilibrium outcome; i.e. “cost plus pricing” [36] is not rationalized by Bertrand competition under conventional Logit models. Second, there exists a portfolio effect for Bertrand-Nash equilibrium under Logit with constant unit costs: equilibrium prices for an identical product offered at the same unit cost by two distinct firms depends on the profitability of the entire portfolio of products offered by these firms. In other words, heterogeneous portfolios can lead to distinct equilibrium prices for otherwise identical products. This property would not be observed from analyses that assume firms are homogeneous. The most limiting assumption in this article is the absence of consumer heterogeneity in the choice model. The Logit RUM model does allows some random variance in the utilities individuals in the population derive from the differentiated products, and thus contains some degree of population heterogeneity. However, the degree to which this is expressed with the Logit model has long been known to generate patterns of substitution that are unrealistic form many empirical applications [10, 50]. The techniques used in this paper to prove the existence of simultaneously stationary prices can be extended to a large class of Mixed Logit models with some ease; see [33, 32]. Moreover, the fixed-point equations used here are very useful in computations of equilibrium prices under such models [33]. However, the central element of the existence results established here is a generalization of quasi-concavity property that ensures profits under Logit have unique profit-maximizing prices. The conditions under which such a condition holds for Mixed Logit models are not obvious and will be nontrivial [33]. A review of mathematical notation, several examples, and additional results are provided in the appendices. ## 2\. Bertrand Competition Under an Arbitrary Random Utility Model This section presents a mathematical framework for Bertrand competition under an arbitrary Random Utility Maximization (RUM) model. This generalizes the discussion in [8] to multi-product firms and RUM demand. Conceptually, a fixed number of firms decide on prices for a fixed set of products prior to some time period in which these prices must remain fixed. During this purchasing period, a fixed number of individuals independently choose to purchase one of the products offered by these firms, or to forgo purchase of any of these products, following a given RUM model. Verboven [51] describes this as a two- stage stochastic game, where in the first stage the firms choose prices and in the second stage individuals choose products to maximize their own utility after sampling, or “drawing,” from the distribution of random utilities.222In the second stage, all consumers have dominant strategies. ### 2.1. Random Utility Models and Demand for Products RUM models provide a means to describe selection from a choice set, a collection of $J\in\mathbb{N}$ products that individuals may choose to purchase along with a no-purchase option (or “outside good”) indexed by 0. Each product $j\in\mathbb{N}(J)$ is characterized by its price $p_{j}\in[0,\infty)$ and vector of characteristics $\mathbf{y}_{j}\in\mathcal{Y}$, where $\mathcal{Y}\subset\mathbb{R}^{K}$ for some $K\in\mathbb{N}$. The random variable $U_{i,j}(\mathbf{y}_{j},p_{j})$ gives the utility individual $i$ receives by purchasing product $j\in\mathbb{N}(J)$, while the random variable $U_{i,0}$ gives the utility received by not purchasing any of the products (i.e. “purchasing the outside good”). Conditional on the values of $\\{U_{i,0}\\}\cup\\{U_{i,j}(\mathbf{y}_{j},p_{j})\\}_{j\in\mathbb{N}(J)}$, individual $i$ chooses the option $j\in\\{0\\}\cup\mathbb{N}(J)$ with the highest utility. The choice variable $C_{i}(\mathbf{Y},\mathbf{p})$ encapsulates this selection, taking values in $\\{0\\}\cup\mathbb{N}(J)$ following the distribution $\mathbb{P}(C_{i}(\mathbf{Y},\mathbf{p})=j)=\left\\{\begin{aligned} &\mathbb{P}\left(U_{i,j}(\mathbf{y}_{j},p_{j})=\max\left\\{\;U_{i,0}\;,\;\max_{k\in\mathbb{N}(J)}U_{i,k}(\mathbf{y}_{k},p_{k})\;\right\\}\right)&&\quad\text{if }j\in\mathbb{N}(J)\\\ &\mathbb{P}\left(U_{i,0}=\max\left\\{\;U_{i,0}\;,\;\max_{k\in\mathbb{N}(J)}U_{i,k}(\mathbf{y}_{k},p_{k})\;\right\\}\right)&&\quad\text{if }j=0\end{aligned}\right.$ The distribution of these random utilities assures that “ties” occur with probability zero. Let $\mathbf{U}_{i}(\mathbf{Y},\mathbf{p})=(U_{i,0},U_{i,1}(\mathbf{Y},\mathbf{p}),\dotsc,U_{i,J}(\mathbf{Y},\mathbf{p})),$ and make the following assumption: ###### Assumption 2.1. For any $(\mathbf{Y},\mathbf{p})\in\mathcal{Y}^{J}\times\mathbb{R}_{+}^{J}$ and $i,i^{\prime}\in\mathbb{N}(I)$, $\mathbf{U}_{i}(\mathbf{Y},\mathbf{p})$ and $\mathbf{U}_{i^{\prime}}(\mathbf{Y},\mathbf{p})$ are independent and identically distributed. Under this common assumption, the individual index on utilities and the choice variable can be dropped. Note also that this does not imply that $U_{j}(\mathbf{y}_{j},p_{j})$ and $U_{k}(\mathbf{y}_{k},p_{k})$ are independent. In practice, models often take the form $U_{0}=\vartheta+\mathcal{E}_{0}\quad\quad\text{and}\quad\quad U_{j}(\mathbf{y}_{j},p_{j})=u(\mathbf{y}_{j},p_{j})+\mathcal{E}_{j}\text{ for all }j\in\mathbb{N}(J)$ for some (conditional, indirect) utility function $u:\mathcal{Y}\times[0,\infty)\to\mathbb{R}$, $\vartheta\in[-\infty,\infty)$ and “error” vector $\boldsymbol{\mathcal{E}}=\\{\mathcal{E}_{j}\\}_{j=0}^{J}$. When $\boldsymbol{\mathcal{E}}$ is given an i.i.d. extreme value distribution, we have the Logit RUM [3, 50].333This independence assumption on $\boldsymbol{\mathcal{E}}$ is distinct from our independence assumption on the random utilities in that now it is independence across products in the choice set, rather than across individuals in the population. Letting $\boldsymbol{\mathcal{E}}$ have a Generalized Extreme Value distribution we have a GEV RUM like the Nested Logit model [50, 13], or taking $\boldsymbol{\mathcal{E}}$ multivariate normal gives the Probit RUM [50]. Either of these latter two forms can have a $\boldsymbol{\mathcal{E}}$ with correlated components.444More generally, we can let $\mathcal{T}$ be a space of individual characteristics or “demographics” and define $U_{0}=\vartheta(\boldsymbol{\Theta})+\mathcal{E}_{0}\quad\quad\text{and}\quad\quad U_{j}(\mathbf{y}_{j},p_{j})=u(\boldsymbol{\Theta},\mathbf{y}_{j},p_{j})+\mathcal{E}_{j}\text{ for all }j\in\mathbb{N}(J)$ where $u:\mathcal{T}\times\mathcal{Y}\times[0,\infty)\to\mathbb{R}$, $\vartheta:\mathcal{T}\to[-\infty,\infty)$ where $\boldsymbol{\Theta}$ is a $\mathcal{T}$-valued random variable with the distribution $\mu$, ostensibly representing the distribution of demographic variables over the population. With the same error distribution, we obtain a “mixed” RUM. Particularly, taking $\boldsymbol{\mathcal{E}}$ i.i.d. extreme value gives the “random coefficients” or Mixed Logit RUM class [50, Chapter 6]. Demands, the total quantity of each product purchased during the purchasing period, must be defined to define firms’ profits. Extrapolating demands from the stochastic choice model above requires the following assumption. ###### Assumption 2.2. Every individual $i\in\mathbb{N}(I)$ observes the same choice set, $\\{0\\}\cup\mathbb{N}(J)$, during the purchase period. Under this assumption, the demand $Q_{j}(\mathbf{Y},\mathbf{p})$ for each product $j\in\mathbb{N}(J)$ can be expressed simply as $Q_{j}(\mathbf{Y},\mathbf{p})=\sum_{i=1}^{I}1_{\\{C_{i}(\mathbf{Y},\mathbf{p})=j\\}}$, where here $\\{C_{i}(\mathbf{Y},\mathbf{p})\\}_{i\in\mathbb{N}(I)}$ are $I$ i.i.d. “copies” of $C(\mathbf{Y},\mathbf{p})$. The primary benefit of Assumption 2.2 is that $\\{Q_{0}(\mathbf{Y},\mathbf{p})\\}\cup\\{Q_{j}(\mathbf{Y},\mathbf{p})\\}_{j\in\mathbb{N}(J)}$ is a multinomial family of variables with parameter $I$ and probabilities $\\{P_{j}(\mathbf{Y},\mathbf{p})\\}_{j=0}^{J}$, and thus expected demands for each product are given simply by $\mathbb{E}[Q_{j}(\mathbf{Y},\mathbf{p})]=IP_{j}(\mathbf{Y},\mathbf{p})$ [17]. A more serious implication of Assumption 2.2 is there must be at least $I$ units of every product available for the individuals to choose during the purchasing period.555We could alternatively interpret this condition in terms of consumers “ordering” products during the purchasing period and assuming delivery schedules do not impact demand. Specifically, no product can “sell out,” or if it does, “backordering” does not impact utilities. If any firm commits (or is forced by capacity constraints) to only produce $I^{\prime}<I$ units of some product they offer during the purchasing period and individuals do not “backorder”, then with some positive probability Assumption 2.2 will be violated. ### 2.2. Firms, Product Portfolios, Costs, and Profits Let $F\in\mathbb{N}$ denote the number of firms. For each $f\in\mathbb{N}(F)$, there exists a set $\mathcal{J}_{f}\subset\mathbb{N}(J)$ of indices that corresponds to the $J_{f}=\left\lvert\mathcal{J}_{f}\right\rvert$ products offered by firm $f$. The collection of all these sets, $\\{\mathcal{J}_{f}\\}_{f=1}^{F}$, forms a partition of $\mathbb{N}(J)$. Subsequently, in writing “$f(j)$” for some $j\in\mathbb{N}(J)$, we mean the unique $f\in\mathbb{N}(F)$ such that $j\in\mathcal{J}_{f}$. The vector $\mathbf{p}_{f}\in\mathbb{R}^{J_{f}}$ refers to the vector of prices of the products offered by firm $f$. Negative subscripts denote competitor’s variables as in, for instance, $\mathbf{p}_{-f}\in\mathbb{R}^{J_{-f}}$, where $J_{-f}=\sum_{g\neq f}J_{g}$, is the vector of prices for products offered by all of firm $f$’s competitors. Firm-specific choice probability functions are denoted by $\mathbf{P}_{f}(\mathbf{p})$. Additional assumptions concerning costs and production are required to complete the definition of firms’ profits. ###### Assumption 2.3. There exists a unit cost function $c_{f}^{U}:\mathcal{Y}\to\mathbb{R}_{+}$ and a fixed cost function $c_{f}^{F}:\mathfrak{F}(\mathcal{Y})\to\mathbb{R}_{+}$ for all $f\in\mathbb{N}(F)$ that depend only on the collection of product characteristics chosen by the firm. Particularly, unit and fixed costs are independent of the quantity sold, ruling out dependence of unit and fixed costs on production volumes. This assumption is relaxed in Section 5 below. Bertrand competition also entails the following “comittment” assumption on the quantities produced [8]. ###### Assumption 2.4 (Bertrand Production Assumption). Each firm commits to producing exactly $Q_{j}(\mathbf{Y},\mathbf{p})$ units of each product $j\in\mathcal{J}_{f}$ during the purchasing period. Again, this implies that the firm has no production capacity constraints that limit a firm’s ability to meet any demands that arise during the purchase period. The random variable $\sum_{j\in\mathcal{J}_{f}}c_{f}^{U}(\mathbf{y}_{j})Q_{j}(\mathbf{Y},\mathbf{p})+c_{f}^{F}(\mathbf{Y}_{f})$ gives the total cost firm $f$ incurs in producing $Q_{j}(\mathbf{Y},\mathbf{p})$ units of product $j$, for all $j\in\mathcal{J}_{f}$, during the purchasing period. We let $\mathbf{c}_{f}^{U}(\mathbf{Y}_{f})$ be the vector of these unit costs for the products offered by firm $f$. Under Assumption 2.4, the random variable $\Pi_{f}(\mathbf{Y},\mathbf{p})=\mathbf{Q}_{f}(\mathbf{Y},\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f}^{U}(\mathbf{Y}_{f}))-c_{f}^{F}(\mathbf{Y}_{f})$ gives firm $f$’s profits for the production period as a function of product characteristics and prices. Following most of the theoretical and empirical literature in both marketing and economics, we assume that firms take expected profits, (1) $\pi_{f}(\mathbf{Y},\mathbf{p})=I\hat{\pi}_{f}(\mathbf{Y},\mathbf{p})-c_{f}^{F}(\mathbf{Y}_{f})\quad\text{where}\quad\hat{\pi}_{f}(\mathbf{Y},\mathbf{p})=\mathbf{P}_{f}(\mathbf{Y},\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f}^{U}(\mathbf{Y}_{f})),$ as the metric by which they optimize their pricing decisions in this stochastic optimization problem. Eqn. (1) demonstrates that neither the total firm fixed costs $c_{f}^{F}$ nor the population size $I$ play a role in determining the prices that maximize expected profits. Therefore we only consider the “population-normalized gross expected profits” $\hat{\pi}_{f}(\mathbf{p})$, referred to below as simply “profits”. We also consider $\mathbf{Y}$ fixed, and cease to include this characteristic matrix as an argument. Finally, we write $\mathbf{c}_{f}=\mathbf{c}_{f}^{U}$ as these are the only relevant costs for the price equilibrium problem. Henceforth we write simply $\hat{\pi}_{f}(\mathbf{p})=\mathbf{P}_{f}(\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f}).$ The following adaptation of well-known necessary conditions for the local maximization of an unconstrained, continuously differentiable function (e.g., [34]) informs our derivation of the Simultaneous Stationarity Condition. ###### Lemma 2.1. Suppose $\mathbf{P}_{f}(\cdot,\mathbf{p}_{-f})$ is continuously differentiable on some open $\mathcal{A}\subset(\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J_{f}}$. If $\mathbf{p}_{f}\in\mathcal{A}$ is a local maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, then (2) $(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f})+\mathbf{P}_{f}(\mathbf{p})=\mathbf{0}.$ ### 2.3. Local Equilibrium and the Simultaneous Stationarity Conditions As in much of the existing literature, our analysis relies on local conditions for optimality of prices and thus must rely on the following local definition of equilibrium. ###### Definition 2.1. A price vector $\mathbf{p}\in[\mathbf{0},\boldsymbol{\infty}]$ is called a local equilibrium if $\mathbf{p}_{f}$ is a local maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ for all $f\in\mathbb{N}(F)$. A price vector $\mathbf{p}\in[\mathbf{0},\boldsymbol{\infty}]$ is called an equilibrium if $\mathbf{p}_{f}$ is a maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ for all $f\in\mathbb{N}(F)$. Finally, the following Simultaneous Stationarity Condition is a generic necessary condition for local equilibrium if the RUM choice probabilities are continuously differentiable in prices. ###### Definition 2.2. Let $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$ denote the “combined gradient” with components $((\tilde{\nabla}\hat{\pi})(\mathbf{p}))_{j}=(D_{j}\hat{\pi}_{f(j)})(\mathbf{p})$. Let $(\tilde{D}\mathbf{P})(\mathbf{p})$ be the sparse matrix corresponding to the intra-firm price derivatives of choice probabilities; that is, $\big{(}(\tilde{D}\mathbf{P})(\mathbf{p})\big{)}_{j,k}=\left\\{\begin{aligned} &(D_{k}P_{j})(\mathbf{p})&&\quad\text{if }f(j)=f(k)\\\ &\quad\quad 0&&\quad\text{if }f(j)\neq f(k)\end{aligned}\right..$ ###### Lemma 2.2 (Simultaneous Stationarity Condition). Suppose $\mathbf{P}$ is continuously differentiable on some open $\mathcal{A}\subset(\mathbf{0},\boldsymbol{\infty})$. If $\mathbf{p}\in\mathcal{A}$ is a local equilibrium, then (3) $(\tilde{\nabla}\hat{\pi})(\mathbf{p})=(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{c})+\mathbf{P}(\mathbf{p})=\mathbf{0}.$ Prices satisfying Eqn. (3) are called “simultaneously stationary.” In principle, simultaneously stationary prices need not be equilibria. Additional analysis is required to link stationarity with local optimality of profits with respect to changes to the prices of a firm’s own products. The necessity of the Simultaneous Stationarity Condition does not depend on the RUM type, but only on the continuous differentiability of the choice probabilities (with respect to price) and the cost assumption. Furthermore, much of this development is the same for an arbitrary demand function, rather than a RUM; see, e.g., [14, 24]. Thus, Eqn. (3) has appeared in many different studies using alternative RUM specifications such as Logit models [12], Generalized Extreme Value models [19, 20, 12, 52], and Mixed Logit models [10, 11, 37, 38, 49, 43, 48, 1, 9, 33]. In most of these studies, Eqn. (3) has not been investigated far beyond Lemma 2.2. Eqn. (3) has been consistently used through the corresponding BLP markup equation $\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ where (4) $\boldsymbol{\eta}(\mathbf{p})=-(\tilde{D}\mathbf{P})(\mathbf{p})^{-\top}\mathbf{P}(\mathbf{p})$ assuming $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ is nonsingular.666For competing single-product firms, this reduces to the famous “negative reciprocal of elasticity” form for the Lerner index (i.e. percent markups); see [41]. This equation is typically used to estimate costs assuming prices are in equilibrium. However, the markup equation $\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ is a fixed-point equation satisfied by all simultaneously stationary prices. In Section 4, we give a specific form for $\boldsymbol{\eta}$ under the Logit model and derive a new fixed-point equation for simultaneously stationary prices by factoring out the gradient of the inclusive value from $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$; this fixed-point equation is a specialization of the $\boldsymbol{\zeta}$-markup equation used by Morrow & Skerlos for large-scale computations of equilibrium prices [33, 32]. ## 3\. Logit Models This section reviews the Logit model, providing the groundwork for the analysis in later sections. Section 3.1 defines a new class of nonlinear utility functions for which equilibrium prices can be shown to exist. Section 3.2 derives the corresponding Logit choice probabilities and their derivatives. Finally, Section 3.3 provides conditions under which profit- maximizing prices are finite, a pre-requisite for the existence of (finite) equilibrium prices. ### 3.1. Systematic Utility Specifications The random utility any individual receives by purchasing any particular product is parameterized by its characteristic vector and price through some function $u:\mathcal{Y}\times[0,\infty)\to[-\infty,\infty)$. We consider specifications of the following form. ###### Assumption 3.1. There are functions $w:\mathcal{Y}\times[0,\infty)\to(-\infty,\infty)$ and $v:\mathcal{Y}\to(-\infty,\infty)$ such that utility can be written $u(\mathbf{y},p)=w(\mathbf{y},p)+v(\mathbf{y})$. Concerning the behavior of $w$, we assume that, for all $\mathbf{y}\in\mathcal{Y}$, $w(\mathbf{y},\cdot):[0,\infty)\to(-\infty,\infty)$ is (a) strictly decreasing, and (b) continuously differentiable on $(0,\infty)$. We also assume that (c) $\lim_{p\uparrow\infty}w(\mathbf{y},p)=-\infty$, and subsequently set $w(\mathbf{y},\infty)=-\infty$. Writing $u(\mathbf{y},p)=w(\mathbf{y},p)+v(\mathbf{y})$ is completely general so long as utility is defined for all $p\in[0,\infty)$. This form is convenient to define the “value” of a product as that component of utility that does not vary with price, and to define “separable” utilities, the most common class of utility functions used in practice. ###### Definition 3.1. We say $v(\mathbf{y})$ is the value of any product with characteristic vector $\mathbf{y}$, and that utility is separable in price and characteristics (or simply separable) if $w(\mathbf{y},p)=w(p)$ for all $\mathbf{y}\in\mathcal{Y}$. We call $\left\lvert(Dw)(\mathbf{y},p)\right\rvert^{-1}$ the (local) willingness to pay (for product value). The class formed by Assm. 3.1 encompasses the majority of utility functions used in the theoretical and empirical literature. Assm. (a) is required of a suitable indirect utility function, and (b) is required for an analysis of equilibrium based on the first-order conditions. The assumption (c) is a natural condition that ensures that the choice probabilities vanish as prices increase without bound. In fact, by including utility functions that are not concave-in-price, this class is larger than that typically studied. A number of examples are given in Appendix B. Concave-in-price utilities are certainly an important special case often considered in economics. However, concavity turns out to be a stronger assumption than is required to ensure the existence of finite equilibrium prices under Logit. Instead, the following weaker property of the utility price derivatives is sufficient. ###### Definition 3.2. $w$ eventually decreases sufficiently quickly at $\mathbf{y}\in\mathcal{Y}$ if there exists some $r(\mathbf{y})>1$ and some $\bar{p}(\mathbf{y})\in[0,\infty)$ such that $(Dw)(\mathbf{y},p)\leq-r(\mathbf{y})/p=-r(\mathbf{y})D[\log p]$ for all $p>\bar{p}(\mathbf{y})$. $w$ itself eventually decreases sufficiently quickly if $w$ eventually decreases sufficiently quickly at all $\mathbf{y}\in\mathcal{Y}$. The most commonly used finite utility functions, particularly strictly decreasing and concave in price utility functions, satisfy $\lim_{p\to\infty}(Dw)(\mathbf{y},p)<0$, and hence eventually decrease sufficiently quickly with any $r$. Note also that if $w$ does not eventually decrease sufficiently quickly at $\mathbf{y}$, then necessarily $\left\lvert(Dw)(\mathbf{y},p)\right\rvert\to 0$ as $p\to\infty$. The example $w(\mathbf{y},p)=-\alpha(\mathbf{y})\log p$ ($\alpha(\mathbf{y})>0$) shows that this does not contradict Assm. 3.1, (c). A distinct requirement on the second derivatives of utility is synonymous with the sufficiency of stationarity under Logit. ###### Definition 3.3. Suppose $w(\mathbf{y},\cdot)$ is twice continuously differentiable for all $\mathbf{y}\in\mathcal{Y}$. We say that $w$ has sub-quadratic second derivatives at $(\mathbf{y},p)\in\mathcal{Y}\times[0,\infty)$ if $\omega(\mathbf{y},p)=(D^{2}w)(\mathbf{y},p)/(Dw)(\mathbf{y},p)^{2}<1$. We say that $w$ itself has sub-quadratic second derivatives if $w$ has sub-quadratic second derivatives at all $(\mathbf{y},p)\in\mathcal{Y}\times[0,\infty)$. Note that if $w(\mathbf{y},\cdot)$ is concave, then $w$ trivially has sub- quadratic second derivatives. However, $w(\mathbf{y},\cdot)$ can be convex and still have sub-quadratic second derivatives. For example, $w(\mathbf{y},p)=-\alpha(\mathbf{y})\log p$ for $\alpha(\mathbf{y})>1$ has $\omega(\mathbf{y},p)=1/\alpha<1$. With any collection of fixed product characteristic vectors $\\{\mathbf{y}_{j}\\}_{j=1}^{J}$, we set $w_{j}(p)=w(\mathbf{y}_{j},p)$ and $v_{j}=v(\mathbf{y}_{j})$ and thus generate a collection of product-specific utility functions, $u_{j}(p)=w_{j}(p)+v_{j}$, that depend on price alone. Vector functions $\mathbf{w}:[0,\infty]^{J}\to[-\infty,\infty)^{J}$ and $\mathbf{u}:[0,\infty]^{J}\to[-\infty,\infty)^{J}$ are constructed from these product-specific components by taking $(\mathbf{w}(\mathbf{p}))_{j}=w_{j}(p_{j})$ and $(\mathbf{u}(\mathbf{p}))_{j}=u_{j}(p_{j})$. In particular, $\mathbf{u}(\mathbf{p})=\mathbf{w}(\mathbf{p})+\mathbf{v}$. Firm-specific product values $\mathbf{v}_{f}$ and utilities $\mathbf{u}_{f}(\mathbf{p}_{f})=\mathbf{w}_{f}(\mathbf{p}_{f})+\mathbf{v}_{f}$ are also defined in the natural way. ### 3.2. Logit Choice Probabilities The Logit model [50, Chapter 3] takes the utility any individual receives when purchasing product $j$ to be the random variable $U_{j}(\mathbf{y}_{j},p_{j})=u(\mathbf{y}_{j},p_{j})+\mathcal{E}_{j}$ and the utility of the outside good to be the random variable $U_{0}=\vartheta+\mathcal{E}_{0}$, where $\boldsymbol{\mathcal{E}}=\\{\mathcal{E}_{j}\\}_{j=0}^{J}$ is a family of i.i.d. standard extreme value variables and $\vartheta\in[-\infty,\infty)$ is a number representing the utility of the outside good. The i.i.d. standard extreme value specification for $\boldsymbol{\mathcal{E}}$ generates the following choice probabilities (see, e.g., [50]): (5) $P_{j}^{L}(\mathbf{p})=\frac{e^{u_{j}(p_{j})}}{e^{\vartheta}+\sum_{k=1}^{J}e^{u_{k}(p_{k})}}$ The equivalent formula $P_{j}^{L}(\mathbf{p})=\frac{e^{(u_{j}(p_{j})-\vartheta)}}{1+\sum_{k=1}^{J}e^{(u_{k}(p_{k})-\vartheta)}}$ corresponding to setting $\vartheta=0$ is often seen in the literature, but offers no substantial advantage to the analysis in this article. When $\vartheta=-\infty$, $P_{j}^{L}(\mathbf{p})=\frac{e^{u_{j}(p_{j})}}{\sum_{k=1}^{J}e^{u_{k}(p_{k})}}.$ The following basic properties of the Logit choice probabilities are used throughout. ###### Lemma 3.1. The following hold under Assumption 3.1, for any $j$ and $f$: (i) $0<P_{j}^{L}(\mathbf{p})<1$ and $\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{1}<1$ for all $\mathbf{p}\in[0,\infty)^{J}$. (iii) If $\vartheta>-\infty$ and $\mathbf{q}\in[0,\infty]^{J}$, $\lim_{\mathbf{p}\to\mathbf{q}}P_{j}^{L}(\mathbf{p})$ exists. Moreover, $\lim_{\mathbf{p}\to\mathbf{q}}P_{j}^{L}(\mathbf{p})=0$ if $q_{j}=\infty$, and $\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{1}<1$ for all $\mathbf{p}\in[0,\infty]^{J}$. (iv) If $\vartheta=-\infty$, then for any $\mathbf{x}\in[0,1]^{J}$, $\sum_{j=1}^{J}x_{j}=1$, there exists some sequence $\\{\mathbf{p}^{(n)}\\}_{n\in\mathbb{N}}\subset[0,\infty)^{J}$ with $\mathbf{p}^{(n)}\to\boldsymbol{\infty}$ such that $\lim_{n\to\infty}\mathbf{P}^{L}(\mathbf{p}^{(n)})=\mathbf{x}$. ###### Proof. (i), (ii), and (iii) follow easily from Eqn. 5. To prove (iv), first note that $\mathbf{P}:[0,\infty)^{J}\to\triangle(J)$ is onto when $\vartheta=-\infty$, where $\triangle(J)=\\{\mathbf{x}\in[0,1]^{J}:\sum_{j=1}^{J}x_{j}=1\\}$. Let $\mathbf{x}\in\triangle(J)$. It suffices to solve $u_{j}(p_{j})=\log x_{j}$ for $p_{j}$, for all $j$, for then $\displaystyle P_{j}^{L}(\mathbf{p})=\frac{e^{u_{j}(p_{j})}}{\sum_{k=1}^{J}e^{u_{k}(p_{k})}}=\frac{e^{\log x_{j}}}{\sum_{k=1}^{J}e^{\log x_{k}}}=\frac{x_{j}}{\sum_{k=1}^{J}x_{k}}=x_{j}.$ So long as $\log x_{j}\geq u_{j}(0)$, such a $p_{j}$ exists and is unique. Assuming, without loss of generality, that $u_{j}(0)\geq 0$ for all $j$ ensures that this condition holds for all $x_{j}\in[0,1]$. The existence of a sequence tending to infinity with $\lim_{n\to\infty}\mathbf{P}^{L}(\mathbf{p}^{(n)})=\mathbf{x}$ then follows from the invariance result in Lemma 3.2 below. ∎ Claim (iv) amounts to the fact that the Logit choice probabilities without an outside good cannot be both continuous and single valued on $[0,\infty]^{J}$, and suggests that the presence of an outside good “purchased” with positive probability is very important to optimization and equilibrium problems under Logit. As noted in the proof, this claim is a consequence of the following generalization of the “invariance of uniform price shifts” property of the linear in price utility Logit model to the class of utility functions specified by Assumption 3.1: ###### Lemma 3.2. Suppose $w$ satisfies Assumption 3.1. For any $p\in(0,\infty)$ and each $j\in\mathbb{N}(J)$, define $\chi_{j,p}:[1,\infty)\to[p,\infty)$ by $\chi_{j,p}(\lambda)=w_{j}^{-1}(w_{j}(p)-\log\lambda)$, and define $\boldsymbol{\chi}_{\mathbf{p}}:[1,\infty)\to[\mathbf{p},\boldsymbol{\infty})$ componentwise by $(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))_{j}=\chi_{j,p_{j}}(\lambda)$. (i) $\boldsymbol{\chi}_{\mathbf{p}}(\lambda)$ is well-defined, strictly increasing, and $\lim_{\lambda\to\infty}\boldsymbol{\chi}_{\mathbf{p}}(\lambda)=\boldsymbol{\infty}$. (ii) If $\vartheta=-\infty$, $\mathbf{P}^{L}$ is invariant on $\boldsymbol{\chi}_{\mathbf{p}}([1,\infty))$; i.e., $\mathbf{P}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))\equiv\mathbf{P}^{L}(\mathbf{p})$. (iii) If $\vartheta>-\infty$, $\mathbf{P}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))$ is strictly decreasing in $\lambda$, and $\mathbf{P}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))\to\mathbf{0}$ as $\lambda\to\infty$. ###### Proof. (i): By definition, $w_{j}(\chi_{j,p}(\lambda))=w_{j}(p)-\log\lambda$. Because $w_{j}$ is strictly decreasing and $w_{j}(\cdot):[p,\infty)\to(-\infty,w_{j}(p)]$ is onto, $\chi_{j,p}(\lambda)$ is uniquely defined for all $\lambda\geq 1$ and strictly increasing. Because $w_{j}(p)-\log\lambda\downarrow-\infty$ as $\lambda\uparrow\infty$, $\lim_{\lambda\uparrow\infty}\chi_{j,p}(\lambda)=\infty$. (ii): Note that $e^{u_{j}(\chi_{j,p}(\lambda))}=e^{w_{j}(\chi_{j,p}(\lambda))+v_{j}}=e^{w_{j}(p)-\log\lambda+v_{j}}=\lambda^{-1}\big{(}e^{w_{j}(p)+v_{j}}\big{)}.$ Thus if $\vartheta=-\infty$, $P_{j}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))=\frac{e^{u_{j}(\chi_{j,p_{k}}(\lambda))}}{\sum_{k=1}^{J}e^{u_{k}(\chi_{k,p_{k}}(\lambda))}}=\frac{\lambda^{-1}e^{w_{j}(p_{j})+v_{j}}}{\lambda^{-1}\sum_{k=1}^{J}e^{w_{k}(p_{k})+v_{k}}}=P_{j}^{L}(\mathbf{p}).$ (iii): Similarly, if $\vartheta>-\infty$, $P_{j}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))=\frac{e^{u_{j}(\chi_{j,p_{k}}(\lambda))}}{e^{\vartheta}+\sum_{k=1}^{J}e^{u_{k}(\chi_{k,p_{k}}(\lambda))}}=\frac{e^{w_{j}(p_{j})+v_{j}}}{\lambda e^{\vartheta}+\sum_{k=1}^{J}e^{w_{k}(p_{k})+v_{k}}}<P_{j}^{L}(\mathbf{p})$ for all $\lambda>1$ and $P_{j}^{L}(\boldsymbol{\chi}_{\mathbf{p}}(\lambda))\to 0$ as $\lambda\to\infty$. ∎ The invariance of the Logit choice probabilities over sequences of prices that tend to infinity should be viewed as an unacceptable property for realistic market models.777Mizuno [30] makes explicit use of this unrealistic property in proving the existence and uniqueness of equilibrium prices under Logit with single-product firms and linear in price utilities. Individuals are sure to make purchasing decisions based on the absolute value of product prices, rather than just the relative value. It is easy also fairly easy to see that Lemma 3.2, (iv) extends beyond Logit to any Generalized Extreme Value model without an outside good. The following form for the price derivatives of the Logit choice probabilities is also required. ###### Lemma 3.3. If $w$ satisfies Assm. 3.1 (b), then $\mathbf{P}^{L}$ is continuously differentiable for all $\mathbf{p}\in(0,\infty)^{J}$ with (6) $\displaystyle(D_{k}P_{j}^{L})(\mathbf{p})=P_{j}^{L}(\mathbf{p})(\delta_{j,k}-P_{k}^{L}(\mathbf{p}))(Dw_{k})(p_{k})=(\delta_{j,k}-P_{j}^{L}(\mathbf{p}))\lambda_{k}(\mathbf{p})$ where $\lambda_{k}(\mathbf{p})=(Dw_{k})(p_{k})P_{k}^{L}(\mathbf{p})$. In other words, (7) $(D_{f}\mathbf{P}^{L}_{f})(\mathbf{p})=\left(\mathbf{I}-\mathbf{P}^{L}_{f}(\mathbf{p})\mathbf{1}^{\top}\right)\boldsymbol{\Lambda}_{f}(\mathbf{p})\quad\text{and}\quad(D\mathbf{P}^{L})(\mathbf{p})=\left(\mathbf{I}-\mathbf{P}^{L}(\mathbf{p})\mathbf{1}^{\top}\right)\boldsymbol{\Lambda}(\mathbf{p})$ where $\boldsymbol{\Lambda}_{f}(\mathbf{p})=\mathrm{diag}(\boldsymbol{\lambda}_{f}(\mathbf{p}))$ and $\boldsymbol{\Lambda}(\mathbf{p})=\mathrm{diag}(\boldsymbol{\lambda}(\mathbf{p}))$. When $w$ is twice differentiable on $(0,\infty)$, $\mathbf{P}^{L}$ is as well and the second derivatives of the Logit choice probabilities are given by (8) $\displaystyle(D_{l}D_{k}P_{j}^{L})(\mathbf{p})$ $\displaystyle=\delta_{k,l}\big{(}(D^{2}w_{k})(p_{k})+(Dw_{k})(p_{k})^{2}\big{)}P_{k}^{L}(\mathbf{p})\big{(}\delta_{j,k}-P_{j}^{L}(\mathbf{p})\big{)}$ $\displaystyle\quad\quad\quad\quad+\lambda_{k}(\mathbf{p})\big{(}2P_{j}^{L}(\mathbf{p})-\delta_{j,k}-\delta_{j,l}\big{)}\lambda_{l}(\mathbf{p}).$ ###### Proof. These follow directly from Eqn. (5). ∎ ### 3.3. Bounded and Vanishing Logit Profits An understanding of when profits are bounded over the set of all non-negative prices is a pre-requisite to a general analysis of profit-optimal prices and corresponding price equilibrium. One might expect that because Assumption 3.1 (c) implies that the choice probabilities vanish as prices increase without bound that profits should also, but this is not true: $w(\mathbf{y},p)=-\alpha\log p$, a specification derived by Allenby & Rossi to represent “asymmetric brand switching under price changes” [2], can generate unbounded profits even though the choice probabilities vanish. The following property of utility functions guarantees not only the finiteness of Logit profits, but that these profits vanish as prices increase without bound.888The constant $\kappa(\mathbf{y})$ is convenient, but not necessary; it is easy to show that $w$ is eventually log bounded with $(r(\mathbf{y}),\bar{p}(\mathbf{y}),\kappa(\mathbf{y}))$ where $\kappa(\mathbf{y})\neq 0$ if and only if it is so with some $(r^{\prime}(\mathbf{y}),\bar{p}^{\prime}(\mathbf{y}),0)$. ###### Definition 3.4. $w$ is eventually log bounded at $\mathbf{y}\in\mathcal{Y}$ if there exists some $r(\mathbf{y})>1$, $\kappa(\mathbf{y})$, and some $\bar{p}(\mathbf{y})\in[0,\infty)$ such that $w(\mathbf{y},p)\leq-r(\mathbf{y})\log p+\kappa(\mathbf{y})$ for all $p>\bar{p}(\mathbf{y})$. $w$ itself is eventually log bounded if $w$ is eventually log bounded at all $\mathbf{y}\in\mathcal{Y}$. Note that if $w$ is eventually log bounded then Assumption 3.1 (c) necessarily holds. Furthermore, if $w$ eventually decreases sufficiently quickly then the fundamental theorem of calculus implies that $w$ is also eventually log bounded. Appendix B contains a somewhat pathological example demonstrating that the converse need not hold. ###### Lemma 3.4. Suppose $w$ satisfies Assumption 3.1. (i) Let $\vartheta>-\infty$, $\mathbf{q}\in[0,\infty]^{J}$, and suppose that there exists $r:\mathcal{Y}\to[1,\infty)$, $\bar{p}:\mathcal{Y}\to[0,\infty)$, and $\kappa:\mathcal{Y}\to\mathbb{R}$ such that $w(\mathbf{y},p)\leq-r(\mathbf{y})\log p+\kappa(\mathbf{y})$ for all $p>\bar{p}(\mathbf{y})$. Then $\lim_{\mathbf{p}\to\mathbf{q}}\hat{\pi}_{f}(\mathbf{p})<\infty$. (ii) If in fact $w$ is eventually log bounded, i.e. $r(\mathbf{y}):\mathcal{Y}\to(1,\infty)$, then $\lim_{\mathbf{p}\to\mathbf{q}}\hat{\pi}_{f}(\mathbf{p})=0$ if $\mathbf{q}_{f}=\boldsymbol{\infty}$. ###### Proof. The following inequality always holds: $P_{j}^{L}(\mathbf{p})p_{j}\leq\frac{e^{u_{j}(p_{j})}}{e^{\vartheta}+e^{u_{j}(p_{j})}}p_{j}=\frac{e^{w_{j}(p_{j})+v_{j}-\vartheta}}{1+e^{w_{j}(p_{j})+v_{j}-\vartheta}}p_{j}=p_{j}e^{w_{j}(p_{j})+v_{j}-\vartheta}.$ Under the hypothesis of (i), $P_{j}^{L}(\mathbf{p})p_{j}\leq p_{j}^{1-r_{j}}e^{\kappa_{j}+v_{j}-\vartheta}\leq e^{\kappa_{j}+v_{j}-\vartheta}$ for all $p_{j}$ sufficiently large. Claim (i) is a consequence of this bound. Moreover, $r_{j}<1$ for all $j$, then $P_{j}^{L}(\mathbf{p})p_{j}\downarrow 0$ as $p_{j}\uparrow\infty$. Claim (ii) is a consequence. ∎ Appendix B contains an example demonstrating that the converse to the second claim is false. That is, bounded and vanishing Logit profits need not imply that $w$ is eventually log bounded. If eventual log boundedness is strongly violated in the sense of the hypothesis in the following lemma, then profits must increase without bound as prices do. ###### Lemma 3.5. Let $\vartheta>-\infty$ and Assumption 3.1 hold. Suppose that for some $\mathbf{y}_{*}\in\mathcal{Y}$ there exists $r(\mathbf{y}_{*})\in(0,1)$, $\kappa(\mathbf{y}_{*})$, and $\bar{p}\in[0,\infty)$ such that for all $p>\bar{p}$, $w(\mathbf{y}_{*},p)\geq-r\log p+\kappa(\mathbf{y}_{*})$. Suppose further that $\mathbf{y}_{j}=\mathbf{y}_{*}$ for some $j\in\mathcal{J}_{f}$. Then $\lim_{\mathbf{p}\to\mathbf{q}}\hat{\pi}_{f}(\mathbf{p})=\infty$ for any $\mathbf{q}\in[0,\infty]^{J}$ with $q_{j}=\infty$. ###### Proof. Under the hypothesis, $p_{j}e^{u_{j}(p_{j})}\geq(p_{j})^{1-r}e^{\kappa+v_{j}}$ for all sufficiently large $p_{j}$. Thus $p_{j}e^{u_{j}(p_{j})}\to\infty$ as $p_{j}\uparrow\infty$ because $r<1$. Clearly then $P_{j}^{L}(p_{j},\mathbf{p}_{-j})p_{j}\to\infty$ as $p_{j}\uparrow\infty$. The claim follows. ∎ The results above establish when optimal profits are positive and finite, and when profit-optimal prices are not all infinite. Showing that profit maximizing prices are all finite is proved in Section 4 with the slightly strengthened hypothesis that $w$ eventually decreases sufficiently quickly. ###### Lemma 3.6. Suppose $w$ satisfies Assumption 3.1 and is eventually log bounded. (i) If $\vartheta>-\infty$ and $\mathbf{p}_{f}\in[0,\infty]^{J_{f}}$ locally maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ for any $\mathbf{p}_{-f}\in[0,\infty]^{J_{-f}}$, then $\mathbf{p}_{f}\neq\boldsymbol{\infty}$. (ii) If $\vartheta=-\infty$ and $\mathbf{p}_{f}\in[0,\infty]^{J_{f}}$ locally maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ for any $\mathbf{p}_{-f}\in[0,\infty]^{J_{-f}}\setminus\\{\boldsymbol{\infty}\\}$, then $\mathbf{p}_{f}\neq\boldsymbol{\infty}$. However, $\hat{\pi}_{f}(\cdot,\boldsymbol{\infty})$ is maximized only by $\mathbf{p}_{f}=\boldsymbol{\infty}$, and thus $\boldsymbol{\infty}$ is always an equilibrium. ###### Proof. (i): Profit maximizing prices $\mathbf{p}_{f}$ are not all infinite because $\hat{\pi}_{f}(\boldsymbol{\infty},\mathbf{p}_{-f})=0$, and any prices $\mathbf{p}_{f}>\mathbf{c}_{f}$ give $\hat{\pi}_{f}(\mathbf{p})>0$. (ii): The same holds when $\vartheta=-\infty$ and $\mathbf{p}_{-f}\neq\boldsymbol{\infty}$, because some product’s utility is finite. However, if $\mathbf{p}_{-f}=\boldsymbol{\infty}$, Lemma 3.2 proves that $\displaystyle\hat{\pi}_{f}(\boldsymbol{\chi}_{f,\mathbf{p}_{f}}(\lambda),\boldsymbol{\infty})=\mathbf{P}_{f}^{L}(\boldsymbol{\chi}_{f,\mathbf{p}}(\lambda),\boldsymbol{\infty})^{\top}(\boldsymbol{\chi}_{f,\mathbf{p}_{f}}(\lambda)-\mathbf{c}_{f})=\mathbf{P}_{f}^{L}(\mathbf{p}_{f},\boldsymbol{\infty})^{\top}(\boldsymbol{\chi}_{f,\mathbf{p}_{f}}(\lambda)-\mathbf{c}_{f})\to\infty$ as $\lambda\uparrow\infty$, for any finite $\mathbf{p}_{f}$. ∎ ## 4\. Equilibrium Prices Under Logit Models This section proves the following theorem regarding equilibrium prices for Bertrand competition under the Logit model as described in Sections 2-3: ###### Theorem 4.1. Suppose that $\vartheta>-\infty$, Assumption 3.1 holds, $w$ eventually decreases sufficiently quickly (Defn. 3.2) and $w$ has sub-quadratic second derivatives (Defn. 3.3). There is at least one equilibrium $\mathbf{p}$, and any equilibrium satisfies $\mathbf{c}<\mathbf{p}<\boldsymbol{\infty}$. For further clarity, the key results for both profit maximization and equilibrium problems are outlined with their assumptions in Tables 2 and 2. Three fixed-point characterizations are applied to prove Theorem 4.1. One fixed-point characterization is a generalization of existing results, while two are apparently novel. The first of these novel characterizations states that markups are equal to profits plus the (local) willingness to pay for product value. In essence, this equation is derived by factoring out the gradient of the “inclusive value,” or expected maximum utility, from $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$. This new fixed-point equation also proves that multi-product firm optimal pricing problems under Logit are always “one-parameter” problems. Specifically, all profit-maximizing prices are determined uniquely from a knowledge of profits a single price. This observation yields our second novel fixed-point characterization, a “reduced- form” characterization in terms of equilibrium profits alone. As is common in theoretical economics, the proof has two parts: First, the existence of simultaneously stationary prices is proved, followed by a proof that simultaneously stationary prices are equilibria. The most general proof of the existence of finite simultaneously stationary prices is accomplished using the Poincare-Hopf theorem [29, 47]. Brouwer’s theorem can also be applied under stronger assumptions. Proving that simultaneously stationary prices are in fact equilibria is somewhat more involved. In the past appeals to quasi-concavity have been used to prove that profits have unique maximizers (see, e.g., [15]). While the multi-product firm Logit profit functions are not quasi-concave [22], under utilities with sub-quadratic second derivatives first-order stationarity of profits in fact implies local concavity, the second-order sufficiency condition. A distinct application of the Poincare- Hopf theorem then implies that Logit profits have unique stationary points which must be unique global profit maximizers for fixed competitor’s prices, effectively circumventing the difficulties with profits that are not quasi- concave. Note that while this argument establishes that simultaneously stationary prices are equilibria, it does not necessarily imply that equilibria are unique. The analysis in this section concerning models with constant unit costs and no finite limit on purchasing power serves as a prototype for the analysis of the more general cases presented in Sections 5 and 6. Table 1. Assumptions required for important profit maximizations results. Stationarity is necessary | Assumption 3.1 ---|--- Stationarity is sufficient | Assumption 3.1, Defn. 3.3 Optimal profits are finite | Defn. 3.4 Profit-maximizing prices are finite | $\vartheta>-\infty$, Assumption 3.1, Defn. 3.2 Profit-maximizing prices are unique | $\vartheta>-\infty$, Assumption 3.1, Defn. 3.2, Defn. 3.3 Table 2. Assumptions required for important equilibrium results. Simultaneous stationarity is necessary | Assumption 3.1 ---|--- Simultaneously stationary prices are local equilibria | Assumption 3.1, Defn. 3.3 Simultaneously stationary prices exist | $\vartheta>-\infty$, Assumption 3.1, Defn. 3.2 Local equilibria are equilibria | Assumption 3.1, Defn. 3.3 ### 4.1. Fixed-Point Characterizations of Price Equilibrium This section characterizes simultaneous stationarity in terms of fixed-point equations. #### 4.1.1. The BLP Markup Equation One fixed-point characterization is derived by noting that $(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}=\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top})$ and hence $(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\mathbf{0}$ for $\mathbf{p}\in(0,\infty)^{J}$ if, and only if, (9) $(\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top})(\mathbf{p}_{f}-\mathbf{c}_{f})=-(D_{f}\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}.$ This is a direct generalization of the fixed-point equations derived under “constant coefficient” linear in price utility (i.e., $w(\mathbf{y},p)=-\alpha p$ for some $\alpha>0$) for single-product firms by Anderson & de Palma [3] and for multi-product firms by Besanko et al [12]. The following statements formalize this result. ###### Lemma 4.2. Suppose $w$ satisfies Assumption 3.1. (i) $(\mathbf{I}-\mathbf{1}\mathbf{P}^{L}_{f}(\mathbf{p})^{\top})^{-1}$ exists whenever $\vartheta>-\infty$ or, if $\vartheta=-\infty$, when $\mathbf{p}_{-f}\neq\boldsymbol{\infty}$, but not otherwise. (ii) If $\mathbf{p}_{f}\in(0,\infty)^{J_{f}}$ locally maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, then $\mathbf{p}_{f}=\mathbf{c}_{f}+\boldsymbol{\eta}_{f}(\mathbf{p})$ where (10) $\boldsymbol{\eta}_{f}(\mathbf{p})=-(\mathbf{I}-\mathbf{1}\mathbf{P}^{L}_{f}(\mathbf{p})^{\top})^{-1}(D\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}.$ (iii) If $\mathbf{p}\in(0,\infty)^{J}$ is a local equilibrium, then $\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ where (11) $\boldsymbol{\eta}_{f}(\mathbf{p})=-(D\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}-\left(\frac{\mathbf{P}^{L}_{f}(\mathbf{p})^{\top}(D\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}}{1-\mathbf{P}^{L}_{f}(\mathbf{p})^{\top}\mathbf{1}}\right)\mathbf{1}$ (iv) For any $\mathbf{p}\in(0,\infty)^{J}$, $\boldsymbol{\eta}_{f}(\mathbf{p})>\mathbf{0}$ and $\boldsymbol{\eta}_{f}(\mathbf{p})>\mathbf{0}$. As a consequence, equilibrium prices have positive markups. The fixed-point equation in (iii) is a specialization of the “markup” equation Eqn. (LABEL:MarkupEqn) popularized for Mixed Logit models by Berry, Levinsohn, & Pakes [10]; see also [31, 33]. ###### Proof. (i): The Sherman-Morrison-Woodbury formula for the inverse of a rank-one perturbation of the identity [39, Chapter 2, pg. 50] implies that (12) $\left(\mathbf{I}-\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\right)^{-1}=\mathbf{I}+\left(\frac{1}{1-\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{1}}\right)\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top};$ so long as $\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{1}<1$. This last condition will hold if either $\vartheta>-\infty$ or, if $\vartheta=-\infty$, $\mathbf{p}_{-f}\neq\boldsymbol{\infty}$. (ii): We can write $\displaystyle(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ $\displaystyle=\boldsymbol{\Lambda}_{f}(\mathbf{p})\Big{(}(\mathbf{I}-\mathbf{1P}_{f}^{L}(\mathbf{p})^{\top})(\mathbf{p}_{f}-\mathbf{c}_{f})+(D_{f}\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}\Big{)}.$ Stationarity then requires $(\mathbf{I}-\mathbf{1P}_{f}^{L}(\mathbf{p})^{\top})(\mathbf{p}_{f}-\mathbf{c}_{f})+(D_{f}\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}=\mathbf{0}$. (iii) is a consequence of (ii). (iv): Eqn. (12) proves that $\boldsymbol{\eta}_{f}(\mathbf{p})>\mathbf{0}$ so long as $(Dw_{j})(p_{j})<0$. ∎ #### 4.1.2. A New Equation Another fixed-point characterization follows by multiplying $\mathbf{p}_{f}-\mathbf{c}_{f}$ through $\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top}$, instead of inverting $\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top}$ as a whole, yielding $\mathbf{p}_{f}-\mathbf{c}_{f}=\hat{\pi}_{f}(\mathbf{p})\mathbf{1}-(D_{f}\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}$ $\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top}$ could be considered the “contractive” part of $(\mathbf{I}-\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top})$ because $\lvert\lvert\mathbf{1}\mathbf{P}_{f}(\mathbf{p})^{\top}\rvert\rvert_{\infty}=\lvert\lvert\mathbf{P}_{f}(\mathbf{p})\rvert\rvert_{1}<1$. This derivation proves the following result. ###### Lemma 4.3. Suppose $w$ satisfies Assumption 3.1 (a) and (b). Define $\boldsymbol{\zeta}:(0,\infty)^{J}\to\mathbb{R}^{J}$ by $\boldsymbol{\zeta}(\mathbf{p})=\tilde{\boldsymbol{\pi}}(\mathbf{p})-(D\mathbf{w})(\mathbf{p})^{-1}\mathbf{1}$ where $\tilde{\boldsymbol{\pi}}(\mathbf{p})\in\mathbb{R}^{J}$ is the vector with components $(\tilde{\boldsymbol{\pi}}(\mathbf{p}))_{j}=\hat{\pi}_{f(j)}(\mathbf{p})$. $\boldsymbol{\zeta}$ has components $\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})-(Dw_{j})(p_{j})^{-1}$ where $j\in\mathcal{J}_{f}$, and “intra-firm” components $\boldsymbol{\zeta}_{f}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})\mathbf{1}-(D\mathbf{w}_{f})(\mathbf{p}_{f})^{-1}\mathbf{1}$. (i) For any $\mathbf{p}\in(0,\infty)^{J}$, $(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\boldsymbol{\Lambda}_{f}(\mathbf{p})\boldsymbol{\varphi}_{f}(\mathbf{p})$ and $(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})\boldsymbol{\varphi}(\mathbf{p})$ where (13) $\boldsymbol{\varphi}_{f}(\mathbf{p})=\mathbf{p}_{f}-\mathbf{c}_{f}-\boldsymbol{\zeta}_{f}(\mathbf{p})\quad\quad\text{and}\quad\quad\boldsymbol{\varphi}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p}).$ (ii) If $\mathbf{p}_{f}\in(0,\infty)^{J_{f}}$ locally maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, then $\boldsymbol{\varphi}_{f}(\mathbf{p})=\mathbf{0}$; i.e., $\mathbf{p}_{f}=\mathbf{c}_{f}+\boldsymbol{\zeta}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})$. (iii) If $\mathbf{p}\in(0,\infty)^{J}$ is a local equilibrium, then $\boldsymbol{\varphi}(\mathbf{p})$; i.e. $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$. In Appendix B, Eqn. (13) is used to show that profits under Logit with $w(\mathbf{y},p)=-\alpha\log p$, a model first posed by Allenby & Rossi [2], have no finite profit-maximizing prices when $\alpha\leq 1$. Sandor [45] has also made this observation. Notably, Allenby & Rossi do undertake price optimization exercises. Such exercises thus rely on estimating a coefficient $\alpha$ that is statistically significantly strictly greater than one, a question not addressed in [2]. Positivity and finiteness of equilibrium prices can be considered important regularity properties, and follow from the $\boldsymbol{\zeta}$ characterization. ###### Lemma 4.4. Suppose $w$ satisfies Assumption 3.1, $w$ eventually decreases sufficiently quickly, and either $\vartheta>-\infty$ or $\mathbf{p}_{-f}\neq\boldsymbol{\infty}$ if $\vartheta=-\infty$. Then no $\mathbf{p}_{f}$ with some $p_{j}<c_{j}$ or $p_{j}=\infty$ maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. ###### Proof. Lemma 4.2, (iv), proves that $p_{j}>c_{j}$ if $p_{j}\neq 0$. We complete the claim by proving that no price $p_{j}=0$ in equilibrium. For $p_{j}>0$, the profit derivatives are $\displaystyle(D_{j}\hat{\pi}_{f})(\mathbf{p})=\lambda_{j}(\mathbf{p})(p_{j}-c_{j}-\hat{\pi}_{f}(\mathbf{p}))+P_{j}^{L}(\mathbf{p})=\lambda_{j}(\mathbf{p})(p_{j}-c_{j}-\zeta_{j}(\mathbf{p}))$ If $\lim_{p\downarrow 0}(Dw_{j})(p)=0$, then the first equation here proves that $\lim_{p_{j}\downarrow 0}(D_{j}\hat{\pi}_{f})(\mathbf{p})=P_{j}^{L}(\mathbf{p})>0$, and thus $p_{j}=0$ cannot be profit-maximizing. If $\lim_{p\downarrow 0}(Dw_{j})(p)>0$, $(D_{j}\hat{\pi}_{f})(\mathbf{p})\leq 0$ if, and only if, $p_{j}-c_{j}-\zeta_{j}(\mathbf{p})\geq 0$. As $p_{j}\downarrow 0$, $p_{j}-c_{j}-\zeta_{j}(\mathbf{p})\geq 0$ if, and only if, $\displaystyle-\hat{\pi}_{f}(\mathbf{p})\geq(1-P_{j}^{L}(\mathbf{p}))c_{j}+\frac{1}{\left\lvert(Dw_{j})(p_{j})\right\rvert}.$ Because profit-optimal prices are positive, the left hand side is negative while the right hand side is positive. By contradiction, $p_{j}=0$ cannot be profit-maximizing. The finiteness of equilibrium prices follows from Lemma 4.10. ∎ #### 4.1.3. A Single-Parameter Equation The $\boldsymbol{\zeta}$ characterization also illustrates that price equilibrium problems with the Logit model and constant unit costs are “single- parameter problems.” Define $\psi_{j}(p)=p-c_{j}+(Dw_{j})(p)^{-1}$, and write $p_{j}=c_{j}+\zeta_{j}(\mathbf{p})$ as $\psi_{j}(p_{j})=\hat{\pi}_{f}(\mathbf{p})$. Note that for fixed $f$, the right hand side of this equation is the same for all $j\in\mathcal{J}_{f}$. Thus if the right hand side is known and $\psi_{j}$ is invertible, all prices are uniquely defined. Conversely if a single price is known the right hand side can be computed, thus generating all prices. The following characteristics of the maps $\psi_{j}:[c_{j},\infty)\to[0,\infty)$ formalize this logic. ###### Lemma 4.5. Suppose Assumption 3.1 holds. (i) $\psi_{j}(c_{j})<0$. (ii) If $w$ also eventually decreases sufficiently quickly, then $\psi_{j}(p)\to\infty$ as $p\to\infty$. (iii) Finally, $\psi_{j}$ is differentiable and strictly increasing if, and only if, $w$ is twice differentiable and has sub-quadratic second derivatives. ###### Proof. (i): $\psi_{j}(c_{j})=(Dw_{j})(c_{j})^{-1}<0$. (ii): By assumption, there exists some $r_{j}>1$ and $\bar{p}_{j}>0$ such that $(Dw_{j})(p_{j})^{-1}\geq- r_{j}/p_{j}$ for all $p_{j}\geq\bar{p}_{j}$. Then $\psi_{j}(p_{j})=p_{j}-c_{j}+\frac{1}{(Dw_{j})(p_{j})}\geq\left(1-\frac{1}{r_{j}}\right)p_{j}-c_{j}\to 0\quad\text{as}\quad p_{j}\uparrow\infty.$ (iii): $\psi_{j}$ is continuously differentiable if $w_{j}$ is twice continuously differentiable and strictly decreasing. Specifically, $(D\psi_{j})(p)=1-\omega_{j}(p)$, and $\psi_{j}$ is increasing if, and only if, $w$ has sub-quadratic second derivatives. ∎ ###### Corollary 4.6. Let $w$ satisfy Assumption 3.1, eventually decrease sufficiently quickly, and be twice continuously differentiable with sub-quadratic second derivatives. Then for all $j$ the equation $\psi_{j}(p)=\pi$ has a unique solution $\Psi_{j}(\pi)>c_{j}$ for any $\pi>0$. Equilibrium prices under Logit models can thus be characterized in terms of a fixed-point equation for equilibrium profits alone. Let $\boldsymbol{\Psi}:\mathcal{P}^{F}\to\prod_{j=1}^{J}[c_{j},\infty)$ be defined component-wise by $(\boldsymbol{\Psi}(\boldsymbol{\pi}))_{j}=\Psi_{j}(\pi_{f(j)})$, where $\boldsymbol{\pi}\in\mathcal{P}^{F}$ and $f(j)\in\\{1,\dotsc,F\\}$ denotes the (unique) index of the firm offering product $j$. Next, let $\hat{\boldsymbol{\pi}}:\prod_{j=1}^{J}[c_{j},\infty)\to\mathcal{P}^{F}$ have profits $\hat{\pi}_{f}$ as component functions; i.e. $\hat{\pi}_{f}:\mathcal{P}^{J}\to\mathbb{R}$. Equilibrium profits satisfy the fixed-point equation $\boldsymbol{\pi}=\hat{\boldsymbol{\pi}}(\boldsymbol{\Psi}(\boldsymbol{\pi}))=(\hat{\boldsymbol{\pi}}\circ\boldsymbol{\Psi})(\boldsymbol{\pi})=\boldsymbol{\phi}(\boldsymbol{\pi}).$ Given any such fixed-point $\boldsymbol{\pi}$, all equilibrium prices can be recovered by evaluating $\boldsymbol{\Psi}$. ### 4.2. Existence of Simultaneously Stationary Prices This section provides three proofs of the existence of simultaneously stationary prices using each of the fixed-point characterizations given above. Brouwer’s fixed-point theorem is the typical tool, and is used for proofs based on the $\boldsymbol{\eta}$ and $\boldsymbol{\phi}$ characterizations. However the most general result applies the $\boldsymbol{\zeta}$ characterization and the Poincare-Hopf theorem. Throughout this section it is assumed that $\vartheta>-\infty$. #### 4.2.1. A proof based on the $\boldsymbol{\eta}$ map. Brouwer’s theorem can be applied to the $\boldsymbol{\eta}$ characterization. First recall that $\boldsymbol{\eta}(\mathbf{p})\geq\mathbf{0}$. Next, assume that $\tau=\sup_{\mathbf{p}\in(\mathbf{0},\boldsymbol{\infty})}\lvert\lvert\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}<\infty$; conditions for this are given below. It then follows that $\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ maps $[\mathbf{c},\tau\mathbf{1}]\subset\mathbb{R}^{J}$, a compact convex set, into itself. Since $\mathbf{c}+\boldsymbol{\eta}(\cdot)$ is also continuous, Brouwer’s fixed-point theorem implies the existence of a fixed-point $\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$. ###### Lemma 4.7. Suppose $\vartheta>-\infty$, Assumption 3.1 holds, and $w$ is concave in price. Then $\tau<\infty$. ###### Proof. Because $\lvert\lvert(\mathbf{I}-\mathbf{1P}_{f}^{L}(\mathbf{p})^{\top})^{-1}\rvert\rvert_{\infty}\leq(1-\lvert\lvert\mathbf{P}_{f}^{L}(\mathbf{p})\rvert\rvert_{1})^{-1}$, the bound $\displaystyle\lvert\lvert\boldsymbol{\eta}_{f}(\mathbf{p})\rvert\rvert_{\infty}$ $\displaystyle=\frac{\max_{j\in\mathcal{J}_{f}}\left\lvert(Dw_{j})(p_{j})\right\rvert^{-1}}{1-\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})}$ controls the growth in $\boldsymbol{\eta}$. If $w$ satisfies Assumption 3.1 and is concave in price, then $\left\lvert(Dw_{j})(p_{j})\right\rvert^{-1}\leq\left\lvert(Dw_{j})(c_{j})\right\rvert^{-1}$ for all $p_{j}\geq c_{j}$. This implies that $\max_{j\in\mathcal{J}_{f}}\left\lvert(Dw_{j})(p_{j})\right\rvert^{-1}$ is bounded over $\mathbf{p}_{f}\in[\mathbf{c}_{f},\boldsymbol{\infty})\subset\mathbb{R}^{J_{f}}$. If $\vartheta>-\infty$, $\sup_{\mathbf{p}\in(\mathbf{0},\boldsymbol{\infty})}(\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p}))<1$ for all $f$. Under these assumptions, $\tau<\infty$. ∎ ###### Corollary 4.8. If $\vartheta>-\infty$ and $w$ satisfies Assumption 3.1 and is concave in price, then there exists a fixed-point $\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$. #### 4.2.2. A proof based on $\boldsymbol{\phi}$. Section 4.1 defined simultaneously stationary profits as a fixed-point of the map $\boldsymbol{\phi}=\hat{\boldsymbol{\pi}}\circ\boldsymbol{\Psi}$. This characterization and Brouwer’s theorem can be used to prove the existence of simultaneously stationary prices. ###### Lemma 4.9. Suppose $w$ satisfies Assumption 3.1, eventually decreases sufficiently quickly, has sub-quadratic second derivatives and $\vartheta>-\infty$. Then there exists at least one fixed-point $\boldsymbol{\pi}=\boldsymbol{\phi}(\boldsymbol{\pi})$. ###### Proof. $\Psi_{j}$ can be continuously extended to $[0,\infty]$ as $\Psi_{j}(\infty)=\infty$ under the condition that $w$ eventually decreases sufficiently quickly. Because $w$ is then also eventually log bounded, letting $\vartheta>-\infty$ ensures that for any $f$, $\hat{\pi}_{f}(\mathbf{p})<\infty$ for all $\mathbf{p}\in[0,\infty]^{J}$. Thus $\boldsymbol{\phi}$ is continuous and maps the compact, convex set $[0,\infty]^{J}$ strictly into itself. By Brouwer’s fixed-point theorem, there exists a fixed-point $\boldsymbol{\pi}$ on $[0,\infty]^{F}$. Furthermore, this fixed-point has no infinite components, by Lemma 3.6. ∎ While the restriction $\vartheta>-\infty$ could be removed through an application of Kakutani’s extension of Brouwer’s theorem [25, 15], the fact that $\boldsymbol{\infty}$ is always an equilibrium makes this approach uninformative. #### 4.2.3. A proof based on the $\boldsymbol{\zeta}$ map. The Poincare-Hopf theorem requires that sum of the indices of the vector field $\boldsymbol{\varphi}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$ over all zeros of $\boldsymbol{\varphi}$ equals one, so long as $\boldsymbol{\varphi}$ points outward on the boundary of some compact hyper- rectangle $[\mathbf{c},\bar{\mathbf{p}}]\subset\mathcal{P}^{J}$ for some $\bar{\mathbf{p}}<\boldsymbol{\infty}$. Particularly, this sum of indices cannot be empty and hence there must be at least one zero of $\boldsymbol{\varphi}$, and thus at least one simultaneously stationary point. The hypotheses required in the Poincare-Hopf Theorem follow from the next lemma. ###### Lemma 4.10. Suppose $\vartheta>-\infty$ and $w$ satisfies Assumption 3.1. (i) If $\mathbf{p}\geq\mathbf{c}$ and $p_{j}=c_{j}$, $\varphi_{j}(\mathbf{p})<0$. (ii) If $w$ eventually decreases sufficiently quickly, there exists some $\bar{\mathbf{p}}\in(\mathbf{c},\boldsymbol{\infty})$ such that $\varphi_{j}(\mathbf{p})>0$ whenever $p_{j}\geq\bar{p}_{j}$, regardless of $\mathbf{p}_{-j}$. ###### Proof. (i): When $\mathbf{p}\geq\mathbf{c}$ and $p_{j}=c_{j}$, $\displaystyle p_{j}-c_{j}-\zeta_{j}(\mathbf{p})=-\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})-\frac{1}{\left\lvert(Dw_{j})(c_{j})\right\rvert}<0.$ (ii): Observe that the following bound is valid for large enough $p_{j}$ because $w$ eventually decreases sufficiently quickly: $\displaystyle p_{j}-c_{j}-\zeta_{j}(\mathbf{p})\geq\left(1-\frac{1}{r_{j}}\right)p_{j}-(c_{j}+\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})).$ Because $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ is bounded ($w$ is eventually log bounded), and $r_{j}>1$, $(1-1/r_{j})p_{j}\to\infty$ as $p_{j}\to\infty$, $p_{j}$ can always be chosen large enough to make $p_{j}-c_{j}-\zeta_{j}(\mathbf{p})>0$. When $\vartheta>-\infty$, $\hat{\pi}_{f}(\cdot)$ itself is bounded and hence $\bar{\mathbf{p}}_{f}$ can be chosen independently of $\mathbf{p}_{-f}$. ∎ ###### Theorem 4.11. Suppose $\vartheta>-\infty$, $w$ satisfies Assumption 3.1 and eventually decreases sufficiently quickly. There exists at least one $\mathbf{p}\in(\mathbf{c},\boldsymbol{\infty})$ such that $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$. ###### Proof. Let $\bar{\mathbf{p}}$ be as in Lemma 4.10. $\boldsymbol{\varphi}(\mathbf{p})$ is a vector field on $[\mathbf{c},\bar{\mathbf{p}}]$ that points outward on the boundary of $[\mathbf{c},\bar{\mathbf{p}}]$. Let the set of zeros of $\boldsymbol{\varphi}$ be denoted by $\mathfrak{Z}=\\{\mathbf{p}\in(\mathbf{c},\bar{\mathbf{p}}):\boldsymbol{\varphi}(\mathbf{p})=\mathbf{0}\\}$ and let $\mathrm{index}_{\mathbf{p}}(\boldsymbol{\varphi})$ denote the index of $\boldsymbol{\varphi}$ at $\mathbf{p}\in\mathfrak{Z}$. The Poincare-Hopf Theorem states that $\sum_{\mathbf{p}\in\mathfrak{Z}}\mathrm{index}_{\mathbf{p}}(\boldsymbol{\varphi})=1$, where the value of the sum on the left is taken to be $0$ if $\mathfrak{Z}=\\{\emptyset\\}$. Hence there is at least one zero of $\boldsymbol{\varphi}$. ∎ Note that it is not required that $w$ be concave in price or have sub- quadratic second derivatives in order for simultaneously stationary prices to exist. Lemma 4.10 also shows that $\mathbf{c}+\boldsymbol{\zeta}(\cdot)$ maps $[\mathbf{c},\bar{\mathbf{p}}+\boldsymbol{\epsilon}]$ into itself, for any $\boldsymbol{\epsilon}\geq\mathbf{0}$. Thus Brouwer’s Theorem could be applied just as easily to achieve this existence result. This is not the case in Section 5 below, however. ### 4.3. Sufficiency of Stationarity A general approach to multi-product firm equilibrium problems can rely on quasi-concavity to establish the uniqueness of profit-maximizing prices [22]. However, something like quasi-concavity is required to be able to connect fixed-points $\mathbf{p}_{f}=\mathbf{c}_{f}+\boldsymbol{\zeta}_{f}(\mathbf{p})$ to profit maximizers, and thus fixed-points $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ to local equilibria. Furthermore, uniqueness of profit maximizing prices is required to ensure the existence of equilibria proper. Logit profits have the surprising property that stationarity, the first-order necessary condition, implies local concavity, the second-order sufficient condition when $w$ has sub-quadratic second derivatives. The Poincare-Hopf theorem again serves to commute this local result on the second derivatives of profits to a global property, the uniqueness of profit-maximizing prices. ###### Lemma 4.12. Suppose $\vartheta>-\infty$ and $w$ satisfies Assumption 3.1 and has sub- quadratic second derivatives. (i) Satisfaction of the first-order condition $(\nabla_{f}\hat{\pi}_{f})(\mathbf{p}_{f},\mathbf{p}_{-f})=\mathbf{0}$ is sufficient for $\mathbf{p}_{f}\in(\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J_{f}}$ to be a local maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. (ii) Satisfaction of the simultaneous stationarity condition $(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{0}$ is sufficient for $\mathbf{p}\in(\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J}$ to be a local equilibrium. (iii) If, in addition, $w$ also eventually decreases sufficiently quickly then there is a unique stationary point that is a finite maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. (iv) When $w$ eventually decreases sufficiently quickly, the simultaneous stationarity condition $(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{0}$ is sufficient for $\mathbf{p}\in(\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J}$ to be an equilibrium. ###### Proof. Claim (ii) is an obvious corollary to (i), and claim (iv) is an obvious corollary to (iii). Claim (i) is a consequence of the following componentwise formula for the intra-firm profit price-Hessians $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ when $\mathbf{p}_{f}$ makes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ stationary: for $k,l\in\mathcal{J}_{f}$, $\displaystyle(D_{l}D_{k}\hat{\pi}_{f}^{L})(\mathbf{p})$ $\displaystyle=(D_{k}\lambda_{k})(\mathbf{p})\varphi_{k}(\mathbf{p})+\lambda_{k}(\mathbf{p})(D_{l}\varphi_{k})(\mathbf{p})$ $\displaystyle=\lambda_{k}(\mathbf{p})\Bigg{(}\delta_{k,l}-(D_{l}\hat{\pi}_{f})(\mathbf{p})-\omega_{k}(p_{k})\delta_{k,l}\Bigg{)}$ $\displaystyle=\lambda_{k}(\mathbf{p})(1-\omega_{k}(p_{k}))\delta_{k,l}$ In matrix form, $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}_{f}))$ where $\boldsymbol{\Omega}_{f}(\mathbf{p}_{f})$ is a diagonal matrix with entries $\omega_{j}(p_{j})=(D^{2}w_{j})(p_{j})/(Dw_{j})(p_{j})^{2}$. Thus, the Hessians are diagonal matrices with negative diagonal entries when $w$ has sub-quadratic second derivatives and $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ is locally concave at any stationary prices. Claim (iii) is a consequence of the Poincare-Hopf Theorem: Because $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ is maximized at $\mathbf{p}_{f}=\mathbf{c}_{f}+\boldsymbol{\zeta}_{f}(\mathbf{p})$, $\displaystyle(-1)^{J_{f}}$ $\displaystyle=\mathrm{index}_{\mathbf{p}_{f}}((\nabla_{f}\hat{\pi}_{f})(\cdot,\mathbf{p}_{-f}))=\mathrm{sign}\det(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ $\displaystyle=\mathrm{sign}\det\boldsymbol{\Lambda}_{f}(\mathbf{p})\cdot\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\mathbf{p})=(-1)^{J_{f}}\cdot\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\mathbf{p});$ see Chapter 6 in [29] for some of the basic results invoked here. Because of these equalities, $\mathrm{index}_{\mathbf{p}_{f}}(\boldsymbol{\varphi}_{f}(\cdot,\mathbf{p}_{-f}))=\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\mathbf{p})=1.$ But the Poincare-Hopf theorem require the sum of indices of all zeros of $\boldsymbol{\varphi}_{f}(\cdot,\mathbf{p}_{-f})$ over $[\mathbf{c}_{f},\bar{\mathbf{p}}_{f}]$ to be 1. The zero must, therefore, be unique. ∎ ## 5\. Quantity-Dependent Unit Costs Much of the theoretical literature allows unit costs to depend on sales volumes. This section extends the techniques used in the previous section to this case. Specifically, the analysis in this section proves the following theorem: ###### Theorem 5.1. Let $\vartheta>-\infty$, Assumption 3.1 holds with a $w$ that eventually decreases sufficiently quickly with sub-quadratic second derivatives, and total costs satisfy Assumption 5.1. Then there exists a vector of equilibrium prices $\mathbf{p}$ satisfying $\mathbf{c}(\mathbf{P}(\mathbf{p}))<\mathbf{p}<\boldsymbol{\infty}$, and no equilibrium prices that do not satisfy these bounds. ### 5.1. Assumptions Assumption 2.3 restricted attention to constant unit costs for the following reason: Suppose that unit costs did depend on the quantity sold, and let $c_{f}^{U}(\mathbf{y}_{j},Q_{j}(\mathbf{Y},\mathbf{p}))$ give the unit costs to firm $f$ for offering product $\mathbf{y}_{j}$ in the market of products with characteristics $\mathbf{Y}$, prices $\mathbf{p}$, and resultant demand $Q_{j}(\mathbf{Y},\mathbf{p})$. Then firm $f$’s random profits are $\Pi_{f}(\mathbf{Y},\mathbf{p})=\mathbf{Q}_{f}(\mathbf{Y},\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{c}_{f}^{U}(\mathbf{Y}_{f},\mathbf{Q}_{f}(\mathbf{Y},\mathbf{p})))-c_{f}^{F}(\mathbf{Y}_{f}).$ Notice that profits are no longer a linear function of the demands, and thus expected profits need not be a linear function of the expected demands. Computing these expected costs could be quite difficult, as this would involve sums over the space of realizable demands. One way to relax the assumption that unit costs do not depend on production volume without overly complicating the resulting equilibrium conditions is to suppose that firms decide on pricing based on the total costs corresponding to expected demands, as opposed to the actual expected total costs. The following assumption then generalizes Assumption 2.3 to unit costs that may depend on the quantity sold: ###### Assumption 5.1. Firm $f$ has a normalized cost function $C_{f}:\mathcal{Y}\times[0,1]\to[0,\infty)$ so that the total cost of offering a product with characteristics $\mathbf{y}\in\mathcal{Y}$ for any demand $q$ is $IC_{f}(\mathbf{y},q/I)$, where again $I$ is the market size. Assume also that for any $\mathbf{y}\in\mathcal{Y}$, $C_{f}(\mathbf{y},\cdot):[0,1]\to\mathbb{R}$ is twice continuously differentiable, strictly increasing and convex on $[0,1)$, $C_{f}(\mathbf{y},0)=0$, and $\sup_{P\in[0,1]}c_{f}(\mathbf{y},P)<\infty$ where $c_{f}(\mathbf{y},P)=(D^{P}C_{f})(\mathbf{y},P)$. The simplest example is, of course, $C_{f}(\mathbf{y},P)=c_{f}^{U}(\mathbf{y})P$ as studied in Sections 2-4. Given $\mathbf{Y}_{f}\in\mathcal{Y}^{J_{f}}$, define $C_{j}:[0,1]\to[0,\infty)$ by $C_{j}(P)=C_{f}(\mathbf{y}_{j},P)$ and $c_{j}(P)=(DC_{f})(\mathbf{y}_{j},P)$ where the derivative is with respect to $P$. Under Assumption 5.1, firms choose prices by solving $\text{maximize}\quad\mathbb{E}\Pi_{f}(\mathbf{p})=I\sum_{j\in\mathcal{J}_{f}}\Big{(}P_{j}^{L}(\mathbf{p})p_{j}-C_{j}(\mathbb{E}[Q_{j}(\mathbf{p})]/I)\Big{)}\quad\mathrm{with\;respect\;to}\quad p_{j}\in\mathcal{J}_{f}$ equivalent to (14) $\text{maximize}\quad\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}\big{(}P_{j}^{L}(\mathbf{p})p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))\big{)}\quad\mathrm{with\;respect\;to}\quad p_{j}\in\mathcal{J}_{f}$ The following observation is an extension of Lemma 3.6. ###### Lemma 5.2. Suppose $\vartheta>-\infty$, Assumption 3.1 holds with a $w$ that is eventually log bounded, and Assumption 5.1 holds. For any $\mathbf{p}_{-f}$, the optimal profits for Prob. (14) are positive and finite. ###### Proof. Note that, for any $j$, $\lim_{p_{j}\uparrow\infty}\left(\frac{C_{j}(P_{j}^{L}(\mathbf{p}))}{P_{j}^{L}(\mathbf{p})}\right)=\lim_{p_{j}\uparrow\infty}\left(\frac{c_{j}(P_{j}^{L}(\mathbf{p}))(D_{j}P_{j}^{L})(\mathbf{p})}{(D_{j}P_{j}^{L})(\mathbf{p})}\right)=c_{j}(0)<\infty$ by L’Hopital’s rule. Thus, as $p_{j}\uparrow\infty$, $p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))/P_{j}^{L}(\mathbf{p})\to\infty$. Thus for all $j$, $p_{j}$ can be chosen large enough so that $p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))/P_{j}^{L}(\mathbf{p})>0$. Writing $\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}\big{(}P_{j}^{L}(\mathbf{p})p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))\big{)}=\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})\left(p_{j}-\frac{C_{j}(P_{j}^{L}(\mathbf{p}))}{P_{j}^{L}(\mathbf{p})}\right)$ proves that there exists $\mathbf{p}_{f}$, for any $\mathbf{p}_{-f}$ such that $\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})>0$. If $w$ is eventually log bounded, then $P_{j}(\mathbf{p})(p_{j}-C_{j}(P_{j}^{L}(\mathbf{p}))/P_{j}^{L}(\mathbf{p}))\to 0$ as $p_{j}\uparrow\infty$. Because $\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})\to 0$ as $\mathbf{p}_{f}\to\boldsymbol{\infty}$, and $\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})$ is finite if $p_{j}<\infty$ for any $j\in\mathcal{J}_{f}$, optimal profits are finite. ∎ Note that this proof does not say that all prices are finite; this is proved below using similar techniques to those used in Section 4. ### 5.2. Stationarity The stationarity conditions for Prob. (14) are $\displaystyle(D_{k}\hat{\pi}_{f})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}(D_{k}P_{j}^{L})(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))+P_{k}^{L}(\mathbf{p})=0$ for all $k\in\mathcal{J}_{f}$ and all $f$, as can be checked. As before, stationarity can be written in either of two fixed-point forms: ###### Lemma 5.3. Suppose $\vartheta>-\infty$ and Assumptions 3.1 and 5.1 hold. At any simultaneously stationary prices $\mathbf{p}\in(0,\infty)^{J}$, $\mathbf{p}=\mathbf{c}(\mathbf{P}(\mathbf{p}))+\boldsymbol{\eta}(\mathbf{p})$ where $\boldsymbol{\eta}$ is as defined above and $\mathbf{p}=\mathbf{c}(\mathbf{P}(\mathbf{p}))+\boldsymbol{\zeta}(\mathbf{p})$ where $\boldsymbol{\zeta}:[0,\infty)^{J}\to\mathbb{R}^{J}$ is defined componentwise by $\displaystyle\zeta_{k}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\mathbf{p})\big{(}p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))\big{)}+\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}$ As in Section 4, these characterizations immediately establishes the positivity of markups in equilibrium: ###### Lemma 5.4. Suppose $\vartheta>-\infty$, Assumption 3.1 holds with $w$ eventually decreasing sufficiently quickly, and Assumption 5.1 holds. If $\mathbf{p}\in[0,\infty]^{J}$ is a vector of equilibrium prices, then $\mathbf{c}(\mathbf{P}(\mathbf{p}))<\mathbf{p}<\boldsymbol{\infty}$. ###### Proof. Lemma 5.6 below proves that there exists some $\bar{p}_{j}$ such that $\hat{\pi}_{f}(\mathbf{p})<0$ whenever $p_{j}>\bar{p}_{j}$. Thus no price can be infinite in equilibrium. Moreover, no price can be zero: Suppose $\lim_{p_{k}\downarrow 0}(Dw_{k})(p_{k})=0$. Then for $p_{k}>0$, $(D_{k}\hat{\pi}_{f})(\mathbf{p})=(Dw_{k})(p_{k})P_{k}^{L}(\mathbf{p})\left(p_{k}-c_{k}-\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\mathbf{p})\big{(}p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))\right)+P_{k}^{L}(\mathbf{p})\to P_{k}^{L}(\mathbf{p})>0$ as $p_{k}\downarrow 0$. Now suppose $\lim_{p_{k}\downarrow 0}(Dw_{k})(p_{k})<0$. For $\mathbf{p}$ with $0<p_{k}<\infty$, $(D_{k}\hat{\pi}_{f})(\mathbf{p})\leq 0$ if, and only if, $\displaystyle p_{k}-c_{k}(P_{k}^{L}(\mathbf{p})))-\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\mathbf{p})\big{(}p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))\big{)}-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}\geq 0.$ Taking the limit as $p_{k}\downarrow 0$ yields $\displaystyle-\sum_{j\in\mathcal{J}_{f(k)}\setminus k}P_{j}^{L}(\mathbf{p})\big{(}p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))\big{)}\geq(1-P_{k}^{L}(\mathbf{p}))c_{k}(P_{k}^{L}(\mathbf{p})))+\frac{1}{\left\lvert(Dw_{k})(0)\right\rvert}.$ The right hand side is positive, but if profits are optimal, the left hand side is negative. Thus $(D_{k}\hat{\pi}_{f})(\mathbf{p})>0$ for all $p_{k}$ sufficiently close to zero and all positive profits. Hence, $p_{k}=0$ cannot be profit-optimal. Knowing then that $\mathbf{p}\in(0,\infty)^{J}$, the equation $\mathbf{p}=\mathbf{c}(\mathbf{P}(\mathbf{p}))+\boldsymbol{\eta}(\mathbf{p})$ applies. Because $\boldsymbol{\eta}(\mathbf{p})$ is positive valued for all $\mathbf{p}\in(0,\infty)^{J}$, $\mathbf{p}>\mathbf{c}(\mathbf{P}(\mathbf{p}))$. ∎ The remainder of the proof of equilibrium existence is analogous to the process for constant unit costs: First simultaneously stationary prices are shown to exist, followed by a proof that such prices are in fact always equilibrium prices. ### 5.3. Existence of Stationary Prices The approach to establishing the existence of simultaneously stationary points in the following lemmas is as follows: First, a homeomorphism, $\boldsymbol{\rho}$, between $\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}$ and $[\mathbf{0},\boldsymbol{\infty})$ is constructed. Second, the vector field $\boldsymbol{\varphi}:[\mathbf{0},\boldsymbol{\infty})$ defined componentwise by $\displaystyle\varphi_{k}(\mathbf{p})=p_{k}-c_{k}(P_{k}^{L}(\mathbf{p}))-\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}$ is transported from $\\{\mathbf{p}:\mathbf{p}\geq(D\mathbf{C})(\mathbf{P}(\mathbf{p}))\\}$ to $[\mathbf{0},\boldsymbol{\infty})$ by defining $\boldsymbol{\psi}:[\mathbf{0},\boldsymbol{\infty})\to\mathbb{R}^{J}$ by $\boldsymbol{\psi}(\boldsymbol{\epsilon})=\boldsymbol{\varphi}(\boldsymbol{\rho}(\boldsymbol{\epsilon}))$ for any $\boldsymbol{\epsilon}\in[\mathbf{0},\boldsymbol{\infty})$. Third, a compact rectangle $[\mathbf{0},\bar{\boldsymbol{\epsilon}}]$ is constructed, on which $\boldsymbol{\psi}$ is continuous and points outward on the boundary. The Poincare-Hopf theorem then proves the existence of a zero $\boldsymbol{\epsilon}_{0}\in(\mathbf{0},\bar{\boldsymbol{\epsilon}})$ for $\boldsymbol{\psi}$, which maps to a zero $\mathbf{p}_{0}=\boldsymbol{\rho}(\boldsymbol{\epsilon}_{0})$ of $\boldsymbol{\varphi}$ such that $\mathbf{p}_{0}>(D\mathbf{C})(\mathbf{P}(\mathbf{p}_{0}))$. Such a point is necessarily simultaneously stationary because $(D_{k}\hat{\pi}_{f})(\mathbf{p})=\lambda_{k}(\mathbf{p})\varphi_{k}(\mathbf{p})$. ###### Lemma 5.5. Suppose $\vartheta>-\infty$, Assumption 3.1 holds with $w$ eventually decreasing sufficiently quickly, and Assumption 5.1 holds. The fixed-point problem $\mathbf{p}=\mathbf{F}_{\boldsymbol{\epsilon}}(\mathbf{p})$ where $\mathbf{F}_{\boldsymbol{\epsilon}}(\mathbf{p})=\mathbf{c}(\mathbf{P}(\mathbf{p}))+\boldsymbol{\epsilon}$ has a unique solution $\boldsymbol{\rho}(\boldsymbol{\epsilon})>\boldsymbol{\epsilon}$ for every $\boldsymbol{\epsilon}\in[\mathbf{0},\boldsymbol{\infty})\subset\mathbb{R}^{J}$. Moreover, the corresponding solution map, $\boldsymbol{\rho}:[\mathbf{0},\boldsymbol{\infty})\to\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}$ is a homeomorphism between $[\mathbf{0},\boldsymbol{\infty})$ and $\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}$. ###### Proof. Suppose that $\mathbf{c}(\mathbf{P}(\mathbf{p}))$ depends on $\mathbf{p}$, otherwise the claims are trivial. Assume the fixed-point $\boldsymbol{\rho}(\boldsymbol{\epsilon})$ is unique. $\boldsymbol{\rho}(\boldsymbol{\epsilon})>\boldsymbol{\epsilon}$ because $c_{j}(q)>0$ for all $q<1$ and $P_{j}^{L}(\mathbf{p})<1$ for all $\mathbf{p}$ when there is an outside good. The implicit function theorem guarantees the continuity of $\boldsymbol{\rho}(\boldsymbol{\epsilon})$, and the inverse map $\mathbf{p}\mapsto\boldsymbol{\epsilon}=\mathbf{p}-\mathbf{c}(\mathbf{P}(\mathbf{p}))$ is continuous because $\mathbf{C}$ and $\mathbf{P}$ are continuously differentiable. $\boldsymbol{\rho}(\boldsymbol{\epsilon})$ is thus a homeomorphism. We now show that $\boldsymbol{\rho}(\boldsymbol{\epsilon})$ does indeed exist, as claimed. Because $\sup_{P\in[0,1]}c_{j}(P)=\kappa_{j}<\infty$, there are no fixed points with $p_{j}\geq\bar{p}_{j}=\kappa_{j}+\epsilon_{j}+\delta$ for any fixed $\delta>0$. Moreover, there are no fixed points with $p_{j}=0$ when $\inf_{\mathbf{p}}c_{j}(P_{j}^{L}(\mathbf{p}))>0$; this will hold when there is an outside good, even if $c_{j}(1)=0$, so long as $c_{j}(P)>0$ for all $P\in[0,1)$. Let $\mathcal{D}=[\mathbf{0},\bar{\mathbf{p}}]$, and consider $(D\mathbf{F}_{\boldsymbol{\epsilon}})(\mathbf{p})=(D^{2}\mathbf{C})(\mathbf{P}(\mathbf{p}))(D\mathbf{P})(\mathbf{p})=(D^{2}\mathbf{C})(\mathbf{P}(\mathbf{p}))(\mathbf{I}-\mathbf{P}(\mathbf{p})\mathbf{1}^{\top})\boldsymbol{\Lambda}(\mathbf{p})$ $(D\mathbf{F}_{\boldsymbol{\epsilon}})(\mathbf{p})$ has one as an eigenvalue only if there exists $\mathbf{x}\neq\mathbf{0}$ such that $\displaystyle(D^{2}\mathbf{C})(\mathbf{P}(\mathbf{p}))(\mathbf{I}-\mathbf{P}(\mathbf{p})\mathbf{1}^{\top})\boldsymbol{\Lambda}(\mathbf{p})\mathbf{x}$ $\displaystyle=\mathbf{x}.$ When $\left\lvert\lambda_{j}(\mathbf{p})\right\rvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))\neq-1$ for all $j$, this holds only if $\displaystyle\sum_{j=1}^{J}\beta_{j}(\mathbf{p})P_{j}^{L}(\mathbf{p})=1\quad\text{where}\quad\beta_{j}(\mathbf{p})=\frac{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))}{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))+1}$ as can be checked. Clearly $\left\lvert\lambda_{j}(\mathbf{p})\right\rvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))\neq-1$ and $0\leq\beta_{j}(\mathbf{p})<1$ for all $j$ when $C_{j}$ is convex. Thus, when $\vartheta\neq-\infty$, $\displaystyle\sum_{j=1}^{J}\beta_{j}(\mathbf{p})P_{j}^{L}(\mathbf{p})<\sum_{j=1}^{J}\beta_{j}(\mathbf{p})\left(\frac{e^{u_{j}(p_{j})}}{\sum_{k=1}^{J}e^{u_{k}(p_{k})}}\right)\leq\max_{j=1,\dotsc,J}\beta_{j}(\mathbf{p})<1.$ Thus $(D\mathbf{F}_{\boldsymbol{\epsilon}})(\mathbf{p})$ cannot have 1 as an eigenvalue. By Kellogg’s Uniqueness Theorem [26], $\boldsymbol{\rho}(\boldsymbol{\epsilon})$ is well-defined. ∎ The assumption of convex costs $C_{j}$ would be difficult to relax in this proof: $\beta_{j}(\mathbf{p})$ cannot always be non-negative for concave costs such as $C_{j}(P_{j}^{L}(\mathbf{p}))=-\kappa_{j}P_{j}^{L}(\mathbf{p})^{2}$ because $\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\to 0$ as $p_{j}\uparrow\infty$, and thus $\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\kappa_{j}\leq 1$ for large enough $p_{j}$. Bounding the sums of $\beta$’s in a similar way can be done if we ensure that $\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\left\lvert(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))\right\rvert>1$, but this suggests assumptions that simultaneously restrict the behavior allowed in the utilities and costs. ###### Lemma 5.6. Suppose $\vartheta>-\infty$ and Assumptions 3.1 and 5.1 hold. (i) If $\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))$ and $p_{k}=c_{k}(P_{k}^{L}(\mathbf{p}))$, then $\varphi_{k}(\mathbf{p})<0$. (ii) When $w$ also eventually decreases sufficiently quickly, there exists $\bar{p}_{k}>0$ such that $\varphi_{k}(\mathbf{p})>0$ for all $p_{k}\geq\bar{p}_{k}$ regardless of $\mathbf{p}_{-k}$. ###### Proof. (i) If $\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))$ and $p_{k}=c_{k}(P_{k}^{L}(\mathbf{p}))$, $\displaystyle\varphi_{k}(\mathbf{p})$ $\displaystyle=-\sum_{j\in\mathcal{J}_{f}\setminus\\{k\\}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}$ $\displaystyle<-\sum_{j\in\mathcal{J}_{f}\setminus\\{k\\}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))\leq 0.$ (ii) Write $\varphi_{k}(\mathbf{p})>0$ as (15) $\displaystyle p_{k}-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}>c_{k}(P_{k}^{L}(\mathbf{p}))+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))$ Because there exists $r_{k}>1$ and $\bar{p}_{k}$ such that $\displaystyle p_{k}-\frac{1}{\left\lvert(Dw_{k})(p_{k})\right\rvert}\geq\left(1-\frac{1}{r_{k}}\right)p_{k}$ for all $p_{k}>\bar{p}_{k}$, the left-hand-side in (15) can be made as large as desired. Similarly, because $\sup_{q\in[0,1]}c_{j}(q)<\infty$ and $w_{k}$ is necessarily eventually log-bounded, the right-hand-side $\displaystyle c_{k}(P_{k}^{L}(\mathbf{p}))+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))\leq c_{k}(P_{k}^{L}(\mathbf{p}))+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})p_{j}$ is bounded over all $\mathbf{p}$. Thus there must exist $\bar{p}_{k}$ so large to make (15) hold for all $p_{k}\geq\bar{p}_{k}$. ∎ ###### Corollary 5.7. Suppose $\vartheta>-\infty$, Assumption 3.1 holds with $w$ eventually decreasing sufficiently quickly, and Assumption 5.1 holds. $\boldsymbol{\psi}=\boldsymbol{\varphi}\circ\boldsymbol{\rho}:[\mathbf{0},\bar{\mathbf{p}}]\to\mathbb{R}^{J}$ (i.e. $\boldsymbol{\psi}(\boldsymbol{\epsilon})=\boldsymbol{\varphi}(\boldsymbol{\rho}(\boldsymbol{\epsilon}))$) points outward on the boundary of $[\mathbf{0},\bar{\mathbf{p}}]$. ###### Proof. This follows immediately from Lemma 5.6, recalling that $\rho_{k}(\boldsymbol{\epsilon})>\bar{p}_{k}$ when $\epsilon_{k}=\bar{p}_{k}$. ∎ ###### Theorem 5.8. There exists a zero, $\mathbf{p}$, of $\boldsymbol{\varphi}$ such that $\mathbf{c}(\mathbf{P}(\mathbf{p}))<\mathbf{p}<\boldsymbol{\infty}$. ###### Proof. Apply the Poincare-Hopf theorem to $\boldsymbol{\psi}$, as described above, to establish the existence of $\boldsymbol{\epsilon}\in(\mathbf{0},\bar{\mathbf{p}})$ such that $\boldsymbol{\psi}(\boldsymbol{\epsilon})=\mathbf{0}$. Define $\mathbf{p}=\boldsymbol{\rho}(\boldsymbol{\epsilon})>\mathbf{c}(\mathbf{P}(\mathbf{p}))$, observing that $\boldsymbol{\varphi}(\mathbf{p})=\boldsymbol{\psi}(\boldsymbol{\epsilon})=\mathbf{0}$. ∎ Note that the Poincare-Hopf Theorem is very useful here, relative to Brouwer’s Theorem. Specifically, there is no obvious fixed-point equation for $\boldsymbol{\epsilon}$, and it is not obvious when $\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}$ (and thus $\\{\mathbf{p}:\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))\\}\cap[\mathbf{0},\bar{\mathbf{p}}]$) would be convex. ### 5.4. Existence of Equilibrium The second component of the proof that equilibrium exists requires demonstrating the “sufficiency of stationarity” and the uniqueness of profit- maximizing prices. As in Section 4, this is accomplished by proving that $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ is negative definite at any stationary prices and then applying the Poincare-Hopf Theorem. ###### Lemma 5.9. Suppose $\vartheta>-\infty$, Assumption 3.1 holds with $w$ twice continuously differentiable, and Assumption 5.1 holds. At any stationary point, $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))-\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{H}_{f}(\mathbf{p})\boldsymbol{\Lambda}_{f}(\mathbf{p})$ where $\mathbf{H}_{f}(\mathbf{p})=\mathbf{K}_{f}(\mathbf{p})-\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}^{L}(\mathbf{p})\mathbf{1}^{\top}-\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})+\left(\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}^{L}(\mathbf{p})\right)\mathbf{1}\mathbf{1}^{\top}$ and $\mathbf{K}_{f}(\mathbf{p})=(D^{2}\mathbf{C}_{f})(\mathbf{P}_{f}^{L}(\mathbf{p}))$. ###### Proof. Because $(D_{k}\hat{\pi}_{f})(\mathbf{p})=\lambda_{k}(\mathbf{p})\varphi_{k}(\mathbf{p})$ and $\varphi_{k}(\mathbf{p})=0$ at any stationary point, $(D_{l}D_{k}\hat{\pi}_{f})(\mathbf{p})=\lambda_{k}(\mathbf{p})(D_{l}\varphi_{k})(\mathbf{p})$ for any $k,l\in\mathcal{J}_{f}$. Moreover, $\displaystyle(D_{l}\varphi_{k})(\mathbf{p})$ $\displaystyle=(1-\omega_{k}(p_{k}))\delta_{k,l}-(D^{2}C_{k})(P_{k}^{L}(\mathbf{p}))(D_{l}P_{k}^{L})(\mathbf{p})$ $\displaystyle\quad\quad\quad\quad\quad\quad\quad-\sum_{j\in\mathcal{J}_{f}}(D_{l}P_{j}^{L})(\mathbf{p}))(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))-P_{l}^{L}(\mathbf{p})$ $\displaystyle\quad\quad\quad\quad\quad\quad\quad+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))(D_{l}P_{j}^{L})(\mathbf{p})$ $\displaystyle=(1-\omega_{k}(p_{k}))\delta_{k,l}-(D^{2}C_{k})(P_{k}^{L}(\mathbf{p}))(D_{l}P_{k}^{L})(\mathbf{p})$ $\displaystyle\quad\quad\quad\quad\quad\quad\quad+\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{p})(D^{2}C_{j})(P_{j}^{L}(\mathbf{p}))(D_{l}P_{j}^{L})(\mathbf{p})$ because $(D_{l}\hat{\pi}_{f})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}(D_{l}P_{j}^{L})(\mathbf{p}))(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))-P_{l}^{L}(\mathbf{p})=0$. The result follows by substituting the definition of $(D_{l}P_{k}^{L})(\mathbf{p})$ into this last equation and re-arranging terms. ∎ A sufficient condition for $\hat{\pi}_{f}(\mathbf{p})$ to be locally concave at any stationary point follows: ###### Lemma 5.10. Suppose $\vartheta>-\infty$, Assumptions 3.1 and 5.1 hold, and $w$ has sub- quadratic second derivatives. (i) $(D_{f}\nabla_{f}\hat{\pi}_{f})(\cdot,\mathbf{p}_{-f})$ is negative definite at any stationary prices $\mathbf{p}_{f}$. (ii) As a consequence, profit- maximizing prices $\mathbf{p}_{f}$ are unique for any competitor’s prices $\mathbf{p}_{-f}$ and (iii) any simultaneously stationary point is an equilibrium. ###### Proof. First note that if $w$ has sub-quadratic second derivatives and $\mathbf{H}_{f}(\mathbf{p})$ is positive semi-definite (at any stationary prices $\mathbf{p}_{f}$) then $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ is negative definite (at any stationary prices $\mathbf{p}_{f}$). For if $w$ has sub-quadratic second derivatives, then $\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))$ is negative definite, and if $\mathbf{H}_{f}(\mathbf{p})$ is positive semi- definite, then $-\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{H}_{f}(\mathbf{p})\boldsymbol{\Lambda}_{f}(\mathbf{p})$ is negative semi-definite. We now show that when $C_{j}$ is convex, $\mathbf{H}_{f}(\mathbf{p})$ is positive semi-definite. Because $\mathbf{K}_{f}(\mathbf{p})$ is positive definite, define an inner product $\langle\mathbf{x},\mathbf{y}\rangle_{f}=\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{y}$ on $\mathbb{R}^{J_{f}}$ with $\lvert\lvert\mathbf{x}\rvert\rvert_{f}=\sqrt{\langle\mathbf{x},\mathbf{x}\rangle_{f}}$ the corresponding norm. The Cauchy-Schwartz inequality states that $\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}^{2}=\left\lvert\langle\mathbf{P}_{f}(\mathbf{p}),\mathbf{x}\rangle_{f}\right\rvert^{2}\leq\lvert\lvert\mathbf{P}_{f}(\mathbf{p})\rvert\rvert_{f}^{2}\lvert\lvert\mathbf{x}\rvert\rvert_{f}^{2}=\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}(\mathbf{p})\big{)}\big{(}\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}$ for any vector $\mathbf{x}\in\mathbb{R}^{J_{f}}$. Note that $\mathbf{x}^{\top}\mathbf{H}_{f}(\mathbf{p})\mathbf{x}=\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}(\mathbf{p})\big{)}\big{(}\mathbf{1}^{\top}\mathbf{x}\big{)}^{2}-2\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}\big{(}\mathbf{1}^{\top}\mathbf{x}\big{)}+\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}$ Because any convex quadratic $q(\xi)=a\xi^{2}-2b\xi+c$ (i.e., where $a>0$) is minimized at $\xi_{*}=b/a$ with value $q(\xi_{*})=c-b^{2}/a$, $\mathbf{x}^{\top}\mathbf{H}_{f}(\mathbf{p})\mathbf{x}\geq 0$ for all $\mathbf{x}$ if $\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}^{2}\leq\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{P}_{f}(\mathbf{p})\big{)}\big{(}\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}\big{)}$ for all $\mathbf{x}$, which follows from the Cauchy-Schwartz inequality. Strictly speaking, this inequality is only required for $\mathbf{x}$ satisfying $\mathbf{1}^{\top}\mathbf{x}=\mathbf{P}_{f}(\mathbf{p})^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}/\mathbf{x}^{\top}\mathbf{K}_{f}(\mathbf{p})\mathbf{x}$ to prove the positive semi-definiteness of $\mathbf{H}_{f}(\mathbf{p})$; however the Cauchy-Schwartz inequality requires this to hold for all $\mathbf{x}$. The discussion above proves that any vector of stationary prices $\mathbf{p}_{f}$ is in fact a local maximizer of firm $f$’s profits; it remains to prove that this is fact a unique, global maximizer of firm $f$’s profits. The homotopic construction in Lemma 5.5 applies equally well to a single firm, given fixed competitor prices $\mathbf{p}_{-f}$. By the analogue of Lemma 5.6, the vector field $\boldsymbol{\psi}_{f}=\boldsymbol{\varphi}_{f}\circ\boldsymbol{\rho}_{f}:[\mathbf{0},\bar{\mathbf{p}}_{f}]\to\mathbb{R}^{J_{f}}$ points outward on the boundary of $[\mathbf{0},\bar{\mathbf{p}}_{f}]$ and has at least one zero $\boldsymbol{\epsilon}_{f}$. Assume that the index of any such zero is one. Then there can be only one such zero, because the Poincare- Hopf Theorem requires the sum of the indices of all zeros to be one. The proof that the index of any zero $\boldsymbol{\epsilon}_{f}$ of $\boldsymbol{\psi}_{f}$ is one begins with the standard index formula: $\mathrm{index}_{\boldsymbol{\epsilon}_{f}}(\boldsymbol{\psi}_{f})=\mathrm{sign}\det(D_{f}\boldsymbol{\psi}_{f})(\boldsymbol{\epsilon}_{f})=\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f}))\cdot\mathrm{sign}\det(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f}).$ Both determinants on the right-hand-side are positive, as we now prove. First, consider $\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f}))$. The zero $\mathbf{p}_{f}=\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f})$ of $\boldsymbol{\varphi}_{f}(\cdot,\mathbf{p}_{-f})$ is a local maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, and thus the index of the gradient vector field $(\nabla_{f}\hat{\pi}_{f})(\cdot,\mathbf{p}_{-f})$ at $\mathbf{p}_{f}$ is $(-1)^{J_{f}}$. But then $(-1)^{J_{f}}=\mathrm{sign}\det(D_{f}\nabla_{f}\hat{\pi}_{f})=\mathrm{sign}\det\Big{(}\boldsymbol{\Lambda}_{f}(D_{f}\boldsymbol{\varphi}_{f})\Big{)}=(-1)^{J_{f}}\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f}),$ and $\mathrm{sign}\det(D_{f}\boldsymbol{\varphi}_{f})(\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f}),\mathbf{p}_{-f})=1$ as claimed. Next, consider $\mathrm{sign}\det(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$. The Jacobian of $(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$ is given by $(D\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})=(\mathbf{I}-(D^{2}\mathbf{C}_{f})(D_{f}\mathbf{P}_{f}^{L}))^{-1}$ (where we neglect the arguments for simplicity); this inverse is well defined because $(D^{2}\mathbf{C}_{f})(D_{f}\mathbf{P}_{f}^{L})$ does not have one as an eigenvalue, as proved in Lemma 5.5. Thus zero is not an eigenvalue of $(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$, and any eigenvalue $\mu$ ($\neq 0$) satisfies $\Big{(}\mathbf{I}-(D^{2}\mathbf{C}_{f})(D_{f}\mathbf{P}_{f}^{L})\Big{)}\mathbf{x}=\left(\frac{1}{\mu}\right)\mathbf{x}.$ Rearranging this equation yields $\left[\left(\frac{1}{\mu}-1\right)\mathbf{I}+(D^{2}\mathbf{C}_{f})\boldsymbol{\Lambda}_{f}\right]\mathbf{x}=\big{(}\mathbf{1}^{\top}\boldsymbol{\Lambda}_{f}\mathbf{x}\big{)}(D^{2}\mathbf{C}_{f})\mathbf{P}_{f}^{L}.$ It is straightforward to see that this can hold only if $1=\theta\left(1-\frac{1}{\mu}\right)\quad\text{where}\quad\theta(\alpha)=\sum_{j\in\mathcal{J}_{f}}\left(\frac{\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})}{\alpha+\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})}\right)P_{j}^{L}.$ Without loss of generality, suppose $\mathcal{J}_{f}=\\{1,\dotsc,J_{f}\\}$, $\left\lvert\lambda_{1}\right\rvert(D^{2}C_{1})\leq\dotsb\leq\left\lvert\lambda_{J_{f}}\right\rvert(D^{2}C_{J_{f}}),$ let $N$ be the number of distinct values of $\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})$, $\kappa_{1}<\dotsb<\kappa_{N}$ these values, and $M_{n}$ be the number of $j\in\mathcal{J}_{f}$ for which $\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})=\kappa_{n}$. Note that $\theta$ is not defined for $\alpha\in\mathcal{S}_{f}=\\{-\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j}):j\in\mathcal{J}_{f}\\}=\\{-\kappa_{n}\\}_{n=1}^{N}$, and $D\theta(\alpha)=-\sum_{j\in\mathcal{J}_{f}}\left(\frac{\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j})}{(\alpha+\left\lvert\lambda_{j}\right\rvert(D^{2}C_{j}))^{2}}\right)P_{j}^{L}<0,$ when defined. Moreover, $\theta(\alpha)<0<1$ for all $\alpha\in(-\infty,\kappa_{N})$; thus no solutions to $\theta(\alpha)=1$ are less than $-\kappa_{N}$. Consider any interval $(-\kappa_{n+1},-\kappa_{n})$, $n\in\\{1,\dotsc,N-1\\}$. In such an interval $\theta$ is strictly decreasing, $\theta(\alpha)\uparrow\infty$ as $\alpha\downarrow-\kappa_{n+1}$, and $\theta(\alpha)\downarrow-\infty$ as $\alpha\uparrow-\kappa_{n}$. There is thus a unique $\alpha_{n}\in(-\kappa_{n+1},-\kappa_{n})$ such that $\theta(\alpha_{n})=1$. Finally, $\theta(\alpha)>0$ for all $\alpha>-\kappa_{1}$, $D\theta(\alpha)<0$ for all $\alpha>-\kappa_{1}$, and $\theta(0)=\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}<1$ imply that there is a unique $\alpha_{N}\in(-\kappa_{1},0)$ such that $\theta(\alpha_{N})=1$. The $N$ solutions to this equation map to the $N$ distinct eigenvalues of $(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$ (with multiplicities $M_{n}$) via $\alpha_{n}=1-1/\mu_{n}$; that is, $\mu=1/(1-\alpha_{n})$. Because each $\alpha_{n}<0$, each distinct eigenvalue of $(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})$ is positive. Thus $\mathrm{sign}\det(D_{f}\boldsymbol{\rho}_{f})(\boldsymbol{\epsilon}_{f})=1$ as claimed. ∎ Consider this proof from the perspective of concave costs $C_{j}$. $\mathbf{x}^{\top}\mathbf{H}_{f}(\mathbf{p})\mathbf{x}=-\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\left\lvert\mathbf{K}_{f}(\mathbf{p})\right\rvert\mathbf{P}_{f}(\mathbf{p})\big{)}\big{(}\mathbf{1}^{\top}\mathbf{x}\big{)}^{2}+2\big{(}\mathbf{P}_{f}(\mathbf{p})^{\top}\left\lvert\mathbf{K}_{f}(\mathbf{p})\right\rvert\mathbf{x}\big{)}\big{(}\mathbf{1}^{\top}\mathbf{x}\big{)}-\mathbf{x}^{\top}\left\lvert\mathbf{K}_{f}(\mathbf{p})\right\rvert\mathbf{x}$ is now a concave quadratic in $\mathbf{1}^{\top}\mathbf{x}$, $q(\xi)=-a\xi^{2}+2b\xi-c$ (where $a,c>0$), maximized at $\xi_{*}=b/a$ with value $q(\xi_{*})=b^{2}/a-c$. However, the Cauchy-Schwartz inequality then requires $q(\xi)\leq q(\xi_{*})\leq 0$, and $\mathbf{H}_{f}(\mathbf{p})$ is negative semi-definite. $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ is negative definite then only if $\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))$ is “more” negative-definite than $\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{H}_{f}(\mathbf{p})\boldsymbol{\Lambda}_{f}(\mathbf{p})$ is, in the sense that $\mathbf{x}^{\top}\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))\mathbf{x}<\mathbf{x}^{\top}\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{H}_{f}(\mathbf{p})\boldsymbol{\Lambda}_{f}(\mathbf{p})\mathbf{x}.$ ## 6\. Finite Purchasing Power This section considers models in which there exists some limit on the population’s purchasing power. That is, there exists $\varsigma\in[0,\infty)$ such that if $p_{j}\geq\varsigma$, no individual can purchase product $j$. However, important empirical examples of Mixed Logit models have finite purchasing power; see, e.g., [10, 43]. Thus it is important to consider this case in the analysis of existence and uniqueness of equilibrium prices. Section 4 above proves that if $\varsigma=\infty$ (no purchasing power limit) and there is an outside good, it is not possible for any product’s price to be $\infty$ in equilibrium. However, when $\varsigma<\infty$, it is possible for some$-$but not all$-$prices to be equal to $\varsigma$; in other words, firms may “price some products out of the market”. Theoretical and computational treatments must be specially adapted to this case to account for this qualitatively different behavior. The analysis in this section proves the following theorem: ###### Theorem 6.1. Suppose $\vartheta>-\infty$, costs satisfy Assumption 5.1 with $(D^{2}C)(\mathbf{y},P)$ finite as $P\downarrow 0$ for any $\mathbf{y}\in\mathcal{Y}$, and Assumption 6.1 holds with $\varsigma<\infty$ and a $w$ that eventually decreases sufficiently quickly, has finite sub- quadratic second derivatives, and $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists for any $\mathbf{y}\in\mathcal{Y}$. Then there exists at least one equilibrium $\mathbf{p}\in[0,\varsigma]^{J}$, and any equilibrium satisfies $\mathbf{p}>\mathbf{c}(\mathbf{P}(\mathbf{p}))$. ### 6.1. Assumptions Assumption 3.1 must be revised to account for finite $\varsigma$: ###### Assumption 6.1. There exists $\varsigma\in(c_{*},\infty]$, where $c_{*}=\max_{j}\\{\sup_{P\in[0,1]}c_{j}(P)\\}<\infty$, and functions $w:\mathcal{Y}\times[0,\varsigma)\to(-\infty,\infty)$ and $v:\mathcal{Y}\to(-\infty,\infty)$ such that the utility $u:\mathcal{Y}\times[0,\infty)\to\mathbb{R}$ can be written $u(\mathbf{y},p)=w(\mathbf{y},p)+v(\mathbf{y})$ for all $p<\varsigma$ and $u(\mathbf{y},p)=-\infty$ for all $p\geq\varsigma$. Moreover, assume that, for all $\mathbf{y}\in\mathcal{Y}$, $w(\mathbf{y},\cdot):[0,\varsigma)\to(-\infty,\infty)$ is (a) strictly decreasing, and (b) continuously differentiable, and (c) $\lim_{p\uparrow\varsigma}w(\mathbf{y},p)=-\infty$. The basic properties so useful in the case of infinite purchasing power must also be generalized: ###### Definition 6.1. If $\varsigma<\infty$, $w(\mathbf{y},\cdot)$ eventually decreases sufficiently quickly if there exists $\delta(\mathbf{y})>0$ and $\alpha(\mathbf{y})>1$ such that $(Dw)(\mathbf{y},p)\leq-\alpha(\mathbf{y})/(\varsigma-p)$ for all $p\in[\varsigma-\delta(\mathbf{y}),\varsigma)$. ###### Definition 6.2. $w(\mathbf{y},\cdot)$ has sub-quadratic second derivatives if $\omega(\mathbf{y},p)<1$ for all $p\in(0,\varsigma)$. Gallego et. al [18] consider equilibrium pricing under the attraction demand model, equivalent to the Logit model with nonlinear utilities, and allow $\varsigma<\infty$. Specifically, Gallego et. al formulate their model in terms of the “attraction function” $a_{j}(p)=e^{u_{j}(p)}$ and make what amount to the following assumptions on $u_{j}$: $u_{j}:[0,\varsigma)\to\mathbb{R}$ is continuously differentiable, strictly decreasing, and $p(D^{2}u_{j})(p)\geq(Du_{j})(p)$. This last assumption is violated by the “BLP”-type utility $u_{j}(p)=\alpha\log(\varsigma-p)+v_{j}$ [10, 31, 33, 32], as can be easily checked. Assumption 6.1 and Defs. 6.1 and 6.2 above are weak enough to allow an analysis of this important case. The following basic observations, stated without proof, are needed: ###### Lemma 6.2. Under Assumption 6.1 with $\vartheta>-\infty$ and $\varsigma<\infty$, * • $P_{j}^{L}$ can be continuously extended to $[0,\varsigma]^{J}$, with $P_{j}^{L}(\mathbf{p})\downarrow 0$ as $p_{j}\uparrow\varsigma$. Specifically, $P_{j}^{L}(\mathbf{p})=\frac{e^{u_{j}(p_{j})}}{e^{\vartheta}+\sum_{k:p_{k}<\varsigma}e^{u_{k}(p_{k})}}$ * • $\hat{\pi}_{f}(\mathbf{p})$ can be continuously extended to $[0,\varsigma]^{J}$, with $\hat{\pi}_{f}(\mathbf{p})\downarrow 0$ as $\mathbf{p}_{f}\uparrow\varsigma\mathbf{1}$ with $\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f},p_{j}<\varsigma}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))\quad\text{and}\quad 0<\sup_{\mathbf{p}_{f}\in[0,\varsigma]^{J_{f}}}\hat{\pi}_{f}(\mathbf{p})<\infty$ * • If $w$ eventually decreases sufficiently quickly, $\lambda_{j}(\mathbf{p})=(Dw_{j})(p_{j})P_{j}^{L}(\mathbf{p})$ can be continuously extended to $[0,\varsigma]^{J}$ with $\lambda_{j}(\mathbf{p})\uparrow 0$ as $p_{j}\uparrow\varsigma$. If $\vartheta=-\infty$, $P_{j}^{L}$ cannot be continuously extended to $[0,\varsigma]^{J}$, for reasons analogous to those discussed in Section 3. The following consequences of Assumption 6.1 and Defs. 6.1 and 6.2 are also used below. ###### Lemma 6.3. Suppose Assumption 6.1 holds with $\varsigma<\infty$. (i) If $\lim_{p\uparrow\varsigma}(Dw)(\mathbf{y},p)$ exists, then $(Dw)(\mathbf{y},p)=-\infty$ as $p\uparrow\varsigma$, and thus $(Dw)(\mathbf{y},p)^{-1}\uparrow 0$ as $p\uparrow\varsigma$. (ii) If $w$ eventually decreases sufficiently quickly, then $(Dw)(\mathbf{y},p)=-\infty$ as $p\uparrow\varsigma$. (iii) If $w(\mathbf{y},\cdot)$ is twice continuously differentiable and $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists, then it is “eventually concave” in the sense that there exists some $\epsilon\in(0,\varsigma)$ such that $(D^{2}w)(\mathbf{y},\cdot)<0$ on $(\varsigma-\epsilon,\varsigma)$. (iv) If $w$ is twice continuously differentiable and $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists, then $\limsup_{p\uparrow\varsigma}\omega(\mathbf{y},p)\leq 0$. (v) If $w$ is twice continuously differentiable and eventually decreases sufficiently quickly, then $\liminf_{p\uparrow\varsigma}\omega(\mathbf{y},p)>-1$. ###### Proof. (ii) is trivial. (i) and (iii) are both consequences of the following technical result: If $f:(0,1)\to(-\infty,0)$ is continuously differentiable, $\lim_{x\uparrow 1}f(x)=-\infty$, and $\lim_{x\uparrow 1}(Df)(x)$ exists, then $\lim_{x\uparrow 1}(Df)(x)=-\infty$. For proof, note that the fundamental theorem of calculus requires that $\lim_{\delta\downarrow 0}\int_{x}^{1-\delta}(Df)(y)dy=-\infty.$ If $\lim_{x\uparrow 1}(Df)(x)$ exists and were finite, then $\limsup_{x\uparrow 1}(Df)(x)>-\infty$ and this integral would be finite. Thus if $\lim_{x\uparrow 1}(Df)(x)$ exists then $\lim_{x\uparrow 1}(Df)(x)=-\infty$. The assumption that the limit exists is required because $(Df)$ could be highly oscillatory (in a manner similar to $\sin(x^{-1})$) and still generate $\lim_{x\uparrow 1}f(x)=-\infty$. (iv): By (iii), $(D^{2}w)(\mathbf{y},p)$ is negative for all $p$ sufficiently close to $\varsigma$. Thus $\omega(\mathbf{y},p)<0$ for all $p$ sufficiently close to $\varsigma$. (v): Note that $-\omega(\mathbf{y},p)=D[(Dw)(\mathbf{y},p)^{-1}]$. The mean value theorem then states that for all $\delta\in(0,\varsigma)$, there exists some $\epsilon\in(0,\delta]$ such that $\frac{1}{(Dw)(\mathbf{y},\varsigma-\delta)}=-\left(\frac{1}{(Dw)(\mathbf{y},\varsigma)}-\frac{1}{(Dw)(\mathbf{y},\varsigma-\delta)}\right)=-\Big{(}-\omega(\mathbf{y},\varsigma-\epsilon)\delta\Big{)}=\omega(\mathbf{y},\varsigma-\epsilon)\delta.$ Because $w$ eventually decreases sufficiently quickly, there exists $\gamma>0$ and $\alpha>1$ such that $\omega(\mathbf{y},\varsigma-\epsilon)\delta=\frac{1}{(Dw)(\mathbf{y},\varsigma-\delta)}\geq-\left(\frac{\delta}{\alpha}\right).$ Thus $\omega(\mathbf{y},\varsigma-\epsilon)\geq-\alpha^{-1}>-1$, and thus $\liminf_{p\uparrow\varsigma}\omega(\mathbf{y},p)>-1$. Note that we have not assumed $\lim_{p\uparrow\varsigma}\omega(\mathbf{y},p)$ exists in proving that $\liminf_{p\uparrow\varsigma}\omega(\mathbf{y},p)>-1$. ∎ ### 6.2. A Variational Approach The natural approach to characterizing profit-maximizing prices when $\varsigma<\infty$ would be to assume firms solve the bound-constrained optimization problem $\displaystyle\text{maximize}\quad\hat{\pi}_{f}(\mathbf{p})\quad\mathrm{with\;respect\;to}\quad\mathbf{p}_{f}\in[0,\varsigma]^{J_{f}}.$ The KKT conditions are the Variational Inequality (VI) (16) $(\nabla_{f}\hat{\pi}_{f})(\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{q}_{f})\geq\mathbf{0}\quad\text{for all}\quad\mathbf{q}_{f}\in[0,\varsigma]^{J_{f}}$ This approach is, unfortunately, not useful because of the following result: ###### Lemma 6.4. Suppose $\vartheta>-\infty$, costs satisfy Assumption 5.1, Assumption 6.1 holds with $\varsigma<\infty$, and $w$ eventually decreases sufficiently quickly. Then $\hat{\pi}_{f}(\mathbf{p})$ can be continuously differentiably extended to $[0,\infty)^{J}$, where $(D_{k}\hat{\pi}_{f})(\mathbf{p})=0$ whenever $p_{k}\geq\varsigma$, $k\in\mathcal{J}_{f}$. ###### Proof. Suppose $\mathbf{q}\in(\mathbf{0},\varsigma\mathbf{1})$. Then $(D_{k}\hat{\pi}_{f})(\mathbf{q})=\lambda_{k}(\mathbf{q})\left(q_{k}-c_{k}(P_{k}(\mathbf{q}))-\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\mathbf{q})(q_{j}-c_{j}(P_{j}(\mathbf{q})))-\frac{1}{\left\lvert(Dw_{k})(q_{k})\right\rvert}\right).$ Each quantity can be extended, continuously, to $[0,\varsigma]^{J}$, and thus so can $(D_{k}\hat{\pi}_{f})$. Note in particular that the term in parentheses tends to $\varsigma- c_{k}(0)-\sum_{j\in\mathcal{J}_{f},p_{j}<\varsigma}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))$ which is finite, and $\lambda_{k}(\mathbf{p})\uparrow 0$ as $p_{k}\uparrow\varsigma$. Thus $(D_{k}\hat{\pi}_{f})(\mathbf{p})=0$ when $p_{k}=\varsigma$. ∎ As a result, the standard VI (16) does not provide any information about profit-optimal prices or equilibria that have some prices equal to $\varsigma$. For an extreme illustration of this fact, note that $\mathbf{p}_{f}=\varsigma\mathbf{1}$ (trivially) solves (16). However, $\varsigma\mathbf{1}$ cannot possible be profit-maximizing, because $\hat{\pi}_{f}(\varsigma,\mathbf{1},\mathbf{p}_{-f})=0$ while $\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})>0$ for any $\mathbf{p}_{f}$ satisfying $\mathbf{c}_{f}(\mathbf{P}_{f}(\mathbf{p}))<\mathbf{p}_{f}<\varsigma\mathbf{1}$. The $\boldsymbol{\zeta}$ fixed-point form introduced above provides a convenient solution to this problem. ###### Theorem 6.5. Suppose $\vartheta>-\infty$, Assumptions 5.1 and 6.1 hold, $w$ eventually decreases sufficiently quickly and has sub-quadratic second derivatives. For any $\mathbf{p}\in[0,\varsigma]^{J}$, let $\mathcal{J}_{f}^{\circ}=\\{j\in\mathcal{J}_{f}:p_{j}<\varsigma\\}$, $\mathcal{J}_{f}^{*}=\\{j\in\mathcal{J}_{f}:p_{j}=\varsigma\\}$ and if $j\in\mathcal{J}_{f}^{*}$, define $\zeta_{j}(\mathbf{p})=\lim_{p_{j}\uparrow\varsigma}\zeta_{j}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p}))).$ (i) If $\mathbf{p}_{f}\in[0,\varsigma]^{J_{f}}$ locally maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ then $p_{j}=c_{j}(P_{j}(\mathbf{p}))+\zeta_{j}(\mathbf{p})$ for all $j\in\mathcal{J}_{f}^{\circ}$ and $\varsigma- c_{j}(0)-\zeta_{j}(\mathbf{p})\leq 0$ for all $j\in\mathcal{J}_{f}^{*}$. (ii) If, in addition, $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},\cdot)$ exists and $(D^{2}C)(\mathbf{y},P)$ is finite as $P\downarrow 0$ for all $\mathbf{y}\in\mathcal{Y}$, then $p_{j}=c_{j}(P_{j}(\mathbf{p}))+\zeta_{j}(\mathbf{p})$ for all $j\in\mathcal{J}_{f}^{\circ}$ and $\varsigma- c_{j}(0)-\zeta_{j}(\mathbf{p})\leq 0$ for all $j\in\mathcal{J}_{f}^{*}$ is sufficient $\mathbf{p}_{f}\in[0,\varsigma]^{J_{f}}$ to maximize $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. ###### Proof. Note that $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ is continuously differentiable on $(0,\infty)^{J_{f}}$, and thus we can apply the vector mean value theorem to $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ on $(0,\varsigma]^{J_{f}}$. We also note that there exist neighborhoods $\mathcal{U}_{f}^{*}$ and $\mathcal{U}_{f}^{\circ}$, of $\mathbf{p}_{f}^{*}$ and $\mathbf{p}_{f}^{\circ}$ respectively, and a map $\mathbf{p}_{f}^{\circ}:(\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}})\to\mathcal{U}_{f}^{\circ}$ such that $(\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{q}_{f}^{*})\in\mathcal{U}_{f}^{\circ}\times(\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}})$ and $\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*})$ is the unique solution to $(\nabla_{f}^{\circ}\hat{\pi}_{f})\big{(}\mathbf{q}_{f}^{\circ},\mathbf{q}_{f}^{*},\mathbf{p}_{-f})=\mathbf{0}$ as a problem in $\mathbf{q}_{f}^{\circ}$ only. This actually follows from the Implicit Function Theorem, applied to the continuously differentiable extension of $\boldsymbol{\varphi}$ to all of $[0,\varsigma]^{J}$ given below in Lemma 6.7. Because of the sufficiency of stationarity, when $w$ has sub- quadratic second derivatives, $\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*})$ is, in fact, the unique local maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{q}_{f}^{*},\mathbf{p}_{-f})$ on $\mathcal{U}_{f}^{\circ}$. Thus $\hat{\pi}_{f}\big{(}\mathbf{q}_{f}^{\circ},\mathbf{q}_{f}^{*},\mathbf{p}_{-f})<\hat{\pi}_{f}\big{(}\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{q}_{f}^{*},\mathbf{p}_{-f})$ for all $\mathbf{q}_{f}^{*}\in\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}}$ and $\mathbf{q}_{f}^{\circ}\in\mathcal{U}_{f}^{\circ}$. Furthermore, $(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\mathbf{q}_{f}^{*})\to\mathbf{0}$ as $\mathbf{q}_{f}^{*}\uparrow\varsigma\mathbf{1}$. For $(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\mathbf{q}_{f}^{*})$ solves $(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\mathbf{q}_{f}^{*})=(D_{f}^{*}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$ while $(D_{f}^{*}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})=\lim_{\mathbf{q}_{f}^{*}\uparrow\varsigma\mathbf{1}}(D_{f}^{*}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})\to\mathbf{0}$, and $(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})$ is nonsingular. (i): The necessity of $\varphi_{j}(\mathbf{p})=p_{j}-c_{j}(P_{j}(\mathbf{p}))-\zeta_{j}(\mathbf{p})=0$ for $j\in\mathcal{J}_{f}^{\circ}$ is obvious. Suppose then that there is some $j\in\mathcal{J}_{f}^{*}$ such that $\varsigma- c_{j}(0)-\zeta_{j}(\mathbf{p})>0$. We can choose the neighborhood $\mathcal{U}_{f}^{*}$ above so that $\varphi_{j}(\mathbf{q}_{f},\mathbf{p}_{-f})>0$ for all $\mathbf{q}_{f}=(\mathbf{q}_{f}^{\circ},\mathbf{q}_{f}^{*})\in\mathcal{U}_{f}^{\circ}\times(\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}})$. Letting $\mathbf{q}_{f}^{*}=\varsigma\mathbf{1}-(\varsigma- q_{j})\mathbf{e}_{j}$ for some $q_{j}<\varsigma$ (that is, changing only the $j^{\text{th}}$ products’ price), the vector mean value theorem states that there exists some $\mathbf{r}_{f}^{*}=\varsigma\mathbf{1}-(\varsigma-\tau)\mathbf{e}_{j}$, $\tau\in(q_{j},\varsigma)$, such that $\displaystyle\hat{\pi}_{f}(\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{q}_{f}^{*},\mathbf{p}_{-f})=\hat{\pi}_{f}(\mathbf{p})+(D_{j}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})(\varsigma- q_{j})>\hat{\pi}_{f}(\mathbf{p}).$ Thus $\mathbf{p}_{f}$ is not locally profit-maximizing for $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. By contraposition, (i) holds. (ii): Define $\hat{\pi}_{f}^{*}(\mathbf{q}_{f}^{*})=\hat{\pi}_{f}(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$. Also let $\nabla_{f}^{*}\hat{\pi}_{f}$ and $\nabla_{f}^{\circ}\hat{\pi}_{f}$ denote the derivatives of firm $f$’s profits with respect to the prices of products in $\mathcal{J}_{f}^{*}$ and $\mathcal{J}_{f}^{\circ}$, respectively. Note that $\displaystyle(\nabla^{*}\hat{\pi}_{f}^{*})(\mathbf{q}_{f}^{*})$ $\displaystyle=(\nabla^{*}\hat{\pi}_{f})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})+(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\mathbf{q}_{f}^{*})^{\top}(\nabla^{\circ}\hat{\pi}_{f})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$ $\displaystyle=(\nabla^{*}\hat{\pi}_{f})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$ because $(\nabla^{\circ}\hat{\pi}_{f})(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})=\mathbf{0}$, by definition. Let $\mathbf{q}_{f}^{*}=\varsigma\mathbf{1}-\boldsymbol{\delta}$ for some $\boldsymbol{\delta}\geq 0$, $\boldsymbol{\delta}\neq\mathbf{0}$. Then the vector mean value theorem states that there exists $\mathbf{r}_{f}^{*}=\varsigma\mathbf{1}-\tau\boldsymbol{\delta}$, $\tau\in(0,1)$, such that $\hat{\pi}_{f}(\mathbf{q}_{f}^{*},\mathbf{q}_{f}^{\circ},\mathbf{p}_{-f})\leq\hat{\pi}_{f}^{*}(\mathbf{q}_{f}^{*})=\hat{\pi}_{f}^{*}(\mathbf{p}_{f}^{*})-(\nabla_{f}^{*}\hat{\pi}_{f}^{*})(\mathbf{r}_{f}^{*})^{\top}\boldsymbol{\delta}=\hat{\pi}_{f}(\mathbf{p})-(\nabla_{f}^{*}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})^{\top}\boldsymbol{\delta}$ Note also that $\displaystyle\boldsymbol{\varphi}_{f}^{*}(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})$ $\displaystyle=\boldsymbol{\varphi}_{f}^{*}(\mathbf{p})-\Big{(}(D_{f}^{*}\boldsymbol{\varphi}_{f}^{*})(\mathbf{p})+(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{*})(\mathbf{p})(D_{f}^{*}\mathbf{p}_{f}^{\circ})(\varsigma\mathbf{1})\Big{)}\boldsymbol{\delta}+\mathcal{O}(\lvert\lvert\boldsymbol{\delta}\rvert\rvert^{2})$ $\displaystyle=\boldsymbol{\varphi}_{f}^{*}(\mathbf{p})-(D_{f}^{*}\boldsymbol{\varphi}_{f}^{*})(\mathbf{p})\boldsymbol{\delta}+\mathcal{O}(\lvert\lvert\boldsymbol{\delta}\rvert\rvert^{2})$ $\displaystyle=\boldsymbol{\varphi}_{f}^{*}(\mathbf{p})-(\mathbf{I}-\boldsymbol{\Omega}_{f}^{*}(\varsigma\mathbf{1}))\boldsymbol{\delta}+\mathcal{O}(\lvert\lvert\boldsymbol{\delta}\rvert\rvert^{2})$ Because $\boldsymbol{\varphi}_{f}^{*}(\mathbf{p})\leq\mathbf{0}$ and $1-\omega_{j}(\varsigma)>0$ for all $j\in\mathcal{J}_{f}^{*}$, $\boldsymbol{\varphi}_{f}^{*}(\mathbf{q}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{q}_{f}^{*}),\mathbf{p}_{-f})<\mathbf{0}$ for all $\boldsymbol{\delta}\neq\mathbf{0}$ sufficiently small. For such $\boldsymbol{\delta}\geq 0$, $\boldsymbol{\delta}\neq\mathbf{0}$, also satisfying $\mathbf{q}_{f}^{*}=\varsigma\mathbf{1}-\boldsymbol{\delta}\in\mathcal{U}_{f}^{*}\cap[0,\varsigma]^{J_{f}^{*}}$, $(\nabla_{f}^{*}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})=\boldsymbol{\Lambda}_{f}^{*}(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})\boldsymbol{\varphi}_{f}^{*}(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})\geq\mathbf{0}$ with at least one positive component. Thus $(\nabla_{f}^{*}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})^{\top}\boldsymbol{\delta}>0$, and $\displaystyle\hat{\pi}_{f}(\mathbf{q}_{f}^{*},\mathbf{q}_{f}^{\circ},\mathbf{p}_{-f})\leq\hat{\pi}_{f}(\mathbf{p})-(\nabla_{f}^{*}\hat{\pi}_{f})(\mathbf{r}_{f}^{*},\mathbf{p}_{f}^{\circ}(\mathbf{r}_{f}^{*}),\mathbf{p}_{-f})^{\top}\boldsymbol{\delta}<\hat{\pi}_{f}(\mathbf{p}).$ $\mathbf{p}_{-f}$ is thus a local maximizer of $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. ∎ ###### Corollary 6.6. Suppose $\vartheta>-\infty$, Assumption 5.1 holds with $D^{2}C$ finite as $P\downarrow 0$, and 6.1 holds with $w$ that eventually decreases sufficiently quickly, has sub-quadratic second derivatives, and $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists. (i) $\mathbf{p}_{-f}$ locally maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$, for any $\mathbf{p}_{-f}\in[0,\varsigma]^{J_{-f}}$, if, and only if, $\mathbf{p}_{-f}$ solves the VI (17) $\boldsymbol{\varphi}_{f}(\mathbf{p})^{\top}(\mathbf{p}_{f}-\mathbf{q}_{f})\leq\mathbf{0}\quad\text{for all}\quad\mathbf{q}_{f}\in[0,\varsigma]^{J_{f}}.$ (ii) $\mathbf{p}$ is a local equilibrium if, and only if, $\mathbf{p}$ solves the VI (18) $\boldsymbol{\varphi}(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{q})\leq\mathbf{0}\quad\text{for all}\quad\mathbf{q}\in[0,\varsigma]^{J}.$ Theorem 6.5 also suggests the following general demonstration that it is possible for prices to be equal to $\varsigma$ in equilibrium if unit costs to be constant and differ within firms. Let $c_{j}=\varsigma-\kappa$ be constant unit costs for some $\kappa>0$ and observe that $\varsigma- c_{j}-\hat{\pi}_{f}(\mathbf{p})<0$ if, and only if, $\kappa<\hat{\pi}_{f}(\mathbf{p})$. Because $\hat{\pi}_{f}(\mathbf{p})$ is independent of $c_{j}$ when $p_{j}=\varsigma$, $\kappa$ can be made small enough so that it is less than any lower bound on the optimal profits for firm $f$ excluding product $j$ from their set of offerings. Setting $c_{j}=\varsigma-\kappa$ with such a value of $\kappa$ then ensures that $p_{j}=\varsigma$ is locally profit-optimal for the original problem including product $j$. ### 6.3. Existence of Equilibrium To prove the existence of equilibrium, it remains to show that profit- maximizing prices are unique. While the modified VI (18) can be used to characterize profit-maximizing prices, smooth nonlinear systems are often easier to analyze. Particularly, establishing the uniqueness of profit- maximizing prices with (17) would traditionally require strict monotonicity of $\boldsymbol{\varphi}_{f}$ [23], a property that may be difficult to verify. These obstacles can be overcome by continuously extending the $\boldsymbol{\zeta}$ map, and thus $\boldsymbol{\varphi}$, to all of $[0,\infty)^{J}$ in such a way that solutions of the nonlinear system with the extended $\boldsymbol{\varphi}$ are solutions to the VIs (17) and (18). This enables an existence and uniqueness proofs using the same process applied above. Another approach, enabled by the analysis below, is to apply a VI uniqueness theorem due to Simsek et. al [47, Proposition 5.1] also based on the Poincare-Hopf Theorem. ###### Lemma 6.7. Suppose $\vartheta>-\infty$, Assumption 5.1 holds with $D^{2}C$ finite as $P\downarrow 0$, and 6.1 holds with $w$ that eventually decreases sufficiently quickly, has sub-quadratic second derivatives, and $\lim_{p\uparrow\varsigma}(D^{2}w)(\mathbf{y},p)$ exists. Define the map $\mathbf{z}:[0,\infty)^{J}\to\mathbb{R}^{J}$ componentwise by $z_{k}(\mathbf{p})=\left\\{\begin{aligned} &\sum_{j\in\mathcal{J}_{f(j)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))+\frac{1}{\left\lvert(Dw_{j})(p_{j})\right\rvert}&&\quad\text{if }p_{k}<\varsigma\\\ &\omega_{k}(\varsigma)(p_{k}-\varsigma)+\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))&&\quad\text{if }p_{k}\geq\varsigma\end{aligned}\right.$ and the map $\boldsymbol{\Phi}:[0,\infty)^{J}\to\mathbb{R}^{J}$ by $\boldsymbol{\Phi}(\mathbf{p})=\mathbf{p}-\mathbf{c}(\mathbf{P}(\mathbf{p}))-\mathbf{z}(\mathbf{p})$. (i) $\mathbf{z}$ (or $\boldsymbol{\Phi}$) is a continuously differentiable extension of $\boldsymbol{\zeta}$ (or $\boldsymbol{\varphi}$) from $[0,\varsigma]^{J}$ to $[0,\infty)^{J}$. (ii) For all $j$, $\Phi_{j}(\mathbf{p})<0$ when $\mathbf{p}\geq\mathbf{c}(\mathbf{P}(\mathbf{p}))$ and $p_{j}=c_{j}(P_{j}(\mathbf{p}))$ and there exists $\bar{p}_{j}$ such that $\Phi_{j}(\mathbf{p})>0$ for all $p_{j}>\bar{p}_{j}$, regardless of $\mathbf{p}_{-j}$. (iii) $\boldsymbol{\Phi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})=\mathbf{0}$ if, and only if, $\mathrm{proj}_{[0,\varsigma]}(\mathbf{p}_{f})$ solves the VI (17), where “$\mathrm{proj}_{[0,\varsigma]}$” denotes the projection onto $[0,\varsigma]^{J_{f}}$. (iv) $\boldsymbol{\Phi}(\mathbf{p})=\mathbf{0}$ if, and only if, $\mathrm{proj}_{[0,\varsigma]}(\mathbf{p})$ solves the VI (18), where “$\mathrm{proj}_{[0,\varsigma]}$” denotes the projection onto $[0,\varsigma]^{J}$. ###### Proof. (i): The claim concerning $\mathbf{z}$ and $\boldsymbol{\zeta}$ follow by taking derivatives for any prices in $(0,\varsigma)^{J}$ and taking limits. Specifically, $(D_{l}\zeta_{k})(\mathbf{p})=\omega_{k}(p_{k})\delta_{k,l}+(D_{l}\bar{\pi}_{f})(\mathbf{p})$. Now, $(D_{l}\bar{\pi}_{f})(\mathbf{p})\to 0$ as $p_{l}\uparrow\varsigma$, from which we can deduce the following: * • $(D_{l}\zeta_{k})(\mathbf{p})=(D_{l}z_{k})(\mathbf{p})$ when $k,l\in\mathcal{J}^{\circ}$, * • $(D_{l}\zeta_{k})(\mathbf{p})\to\omega_{k}(\varsigma)\delta_{k,l}=(D_{l}z_{k})(\mathbf{p})$ as $p_{k},p_{l}\uparrow\varsigma$, * • $(D_{l}\zeta_{k})(\mathbf{p})\to 0=(D_{l}z_{k})(\mathbf{p})$ when $p_{k}<\varsigma$ but $p_{l}\uparrow\varsigma$, and * • $(D_{l}\zeta_{k})(\mathbf{p})\to(D_{l}\bar{\pi}_{f(k)})(\mathbf{p})=(D_{l}z_{k})(\mathbf{p})$ when $p_{k}\uparrow\varsigma$ but $p_{l}<\varsigma$. The claim concerning $\boldsymbol{\Phi}$ and $\boldsymbol{\varphi}$ is an obvious consequence, noting that $D_{l}[c_{k}(P_{k}(\mathbf{p}))]\downarrow 0$ as $p_{k}$ or $p_{l}\uparrow\varsigma$, and thus $\mathbf{c}(\mathbf{P}(\mathbf{p}))$ is continuously differentiable on $[0,\infty)^{J}$. (ii): The first part of this claim follows from the corresponding result for $\varphi_{j}$. To prove the second part, note that, by definition, $\Phi_{k}(\mathbf{p})=(1-\omega_{k}(\varsigma))(p_{k}-\varsigma)+\varsigma- c_{k}(0)-\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))$ for all $p_{j}\geq\varsigma$. Because $\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))$ is bounded over $[0,\infty)^{J}$ and $w_{k}$ has finite sub-quadratic second derivatives, $\bar{p}_{k}\geq\varsigma$ can be chosen large enough so that $\Phi_{k}(\mathbf{p})>0$ for all $p_{k}\geq\bar{p}_{k}$, regardless of $\mathbf{p}_{-k}$. (iii) and (iv): We prove (iv), the proof for $(iii)$ being nearly identical. Let $\boldsymbol{\Phi}(\mathbf{p})=\mathbf{0}$. Then $\varphi_{j}(\mathbf{p})=0$ for all $j\in\mathcal{J}^{\circ}$ and $p_{k}-c_{k}(0)-\omega_{k}(\varsigma)(p_{k}-\varsigma)-\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))=0.$ Therefore $\varsigma- c_{k}(0)-\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{p})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{p})))=-(1-\omega_{k})(p_{k}-\varsigma)\leq 0.$ This implies that $\mathbf{p}$ solves the VI (18). Conversely, suppose $\mathbf{q}\in[0,\varsigma]^{J}$ solves the VI (18). Define $p_{k}=q_{k}$ for all $k\in\mathcal{J}^{\circ}$ and $p_{k}=\varsigma-\varphi_{k}(\mathbf{q})/(1-\omega_{k}(\varsigma))$ for all $k\in\mathcal{J}^{*}$. Note that $\Phi_{k}(\mathbf{p})=\varphi_{k}(\mathbf{q})=0$, $p_{k}\geq\varsigma$ for all $k\in\mathcal{J}^{*}$ (because $\varphi_{k}(\mathbf{q})\leq 0$ for all such $k$), and, for all $k\in\mathcal{J}^{*}$, $\displaystyle\Phi_{k}(\mathbf{p})$ $\displaystyle=p_{k}-c_{k}(0)-\omega_{k}(\varsigma)(p_{k}-\varsigma)-\sum_{j\in\mathcal{J}_{f(k)}^{\circ}}P_{j}^{L}(\mathbf{q})(p_{j}-c_{j}(P_{j}^{L}(\mathbf{q})))$ $\displaystyle=(1-\omega_{k}(\varsigma))(p_{k}-\varsigma)+\varphi_{k}(\mathbf{q})=0.$ Thus $\mathbf{q}=\mathrm{proj}_{[0,\varsigma]}(\mathbf{p})$ and $\boldsymbol{\Phi}(\mathbf{p})=\mathbf{0}$. ∎ The obvious corollary is as follows: ###### Corollary 6.8. Assume the hypotheses of Lemma 6.7, and let $\mathbf{p}\in[0,\infty)^{J}$. (i) $\mathrm{proj}_{[0,\varsigma]}(\mathbf{p}_{f})$ locally maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$ if, and only if, $\boldsymbol{\Phi}_{f}(\mathbf{p})=\mathbf{0}$. (ii) $\mathrm{proj}_{[0,\varsigma]}(\mathbf{p})$ is a local equilibrium if, and only if, $\boldsymbol{\Phi}(\mathbf{p})=\mathbf{0}$. An adaptation of the techniques in Sections 4 and 5 again establishes the uniqueness of profit-maximizing prices. ###### Lemma 6.9. Assume the hypotheses of Lemma 6.7. For any $\mathbf{p}_{-f}$, the nonlinear system $\boldsymbol{\Phi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})=\mathbf{0}$ has a unique solution $\mathbf{p}_{f}\in[0,\infty)^{J_{f}}$ satisfying $c_{j}(P_{j}(\mathbf{p}))<p_{j}$ for all $j\in\mathcal{J}_{f}$ with $p_{j}<\varsigma$ for at least one $j\in\mathcal{J}_{f}$. ###### Proof. For any $\mathbf{p}_{-f}$, $\boldsymbol{\Phi}_{f}(\cdot,\mathbf{p}_{-f})$ has a zero in the interior of $\\{\mathbf{p}_{f}:\mathbf{p}_{f}\geq\mathbf{c}_{f}(\mathbf{P}_{f}(\mathbf{p}))\\}$; the proof is exactly analogous to the proof of existence in the case $\varsigma=\infty$: We first show that the homotopy $\boldsymbol{\rho}_{f}$ between $[0,\infty)^{J_{f}}$ and $\\{\mathbf{p}_{f}:\mathbf{p}_{f}\geq\mathbf{c}_{f}(\mathbf{P}_{f}(\mathbf{p}))\\}$ still exists. The inverse map $\mathbf{p}_{f}\mapsto\boldsymbol{\epsilon}_{f}=\mathbf{p}_{f}-\mathbf{c}_{f}(\mathbf{P}_{f}(\mathbf{p}))$ is again well-defined and continuous, trivially. $\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f})$ is ostensibly defined by $\mathbf{p}_{f}=\mathbf{c}_{f}(\mathbf{P}_{f}^{L}(\mathbf{p}))+\boldsymbol{\epsilon}_{f}$. For $\epsilon_{k}=\varsigma-c_{k}(0)+\delta$, $\delta\geq 0$, $\rho_{k}(\boldsymbol{\epsilon})=\varsigma+\delta$ solves this fixed-point equation regardless of $\epsilon_{j}$, $j\neq k$: $p_{k}=\varsigma+\delta=c_{k}(0)+(\varsigma- c_{k}(0)+\delta)=c_{k}(P_{k}(\mathbf{p}))+\epsilon_{k}$ Note that $(D_{k}^{\epsilon}\rho_{k})(\boldsymbol{\epsilon}_{f})=\delta_{k,l}$ for $\epsilon_{k}>\varsigma$. Supposing $\mathbf{p}_{f}<\varsigma\mathbf{1}$, $D_{f}\Big{[}\mathbf{c}_{f}(\mathbf{P}_{f}^{L}(\mathbf{p}))+\boldsymbol{\epsilon}_{f}\Big{]}=(D^{2}\mathbf{C}_{f})(\mathbf{P}_{f}^{L}(\mathbf{p}))\boldsymbol{\Lambda}_{f}(\mathbf{p})-(D^{2}\mathbf{C}_{f})(\mathbf{P}_{f}^{L}(\mathbf{p}))\mathbf{P}_{f}^{L}(\mathbf{p})\boldsymbol{\lambda}_{f}(\mathbf{p})^{\top}.$ As $p_{k}\uparrow\varsigma$, the $k^{\text{th}}$ row and $k^{\text{th}}$ column of this matrix vanish because $\lim_{P\downarrow 0}(D^{2}C_{j})(P)<\infty$. In particular, the proof that the spectrum of the fixed-point map does not contain 1 given before holds on $[0,\infty)^{J_{f}}$. Constructing an upper bound on the magnitude of the fixed point for any $\boldsymbol{\epsilon}$ then proves the fixed point is unique, again by Kellogg’s uniqueness theorem. The vanishing of the derivatives also proves that $(D_{l}\rho_{k})(\boldsymbol{\epsilon}_{f})\to\delta_{k,l}$ as either $p_{k}$ or $p_{l}$ $\uparrow\varsigma$, $k,l\in\mathcal{J}_{f}$, and thus $\boldsymbol{\rho}_{f}$ is continuously differentiable on $[0,\infty)^{J_{f}}$. $\boldsymbol{\rho}_{f}$ must then be continuous on $[0,\infty)^{J_{f}}$. As before, consider the vector field $\boldsymbol{\Psi}_{f}=\boldsymbol{\Phi}_{f}\circ\boldsymbol{\rho}_{f}:[0,\infty)^{J_{f}}\to\mathbb{R}^{J_{f}}$. This vector field points outward on the boundary of $[\mathbf{0},\bar{\mathbf{p}}_{f}]$, where the existence of $\bar{\mathbf{p}}_{f}$ was established in Lemma 6.7. By the Poincare-Hopf Theorem, $\boldsymbol{\Psi}_{f}$ has a zero $\boldsymbol{\epsilon}_{f}\in(\mathbf{0},\bar{\mathbf{p}}_{f})$, which is uniquely related to a zero $\mathbf{p}_{f}=\boldsymbol{\rho}_{f}(\boldsymbol{\epsilon}_{f})$ satisfying $c_{j}(P_{j}(\mathbf{p}))<p_{j}$ for all $j\in\mathcal{J}_{f}$. Note that this zero cannot have $p_{j}\geq\varsigma$ for all $j\in\mathcal{J}_{f}$: For if that were true, then $\varsigma\leq\left(\frac{1}{1-\omega_{j}(\varsigma)}\right)c_{j}(0)-\left(\frac{\omega_{j}(\varsigma)}{1-\omega_{j}(\varsigma)}\right)\varsigma=-\left(\frac{1}{1-\omega_{j}(\varsigma)}\right)(\varsigma- c_{j}(0))+\varsigma<\varsigma.$ By contradiction, there must exist some $j\in\mathcal{J}_{f}$ such that $p_{j}<\varsigma$. Uniqueness of profit-maximizing prices follows from a very similar approach to that used previously. Here we show that any zero of $\boldsymbol{\Phi}_{f}$ has index one, and the rest of the proof proceeds exactly the same way. Note that * • $(D_{l}\Phi_{k})(\mathbf{p})=(D_{l}\varphi_{k})(\mathbf{p})$ when $k,l\in\mathcal{J}_{f}^{\circ}$, * • $(D_{l}\Phi_{k})(\mathbf{p})=(1-\omega_{k}(\varsigma))\delta_{k,l}$ when $k,l\in\mathcal{J}_{f}^{*}$ * • $(D_{l}\Phi_{k})(\mathbf{p})=0$ when $k\in\mathcal{J}_{f}^{\circ}$ but $l\in\mathcal{J}_{f}^{*}$ These relations imply that there exists a symmetric permutation $\mathbf{T}_{f}$ to make $(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})$ block- triangular: $(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})=\mathbf{T}_{f}\begin{bmatrix}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})&\mathbf{0}\\\ \mathbf{A}&\mathbf{I}-\boldsymbol{\Omega}_{f}^{*}(\varsigma)\end{bmatrix}\mathbf{T}_{f}$ where $\mathbf{A}$ is some matrix. Thus $\det(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})=(\det\mathbf{T}_{f})^{2}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}\det\Big{(}\mathbf{I}-\boldsymbol{\Omega}_{f}^{*}(\varsigma)\Big{)}$ and, because $1-\omega_{k}(\varsigma)>0$, $\mathrm{sign}\det(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})=\mathrm{sign}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}.$ But $\boldsymbol{\varphi}_{f}^{\circ}$ is identical to the $\boldsymbol{\varphi}_{f}$ map if we exclude the products $\mathcal{J}_{f}^{*}$ from $\mathcal{J}_{f}$, and $\mathbf{p}_{f}^{\circ}$ are profit-maximizing prices strictly in the interior of $[0,\varsigma]^{J_{f}^{\circ}}$ where $J_{f}^{\circ}=|\mathcal{J}_{f}^{\circ}|$. Then $(-1)^{J_{f}^{\circ}}=\mathrm{sign}\det(D_{f}^{\circ}\hat{\pi}_{f}^{\circ})(\mathbf{p})=\mathrm{sign}\det\boldsymbol{\Lambda}_{f}^{\circ}(\mathbf{p})\mathrm{sign}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}=(-1)^{J_{f}^{\circ}}\mathrm{sign}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}$ implies $\mathrm{sign}\det(D_{f}\boldsymbol{\Phi}_{f})(\mathbf{p})=\mathrm{sign}\det\Big{(}(D_{f}^{\circ}\boldsymbol{\varphi}_{f}^{\circ})(\mathbf{p})\Big{)}=1.$ ∎ This result can also be proved using the uniqueness theorem for VI’s given by Simsek et. al [47, Proposition 5.1]. ## 7\. Properties of Equilibrium Prices This section establishes properties that the finite prices and markups of any equilibrium must satisfy based on properties of $\boldsymbol{\zeta}$. The most general result is Corollary 7.1, which states that the difference between profit optimal markups for two products offered by the same firm with prices less than $\varsigma$ depends only on the prices and characteristics of those two products when unit costs are constant. This property is very similar to the embodiment of the “Independence of Irrelevant Alternatives” (IIA) property in Logit models: the ratio of choice probabilities depends only on the characteristics and prices of those two products [50]. In Corollaries 7.5 through 7.8 this result is applied to concave-in-price utility functions under hypotheses on the unit cost and value functions to illuminate some counterintuitive properties of equilibrium prices under Logit. This is the only section of the article that focuses somewhat on concave-in-price utility functions. ### 7.1. Properties of Profit-Optimal Prices For this subsection, we focus on a single firm $f\in\mathbb{N}(F)$ and derive our results as properties of locally profit-optimal prices. Naturally, these properties will be manifest in locally equilibrium prices as well. This section is also the only portion of this article in which we focus heavily on concave in price utilities, which will satisfy our existence conditions. The basic observation is as follows. ###### Corollary 7.1. Suppose Assumptions 5.1 and 6.1 hold. Let $\mathbf{p}_{f}$ be profit maximizing. For any $j,k\in\mathcal{J}_{f}^{\circ}$, (19) $\Big{(}p_{j}-c_{j}(P_{j}(\mathbf{p}))\Big{)}-\Big{(}p_{k}-c_{k}(P_{k}(\mathbf{p}))\Big{)}=-\left(\frac{1}{(Dw_{j})(p_{j})}-\frac{1}{(Dw_{k})(p_{k})}\right).$ That is, the difference between profit optimal markups for any two products offered by a single firm depends only on the corresponding utility derivatives. If unit costs are constant, this implies that the difference between profit optimal markups for any two products offered by a single firm depends only on the characteristics and prices of those products. ###### Proof. Eqn. (19) follows immediately from the fixed-point equation $p_{j}=c_{j}(P_{j}(\mathbf{p}))+\bar{\pi}_{f}(\mathbf{p})+\left\lvert(Dw_{j})(p_{j})\right\rvert^{-1}$ for all $j\in\mathcal{J}_{f}^{\circ}$. ∎ One application is motivated by the frequent application of constant coefficient linear in price utility functions. ###### Corollary 7.2. Suppose Assumption 5.1 holds. If $w(\mathbf{y},p)\equiv-\alpha p$ for some $\alpha>0$, $p_{j}-c_{j}(P_{j}(\mathbf{p}))=p_{k}-c_{k}(P_{k}(\mathbf{p}))$ for all $j,k\in\mathcal{J}_{f}$ when $\mathbf{p}_{f}$ is profit-maximizing. In other words, profit-optimal markups are constant regardless of product costs or the value of product characteristics. Constant intra-firm markups have appeared as an assumption [44, 16], but not often proven to be an equilibrium outcome. The following example motivates the more general propositions on profit- optimal markups given below. Consider the quadratic in price utility $w(\mathbf{y},p)\equiv w(p)=-\alpha p^{2}$ and constant unit costs. Then $(p_{j}-c_{j})-(p_{k}-c_{k})=\left(\frac{1}{2\alpha}\right)\left(\frac{1}{p_{j}}-\frac{1}{p_{k}}\right),$ demonstrating that locally profit optimal markups decrease with the corresponding prices (i.e., $p_{j}-c_{j}>p_{k}-c_{k}$ if and only if $p_{j}<p_{k}$). Rearranging and setting $\lambda=1/(2\alpha)$, we obtain $\left(p_{j}-\frac{\lambda}{p_{j}}\right)-\left(p_{k}-\frac{\lambda}{p_{k}}\right)=c_{j}-c_{k}.$ The function $\eta_{\lambda}(p)=p-\lambda/p$ is strictly increasing in $p$ for non-negative $\lambda$, and thus $c_{j}>c_{k}$ implies $p_{j}>p_{k}$. Thus, locally profit optimal prices increase with costs while the corresponding markups decrease with costs. Additionally, we note that if $c_{j}=c_{k}$ then $p_{j}=p_{k}$, even if $\mathbf{y}_{j}\neq\mathbf{y}_{k}$; that is, profit- optimal prices reflect only product costs, not value. We first generalize the counterintuitive property that differences in characteristics that do not impact costs or (local) willingness to pay do not impact prices, even if they impact product value. ###### Corollary 7.3. Suppose unit costs are constant, 6.1 holds and $w$ has sub-quadratic second derivatives. Let $\mathbf{p}_{f}$ be profit-maximizing, and suppose that $c_{j}=c_{k}$ and $(Dw_{j})(p)=(Dw_{k})(p)$ for all $p\in(0,\varsigma)$ for some $j,k\in\mathcal{J}_{f}^{\circ}$, even if $\mathbf{y}_{j}\neq\mathbf{y}_{k}$. Then $p_{j}=p_{k}$. In other words, for any separable utility with sub-quadratic second derivatives, profit-optimal prices are determined by costs, not value. One would expect that real firms would not follow this rule, charging higher markups for the more valued product. ###### Proof. Corollary 7.1 implies $p_{j}-\frac{1}{\left\lvert(Dw_{j})(p_{j})\right\rvert}=\theta(p_{j})=\theta(p_{k})=p_{k}-\frac{1}{\left\lvert(Dw_{k})(p_{j})\right\rvert}.$ Because the map $\theta(p)=p-1/\left\lvert(Dw_{j})(p)\right\rvert=p-1/\left\lvert(Dw_{k})(p)\right\rvert$ is strictly increasing when $w$ has sub-quadratic second derivatives, $p_{j}=p_{k}$. ∎ This proposition, as stated, must be restricted to constant unit costs. A weaker result applies for non-constant unit costs: ###### Corollary 7.4. Assume Assumption 5.1 holds, unit costs are strictly convex, 6.1 holds, $w$ has sub-quadratic second derivatives. Let $\mathbf{p}_{f}$ be profit-optimal, and suppose that $c_{j}(P)=c_{k}(P)=c(P)$ for all $P\in[0,1]$ and $(Dw_{j})(p)=(Dw_{k})(p)$ for all $p\in[0,\varsigma)$ for some $j,k\in\mathcal{J}_{f}^{\circ}$, even if $\mathbf{y}_{j}\neq\mathbf{y}_{k}$. Then $p_{j}>p_{k}$ if, and only if, $P_{j}(\mathbf{p})>P_{k}(\mathbf{p})$, $p_{j}=p_{k}$ if, and only if, $P_{j}(\mathbf{p})=P_{k}(\mathbf{p})$, and $p_{j}<p_{k}$ if, and only if, $P_{j}(\mathbf{p})<P_{k}(\mathbf{p})$. This result may also be seen as slightly counterintuitive, as higher-priced products are intuitively associated with lower choice probabilities. ###### Proof. Corollary 7.1 implies $\theta(p_{j})-\theta(p_{k})=c(P_{j}(\mathbf{p}))-c(P_{k}(\mathbf{p}))$ Because total costs are strictly convex, unit costs are strictly increasing in $P$. Thus, $\displaystyle\begin{matrix}p_{j}>p_{k}&\iff&\theta(p_{j})>\theta(p_{k})&\iff&c(P_{j}(\mathbf{p}))>c(P_{k}(\mathbf{p}))&\iff&P_{j}(\mathbf{p})>P_{k}(\mathbf{p})\\\ p_{j}=p_{k}&\iff&\theta(p_{j})=\theta(p_{k})&\iff&c(P_{j}(\mathbf{p}))=c(P_{k}(\mathbf{p}))&\iff&P_{j}(\mathbf{p})=P_{k}(\mathbf{p})\\\ p_{j}<p_{k}&\iff&\theta(p_{j})<\theta(p_{k})&\iff&c(P_{j}(\mathbf{p}))<c(P_{k}(\mathbf{p}))&\iff&P_{j}(\mathbf{p})<P_{k}(\mathbf{p})\end{matrix}.$ ∎ Corollary 7.1 also implies the second counterintuitive property of locally profit optimal markups $-$ that they decrease with costs $-$ under Logit with any utility function that is both strictly concave in price and separable in price and characteristics. ###### Corollary 7.5. Suppose that $w$ is separable in price and characteristics and strictly concave in price. Then firm $f$’s higher unit cost products (at optimality) have lower locally profit optimal markups. That is, if $j,k\in\mathcal{J}_{f}^{\circ}$ and $c_{j}(P_{j}(\mathbf{p}))>c_{k}(P_{k}(\mathbf{p}))$, then $p_{j}-c_{j}(P_{j}(\mathbf{p}))<p_{k}-c_{k}(P_{k}(\mathbf{p}))$. ###### Proof. We prove that $p_{j}-c_{j}(P_{j}(\mathbf{p}))\geq p_{k}-c_{k}(P_{k}(\mathbf{p}))$ implies $c_{j}(P_{j}(\mathbf{p}))\leq c_{k}(P_{k}(\mathbf{p}))$. By Corollary 7.1, $p_{j}-c_{j}(P_{k}(\mathbf{p}))\geq p_{k}-c_{k}(P_{k}(\mathbf{p}))$ implies $(Dw)(p_{j})^{-1}\leq(Dw)(p_{k})^{-1}$ or, equivalently, $(Dw)(p_{j})\geq(Dw)(p_{k})$. By strict concavity, this implies that $p_{j}\leq p_{k}$. But then $p_{j}-c_{j}(P_{j}(\mathbf{p}))\geq p_{k}-c_{k}(P_{k}(\mathbf{p}))$ implies that $c_{j}(P_{j}(\mathbf{p}))-c_{k}(P_{k}(\mathbf{p}))\leq p_{j}-p_{k}\leq 0$ ∎ When unit costs are constant, these propositions can be easily connected to value. Intuition holds that both locally profit optimal markups and costs should increase with value, if not costs. The following assumption makes this connection explicit. ###### Assumption 7.1 (Value Costs Hypothesis). More valued products cost more per unit to offer; that is, $v(\mathbf{y})>v(\mathbf{y}^{\prime})$ implies that $c(\mathbf{y})>c(\mathbf{y}^{\prime})$ for all $\mathbf{y},\mathbf{y}^{\prime}\in\mathcal{Y}$. Mussa & Rosen [35] include this as a basic feature of cost functions. Bresnahan [14] has also remarked that this is a natural condition. When considering equilibrium prices, this assumption need only be applied within firms. That is, there may be firm-specific cost functions each independently satisfying the value costs hypothesis, while the value costs hypothesis is violated across firms. This states that two distinct firms can produce a value-equivalent product at distinct unit costs without violating the results that apply this hypothesis. With this definition, we provide the following restatement of Corollary 7.5. ###### Corollary 7.6. Suppose unit costs are constant, Assumption 6.1 holds, $w$ is separable in price and characteristics, strictly concave in price, and that the value costs hypothesis holds. Then firm $f$’s higher value products have lower locally profit optimal markups. That is, if $j,k\in\mathcal{J}_{f}$ and $v_{j}>v_{k}$, then $p_{j}-c_{j}<p_{k}-c_{k}$. ###### Proof. $v_{j}>v_{k}$ implies $c_{j}>c_{k}$, and the result follows from Corollary 7.5. ∎ Markups can increase with value when $w$ is convex in price. Consider $w(\mathbf{y},p)\equiv w(p)=-\alpha\log p$ and constant unit costs, for which $(p_{j}-c_{j})-(p_{k}-c_{k})=\left(\frac{1}{\alpha}\right)\left(p_{j}-p_{k}\right).$ Thus, locally profit optimal markups increase with the corresponding prices. This implies $\displaystyle p_{j}-p_{k}$ $\displaystyle=\left(\frac{1}{\alpha-1}\right)(c_{j}-c_{k}).$ Hence if $\alpha>1$, locally profit optimal prices increase with costs, and locally profit optimal markups increase with costs. While this is a more intuitive outcome, it comes from a less intuitive utility specification. Another assumption, the “unique value hypothesis,” further connects value with profit-optimal prices. As defined by Nagle [36], the unique value hypothesis postulates that as a product’s combination of characteristics becomes more valued, individuals are less sensitive to price changes. This is transcribed to our framework as follows. ###### Assumption 7.2 (Unique Value Hypothesis). For any $\mathbf{y},\mathbf{y}^{\prime}\in\mathcal{Y}$, $v(\mathbf{y})>v(\mathbf{y}^{\prime})$ implies $\left\lvert(Dw)(\mathbf{y},p)\right\rvert\leq\left\lvert(Dw)(\mathbf{y}^{\prime},p)\right\rvert\quad\text{ for all }\quad p\in(0,\varsigma).$ This definition suggests an example of a non-separable but convex in price utility for which markups increase with value. Consider $w(\mathbf{y},p)=-\alpha(\mathbf{y})p$, where $\alpha:\mathcal{Y}\to(0,\infty)$, and assume unit costs are constant. Then $(p_{j}-c(\mathbf{y}_{j}))-(p_{k}-c(\mathbf{y}_{k}))=\left(\frac{1}{\alpha(\mathbf{y}_{j})}-\frac{1}{\alpha(\mathbf{y}_{k})}\right),$ and $(p_{j}-c(\mathbf{y}_{j}))\geq(p_{k}-c(\mathbf{y}_{k}))$ if and only if $\alpha(\mathbf{y}_{j})\leq\alpha(\mathbf{y}_{k})$. The unique value hypothesis mandates that $v(\mathbf{y}_{j})>v(\mathbf{y}_{k})$ implies $\alpha(\mathbf{y}_{j})\leq\alpha(\mathbf{y}_{k})$, and hence markups do not decrease with value if this hypothesis holds. Note that this is consistent with our previous result for constant coefficient linear in price utility where $\alpha(\mathbf{y})\equiv\alpha\in(0,\infty)$. Whenever $\alpha(\mathbf{y}_{j})<\alpha(\mathbf{y}_{k})$, that is whenever the unique value hypothesis holds in a non-trivial way, markups can strictly increase with value. A related and important question is whether higher value products have higher locally profit optimal prices. By Corollary 7.4, this cannot hold without an additional hypothesis. ###### Corollary 7.7. Suppose unit costs are constant, Assumption 6.1 holds, $w$ has sub-quadratic second derivatives, satisfies the unique value hypothesis, and the value costs hypothesis holds. Then firm $f$’s higher value products have higher locally profit optimal prices. That is, for any $j,k\in\mathcal{J}_{f}^{\circ}$, $v_{j}>v_{k}$ implies that $p_{j}>p_{k}$ when $\mathbf{p}_{f}$ are profit- maximizing. ###### Proof. The unique value hypothesis implies that when $v(\mathbf{y})>v(\mathbf{y}^{\prime})$, $\theta(\mathbf{y},p)\leq\theta(\mathbf{y}^{\prime},p)$ for all $p\in(0,\varsigma)$, where $\theta(\mathbf{y},p)=p-\left\lvert(Dw)(\mathbf{y},p)\right\rvert^{-1}$. Specifically, if $v(\mathbf{y}_{j})>v(\mathbf{y}_{k})$ then $\theta_{j}(p_{k})\leq\theta_{k}(p_{k})$. Suppose that $v(\mathbf{y}_{j})>v(\mathbf{y}_{k})$ while $p_{j}\leq p_{k}$. Because $\theta_{j}(p)$ is a strictly increasing function of $p$, we have $\theta_{j}(p_{j})\leq\theta_{j}(p_{k})\leq\theta_{k}(p_{k}).$ Thus Eqn. (19) implies that $c_{j}-c_{k}=\theta_{j}(p_{j})-\theta_{k}(p_{k})\leq 0$, in contradiction to the value costs hypothesis. ∎ Because any separable utility trivially satisfies the unique value hypotheses, the following is a direct consequence of Corollary 7.7. ###### Corollary 7.8. Suppose unit costs are constant, Assumption 6.1 holds, $w$ is separable in price and characteristics, has sub-quadratic second derivatives, and that the value costs hypothesis holds. Then firm $f$’s higher value products have higher locally profit optimal prices. That is, $v_{j}>v_{k}$ implies that $p_{j}>p_{k}$ for any $j,k\in\mathcal{J}_{f}^{\circ}$. ### 7.2. An Inter-Firm Property of Equilibrium Prices Eqn. (19) is a special case of the following: ###### Corollary 7.9. Suppose Assumptions 5.1 and 6.1 hold, and let $\mathbf{p}\in(0,\infty)^{J}$ be equilibrium prices. For any $f,g\in\mathbb{N}(F)$, $j\in\mathcal{J}_{f}^{\circ}$, and $k\in\mathcal{J}_{g}^{\circ}$, (20) $(p_{j}-c_{j}(P_{j}(\mathbf{p})))-(p_{k}-c_{k}(P_{k}(\mathbf{p})))=\left(\bar{\pi}_{f}(\mathbf{p})-\frac{1}{(Dw_{j})(p_{j})}\right)-\left(\bar{\pi}_{g}(\mathbf{p})-\frac{1}{(Dw_{k})(p_{k})}\right).$ This equation expresses the existence of a portfolio effect present in equilibrium pricing with multi-product firms, constant unit costs, and even the simplest Logit model. ###### Corollary 7.10. Assume unit costs are constant, Assumption 6.1 holds, $w$ have sub-quadratic second derivatives, and $\mathbf{p}\in(0,\infty)^{J}$ are equilibrium prices. Suppose that $\mathbf{y}_{j}=\mathbf{y}_{k}$ and $c_{f}(\mathbf{y}_{j})=c_{g}(\mathbf{y}_{k})$ for some $j\in\mathcal{J}_{f}^{\circ}$ and $k\in\mathcal{J}_{g}^{\circ}$. Then $p_{j}>p_{k}$ if, and only if, $\hat{\pi}_{f}(\mathbf{p})>\hat{\pi}_{g}(\mathbf{p})$, $p_{j}<p_{k}$ if, and only if, $\hat{\pi}_{f}(\mathbf{p})<\hat{\pi}_{g}(\mathbf{p})$, and $p_{j}=p_{k}$ if, and only if, $\hat{\pi}_{f}(\mathbf{p})=\hat{\pi}_{g}(\mathbf{p})$. That is, equilibrium prices for the same product offered at the same cost but by different firms are influenced by the profitability of other products in these firms’ portfolios. Stated another way, if the other products offered by a particular firm did not matter in determining equilibrium prices, then we would expect $\mathbf{y}_{j}=\mathbf{y}_{k}$ and $c_{f}(\mathbf{y}_{j})=c_{g}(\mathbf{y}_{k})$ for some $j\in\mathcal{J}_{f}$ and $k\in\mathcal{J}_{g}$ to imply that $p_{j}=p_{k}$. ###### Proof. The proof follows by observing that Eqn. (20) can be written $\theta_{j}(p_{j})-\theta_{k}(p_{k})=\hat{\pi}_{f}(\mathbf{p})-\hat{\pi}_{g}(\mathbf{p})$, and $\theta_{j}=\theta_{k}$. The result follows. ∎ ## 8\. Conclusions This article has proved the existence of equilibrium prices for Bertrand competition with multi-product firms under the Logit model without restrictive assumptions on the firms or their products. Instead of studying a particular utility function, general conditions on the utility function are identified under which existence holds. The proofs circumvents fundamental obstacles to the extension of existing equilibrium proofs for single-product firms by applying the Poincare-Hopf theorem. One of the fixed-point equations explicitly demonstrates that Logit price equilibrium problems are “single- parameter problems” when unit costs are constant, even when firms offer many products. By invoking the conventional assumption that utility is concave in price and separable in price and characteristics along with the reasonable assumption that more valued products always cost more to make per unit, a counterintuitive result is obtained: the more the population values a product’s characteristics, the lower its profit-optimal markup. There are at least two important areas for future research. One is establishing the uniqueness of equilibrium prices. Kellogg’s uniqueness condition for Brouwer-Schauder fixed-point theorem [26], used in Section 5, can be applied to show that equilibrium prices under linear-in-price utility Logit models are unique, a result already known for both single-product [28, 15] and multi-product [45, 27] firms. Generalizing this analysis to nonlinear utility functions and non-constant costs may be a promising direction. As suggested in the introduction, another important area is the extension of this analysis to non-Logit RUMs, especially those with heterogeneity. Formally the $\boldsymbol{\eta}$ and $\boldsymbol{\zeta}$ characterizations presented in this article extend to both any GEV and Mixed Logit models; see [31, 33, 32] for the extension to Mixed-Logit models, subsequent analysis, and application in large-scale computations of equilibrium prices. Establishing the existence of simultaneously stationary prices using these characterizations is straightforward, but not enough to ensure the existence of equilibrium [32]. ## Appendix A Mathematical Notation Sets. $\mathbb{N}$ denotes the natural numbers $\\{1,2,\dotsc\\}$, and $\mathbb{N}(N)$ denotes the natural numbers up to $N$, that is, $\mathbb{N}(N)=\\{1,\dotsc,N\\}$. $\mathbb{R}$ denotes the set of real numbers $(-\infty,\infty)$, $[0,\infty)$ denotes the non-negative real numbers, and $[0,\infty]$ denotes the extended non-negative half-line. We denote the $(J-1)$-dimensional simplex $\\{(x_{1},\dotsc,x_{N})\in[0,1]^{N}:\sum_{n=1}^{N}x_{n}=1\\}$ by $\mathbb{S}(N)$, and the $J$-dimensional “pyramid” $\\{(x_{1},\dotsc,x_{N})\in[0,1]^{N}:\sum_{n=1}^{N}x_{n}\leq 1\\}$ by $\triangle(J)$. Hyper-rectangles in $\mathbb{R}^{N}$, i.e. sets of the form $[a_{1},b_{1}]\times\dotsb\times[a_{N},b_{N}]$ for some $a_{n},b_{n}\in\mathbb{R}$ with $a_{n}<b_{n}$ for all $n\in\mathbb{N}(N)$, are denoted by $[\mathbf{a},\mathbf{b}]$ where $\mathbf{a}=(a_{1},\dotsc,a_{N})$ and $\mathbf{b}=(b_{1},\dotsc,b_{N})$. For other sets, we typically use calligraphic upper case letters such as “$\mathcal{A}$”. For any set $\mathcal{A}$, $\left\lvert\mathcal{A}\right\rvert$ denotes its cardinality. For any $\mathcal{B}\subset\mathcal{A}$, $\mathcal{A}\setminus\mathcal{B}$ denotes the set $\\{b\in\mathcal{A}:b\notin\mathcal{B}\\}$. For any set $\mathcal{A}$, $\mathfrak{F}(\mathcal{A})$ denotes the collection of finite subsets of $\mathcal{A}$. Symbols. Bold, un-italicized symbols (e.g., “$\mathbf{x}$”) denote vectors and matrices; typically we reserve lower case letters to refer to vectors and use upper case letters to refer to matrices; the vector of choice probabilities “$\mathbf{P}$” is an exception made to conform with existing notation of these quantities. Throughout we use $\mathbf{1}$ to denote a vector of ones of the appropriate size for the context in which it appears. $\mathbf{I}$ always denotes the identity matrix of a size appropriate for the context. For any $\mathbf{x}\in\mathbb{R}^{N}$, $\mathrm{diag}(\mathbf{x})$ denotes the $N\times N$ diagonal matrix whose diagonal is $\mathbf{x}$. Any vector inequalities between vectors are to be taken componentwise: for example, $\mathbf{x}<\mathbf{y}$ means $x_{n}<y_{n}$ for all $n$. Random variables are denoted with capital letters “$X$”, with random vectors being denoted with bold capital letters (e.g., “$\mathbf{Q}$”). While this overlaps with our notation for matrices, it should not cause any confusion. $\mathbb{P}$ denotes a probability and $\mathbb{E}$ denotes an expectation. “$\log$” always denotes the natural (base $e$) logarithm. “$\mathrm{ess}\sup$” denotes the essential supremum of a measurable function, where the measure on measurable subsets of the domain should always be clear. Differentiation. Our conventions for denoting differentiation follow [34]. We use the symbol “$D$” to denote differentiation using subscripts to invoke additional specificity. Letting $\mathbf{f}:\mathbb{R}^{M}\to\mathbb{R}^{N}$, $(D_{m}f_{n})(\mathbf{x})$ denotes the derivative of the $n^{\text{th}}$ component function with respect to the $m^{\text{th}}$ variable and $(D\mathbf{f})(\mathbf{x})$ is the $N\times M$ derivative matrix of $\mathbf{f}$ at $\mathbf{x}$ with components $((D\mathbf{f})(\mathbf{x}))_{n,m}=(D_{m}f_{n})(\mathbf{x})$. Thus for $f:\mathbb{R}^{M}\to\mathbb{R}$, $(Df)(\mathbf{x})$ is a row vector. If $f:\mathbb{R}^{M}\to\mathbb{R}$, we define the gradient $(\nabla f)(\mathbf{x})\in\mathbb{R}^{M}$ as the transposed derivative: $(\nabla f)(\mathbf{x})=(Df)(\mathbf{x})^{\top}$. Other Definitions. Let $\mathcal{X}$ be any topological space and let $f:\mathcal{X}\to\mathbb{R}$. We say $\mathbf{x}^{*}\in\mathcal{X}$ is a local maximizer (over $\mathcal{X}$) of $f$ if there exists a neighborhood of $\mathbf{x}^{*}$, say $\mathcal{U}$, such that $f(\mathbf{x}^{*})\geq f(\mathbf{x})$ for all $\mathbf{x}\in\mathcal{U}$. We say $\mathbf{x}^{*}\in\mathcal{X}$ is a maximizer (over $\mathcal{X}$) of $f$ if $f(\mathbf{x}^{*})\geq f(\mathbf{x})$ for all $\mathbf{x}\in\mathcal{X}$. ## Appendix B Examples for the Logit Model We first provide some examples of indirect utilities to illustrate properties (a-c). A linear in price utility, given by $w(\mathbf{y},p)=-\alpha(\mathbf{y})p$ for some $\alpha:\mathcal{Y}\to(0,\infty)$, satisfies (a-c). More generally, any “Cobb- Douglas” in price utility, given by $w(\mathbf{y},p)=-\alpha(\mathbf{y})p^{\beta(\mathbf{y})}$ with $\alpha,\beta:\mathcal{Y}\to(0,\infty)$, satisfies (a-c). A “Cobb-Douglas” specification for “remaining income,” $w(\mathbf{y},p)=\alpha(\mathbf{y})(\varsigma-p)^{\beta(\mathbf{y})}$ is a bit more complicated, being a function finite for all finite prices and satisfying (a-c) only for $\beta:\mathcal{Y}\to(2\mathbb{N}+1)$, where $2\mathbb{N}+1$ denotes the set of odd positive integers: if $\beta(\mathbf{y}):\mathcal{Y}\to(-\infty,0)$ then $w$ is not finite for all finite $p$; clearly $w$ violates (a) if $\beta(\mathbf{y})=0$; if $\beta(\mathbf{y})>0$ is not an integer, then $w$ is complex for $p>\varsigma$; finally, if $\beta(\mathbf{y})\in\mathbb{N}$ is not an odd positive integer then $w$ violates (a). The common “log-transformed” Cobb- Douglas in “remaining income” utility $w(\mathbf{y},p)=\alpha(\mathbf{y})\log(\varsigma-p)$ for $p<\varsigma<\infty$, $\alpha:\mathcal{Y}\to(0,\infty)$,999This log transformation usually occurs (see [10], [44]) based on the observation that choices are invariant over increasing utility transformations, so that $u^{\prime}(\mathbf{y},p)=e^{w(\mathbf{y},p)}e^{v(\mathbf{y})}$ yields the same random choices as the specification introduced in the text, with the caveat that the additive errors introduced in the text are taken as multiplicative errors (with a related distribution) in the former specification. In a Cobb-Douglas specification for the former, $u^{\prime}(\mathbf{y},p)\propto(\varsigma-p)^{\alpha(\mathbf{y})}=e^{\alpha(\mathbf{y})\log(\varsigma-p)}$, illustrating that the logarithm of this utility has the log-transformed specification for the price component. is not finite for all finite prices. Allenby & Rossi’s negative log of price utility, given by $w(\mathbf{y},p)=-\alpha(\mathbf{y})\log p$ for $\alpha:\mathcal{Y}\to(0,\infty)$ satisfies (a-c) [2]. Finally, the utility $w(p)=-\alpha(\log p-\varepsilon\sin\log p)$, where $\alpha>1$ and $\varepsilon\in(0,1)$, satisfies (a-c). We now demonstrate which of these utility functions is eventually log bounded and/or eventually decreases sufficiently quickly. Any linear in price or Cobb- Douglas in price utility is both eventually log bounded and eventually decreases sufficiently quickly. For if $\beta(\mathbf{y})\geq 1$, $(Dw)(\mathbf{y},p)=-\alpha(\mathbf{y})\beta(\mathbf{y})p^{\beta(\mathbf{y})-1}\downarrow-\infty$ as $p\to\infty$. If $\beta(\mathbf{y})<1$, then although $(Dw)(\mathbf{y},p)=-\alpha(\mathbf{y})\beta(\mathbf{y})p^{-(1-\beta(\mathbf{y}))}\uparrow 0$ as $p\to\infty$, $(Dw)(\mathbf{y},p)-\frac{r}{p}=-\alpha(\mathbf{y})\beta(\mathbf{y})\frac{1}{p^{1-\beta(\mathbf{y})}}+\frac{r}{p}=\left(\frac{1}{p}\right)\left[r-\alpha(\mathbf{y})\beta(\mathbf{y})p^{\beta(\mathbf{y})}\right]\leq 0$ if $p\geq\sqrt[\beta(\mathbf{y})]{\alpha(\mathbf{y})\beta(\mathbf{y})/r}$ and hence $w(\mathbf{y},p)$ eventually decreases sufficiently quickly for any $r$. The class of negative log of price utility functions contain the most obvious examples of utilities that are neither eventually log bounded nor eventually decrease sufficiently quickly; particularly $w(\mathbf{y},p)\leq-\alpha(\mathbf{y})\log p$ with $\alpha(\mathbf{y})\leq 1$. If $\alpha(\mathbf{y})<1$ there are no finite profit maximizing prices under this utility. In the text we defined utilities with sub-quadratic second derivatives. Any linear in price utility has sub-quadratic second derivatives, since $(D^{2}w)(\mathbf{y},p)\equiv 0$. More generally, under any Cobb-Douglas in price utility $\frac{(D^{2}w)(\mathbf{y},p)}{(Dw)(\mathbf{y},p)^{2}}=-\left(\frac{1}{\alpha(\mathbf{y})}\right)\left(\frac{\beta(\mathbf{y})-1}{\beta(\mathbf{y})}\right)\left(\frac{1}{p^{\beta(\mathbf{y})}}\right)\\\ =\left(\frac{\beta(\mathbf{y})-1}{\beta(\mathbf{y})}\right)\left(\frac{1}{w(\mathbf{y},p)}\right),$ and hence $w$ has sub-quadratic second derivatives if $\beta(\mathbf{y})\geq 1$. If $\beta(\mathbf{y})<1$, then $w$ has sub-quadratic second derivatives only at $(\mathbf{y},p)$ such that $\left\lvert w(\mathbf{y},p)\right\rvert>(1-\beta(\mathbf{y}))/\beta(\mathbf{y})$, i.e. $p>\sqrt[\beta(\mathbf{y})]{(1-\beta(\mathbf{y}))/(\alpha(\mathbf{y})\beta(\mathbf{y}))}$. Finally, if $w(\mathbf{y},p)=-\alpha(\mathbf{y})\log p$ then $(D^{2}w)(\mathbf{y},p)/(Dw)(\mathbf{y},p)^{2}\equiv 1/\alpha(\mathbf{y})$ and hence $w$ has sub-quadratic second derivatives if $\alpha(\mathbf{y})>1$. Hence far from requiring concavity, some convex utility functions have sub- quadratic second derivatives. Let $\alpha(\mathbf{y})\equiv\alpha>0$. For the linear-in-price utility, $\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})+1/\alpha$ with the fixed- point equation being $p_{j}=c_{j}+\hat{\pi}_{f}(\mathbf{p})+1/\alpha$. For any Cobb-Douglas in price utility, $\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})+(1/(\alpha\beta))p_{j}^{1-\beta}$ with the fixed-point equation being $p_{j}=c_{j}+\hat{\pi}_{f}(\mathbf{p})+(1/(\alpha\beta))p_{j}^{1-\beta}$. For negative log of price, $\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})+(1/\alpha)p_{j}$ with the fixed-point equation being $p_{j}=c_{j}+\hat{\pi}_{f}(\mathbf{p})+(1/\alpha)p_{j}$. Our proof that the negative log of price utility has no finite profit maximizing prices can be strengthened using the relationship between $\boldsymbol{\zeta}$ and the profit gradients. We already know that $w(\mathbf{y},p)/1=-(\alpha(\mathbf{y})/1)\log p$ does not eventually decrease sufficiently quickly when $\alpha(\mathbf{y})\leq 1$. We have also observed that $\zeta_{j}(\mathbf{p})=\hat{\pi}_{f}(\mathbf{p})+(1/\alpha_{j})p_{j}$, which implies that $\zeta_{j}(\mathbf{p})-(p_{j}-c_{j})=(\hat{\pi}_{f}(\mathbf{p})+c_{j})+\left(\frac{1}{\alpha_{j}}-1\right)p_{j}=(\hat{\pi}_{f}(\mathbf{p})+c_{j})+\left(\frac{1-\alpha_{j}}{\alpha_{j}}\right)p_{j}.$ Thus, if $\alpha_{j}=\alpha(\mathbf{y}_{j})\leq 1$, the $j^{\text{th}}$ price derivative of profit is always positive. While we have already shown that only infinite prices maximize profits under this utility when $\alpha_{j}<1$, this shows the same holds for $\alpha_{j}=1$ as well even though the corresponding maximal profits are finite. We now present an example of a utility function for which has finite profit- maximizing prices but for which a “local” criterion restricting profit maximization at infinity fails. This local criterion is simply that profits decrease for all sufficiently large prices. Let $w(p)=-\alpha(\log p-\varepsilon\sin\log p)$ with $\alpha>1$ and $\varepsilon\in[1-\alpha^{-1},1)$. Then $p_{j}-c_{j}-\zeta_{j}(\mathbf{p})\geq 0$ if and only if (21) $p_{j}\left(1-\frac{1}{\alpha(1-\varepsilon\cos\log p_{j})}\right)\geq c_{j}+\hat{\pi}_{f}(\mathbf{p}).$ But based on our choice of $\varepsilon$, there exist arbitrarily large $p_{j}$ such that the left hand side above is non-positive: For all $\bar{p}$ there exists some $p_{j}>\bar{p}$ such that $\alpha(1-\varepsilon\cos\log p_{j})=\alpha(1-\varepsilon)\leq 1$, which implies the claim. Since $c_{j}+\hat{\pi}_{f}(\mathbf{p})$ is positive (or rather is for all $\mathbf{p}$ that matter), the inequality (21) is violated and there exist arbitrarily large $p_{j}$ such that profits increase, locally, with $p_{j}$, despite the fact that profits must vanish as $\mathbf{p}_{f}\to\boldsymbol{\infty}$ since this utility is eventually log bounded. That is, the local criterion for finite profit maximizing prices is violated. ## Appendix C Inapplicability of Supermodularity This appendix states a generalization of Sandor’s [45] result that profits are neither supermodular nor log-supermodular arbitrarily close to equilibrium prices under Logit with linear in price utility [45, Chapter 4]. Such a result rules out the applicability of the approach developed by Milgrom & Roberts [28] to proving existence of equilibrium prices in the multi-product firm setting by implying that there cannot exist a compact set with non-empty interior containing any equilibrium on which Logit profits are supermodular or log-supermodular. ###### Lemma C.1. Let $\vartheta>-\infty$, unit costs be constant, and Assumption 3.1 hold with a $w$ with sub-quadratic second derivatives. Suppose $\mathbf{p}_{f}^{*}\in(0,\infty)^{J_{f}}$ maximizes $\hat{\pi}_{f}(\cdot,\mathbf{p}_{-f})$. Then for any $\varepsilon>0$, there exists a $\mathbf{p}_{f}$ such that $\lvert\lvert\mathbf{p}_{f}-\mathbf{p}_{f}^{*}\rvert\rvert<\varepsilon$, and $(D_{l}D_{k}\hat{\pi}_{f})(\mathbf{p})<0$, and $(D_{l}D_{k}\log\hat{\pi}_{f})(\mathbf{p})<0$ for all $k,l\in\mathcal{J}_{f}$, $k\neq l$, where $\mathbf{p}=(\mathbf{p}_{f},\mathbf{p}_{-f})$. Naturally, because supermodularity has been used to prove the existence of equilibrium prices under Logit for single-product firms, the proof relies on the fact that firms produce more than one product. ###### Proof. It can be shown that when $k,l\in\mathcal{J}_{f}$ and $k\neq l$, the second derivatives of profits are given by $\displaystyle(D_{l}D_{k}\hat{\pi}_{f})(\mathbf{q})$ $\displaystyle\quad\quad=\left\lvert(Dw_{k})(q_{k})\right\rvert P_{k}^{L}(\mathbf{q})\big{(}\hat{\pi}_{f}(\mathbf{q})-(q_{k}-c_{k})-(Dw_{k})(q_{k})^{-1}\big{)}P_{l}^{L}(\mathbf{q})\left\lvert(Dw_{l})(q_{l})\right\rvert$ $\displaystyle\quad\quad\quad\quad+\left\lvert(Dw_{k})(q_{k})\right\rvert P_{k}^{L}(\mathbf{q})\big{(}\hat{\pi}_{f}(\mathbf{q})-(q_{l}-c_{l})-(Dw_{l})(q_{l})^{-1}\big{)}P_{l}^{L}(\mathbf{q})\left\lvert(Dw_{l})(q_{l})\right\rvert$ for any $\mathbf{q}$. The goal is to choose $\mathbf{q}$, $\lvert\lvert\mathbf{q}-\mathbf{p}\rvert\rvert<\varepsilon$, so that $\hat{\pi}_{f}(\mathbf{q})-(q_{k}-c_{k})-(Dw_{k})(q_{k})^{-1}<0$ and $\hat{\pi}_{f}(\mathbf{q})-(q_{l}-c_{l})-(Dw_{l})(q_{l})^{-1}<0$ for any $k,l\in\mathcal{J}_{f}$, $k\neq l$. By the $\boldsymbol{\zeta}$ fixed-point characterization, $\displaystyle\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})-(p_{k}-c_{k})-(Dw_{k})(p_{k})^{-1}$ $\displaystyle\quad\quad\quad\quad<\hat{\pi}_{f}(\mathbf{p}_{f}^{*},\mathbf{p}_{-f})-(p_{k}-c_{k})-(Dw_{k})(p_{k})^{-1}$ $\displaystyle\quad\quad\quad\quad=\hat{\pi}_{f}(\mathbf{p}_{f}^{*},\mathbf{p}_{-f})-(p_{k}^{*}-c_{k})-(p_{k}-p_{k}^{*})-(Dw_{k})(p_{k})^{-1}$ $\displaystyle\quad\quad\quad\quad=(Dw_{k})(p_{k}^{*})^{-1}-(Dw_{k})(p_{k})^{-1}-(p_{k}-p_{k}^{*}).$ Thus, $\hat{\pi}_{f}(\mathbf{p}_{f},\mathbf{p}_{-f})-(p_{k}-c_{k})-(Dw_{k})(p_{k})^{-1}<0$ if $\theta_{k}(p_{k})\leq\theta_{k}(p_{k}^{*})$. Because $w$ has sub-quadratic second derivatives, $\theta_{k}$ is strictly increasing and any $p_{k}<p_{k}^{*}-\varepsilon$ will do. The same logic goes for $l\in\mathcal{J}_{f}$, and the claim follows. For the second claim, note that $\displaystyle(D_{l}D_{k}\log\hat{\pi}_{f})(\mathbf{p})$ $\displaystyle=\frac{(D_{l}D_{k}\hat{\pi}_{f})(\mathbf{p})\hat{\pi}_{f}(\mathbf{p})-(D_{k}\hat{\pi}_{f})(\mathbf{p})(D_{l}\hat{\pi}_{f})(\mathbf{p})}{\hat{\pi}_{f}(\mathbf{p})^{2}}.$ We have already established that the first term in the numerator is negative at $\mathbf{p}$ as defined above. Furthermore, $(D_{k}\hat{\pi}_{f})(\mathbf{p})=\left\lvert(Dw_{k})(p_{k})\right\rvert P_{k}^{L}(\mathbf{p})(\hat{\pi}_{f}(\mathbf{p})-(p_{k}-c_{k})-(Dw_{k})(p_{k})^{-1})<0$ by the same argument and hence $(D_{k}\hat{\pi}_{f})(\mathbf{p})(D_{l}\hat{\pi}_{f})(\mathbf{p})>0$, making the second term in the numerator also negative. 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arxiv-papers
2010-12-28T20:22:42
2024-09-04T02:49:16.003021
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "W. Ross Morrow and Steven J. Skerlos", "submitter": "William Morrow", "url": "https://arxiv.org/abs/1012.5832" }
1012.5836
# Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed-Logit Demand: A Technical Framework for Analysis and Efficient Computational Methods. W. Ross Morrow Departments of Mechanical Engineering and Economics, Iowa State University, Ames IA 50011 and Steven J. Skerlos Department of Mechanical Engineering, University of Michigan, Ann Arbor MI 48104 ###### Contents 1. 1 Introduction 2. 2 A Technical Framework 1. 2.1 Mathematical Notation 1. 2.1.1 Sets. 2. 2.1.2 Symbols. 3. 2.1.3 Differentiation. 2. 2.2 Consumers, Products, and Choice Probabilities 3. 2.3 Utility Specification 4. 2.4 Profits 1. 2.4.1 Quantity-Dependent Costs 2. 2.4.2 Bounded and Vanishing Profits 5. 2.5 Local Equilibrium and the Simultaneous Stationarity Conditions 6. 2.6 Choice Probability Derivatives 7. 2.7 The BLP-Markup Equation 8. 2.8 The $\zeta$-Markup function 9. 2.9 Existence of Simultaneously Stationary Prices 3. 3 Computational Methods 1. 3.1 Newton’s Method 2. 3.2 Newton’s Method on the Combined Gradient 3. 3.3 Newton’s Method and the Markup Equations 4. 3.4 Fixed-Point Iteration 1. 3.4.1 $\zeta$ Fixed-Point Iteration 2. 3.4.2 $\eta$ Fixed-Point Iteration 5. 3.5 Practical Considerations 1. 3.5.1 Simulation 2. 3.5.2 Truncation of Low Purchase Probability Products 3. 3.5.3 Termination Conditions 4. 3.5.4 Second-Order Conditions. 5. 3.5.5 Computational Burden 6. 3.6 Computing Jacobian Matrices for Newton’s Method 1. 3.6.1 Jacobian of the Combined Gradient 2. 3.6.2 The $\boldsymbol{\eta}$ map. 3. 3.6.3 The $\boldsymbol{\zeta}$ map. 4. 4 The GMRES-Newton Hookstep Method 1. 4.1 Inexact Newton Methods 2. 4.2 GMRES 1. 4.2.1 Householder GMRES 2. 4.2.2 Preconditioning 3. 4.3 The GMRES Hookstep 1. 4.3.1 Model Trust Region Methods. 2. 4.3.2 Model Trust Region Methods on a Subspace 3. 4.3.3 The GMRES-Newton Hookstep 4. 4.3.4 Directional Finite Differences 5. 5 Other Methods 1. 5.1 Variational Methods 1. 5.1.1 The VI formulation is poorly posed 2. 5.1.2 General Results. 3. 5.1.3 The Resolution of Equilibria with $\boldsymbol{\zeta}$ 2. 5.2 Tatonnement 3. 5.3 Least-Squares Minimization and the Gauss-Newton Method 6. 6 Acknowledgements ## 1\. Introduction Bertrand competiton has been a prominent paradigm for the empirical study of differentiated product markets for at least twenty years. Firms engaged in Bertrand competition maximize profits by choosing prices for portfolios of differentiated products, and Bertrand-Nash equilibrium prices simultaneously maximize profits for all firms. Models combining Bertrand competition with the Mixed Logit discrete choice model of consumer demand have been used to study the automotive industry, electronics, entertainment, and food products and services; see Dube et al. (2002). Many applications of Bertrand competition rely on counterfactual experiments: exercises in which hypothetical market conditions are simulated with an estimated model. Such experiments have been used to study corporate mergers (Nevo, 2000a), novel products and services (Petrin, 2002; Goolsbee and Petrin, 2004; Beresteanu and Li, 2008), store locations (Thomadsen, 2005), and regulatory policy changes (Goldberg, 1995, 1998; Beresteanu and Li, 2008). By definition, simulating market outcomes in counterfactual experiments requires computing equilibrium prices after changing the values of exogenous variables such as the number of firms or the products offered. Numerical methods for computing equilibrium prices have not yet received a thorough treatment in the literature, which currently focuses on model specification and estimation; see Knittel and Metaxoglou (2008); Dube et al. (2008); Su and Judd (2008) for recent developments in estimation. Morrow and Skerlos (2010) fills this gap with a detailed investigation of four approaches for computing Bertrand-Nash equilibrium prices in single-period, multi-firm models with Mixed Logit demand. This working paper provides most of the technical background for that investigation. Applying Newton’s method to some form of the first-order or “simultaneous stationarity” condition is currently the de facto approach for computing equilibrium prices; see, for example, Nevo (1997, 2000a); Petrin (2002); Smith (2004); Doraszelski and Draganska (2006); Jacobsen (2006). Newton’s method applied directly to the first-order condition may converge when started at observed prices if changes in exogenous variables have a marginal impact on equilibrium prices. However, when the changes to exogenous variables imply significant changes in product prices Newton’s method applied directly to the first-order conditions may fail to compute equilibrium prices. Furthermore analyses that do not have observed prices to use as an initial guess will require methods with greater reliability. Morrow and Skerlos (2010) demonstrate that solving fixed-point equations equivalent to the first-order condition for equilibrium is more reliable and efficient than solving the first-order condition itself. One fixed-point equation equivalent to the first-order conditions is the BLP-markup equation popularized by Berry et al. (1995). A second fixed-point equation, here termed the $\boldsymbol{\zeta}$-markup equation, is a novel way to write the same condition on markups. Both markup equations lead to more robust numerical methods than found with a simple application of Newton’s method to the first- order condition. Using the fixed-point expressions in this way can be considered “nonlinearly” or “analytically” pre-conditioning the first-order condition satisfied by equilibrium prices, a technique well-known in applied mathematics (Brown and Saad, 1990; Cai and Keyes, 2002). The existence of fixed-point equations for equilibrium suggests applying fixed-point iteration (Judd, 1998) to compute equilibrium prices, instead of Newton’s method. The BLP-markup equation does not appear to be well-suited to fixed-point iteration. Example 7 in Morrow and Skerlos (2010) provides a case in which iterating on the BLP-markup equation is not necessarily locally convergent, while iterating on the $\boldsymbol{\zeta}$-markup equation is superlinearly locally convergent. Iterating on the $\boldsymbol{\zeta}$-markup equation also eliminates the need to solve linear systems, required to implement Newton’s method and to iterate on the BLP-markup equation. This property makes fixed-point steps based on the $\zeta$-markup equation very inexpensive relative to Newton steps, an essential property to obtaining fast computations from generally linearly convergent fixed-point iterations. Besides Newton’s method and fixed-point iteration, few other practical approaches to the computation of equilibrium prices exist. Variational formulations, widely applied in economic and engineering problems (Ferris and Pang, 1997), contain many solutions that need not be equilibria of the original problem. Explicit least-square minimization or Gauss-Newton methods can also be implemented, but are computational disadvantages relative to applications of standard Newton-type methods for nonlinear systems. Some authors apply tattonement $-$ iterating on a game’s best response correspondence $-$ to compute equilibrium in prices or other strategic variables including product mix (Choi et al., 1990), product characteristics (CBO, 2003; Austin and Dinan, 2005; Bento et al., 2005), and engineering variables (Michalek et al., 2004). Tattonement, however, has three issues: it requires the iterative computation of profit-optimal prices (a special case of the problem discussed in this article), should be inefficient relative to direct methods whenever optimal strategies are coupled, and lacks the global convergence guarantees of contemporary Newton solvers. Section 5 reviews these conclusions in more detail. This article should be viewed as a companion to Morrow and Skerlos (2010); some of our notation and text may seem out of place without first reviewing that article. In several places, text from Morrow and Skerlos (2010) is repeated. ## 2\. A Technical Framework This section describes the mathematical framework employed in Morrow and Skerlos (2010). Several key assumptions are introduced and summarized in Table 1. Table 1. List of important assumptions used in this section. Assumption | Purpose ---|--- 2.1 | To provide a general form for utility functions 2.2 | To ensure profits are bounded and vanish as prices increase without bound 2.3 | To ensure the Leibniz Rule holds, validating Eqn. (9) 2.4 | To ensure that $\boldsymbol{\eta}$ is bounded. Implies the coercivity of $\mathbf{F}_{\eta},\mathbf{F}_{\zeta}$ | and the existence of simultaneously stationary prices. 2.5 | To ensure that $\boldsymbol{\zeta}$ is bounded. Implies the coercivity of $\mathbf{F}_{\zeta}$ | and the existence of simultaneously stationary prices. 3.1 | To ensure that the derivatives of profit vanish as prices increase without bound 3.2 | To ensure the coercivity of $\mathbf{F}_{\eta},\mathbf{F}_{\zeta}$ under weaker conditions than | Assumption 2.4. ### 2.1. Mathematical Notation #### 2.1.1. Sets. Table 2 lists some important sets and the symbols used for them. $\mathbb{N}$ denotes the natural numbers $\\{1,2,\dotsc\\}$, and $\mathbb{N}(N)$ denotes the natural numbers up to $N$, that is, $\mathbb{N}(N)=\\{1,\dotsc,N\\}$. $\mathbb{R}$ denotes the set of real numbers $(-\infty,\infty)$, $[0,\infty)$ denotes the non-negative real numbers, and $[0,\infty]$ denotes the extended non-negative half-line. We denote the $(J-1)$-dimensional simplex $\\{(x_{1},\dotsc,x_{N})\in[0,1]^{N}:\sum_{n=1}^{N}x_{n}=1\\}$ by $\mathbb{S}(N)$, and the $J$-dimensional “pyramid” $\\{(x_{1},\dotsc,x_{N})\in[0,1]^{N}:\sum_{n=1}^{N}x_{n}\leq 1\\}$ by $\triangle(J)$. Hyper-rectangles in $\mathbb{R}^{N}$, i.e. sets of the form $[a_{1},b_{1}]\times\dotsb\times[a_{N},b_{N}]$ for some $a_{n},b_{n}\in\mathbb{R}$ with $a_{n}<b_{n}$ for all $n\in\mathbb{N}(N)$, are denoted by $[\mathbf{a},\mathbf{b}]$ where $\mathbf{a}=(a_{1},\dotsc,a_{N})$ and $\mathbf{b}=(b_{1},\dotsc,b_{N})$. $\mathcal{P}$ always denotes the non- negative numbers: $\mathcal{P}=[0,\infty)$. For other sets, we typically use calligraphic upper case letters such as “$\mathcal{A}$”. For any set $\mathcal{A}$, $\left\lvert\mathcal{A}\right\rvert$ denotes its cardinality. For any $\mathcal{B}\subset\mathcal{A}$, $\mathcal{A}\setminus\mathcal{B}$ denotes the set $\\{b\in\mathcal{A}:b\notin\mathcal{B}\\}$. Table 2. Important sets. Symbol | Description ---|--- $\mathbb{N}$ | $=$ | $\\{1,2,\dotsc\\}$ | Natural numbers $\mathbb{R}$ | $=$ | $(-\infty,\infty)$ | Real numbers $\mathcal{P}$ | $=$ | $[0,\infty)$ | Non-negative real numbers $\mathcal{J}$ | $=$ | $\\{1,\dotsc,J\\}$ | Set of product indices $\mathcal{X}$ | $\subset$ | $\mathbb{R}^{K}$ | Set of product characteristics $\mathcal{T}$ | $\subset$ | $\mathbb{R}^{L}$ | Set of individual characteristics #### 2.1.2. Symbols. Table 3 itemizes specific symbols used in the text. Table 3. Summary of important symbols. Symbol | Description | Defined in ---|---|--- Products (see Section 2.2) $J$ | $\in$ | $\mathbb{N}$ | number of products | $K$ | $\in$ | $\mathbb{N}$ | number of non-price product characteristics | $\mathbf{x}_{j}$ | $\in$ | $\mathcal{X}$ | non-price characteristics of product $j$ | $p_{j}$ | $\in$ | $\mathcal{P}$ | price of product $j$ | $\mathbf{p}$ | $\in$ | $\mathcal{P}^{J}$ | vector of all product prices | Individual Characteristics (see Section 2.2) $\boldsymbol{\theta}$ | $\in$ | $\mathcal{T}$ | individual characteristics, including observed | | | | demographics and “random coefficients” | $\mu$ | $-$ | $-$ | distribution of individual characteristics | Choice Probabilities (see Section 2.2) $u_{j}(\boldsymbol{\theta},p_{j})$ | $\in$ | $[-\infty,\infty)$ | utility of product $j$ | $\vartheta(\boldsymbol{\theta})$ | $\in$ | $[-\infty,\infty)$ | utility of the outside good | $P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})$ | $\in$ | [0,1] | Logit choice probability for product $j$ | Eqn. (1) $P_{j}(\mathbf{p})$ | $\in$ | [0,1] | Mixed Logit choice probability for product $j$ | $\mathbf{P}(\mathbf{p})$ | $\in$ | $[0,1]^{J}$ | vector of Mixed Logit choice probabilities for all | | | | products | Firms, Costs, Profits, and Stationarity (see Section 2.4, 2.5) $F$ | $\in$ | $\mathbb{N}$ | number of firms | $\mathcal{J}_{f}$ | $\subset$ | $\mathcal{J}$ | indices of the products offered by firm $f$ | $c_{j}$ | $\in$ | $\mathcal{P}$ | (fixed) unit cost of product $j$ | $\mathbf{c}$ | $\in$ | $\mathcal{P}^{J}$ | vector of all (fixed) unit costs | $\hat{\pi}_{f}(\mathbf{p})$ | $\in$ | $\mathbb{R}$ | expected profits for firm $f$ | Eqn. (2) $(D_{k}\hat{\pi}_{f})(\mathbf{p})$ | $\in$ | $\mathbb{R}$ | derivative of firm $f$’s profits, with respect to the | Eqn. (6) | | | price of product $k$ | $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J}$ | Combined Gradient of profits | Prop. 2.2, Eqn. (7) Choice Probability Derivatives (see Sections 2.5, 2.8) $(D_{k}P_{j})(\mathbf{p})$ | $\in$ | $\mathbb{R}$ | derivative of product $j$’s choice probability | | | | with respect to the price of product $k$ | $(\tilde{D}\mathbf{P})(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J\times J}$ | “intra-firm” Jacobian matrix of the choice | Eqn. (8) | | | probability vector | $\boldsymbol{\Lambda}(\mathbf{p})$, $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J\times J}$ | matrices appearing in our decomposition of $(\tilde{D}\mathbf{P})(\mathbf{p})$ | Eqn. (9), Fixed-Point Equations (see Sections 2.7, 2.8) $\boldsymbol{\eta}(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J}$ | the BLP-markup function (Berry et al., 1995) | Eqn. (13) $\boldsymbol{\zeta}(\mathbf{p})$ | $\in$ | $\mathbb{R}^{J}$ | our $\boldsymbol{\zeta}$-markup function | Eqn. (18) Bold, un-italicized symbols (e.g., “$\mathbf{x}$”) denote vectors and matrices; typically we reserve lower case letters to refer to vectors and use upper case letters to refer to matrices; the vector of choice probabilities “$\mathbf{P}$” is an exception. Throughout we use $\mathbf{1}$ to denote a vector of ones of the appropriate size for the context in which it appears. $\mathbf{I}$ always denotes the identity matrix of a size appropriate for the context. For any $\mathbf{x}\in\mathbb{R}^{N}$, $\mathrm{diag}(\mathbf{x})$ denotes the $N\times N$ diagonal matrix whose diagonal is $\mathbf{x}$. Any vector inequalities between vectors are to be taken componentwise: for example, $\mathbf{x}<\mathbf{y}$ means $x_{n}<y_{n}$ for all $n$. Random variables are denoted with capital letters “$X$”, with random vectors being denoted with bold capital letters (e.g., “$\mathbf{Q}$”). While this overlaps with our notation for matrices, it should not cause any confusion. $\mathbb{P}$ denotes a probability and $\mathbb{E}$ denotes an expectation. $\mathrm{ess}\sup_{\mu}f$ denotes the essential supremum of the (measurable) function $f$ over $\mathcal{T}$, with respect to the measure $\mu$; see, e.g., Bartle (1966). $\log$ always denotes the natural (base $e$) logarithm. We use the “Big-O” notation $\mathcal{O}(g)$ as follows: If there exists some $M<\infty$ such that $\lim_{p\to q}[f(p)/g(p)]\leq M$, we say $f\in\mathcal{O}(g)$; the point $q$ is left implicit. #### 2.1.3. Differentiation. Our conventions for denoting differentiation follow Munkres (1991). We use the symbol “$D$” to denote differentiation using subscripts to invoke additional specificity. Letting $\mathbf{f}:\mathbb{R}^{M}\to\mathbb{R}^{N}$, $(D_{m}f_{n})(\mathbf{x})$ denotes the derivative of the $n^{\text{th}}$ component function with respect to the $m^{\text{th}}$ variable and $(D\mathbf{f})(\mathbf{x})$ is the $N\times M$ derivative matrix of $\mathbf{f}$ at $\mathbf{x}$ with components $((D\mathbf{f})(\mathbf{x}))_{n,m}=(D_{m}f_{n})(\mathbf{x})$. Thus for $f:\mathbb{R}^{M}\to\mathbb{R}$, $(Df)(\mathbf{x})$ is a row vector. If $f:\mathbb{R}^{M}\to\mathbb{R}$, we define the gradient $(\nabla f)(\mathbf{x})\in\mathbb{R}^{M}$ as the transposed derivative: $(\nabla f)(\mathbf{x})=(Df)(\mathbf{x})^{\top}$. ### 2.2. Consumers, Products, and Choice Probabilities A collection of $F\in\mathbb{N}$ firms offer a total of $J\in\mathbb{N}$ products to a population of individuals (or households). Each product $j\in\mathcal{J}=\\{1,\dotsc,J\\}$ is defined by a price, $p_{j}\in\mathcal{P}=[0,\infty)$, and a vector of $K\in\mathbb{N}$ product “characteristics” $\mathbf{x}_{j}\in\mathcal{X}\subset\mathbb{R}^{K}$. Individuals are identified by a vector of characteristics $\boldsymbol{\theta}$ from some set $\mathcal{T}$. These individual characteristics can include both observed demographics and “random coefficients” (Berry et al., 1995; Nevo, 2000b; Train, 2003) that characterize unobserved individual-specific heterogeneity with respect to preference for product characteristics. The relative density of individual characteristic vectors in the population is described by a probability distribution $\mu$ over $\mathcal{T}$. An individual identified by $\boldsymbol{\theta}\in\mathcal{T}$ receives the (random) utility $U_{j}(\boldsymbol{\theta},\mathbf{x}_{j},p_{j})=u(\boldsymbol{\theta},\mathbf{x}_{j},p_{j})+\mathcal{E}_{j}$ from purchasing product $j\in\mathcal{J}$, and $U_{0}(\boldsymbol{\theta})=\vartheta(\boldsymbol{\theta})+\mathcal{E}_{0}$ for forgoing purchase of any of these products; i.e. “purchasing the outside good.” Individuals choose the “product” $j\in\\{0,\dotsc,J\\}$ with maximum utility. Here $u:\mathcal{T}\times\mathcal{X}\times\mathcal{P}\to[-\infty,\infty)$ is a systematic utility function, $\vartheta:\mathcal{T}\to\mathbb{R}$ is a valuation of the no-purchase option or “outside good,” and $\boldsymbol{\mathcal{E}}=\\{\mathcal{E}_{j}\\}_{j=0}^{J}$ is a random vector of i.i.d. standard extreme value variables. Section 2.3 below gives a general specification of utility functions appropriate for equilibrium pricing. The basic requirements are that $u$ is continuously differentiable and strictly decreasing in price, and without lower bound as prices increase. Demand for each product $j$ is characterized by choice probabilities $P_{j}:\mathcal{P}^{J}\to[0,1]$ derived from (random) utility maximization. Given the distributional assumption on $\boldsymbol{\mathcal{E}}$, the choice probabilities for an individual characterized by $\boldsymbol{\theta}\in\mathcal{T}$ are those of the Logit model (Train, 2003, Chapter 3): (1) $P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})=\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{k=1}^{J}e^{u_{k}(\boldsymbol{\theta},p_{k})}}.$ The vector $\mathbf{p}\in\mathcal{P}^{J}$ denotes the vector of all product prices. Product-specific utility functions $u_{j}:\mathcal{T}\times\mathcal{P}\to[-\infty,\infty)$ for all $j$, defined by $u_{j}(\boldsymbol{\theta},p)=u(\boldsymbol{\theta},\mathbf{x}_{j},p)$ for all $(\boldsymbol{\theta},p)\in\mathcal{T}\times\mathcal{P}$, are used in Eqn. (1) and in the following sections. The Mixed Logit choice probabilities $P_{j}(\mathbf{p})=\int P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$ follow from integrating over the distribution of individual characteristics (Train, 2003, Chapter 6). The vector of Mixed Logit choice probabilities for all products is denoted by $\mathbf{P}(\mathbf{p})\in[0,1]^{J}$. The examples below review several instances of this choice model. Examples 1 and 2 are used in Morrow and Skerlos (2010). Example 3 illustrates the type general specifications used in estimation. Example 4 describes one kind of “simulation” of a Mixed Logit model (Train, 2003). ###### Example 1. (Boyd and Mellman, 1980) Take $\mathcal{T}=\mathcal{P}\times\mathbb{R}^{K}$, denoting $\boldsymbol{\theta}=(\alpha,\boldsymbol{\beta})$ for $\alpha\in\mathcal{P}$ and $\boldsymbol{\beta}\in\mathbb{R}^{K}$. Set $u(\alpha,\boldsymbol{\beta},\mathbf{x},p)=-\alpha p+\boldsymbol{\beta}^{\top}\mathbf{x}$ and $\vartheta(\alpha,\boldsymbol{\beta})=-\infty$ for all $(\alpha,\boldsymbol{\beta})\in\mathcal{P}\times\mathbb{R}^{K}$. $\mu$ is defined by specifying that $\alpha$ and $\boldsymbol{\beta}$ are independently lognormally distributed (with appropriately chosen signs, means, and variances). ###### Example 2. (Berry et al., 1995) Take $\mathcal{T}=\mathcal{P}\times\mathbb{R}^{K}\times\mathbb{R}$, denoting $\boldsymbol{\theta}=(\phi,\boldsymbol{\beta},\beta_{0})$ for $\phi\in\mathcal{P}$, $\boldsymbol{\beta}\in\mathbb{R}^{K}$, and $\beta_{0}\in\mathbb{R}$. Set $u(\phi,\boldsymbol{\beta},\mathbf{x},p)=\left\\{\begin{aligned} &\alpha\log(\phi-p)+\boldsymbol{\beta}^{\top}\mathbf{x}&&\quad\text{if }p<\phi\\\ &-\infty&&\quad\text{otherwise}\end{aligned}\right.\quad\quad\text{and}\quad\quad\vartheta(\phi,\beta_{0})=\alpha\log\phi+\beta_{0}$ for some fixed coefficient $\alpha>0$. $\phi$ represents income and is given a lognormal distribution, while the random coefficients $\boldsymbol{\beta},\beta_{0}$ are independently normally distributed with some mean and variance. Note that income ($\phi$) serves as an upper bound on the price an individual can pay for any product. ###### Example 3. (Nevo, 2000b) Take $\mathcal{T}=\mathcal{P}\times\mathbb{R}^{D}\times\mathbb{R}^{K+2}$, denoting $\boldsymbol{\theta}=(\phi,\mathbf{d},\boldsymbol{\nu})$ for $\phi\in\mathcal{P}$, $\mathbf{d}\in\mathbb{R}^{D}$, and $\boldsymbol{\nu}\in\mathbb{R}^{K+2}$. Again, $\phi$ represents income; $\mathbf{d}\in\mathbb{R}^{D}$ represents a vector of $D$ observed demographic variables (which may include income); $\boldsymbol{\nu}\in\mathbb{R}^{K+2}$ represents a vector of $K+2$ random coefficients: one for each product characteristic, one for price, and one for the outside good. Set $\displaystyle u(\phi,\mathbf{d},\boldsymbol{\nu},\mathbf{x},p)$ $\displaystyle=(\alpha+\boldsymbol{\pi}_{p}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{p}^{\top}\boldsymbol{\nu})(\phi-p)+\left(\boldsymbol{\beta}+\boldsymbol{\Pi}\mathbf{d}+\boldsymbol{\Sigma}\boldsymbol{\nu}\right)^{\top}\mathbf{x}$ $\displaystyle\vartheta(\phi,\mathbf{d},\boldsymbol{\nu})$ $\displaystyle=(\alpha+\boldsymbol{\pi}_{p}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{p}^{\top}\boldsymbol{\nu})\phi+\boldsymbol{\pi}_{0}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{0}^{\top}\boldsymbol{\nu}$ where $\alpha\in\mathbb{R}$, $\boldsymbol{\beta}\in\mathbb{R}^{K}$, $\boldsymbol{\pi}_{p},\boldsymbol{\pi}_{0}\in\mathbb{R}^{D}$, $\boldsymbol{\Pi}\in\mathbb{R}^{K\times D}$, $\boldsymbol{\sigma}_{p},\boldsymbol{\sigma}_{0}\in\mathbb{R}^{K+2}$, and $\boldsymbol{\Sigma}\in\mathbb{R}^{K\times(K+2)}$ are coefficients. The distribution of $\mathbf{d}$ is estimated from available data (e.g., Census data) and $\boldsymbol{\nu}$ is assumed to be standard independent multivariate normal. When $\alpha+\boldsymbol{\pi}_{p}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{p}^{\top}\boldsymbol{\nu}$, the coefficient on price, is positive, an individual prefers higher prices. Petrin (2002) and Berry et al. (2004) adopt similar specifications that eliminate this counterintuitive property. Petrin (2002) takes the price component of utility to be $\alpha(\phi)\log(\phi-p)$, where $\alpha:\mathcal{P}\to\mathcal{P}$ is a step function. Berry et al. (2004) take the price component of utility to be $\alpha p$, but define $\alpha=-e^{-(\alpha+\boldsymbol{\pi}_{p}^{\top}\mathbf{d}+\boldsymbol{\sigma}_{p}^{\top}\boldsymbol{\nu})}$. ###### Example 4. (Simulation). Take any of the examples above, and draw $S\in\mathbb{N}$ vectors $\boldsymbol{\theta}_{s}\in\mathcal{T}$ according to the distribution $\mu$. Let $\mathcal{T}^{\prime}=\\{\boldsymbol{\theta}_{s}\\}_{s=1}^{S}$ and define a probability measure $\mu^{\prime}$ over $\mathcal{T}^{\prime}$ by $\mu^{\prime}(\boldsymbol{\theta}_{s})=1/S$ for all $s$. Then $(u,\vartheta,\mathcal{T}^{\prime},\mu^{\prime})$ defines a simulator of the “full” Mixed Logit model with $(u,\vartheta,\mathcal{T},\mu)$; see Train (2003). These approximations are essential in estimation of Mixed Logit models and in computations of equilibrium prices. ### 2.3. Utility Specification This section presents a generalization of the systematic utility functions used in the examples given in the text, a specification closely related to the one introduced by Caplin and Nalebuff (1991). Morrow (2008); Morrow and Skerlos (2008) use a similar specification to analyze equilibrium prices in simple Logit models. ###### Assumption 2.1. For all $j$, there exist functions $w_{j}:\mathcal{T}\times\mathcal{P}\to[-\infty,\infty)$ and $v_{j}:\mathcal{T}\to(-\infty,\infty)$ such that the systematic utility function $u_{j}:\mathcal{T}\times\mathcal{P}\to[-\infty,\infty)$ can be written $u_{j}(\boldsymbol{\theta},p)=w_{j}(\boldsymbol{\theta},p)+v_{j}(\boldsymbol{\theta})$. Furthermore there exists $\varsigma:\mathcal{T}\to(0,\infty]$ such that $w_{j}:\mathcal{T}\times[0,\infty]\to[-\infty,\infty)$ satisfies, for all $j$ and $\mu$-almost every (a.e.) $\boldsymbol{\theta}\in\mathcal{T}$, * (a) $w_{j}(\boldsymbol{\theta},\cdot):(0,\varsigma(\boldsymbol{\theta}))\to[-\infty,\infty)$ is continuously differentiable, strictly decreasing, and finite * (b) $w_{j}(\boldsymbol{\theta},p)=-\infty$ for all $p\geq\varsigma(\boldsymbol{\theta})$, and * (c) $w_{j}(\boldsymbol{\theta},p)\downarrow-\infty$ as $p\uparrow\varsigma(\boldsymbol{\theta})$. $v_{j}:\mathcal{T}\to(-\infty,\infty)$ is arbitrary. Note that we have not restricted $\mu$, the distribution of individual characteristics, with Assumption 2.1. Important examples of $\mu$ from the econometrics and marketing literature include finitely supported distributions (often empirical frequency distributions for integral observed demographic variables), standard continuous distributions (e.g. normal, lognormal and $\chi^{2}$), truncated standard continuous distributions, finite mixtures of standard continuous distributions, and independent products of any of these types of distributions. This generality allows us to address a wide variety of otherwise disparate examples with a single notation. In particular, this generality allows us to use a single framework to treat both “full” Mixed Logit models defined by some $\mu$ with uncountable support and simulation- based approximations to such models. Some existing empirical specifications violate Assumption 2.1 by admiting positive price coefficients for $\boldsymbol{\theta}\in\mathcal{T}^{\prime}\subset\mathcal{T}$, where $\mathcal{T}^{\prime}\subset\mathcal{T}$ has nonzero $\mu$-measure. See, for example, Nevo (2000a) (Example 3) or Brownstone et al. (2000). This implies that $w(\boldsymbol{\theta},\cdot)$ is increasing on $\mathcal{T}^{\prime}$. If $w(\boldsymbol{\theta},\cdot)$ is not decreasing for $\mu$-a.e. $\boldsymbol{\theta}$, or at least eventually decreasing for $\mu$-a.e. $\boldsymbol{\theta}$ in the sense that there are always prices large enough to ensure that $w(\boldsymbol{\theta},\cdot)$ is decreasing for $\mu$-a.e. $\boldsymbol{\theta}$, then profit-optimal pricing is not a well-posed problem and finite equilibrium prices will not exist. The variable $\Sigma=\varsigma(\boldsymbol{\theta})$ represents an individual- specific reservation price. As in the Berry et al. (1995) model of Example 2, this reservation price is most often derived from household or individual income. Correspondingly, $\Sigma$ is often given a lognormal distribution to (roughly) fit empirical income data. In principle, this reservation price could be related to purchasing power derived from observed demographic variables other than income, or unobserved demographic variables such as family wealth. Thus we allow this reservation price to be specified as a function of all “demographic” characteristics, $\boldsymbol{\theta}$. Conditions (b) and (c) in Assumption 2.1 imply that the probability an individual characterized by $\boldsymbol{\theta}$ will purchase a product is zero for any price above $\varsigma(\boldsymbol{\theta})$ and vanishes as the price approaches $\varsigma(\boldsymbol{\theta})$. We set $\varsigma_{*}=\mathrm{ess}\sup\varsigma$ and allow, but do not require, $\varsigma_{*}=\infty$. For example, simulation-based approximations to the Berry et al. demand model have $\varsigma_{*}<\infty$, as can be easily checked. Note also that Condition (c) in Assumption 2.1 ensures the continuity of $P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})$ at any vector of prices with some component equal to $\varsigma(\boldsymbol{\theta})$. We must require this of the Logit choice probabilities to obtain Mixed Logit choice probabilities that are continuous on $(0,\varsigma_{*})^{J}$ for the important class of simulation-based approximations with finitely supported $\mu$. Continuous Logit choice probabilities also imply continuous Mixed Logit choice probabilities, by the Dominated Convergence Theorem. ### 2.4. Profits To describe the optimal pricing problems faced by each firm we use the following notation. Let $F\in\mathbb{N}$ denote the number of firms. For each $f\in\\{1,\dotsc,F\\}$, there exists a set $\mathcal{J}_{f}\subset\mathcal{J}$ of indices that corresponds to the $J_{f}=\left\lvert\mathcal{J}_{f}\right\rvert$ products offered by firm $f$. The collection of all these sets, $\\{\mathcal{J}_{f}\\}_{f=1}^{F}$, forms a partition of $\mathcal{J}$. Subsequently, in writing “$f(j)$” for some $j\in\mathcal{J}$, we mean the unique $f\in\\{1,\dotsc,F\\}$ such that $j\in\mathcal{J}_{f}$. The vector $\mathbf{p}_{f}\in\mathbb{R}^{J_{f}}$ refers to the vector of prices of the products offered by firm $f$. Negative subscripts denote competitor’s variables as in, for instance, $\mathbf{p}_{-f}\in\mathbb{R}^{J_{-f}}$, where $J_{-f}=\sum_{g\neq f}J_{g}$, is the vector of prices for products offered by all of firm $f$’s competitors. Firm-specific choice probability functions are denoted by $\mathbf{P}_{f}(\mathbf{p})\in\mathbb{R}^{J_{f}}$. Two additional assumptions are required to complete the definition of firms’ profits in a manner consistent with empirical applications of Bertrand competition. First, we must specify unit and fixed costs: for each product $j$ there exists a unit cost $c_{j}\in\mathcal{P}$ and for each firm there exists a fixed cost $c_{f}^{F}\in\mathcal{P}$. Both $c_{j}$ and $c_{f}^{F}$ depend only on the collection of product characteristics chosen by the firm, and not on the quantity sold by the firm during the purchasing period for the reasons discussed below. We let $\mathbf{c}_{f}\in\mathcal{P}^{J_{f}}$ denote the vector of unit costs for the products offered by firm $f$, and $\mathbf{c}\in\mathcal{P}^{J}$ denote the vector of unit costs for all products. Second, Bertrand competition entails the following “comittment” assumption on the quantities produced (Baye and Kovenock, 2008). Let $Q_{j}(\mathbf{p})$ denote the (random) quantity of product $j$ that the population will demand during the purchasing period, given prices for all products $\mathbf{p}$. These random demands are derived from random utility maximization. We assume each firm commits to producing exactly $Q_{j}(\mathbf{p})$ units of each product $j\in\mathcal{J}_{f}$ during the purchasing period. This implies either that there are no production capacity constraints that limit a firm’s ability to meet any demands that arise during the purchase period, or that production backlogs do not affect demand. With the commitment and constant costs assumptions, the total cost firm $f$ incurs in producing (and selling) $Q_{j}(\mathbf{p})$ units of product $j$ during the purchasing period are given by the random variable $\sum_{j\in\mathcal{J}_{f}}c_{j}Q_{j}(\mathbf{p})+c_{f}^{F}.$ Random revenues are, of course, given by $\sum_{j\in\mathcal{J}_{f}}Q_{j}(\mathbf{p})p_{j}$. The random variable $\Pi_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}Q_{j}(\mathbf{p})(p_{j}-c_{j})-c_{f}^{F}$ then gives firm $f$’s (random) profits for the purchasing period as a function of all product prices. Following most of the theoretical and empirical literature in both marketing and economics, we assume that firms take expected profits, (2) $\mathbb{E}[\Pi_{f}(\mathbf{p})]=I\hat{\pi}_{f}(\mathbf{p})-c_{f}^{F}\quad\text{where}\quad\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}P_{j}(\mathbf{p})(p_{j}-c_{j})$ as the metric by which they optimize their pricing decisions in this stochastic optimization problem. Here $I\in\mathbb{N}$ denotes the number of individuals in the population. Eqn. (2) demonstrates that neither the total firm fixed costs $c_{f}^{F}$ nor the population size $I$ play a role in determining the prices that maximize expected profits under the assumptions described above. Henceforth we focus on the “population-normalized gross expected profits” $\hat{\pi}_{f}(\mathbf{p})$, referred to in the text and below as simply “profits”. Firms thus solve (3) maximize $\displaystyle\quad\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}P_{j}(\mathbf{p})(p_{j}-c_{j})$ $\displaystyle\mathrm{with\;respect\;to}$ $\displaystyle\quad\mathbf{p}_{f}\in\mathcal{P}^{J_{f}}$ Before continuing with our framework, we discuss quantity-dependent costs and clarify when profits are bounded. #### 2.4.1. Quantity-Dependent Costs Including costs that depend on quantities produced is certainly possible, though this should introduce extra terms into the first-order equations presented below (Eqn. (7)). Generally speaking, unit costs that depend on the quantity produced would be expressed as $c_{j}:\mathbb{Z}_{+}\to\mathcal{P}$, and unit costs that depended on the expected quantity produced would be expressed as $c_{j}:\mathcal{P}\to\mathcal{P}$. If unit costs depend on the quantity produced, then product $j$’s unit costs for the purchasing period (i) are random and (ii) depend on prices. To see this, simply note that product $j$’s unit costs for the purchasing period are $c_{j}(Q_{j}(\mathbf{p}))$. Assuming quantity-dependent costs also obscures expected profits, since there are now nonlinear terms $Q_{j}(\mathbf{p})c_{j}(Q_{j}(\mathbf{p}))$ in the formula for random profits. If unit costs depend only on the expected quantity produced, then unit costs are not random but still depend on prices: $c_{j}(\mathbb{E}[Q_{j}(\mathbf{p})])=c_{j}(IP_{j}(\mathbf{p}))$. In either case the derivatives of unit costs with respect to prices should appear in the first-order conditions. This is acknowledged in the theoretical literature. As these terms have not yet been included in the empirical literature, even when costs are assumed to depend on quantities produced (Berry et al., 1995; Petrin, 2002), we focus on costs that are independent of the quantity produced. #### 2.4.2. Bounded and Vanishing Profits Here we present a technical assumption that ensures that profits are not only bounded, but vanish as all prices approach $\varsigma_{*}$. ###### Assumption 2.2. For all $j$ there exists some $r_{j}:\mathcal{T}\to(1,\infty)$ and some $\bar{p}_{j}:\mathcal{T}\to\mathcal{P}$ satisfying (4) $\sup\Big{\\{}\;p\mu(\\{\boldsymbol{\theta}:\bar{p}(\boldsymbol{\theta})>p\\})\;:\;p\in(0,\varsigma_{*})\;\Big{\\}}<\infty.$ such that (5) $u_{j}(\boldsymbol{\theta},p)\leq-r_{j}(\boldsymbol{\theta})\log p+\vartheta(\boldsymbol{\theta})$ for all $p\geq\bar{p}_{j}(\boldsymbol{\theta})$, $\mu$-a.e. ###### Lemma 2.1. If Assumption 2.2 holds, then $\hat{\pi}_{f}(\mathbf{p})$ is bounded in $\mathbf{p}$ and vanishes as $\mathbf{p}_{f}\to\varsigma_{*}\mathbf{1}\in\mathbb{R}^{J_{f}}$. ###### Proof. We use the Dominated Convergence Theorem. Eqn. (5) ensures that $p_{j}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})$ vanishes $\mu$-a.e. as $p_{j}\uparrow\varsigma_{*}$; see also Morrow and Skerlos (2008). Eqn. (4) ensures that $\hat{\pi}_{f}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}\int p_{j}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})-\sum_{j\in\mathcal{J}_{f}}c_{j}\int P_{j}(\mathbf{p})$ is bounded as prices approach $\varsigma_{*}$, as we now show. The key quantities in this integral are $\int p_{j}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})\leq\int p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)d\mu(\boldsymbol{\theta});$ the $c_{j}P_{j}(\mathbf{p})$ terms vanish if $p_{j}\uparrow\varsigma_{*}$ since $P_{j}(\mathbf{p})$ vanishes. We must show that these terms are bounded as $p_{j}\uparrow\varsigma_{*}$. By assumption, $p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)\leq\left(\frac{1}{p_{j}}\right)^{r_{j}(\boldsymbol{\theta})-1}\left(\frac{1}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)$ for all $p_{j}\geq\bar{p}_{j}(\boldsymbol{\theta})$. Thus we write $\displaystyle\int p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)d\mu(\boldsymbol{\theta})$ $\displaystyle=\int_{\\{\boldsymbol{\theta}:p_{j}<\bar{p}_{j}(\boldsymbol{\theta})\\}}p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)d\mu(\boldsymbol{\theta})$ $\displaystyle\quad\quad\quad\quad\quad\quad+\int_{\\{\boldsymbol{\theta}:p_{j}\geq\bar{p}_{j}(\boldsymbol{\theta})\\}}p_{j}\left(\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}{1+e^{u_{j}(\boldsymbol{\theta},p_{j})-\vartheta(\boldsymbol{\theta})}}\right)d\mu(\boldsymbol{\theta})$ $\displaystyle\leq p_{j}\mu{\\{\boldsymbol{\theta}:p_{j}<\bar{p}_{j}(\boldsymbol{\theta})\\}}$ $\displaystyle\quad\quad\quad\quad\quad\quad+\int_{\\{\boldsymbol{\theta}:p_{j}\geq\bar{p}_{j}(\boldsymbol{\theta})\\}}\left(\frac{1}{p_{j}}\right)^{r_{j}(\boldsymbol{\theta})-1}d\mu(\boldsymbol{\theta}).$ By Eqn. (4), the first term is bounded. We take $p_{j}>1$, without loss of generality, so that $1/p^{r_{j}(\boldsymbol{\theta})-1}\leq 1$ for $\mu$-a.e. $\boldsymbol{\theta}$ and the second term is bounded. ∎ We now make some remarks regarding Assumption 2.2. Note that if $\varsigma(\boldsymbol{\theta})<\infty$ then Eqn. (5) holds for any $r(\boldsymbol{\theta})>1$ by taking $\bar{p}(\boldsymbol{\theta})=\varsigma(\boldsymbol{\theta})$. If $\varsigma(\boldsymbol{\theta})=\infty$, Eqn. (5) admits any utility function $u(\boldsymbol{\theta},\cdot)$ that is (eventually) concave in price. If $\varsigma_{*}<\infty$, then $\varsigma(\boldsymbol{\theta})<\infty$ for $\mu$-a.e. $\boldsymbol{\theta}$. Furthermore, Eqn. (4) is trivial. To further analyze Eqn. (4), we assume $\varsigma_{*}=\infty$. We define $Z=\bar{p}(\boldsymbol{\Theta})$, where $\boldsymbol{\Theta}$ is the $\mathcal{T}$-valued random variable with $\mathbb{P}(\boldsymbol{\Theta}\in\mathcal{A})=\mu(\mathcal{A})=\int_{\mathcal{A}}d\mu(\boldsymbol{\theta})$. If $\varsigma(\boldsymbol{\theta})<\infty$ for $\mu$-a.e. $\boldsymbol{\theta}$, then we can take $Z=\Sigma=\varsigma(\boldsymbol{\Theta})$. Eqn. (4) can be re-written as $\sup\\{p\mathbb{P}(Z>p):p\in(0,\infty)\\}<\infty$, or equivalently $\lim_{p\to\infty}[p\mathbb{P}(Z>p)]<\infty$. Eqn. (4) admits any $Z$ with finite expectation, and even admits any $Z$ with a “fat-tailed” distribution satisfying $p^{1+\beta}\mathbb{P}(Z>p)\to 1$ as $p\to\infty$ for some $\beta>0$. Eqn. (4) can be written $\mathbb{P}(Z>p)=\mathcal{O}(1/p)$. ### 2.5. Local Equilibrium and the Simultaneous Stationarity Conditions Assuming that the choice probabilities are continuously differentiable in prices, at equilibrium each firm’s prices satisfy the stationarity condition (6) $(D_{k}\hat{\pi}_{f})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f}}(D_{k}P_{j})(\mathbf{p})(p_{j}-c_{j})+P_{k}(\mathbf{p})\quad\quad\text{for all }k\in\mathcal{J}_{f}.$ Combining the stationarity condition for each firm we obtain the Simultaneous Stationarity Condition, a first-order (necessary) condition for local equilibrium prices. ###### Proposition 2.2 (Simultaneous Stationarity Condition). Suppose $\mathbf{P}$ is continuously differentiable. Let $(\tilde{\nabla}\hat{\pi})(\mathbf{p})\in\mathbb{R}^{J}$ denote the “combined gradient” with components $((\tilde{\nabla}\hat{\pi})(\mathbf{p}))_{j}=(D_{j}\hat{\pi}_{f(j)})(\mathbf{p})$ where $f(j)$ denotes the index of the firm offering product $j$. If $\mathbf{p}$ is a local equilibrium, then (7) $(\tilde{\nabla}\hat{\pi})(\mathbf{p})=(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{c})+\mathbf{P}(\mathbf{p})=\mathbf{0}.$ where $(\tilde{D}\mathbf{P})(\mathbf{p})\in\mathbb{R}^{J\times J}$ is the “intra-firm” Jacobian matrix of price derivatives of the choice probabilities defined by (8) $\big{(}(\tilde{D}\mathbf{P})(\mathbf{p})\big{)}_{j,k}=\left\\{\begin{aligned} &(D_{k}P_{j})(\mathbf{p})&&\quad\text{if products $j$ and $k$ are offered by the same firm }\\\ &\quad\quad 0&&\quad\text{otherwise }\end{aligned}\right.$ Prices $\mathbf{p}$ satisfying Eqn. (7) are called “simultaneously stationary.” The matrix $-(\tilde{D}\mathbf{P})(\mathbf{p})$ has previously been denoted by “$\triangle$” (Berry et al., 1995; Petrin, 2002; Beresteanu and Li, 2008), “$\boldsymbol{\Omega}$” (Nevo, 2000a), and “$\boldsymbol{\Phi}$” (Dube et al., 2002). We prefer the “$D$” notation to emphasize the relationship of $(\tilde{D}\mathbf{P})(\mathbf{p})$ to the Jacobian matrix of the choice probabilities $\mathbf{P}$, while using the superscript “$\sim$” to denote the intra-firm sparsity structure. A set of simultaneously stationary prices are a local equilibrium only if every firm’s profits are locally maximized at those prices; this can be verified by confirming that every firm’s profits are locally concave (Section LABEL:SUBSECSufficiency). Note that there is no convenient condition to verify that every firm’s profits are globally maximized at a particular local equilibrium. That is, there is no convenient condition to ensure that certain prices are a proper equilibrium. ### 2.6. Choice Probability Derivatives In this section we examine the price derivatives of Mixed Logit choice probabilities. In what follows, $(Dw_{j})(p_{j})$ denotes the derivative of the price component of utility, $w_{j}$, with respect to price. ###### Proposition 2.3. Fix $\mathbf{p}\in(0,\varsigma_{*})^{J}$, let $u_{j}$ be given as in Assumption 2.1 for all $j$, and suppose the Leibniz Rule holds for the Mixed Logit choice probabilities $P_{j}(\mathbf{p})=\int P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$; that is, $(D_{k}P_{j})(\mathbf{p})=\int(D_{j}P_{k}^{L})(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$. Then the Jacobian matrix of $\mathbf{P}$ is given by (9) $(D\mathbf{P})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})-\boldsymbol{\Gamma}(\mathbf{p})$ where $\boldsymbol{\Lambda}(\mathbf{p})\in\mathbb{R}^{J\times J}$ is the diagonal matrix with diagonal entries $\lambda_{j}(\mathbf{p})=\int_{\mathcal{L}(p_{j})}(Dw_{j})(\boldsymbol{\theta},p_{j})P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta}),\quad\quad\mathcal{L}(p)=\\{\boldsymbol{\theta}:\varsigma(\boldsymbol{\theta})>p\\}$ and $\boldsymbol{\Gamma}(\mathbf{p})$ is the full $J\times J$ matrix with entries $\gamma_{j,k}(\mathbf{p})=\int_{\mathcal{G}(p_{j},p_{k})}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{k})(\boldsymbol{\theta},p_{k})d\mu(\boldsymbol{\theta}),\quad\quad\mathcal{G}(p,q)=\mathcal{L}(p)\cap\mathcal{L}(q).$ The intra-firm price derivatives of the Mixed Logit choice probabilities are given by $(\tilde{D}\mathbf{P})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})-\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ where $\big{(}\tilde{\boldsymbol{\Gamma}}(\mathbf{p})\big{)}_{j,k}=\gamma_{j,k}(\mathbf{p})$ if $f(j)=f(k)$ and $\big{(}\tilde{\boldsymbol{\Gamma}}(\mathbf{p})\big{)}_{j,k}=0$ otherwise. ###### Proof. We first characterize the Logit choice probabilities. For all $j,k$ we have $\displaystyle(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})$ $\displaystyle=P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})(\delta_{j,k}-P_{k}^{L}(\boldsymbol{\theta},\mathbf{p}))(Dw_{k})(\boldsymbol{\theta},p_{k})$ $\displaystyle=\delta_{j,k}P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{k})(\boldsymbol{\theta},p_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{k})(\boldsymbol{\theta},p_{k})$ for any $\boldsymbol{\theta}\in\mathcal{L}(p_{k})$ and $(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})=0$ for any $\boldsymbol{\theta}\in\\{\boldsymbol{\theta}^{\prime}\in\mathcal{T}:p_{k}>\varsigma(\boldsymbol{\theta}^{\prime})\\}$ (because $P_{j}^{L}(\boldsymbol{\theta},\cdot)$ is identically zero in a neighborhood of $\mathbf{p}$). Neglecting values $\boldsymbol{\theta}\in\varsigma^{-1}(p_{k})$ for the moment, we observe that these formulae and the Leibniz rule generate the desired expression for the Mixed Logit choice probabilities. We complete the proof by considering $\boldsymbol{\theta}\in\varsigma^{-1}(p_{k})$. If $\varsigma^{-1}(p_{k})$ has $\mu$-measure zero for any $p_{k}$, then we do not need to worry about Logit choice probability derivatives at $\boldsymbol{\theta}\in\varsigma^{-1}(p_{k})$. On the other hand if $\varsigma^{-1}(p_{k})$ has positive $\mu$-measure for some $p_{k}$, we must assume continuity of the Logit choice probability derivatives: i.e. $(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})\to 0$ as $p_{k}\uparrow\varsigma(\boldsymbol{\theta})$. Otherwise, the Logit choice probability derivative is not defined on a set of demographics with positive measure. ∎ $\boldsymbol{\lambda}$ is closely related to a familiar economic quantity. Recall that the “inclusive value,” or expected maximum utility, conditional on demographics is given by (Small and Rosen, 1981; Train, 2003) $\iota^{L}(\boldsymbol{\theta},\mathbf{p})=\log\left(e^{\vartheta(\boldsymbol{\theta})}+\sum_{j=1}^{J}e^{u_{j}(\boldsymbol{\theta},p_{j})}\right)$ It is easy to see that $\lambda_{k}$ is the derivative of the “aggregate inclusive value” $\iota(\mathbf{p})=\int\iota^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$ with respect to the $k^{\text{th}}$ price: $\lambda_{k}(\mathbf{p})=(D_{k}\iota)(\mathbf{p})=\int(D_{k}\iota^{L})(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})$. Note that $\boldsymbol{\Gamma}(\mathbf{p})$ and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ are not necessarily symmetric for all $\mathbf{p}$. If $(Dw_{k})(\boldsymbol{\theta},p)$ is independent of both $k$ and $p$, as in the case of the Boyd and Mellman (1980) model presented in Example 1 above, then $\boldsymbol{\Gamma}(\mathbf{p})$ (and thus $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$) is symmetric for all $\mathbf{p}$. On the other hand if $(Dw_{k})(\boldsymbol{\theta},\cdot)$ is independent of $k$ and strictly monotone in $p$, as is the case of the strictly concave in price utility from Berry et al. (1995), then $\gamma_{j,k}(\mathbf{p})=\gamma_{k,j}(\mathbf{p})$ if and only if $p_{j}=p_{k}$. The following assumption gives a simple, abstract condition on $(\mathbf{u},\vartheta,\mu)$ that guarantees the Leibniz Rule holds and defines continuously differentiable choice probabilities. ###### Assumption 2.3. Let $k$ be arbitrary and define $\psi_{k}:\mathcal{T}\times\mathcal{P}\to\mathcal{P}$ by $\psi_{k}(\boldsymbol{\theta},p)=\left\\{\begin{aligned} &\left\lvert(Dw_{k})(\boldsymbol{\theta},p)\right\rvert\left(\frac{e^{u_{k}(\boldsymbol{\theta},p)}}{e^{\vartheta(\boldsymbol{\theta})}+e^{u_{k}(\boldsymbol{\theta},p)}}\right)&&\quad\text{if }p<\varsigma(\boldsymbol{\theta})\\\ &\quad\quad\quad\quad\quad 0&&\quad\text{if }p\geq\varsigma(\boldsymbol{\theta})\\\ \end{aligned}\right.$ Assume (i) $\psi_{k}(\boldsymbol{\theta},\cdot):(0,\varsigma_{*})\to\mathcal{P}$ is continuous for $\mu$-a.e. $\boldsymbol{\theta}\in\mathcal{T}$; that is, $\psi_{k}(\boldsymbol{\theta},q)\to\psi_{k}(\boldsymbol{\theta},p)$ as $q\to p$ for any $p\in(0,\varsigma_{*})$. (ii) $\psi_{k}(\cdot,p^{\prime}):\mathcal{T}\to\mathcal{P}$ is uniformly $\mu$-integrable for all $p^{\prime}$ in some neighborhood of any $p\in(0,\varsigma_{*})$; that is, there exists some $\varphi:\mathcal{T}\to[0,\infty)$ with $\int\varphi(\boldsymbol{\theta})d\mu(\boldsymbol{\theta})<\infty$ (that may depend on $k$ and $p$), such that $\psi_{k}(\boldsymbol{\theta},p^{\prime})\leq\varphi(\boldsymbol{\theta})$ for all $p^{\prime}$ in some neighborhood of $p$. Note that under Assumption 2.1, (i) requires only that $\psi_{k}(\boldsymbol{\theta},p)\to 0$ as $p\uparrow\varsigma(\boldsymbol{\theta})$ for $\mu$-a.e. $\boldsymbol{\theta}$. ###### Proposition 2.4. If Assumption 2.3 holds, then the Leibniz Rule holds for the Mixed Logit choice probabilities which are also continuously differentiable on $(0,\varsigma_{*})^{J}$. ###### Proof. Taking for granted that $(D_{k}P_{j}^{L})(\boldsymbol{\theta},\cdot)$ is continuous at $\mathbf{p}$ and the differences (10) $h^{-1}\big{(}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}$ are uniformly $\mu$-integrable for small enough $h$, the Dominated Convergence Theorem implies that $\displaystyle\lim_{h\to 0}h^{-1}\Big{(}\int P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})d\mu(\boldsymbol{\theta})-\int P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta})\Big{)}$ $\displaystyle\quad\quad\quad\quad=\lim_{h\to 0}\int h^{-1}\big{(}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}d\mu(\boldsymbol{\theta})$ $\displaystyle\quad\quad\quad\quad=\int\lim_{h\to 0}h^{-1}\big{(}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}d\mu(\boldsymbol{\theta})$ $\displaystyle\quad\quad\quad\quad=\int(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta}).$ This validates the Leibniz Rule. This proof is essentially that given in a general setting by (Bartle, 1966, Chapter 5, pg. 46). To complete the proof we must validate that $(D_{k}P_{j}^{L})(\boldsymbol{\theta},\cdot)$ is continuous in $p_{k}$ and the differences in Eqn. (10) are uniformly $\mu$-integrable in a neighborhood of $p_{k}$. It is easy to see that the desired continuity follows from Assumption 2.1 and Assumption 2.3, Condition (i). Specifically, note that $(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})=0=\psi_{k}(\boldsymbol{\theta},p_{k})$ for $\boldsymbol{\theta}\in\\{\boldsymbol{\theta}^{\prime}\in\mathcal{T}:p_{k}>\varsigma(\boldsymbol{\theta}^{\prime})\\}$ and $\displaystyle(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})$ $\displaystyle=\big{(}\delta_{j,k}-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{k})(\boldsymbol{\theta},p_{k})$ $\displaystyle=\big{(}\delta_{j,k}-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}\left(\frac{e^{\vartheta(\boldsymbol{\theta})}+e^{u_{k}(\boldsymbol{\theta},p_{k})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i=1}^{J}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\left(\frac{e^{u_{k}(\boldsymbol{\theta},p_{k})}}{e^{\vartheta(\boldsymbol{\theta})}+e^{u_{k}(\boldsymbol{\theta},p_{k})}}\right)(Dw_{k})(\boldsymbol{\theta},p_{k})$ $\displaystyle=\big{(}\delta_{j,k}-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}\left(\frac{e^{\vartheta(\boldsymbol{\theta})}+e^{u_{k}(\boldsymbol{\theta},p_{k})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i=1}^{J}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\psi_{k}(\boldsymbol{\theta},p_{k})$ for $\boldsymbol{\theta}\in\mathcal{L}(p_{k})$. Suppose $p_{k}=\varsigma(\boldsymbol{\theta})$. By Assumption 2.1 (a) and (b), the first two terms are continuous. By Assumption 2.1 (c), $\displaystyle\lim_{\mathbf{q}\to\mathbf{p},q_{k}<\varsigma(\boldsymbol{\theta})}(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})$ $\displaystyle=\left(\delta_{j,k}-\frac{e^{u_{j}(\boldsymbol{\theta},p_{j})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i\neq k}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\left(\frac{e^{\vartheta(\boldsymbol{\theta})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i\neq k}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\lim_{q_{k}\uparrow\varsigma(\boldsymbol{\theta})}\psi_{k}(\boldsymbol{\theta},q_{k})$ Assumption 2.3, Condition (i) is then necessary for the continuity of $(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})$ for all $j,k$ and $\mathbf{p}\in(\mathbf{0},\varsigma_{*}\mathbf{1})$. Specifically if $\psi_{k}(\boldsymbol{\theta},\cdot)$ is discontinuous at $\varsigma(\boldsymbol{\theta})$, then $\displaystyle\lim_{\mathbf{q}\to\mathbf{p},q_{k}<\varsigma(\boldsymbol{\theta})}(D_{k}P_{k}^{L})(\boldsymbol{\theta},\mathbf{p})$ $\displaystyle=\left(\frac{e^{\vartheta(\boldsymbol{\theta})}}{e^{\vartheta(\boldsymbol{\theta})}+\sum_{i\neq k}e^{u_{i}(\boldsymbol{\theta},p_{i})}}\right)\lim_{q_{k}\uparrow\varsigma(\boldsymbol{\theta})}\psi_{k}(\boldsymbol{\theta},q_{k})$ To prove the integrability, we first note that for all $j,k$ and $\mathbf{p}$ we have $\left\lvert(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})\right\rvert\leq\psi_{k}(\boldsymbol{\theta},p_{k})$. This bound is a consequence of the formula above, and is tight as $\mathbf{p}_{-k}$ varies. The mean value theorem for functions of a single real variable states that $\displaystyle h^{-1}(P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}))=(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p}+\eta\mathbf{e}_{k})$ for some $\eta$ such that $\left\lvert\eta\right\rvert<\left\lvert h\right\rvert$, and thus $\displaystyle\left\lvert h\right\rvert^{-1}\left\lvert P_{j}^{L}(\boldsymbol{\theta},\mathbf{p}+h\mathbf{e}_{k})-P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\right\rvert\leq\psi_{k}(\boldsymbol{\theta},p_{k}+\eta)\leq\varphi(\boldsymbol{\theta})$ for $\mu$-a.e. $\boldsymbol{\theta}\in\mathcal{T}$ and small enough $h$. Thus, the desired uniform $\mu$-integrability follows from Assumption 2.3, Condition (ii). ∎ An “easier” bound is simply $\left\lvert(D_{k}P_{j}^{L})(\boldsymbol{\theta},\mathbf{p})\right\rvert\leq\left\lvert(Dw_{k})(\boldsymbol{\theta},p_{k})\right\rvert$, and thus we might consider changing the statement of Proposition 2.4 to hypothesize only the uniform $\mu$-integrability of the utility price derivatives. In fact, this bound can be used to validate the Leibniz Rule for the Boyd and Mellman model of Example 1 that lacks an outside good. However, this bound fails to be useful for the Berry et al. model of Example 2, since $w(p)=\alpha\log(\varsigma(\boldsymbol{\theta})-p)$ and $\left\lvert(Dw_{k})(\boldsymbol{\theta},p_{k})\right\rvert=\alpha/(\varsigma(\boldsymbol{\theta})-p)$ is singular on $\varsigma^{-1}(p)$. In empirical applications, $\varsigma$ is onto, generating a singularity somewhere in $\mathcal{T}$ for all $p$; this singularity cannot be “controlled” for all $p$ by choosing the measure $\mu$. In this case, a hypothesis only about the price derivatives of utility is not useful. We close this section by stating some basic results concerning $(\tilde{D}\mathbf{P})(\mathbf{p})$ that are used below. ###### Lemma 2.5. Under Assumption 2.1, $P_{j}(\mathbf{p})$ and $\lambda_{j}(\mathbf{p})$ are never zero on $(0,\varsigma_{*})^{J}$. Thus $\boldsymbol{\Lambda}(\mathbf{p})$ is nonsingular for all $\mathbf{p}\in(0,\varsigma_{*})^{J}$. ###### Proof. Note that $\mathcal{L}(p_{j})$ is nonempty and has positive $\mu$-measure, $P_{j}^{L}(\cdot,\mathbf{p})$ is strictly positive on $\mathcal{L}(p_{j})$, and $(Dw_{j})(\cdot,p_{j})P_{j}^{L}(\cdot,\mathbf{p})$ is strictly negative on $\mathcal{L}(p_{j})$. It follows that $P_{j}(\mathbf{p})$ and $\lambda_{j}(\mathbf{p})$ are nonzero. ∎ ###### Lemma 2.6. Let $\mathbf{p}\in(0,\varsigma_{*})^{J}$, suppose $\vartheta:\mathcal{T}\to(-\infty,\infty)$, and define (11) $\displaystyle\boldsymbol{\Omega}_{f}(\mathbf{p})$ $\displaystyle=\boldsymbol{\Lambda}_{f}(\mathbf{p})^{-1}\boldsymbol{\Gamma}_{f}(\mathbf{p})^{\top}\in\mathbb{R}^{J_{f}\times J_{f}}\text{ for all }f$ (12) $\displaystyle\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ $\displaystyle=\boldsymbol{\Lambda}(\mathbf{p})^{-1}\tilde{\boldsymbol{\Gamma}}(\mathbf{p})^{\top}\in\mathbb{R}^{J\times J}.$ These matrices are well-defined by Lemma 2.5, and have the following properties: * (i) $(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}=\boldsymbol{\Lambda}_{f}(\mathbf{p})(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))$ and $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}=\boldsymbol{\Lambda}(\mathbf{p})(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))$. * (ii) $\lvert\lvert\boldsymbol{\Omega}_{f}(\mathbf{p})\rvert\rvert_{\infty}<1$ and $\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}<1$. * (iii) $\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p})\in\mathbb{R}^{J_{f}\times J_{f}}$ and $\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})\in\mathbb{R}^{J\times J}$ are strictly diagonally dominant and nonsingular. * (iv) $(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))^{-1}\in\mathbb{R}^{J_{f}\times J_{f}}$ and $(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))^{-1}\in\mathbb{R}^{J\times J}$ map positive vectors to positive vectors. ###### Proof. * (i) This follows immediately from Prop. 2.3. * (ii) We note that $\displaystyle\omega_{k,l}(\mathbf{p})=\frac{\gamma_{l,k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})}=\int_{\mathcal{L}(p_{k})}P_{l}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})$ where $\mu_{k,\mathbf{p}}$ is the probability distribution with density, with respect to $\mu$, given by $d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})=\frac{P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})\left\lvert(Dw_{k})(\boldsymbol{\theta},p_{k})\right\rvert d\mu(\boldsymbol{\theta})}{\int_{\mathcal{L}(p_{k})}P_{k}^{L}(\boldsymbol{\phi},\mathbf{p})\left\lvert(Dw_{k})(\boldsymbol{\phi},p_{k})\right\rvert d\mu(\boldsymbol{\phi})}.$ Thus $\boldsymbol{\Lambda}_{f}(\mathbf{p})^{-1}\boldsymbol{\Gamma}_{f}(\mathbf{p})^{\top}$ has row sums $\displaystyle\int\left(\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\right)d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})<1.$ The additional assumption that $\vartheta:\mathcal{T}\to(-\infty,\infty)$ plays a role in establishing this inequality because then there is always a set $\mathcal{T}_{k}^{\prime}\subset\mathcal{T}$ with $\mu_{k,\mathbf{p}}(\mathcal{T}_{k}^{\prime})>0$ on which $\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})<1$. * (iii) The inequality $\displaystyle 1>\int\left(\sum_{j\in\mathcal{J}_{f}}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\right)d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})$ is equivalent to $\displaystyle\left\lvert 1-\omega_{k,k}(\mathbf{p})\right\rvert$ $\displaystyle=1-\int P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})$ $\displaystyle>\int\left(\sum_{j\in\mathcal{J}_{f}\setminus k}P_{j}^{L}(\boldsymbol{\theta},\mathbf{p})\right)d\mu_{k,\mathbf{p}}(\boldsymbol{\theta})=\sum_{l\neq k}\omega_{k,l}(\mathbf{p}).$ The claim follows. * (iii) Because $\boldsymbol{\Omega}_{f}(\mathbf{p})$ maps positive vectors to positive vectors, so does its power series $\sum_{n=1}^{\infty}\boldsymbol{\Omega}_{f}(\mathbf{p})^{n}=(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))^{-1}.$ ∎ ###### Corollary 2.7. $(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}$ and $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ are strictly diagonally dominant and nonsingular for $\mathbf{p}\in(\mathbf{0},\varsigma_{*}\mathbf{1})$. ###### Proof. This follows directly from Lemma 2.6, claims (i) and (iii). ∎ ### 2.7. The BLP-Markup Equation A prominent form of the first-order conditions Eqn. (7) is the BLP-markup equation: (13) $\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})\quad\text{where}\quad\boldsymbol{\eta}(\mathbf{p})=-(\tilde{D}\mathbf{P})(\mathbf{p})^{-\top}\mathbf{P}(\mathbf{p}).$ Note that $\boldsymbol{\eta}$ is defined for any continuously differentiable choice probabilities with nonsingular $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$. We have shown above that this applies to certain Mixed Logit models (Section 2.6). Eqn. (13) and Corollary 2.7 show that $\boldsymbol{\eta}$ is well-defined and continuous, at least for $\mathbf{p}\in(0,\varsigma_{*})^{J}$. Traditionally, the BLP-markup equation (13) has been used to estimate costs assuming observed prices are in equilibrium via the formula $\mathbf{c}=\mathbf{p}-\boldsymbol{\eta}(\mathbf{p})$; see, e.g., Berry et al. (1995) or Nevo (2000a). These cost estimates form the basis of counterfactual experiments with an estimated demand model. Beresteanu and Li (2008) have recently suggested that the BLP-markup equation is also useful for computing equilibrium prices, a suggestion we explore further below. Note that the BLP- markup equation must be interpreted as a nonlinear fixed-point equation when applied to compute equilibrium prices. We now derive several important properties of $\boldsymbol{\eta}$ from an alternative form of Eqn. (13) based on Lemma 2.6, valid when $\mathbf{p}\in(0,\varsigma_{*})^{J}$: (14) $\Big{(}\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})\Big{)}\boldsymbol{\eta}(\mathbf{p})=-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p}).$ First, Eqn. (14) proves that profit-optimal markups are positive for the class of Mixed Logit models we consider, thanks to Lemma 2.6, claim (iv). ###### Corollary 2.8. Suppose Assumptions 2.1-2.3 hold. Then $\boldsymbol{\eta}(\mathbf{p})>\mathbf{0}$ for all $\mathbf{p}\in(0,\varsigma_{*})^{J}$. Hence if $\mathbf{p}\in(0,\varsigma_{*})^{J}$ is a local equilibrium, then $\mathbf{p}>\mathbf{c}$. Second, Eqn. (14), rather than Eqn. (13), should be used to actually compute $\boldsymbol{\eta}$. Recall that $\kappa_{2}(\mathbf{A})$ denotes the 2-norm condition number of the matrix $\mathbf{A}$ (Trefethen and Bau, 1997). ###### Lemma 2.9. Suppose Assumptions 2.1-2.3 hold. Eqn. (14) is better conditioned than Eqn. (13), in the sense that $\kappa_{2}\big{(}\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})\big{)}\leq\left(\frac{\min_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}{\max_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}\right)\kappa_{2}\big{(}(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}\big{)}$ for all $\mathbf{p}\in(\mathbf{0},\varsigma_{*}\mathbf{1})$. ###### Proof. This follows from Lemma 2.6, claim (i), the inequality $\kappa_{2}(\mathbf{AB})\leq\kappa_{2}(\mathbf{A})\kappa_{2}(\mathbf{B})$ valid for any matrices $\mathbf{A}$ and $\mathbf{B}$, and the formula $\kappa_{2}(\boldsymbol{\Lambda}(\mathbf{p})^{-1})=\frac{\min_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}{\max_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}.$ ∎ Lemma 2.9 states that the greater the variation in aggregate absolute rate of change in inclusive values, the more poorly conditioned $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ is relative to $\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})$. Because $\boldsymbol{\Lambda}(\mathbf{p})$ is diagonal, $\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{1}=\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{2}=\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{\infty}$ and thus the same bound applies for condition numbers in norms other than the 2-norm. Third, Eqn. (14) also provides bounds on the magnitude of values taken by $\boldsymbol{\eta}$: ###### Lemma 2.10. Suppose Assumptions 2.1-2.3 hold. For all $\mathbf{p}\in(0,\varsigma_{*})^{J}$, $\boldsymbol{\eta}$ satisfies (15) $\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{1+\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}}\leq\lvert\lvert\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\leq\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{1-\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}}.$ ###### Proof. This follows immediately from Eqn. (14), using the triangle inequality. ∎ The upper bound suggests the following assumptions to ensure that $\boldsymbol{\eta}$ itself is bounded: ###### Assumption 2.4. Suppose there exist $M\in(0,\infty)$ and $\varepsilon\in(0,1)$ such that (16) $\displaystyle\sup\big{\\{}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}\;:\;\mathbf{p}\in(0,\varsigma_{*})^{J}\big{\\}}$ $\displaystyle=M<\infty$ (17) $\displaystyle\sup\big{\\{}\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}\;:\;\mathbf{p}\in(0,\varsigma_{*})^{J}\big{\\}}$ $\displaystyle=1-\varepsilon<1.$ Under simple Logit, $P_{k}^{L}(\mathbf{p})/\left\lvert\lambda_{k}(\mathbf{p})\right\rvert=\left\lvert(Dw_{k})(p_{k})\right\rvert^{-1}$ and $\boldsymbol{\Omega}_{f}(\mathbf{p})=\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}$. Thus Eqn. (16) is akin to concavity of $w_{k}$ for all sufficiently large $p_{k}$, and Eqn. (17) is implied by $\vartheta>-\infty$, i.e. the existence of an outside good with positive purchase probability. ###### Lemma 2.11. Suppose Assumptions 2.1-2.3 hold. * (i) If Assumption 2.4 also holds, $N=\sup\\{\lvert\lvert\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}:\mathbf{p}\in(0,\varsigma_{*})^{J}\\}<\infty$. * (ii) Moreover Eqn. (16) in Assumption 2.4 is necessary for $N$ to be finite. Unfortunately some simple models do not satisfy Assumption 2.4. A simple Logit model with $w(p)=-\alpha\log p$ for some $\alpha>0$ violates Eqn. (16). More generally, the Boyd and Mellman (1980) model of Example 1 does not satisfy Eqn. (16). This is most easily seen by noting that finite-sample approximations to this model have $\lim_{p_{k}\to\infty}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}=\max_{s=1,\dotsc,S}\left\\{\frac{1}{\alpha_{s}}\right\\}$ where $\\{\alpha_{s}\\}_{s=1}^{S}$ are the sampled price coefficients. Of course, as $S\to\infty$, $\min_{s=1,\dotsc,S}\\{\alpha_{s}\\}\to 0$, and thus $\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}\to\infty$. ### 2.8. The $\zeta$-Markup function Substituting Eqn. (9) into Eqn. (7) yields the $\boldsymbol{\zeta}$-markup equation introduced in Morrow and Skerlos (2010): (18) $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})\quad\text{where}\quad\boldsymbol{\zeta}(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})^{-1}\tilde{\boldsymbol{\Gamma}}(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{c})-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$ when $\boldsymbol{\Lambda}(\mathbf{p})$ is nonsingular (Section 2.6, 2.8). Thus the $\boldsymbol{\zeta}$-markup equation is specific to Mixed Logit models, unlike the BLP-markup equation. We observe a relationship between the maps $\boldsymbol{\zeta}$ and $\boldsymbol{\eta}$. ###### Proposition 2.12. Suppose Assumption 2.1-2.3 hold. For any $\mathbf{p}\in(0,\varsigma_{*})^{J}$, $\boldsymbol{\zeta}(\mathbf{p})=\tilde{\boldsymbol{\Omega}}(\mathbf{p})(\mathbf{p}-\mathbf{c})+(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))\boldsymbol{\eta}(\mathbf{p})$. ###### Proof. This follows directly from Eqns. (14) and (18). ∎ In so far as $\boldsymbol{\eta}$ and $\boldsymbol{\zeta}$ are distinct maps, they can generate numerical methods for the computation of equilibrium prices with entirely different properties. The equation above implies that $\boldsymbol{\zeta}(\mathbf{p})=\boldsymbol{\eta}(\mathbf{p})$ if, and only if, $\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$ lies in the null space of $\tilde{\boldsymbol{\Omega}}(\mathbf{p})$. Thus if $\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ is full-rank, $\boldsymbol{\zeta}$ and $\boldsymbol{\eta}$ coincide only at simultaneously stationary prices. Simple examples of Mixed Logit models can be constructed that always have $\mathrm{rank}(\tilde{\boldsymbol{\Omega}}(\mathbf{p}))=J$. For Logit, $\boldsymbol{\Omega}_{f}(\mathbf{p})=\mathbf{1}\mathbf{P}_{f}^{L}(\mathbf{p})^{\top}$ for all $f$ and $\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ always has rank $F\leq J$. However the analysis in Morrow and Skerlos (2008) can be used to show that $\boldsymbol{\zeta}$ and $\boldsymbol{\eta}$ coincide only at simultaneously stationary prices. We now explore $\boldsymbol{\zeta}$’s asymptotic properties. ###### Lemma 2.13. Under Assumption 2.4 $\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}<\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}$ whenever $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}>M/\varepsilon$. Moreover $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\to\infty$ as $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}\to\infty$. ###### Proof. We simply note that $\displaystyle\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}$ $\displaystyle\leq\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}+\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}$ $\displaystyle\leq(1-\varepsilon)\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}+M$ $\displaystyle\leq\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\big{(}\varepsilon\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-M\big{)}$ $\displaystyle\leq\left[1-\varepsilon+\frac{M}{\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}}\right]\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}$ Now if $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}>M/\varepsilon$, then $M/\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}<\varepsilon$. Thus $\displaystyle\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}<\left[1-\varepsilon+\varepsilon\right]\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}=\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}.$ To prove that $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\to\infty$, note that $\displaystyle\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\geq\left(\varepsilon-\frac{M}{\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}}\right)\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}$ For all $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}>M/\varepsilon$, the term in parentheses is positive. Furthermore, this term approaches $\varepsilon$ as $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}\to\infty$. Thus $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\to\infty$ as $\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}\to\infty$. ∎ A slightly different assumption concerning $\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ is useful when analyzing the $\boldsymbol{\zeta}$ map. ###### Assumption 2.5. Suppose (19) $\sup\Big{\\{}\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})(\mathbf{p}-\mathbf{c})\rvert\rvert_{\infty}:\mathbf{p}\in(0,\varsigma_{*})^{J}\Big{\\}}<\infty,$ ###### Lemma 2.14. Suppose Assumption 2.1-2.3 holds and $\varsigma_{*}=\infty$. Then $\boldsymbol{\zeta}$ is bounded if, and only if, Eqn. (16) and Eqn. (19) hold. ###### Proof. This follows directly from the triangle inequality and the non-negativity of $\tilde{\boldsymbol{\Omega}}(\mathbf{p})(\mathbf{p}-\mathbf{c})$ and $-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$ for all $\mathbf{p}\geq\mathbf{c}$. ∎ For future reference, we prove that Eqn. (19) strengthens Eqn. (17). ###### Lemma 2.15. If Eqn. (19) holds, then Eqn. (17) holds. ###### Proof. Note that Eqn. (19) implies that for any $k$, $\lim_{p_{j}\to\infty}\big{(}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})\big{)}<\infty\quad\text{for all}\quad j\in\mathcal{J}_{f(k)}.$ This, in turn, implies that $\omega_{k,j}(\mathbf{p})\to 0$ as $p_{j}\to\infty$. Now Eqn. (17) fails only if $\lim_{\mathbf{p}\to\mathbf{q}}\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}=1$ where $\mathbf{q}$ has some $q_{j}=\infty$. But the row sums of $\tilde{\boldsymbol{\Omega}}(\mathbf{p})$ satisfy $\lim_{\mathbf{p}\to\mathbf{q}}\left[\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})\right]=\sum_{j\in\mathcal{J}_{f(k)}\cup\\{j:q_{j}<\infty\\}}\omega_{k,j}(\mathbf{q})<1.$ Thus if Eqn. (19) holds, Eqn. (17) cannot fail. ∎ ### 2.9. Existence of Simultaneously Stationary Prices This section provides two existence results. Neither establish the existence of a local equilibrium, or the uniqueness of simultaneously stationary points. To address the existence of local equilibrium will require additional conditions to ensure that each firm’s profits are locally concave at the simultaneously stationary prices whose existence can be ensured (Morrow, 2008). Little is known about how to address the uniqueness of simultaneously stationary points. Indeed, Morrow and Skerlos (2010) provide an example of a Mixed Logit model with 9 simultaneously stationary prices, 4 of which are local equilibria and 2 of which are proper equilibria. Assumption 2.4 ensures the existence of finite simultaneously stationary prices when $\varsigma_{*}=\infty$. ###### Corollary 2.16. Suppose $\varsigma_{*}=\infty$ and Assumptions 2.1-2.3, 2.4 hold. Then there exists at least one vector of finite simultaneously stationary prices. ###### Proof. This is a direct consequence of Brouwer’s fixed-point theorem. $\mathbf{c}+\boldsymbol{\eta}(\cdot)$ is a continuous map that takes the compact, convex set $[0,M/\varepsilon]^{J}$ into itself, and thus there is at least one fixed-point $\mathbf{p}=\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})\in[0,M/\varepsilon]^{J}$. ∎ To apply Corollary 2.16 to cases when $\varsigma_{*}<\infty$, $\boldsymbol{\eta}$ must be extended from $(0,\varsigma_{*})$ to all of $\mathcal{P}^{J}$ preserving the bounds (15). This is easy for many of the simulation-based approximations encountered in practice, but difficult for the general case. We can extend this existence result using Eqn. (22) and the $\boldsymbol{\zeta}$ map. ###### Lemma 2.17. Suppose $\varsigma_{*}=\infty$, Assumptions 2.1-2.3, Eqn. (19), and Eqn. (22) hold. Then there exists some $\bar{q}_{k}>c_{k}$ such that $p_{k}-c_{k}-\zeta_{k}(\mathbf{p})>0$ for all $\mathbf{p}$ with $p_{k}\geq\bar{q}_{k}$. ###### Proof. The assumed bound implies $\left\lvert\sum_{j\in\mathcal{J}_{f}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})\right\rvert\leq L<\infty$ for any $k$ and any $\mathbf{p}\in(0,\infty)^{J}$. Consider $\displaystyle p_{k}-c_{k}-\zeta_{k}(\mathbf{p})$ $\displaystyle=(p_{k}-c_{k})-\sum_{j\in\mathcal{J}_{f}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})+\frac{P_{k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})}$ $\displaystyle=\left(1-\frac{P_{k}(\mathbf{p})}{\left\lvert\lambda_{k}(\mathbf{p})\right\rvert(p_{k}-c_{k})}\right)(p_{k}-c_{k})-\sum_{j\in\mathcal{J}_{f}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})$ If Eqn. (22) holds, then $0\leq\lim_{p_{k}\to\infty}\left[\frac{P_{k}(\mathbf{p})}{\left\lvert\lambda_{k}(\mathbf{p})\right\rvert(p_{k}-c_{k})}\right]\leq\delta<1.$ Thus for any $\epsilon>0$, there exists some $\bar{p}_{k}>0$ and $\triangle(\mathbf{p})$ with $\left\lvert\triangle(\mathbf{p})\right\rvert<\epsilon$ such that $\frac{P_{k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})(p_{k}-c_{k})}\leq\delta+\triangle(\mathbf{p})\quad\quad\text{for all}\quad\quad p_{k}>\bar{p}_{k}.$ Thus $\displaystyle p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\geq(1-\delta+\triangle(\mathbf{p}))(p_{k}-c_{k})-L\quad\quad\text{for all}\quad\quad p_{k}>\bar{p}_{k}.$ In particular, if we choose $\epsilon\leq(1-\delta)/2$ we have $\displaystyle p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\geq\left(\frac{1-\delta}{2}\right)(p_{k}-c_{k})-L=\left(\frac{1-\delta}{2}\right)\left(p_{k}-c_{k}-\frac{2L}{1-\delta}\right)>0$ for all $p_{k}\geq\bar{q}_{k}=\max\\{c_{k}+2L/(1-\delta),\bar{p}_{k}\\}$. ∎ One consequence of this lemma is that infinite prices are never an equilibrium. ###### Corollary 2.18. Under the assumptions of Lemma 2.17, any profit derivative is eventually negative. ###### Proof. Note that $(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=-\left\lvert\lambda_{k}(\mathbf{p})\right\rvert(p_{k}-c_{k}-\zeta_{k}(\mathbf{p})).$ Since $p_{k}-c_{k}-\zeta_{k}(\mathbf{p})$ is positive for all large enough $p_{k}$, $(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})$ is negative for all large enough $p_{k}$, regardless of $\mathbf{p}_{-k}$. ∎ Another consequence of Lemma 2.17 is an alternative existence result. ###### Corollary 2.19. Under the assumptions of Lemma 2.17 there exists at least one simultaneously stationary point. ###### Proof. Following Morrow and Skerlos (2008), we prove this proposition using the Poincare-Hopf Theorem (Milnor, 1965). The logic is simple: We will consider the vector field $\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$ on a hyper-rectangle $[\mathbf{c},\bar{\mathbf{q}}]$ whose critical points are simultaneously stationary; $\bar{\mathbf{q}}$ has components $\bar{q}_{k}$ defined in Lemma 2.17. The Poincare-Hopf Theorem then states that the sum of the indices of all the critical points of this vector field equals one, the Euler characteristic of $[\mathbf{c},\bar{\mathbf{q}}]$. Thus it is not possible that the vector field have no critical points, for then the sum of indices would be zero. We must only prove one property of $\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$: that this vector field points outward on the boundary of the chosen hyper-rectangle. Half of this proof is Lemma 2.17, in which we prove that $p_{k}-c_{k}-\zeta_{k}(\mathbf{p})>0$ if $\mathbf{p}\in[\mathbf{c},\bar{\mathbf{q}}]$ with $p_{k}=\bar{q}_{k}$. We must also show that $p_{k}-c_{k}-\zeta_{k}(\mathbf{p})<0$ if $\mathbf{p}\in[\mathbf{c},\bar{\mathbf{q}}]$ with $p_{k}=c_{k}$. But $\displaystyle c_{k}-c_{k}-\zeta_{k}(\mathbf{p})=-\left(\sum_{j\in\mathcal{J}_{f}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})+\frac{P_{k}(\mathbf{p})}{\left\lvert\lambda_{k}(\mathbf{p})\right\rvert}\right)<0.$ ∎ This proof does not need to make any claims about the number of critical points, or of their indices. If it can be shown that any critical point of $\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$ cannot have a zero or negative index, then the simultaneously stationary point is unique. ## 3\. Computational Methods This section provides details for the four approaches to computing equilibrium prices described in Morrow and Skerlos (2010); see Table 4. Section 3.1 briefly reviews Newton’s method, followed by application of Newton’s method to solve Eqn. (7) in Section 3.2. Newton’s method applied directly to Eqn. (7) may compute “spurious” solutions with infinite prices because the combined gradient vanishes as prices increase without bound. Section 3.3 avoids this difficulty by applying Newton’s method to the two markup equations instead of Eqn. (7) itself. Section 3.4 discusses fixed-point iterations based on the BLP- and $\boldsymbol{\zeta}$-markup equations, and Section 3.5 reviews a number of practical considerations. Table 4. Summary of the numerical methods examined in this article. Newton Methods (NM) --- Abbr. | Method | Section | Advantage | Our Experience${}^{\text{(a)}}$ CG-NM | Solve $\mathbf{F}_{\pi}(\mathbf{p})=(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{0}$ | 3.2 | $-$ | Unreliable, slow $\boldsymbol{\eta}$-NM | Solve $\mathbf{F}_{\eta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})=\mathbf{0}$ | 3.3 | Coercive | Reliable, slow $\boldsymbol{\zeta}$-NM | Solve $\mathbf{F}_{\zeta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})=\mathbf{0}$ | 3.3 | Coercive | Reliable, slow Fixed-Point Iterations (FPI) Abbr. | Method | Section | Advantage | Our Experience $\boldsymbol{\zeta}$-FPI | Iterate $\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ | 3.4 | Easy to evaluate | Reliable, fast $\boldsymbol{\eta}$-FPI | Iterate $\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$ | 3.4 | $-$ | Not convergent (a) Conclusions on behavior of these methods is based on the numerical experiments described in Morrow and Skerlos (2010), using a novel GMRES-Newton method with Levenberg-Marquardt style trust-region global convergence strategy. ### 3.1. Newton’s Method Newton’s method, a classical technique to compute a zero of an arbitrary function $\mathbf{F}:\mathbb{R}^{J}\to\mathbb{R}^{J}$, is now a portfolio of related approaches to solve nonlinear systems (Ortega and Rheinboldt, 1970; Kelley, 1995; Dennis and Schnabel, 1996; Judd, 1998; Kelley, 2003). Generally speaking, Newton-type methods are differentiated in two relatively independent directions: (i) the technique used to approximate the Jacobian matrices $(D\mathbf{F})$ and solve for the Newton step and (ii) the technique used to enforce convergence from arbitrary initial conditions. See Dennis and Schnabel (1996), Judd (1998), or Kelley (2003) for good treatments of these issues. Choosing the right variant of Newton’s method determines the reliability and efficiency of equilibrium price computations. Problem formulation also determines the reliability and efficiency of equilibrium price computations using Newton’s method. Scalings of the variables and function values are one prominent example of a problem transformation that improves the performance of Newton’s method (Dennis and Schnabel, 1996). Nonlinear problem preconditioning can also be important (Cai and Keyes, 2002), as the following example demonstrates. ###### Example 5. Let $\mathbf{F}:\mathbb{R}^{N}\to\mathbb{R}^{N}$ be defined by $\mathbf{F}(\mathbf{x})=\mathbf{x}/(1+\lvert\lvert\mathbf{x}\rvert\rvert_{2}^{2})$. Iterating Newton steps converges to the unique (finite) zero $\mathbf{x}_{*}=\mathbf{0}$ only from initial conditions $\mathbf{x}_{0}$ with $\lvert\lvert\mathbf{x}_{0}\rvert\rvert_{2}<1/\sqrt{3}$. Newton’s method diverges or fails for all other starting points. Standard global convergence strategies for Newton’s method (line search, trust region methods) cannot improve this poor global convergence behavior because $\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}$ has unbounded level sets; see Morrow and Skerlos (2010) for details. A simple nonlinear transformation overcomes this poor global convergence behavior. Note that $\mathbf{F}(\mathbf{x})=\mathbf{A}(\mathbf{x})\mathbf{f}(\mathbf{x})$ where $\mathbf{A}(\mathbf{x})=(1+\lvert\lvert\mathbf{x}\rvert\rvert_{2}^{2})^{-1}\mathbf{I}$ and $\mathbf{f}(\mathbf{x})=\mathbf{x}$. Because $\mathbf{A}(\mathbf{x})$ is nonsingular for all $\mathbf{x}$, the problems $\mathbf{F}(\mathbf{x})=\mathbf{0}$ and $\mathbf{f}(\mathbf{x})=\mathbf{0}$ have identical solution sets. However applying Newton’s method to the problem $\mathbf{f}(\mathbf{x})=\mathbf{0}$ trivially converges in a single step from any initial condition without a global convergence strategy. Example 5 illustrates why computing equilibrium prices based on the markup equations is more reliable and efficient than using Eqn. (7) directly. The following two sections echo the pattern of this example to provide the details. ### 3.2. Newton’s Method on the Combined Gradient The most direct approach to compute equilibrium prices using Newton’s method is to solve $\mathbf{F}_{\pi}(\mathbf{p})=(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{0}$, abbreviated CG-NM in Table 4. This approach works well when the initial condition is near an equilibrium, as required by theory (Ortega and Rheinboldt, 1970; Kelley, 1995; Dennis and Schnabel, 1996). In practice, computing counterfactual equilibrium prices starting with the observed prices may exploit this local convergence if changes to exogeneous variables have a relatively small impact on equilibrium prices. On the other hand, CG-NM can be unreliable when started “far” from equilibrium. The challenge is the tendency for the derivatives of profits to vanish as prices become large Morrow and Skerlos (2010), as demonstrated in Example 6 below. ###### Example 6. Consider a simple Logit model with linear in price utility and an outside good: $u(p)=-\alpha p+v$ for some $\alpha>0$ and any $v\in\mathbb{R}$, and $\vartheta>-\infty$. The derivative of firm $f$’s profit function with respect to the price of product $k\in\mathcal{J}_{f}$ is $(D_{k}\hat{\pi}_{f})(\mathbf{p})=-\alpha P_{k}^{L}(\mathbf{p})(p_{k}-c_{k})+\alpha P_{k}^{L}(\mathbf{p})\hat{\pi}_{f}(\mathbf{p})+P_{k}^{L}(\mathbf{p}).$ Since $P_{k}^{L}(\mathbf{p})$ and $P_{k}^{L}(\mathbf{p})(p_{k}-c_{k})$ both vanish as $p_{k}\to\infty$ (as is easily checked), $\hat{\pi}_{f}(\mathbf{p})$ is bounded in $\mathbf{p}$. Thus $(D_{k}\hat{\pi}_{f})(\mathbf{p})\to 0$ as $p_{k}\to\infty$. We now provide a general assumption under which $(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})\to 0$ as $p_{k}\uparrow\varsigma_{*}$. ###### Assumption 3.1. Let $\psi_{k}$ be defined as in Assumption 2.3. Assume: (i) $p_{k}\psi_{k}(\boldsymbol{\theta},p_{k})\to 0$ as $p_{k}\uparrow\varsigma(\boldsymbol{\theta})$ for $\mu$-a.e. $\boldsymbol{\theta}$. (ii) There exists $M<\infty$ and $\bar{p}_{k}\in[0,\varsigma_{*})$ such that $p_{k}\int\psi_{k}(\boldsymbol{\theta},p_{k})d\mu(\boldsymbol{\theta})\leq M$ for all $p_{k}\in(\bar{p}_{k},\varsigma_{*})$. As with Assumption 2.3 above, (i) and (ii) are essentially conditions for the Dominated Convergence Theorem. Assumption 3.1 (i) extends Assumption 2.3 (i) to include a neighborhood of $\varsigma_{*}$. Note that if $\varsigma(\boldsymbol{\theta})<\infty$ then (i) holds if, and only if, $\psi_{k}(\boldsymbol{\theta},p_{k})\to 0$ as $p_{k}\uparrow\varsigma(\boldsymbol{\theta})$; i.e. $\psi_{k}(\boldsymbol{\theta},\cdot)$ is continuous at $\varsigma(\boldsymbol{\theta})$. Thus if $\varsigma(\boldsymbol{\theta})<\infty$ Assumption 3.1 (i) and Assumption 2.3 (i) are the same. If $\varsigma(\boldsymbol{\theta})=\infty$ and $p_{k}\psi_{k}(\boldsymbol{\theta},p_{k})\to 0$ as $p_{k}\uparrow\infty$, then necessarily $\psi_{k}(\boldsymbol{\theta},p_{k})\to 0$ as $p_{k}\uparrow\infty$. The converse, however, need not hold. If $\varsigma_{*}<\infty$, Assumption 3.1 (ii) simply says that $\int\psi_{k}(\boldsymbol{\theta},p_{k})d\mu(\boldsymbol{\theta})$ is bounded as $p_{k}\uparrow\varsigma_{*}$. This is not implied by Assumption 2.3 (ii), but is a natural extension of it. ###### Lemma 3.1. Suppose Assumptions 2.1-3.1 hold. Then $p_{k}\left\lvert\lambda_{k}(\mathbf{p})\right\rvert\to 0$ as $p_{k}\uparrow\varsigma_{*}$ for all $k$. Additionally, $p_{k}\left\lvert\gamma_{j,k}(\mathbf{p})\right\rvert\to 0$ as $p_{k}\uparrow\varsigma_{*}$ for all $j$. Subsequently, $(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})\to 0$ as $p_{k}\uparrow\varsigma_{*}$. ###### Proof. Let $\\{p_{k}^{(n)}\\}_{n\in\mathbb{N}}\subset(0,\varsigma_{*})$ be any sequence converging to $\varsigma_{*}$. Define $\Psi_{k}^{(n)}:\mathcal{T}\to\mathcal{P}$ by $\Psi_{k}^{(n)}(\boldsymbol{\theta})=p_{k}^{(n)}\psi_{k}(\boldsymbol{\theta},p_{k}^{(n)})$. The functions $\\{\Psi_{k}^{(n)}\\}_{n\in\mathbb{N}}$ converge pointwise to zero and have integrals uniformly bounded by the constant $M$. By the Dominated Convergence Theorem $\lim_{n\to\infty}\int\Psi_{k}^{(n)}(\boldsymbol{\theta})d\mu(\boldsymbol{\theta})=\int\Big{(}\lim_{n\to\infty}\Psi_{k}^{(n)}(\boldsymbol{\theta})\Big{)}d\mu(\boldsymbol{\theta})=0.$ ∎ In other words, under Assumption 3.1 the components of $\mathbf{F}_{\pi}$ vanish as the corresponding price tends to $\varsigma_{*}$ even though this may not mean that $\varsigma_{*}$ maximizes profits. Because of this, CG-NM may converge to a zero of $\mathbf{F}_{\pi}$ at $\varsigma_{*}\mathbf{1}$, or with some components equal to $\varsigma_{*}$, that is not an equilibrium. Note that even though the price derivatives vanish at infinity, this does not mean that infinite prices maximize profits. Nonetheless, CG-NM may converge to a zero of $\mathbf{F}_{\pi}$ with some components equal to infinity that is not an equilibrium. Moreover, because the components of $\mathbf{F}_{\pi}(\mathbf{p})$ can vanish over some divergent sequences, standard global convergence strategies based on minimizing $\lvert\lvert\mathbf{F}_{\pi}(\mathbf{p})\rvert\rvert_{2}$ will not be effective ways of avoiding this behavior. As in Example 5, we must reformulate the problem to obtain reliable and efficient approaches for computing equilibrium prices. ### 3.3. Newton’s Method and the Markup Equations Reliable and efficient implementations of Newton’s method are found by observing that the combined gradient, $\mathbf{F}_{\pi}$, can be written as follows: (20) $\displaystyle\mathbf{F}_{\pi}(\mathbf{p})$ $\displaystyle=(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}\mathbf{F}_{\eta}(\mathbf{p})$ where $\displaystyle\mathbf{F}_{\eta}(\mathbf{p})$ $\displaystyle=\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})$ (21) $\displaystyle\mathbf{F}_{\pi}(\mathbf{p})$ $\displaystyle=\boldsymbol{\Lambda}(\mathbf{p})\mathbf{F}_{\zeta}(\mathbf{p})$ where $\displaystyle\mathbf{F}_{\zeta}(\mathbf{p})$ $\displaystyle=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p}).$ Either $\mathbf{F}_{\eta}$ or $\mathbf{F}_{\zeta}$ can be used to compute simultaneously stationary prices when $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ and $\boldsymbol{\Lambda}(\mathbf{p})$, respectively, are nonsingular (Morrow and Skerlos, 2010). Of course, $\mathbf{F}_{\eta}$ and $\mathbf{F}_{\zeta}$ recast the first-order condition as a fixed-point problem: $\mathbf{F}_{\eta}$ is zero if and only if the BLP-markup equation holds, and $\mathbf{F}_{\zeta}$ is zero if and only if the $\boldsymbol{\zeta}$-markup equation holds. Solving $\mathbf{F}_{\eta}(\mathbf{p})=\mathbf{0}$ or $\mathbf{F}_{\zeta}(\mathbf{p})=\mathbf{0}$, abbreviated $\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM respectively in Table 4, requires the solution of nontrivial nonlinear systems with Newton’s method. $\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM, however, are less likely to have the computational problems that CG-NM exhibits because they exploit norm-coercivity of the maps $\mathbf{F}_{\eta}$ and $\mathbf{F}_{\zeta}$ (Morrow and Skerlos, 2010). A norm-coercive map has a norm that tends to infinity with the norm of its argument (Ortega and Rheinboldt, 1970; Harker and Pang, 1990). Globally convergent implementations of Newton’s method that decrease the value of $\lvert\lvert\mathbf{F}(\mathbf{p})\rvert\rvert_{2}$ in each step produce bounded sequences of iterates when $\mathbf{F}$ is norm- coercive. Thus, solving the BLP- or $\boldsymbol{\zeta}$-markup equation instead of the literal first-order condition removes the tendency for applications of Newton’s method to compute “spurious” solutions at infinity. We now prove that the maps $\mathbf{F}_{\eta}$ and $\mathbf{F}_{\zeta}$ are indeed coercive. ###### Lemma 3.2. Suppose $\varsigma_{*}=\infty$ and Assumption 2.1-2.3 hold. * (i) Norm-coercivity of $\mathbf{F}_{\zeta}(\mathbf{p})$ implies that of $\mathbf{F}_{\eta}(\mathbf{p})$. * (ii) If Eqn. (17) holds, then norm-coercivity of $\mathbf{F}_{\eta}(\mathbf{p})$ implies that of $\mathbf{F}_{\zeta}(\mathbf{p})$. ###### Proof. Proposition 2.12 implies that $\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})=\big{(}\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})\big{)}(\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})).$ To prove (i), note that $\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\geq\left(\frac{1}{1+\lvert\lvert\tilde{\boldsymbol{\Omega}}(\mathbf{p})\rvert\rvert_{\infty}}\right)\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\geq\left(\frac{1}{2}\right)\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}.$ To prove (ii), note that if Eqn. (17) holds, $\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\geq\big{(}1-\lvert\lvert\boldsymbol{\Omega}(\mathbf{p})\rvert\rvert_{\infty}\big{)}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\geq\varepsilon\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}.$ ∎ ###### Lemma 3.3. Suppose $\varsigma_{*}=\infty$ and Assumption 2.1-2.3 and 2.4 hold. Then $\lim_{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\to\infty}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}=\infty=\lim_{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\to\infty}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}.$ ###### Proof. The norm-coercivity of $\boldsymbol{\eta}$ is a trivial consequence of the boundedness of $\boldsymbol{\eta}$ under Assumption 2.4. The norm-coercivity of $\boldsymbol{\zeta}$ then follows from Lemma 3.2. ∎ We now weaken Assumption 2.4’s Eqn. (16). ###### Assumption 3.2. Suppose that $\varsigma_{*}=\infty$ and (22) $\displaystyle\lim_{M\uparrow\infty}\sup\left\\{\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}}:\mathbf{p}\in\mathcal{P}^{J},\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\geq M\right\\}=\delta\in[0,1).$ Note that the limit is of a non-increasing sequence of non-negative numbers, and thus exists. ###### Lemma 3.4. Assuming Eqn. (22) is equivalent to assuming that for any sequence $\mathbf{p}_{n}$ with $\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}\to\infty$, $\lim_{n\to\infty}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{n})^{-1}\mathbf{P}(\mathbf{p}_{n})\rvert\rvert_{\infty}/\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}\leq\delta$. ###### Proof. If Eqn. (22) holds, then for any $\varepsilon>0$ there exists an $M>0$ such that $\sup\left\\{\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}}:\mathbf{p}\in\mathcal{P}^{J},\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\geq M\right\\}<\delta+\varepsilon.$ If $\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}\to\infty$, then there is also an $N_{\epsilon}$ such that $\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}\geq M$ for all $n>N_{\epsilon}$. Thus $\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{n})^{-1}\mathbf{P}(\mathbf{p}_{n})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}}<\delta+\varepsilon$ for all $n>N_{\epsilon}$, and thus $\lim_{n\to\infty}\left[\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{n})^{-1}\mathbf{P}(\mathbf{p}_{n})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{n}\rvert\rvert_{\infty}}\right]\leq\delta.$ Conversely, if Eqn. (22) fails, then there is a $\bar{M}>0$ such that $S(M)=\sup\left\\{\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}}:\mathbf{p}\in\mathcal{P}^{J},\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\geq M\right\\}\geq 1$ for all $M\geq\bar{M}$. We can thus choose $\mathbf{p}_{M}$ with $\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}\geq M$ satisfying $1-\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{M})^{-1}\mathbf{P}(\mathbf{p}_{M})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}}\leq S(M)-\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{M})^{-1}\mathbf{P}(\mathbf{p}_{M})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}}\leq\frac{1}{M}.$ In other words, $\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{M})^{-1}\mathbf{P}(\mathbf{p}_{M})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}}\geq 1-\frac{1}{M}$ for all $M\geq\bar{M}$, and thus $\lim_{M\to\infty}\left[\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p}_{M})^{-1}\mathbf{P}(\mathbf{p}_{M})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}_{M}\rvert\rvert_{\infty}}\right]\geq 1.$ Hence the “sequence version” of Eqn. (22) fails, and thus by contraposition the sequence version and Eqn. (22) are identical. ∎ Next we note that Eqn. (22) weakens Eqn. (16). ###### Lemma 3.5. If Eqn. (16) holds, then Eqn. (22) holds. ###### Proof. If $\sup\\{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}:\mathbf{p}\in\mathcal{P}^{J}\\}\leq M$, then $S(L)=\sup\left\\{\frac{\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})\rvert\rvert_{\infty}}{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}}:\mathbf{p}\in\mathcal{P}^{J},\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\geq L\right\\}\leq\frac{M}{L}.$ Thus $\lim_{L\to\infty}S(L)=0$, a special case of Eqn. (22). ∎ Now we prove the alternative coercivity result. ###### Lemma 3.6. Suppose $\varsigma_{*}=\infty$ and Assumptions 2.1-2.3, 2.5 and 3.2 hold. Then $\lim_{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\to\infty}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}=\infty=\lim_{\lvert\lvert\mathbf{p}\rvert\rvert_{\infty}\to\infty}\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}.$ ###### Proof. We prove the claim for $\boldsymbol{\zeta}$; the result for $\boldsymbol{\eta}$ then follows from Lemma 2.15. Note that $\displaystyle\left\lvert p_{k}-c_{k}-\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})-\frac{P_{k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})}\right\rvert$ $\displaystyle\quad\quad\quad\quad\quad\quad=\left\lvert\left(1-\frac{P_{k}(\mathbf{p})}{p_{k}\lambda_{k}(\mathbf{p})}\right)p_{k}-c_{k}-\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})\right\rvert$ $\displaystyle\quad\quad\quad\quad\quad\quad\geq\left\lvert\;\left\lvert 1-\frac{P_{k}(\mathbf{p})}{p_{k}\lambda_{k}(\mathbf{p})}\right\rvert p_{k}-\Bigg{|}c_{k}+\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})\Bigg{|}\;\right\rvert$ Suppose that $p_{k}\to\infty$. By assumption, $\lim_{n\to\infty}\left[1-\frac{P_{k}(\mathbf{p})}{p_{k}\lambda_{k}(\mathbf{p})}\right]\geq 1-\delta>0$ while the second term is bounded. Thus $\displaystyle\left\lvert p_{k}-c_{k}-\sum_{j\in\mathcal{J}_{f(k)}}\omega_{k,j}(\mathbf{p})(p_{j}-c_{j})-\frac{P_{k}(\mathbf{p})}{\lambda_{k}(\mathbf{p})}\right\rvert\to\infty.$ ∎ Note that since we did not require that Eqn. (17) held, $\boldsymbol{\zeta}$ need not be bounded for $\mathbf{F}_{\eta}$ and $\mathbf{F}_{\zeta}$ to be coercive. ### 3.4. Fixed-Point Iteration In addition to applications of Newton’s method, the BLP- and $\boldsymbol{\zeta}$-markup equations suggest applying fixed-point iteration to solve for equilibrium prices. We examine fixed-point iterations based on both equations. #### 3.4.1. $\zeta$ Fixed-Point Iteration The fixed-point iteration $\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ based on the $\boldsymbol{\zeta}$-markup equation, here abbreviated $\boldsymbol{\zeta}$-FPI, can efficiently compute equilibrium prices for some problems. $\boldsymbol{\zeta}$-FPI has relatively efficient steps because no linear systems need to be solved, unlike every other method listed in Table 4. While we are not aware of a general convergence proof for $\boldsymbol{\zeta}$-FPI, this iteration has converged reliably on test problems including the examples in Morrow and Skerlos (2010). The first observation we make is that the $\boldsymbol{\zeta}$-FPI steps always point in directions of “myopic gradient ascent.” ###### Lemma 3.7. Let $\mathbf{p}\in(0,\varsigma_{*})^{J}$, and let $\boldsymbol{\delta}\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})-\mathbf{p}$ denote the $\boldsymbol{\zeta}$-FPI step. Then $\frac{1}{\max_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}\leq\frac{(\tilde{\nabla}\hat{\pi})(\mathbf{p})^{\top}\boldsymbol{\delta}\mathbf{p}}{(\tilde{\nabla}\hat{\pi})(\mathbf{p})^{\top}(\tilde{\nabla}\hat{\pi})(\mathbf{p})}\leq\frac{1}{\min_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}.$ Similarly, let $\theta(\mathbf{p})$ denote the angle between $\boldsymbol{\delta}\mathbf{p}$ and $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$, and suppose $\mathbf{p}$ is not simultaneously stationary. Then $\cos\theta(\mathbf{p})\geq\frac{\min_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}{\max_{j}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}.$ ###### Proof. Both results follows directly from the equation $(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\left\lvert\boldsymbol{\Lambda}(\mathbf{p})\right\rvert\boldsymbol{\delta}\mathbf{p}$ where $\left\lvert\boldsymbol{\Lambda}(\mathbf{p})\right\rvert$ denotes the absolute value of the components of $\boldsymbol{\Lambda}(\mathbf{p})$. ∎ Specifically, the $\boldsymbol{\zeta}$-FPI steps have a positive projection onto the combined gradient, and cannot become orthogonal to the combined gradient over any sequence of non-simultaneously stationary prices that stay in $(0,\varsigma_{*})^{J}$. If $F=1$, and the equilibrium problem is an optimization problem, this implies $\boldsymbol{\zeta}$-FPI has steps that point in gradient ascent directions and, when properly scaled, converge to local maximizers of profit. More specifically, $\boldsymbol{\zeta}$-FPI cannot converge to minimizers of profits. This may generate the properties of $\boldsymbol{\zeta}$-FPI observed in Example 10 from Morrow and Skerlos (2010). ###### Corollary 3.8. Let Assumptions 2.1-2.4 hold, and suppose $\\{\mathbf{p}^{(n)}\\}_{n=1}^{\infty}$ is the $\boldsymbol{\zeta}$-FPI sequence. Then $\\{\mathbf{p}^{(n)}\\}_{n=1}^{\infty}$ is bounded. ###### Proof. By Lemma 2.13, for any sufficiently large $M>0$ we can find some $L>0$ such that $\lvert\lvert\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}<\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}-M\quad\text{for all}\quad\lvert\lvert\mathbf{p}-\mathbf{c}\rvert\rvert_{\infty}>L.$ If the $\boldsymbol{\zeta}$-FPI sequence diverges, then for any such $L$ there is an $N$ such that $\lvert\lvert\mathbf{p}^{(n)}-\mathbf{c}\rvert\rvert_{\infty}>L\quad\text{for all}\quad n>N.$ But then $\lvert\lvert\mathbf{p}^{(n+1)}-\mathbf{c}\rvert\rvert_{\infty}=\lvert\lvert\boldsymbol{\zeta}(\mathbf{p}^{(n)})\rvert\rvert_{\infty}<\lvert\lvert\mathbf{p}^{(n)}-\mathbf{c}\rvert\rvert_{\infty}-M<\lvert\lvert\mathbf{p}^{(n)}-\mathbf{c}\rvert\rvert_{\infty}\quad\text{for all}\quad n>N,$ which states that the $\boldsymbol{\zeta}$-FPI sequence is decreasing. This is a contradiction of the hypothesis that the $\boldsymbol{\zeta}$-FPI sequence diverges. ∎ To implement $\boldsymbol{\zeta}$-FPI, one simply needs to iterate the assignment $\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ where Eqn. (18) defines $\boldsymbol{\zeta}(\mathbf{p})$. As shown in Table 5 below, integral approximations, rather than the actual computation of the step, drive the computational burden. Given a price vector, utilities, and utility derivatives, computing $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ for a set of $S$ samples requires $\mathcal{O}(S\sum_{f=1}^{F}J_{f}^{2})$ floating point operations (flops), while the fixed-point step itself only requires $\mathcal{O}(\sum_{f=1}^{F}J_{f}^{2})$ flops. Note that computing the fixed- point step $\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ requires an equivalent amount of work as computing the combined gradient $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$. Furthermore, because $\boldsymbol{\Lambda}(\mathbf{p})$ is a diagonal matrix, no serious obstacles to computing the fixed point step arise as $J$ becomes large. #### 3.4.2. $\eta$ Fixed-Point Iteration The fixed-point iteration $\mathbf{p}\leftarrow\mathbf{c}+\boldsymbol{\eta}(\mathbf{p})$, abbreviated $\boldsymbol{\eta}$-FPI, based on the BLP-markup equation need not converge. Example 7 below, repeated from Morrow and Skerlos (2010), gives a case in which $\boldsymbol{\eta}$ can fail to be even locally convergent. ###### Example 7. Consider multi-product monopoly pricing with a simple Logit model having $u_{j}(p)=-\alpha p+v_{j}$ for some $\alpha>0$, any $v_{j}\in\mathbb{R}$, and $\vartheta>-\infty$. It is well known that for a single-product firm, unique profit-maximizing prices exist (Anderson and de Palma, 1988; Milgrom and Roberts, 1990; Caplin and Nalebuff, 1991). Morrow (2008) proves that profit- optimal prices $\mathbf{p}_{*}$ are unique for the multi-product case $-$ and even so with multiple firms $-$ even though profits are not quasi-concave (Hanson and Martin, 1996). In this example, $\boldsymbol{\eta}$-FPI is not always locally convergent near $\mathbf{p}_{*}$, while $\boldsymbol{\zeta}$-FPI is always superlinearly locally convergent. For an arbitrary continuously differentiable function $\mathbf{F}$ and $\mathbf{p}_{*}=\mathbf{F}(\mathbf{p}_{*})$, $\mathbf{F}$ is contractive on some neighborhood of $\mathbf{p}_{*}$ in some norm $\lvert\lvert\cdot\rvert\rvert$ if $\rho((D\mathbf{F})(\mathbf{p}_{*}))<1$ where $\rho(\mathbf{A})$ (Ortega and Rheinboldt, 1970). We show that $\rho((D\boldsymbol{\eta})(\mathbf{p}_{*}))>1$ may hold while $\rho((D\boldsymbol{\zeta})(\mathbf{p}_{*}))=0$, where $\rho(\mathbf{A})$ denotes the spectral radius of the matrix $\mathbf{A}$. The components of the BLP-markup function $\boldsymbol{\eta}$ are given by $\eta_{k}(\mathbf{p})=\alpha^{-1}(1-\sum_{j=1}^{J}P_{j}^{L}(\mathbf{p}))^{-1}$ for all $k$. From this formula the equation $\rho((D\boldsymbol{\eta})(\mathbf{p}_{*}))=\frac{\sum_{j=1}^{J}P_{j}^{L}(\mathbf{p}_{*})}{1-\sum_{j=1}^{J}P_{j}^{L}(\mathbf{p}_{*})}=\sum_{j=1}^{J}e^{u_{j}(p_{j,*})-\vartheta}$ can be derived. For valuations of the outside good, $\vartheta$, sufficiently close to $-\infty$, $\rho((D\boldsymbol{\eta})(\mathbf{p}_{*}))>1$ can hold; see Morrow and Skerlos (2010) for details. To prove the claim regarding $\rho((D\boldsymbol{\zeta})(\mathbf{p}_{*}))$, note that $\zeta_{k}(\mathbf{p})=\hat{\pi}(\mathbf{p})+1/\alpha$, and thus $(D_{l}\zeta_{k})(\mathbf{p}_{*})=(D_{l}\hat{\pi})(\mathbf{p}_{*})=0$ for all $k,l$. Even if the BLP-markup equation does generate a convergent fixed-point iteration, evaluating $\boldsymbol{\eta}$ involves the solution of $F$ linear systems that grow in size with the number of products offered by the firms. The work required to evaluate $\boldsymbol{\eta}$ using a direct method like PLU or QR factorization is $\mathcal{O}([\max_{f}J_{f}]^{3})$, given values of $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ as approximated using simulation. The work to evaluate $\boldsymbol{\zeta}$ is only $\mathcal{O}([\max_{f}J_{f}]^{2})$ given $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ (Table 5). Generally speaking, function evaluations must be cheap for the linear convergence of fixed-point iterations to result in faster computations than the superlinearly or quadratically convergent variants of Newton’s method. ### 3.5. Practical Considerations This section addresses several practical considerations. #### 3.5.1. Simulation Any method for computing equilibrium prices under Mixed Logit models faces a common obstacle: the integrals that define the choice probabilities ($\mathbf{P}$) and their derivatives ($\boldsymbol{\Lambda},\tilde{\boldsymbol{\Gamma}}$) cannot be computed exactly. We employ finite-sample versions of the methods discussed below by drawing $S\in\mathbb{N}$ samples from the demographic distribution and applying the method to the finite-sample model thus generated. Particularly, these samples are used to compute approximate $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$; see Table 5. These samples are kept fixed for all steps of the method and, in principle, can be generated in any way. We draw directly from the demographic distribution, although importance and quasi-random sampling (e.g., see Train (2003)) can also be employed. The Law of Large Numbers motivates this widely-used approach to econometric analysis (e.g., see McFadden (1989) and Draganska and Jain (2004)). While all numerical approaches for computing equilibrium prices described here rely on a Law of Large Numbers for simultaneously stationary prices, we do not provide a formal convergence theorem. We do provide numerical evidence that computed equilibrium prices based on the fixed-point iteration for our examples do indeed follow such a law. #### 3.5.2. Truncation of Low Purchase Probability Products All of the methods we implement can be built to ignore products with excessively low choice probabilities. That is, one can ignore price updates for all products with $P_{j}(\mathbf{p})\leq\varepsilon_{P}$, where $\varepsilon_{P}$ is some small value (say $10^{-10}$). Products with a choice probability this small (or smaller) need not be considered a part of the market in the price equilibrium computations. For example, Wards (2007) reports total sales of cars and light trucks during 2005 as $N=16,947,754$. Particularly, 7,667,066 cars and 9,280,688 light trucks. Because expected demand is defined by $\mathbb{E}[Q_{j}(\mathbf{p})]=NP_{j}(\mathbf{p})$, any $\varepsilon_{P}\leq 0.5*N^{-1}\approx 3\times 10^{-8}$ ignores any vehicle that, as priced, is not expected to have a single customer out of the millions of customers that bought or considered buying new vehicles. There are also technical reasons for this truncation. Particularly, $\boldsymbol{\Lambda}(\mathbf{p})$ and $(D\tilde{\nabla}\hat{\pi})(\mathbf{p})$ become singular as $P_{j}(\mathbf{p})\to 0$, for any $j$. Truncating avoids this non-singularity and hopefully helps conditioning. #### 3.5.3. Termination Conditions We terminate all iterations with the numerical simultaneous stationarity condition $\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$ where $\varepsilon_{T}$ is some small number (e.g., $10^{-6}$). Note that a standard application of Newton’s method to solve $\mathbf{F}_{\eta}(\mathbf{p})=\mathbf{0}$ or $\mathbf{F}_{\zeta}(\mathbf{p})=\mathbf{0}$ would terminate when either (23) $\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}\quad\quad\text{or}\quad\quad\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T},$ respectively. For example, Aguirregabiria and Vicentini (2006) use the condition $\lvert\lvert\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$. Ensuring that Eqn. (23) holds does not necessarily imply that $\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$, the strictly interpreted first-order condition. Because $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p}))=(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})(\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})),$ it is easy to terminate all methods, CG-NM, $\boldsymbol{\eta}$-NM, $\boldsymbol{\zeta}$-NM, and $\boldsymbol{\zeta}$-FPI, when $\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$. While this is done here to ensure consistency in our comparisons of different methods, $\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{\infty}\leq\varepsilon_{T}$ should always be the termination condition for price equilibrium computations. Three other standard termination conditions are used (Brown and Saad, 1990; Dennis and Schnabel, 1996). We terminate the iteration if the (relative) step length becomes too small, if a maximum number of iterations is exceeded, or if an exceptional event occurs (e.g. division by zero). These three conditions are considered “failure” as the iteration has failed to compute a numerically simultaneously stationary point in the sense of the first termination condition. #### 3.5.4. Second-Order Conditions. Each method in Table 4 finds simultaneously stationary points, rather than local equilibria. Unlike in optimization, there is no a priori assurance that first-order iterative methods for equilibrium problems will converge to certain types of stationary points. Thus in computing equilibria it is vitally important to check the second-order sufficient conditions to verify that a local equilibrium has indeed been found. In local equilibrium every firm’s profit Hessian, $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$, should also be negative definite. The formulas given in Proposition 3.9 below provide an expression for $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ that we use to check the second-order sufficient condition. Cholesky factorization, rather than direct approximation of the spectrum, is used to test the negative definiteness of $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})$ (Golub and Loan, 1996). #### 3.5.5. Computational Burden Table 5 reviews the formulae and computational burden of computing $(\tilde{\nabla}\hat{\pi})$, $\boldsymbol{\eta}$, and $\boldsymbol{\zeta}$. Table 5. Work required to evaluate $(\tilde{\nabla}\hat{\pi})$, $\boldsymbol{\eta}$, and $\boldsymbol{\zeta}$ given $S$ samples $\\{\boldsymbol{\theta}_{s}\\}_{s=1}^{S}\subset\mathcal{T}$, an $S\times J$ matrix $\mathbf{L}(\mathbf{p})$ of Logit choice probabilities ($(\mathbf{L}(\mathbf{p}))_{s,j}=P_{j}^{L}(\boldsymbol{\theta}_{s},\mathbf{p})$), and an $S\times J$ matrix of utility derivatives $\mathbf{D}(\mathbf{p})$ ($(\mathbf{D}(\mathbf{p}))_{s,j}=(Dw_{j})(\boldsymbol{\theta}_{s},p_{j})$). The first section gives work required for sample-average approximations to $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$. The second section takes $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ as given. Quantity | Formula | flops ---|---|--- $\mathbf{P}(\mathbf{p})$ | $S^{-1}\mathbf{L}(\mathbf{p})^{\top}\mathbf{1}$ | $SJ$ $\mathbf{V}(\mathbf{p})$ | $\mathbf{L}(\mathbf{p})\cdot\mathbf{D}(\mathbf{p})$${}^{\text{(a)}}$ | $SJ$ $\boldsymbol{\Lambda}(\mathbf{p})$ | $S^{-1}\mathbf{V}(\mathbf{p})^{\top}\mathbf{1}$ | $SJ$ $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ | $S^{-1}\mathbf{L}(\mathbf{p})^{\top}\mathbf{V}(\mathbf{p})$ | $2S\sum_{f=1}^{F}J_{f}^{2}$ Total work to compute $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ | $S\left(3J+2\sum_{f=1}^{F}J_{f}^{2}\right)$ $\boldsymbol{\zeta}(\mathbf{p})$ | $\tilde{\boldsymbol{\Omega}}(\mathbf{p})(\mathbf{p}-\mathbf{c})-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$ | $2\sum_{f=1}^{F}J_{f}^{2}+4J$ $\boldsymbol{\eta}(\mathbf{p})$ | $(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))\boldsymbol{\eta}(\mathbf{p})=-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$ | $\left(\frac{4}{3}\right)\sum_{f=1}^{F}J_{f}^{3}+\left(\frac{7}{2}\right)\left(\sum_{f=1}^{F}J_{f}^{2}+J\right)-2$ $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$ | $(\boldsymbol{\Lambda}(\mathbf{p})-\tilde{\boldsymbol{\Gamma}}(\mathbf{p})^{\top})(\mathbf{p}-\mathbf{c})+\mathbf{P}(\mathbf{p})$ | $2\sum_{f=1}^{F}J_{f}^{2}+5J$ | $=\boldsymbol{\Lambda}(\mathbf{p})(\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p}))$ | $2\sum_{f=1}^{F}J_{f}^{2}+6J$ (a) “$\cdot$” here denotes element-by-element multiplication. Computing $\boldsymbol{\eta}$ and applying Newton’s method to $\mathbf{F}_{\eta}$ requires solving linear systems. We give some more details regarding these computations here. As stated above, the linear system $(\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p}))\boldsymbol{\eta}(\mathbf{p})=-\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{P}(\mathbf{p})$ should be used to solve for $\boldsymbol{\eta}(\mathbf{p})$. Note also that only the systems $(\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p}))\boldsymbol{\eta}_{f}(\mathbf{p})=-\boldsymbol{\Lambda}_{f}(\mathbf{p})^{-1}\mathbf{P}_{f}(\mathbf{p})$ for all $f$ need be solved. Of course, our condition bound applies within firms as well: $\kappa_{2}\big{(}(D_{f}\mathbf{P}_{f})(\mathbf{p})^{\top}\big{)}\geq\left(\frac{\max_{j\in\mathcal{J}_{f}}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}{\min_{j\in\mathcal{J}_{f}}\left\lvert\lambda_{j}(\mathbf{p})\right\rvert}\right)\kappa_{2}\big{(}\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p})\big{)}.$ If Householder QR factorization is used to solve these systems, then computing $\boldsymbol{\eta}(\mathbf{p})$ from $\mathbf{P}(\mathbf{p})$, $\boldsymbol{\Lambda}(\mathbf{p})$, and $\tilde{\boldsymbol{\Gamma}}(\mathbf{p})$ requires $\mathcal{O}(\sum_{f=1}^{F}J_{f}^{3})$ flops (Table 5). This is a significant increase in computational effort relative to computing $\boldsymbol{\zeta}(\mathbf{p})$ or $(\tilde{\nabla}\hat{\pi})(\mathbf{p})$. The diagonal dominance of $\mathbf{I}-\tilde{\boldsymbol{\Omega}}(\mathbf{p})$, indeed of $(\tilde{D}\mathbf{P})(\mathbf{p})$ itself, suggests that Jacobi, Gauss- Seidel, and Successive Over-Relaxation (SOR) iterations (Golub and Loan, 1996) may be a relatively efficient way to compute $\boldsymbol{\eta}$. Additional work is required to compute $(D\boldsymbol{\eta})(\mathbf{p})$, if this is to be used in Newton’s method. Though it requires solving a matrix- linear system of the type $(\tilde{D}\mathbf{P})(\mathbf{p})(D\boldsymbol{\eta})(\mathbf{p})=\mathbf{B}(\mathbf{p})$, the required matrix factorizations of $\mathbf{I}-\boldsymbol{\Omega}_{f}(\mathbf{p})$ need only be computed once to compute both $\boldsymbol{\eta}$ and $(D\boldsymbol{\eta})$, but must be updated for each vector of prices. ### 3.6. Computing Jacobian Matrices for Newton’s Method Standard “exact” or Quasi-Newton methods to solve $\mathbf{F}(\mathbf{x})=\mathbf{0}$ either always or periodically require the Jacobian matrix $(D\mathbf{F})(\mathbf{x})$. Using finite differences to approximate Jacobian matrices requires $J$ evaluations of the function $\mathbf{F}$, an unacceptable workload. In the 993 vehicle example from Morrow and Skerlos (2010), approximating $(D\mathbf{F})(\mathbf{x})$ once with finite differences would take roughly 993 evaluations of $\mathbf{F}$, when the work of less than 50 evaluations appears to sufficient to converge to equilibrium prices using the $\boldsymbol{\zeta}$-FPI. We recommend directly approximating $(D\mathbf{F})(\mathbf{x})$ using integral expressions for $(D\tilde{\nabla}\hat{\pi})(\mathbf{p})$, $(D\boldsymbol{\eta})(\mathbf{p})$, and $(D\boldsymbol{\zeta})(\mathbf{p})$ provided below. An alternative is to use automatic differentiation, but we are skeptical that this would in fact be faster than the direct formulae provided here. #### 3.6.1. Jacobian of the Combined Gradient Assuming a second application of the Leibniz Rule holds, we can derive integral expressions for the second derivatives $(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})$ through $\big{(}(D\tilde{\nabla}\hat{\pi})(\mathbf{p})\big{)}_{k,l}=(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=\int(D_{l}D_{k}\hat{\pi}_{f(k)}^{L})(\boldsymbol{\theta},\mathbf{p})d\mu(\boldsymbol{\theta}).$ ###### Proposition 3.9. Let $w$ be twice continuously differentiable in $p$ and suppose a second application of the Leibniz Rule holds for the Mixed Logit choice probabilities at $\mathbf{p}$. Set $\displaystyle\phi_{k,l}(\mathbf{p})$ $\displaystyle=\int(Dw_{k})(\boldsymbol{\theta},p_{k})P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})P_{l}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{l})(\boldsymbol{\theta},p_{l})d\mu(\boldsymbol{\theta})$ $\displaystyle\psi_{k,l}(\mathbf{p})$ $\displaystyle=\int(Dw_{k})(\boldsymbol{\theta},p_{k})P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})\hat{\pi}_{f(k)}^{L}(\boldsymbol{\theta},\mathbf{p})P_{l}^{L}(\boldsymbol{\theta},\mathbf{p})(Dw_{l})(\boldsymbol{\theta},p_{l})d\mu(\boldsymbol{\theta})$ $\displaystyle\chi_{k}(\mathbf{p})$ $\displaystyle=\left(\frac{1}{2}\right)\int\big{(}(D^{2}w_{k})(\boldsymbol{\theta},p_{k})+(Dw_{k})(\boldsymbol{\theta},p_{k})^{2}\big{)}$ $\displaystyle\quad\quad\quad\quad\quad\quad\times P_{k}^{L}(\boldsymbol{\theta},\mathbf{p})\big{(}(p_{k}-c_{k})-\hat{\pi}_{f(k)}^{L}(\boldsymbol{\theta},\mathbf{p})\big{)}d\mu(\boldsymbol{\theta})$ * (i) Component form: Setting $\xi_{k,l}(\mathbf{p})=\delta_{k,l}(\lambda_{k}(\mathbf{p})+\chi_{k}(\mathbf{p}))-\gamma_{k,l}(\mathbf{p})-(p_{k}-c_{k})\varphi_{k,l}(\mathbf{p})$ we have $\displaystyle(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=\xi_{k,l}(\mathbf{p})+2\psi_{k,l}(\mathbf{p})+\delta_{f(k),f(l)}\xi_{l,k}(\mathbf{p})$ * (ii) Matrix form: Let $\boldsymbol{\Phi}(\mathbf{p})$, $\boldsymbol{\Psi}(\mathbf{p})$ and $\mathbf{X}(\mathbf{p})=\mathrm{diag}(\boldsymbol{\chi}(\mathbf{p}))$ be the matrices of these quantities. Also set $\boldsymbol{\Xi}(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})-\boldsymbol{\Gamma}(\mathbf{p})-\mathrm{diag}(\mathbf{p}-\mathbf{c})\boldsymbol{\Phi}(\mathbf{p})+\mathbf{X}(\mathbf{p}).$ and $(\tilde{\boldsymbol{\Xi}}(\mathbf{p}))_{k,l}=\left\\{\begin{aligned} &\xi_{k,l}(\mathbf{p})&&\quad\text{if }f(k)=f(l)\\\ &\quad 0&&\quad\text{if }f(k)\neq f(l)\end{aligned}\right.$ Then (24) $(D\tilde{\nabla}\hat{\pi})(\mathbf{p})=\boldsymbol{\Xi}(\mathbf{p})+2\boldsymbol{\Psi}(\mathbf{p})+\tilde{\boldsymbol{\Xi}}(\mathbf{p})^{\top}.$ ###### Proof. To see that this only relies on a second application of the Leibniz Rule to the choice probabilities, note that $(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}(D_{l}D_{k}P_{j})(\mathbf{p})(p_{j}-c_{j})+\delta_{f(k),f(l)}(D_{k}P_{l})(\mathbf{p})+(D_{l}P_{k})(\mathbf{p})$ and thus the continuous second-order differentiability of $\hat{\pi}_{f}(\mathbf{p})$ depends only on the second-order continuous differentiability of $\mathbf{P}_{f}$. This result is then an immediate consequence of the validity of the Leibniz Rule, if a bit tedious to derive. ∎ The validity of a second application of the Leibniz Rule to the choice probabilities is ensured by the following condition. ###### Proposition 3.10. Let $(u,\vartheta,\mu)=(w+v,\vartheta,\mu)$ be such that * (i) $w(\boldsymbol{\theta},\mathbf{y},\cdot):(0,\varsigma_{*})\to\mathbb{R}$ is twice continuously differentiable for all $\mathbf{y}\in\mathcal{Y}$ and $\mu$-a.e. $\boldsymbol{\theta}\in\mathcal{T}$ * (ii) for all $(\mathbf{y},p)\in\mathcal{Y}\times(0,\varsigma_{*})$, $\left\lvert(D^{2}w)(\cdot,\mathbf{y},q)+(Dw)(\cdot,\mathbf{y},q)^{2}\right\rvert e^{u(\cdot,\mathbf{y},q)-\vartheta(\cdot)}:\mathcal{T}\to[0,\infty)$ is uniformly $\mu$-integrable for all $q$ in some neighborhood of $p$. * (iii) for all $(\mathbf{y},p),(\mathbf{y}^{\prime},p^{\prime})\in\mathcal{Y}\times(0,\varsigma_{*})$, $\left\lvert(Dw)(\cdot,\mathbf{y},q)\right\rvert e^{u(\cdot,\mathbf{y},q)-\vartheta(\cdot)}e^{u(\cdot,\mathbf{y}^{\prime},q^{\prime})-\vartheta(\cdot)}\left\lvert(Dw)(\cdot,\mathbf{y}^{\prime},q^{\prime})\right\rvert:\mathcal{T}\to[0,\infty)$ is uniformly $\mu$-integrable for all $(q,q^{\prime})$ in some neighborhood of $(p,p^{\prime})$. Then a second application of the Leibniz Rule holds for the Mixed Logit choice probabilities, which are also continuously differentiable on $(\mathbf{0},\varsigma_{*}\mathbf{1})$. This is proved in the same manner as Proposition 2.4. We also observe the following. ###### Proposition 3.11. If $P_{k}(\mathbf{p})=0$ then $(D_{l}D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=(D_{k}D_{l}\hat{\pi}_{f(l)})(\mathbf{p})=0$ for all $l\in\mathbb{N}(J)$. The proof follows from the derivative formulae given above. Of course, if $P_{k}(\mathbf{p})=0$ then $(D_{k}\hat{\pi}_{f(k)})(\mathbf{p})=0$ as well and we have the following situation: (i) the Newton system is consistent for any $s_{k}^{N}(\mathbf{p})\in\mathbb{R}$ and (ii) $s_{l}^{N}(\mathbf{p})$ does not depend on $s_{k}^{N}(\mathbf{p})$ for all $l\in\mathbb{N}(J)\setminus\\{k\\}$. Thus, in practice one can restrict attention to the Newton step defined by the submatrix of $(D\tilde{\nabla}\hat{\pi})(\mathbf{p})$ formed by rows and columns indexed by $\\{j:P_{j}(\mathbf{p})>\varepsilon_{P}\\}$. The formulae above give the following expression of the profit Hessians. ###### Corollary 3.12. Let $w$ be twice continuously differentiable in $p$ and suppose a second application of the Leibniz Rule holds for the Mixed Logit choice probabilities. Firm $f$’s profit Hessian is given by $(D_{f}\nabla_{f}\hat{\pi}_{f})(\mathbf{p})=\boldsymbol{\Xi}_{f,f}(\mathbf{p})+2\boldsymbol{\Psi}_{f,f}(\mathbf{p})+\boldsymbol{\Xi}_{f,f}(\mathbf{p})^{\top}.$ #### 3.6.2. The $\boldsymbol{\eta}$ map. For $\mathbf{F}_{\eta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\eta}(\mathbf{p})$, we have $(D\mathbf{F}_{\eta})(\mathbf{p})=\mathbf{I}-(D\boldsymbol{\eta})(\mathbf{p})$ where $(D\boldsymbol{\eta})(\mathbf{p})$ solves the linear matrix equation $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}(D\boldsymbol{\eta})(\mathbf{p})=-(\mathbf{A}(\mathbf{p})+(D\mathbf{P})(\mathbf{p})).$ Here $(\mathbf{A}(\mathbf{p}))_{k,l}=\sum_{j\in\mathcal{J}_{f(k)}}(D_{l}D_{k}P_{j})(\mathbf{p})\eta_{j}(\mathbf{p})$. This is easily derived from the defining formula $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}\boldsymbol{\eta}(\mathbf{p})=-\mathbf{P}(\mathbf{p})$. #### 3.6.3. The $\boldsymbol{\zeta}$ map. For $\mathbf{F}_{\zeta}(\mathbf{p})=\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})$, we have $(D\mathbf{F}_{\zeta})(\mathbf{p})=\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p})$ where $(D\boldsymbol{\zeta})(\mathbf{p})$ can be computed using the following formula: $\displaystyle(D_{l}\zeta_{k})$ $\displaystyle=\lambda_{k}^{-1}\Bigg{[}\delta_{k,l}\left[\int P_{k}^{L}\big{(}(D^{2}w_{k})+(Dw_{k})^{2}\big{)}\left(\hat{\pi}_{f(k)}^{L}-\zeta_{k}\right)-\lambda_{k}\right]$ $\displaystyle\quad\quad\quad\quad\quad\quad+\zeta_{k}\phi_{k,l}+\gamma_{k,l}+\delta_{f(k),f(l)}\phi_{k,l}(p_{l}-c_{l})+\delta_{f(k),f(l)}\gamma_{l,k}-2\psi_{k,l}\Bigg{]}.$ ## 4\. The GMRES-Newton Hookstep Method In this section we provide some details regarding the GMRES-Newton Hookstep method employed in Morrow and Skerlos (2010). For complete details, see Morrow10b. ### 4.1. Inexact Newton Methods A strong theory of “Inexact” Newton methods exists for the solution of systems of nonlinear equations when there are “many” variables. Inexact Newton steps are simply “inexact” solutions to the Newton system; that is, an inexact Newton step $\mathbf{s}^{IN}$ is any vector that satisfies (25) $\lvert\lvert\mathbf{F}(\mathbf{x})+(D\mathbf{F})(\mathbf{x})\mathbf{s}^{IN}\rvert\rvert\leq\delta\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert$ for some fixed $\delta\in(0,1)$ (Dembo et al., 1982; Brown and Saad, 1990; Eisenstat and Walker, 1994, 1996; Pernice and Walker, 1998). The name “truncated” Newton method has also been used for the specific case when the inexactness comes from the use of iterative linear system solvers like GMRES (Saad and Schultz, 1986; Walker, 1988) or BiCGSTAB (van der Vorst, 1992; Sleijpen and Fokkema, 1993). We focus on GMRES, a particularly simple yet strong iterative method for general linear systems that has been consistently used in the context of solving nonlinear systems (Brown and Saad, 1990). By appropriately choosing a sequence of $\delta$’s, the local asymptotic convergence rate of an inexact Newton’s method can be fully quadratic (Dembo et al., 1982; Eisenstat and Walker, 1994). Of course, taking $\delta\to 0$ to achieve the quadratic convergence rate will also require increasingly burdensome computations of inexact Newton steps that satisfy increasingly strict inexact Newton conditions. On the other hand, $\delta$ can be chosen to be a constant if a linear locally asymptotic convergence rate is suitable (Pernice and Walker, 1998). Generally speaking there are three reasons to adopt the inexact perspective. First, direct methods like QR factorization may not be the most effective means to solve the Newton system when this system is large, because of computational burden and accumulation of roundoff errors. Instead, iterative solution methods are often used to solve linear systems with many variables; see, e.g. Trefethen and Bau (1997). Second, iterative methods like GMRES require only matrix-vector products $(D\mathbf{F})(\mathbf{p})\mathbf{s}$ that can be approximated with finite directional derivatives (Brown and Saad, 1990; Pernice and Walker, 1998). Thus inexact Newton’s methods can be “matrix-free”; see Section 4.3.4 below. Third, Newton steps often point in inaccurate directions when far from a solution (Pernice and Walker, 1998). Thus solving for exact Newton steps may involve wasted effort, especially when there are many variables. matlab’s fsolve function implements a related approach using the (preconditioned) Conjugate Gradient (CG) method applied to the normal equation for the Newton system, $(D\mathbf{F})(\mathbf{p})^{\top}(D\mathbf{F})(\mathbf{p})\mathbf{s}^{IN}=-(D\mathbf{F})(\mathbf{p})^{\top}\mathbf{F}(\mathbf{p})$. Use of the normal equations is required because CG is applicable only to symmetric systems (Trefethen and Bau, 1997). Note that this requires that the Jacobian $(D\mathbf{F})$ is explicitly available. Although this holds for price equilibrium problems under Mixed Logit models, it can be a significant restriction for general problems. By requiring products $(D\mathbf{F})(\mathbf{p})^{\top}\mathbf{h}$ in each step of the iterative linear solver, this approach also increases the work by $\mathcal{O}(NJ^{2})$ flops where the solver takes $N$ steps. Finally, this approach can also be less accurate: using the normal equation squares the linear problem’s condition number, and thus risks serious degradation in solution quality (Trefethen and Bau, 1997). Pernice and Walker (1998) describe a similar approach using BiCGSTAB: the extension of CG to non-symmetric systems. ### 4.2. GMRES The “Generalized Minimum Residuals” or GMRES method (Saad and Schultz, 1986) solves a linear system $\mathbf{Ax}=\mathbf{b}$ by using the Arnoldi process to compute an orthonormal basis of the successive Krylov subspaces $\mathcal{K}^{(n)}$ and then takes approximate solutions from those subspaces having least squares residuals. See Trefethen and Bau (1997) for a good introduction to Krylov methods in general, including the Arnoldi process and GMRES. In the $n^{\text{th}}$ stage, GMRES “factors” $\mathbf{A}$ as $\mathbf{A}\mathbf{Q}^{(n)}=\mathbf{Q}^{(n+1)}\tilde{\mathbf{H}}^{(n)}$ where $\mathbf{Q}^{(n)}\in\mathbb{R}^{N\times n}$ is an orthonormal basis for $\mathcal{K}^{(n)}$, $\mathbf{Q}^{(n+1)}\in\mathbb{R}^{N\times(n+1)}$ is an orthonormal basis for $\mathcal{K}^{(n+1)}\supset\mathcal{K}^{(n)}$, and $\tilde{\mathbf{H}}^{(n)}\in\mathbb{R}^{(n+1)\times n}$ is upper-Hessenberg. Any vector $\mathbf{x}\in\mathcal{K}^{(n)}\subset\mathbb{R}^{N}$ can be written $\mathbf{x}=\mathbf{Q}^{(n)}\mathbf{y}$ for some $\mathbf{y}\in\mathbb{R}^{n}$ and thus the least-squares residual problem becomes $\min_{\mathbf{x}\in\mathcal{K}^{(n)}}\lvert\lvert\mathbf{As}-\mathbf{b}\rvert\rvert_{2}=\min_{\mathbf{y}\in\mathbb{R}^{n}}\lvert\lvert\mathbf{A}\mathbf{Q}^{(n)}\mathbf{y}-\mathbf{b}\rvert\rvert_{2}=\min_{\mathbf{y}\in\mathbb{R}^{n}}\lvert\lvert\tilde{\mathbf{H}}^{(n)}\mathbf{y}-(\mathbf{Q}^{(n+1)})^{\top}\mathbf{b}\rvert\rvert_{2}.$ The orthonormal basis is typically chosen so that $(\mathbf{Q}^{(n+1)})^{\top}\mathbf{b}=\beta\mathbf{e}_{1}$ for some $\beta\in\mathbb{R}$, and hence the GMRES solution $\mathbf{x}^{(n)}=\mathbf{Q}^{(n)}\mathbf{y}$ where $\mathbf{y}$ solves $\min_{\mathbf{q}\in\mathbb{R}^{n}}\lvert\lvert\tilde{\mathbf{H}}^{(n)}\mathbf{y}-\beta\mathbf{e}_{1}\rvert\rvert_{2}$. This least squares problem can be solved using the QR factorization of $\tilde{\mathbf{H}}^{(n)}$. Furthermore this factorization can be efficiently updated in each iteration, instead of computed from scratch. Moreover the actual solution vector need not be formed until the residual is suitably small. #### 4.2.1. Householder GMRES We have implemented a variant of GMRES based on Householder transformations due to Walker (1988); this is also the version implemented in matlab’s gmres code. We have verified that our implementation generates results matching matlab’s implementation. In this version of the GMRES process applied to the generic problem $\mathbf{Ax}=\mathbf{b}$, Householder reflectors $\mathbf{P}^{(n)}\in\mathbb{R}^{N\times N}$ are used to generate the orthonormal matrices $\mathbf{Q}^{(n)}=\mathbf{P}^{(1)}\dotsb\mathbf{P}^{(n)}\begin{bmatrix}\mathbf{I}\\\ \mathbf{0}\end{bmatrix}\in\mathbb{R}^{N\times n}\quad\quad(\mathbf{I}\in\mathbb{R}^{n\times n},\;\mathbf{0}\in\mathbb{R}^{(N-n)\times n})$ satisfying $\mathbf{A}\mathbf{Q}^{(n)}=\mathbf{P}^{(1)}\dotsb\mathbf{P}^{(n+1)}\mathbf{H}^{(n)}=\mathbf{Q}^{(n+1)}\tilde{\mathbf{H}}^{(n)}$ where $\mathbf{H}^{(n)}\in\mathbb{R}^{N\times n}$ is $\mathbf{H}^{(n)}=\begin{bmatrix}\tilde{\mathbf{H}}^{(n)}\\\ \mathbf{0}\end{bmatrix}$ for upper Hessenberg $\tilde{\mathbf{H}}^{(n)}\in\mathbb{R}^{(n+1)\times n}$ and $\mathbf{0}\in\mathbb{R}^{(N-n-1)\times n}$. $\mathbf{P}^{(1)}$ is chosen to satisfy $\mathbf{P}^{(1)}\mathbf{b}=-\beta\mathbf{e}_{1}$ where $\beta=\mathrm{sign}(b_{1})\lvert\lvert\mathbf{b}\rvert\rvert_{2}$, and hence $(\mathbf{Q}^{(n)})^{\top}\mathbf{b}=-\beta\mathbf{e}_{1}$. The $n^{\text{th}}$ approximate solution $\mathbf{x}^{(n)}$ is taken to be $\mathbf{x}^{(n)}=\mathbf{Q}^{(n)}\mathbf{y}^{(n)}$ where $\mathbf{y}^{(n)}\in\mathbb{R}^{n}$ solves $\min_{\mathbf{y}\in\mathbb{R}^{n}}\lvert\lvert\tilde{\mathbf{H}}^{(n)}\mathbf{y}-\beta\mathbf{e}_{1}\rvert\rvert_{2}.$ Again these problems can be solved cheaply by updating QR factorizations with Givens rotations. Neither the solution vector nor the residual vector be formed until GMRES converges. An efficient implementation requires $\mathcal{O}(Jn)$ flops and a matrix multiply in the $n^{\text{th}}$ iteration, so that taking $N$ iterations requires $\mathcal{O}(JN^{2})$ of “overhead” in addition to the $\mathcal{O}(NJ^{2})$ work required for the matrix multiplications (using the actual Jacobians). So long as $N<J$, using GMRES with the actual Jacobians is cheaper than solving for the actual Jacobian with QR. With small $N$, as we achieve using $\boldsymbol{\eta}$ and $\boldsymbol{\zeta}$, the savings is quite substantial. We note the following formulae specific to the Newton system case. For $\mathbf{A}=(D\mathbf{F})(\mathbf{x})$ and $\mathbf{b}=-\mathbf{F}(\mathbf{x})$, $\beta=-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}$ and $-\beta\mathbf{e}_{1}=\mathbf{P}^{(1)}\mathbf{b}=-\mathbf{P}^{(1)}\mathbf{F}(\mathbf{x})$ so that $\mathbf{P}^{(1)}\mathbf{F}(\mathbf{x})=\beta\mathbf{e}_{1}=-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}.$ Moreover, $\mathbf{P}^{(n)}\mathbf{e}_{1}=\mathbf{e}_{1}$ for all $n>1$ so that $(\mathbf{Q}^{(n)})^{\top}\mathbf{F}(\mathbf{x})=-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}.$ #### 4.2.2. Preconditioning As is well known, preconditioning is key to the effectiveness of iterative linear solvers; see Golub and Loan (1996). We have not found the linear systems in $\boldsymbol{\eta}$-NM or $\boldsymbol{\zeta}$-NM to need preconditioning. However we have found the preconditioned system (26) $\boldsymbol{\Lambda}(\mathbf{p})^{-1}(D\tilde{\nabla}\hat{\pi})(\mathbf{p})\mathbf{s}^{IN}=-\boldsymbol{\Lambda}(\mathbf{p})^{-1}(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})-\mathbf{p}$ to be very necessary for rapid solution of the Newton system in CG-NM. This preconditioner is motivated by the following relationship of the Jacobian of $(\tilde{\nabla}\hat{\pi})$ to the Jacobian of $\boldsymbol{\zeta}$ in equilibrium. ###### Lemma 4.1. $\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})^{-1}(D\tilde{\nabla}\hat{\pi})(\mathbf{p})$ for any simultaneously stationary $\mathbf{p}$. ###### Proof. This follows from differentiating $(\tilde{\nabla}\hat{\pi})(\mathbf{p})=\boldsymbol{\Lambda}(\mathbf{p})(\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p}))$ via the product rule, recognizing that $\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})=\mathbf{0}$ in equilibrium and $D[\mathbf{p}-\mathbf{c}-\boldsymbol{\zeta}(\mathbf{p})]=\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p})$. ∎ In other words, Newton’s methods applied to $\mathbf{F}_{\pi}(\mathbf{p})$ preconditioned as above ends up being essentially the same iteration as $\mathbf{F}_{\zeta}(\mathbf{p})$, close enough to equilibrium. GMRES, if used successfully on this preconditioned system Eqn. (26), will ensure that (27) $\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}(\tilde{\nabla}\hat{\pi})(\mathbf{p})+\boldsymbol{\Lambda}(\mathbf{p})^{-1}(D\tilde{\nabla}\hat{\pi})(\mathbf{p})\mathbf{s}^{IN}\rvert\rvert\leq\delta^{\prime}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert$ for some $\delta^{\prime}$. This is distinct from the inexact Newton condition Eqn. (25). The following proposition gives modified tolerances for the preconditioned system to ensure satisfaction of the original system. ###### Proposition 4.2. Let $\delta>0$ be given. If Eqn. (27) is satisfied with $\delta^{\prime}(\mathbf{p},\delta)\leq\delta$ given by (28) $\delta^{\prime}(\mathbf{p},\delta)=\left(\frac{\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{2}}{\max_{j}\left\\{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\right\\}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{2}}\right)\delta,$ then Eqn. (25) is satisfied. This is a consequence of the following general result, which we state without proof. ###### Lemma 4.3. Let $\mathbf{b}\in\mathbb{R}^{N}$ and $\mathbf{A},\mathbf{M}\in\mathbb{R}^{N\times N}$ be nonsingular. Then (29) $\frac{\lvert\lvert\mathbf{Ax}-\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{b}\rvert\rvert}\leq\alpha\left(\frac{\lvert\lvert\mathbf{M}^{-1}\mathbf{Ax}-\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}\right)$ where $\alpha\in[1,\kappa(\mathbf{M})]$ is given by $\alpha=\frac{\lvert\lvert\mathbf{M}\rvert\rvert\lvert\lvert\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{b}\rvert\rvert}=\lvert\lvert\mathbf{M}\rvert\rvert\left\lvert\left\lvert\mathbf{M}^{-1}\left(\frac{\mathbf{b}}{\lvert\lvert\mathbf{b}\rvert\rvert}\right)\right\rvert\right\rvert.$ This implies that $\frac{\lvert\lvert\mathbf{Ax}-\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{b}\rvert\rvert}\leq\delta\quad\quad\text{if}\quad\quad\frac{\lvert\lvert\mathbf{M}^{-1}\mathbf{Ax}-\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}{\lvert\lvert\mathbf{M}^{-1}\mathbf{b}\rvert\rvert}\leq\frac{\delta}{\alpha}.$ Note that the preconditioned system must always be solved to a stricter tolerance than is desired for the un-preconditioned system using this bound. Additionally, computing $\alpha$ for a generic preconditioner $\mathbf{M}$ relies on the ability to compute $\lvert\lvert\mathbf{M}\rvert\rvert$. Eqn. (28) simply adopts the 2-norm and applies the formula (Golub and Loan, 1996) $\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{2}=\sqrt{\max_{j}\\{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert^{2}\\}}=\max_{j}\\{\left\lvert\lambda_{j}(\mathbf{p})\right\rvert\\}$ Eqn. (29) also implies that if Eqn. (27) holds with $\delta^{\prime}>0$, then $\frac{\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})+(D\tilde{\nabla}\hat{\pi})(\mathbf{p})\mathbf{s}^{IN}\rvert\rvert_{2}}{\lvert\lvert(\tilde{\nabla}\hat{\pi})(\mathbf{p})\rvert\rvert_{2}}\leq\kappa_{2}(\boldsymbol{\Lambda}(\mathbf{p}))\delta^{\prime}$ where $\kappa_{2}(\boldsymbol{\Lambda}(\mathbf{p}))=\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})\rvert\rvert_{2}\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\rvert\rvert_{2}$ is the (2-norm) condition number of $\boldsymbol{\Lambda}(\mathbf{p})$. This equation, while the more compact representation, can also be overly conservative as clearly illustrated in Fig. 1. It is unlikely that $\kappa(\boldsymbol{\Lambda}(\mathbf{p}))$ is a tight upper bound on the multiplier in Eqn. (28). In fact, the multiplier on $\delta$ depends only on the norm of $\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{x}$ at a single point on the surface of the unit sphere in $\mathbb{R}^{J}$ rather than $\lvert\lvert\boldsymbol{\Lambda}(\mathbf{p})^{-1}\rvert\rvert_{2}$, the maximum norm of $\boldsymbol{\Lambda}(\mathbf{p})^{-1}\mathbf{x}$ over this entire sphere. Our examples in Fig. 1 bear this out, having condition numbers many orders of magnitude larger than the multiplier in Eqn. (28). The power of the preconditioning is that the preconditioned system Eqn. (27) appears to be solved to a relative error of $\delta^{\prime}(\mathbf{p},\delta)$ much faster than the original system can be solved to a relative error of $\delta$, even though $\delta^{\prime}(\mathbf{p},\delta)\leq\delta$. As can be seen in Fig. 1, solving the preconditioned system to $\delta^{\prime}(\mathbf{p},\delta)$ can achieve a relative error in the original system below $\delta=10^{-10}$ in roughly four orders of magnitude fewer iterations than solving the original system to this same relative error for prices near equilibrium. Away from equilibrium, GMRES may not be able to solve the original system to small relative errors like $10^{-6}$ at all. Thus using the original system would appear to slow, if not halt, an implementation of the inexact Newton’s method. Figure 1. Relative error in computed solutions to the CG-NM Newton system and its preconditioned form using GMRES in the vehicle example from Morrow and Skerlos (2010) using the Berry et al. (1995) model. On the top, prices are $\mathbf{p}=\mathbf{p}^{*}+100\boldsymbol{\nu}$ where $\mathbf{p}^{*}$ are equilibrium prices and $\boldsymbol{\nu}\in[-\mathbf{1},\mathbf{1}]$ is a sample from a uniformly distributed random vector. For this case $\kappa(\boldsymbol{\Lambda}(\mathbf{p}))=1.56\times 10^{11}$ while the multiplier in Eqn. (29) is only 106.41. On the bottom, prices are $\mathbf{p}=20,000\boldsymbol{\nu}+5,000$ where $\boldsymbol{\nu}$ is a sample from a random vector uniformly distributed on $[\mathbf{0},\mathbf{1}]$. For this case $\kappa(\boldsymbol{\Lambda}(\mathbf{p}))=4.6\times 10^{4}$ while the multiplier in Eqn. (29) is only 10.73. Abbreviations are as follows. REL: relative error in the Newton System; PREL: relative error in the pre- conditioned Newton System; OBREL: our bound, Eqn. (29), on the relative error in the Newton System as determined from the relative error in the preconditioned Newton system; CNBREL: condition number bound on the relative error in the Newton System as determined from the relative error in the preconditioned Newton system. ### 4.3. The GMRES Hookstep Suitable modifications of each of the globalization strategies originally developed for “exact” Newton methods can be applied in the inexact context. Brown and Saad (1990) directly extend line search and a dogleg steps to GMRES- Newton methods. Eisenstat and Walker (1996) and Pernice and Walker (1998) apply a safeguarded backtracking line search to facilitate global convergence. More recently, Pawlowski et al. (2006, 2008) have studied dogleg steps suitable for GMRES-Newton methods in some detail. Finally Viswanath (2007) has derived an elegant version of the hookstep method suitable for GMRES-Newton methods. In contrast with the hookstep approach for the “exact” Newton method with Jacobian $(D\mathbf{F})(\mathbf{p})$, Viswanath’s approach requires computing the SVD only of a matrix whose size is determined by the number of iterations taken by GMRES. For reasonable applications of GMRES, this can be far less than the size of $(D\mathbf{F})(\mathbf{p})$ itself. For the examples in Morrow and Skerlos (2010), the size difference is roughly two orders of magnitude: the GMRES-Newton hookstep worked with roughly $10\times 10$ instead of $1,000\times 1,000$ matrices. Thus, the GMRES-hookstep can accumulate a tremendous savings over an exact-Newton implementation of the hookstep method. Again, each of these approaches iterates until an acceptable step is found, and can, in principle, involve many additional evaluations of $\mathbf{F}$ or fail to find an acceptable step altogether. Here we describe an implementation of the Levenberg-Marquardt method or “hookstep” (Dennis and Schnabel, 1996) suitable for GMRES as first suggested by Viswanath (2007). See also Viswanath (2009); Viswanath and Cvitanovic (2009); Halcrow et al. (2009). First, we recall the basic structure of model trust region methods; see (Dennis and Schnabel, 1996, Chapter 6, Section 4). We then adopt this structure to the case of Krylov subspace methods, particularly GMRES. Again, see Morrow10b for a more detailed discussion of this method. #### 4.3.1. Model Trust Region Methods. Trust region methods assume that for steps $\mathbf{s}$ satisfying $\lvert\lvert\mathbf{s}\rvert\rvert_{2}\leq\delta$, the function $\hat{m}_{\mathbf{x}}(\mathbf{s})=\left(\frac{1}{2}\right)\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}+((D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x}))^{\top}\mathbf{s}+\left(\frac{1}{2}\right)\mathbf{s}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{s}$ is an accurate local model of $f(\mathbf{x})=\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}/2$ for suitably small steps. Note that $\hat{m}_{\mathbf{x}}$ is not the usual, quadratic model of $f$ derived from a Taylor series because $(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\neq(D\nabla f)(\mathbf{x})$ (Dennis and Schnabel, 1996, pg. 149). The idea is to solve (30) $\min_{\lvert\lvert\mathbf{s}\rvert\rvert_{2}\leq\delta}\hat{m}_{\mathbf{x}}(\mathbf{s}).$ The solution $\mathbf{s}_{*}$ is given as follows: take $\mathbf{s}_{*}=\mathbf{s}^{N}=-(D\mathbf{F})(\mathbf{x})^{-1}\mathbf{F}(\mathbf{x})$ if $\lvert\lvert\mathbf{s}^{N}\rvert\rvert_{2}\leq\delta$; if $\lvert\lvert\mathbf{s}^{N}\rvert\rvert_{2}>\delta$, take $\mathbf{s}_{*}=\mathbf{s}(\mu_{*})$ where $\mathbf{s}(\mu)=-\big{(}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})+\mu\mathbf{I}\big{)}^{-1}(D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x})$ and $\mu_{*}>0$ is the unique $\mu>0$ such that $\lvert\lvert\mathbf{s}(\mu)\rvert\rvert_{2}=\delta$. These follow from the standard optimality conditions, or rather that the gradient $(\nabla\hat{m}_{\mathbf{x}})(\mathbf{s})$ must lie in the negative normal cone to $\bar{\mathbb{B}}_{\delta}(\mathbf{0})=\\{\mathbf{y}\in\mathbb{R}^{N}:\lvert\lvert\mathbf{y}\rvert\rvert_{2}\leq\delta\\}$ at $\mathbf{x}$ (Clarke, 1975); see (Dennis and Schnabel, 1996, Lemma 6.4.1, pg. 131). Solving the problem above exactly generates the Levenberg-Marquardt method (Levenberg, 1944; Marquardt, 1963) or “hookstep.” By computing the SVD of $(D\mathbf{F})(\mathbf{x})$ we can easily solve for $\mathbf{s}(\mu)$ when $\lvert\lvert\mathbf{s}^{N}\rvert\rvert_{2}>\delta$ (Dennis and Schnabel, 1996); see (Golub and Loan, 1996, Section 12.1, pgs. 580-583) for closely related results. Let $(D\mathbf{F})(\mathbf{x})=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{\top}$. We can then set $\mathbf{s}(\mu)=\mathbf{V}\mathbf{r}(\mu)$ where $\mathbf{r}(\mu)=-(\boldsymbol{\Sigma}^{2}+\mu\mathbf{I})^{-1}\boldsymbol{\Sigma}\mathbf{U}^{\top}\mathbf{F}(\mathbf{x}).$ A simple single-dimensional iteration can then be used to solve for the unique $\mu_{*}$ such that $\lvert\lvert\mathbf{s}(\mu_{*})\rvert\rvert_{2}=\delta$. Morrow10b derives two globally convergent methods for this task using Newton’s method and a nonlinear local model (Dennis and Schnabel, 1996). The difficulty here is computing the SVD of $(D\mathbf{F})(\mathbf{x})$, requiring $\mathcal{O}(J^{3})$ flops (Golub and Loan, 1996, Chapter 5, pg. 254). The step $\mathbf{s}_{*}$ computed by either approach is acceptable if it generates sufficient decrease in the squared 2-norm of $\mathbf{F}$. Specifically, fix $\rho\in(0,1)$, $\alpha>1$, and $\beta_{2}\leq\beta_{1}<1$. If $\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}-\lvert\lvert\mathbf{F}(\mathbf{x}+\mathbf{s}_{*})\rvert\rvert_{2}^{2}\geq\rho(\lvert\lvert(D\mathbf{F})(\mathbf{x})\rvert\rvert_{2}^{2}-\lvert\lvert\mathbf{F}(\mathbf{x})+(D\mathbf{F})(\mathbf{x})\mathbf{s}_{*}\rvert\rvert_{2}^{2})$ then $\mathbf{p}\leftarrow\mathbf{p}+\mathbf{s}_{*}$ and a the step length bound is expanded to $[\delta,\alpha\delta]$ for the next iteration. Otherwise, $\delta$ is chosen from $[\beta_{1}\delta,\beta_{2}\delta]$ and the corresponding $\mathbf{s}_{*}$ is computed. While this process of specifying an acceptable $\mathbf{s}_{*}$ is iterative, much of the work required to build a trial step does not need to be repeated. Specifically the SVD required for the hookstep does not change (so long as it was computed in a previous iteration) while in the doglep step the Newton and Cauchy steps remain the same. However every time the step size bound is decreased $\mathbf{F}$ must be re-evaluated at the new trial step, with a computational burden equivalent to taking a fixed-point step. #### 4.3.2. Model Trust Region Methods on a Subspace A Krylov method for solving $(D\mathbf{F})(\mathbf{x})\mathbf{s}^{N}=-\mathbf{F}(\mathbf{x})$ builds approximate solutions in the successive Krylov subspaces $\mathcal{K}^{(n)}$. This has the effect of further constraining the local model problem (30) to (31) $\min_{\mathbf{s}\in\mathcal{K}^{(n)},\;\lvert\lvert\mathbf{s}\rvert\rvert_{2}\leq\delta}\hat{m}_{\mathbf{x}}(\mathbf{s}).$ For any $\mathbf{Q}\in\mathbb{R}^{J\times n}$ with orthonormal columns (generated by GMRES or not) we can set $\hat{m}_{\mathbf{x},\mathbf{Q}}(\mathbf{y})=\hat{m}_{\mathbf{x}}(\mathbf{Q}\mathbf{y})$ and restrict attention to the trust region problem $\min_{\lvert\lvert\mathbf{y}\rvert\rvert_{2}\leq\delta}\hat{m}_{\mathbf{x},\mathbf{Q}}(\mathbf{y})$. See (Brown and Saad, 1990, pgs. 149-150). The first-order conditions for this problem are equivalent to either * (i) $(\nabla\hat{m}_{\mathbf{x},\mathbf{Q}})(\mathbf{y})=\mathbf{0}$ and $\lvert\lvert\mathbf{y}\rvert\rvert_{2}\leq\delta$ * (ii) or $(\nabla\hat{m}_{\mathbf{x},\mathbf{Q}})(\mathbf{y})+\mu\mathbf{y}=\mathbf{0}$ for $\lvert\lvert\mathbf{y}\rvert\rvert_{2}=\delta$ and some $\mu>0$. By the definition of $\hat{m}_{\mathbf{x},\mathbf{Q}}$, (i) implies $\mathbf{Q}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{Q}\mathbf{y}+\mathbf{Q}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x})=\mathbf{0}$ and (ii) implies $\left(\mathbf{Q}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{Q}+\mu\mathbf{I}\right)\mathbf{y}+\mathbf{Q}^{\top}(D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x})=\mathbf{0}.$ Note that these are square problems that can be solved exactly. #### 4.3.3. The GMRES-Newton Hookstep Using GMRES started at zero, $(D\mathbf{F})(\mathbf{x})\mathbf{Q}^{(n)}=\mathbf{Q}^{(n+1)}\tilde{\mathbf{H}}^{(n)}$ and $(\mathbf{Q}^{(n+1)})^{\top}\mathbf{F}(\mathbf{x})=-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}$. Thus we consider the family of $n\times n$ linear systems $\displaystyle(\mathbf{Q}^{(n)})^{\top}(D\mathbf{F})(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{Q}^{(n)}\mathbf{q}+\mu\mathbf{q}+(\mathbf{Q}^{(n)})^{\top}(D\mathbf{F})(\mathbf{x})^{\top}\mathbf{F}(\mathbf{x})$ $\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad=\big{(}(\tilde{\mathbf{H}}^{(n)})^{\top}\tilde{\mathbf{H}}^{(n)}+\mu\mathbf{I}\big{)}\mathbf{q}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}(\tilde{\mathbf{H}}^{(n)})^{\top}\mathbf{e}_{1}=\mathbf{0}$ defined for all $\mu\geq 0$. By computing the (“thin”) Singular Value Decomposition of $\tilde{\mathbf{H}}^{(n)}$, $\tilde{\mathbf{H}}^{(n)}=\tilde{\mathbf{U}}\boldsymbol{\Sigma}\mathbf{V}^{\top}$ where $\tilde{\mathbf{U}}\in\mathbb{R}^{(n+1)\times n}$, $\mathbf{V}\in\mathbb{R}^{n\times n}$, and $\boldsymbol{\Sigma}\in\mathbb{R}^{n\times n}$, we can easily solve each such problem. See (Golub and Loan, 1996, Section 12.1, pgs. 580-583) for closely related results. Particularly, $\displaystyle((\tilde{\mathbf{H}}^{(n)})^{\top}\tilde{\mathbf{H}}^{(n)}+\mu\mathbf{I})\mathbf{q}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}(\tilde{\mathbf{H}}^{(n)})^{\top}\mathbf{e}_{1}=\mathbf{0}$ is solved by $\mathbf{q}(\mu)=\mathbf{V}\mathbf{r}(\mu)$ where $\mathbf{r}(\mu)=\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}(\boldsymbol{\Sigma}^{2}+\mu\mathbf{I})^{-1}\boldsymbol{\Sigma}\tilde{\mathbf{U}}^{\top}\mathbf{e}_{1}.$ Because the diagonal elements of $\boldsymbol{\Sigma}^{2}$ are positive, $\mathbf{r}(\mu)$ is well defined for all $\mu\geq 0$. Note also that we only need the first row of $\mathbf{U}$, but all of $\mathbf{V}$, to compute $\mathbf{q}(\mu)$. In particular, $\mathbf{q}(0)=\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{V}\boldsymbol{\Sigma}^{-1}\mathbf{U}^{\top}\mathbf{e}_{1}$. Invoking the full SVD of $\tilde{\mathbf{H}}^{(n)}$, $\displaystyle\tilde{\mathbf{H}}^{(n)}=\begin{bmatrix}\tilde{\mathbf{U}}&\mathbf{u}_{n+1}\end{bmatrix}\begin{bmatrix}\boldsymbol{\Sigma}\\\ \mathbf{0}^{\top}\end{bmatrix}\mathbf{V}^{\top}$ for some $\mathbf{u}_{n+1}\perp\mathrm{span}\\{\mathbf{u}_{i}\\}_{i=1}^{n}$, we can write $\displaystyle\lvert\lvert\tilde{\mathbf{H}}^{(n)}\mathbf{q}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}\rvert\rvert_{2}$ $\displaystyle=\left\lvert\left\lvert\begin{bmatrix}\tilde{\boldsymbol{\Sigma}}\mathbf{V}^{\top}\mathbf{q}\\\ 0\end{bmatrix}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\begin{bmatrix}\tilde{\mathbf{U}}^{\top}\mathbf{e}_{1}\\\ u_{1,n+1}\end{bmatrix}\right\rvert\right\rvert_{2}.$ We thus see that $\mathbf{q}(0)$ solves the $(n+1)\times n$ GMRES least squares problem $\min_{\mathbf{q}}\lvert\lvert\mathbf{H}^{(n+1,n)}\mathbf{q}-\mathrm{sign}(F_{1}(\mathbf{x}))\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}\mathbf{e}_{1}\rvert\rvert_{2}.$ with residual $\left\lvert u_{1,n+1}\right\rvert\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}$. $\left\lvert u_{1,n+1}\right\rvert$ is unique: First, note that $\mathbf{u}_{n+1}$ is a unit vector in the span of a single vector, say $\mathbf{v}$, that is orthogonal to the span of the columns of $\tilde{\mathbf{U}}$. There are only two unit vectors in this span, specifically $\pm\mathbf{v}/\lvert\lvert\mathbf{v}\rvert\rvert_{2}$, and thus $\mathbf{u}_{n+1}\in\\{\pm\mathbf{v}/\lvert\lvert\mathbf{v}\rvert\rvert_{2}\\}$. Thus $\left\lvert u_{1,n+1}\right\rvert\in\left\lvert\pm v_{1}/\lvert\lvert\mathbf{v}\rvert\rvert_{2}\right\rvert=\left\lvert v_{1}\right\rvert/\lvert\lvert\mathbf{v}\rvert\rvert_{2}$. It is also easy to see that $\displaystyle\mathbf{F}(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{s}^{(n)}(\mu)$ $\displaystyle=\mathbf{F}(\mathbf{x})^{\top}(D\mathbf{F})(\mathbf{x})\mathbf{Q}^{(n)}\mathbf{q}^{(n)}(\mu)$ $\displaystyle=\left(\big{(}\mathbf{Q}^{(n+1)}\big{)}^{\top}\mathbf{F}(\mathbf{x})\right)^{\top}\tilde{\mathbf{H}}^{(n)}\mathbf{q}^{(n)}(\mu)$ $\displaystyle=-\beta^{2}\left(\boldsymbol{\nu}_{1}^{\top}\mathbf{D}(\mu)\boldsymbol{\nu}_{1}\right)$ $\displaystyle=-\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}\left(\boldsymbol{\nu}_{1}^{\top}\mathbf{D}(\mu)\boldsymbol{\nu}_{1}\right)<0$ where $\boldsymbol{\nu}_{1}$ is the first row of $\tilde{\mathbf{U}}$ and $\mathbf{D}(\mu)=\mathrm{diag}(d_{1}(\mu),\dotsc,d_{n}(\mu))$ for $d_{i}(\mu)=\sigma_{i}^{2}/(\sigma_{i}^{2}+\mu)$. That is, the Householder GMRES-Newton Hookstep always lies in a descent direction for the globalizing objective $f(\mathbf{x})=\lvert\lvert\mathbf{F}(\mathbf{x})\rvert\rvert_{2}^{2}/2$. #### 4.3.4. Directional Finite Differences Recall that one advantage to using an iterative solver like GMRES to solve the Newton system is that only products of the type $(D\mathbf{F})(\mathbf{p})\mathbf{s}$ will be required to solve the Newton system for $\mathbf{F}$ at $\mathbf{p}$ (Brown and Saad, 1990; Pernice and Walker, 1998). Such products can be approximated by a single additional evaluation of $\mathbf{F}$ in a “directional” finite difference (Brown and Saad, 1990; Pernice and Walker, 1998). For example, the first-order formula $(D\mathbf{F})(\mathbf{x})\mathbf{s}\approx h^{-1}\big{(}\mathbf{F}(\mathbf{x}+h\mathbf{s})-\mathbf{F}(\mathbf{x})\big{)},$ requires only a single additional evaluation of $\mathbf{F}$ per (approximate) evaluation of $(D\mathbf{F})(\mathbf{x})\mathbf{s}$. Higher-order formulae requiring 2 and 4 additional evaluations of $\mathbf{F}$ are easy to derive; see Pernice and Walker (1998). In their implementation of the GMRES method in the context of an inexact Newton method, Pernice and Walker (1998) only use higher order finite-differencing formulas at restarts. Brown and Saad (1990) provide a practical formula for computing an appropriate value of $h$. Since directional finite derivatives must be repeated at each step of iterative linear solvers, each step of an iterative Newton system solver using directional finite differences could be at least as expensive as a $\boldsymbol{\zeta}$-FPI step. That is, if an iterative solver should take 100 steps to compute an inexact Newton step having small enough residual to satisfy the inexact Newton condition, then we could have equivalently taken 100, 200, and 400 $\boldsymbol{\zeta}$-FPI steps with the first, second, and fourth order formulae available in Pernice and Walker (1998). In our examples, using GMRES regularly solves the $\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM Newton systems in approximately 10 steps. This implies that each $\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM step is roughly equivalent to $10$ $\boldsymbol{\zeta}$-FPI steps. In the Newton context, whether efficiency is ultimately gained by using directional finite differences instead of computing the Jacobian matrices and using standard matrix-vector products depends on the number of steps taken by the iterative linear solver. If GMRES takes $N\in\mathbb{N}$ iterations to find an inexact Newton step for $\mathbf{F}$, computing and using the Jacobian requires $\mathcal{O}((S+N)J^{2})$ flops while using directional finite differences requires $\mathcal{O}(SN\sum_{f=1}^{F}J_{f}^{2})$ flops. We have observed that for $\boldsymbol{\eta}$-NM and $\boldsymbol{\zeta}$-NM, using the actual Jacobian takes roughly half the computation time than using directional finite differences, even though GMRES converges in very few iterations ($N\sim 10$). Fig. 2 plots the sample trials for the Boyd and Mellman (1980) model provided in Morrow and Skerlos (2010) using both analytical Jacobians and directional finite differences. First note that the $\boldsymbol{\zeta}$-FPI regularly takes about 1 s per iteration. For $\kappa=1$ USD, the single-step convergence of the GMRES-Newton Hookstep method translates into about 10 $\boldsymbol{\zeta}$-FPI steps, or about $10$ s. Because GMRES itself requires some small overhead ($\mathcal{O}(Jn)$ in the $n^{\text{th}}$ step), this is a somewhat reasonable estimate of the work required. Two GMRES-Newton steps are required with $\kappa=10$ USD and we would expect about $20$ s, a somewhat less sound estimate of the time required. Three GMRES-Newton steps are required with $\kappa=100$ USD, leading us to expect about $20$ s, a further less sound estimate of the time required. (These observations can be matched with an asymptotic analysis of the work required.) Note also that the $\boldsymbol{\eta}$-NM has the greatest increase in time as a consequence of using the directional finite differences. This is a consequence of having to repeat block QR factorizations when evaluating $\boldsymbol{\eta}$ at different points, while evaluating $(D\boldsymbol{\eta})$ requires only a single factorization. Figure 2. Typical convergence curves for perturbation trials under the Boyd and Mellman (1980) model using both analytical and directional finite difference Jacobians. See also Fig. LABEL:FIG_BM80_BestCasePert. Convergence curves for analytical Jacobian are drawn with solid lines, whereas convergence curves for directional finite differences are drawn with dashed lines of the same color. Fig. 3 plots the sample trials for the Berry et al. (1995) model provided in Morrow and Skerlos (2010) using both analytical Jacobians and directional finite differences. Interestingly, in this case use of the directional finite differences appears to generate a convergence rate improvement. Otherwise, the story remains much the same as that discussed above for the Boyd and Mellman (1980) model. Figure 3. Typical convergence curves for perturbation trials under the Boyd and Mellman (1980) model using both analytical and directional finite difference Jacobians. See also Fig. LABEL:FIG_BLP95_BestCasePert. Convergence curves for analytical Jacobian are drawn with solid lines, whereas convergence curves for directional finite differences are drawn with dashed lines of the same color. ## 5\. Other Methods ### 5.1. Variational Methods Equilibrium problems are commonly formulated as variational inequalities or complementarity problems (Harker and Pang, 1990; Ferris and Pang, 1997). To be nontrivially distinct from nonlinear equations, such formulations require restricting the variables to a proper, convex subset of $\mathbb{R}^{J}$. When $\varsigma_{*}<\infty$ there is an appropriate variational formulation of the equilibrium pricing problem: (32) $\text{find}\quad\mathbf{p}\in[0,\varsigma_{*}]^{J}\quad\text{such that}\quad(\tilde{\nabla}\hat{\pi})(\mathbf{p})^{\top}(\mathbf{p}-\mathbf{q})\geq 0\quad\text{for all}\quad\mathbf{q}\in[0,\varsigma_{*}]^{J}.$ #### 5.1.1. The VI formulation is poorly posed Unfortunately, the Variational Inequality (32) is poorly posed when the derivatives of profit vanish as prices approach $\varsigma_{*}<\infty$. There are two specific issues with Eqn. (32) in this case. First, $\varsigma_{*}\mathbf{1}\in\mathcal{P}^{J}$ is always a solution but never an equilibrium when profits vanish as all prices approach $\varsigma_{*}$; see Section LABEL:ECSUBSEC:Profits and Lemma 5.1. Second, Eqn. (32) can be solved by any equilibrium of any differentiated product market model constructed with a subset of the products offered (Prop. 5.2). Equilibria of such “sub- problems” are not necessarily equilibria of the original problem, as demonstrated in Example 8 below. This issue with Eqn. (32) is, in fact, equivalent to the problem with CG-NM discussed in Section 3.2. These issues imply that variational methods can compute many “spurious” solutions. If an equilibrium problem and all its sub-problems have unique equilibria with all prices less than $\varsigma_{*}$, Eqn. (32) has $2^{J}$ solutions that might be recovered by a global method such as PATH (Ralph, 1994; Dirkse and Ferris, 1995). However, only one of these solutions is an equilibrium of the original problem, by assumption. A simple example demonstrates this phenomenon. ###### Example 8. Consider a monopoly with two products produced at the same unit cost $c$. Demand is given by a simple Logit model with product-specific utility functions $u_{j}(p_{j})=\alpha\log(\varsigma-p_{j})+v_{j}$ for $j\in\\{1,2\\}$, where $\varsigma\in(c,\infty)$, $v_{1},v_{2}\in\mathbb{R}$, and $\vartheta>-\infty$. The firm has unique profit-maximizing prices $(p_{1}^{*},p_{2}^{*})$. Furthermore $p_{1}^{*},p_{2}^{*}<\varsigma$, and $(p_{1}^{*},p_{2}^{*})$ is the unique fixed-point of the map $\mathbf{c}+\boldsymbol{\zeta}(\cdot)$ on all of $\mathcal{P}^{2}$. However the variational inequality formulation contains four distinct solutions, only one of which is profit-maximizing. These four solutions are $(p_{1}^{*},p_{2}^{*})$, $(\varsigma,\varsigma)$, $(q_{1}^{*},\varsigma)$, and $(\varsigma,q_{2}^{*})$, where $q_{j}^{*}<\varsigma$ for $j\in\\{1,2\\}$ are the unique profit-maximizing prices that exist should the firm offer only product $1$ or $2$. Only the first solution, $(p_{1}^{*},p_{2}^{*})$, is profit-maximizing. ###### Proof. We complete the details of Example 8. Consider a monopoly with two products produced at the same unit cost ($c=c_{1}=c_{2}>0$), $\vartheta>-\infty$, and simple Logit model with utility $u_{1}(p_{1})=\alpha\log(\varsigma-p_{1})+v_{1}\quad\text{and}\quad u_{2}(p_{2})=\alpha\log(\varsigma-p_{2})+v_{2}$ for some fixed $\varsigma\in(c,\infty)$, $\alpha>1$, and arbitrary $v_{1},v_{2}\in\mathbb{R}$. Let $p_{2}\leq\varsigma$, and observe that $\lim_{p_{1}\uparrow\varsigma}\Big{(}p_{1}-c-\zeta_{1}(p_{1},p_{2})\Big{)}=\varsigma- c- P_{2}(\varsigma,p_{2})(p_{2}-c)=(\varsigma-c)\left[1-P_{2}(\varsigma,p_{2})\left(\frac{p_{2}-c}{\varsigma-c}\right)\right].$ Since $p_{2}\leq\varsigma$ and $P_{2}(p_{1},p_{2})<1$ for all $p_{1},p_{2}$, we have $\lim_{p_{1}\uparrow\varsigma}(p_{1}-c-\zeta_{1}(p_{1},p_{2}))>0$. Thus $(D_{1}\hat{\pi})(p_{1},p_{2})<0$ for all $p_{1}$ sufficiently close to $\varsigma$. A similar argument can be made for $(D_{2}\hat{\pi})(p_{1},p_{2})$. Note also that this proves that $\varsigma+\epsilon>c+\zeta_{1}(\varsigma+\epsilon,p_{2})$ for any $\epsilon\geq 0$ and $p_{2}$, where $\zeta_{1}$ is the extended map. A similar result holds for $\zeta_{2}$, instead of $\zeta_{1}$. Thus no $(p_{1},p_{2})$ outside of $(0,\varsigma)$ is fixed for the extended map $\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$. We now prove that there exists a unique pair of profit-maximizing prices $\mathbf{p}^{*}=(p_{1}^{*},p_{2}^{*})\in(0,\varsigma)^{2}$. Since $\lim_{p_{j}\uparrow\varsigma}\Big{(}p_{j}-c-\zeta_{j}(p_{1},p_{2})\Big{)}<\infty$ for $j\in\\{1,2\\}$, $\boldsymbol{\zeta}=(\zeta_{1},\zeta_{2})$ is bounded and continuous on $\mathcal{P}^{2}$. By Brower’s fixed-point theorem, there exists a stationary point $\mathbf{p}^{*}=(p_{1}^{*},p_{2}^{*})$. Both prices must both be less than $\varsigma$, since profits decrease for all prices sufficiently close to $\varsigma$. We now show that these prices are also unique, borrowing a technique from Morrow and Skerlos (2008). The first step is to prove that $(D\nabla\hat{\pi})(\mathbf{p}^{*})$ is negative definite at any stationary $\mathbf{p}^{*}$. Note that $(D\nabla\hat{\pi})(\mathbf{p}^{*})=\boldsymbol{\Lambda}(\mathbf{p}^{*})(\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p}^{*}))$; this relationship is valid for Mixed Logit models with multiple firms as well. Furthermore $\zeta_{j}(\mathbf{p})=\hat{\pi}(\mathbf{p})-(Dw_{k})(p_{k})^{-1}$ for any simple Logit model and any number of products. Hence $(D_{k}\zeta_{j})(\mathbf{p})=(D_{k}\hat{\pi})(\mathbf{p})+\delta_{j,k}\left(\frac{(D^{2}w_{k})(p_{k})}{(Dw_{k})(p_{k})^{2}}\right),$ and $\mathbf{I}-(D\boldsymbol{\zeta})(\mathbf{p}^{*})$ is a diagonal matrix with elements $1-\frac{(D^{2}w_{k})(p_{k})}{(Dw_{k})(p_{k})^{2}}.$ In the case of this example, $1-\frac{(D^{2}w_{1})(p_{1})}{(Dw_{1})(p_{1})^{2}}=1-\frac{(D^{2}w_{2})(p_{2})}{(Dw_{2})(p_{2})^{2}}=1+\frac{1}{\alpha}>0.$ Thus $(D\nabla\hat{\pi})(\mathbf{p}^{*})$ is negative definite at any stationary point, and any stationary point maximizes profits. The next step is to prove that the existence of only maximizers of profits proves that there is a unique pair of profit-maximizing prices. Morrow and Skerlos (2008) accomplish this with an application of the Poincare-Hopf theorem (Milnor, 1965), as follows. Consider $-\hat{\pi}(\mathbf{p})$. This function is minimized at any stationary $\mathbf{p}^{*}=(p_{1},p_{2})$, and thus the gradient vector field $-(\nabla\hat{\pi})(\mathbf{p})$ has index $1$ at any stationary point $\mathbf{p}^{*}$ (Milnor, 1965). Note also that $\displaystyle\mathrm{sign}\\{-(D_{j}\hat{\pi})(p_{1},p_{2})\\}$ $\displaystyle=\mathrm{sign}\left\\{p_{j}-c-\hat{\pi}(p_{1},p_{2})-\frac{\varsigma- p_{j}}{\alpha}\right\\}$ for $j\in\\{1,2\\}$. This equation shows that the gradient vector field $-(\nabla\hat{\pi})(\mathbf{p})$ points outward on the boundary of the compact, convex set $[c,\varsigma]^{2}$, as can be checked. Thus the Poincare- Hopf theorem states that the sum of the indices of the critical (stationary) points equals one, the Euler characteristic of $[c,\varsigma]^{2}$. Since the index of any critical (stationary) point of $-(\nabla\hat{\pi})(\mathbf{p})$ is one, there can only be one stationary point. Using similar arguments, we see that the sub-problems formed by offering product 1 or product 2 alone also have unique profit-maximizing prices $q_{1}^{*}$ and $q_{2}^{*}$, respectively. Because $v_{1}$ and $v_{2}$ may be distinct, these prices need not be the same. We have claimed that variational formulation of this problem has four solutions, only one of which is an equilibrium. Indeed, these four solutions are $(p_{1}^{*},p_{2}^{*})$, $(q_{1}^{*},\varsigma)$, $(\varsigma,q_{2}^{*})$, and $(\varsigma,\varsigma)$ but, as shown above, only $(p_{1}^{*},p_{2}^{*})$ is an equilibrium. While this follows from Props. 5.1 and 5.2 above, we prove it directly here. Of course, $(p_{1}^{*},p_{2}^{*})$ is a solution since $(\nabla\hat{\pi})(p_{1}^{*},p_{2}^{*})=(0,0)$. Since $\lim_{p_{j}\uparrow\varsigma}\lambda_{j}(p_{1},p_{2})=\lim_{p_{j}\uparrow\varsigma}\left[\left(\frac{\varsigma- p_{j}}{\alpha}\right)P_{j}(p_{1},p_{2})\right]=0$ for $j\in\\{1,2\\}$, $\lim_{p_{j}\uparrow\varsigma}(D_{j}\hat{\pi})(p_{1},p_{2})=0$ (i.e., Assumption 3.1 holds). Thus $(\nabla\hat{\pi})(\varsigma,\varsigma)=(0,0)$, and the variational inequality is satisfied at $(\varsigma,\varsigma)$. Furthermore, $(D_{1}\hat{\pi})(\varsigma,p_{2})(\varsigma- q_{1})+(D_{2}\hat{\pi})(\varsigma,p_{2})(p_{2}-q_{2})=(D_{2}\hat{\pi})(\varsigma,p_{2})(p_{2}-q_{2})$ and thus $(\varsigma,q_{2}^{*})$ is also a solution to the variational inequality. Similarly, $(q_{1}^{*},\varsigma)$ is also a solution. This completes the proof. ∎ Example 8 is easily generalized to include $J>2$ products and a variational inequality with $2^{J}$ solutions. One of these solutions is the unique vector of profit-maximizing prices for the original problem, one is $\varsigma\mathbf{1}\in\mathcal{P}^{J}$ and is not profit-maximizing for any sub-problem, and the rest are profit-maximizing for some sub-problem but not profit-maximizing for the original problem. This property of variational formulations is especially problematic since computations of equilibrium prices must often be performed using models with $\varsigma_{*}<\infty$. Such models may be derived from simulation-based approximations to Mixed Logit models with reservation prices that are finite $\mu$-a.e., as in the Berry et al. (1995) model of Example 2. Fortunately methods based on the $\boldsymbol{\zeta}$ map resolve only equilibria of the original problem. In Section 5.1.3 we consider the important class of simulation-based approximations to Mixed Logit models like those from Example 2 and prove that fixed-points of $\mathbf{c}+\boldsymbol{\zeta}(\cdot)$ cannot be equilibria of a sub-problem that is not an equilibria of the original model. This is essentially a consequence of Eqn. (21), which connects the sign of $(D_{k}\hat{\pi}_{f})(\mathbf{p})$ directly to the sign of $p_{k}-c_{k}-\zeta_{k}(\mathbf{p})$. Similar results may apply to the markup equation. However because Eqn. (20) involves $(\tilde{D}\mathbf{P})(\mathbf{p})^{\top}$ instead of simply the diagonal matrix $\boldsymbol{\Lambda}(\mathbf{p})$, the relationship between the sign of $p_{k}-c_{k}-\eta_{k}(\mathbf{p})$ and the sign of $(D_{k}\hat{\pi}_{f})(\mathbf{p})$ is not clear. #### 5.1.2. General Results. We now prove the results stated above concerning a variational formulation of the price equilibrium problem when $\varsigma_{*}<\infty$. ###### Proposition 5.1. Suppose $\varsigma_{*}<\infty$ and Assumptions 2.1-3.1 hold. Then the variational inequality (32) always contains $\varsigma_{*}\mathbf{1}\in\mathcal{P}^{J}$ as a solution. ###### Proof. Since $(\tilde{\nabla}\hat{\pi})(\varsigma_{*}\mathbf{1})=\mathbf{0}$, Eqn. (32) is trivially satisfied. ∎ The following proposition states that this variational formulation is poorly posed in the sense that it contains solutions to all sub-problems. ###### Proposition 5.2. Let $\varsigma_{*}<\infty$ and Assumptions 2.1-3.1 hold. Consider a proper subset $\mathcal{J}^{\prime}\subset\mathbb{N}(J)$ of $J^{\prime}=\left\lvert\mathcal{J}^{\prime}\right\rvert$ product indices, and any solution $\mathbf{p}_{\mathcal{J}^{\prime}}^{*}=\\{p_{j}^{*}:j\in\mathcal{J}^{\prime})$ to the sub-variational inequality $\sum_{j\in\mathcal{J}^{\prime}}(D_{j}\hat{\pi}_{f(j)})(\mathbf{p}_{\mathcal{J}^{\prime}}^{*})(p_{j}^{*}-q_{j})\geq 0\quad\text{for all}\quad\mathbf{q}_{\mathcal{J}^{\prime}}=\\{q_{j}:j\in\mathcal{J}^{\prime}\\}\subset[0,\varsigma_{*}]^{J^{\prime}}.$ If we define $\mathbf{p}\in[0,\varsigma_{*}]^{J}$ by $p_{j}=p_{j}^{*}$ for all $j\in\mathcal{J}^{\prime}$ and $p_{k}=\varsigma_{*}$ for all $k\notin\mathcal{J}^{\prime}$ then $\mathbf{p}$ solves the full variational inequality (32). ###### Proof. Because $(D_{j}\hat{\pi}_{f(j)})(\mathbf{p})=\left\\{\begin{aligned} &(D_{j}\hat{\pi}_{f(j)})(\mathbf{p}_{\mathcal{J}^{\prime}}^{*})&&\quad\text{if }j\in\mathcal{J}^{\prime}\\\ &\quad\quad 0&&\quad\text{if }j\notin\mathcal{J}^{\prime}\\\ \end{aligned}\right.$ we have $\sum_{j=1}^{J}(D_{j}\hat{\pi}_{f(j)})(\mathbf{p})(p_{j}-q_{j})=\sum_{j\in\mathcal{J}^{\prime}}(D_{j}\hat{\pi}_{f(j)})(\mathbf{p}_{\mathcal{J}^{\prime}}^{*})(p_{j}^{*}-q_{j})\geq 0$ for all $\mathbf{q}\in[0,\varsigma_{*}]^{J}$. ∎ #### 5.1.3. The Resolution of Equilibria with $\boldsymbol{\zeta}$ We have shown that variational formulations of the equilibrium problem nest equilibria of all sub-problems, which may not be equilibria of the original problem as Example 8. In this section we show that methods based on the $\boldsymbol{\zeta}$ map need not have this unfortunate shortcoming. This result strongly distinguishes nonlinear system methods based on the $\boldsymbol{\zeta}$ map from variational approaches. We motivate this result with an example. ###### Example 9. Consider a finite-sample approximation to the Berry et al. (1995) model of Example 2. That is, choose $S\in\mathbb{N}$ and draw $\\{\boldsymbol{\theta}_{s}\\}_{s=1}^{S}$ where $\boldsymbol{\theta}_{s}=(\phi_{s},\boldsymbol{\beta}_{s},\beta_{0,s})$. These samples could be drawn via standard sampling from $\mu$ or from another technique like importance or quasi-random sampling. In any case, suppose that the $\phi$’s drawn are distinct with probability one: $\phi_{s}\neq\phi_{r}$ for all $s,r\in\mathbb{N}(S)$ with probability one. Without loss of generality we take $\phi_{1}<\phi_{2}<\dotsb<\phi_{S}$, and note that $\varsigma_{*}=\phi_{S}<\infty$. If $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ and $p_{k}>\varsigma_{*}$, then firm $f(k)$’s profits increase with the price of the $k^{\text{th}}$ product in some neighborhood of $\varsigma_{*}$. Thus if we compute some fixed-point $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ with $p_{k}>\varsigma_{*}$, we know that excluding product $k$ is profit-optimal for firm $f(k)$. As shown in Example 8, this is not the case with the VI formulation. ###### Proof. We will first define $\boldsymbol{\zeta}$ on all of $\mathcal{P}^{J}$, and then consider fixed-points $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ with $p_{k}\geq\phi_{S}=\varsigma_{*}$. To extend $\boldsymbol{\zeta}$, we define $\zeta_{k}(p_{1},\dotsc,p_{k},\dotsc,p_{J})=\zeta_{k}(p_{1},\dotsc,\varsigma_{*},\dotsc,p_{J})=\lim_{q\to\varsigma_{*}}\zeta_{k}(p_{1},\dotsc,q,\dotsc,p_{J}).$ when $p_{k}\geq\varsigma_{*}$. Note that for all $k$ and all $\mathbf{p}\in(0,\varsigma_{*})^{J}$ we can write $\zeta_{k}(\mathbf{p})=\sum_{s:\phi_{s}>p_{k}}\left(\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{s},\mathbf{p})(p_{j}-c_{j})+\frac{\phi_{s}-p_{k}}{\alpha}\right)\left(\frac{P_{k}^{L}(\boldsymbol{\theta}_{s},\mathbf{p})/(\phi_{s}-p_{k})}{\sum_{r:\phi_{r}>p_{k}}P_{k}^{L}(\boldsymbol{\theta}_{r},\mathbf{p})/(\phi_{r}-p_{k})}\right)$ We first define $\lim_{p_{k}\uparrow\phi_{S}}\zeta_{k}(\mathbf{p})$, we first note that for all $p_{k}\in(\phi_{S-1},\phi_{S})\neq\\{\emptyset\\}$, we have $\zeta_{k}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})+\frac{\phi_{S}-p_{k}}{\alpha}$ since $p_{k}>\phi_{s}$ for all $s\in\\{1,\dotsc,S-1\\}$. Thus $\lim_{p_{k}\uparrow\phi_{S}}\zeta_{k}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}\setminus\\{k\\}}\left[\lim_{p_{k}\uparrow\phi_{S}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})\right](p_{j}-c_{j}).$ In other words, as $p_{k}$ approaches $\phi_{S}=\varsigma_{*}$, $\zeta_{k}$ approaches the profits firm $f(k)$ accrues from selling all products other than $p_{k}$ to the sampled individual with the highest income. This establishes that the extended $\boldsymbol{\zeta}$ is well-defined and continuous. Now suppose $p_{k}=c_{k}+\zeta_{k}(\mathbf{p})$, where $p_{k}>\phi_{S}=\varsigma_{*}$. Thus $0=p_{k}-c_{k}-\zeta_{k}(\mathbf{p})>\phi_{S}-c_{k}-\zeta_{k}(\mathbf{p})=\lim_{q_{k}\uparrow\phi_{S}}\Big{(}q_{k}-c_{k}-\zeta_{k}(p_{1},\dotsc,q_{k},\dotsc,p_{J})\Big{)},$ and there must exist some $\delta>0$ such that $q_{k}-c_{k}-\zeta_{k}(p_{1},\dotsc,q_{k},\dotsc,p_{J})<0$ for all $q_{k}\in(\varsigma_{*}-\delta,\varsigma_{*})$. Hence $(D_{k}\hat{\pi}_{f(k)})(p_{1},\dotsc,q_{k},\dotsc,p_{J})>0$ $(D_{k}\hat{\pi}_{f(k)})(p_{1},\dotsc,q_{k},\dotsc,p_{J})=\lambda_{k}(p_{1},\dotsc,q_{k},\dotsc,p_{J})\big{(}q_{k}-c_{k}-\zeta_{k}(p_{1},\dotsc,q_{k},\dotsc,p_{J})\big{)}>0.$ In other words, if $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ and $p_{k}>\varsigma_{*}$, then firm $f(k)$’s profits increase with the price of the $k^{\text{th}}$ product in some neighborhood of $\varsigma_{*}$. ∎ Fortunately this example is fairly general. In the following proposition we prove that all finite-sample simulators generate $\boldsymbol{\zeta}$ maps that do not have equilibria of sub-problems as fixed points unless they are, in fact, equilibria of the original problem. Three assumptions are added: utilities must be twice continuously differentiable in prices, $\varsigma(\boldsymbol{\theta})$ is finite $\mu$-a.e. as in the Berry et al. (1995) model, and the sampled values $\varsigma(\boldsymbol{\theta}_{s})$ must be distinct with probability one. ###### Proposition 5.3. Consider a Mixed Logit model satisfying Assumptions 2.1, 2.3, and 3.1 with $w_{j}(\boldsymbol{\theta},\cdot):(0,\varsigma(\boldsymbol{\theta}))\to\mathbb{R}$ twice continuously differentiable in price and $\varsigma:\mathcal{T}\to\mathcal{P}$ finite $\mu$-a.e.. Generate a finite-sample simulator to this Mixed Logit model with $\\{\boldsymbol{\theta}_{s}\\}_{s=1}^{S}$ for some $S\in\mathbb{N}$. Let $\varsigma_{s}=\varsigma(\boldsymbol{\theta}_{s})$, and assume that $\varsigma_{s}\neq\varsigma_{r}$ with probability one for any $s\neq r$. Subsequently, order the samples so that $\varsigma_{1}<\dotsb<\varsigma_{S}=\varsigma_{*}$. Suppose that $\mathbf{p}\in\mathcal{P}^{J}$ satisfies $\mathbf{p}=\mathbf{c}+\boldsymbol{\zeta}(\mathbf{p})$ where $\boldsymbol{\zeta}$ is the extended map as in Example 9. If $p_{k}\geq\varsigma_{S}$, then excluding product $k$ is profit-optimal for firm $f=f(k)$; particularly, there exists $\delta>0$ such that $(D_{k}\hat{\pi}_{f(k)})(p_{1},\dotsc,p_{k},\dotsc,p_{J})>0$ for all $p_{k}\in(\varsigma_{S}-\delta,\varsigma_{S})$. ###### Proof. The case $p_{k}>\varsigma_{S}$ is handled exactly as in Example 9. We must only consider the case where $0=\varsigma_{S}-c_{k}-\lim_{p_{k}\uparrow\varsigma_{S}}\zeta_{k}(\mathbf{p})=\lim_{p_{k}\uparrow\varsigma_{S}}\Big{[}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{]}.$ Our approach is to show that $D_{k}[p_{k}-c_{k}-\zeta_{k}(\mathbf{p})]>0$ for all $p_{k}$ near enough to $\varsigma_{S}$, and thus $\displaystyle p_{k}-c_{k}-\zeta_{k}(\mathbf{p})$ $\displaystyle=p_{k}-c_{k}-\zeta_{k}(\mathbf{p})-\Big{[}\varsigma_{S}-c_{k}-\lim_{p_{k}\uparrow\varsigma_{S}}\zeta_{k}(\mathbf{p})\Big{]}$ $\displaystyle\quad\quad\quad\quad=-\int_{p_{k}}^{\varsigma_{S}}D_{k}[p_{k}-c_{k}-\zeta_{k}(\mathbf{p})]dp_{k}<0$ (with a slight abuse of notation in the integral). More specifically, we prove that $\lim_{p_{k}\uparrow\varsigma_{S}}D_{k}[p_{k}-c_{k}-\zeta_{k}(\mathbf{p})]>0$, which implies that $D_{k}[p_{k}-c_{k}-\zeta_{k}(\mathbf{p})]>0$ for all $p_{k}$ near enough to $\varsigma_{S}$. Because $p_{k}-c_{k}-\zeta_{k}(\mathbf{p})<0$ for $p_{k}$ near enough to $\varsigma_{S}$, $(D_{k}\hat{\pi}_{f(k)})(p_{1},\dotsc,q_{k},\dotsc,p_{J})=\lambda_{k}(\mathbf{p})\Big{(}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{)}>0.$ As in Example 9, note that for all $p_{k}\in(\varsigma_{S-1},\varsigma_{S})$ we have $\zeta_{k}(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})-\frac{1}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})}$ since $p_{k}>\varsigma_{s}$ for all $s\in\\{1,\dotsc,S-1\\}$. From this equation we derive $(D_{k}\zeta_{k})(\mathbf{p})=\sum_{j\in\mathcal{J}_{f(k)}}(D_{k}P_{j}^{L})(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})+P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})+\frac{(D^{2}w_{k})(\boldsymbol{\theta}_{S},p_{k})}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})^{2}}$ and thus $\displaystyle D_{k}\Big{[}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{]}$ $\displaystyle\quad\quad=1-(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})-P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})-\frac{(D^{2}w_{k})(\boldsymbol{\theta}_{S},p_{k})}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})^{2}}$ Now $\lim_{p_{k}\uparrow\varsigma_{S}}P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})=0$, we have assumed that $\lim_{p_{k}\uparrow\varsigma_{S}}\big{[}(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})P_{k}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})\big{]}=0$ (Assumption 3.1), and $\lim_{p_{k}\uparrow\varsigma_{S}}\left[\sum_{j\in\mathcal{J}_{f(k)}}P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})(p_{j}-c_{j})\right]=\sum_{j\in\mathcal{J}_{f(k)}\setminus\\{k\\}}\lim_{p_{k}\uparrow\varsigma_{S}}\left[P_{j}^{L}(\boldsymbol{\theta}_{S},\mathbf{p})\right](p_{j}-c_{j})<\infty,$ we have $\displaystyle\lim_{p_{k}\uparrow\varsigma_{S}}D_{k}\Big{[}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{]}=1-\lim_{p_{k}\uparrow\varsigma_{S}}\left[\frac{(D^{2}w_{k})(\boldsymbol{\theta}_{S},p_{k})}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})^{2}}\right]$ So long as $\displaystyle\lim_{p_{k}\uparrow\varsigma_{S}}\left[\frac{(D^{2}w_{k})(\boldsymbol{\theta}_{S},p_{k})}{(Dw_{k})(\boldsymbol{\theta}_{S},p_{k})^{2}}\right]<1$ we have $\lim_{p_{k}\uparrow\varsigma_{S}}D_{k}\Big{[}p_{k}-c_{k}-\zeta_{k}(\mathbf{p})\Big{]}>0$. This must be true, as Claim 1 below demonstrates. This completes the proof. ∎ ###### Claim 1. Let $w:(0,\varsigma)\to\mathbb{R}$ be twice continuously differentiable, with $(Dw)(p)<0$ for all $p\in(0,\varsigma)$ and $(Dw)(p)\downarrow-\infty$ as $p\uparrow\varsigma$. Then $\lim_{p\uparrow\varsigma}\left[\frac{(D^{2}w)(p)}{(Dw)(p)^{2}}\right]<1.$ ###### Proof. Proof We prove this by contradiction. Note that $D\left[\frac{1}{\left\lvert(Dw)(p)\right\rvert}\right]=\frac{(D^{2}w)(p)}{(Dw)(p)^{2}}.$ Now if $\displaystyle\lim_{p\uparrow\varsigma_{S}}\left[\frac{(D^{2}w)(p)}{(Dw)(p)^{2}}\right]\geq 1,$ there must exist some $\bar{p}\in(0,\varsigma)$ such that $\displaystyle\frac{(D^{2}w)(p)}{(Dw)(p)^{2}}>0\quad\text{for all}\quad p\in[\bar{p},\varsigma).$ But then $0\leq\int_{p}^{\varsigma}\frac{(D^{2}w)(q)}{(Dw)(q)^{2}}dq=\int_{p}^{\varsigma}D\left[\frac{1}{\left\lvert(Dw)(q)\right\rvert}\right]dq=\lim_{q\uparrow\varsigma}\left[\frac{1}{\left\lvert(Dw)(q)\right\rvert}\right]-\frac{1}{\left\lvert(Dw)(p)\right\rvert}=-\frac{1}{\left\lvert(Dw)(p)\right\rvert}<0,$ a contradiction. ∎ ### 5.2. Tatonnement Some authors iterate best responses $-$ i.e. tatonnement $-$ to compute equilibria. See, for example, Choi et al. (1990); CBO (2003); Michalek et al. (2004); Austin and Dinan (2005); Bento et al. (2005); Hu and Ralph (2007). For this process Newton’s method, or another algorithm of (unconstrained) optimization, will be required. Tatonnement should be an efficient way to compute “equilibrium” if all firm’s profit-maximizing prices are independent of their competitor’s decisions, but wasteful if some firm’s optimal pricing depends heavily on their competitors’ prices. Furthermore no convergence guarantees exist for tatonnement while there are at least theoretical guarantees that Newton’s method, properly constructed, will converge to simultaneously stationary prices. ### 5.3. Least-Squares Minimization and the Gauss-Newton Method In principle one could also use optimization methods to explicitly minimize $f(\mathbf{p})=\lvert\lvert\mathbf{F}(\mathbf{p})\rvert\rvert_{2}^{2}/2$ for any of our choices of $\mathbf{F}$. In fact, line search and trust-region strategies for global convergence implicitly minimize this function (Dennis and Schnabel, 1996). Computations of equilibrium prices benefit from leaving this implicit, as explicit minimization via Newton’s method requires third- order derivatives of $\mathbf{F}$, increasing both differentiability requirements and computational burden. The Gauss-Newton method (Ortega and Rheinboldt, 1970) is obtained by neglecting the influence of the third-order derivatives of $\mathbf{F}$. This defines the Gauss-Newton step as a solution to the (symmetric) normal equation $(D\mathbf{F})(\mathbf{p})^{\top}(D\mathbf{F})(\mathbf{p})\mathbf{s}=-(D\mathbf{F})(\mathbf{p})^{\top}\mathbf{F}(\mathbf{p})$; note that the same problem arises should one wish to use the Conjugate Gradient method to solve the Newton system. So long as $(D\mathbf{F})(\mathbf{p})$ is nonsingular the standard Newton steps will be recovered from the Gauss-Newton method. However they are explicitly formulated as solutions to linear systems that are more poorly conditioned (Golub and Loan, 1996; Trefethen and Bau, 1997) and thus we should at least expect to accumulate more error in the process of solving for the same steps. The burden of computing these steps also increases because of the requirement to multiply by the transpose of the Jacobian of $\mathbf{F}$. ## 6\. Acknowledgements This research was supported by the National Science Foundation, the University of Michigan Transportation Research Institute’s Doctoral Studies Program, and a research fellowship at the Belfer Center for Science and International Affairs at the Harvard Kennedy School. Both authors wish to thank Walter McManus, Brock Palen, Divakar Viswanath, Erin MacDonald, the editors, and the three anonymous reviewers for their contributions to this research. 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arxiv-papers
2010-12-28T20:53:11
2024-09-04T02:49:16.026736
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "W. Ross Morrow and Steven J. Skerlos", "submitter": "William Morrow", "url": "https://arxiv.org/abs/1012.5836" }
1012.5894
2010 Vol. XX No. XX, 000–000 11institutetext: Department of Astronomy, Peking University, Beijing 100871, China; fanzuhui@pku.edu.cn Received [year] [month] [day]; accepted [year] [month] [day] # Comparison of Halo Detection from Noisy Weak Lensing Convergence Maps with Gaussian Smoothing and MRLens Treatment Y.-X. Jiao H.-Y. Shan and Z.-H. Fan ###### Abstract Taking into account the noise from intrinsic ellipticities of source galaxies, we study the efficiency and completeness of halo detections from weak lensing convergence maps. Particularly, with numerical simulations, we compare the Gaussian filter with the so called MRLens treatment based on the modification of the Maximum Entropy Method. For a pure noise field without lensing signals, a Gaussian smoothing results a residual noise field that is approximately Gaussian in statistics if a large enough number of galaxies are included in the smoothing window. On the other hand, the noise field after the MRLens treatment is significantly non-Gaussian, resulting complications in characterizing the noise effects. Considering weak-lensing cluster detections, although the MRLens treatment effectively deletes false peaks arising from noise, it removes the real peaks heavily due to its inability to distinguish real signals with relatively low amplitudes from noise in its restoration process. The higher the noise level is, the larger the removal effects are for the real peaks. For a survey with a source density $n_{g}\sim 30\hbox{ arcmin}^{-2}$, the number of peaks found in an area of $3\times 3\hbox{ deg}^{2}$ after MRLens filtering is only $\sim 50$ for the detection threshold $\kappa=0.02$, while the number of halos with $M>5\times 10^{13}\hbox{ M}_{\odot}$ and with redshift $z\leq 2$ in the same area is expected to be $\sim 530$. For the Gaussian smoothing treatment, the number of detections is $\sim 260$, much larger than that of the MRLens. The Gaussianity of the noise statistics in the Gaussian smoothing case adds further advantages for this method to circumvent the problem of the relatively low efficiency in weak- lensing cluster detections. Therefore, in studies aiming to construct large cluster samples from weak-lensing surveys, the Gaussian smoothing method performs significantly better than the MRLens. ###### keywords: cosmology: theory - gravitation - dark matter - gravitational lensing ## 1 Introduction The weak gravitational lensing effect provides a unique tool in measuring the matter distribution in the universe (e.g., Bartelmann & Schneider 2001; Hoekstra et al. 2006; Massey et al. 2007). Its additional dependence on the distances to the source, to the lens and between the source and lens makes it an excellent probe in cosmological studies of dark energy (e.g., Albrecht et al. 2006; Benjamin et al. 2007; Kilbinger et al. 2009; Li et al. 2009). On the other hand, however, different observational and physical effects can affect the weak lensing analyses significantly. Being extracted from shape distortion of background galaxies, the weak lensing effect on individual source galaxies is severely contaminated by their intrinsic ellipticities. Therefore statistical analyses on a large number of galaxies are necessary in weak lensing studies. Even so, intrinsic shape alignments of galaxies, including intrinsic-intrinsic and shear-intrinsic correlations, can be an important source of error in cosmic shear correlation analyses. For cluster detections from weak lensing convergence maps reconstructed from shear measurements (e.g., Kaiser & Squires 1993; Bartelmann 1995; Kaiser 1995; Schneider & Seitz 1995; Squires & Kaiser 1996; Bridle et al.1998; Marshall et al. 2002), even randomly orientated intrinsic ellipticities can result false peaks by their chance alignments, which can reduce the efficiency of cluster detections significantly (e.g., Schneider 1996; van Waerbeke 2000; White et al. 2002; Hamana et al. 2004; Fan 2007). Thus further treatments for a convergence map are normally required to suppress the noise effects. The noise from intrinsic ellipticities of source galaxies is essentially shot noise, and thus by averaging over a relatively large number of source galaxies in weak lensing analyses, the residual noise can be effectively reduced. This leads to the normal smoothing treatment. It is clear that the residual noise depends on the form of the window function and the smoothing scale. For a Gaussian smoothing with a window function of the form $W(\theta)\propto\exp(-\theta^{2}/\theta_{G}^{2})$, the residual noise can be estimated by $\sigma^{2}_{0}\approx{(\sigma^{2}_{\epsilon}/2)}{[1/(2\pi\theta_{G}^{2}n_{g})]}$, where $\sigma_{\epsilon}$ is the rms of the intrinsic ellipticity of individual source galaxies, $\theta_{G}$ is the smoothing scale, and $n_{g}$ is the surface number density of source galaxies. For $\sigma_{\epsilon}=0.3$, $n_{g}=30\hbox{ arcmin}^{-2}$ and $\theta_{G}=1\hbox{ arcmin}$, we have $\sigma_{0}\approx 0.015$. Recently, Starck et al. (2006) proposed the MRLens filtering technique, which is based on the Bayesian analyses with a multi-scale entropy prior applied. The False Detection Rate (FDR) method is used to select significant/non- significant wavelet coefficients (e.g., Starck et al. 2006; Pires et al. 2009). The MRLens method suppresses noise adaptively according to the strength of the noise itself. A more detailed description of the method is given in §4. In this paper, with numerical simulations, we compare Gaussian smoothing with MRLens treatment, paying particular attention to the completeness and the efficiency of weak lensing halo detections from convergence maps. The rest of the paper is organized as follows. In §2, we describe briefly the weak-lensing convergence reconstruction and the Gaussian smoothing. In §3, we present the important aspects of the MRLens treatment. Results are shown in §4. Section 5 contains summaries and discussions. ## 2 Weak lensing convergence reconstruction In the weak lensing regime, the convergence $\kappa(\vec{\theta})$ is essentially related to the weighted projection of density fluctuations $\delta$ along the unperturbed light path. Specifically, we have $\kappa(\vec{\theta})={3H_{0}^{2}\Omega_{0}\over 2}\int_{0}^{w_{H}}dw\bar{W}(w)f_{K}(w){\delta[f_{K}(w)\vec{\theta},w]\over a(w)}$ (1) where $H_{0}$ is the present Hubble constant, $\Omega_{0}$ is the present matter density of the universe in unit of the critical density, ${w}$ is the radial coordinate, $a(w)$ is the scale factor of the universe, and, with $K$ being the spatial curvature of the universe, $\displaystyle f_{K}(w)$ $\displaystyle=|K|^{-1/2}\sin(|K|^{1/2}w)\quad\quad\ \ \ (K>0)$ (2) $\displaystyle=w\qquad\qquad\qquad\qquad\qquad\quad(K=0)$ $\displaystyle=|K|^{-1/2}\sinh(|K|^{1/2}w)\qquad(K<0)\ .$ The factor $\bar{W}(w)$ is the weighting function that is related to the source galaxy distribution $G(w)$ by $\bar{W}(w)=\int_{w}^{w_{H}}dw^{\prime}G(w^{\prime}){f_{K}(w^{\prime}-w)\over f_{K}(w^{\prime})}\ .$ (3) The lensing potential $\phi$ is related to $\kappa$ by $\kappa={\nabla^{2}\phi\over 2}\ ,$ (4) and the shears $\gamma_{1}$ and $\gamma_{2}$ are $\gamma_{1}={\partial_{11}\phi-\partial_{22}\phi\over 2}\ ,\quad\gamma_{2}=\partial_{12}\phi.$ (5) Since both $\kappa$ and $\gamma_{i}$ are determined by the lensing potential, they are mutually dependent of each other. In the Fourier space, we have (Kaiser & Squires 1993) $\kappa(\vec{k})=c_{1}(k)\gamma_{1}(\vec{k})+c_{2}(k)\gamma_{2}(\vec{k}),$ (6) where $[c_{1},c_{2}]=[\cos(2\phi),\sin(2\phi)]$ with $\vec{k}=k(\cos\phi,\sin\phi)$. Observationally, the shear $\gamma$ can be extracted from the shape measurement of source galaxy images. Under the condition $\kappa<<1$, we have $\vec{e}^{\rm obs}\approx\vec{\gamma}+\vec{e}^{S}\ ,$ (7) where $\vec{e}^{\rm obs}$ and $\vec{e}^{S}$ are the observed ellipticity, and the intrinsic ellipticity of a source galaxy, respectively. Reconstructed from $\vec{e}^{\rm obs}$, the convergence $\kappa_{n}(\vec{k})$ then contains noise from the intrinsic part, i.e., $\kappa_{n}({\vec{k}})=c_{\alpha}(k)e^{\rm obs}_{\alpha}({\vec{k}})=\kappa({\vec{k}})+c_{\alpha}(k)e^{S}_{\alpha}({\vec{k}}).$ (8) With the transformation back to the real 2-D space and applying a smoothing with the window function $W(\vec{\theta})$, we can obtain the smoothed quantities (e.g., van Waerbeke 2000) $\Sigma^{\rm obs}(\vec{\theta})=\Gamma(\vec{\theta})+\frac{1}{n_{g}}\Sigma^{N_{g}}_{i=1}W(\vec{\theta}-\vec{\theta}_{i})e^{\rm S}(\vec{\theta}_{i})$ (9) and $K_{N}(\vec{\theta})=\int d\vec{k}e^{-i\vec{k}\cdot\vec{\theta}}c_{\alpha}(k)\Sigma_{\alpha}^{\rm obs}(\vec{k}),$ (10) where $\Sigma^{\rm obs}$, $\Gamma$, and $K_{N}$ are the smoothed $e^{\rm obs}$, $\gamma$ and $\kappa_{n}$, respectively, and $n_{g}$ and $N_{g}$ are the surface number density and the total number of source galaxies in the field. The noise part of $K_{N}$ due to the intrinsic ellipticities is then $N(\vec{\theta})=\frac{1}{n_{g}}\Sigma^{N_{g}}_{i=1}\int d\vec{k}W(\vec{k})e^{-i\vec{k}\cdot(\vec{\theta}-\vec{\theta}_{i})}c_{\alpha}(k)e^{S}_{\alpha}(\vec{\theta}_{i}),$ (11) where $W(\vec{k})$ is the Fourier transformation of the window function with the form $W(\vec{k})=\frac{1}{(2\pi)^{2}}\int d\vec{\theta}e^{i\vec{k}\cdot\vec{\theta}}W(\vec{\theta}).$ (12) Without considering the intrinsic alignment of $e^{S}$, it is expected from the central limit theorem that the smoothed noise field $N(\vec{\theta})$ is approximately Gaussian in statistics if the effective number of galaxies included in the smoothing window is larger than about $10$ (e.g., van Waerbeke 2000). In this case, smoothing leads to correlations in $N(\vec{\theta})$, and its two-point correlation function is approximately $<N(\vec{\theta})N(\vec{\theta}^{\prime})>=\frac{\sigma_{\epsilon^{2}}}{2n_{g}}(2\pi)^{2}\int d\vec{k}e^{i\vec{k}\cdot(\vec{\theta}^{\prime}-\vec{\theta})}|W(\vec{k})|^{2},$ (13) where $\sigma_{\epsilon}$ is the intrinsic dispersion of $e^{\rm obs}$. The approximate Gaussianity of $N(\vec{\theta})$ allows us to quantify the noise effects straightforwardly. The noise effects on cluster mass reconstruction and the noise peak statistics are analyzed in van Waerbeke (2000). Even with weak alignments of intrinsic ellipticities, $N(\vec{\theta})$ can still be approximately described by a Gaussian random field with a modified two-point correlation function including the effects of intrinsic alignments. The enhancement of the noise peak abundance due to the weakly intrinsic alignments are analyzed in Fan (2007). In Fan et al. (2010), the effects of the presence of real dark matter halos on the noise peak statistics around them as well as the effects of the noise on the peak height of real halos are investigated in detail. They further present a model to calculate the total peak abundance in a large-scale convergence map, including the peaks corresponding to real halos and the noise peaks from the chance alignment of the intrinsic ellipticities of source galaxies. Such a model makes it possible for us to use directly the peaks from convergence maps as cosmological probes without the need to differentiate real and false peaks. Due to its simple operational procedure and the Gaussian statistics of the residual noise field, the smoothing treatment has been widely applied in weak lensing analyses. Different smoothing functions have been used in different studies. In this paper, we consider the Gaussian smoothing function $W_{G}$, which is one of the most commonly adopted window functions. Specifically, we have $W_{G}=\frac{1}{\pi\theta_{G}^{2}}\exp\left(-\frac{\theta^{2}}{\theta_{G}^{2}}\right)\ ,$ (14) where $\theta_{G}$ is the smoothing scale. Then from Eq. (13), the rms of the noise $\sigma_{0}$ after smoothing is given by $\sigma_{0}^{2}={\sigma_{\epsilon}^{2}\over 2}{1\over 2\pi\theta_{G}^{2}n_{g}}\ .$ (15) In our analyses, we choose $\sigma_{\epsilon}=0.3$, the typical value for lensing source galaxies, and $\theta_{G}=1\hbox{ arcmin}$, which is the optimal smoothing scale considering cluster-sized halos. Then for a lensing survey with $n_{g}=30\hbox{ arcmin}^{-2}$, $\sigma_{0}\approx 0.015$, which is about $20$ times lower than $\sigma_{\epsilon}$. ## 3 MRLens method Starck et al. (2006) introduce a new reconstruction and filtering method, namely, Multi-scale Entropy Restoration (MRLens). It is developed from the Maximum Entropy Method. The basic idea is to use only ‘signals’ selected by the so called False Discovery Rate (FDR) (Benjamini & Hochberg 1995) to reconstruct the convergence field through a Multi-scale Entropy prior. In the following, we present specific steps of MRLens. ### 3.1 Wavelet decomposition For an original convergence map $\kappa_{obs}$ with $N=n\times n$ pixels, the first step of MRLens is to decompose the image map into different components representing fine structures of different scales. To do this, we first initialize $j=0$ and set $C_{0}(k,l)=\kappa_{obs}(k,l)$, i.e., $j=0$ corresponds to the unprocessed map with detailed structures. Then we progressively go to higher $j$ to obtain smoother maps through (Starck et al. 2001) $C_{j+1}(k,l)=\sum_{m}\sum_{n}h_{1D}(m)h_{1D}(n)C_{j}(k+2^{j}m,l+2^{j}n),$ (16) where $h_{1D}(m)={[1/16,4/16,6/16,4/16,1/16]}$ for $m=-2,-1,0,1,2$, respectively. Defining $w_{j+1}(k,l)=C_{j}(k,l)-C_{j+1}(k,l),$ (17) we finally obtain $\kappa_{obs}(k,l)=C_{J}(k,l)+\sum_{j=1}^{J}w_{j}(k.l),$ (18) where $J$ is a chosen number determined by specific considerations on how smooth we want to go. Here we set $J=7$. In our following analyses, each map is $3\times 3\hbox{ deg}^{2}$ discretized into $1024\times 1024$ pixels. Thus $2^{J}=128$ pixels corresponding to $\sim 22\hbox{ arcmin}$. Because we do not expect to see significant structures resulting purely from noise on such a large scale, $J=7$ is an appropriate choice. It can be seen from Eq. (18) that $C_{J}(k,l)$ is the most smoothed version of the original map $\kappa_{obs}$, and the terms in the summation contain ever smaller-scale information with smaller $j$. ### 3.2 Multiscale Entropy With the multi-scale wavelet decomposition, one can then construct an entropy with the obtained wavelet coefficients $w_{j}(k,l)$ at each grid $(k,l)$ with $j=1,2,...,J$. It can generally be written as $H(\kappa)=\sum_{k,l}h[C_{J}(k,l)]+\sum_{j=1}^{J}\sum_{k,l}h[w_{j}(k,l)].$ (19) For $h$, there are different definitions (e.g., Starck et al. 2006). Here we follow Starck et al. (2001) to choose the entropy of NOISE-MSE $h_{n}$ in our considerations. At each scale $j$, the noise entropy at each grid $(k,l)$ is derived by weighting the entropy with a probability that $w_{j}(k,l)$ is contributed by noise. Specifically, we have $h_{n}[w_{j}(k,l)]=\int_{0}^{\mid w_{j}(k,l)\mid}P_{n}[\mid w_{j}(k,l)\mid-u]\frac{\partial h(x)}{\partial x}|_{x=u}du,$ (20) where $P_{n}[w_{j}(k,l)]$ is the probability that the coefficient $w_{j}(k,l)$ can be due to noise, and is given by $P_{n}[w_{j}(k,l)]=\mathrm{Prob}[W>\mid w_{j}(k,l)\mid].$ (21) Eq. (20) essentially regards the information contained in $w_{j}(k,l)$ to be built up from the summation of $dh(u)$. For each newly added $dh(u)$, depending on the difference $|w_{j}(k,l)|-u$, there is a probability that it is due to noise. For Gaussian noise with rms $\sigma_{j}$ at scale $j$, we have $\displaystyle P_{n}[w_{j}(k,l)]$ $\displaystyle=$ $\displaystyle\frac{2}{\sqrt{2\pi}\sigma_{j}}\int_{\mid w_{j}(k,l)\mid}^{+\infty}\exp(-W^{2}/2\sigma^{2}_{j})dW$ (22) $\displaystyle=$ $\displaystyle\mbox{erfc}\bigg{(}\frac{\mid w_{j,k,l}\mid}{\sqrt{2}\sigma_{j}}\bigg{)}$ and thus $h_{n}[w_{j}(k,l)]=\frac{1}{\sigma_{j}^{2}}\int_{0}^{\mid w_{j}(k,l)\mid}u\mbox{ erfc}\bigg{(}\frac{\mid w_{j}(k,l)\mid-u}{\sqrt{2}\sigma_{j}}\bigg{)}du.$ (23) ### 3.3 Selecting significant wavelet coefficients using the False Discovery Rate (FDR) The Multiscale Entropy method applies regularizations on wavelet coefficients to minimize noise contributions while keeping the signal information. Thus for those coefficients which are clearly signals, they should be kept unchanged. Then a new Multiscale Entropy is defined as (e.g., Starck et al. 2006) $\tilde{h}_{n}[w_{j}(k,l)]=\bar{M}_{j}(k,l)h_{n}[w_{j}(k,l)],$ (24) where $\bar{M}_{j}(k,l)=1-M_{j}(k,l),$ (25) and $M$ is the multi-resolution support defined as (Starck et al. 1995) $M_{j}(k,l)=\\{\begin{array}[]{ll}1&\textrm{if $w_{j}(k,l)$ is significant}\\\ 0&\textrm{if $w_{j}(k,l)$ is not significant}\end{array}.$ (26) Therefore $\tilde{h}_{n}$ means that we only need to regularize those wavelet coefficients which are ′not significant′, that is, they are likely due to noise. For judging the significance of a wavelet coefficient, a commonly used criterion is a ‘$k\sigma$’ threshold. If a coefficient is above the threshold, it is defined to be ′significant′. This is equivalent to set a threshold for the ratio of ′significant′ detections over the total number of pixels being analyzed. Considering a Gaussian noise, a $2\sigma$ criterion corresponds to a probability of $0.05$ for a noise coefficient being mis-classified as ′significant′. If we have totally $N$ pixels to consider, the number of false discoveries is then on average $0.05N$. If the number of pixels related to real signals in the analyses is comparable to $0.05N$, the false discovery rate with respect to the number of real signals can be much higher than $0.05$. Increasing $k$ can lower the number of false detections at the expense, however, of the power of real detections. To overcome such difficulties, an alternative thresholding technique, the False Discovery Rate (FDR), has been proposed (Benjamini & Hochberg 1995; Miller et al. 2001; Hopkins et al. 2002; Starck et al. 2006). This method can effectively control, in an adaptive manner, the fraction of false discoveries over the total number of discoveries, rather than over the total number of pixels analyzed. Let $P_{1},\ldots,P_{N}$ denote the p-values ordered from low to high for the N pixels, where p-value is defined as $p_{value}=\frac{1}{\sqrt{2\pi}\sigma_{j}}\int_{w_{j}(k,l)}^{\infty}\exp[-(w-\bar{w}_{j})^{2}/2\sigma^{2}_{j}]dw,$ (27) with $\bar{w}_{j}$ the average of $w_{j}(k,l)$ for the scale $j$ over all the pixels. Define $d_{j}=\max\left\\{k_{j}:\ P_{k_{j}}<\frac{k_{j}\alpha_{j}}{c_{N}N}\right\\},$ (28) then all the $w_{j}(k,l)$ with their values larger than $d_{j}$ are classified as ′significant′. Here $c_{N}=1$ if all the pixels are statistically independent. The meaning of $\alpha_{j}$ is approximately the pre-defined false discovery rate at scale $j$ with respect to the total number of detections. The larger the $\alpha_{j}$ value, the larger the fraction of $w_{j}(k,l)$ defined to be ′significant′. In our analyses, we adopt FDR to find the values of $M$ in Eq. (26). In the MRLens program, $\alpha_{0}$ is an adjustable parameter, and $\alpha_{j}=\alpha_{0}\times 2^{j}$ (Starck et al. 2006). ### 3.4 Multi-scale Entropy Filtering algorithm Given the discussions in the previous subsections, the Multi-scale Entropy restoration method reduces to find the reconstructed $\kappa_{f}$ that minimizes $I(\kappa_{f})$ defined as $I({\kappa_{f}})=\frac{\parallel{\kappa_{obs}-\kappa_{f}\parallel^{2}}}{2\sigma_{n}^{2}}+\beta\sum_{j=1}^{J}\sum_{k,l}\tilde{h}_{n}[({\cal W}{\kappa_{f}})_{j}(k,l)],$ (29) where $\sigma_{n}$ is the rms of noise in the original convergence map $\kappa_{obs}$, $J$ is the number of wavelet scales, $\cal W$ is the wavelet transform operator and $\tilde{h}_{n}[({\cal W}{\kappa_{f}})_{j}(k,l)]$ is the multi-scale entropy defined only for non-significant coefficients selected by the FDR method. The $\beta$ parameter is calculated under the restriction that the residual should have a standard deviation equal to the rms of noise. The best $\kappa_{f}$ is then obtained by iterative calculations. Full details of the minimization algorithm can be found in Starck et al. (2001). It can be seen that the two terms in the right of Eq. (29) are balancing each other. While the first term tends to keep the information in $\kappa_{f}$ the most, the second term has the effect to lower the noise as much as possible. ## 4 Results In this section, we present the results of our analyses. For weak-lensing effects from large-scale structures in the universe, we use the publicly available ray-tracing weak-lensing maps provided kindly by White & Vale (2004). The specific set of lensing maps we analyze are generated from large- scale N-body simulations with cosmological parameters $\Omega_{M}=0.296$, $\Omega_{\Lambda}=0.704$, $w=-1.0$, $h=0.7$, and $\sigma_{8}=0.93$. The box size is $300\hbox{ Mpc}~{}h^{-1}$, the number of particles is $512^{3}$ with $m\approx 1.7\times 10^{10}M_{\odot}~{}h^{-1}$ for each, and the softening length is $\approx 20\hbox{ kpc}~{}h^{-1}$. There are totally $16$ convergence maps and each has a size of $3\times 3\hbox{ deg}^{2}$ pixelized into $1024\times 1024$ pixels. The redshift distribution of source galaxies follows $p(z)\propto z^{2}\exp[-(z/z_{0})^{3/2}]$ with $z_{0}=2/3$. For each map, we add in Gaussian noise due to the intrinsic ellipticities of source galaxies with the variance given by (e.g., Hamana et al. 2004), $\sigma_{\rm pix}^{2}=\frac{\sigma_{\epsilon}^{2}}{2}\frac{1}{n_{g}\theta_{\rm pix}^{2}},$ (30) where $\theta_{\rm pix}$ is the pixel size of the simulated convergence-$\kappa$ map, and $\sigma_{\epsilon}$ is the rms of the intrinsic ellipticites taken to be $\sigma_{\epsilon}=0.3$. The surface number density $n_{g}$ depends on specific observations. Here we consider $n_{g}=30\hbox{ arcmin}^{-2}$ which is typical for ground-based observations, and $n_{g}=100\hbox{ arcmin}^{-2}$ expected from space observations, respectively. Figure 1 presents one set of convergence maps without (left) and with (right) noise. It can be seen very clearly that the noise from intrinsic ellipticities of source galaxies dominates the map, and certain post-processing procedures are necessary in order to extract weak-lensing signals embedded under noise. Here we compare two such methods, namely, the normal smoothing method with a Gaussian smoothing function, and the MRLens treatment, paying particular attention to their effects on weak-lensing peak statistics. Figure 1: The convergence maps of $3\times 3\hbox{ deg}^{2}$. The left panel is the noise-free map, and the right panel is the noisy convergence map with $n_{g}=30\hbox{ arcmin}^{-2}$. ### 4.1 Statistical properties of residual noise Post-processing procedures can reduce noise effectively. However, certain levels of residual noise inevitably remain. It is thus important to understand the statistical properties of the residual noise so that we can quantify their effects on weak-lensing cosmological studies properly. For that, we first in this subsection consider pure noise maps without including weak-lensing signals. After applying Gaussian smoothing and MRLens, respectively, we compare the residual noise-peak statistics in the two cases. This is highly relevant to cosmological applications of weak-lensing cluster statistics, in which, high peaks in convergence maps are thought to be related to clusters of galaxies and their abundances contain important cosmological information. The existence of residual noise can generate false peaks in convergence maps, which in turn can contaminate the weak-lensing peak statistics significantly. With Eq. (30), we generate a $3\times 3\hbox{ deg}^{2}$ noise map containing $1024\times 1024$ pixels with $\theta_{pix}=0.176\hbox{ arcmin}$ and the corresponding $\sigma_{pix}=0.22$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and $\sigma_{pix}=0.12$ for $n_{g}=100\hbox{ arcmin}^{-2}$. For Gaussian smoothing, we take $\theta_{G}=1\hbox{ arcmin}$. For MRLens, we take $\alpha_{0}=0.01$. In a smoothed map, a positive (maximum)/negative (minimum) peak position is located if its value is above/below those of its eight neighboring pixels (e.g., Jain & Van Waerbeke 2000; Miyazaki et al. 2002). Figure 2 shows the probability distribution function (PDF) of peaks in the residual noise field for the two cases, respectively, with the left for the Gaussian smoothing and the right for the MRLens. In each panel, the solid and dashed lines correspond to the results with $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The bin size is $\Delta\kappa=0.005$. Both the positive and the negative peaks are counted in. Two distinctly different distributions are seen. For the Gaussian smoothing case, the peak number distribution has a double-peak behavior at $\kappa/\sigma_{0}\sim\pm 1$, in good agreement with that expected for a Gaussian random field (Bond & Efstathiou 1987; Van Waerbeke 2000). The rms of the residual noise in this case is $\sigma_{0}\approx 0.016$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and $\sigma_{0}\approx 0.009$ for $n_{g}=100\hbox{ arcmin}^{-2}$, in excellent agreement with the theoretical value $0.015$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and $\sigma_{0}\approx 0.008$ for $n_{g}=100\hbox{ arcmin}^{-2}$ calculated from Eq. (15). Considering positive peaks that are relevant for weak-lensing analyses, the noise peaks with $\kappa/\sigma_{0}\sim 1$, rather than with $\kappa/\sigma_{0}=0$, have the highest occurrence probability. Such a property of noise can cause statistically a positive shift for the peak height of a cluster measured from noisy convergence maps. The shift depends on the density profile of the cluster. This noise-induced shift can bias the cluster mass estimation from weak-lensing observations. On the other hand, it can increase the weak-lensing detectability of clusters, and thus affect the corresponding cosmological studies significantly (Fan et al. 2010). For the MRLens case, the residual noise after restoration treatment is low with $\sigma_{0}\approx 0.0029$ for $n_{g}=30\hbox{ arcmin}^{-2}$ and $\sigma_{0}\approx 0.0016$ for $n_{g}=100\hbox{ arcmin}^{-2}$, much less than those of the Gaussian smoothing. However, the noise statistics is highly non- Gaussian, which results significant complications in quantifying the noise effect on weak-lensing signals. The number distribution of noise peaks is narrowly concentrated around $\kappa=0$. Thus unlike the Gaussian smoothing, it seems that we do not expect a systematic shift due to noise in weak-lensing cluster peak measurement. It should be noted, however, in MRLens, the noise filtering involves restoration procedures based on NOISE-MSE of Eq. (29). The results depend on the noise properties [the second term in Eq. (29)] as well as on the properties of signals we would like to detect [the first term in Eq. (29)]. The higher the original noise is, the larger the fraction of the wavelet coefficients that are suppressed. In such a treatment, the signals are changed depending on the original noise level and their own properties. Therefore considering the convergence peak for a cluster, the results after MRLens restoration in the cases with and without noise are different. In this sense, the existence of noise also induces a systematic bias for the peak value of a cluster, though for a reason different from and much more complicated than that of the Gaussian smoothing case. The quantitative modeling of such a bias for MRLens needs to be further explored. Figure 2: The probability distribution functions of peaks in pure noise maps. The solid and dashed lines are for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The left panel is for the Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$, and the right panel is for the MRLens result with $\alpha_{0}=0.01$. For MRLens, the $\alpha_{0}$ parameter in FDR affects the classification of significant and non-significant wavelet coefficients. A smaller $\alpha_{0}$ results a more stringent criteria for the definition of a significant wavelet coefficient, and thus stronger suppressions of noise. To test the $\alpha_{0}$-dependence, we vary its value to obtain different restoration results for pure noise maps. In Table 1, the rms of the residual noise for different $\alpha_{0}$ and different $n_{g}$ are shown. With the increase of $n_{g}$, the original noise level decreases with $(n_{g})^{-1/2}$. It is noted that after MRLens treatment, the rms of the residual noise also approximately follows $\sigma_{0}\propto(n_{g})^{-1/2}$. For the $\alpha_{0}$-dependence, as expected, the residual noise decreases with the decrease of $\alpha_{0}$. However, this dependence is rather weak. Changing $\alpha_{0}$ from $0.1$ to $0.01$ only decreases $\sigma_{0}$ by $\sim 20\%$. Table 1: Standard deviation of the reconstruction error with MRLens $\alpha_{0}$ | $\sigma_{0}$($n_{g}=15$) | $\sigma_{0}$($n_{g}=30$) | $\sigma_{0}$($n_{g}=50$) | $\sigma_{0}$($n_{g}=100$) ---|---|---|---|--- 0.001 | 0.0038 | 0.0026 | 0.0021 | 0.0015 0.01 | 0.0041 | 0.0029 | 0.0023 | 0.0016 0.02 | 0.0041 | 0.0030 | 0.0023 | 0.0016 0.04 | 0.0044 | 0.0032 | 0.0024 | 0.0017 0.06 | 0.0045 | 0.0033 | 0.0026 | 0.0019 0.08 | 0.0047 | 0.0033 | 0.0027 | 0.0020 0.1 | 0.0050 | 0.0035 | 0.0027 | 0.0021 0.2 | 0.0051 | 0.0036 | 0.0029 | 0.0023 ### 4.2 Peak statistics in noisy convergence maps Now we consider peak statistics of noisy convergence maps. Figure 3 shows the post-processed maps of the right panel of Figure 1 with Gaussian smoothing for $\theta_{G}=1\hbox{ arcmin}$ (upper) and with MRLens for $\alpha_{0}=0.01$ (lower), respectively. The left panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and the right panels are for $n_{g}=100\hbox{ arcmin}^{-2}$. Figure 3: Noisy convergence maps of $3\times 3\hbox{ deg}^{2}$. The upper and lower panels are for the Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$ and MRLens with $\alpha_{0}=0.01$, respectively. The left and right panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. Comparing to the maps in Figure 1, we see that the post-processing procedures can indeed filter out much of the noise so that the real structures in the large-scale mass distribution can be detected. For $n_{g}=30\hbox{ arcmin}^{-2}$, the MRLens map looks very smooth with only very massive structures left. On the other hand, in the Gaussian smoothing case, small structures can also be seen. However, it contains many more noise peaks than that of the MRLens case. For $n_{g}=100\hbox{ arcmin}^{-2}$, the map is smoother for the Gaussian smoothing case than that with $n_{g}=30\hbox{ arcmin}^{-2}$. The MRLens map, however, appears lumpier for the lower noise case. Such opposite trends seen in the Gaussian smoothing and in MRLens reflect clearly the different underlying filtering mechanisms between the two smoothing schemes. For the Gaussian smoothing, the filtering is mainly performed through an averaging procedure. Given a smoothing scale, the peak signals of real clusters are more or less similar regardless the noise level. Meanwhile, the noise peaks with relatively high $\kappa$ values are significantly reduced if the noise level is lowered. Thus the smoother appearance of the upper right panel is mainly due to the less number of high noise peaks than that of the upper left panel. For MRLens, it involves a restoration procedure that depends on the original noise level. The smaller the original noise is, the lower the fraction is for the wavelet coefficients to be suppressed. It is important to note that the suppression leads to the removal of both noise peaks and true peaks of relatively low amplitudes. Thus the lumpier structures seen in the lower right panel is largely attributed to the lower level of removal of real structures than that of the lower left panel. In Figure 4 and Figure 5, we show the probability distribution function of peaks for Gaussian smoothing and for MRLens, respectively. The results for each case are obtained by averaging over $16$ simulated maps with noise added. Figure 4: The probability distribution function of convergence peaks for the case of Gaussian smoothing with $\theta_{G}=1\hbox{ arcmin}$. The black solid line is for the result of the noise-free convergence peaks, the red dashed and red solid lines are for the pure noise peaks and noisy convergence peaks with $n_{g}=30\hbox{ arcmin}^{-2}$, respectively, and the blue dashed and blue solid lines are for the pure noise peaks and noisy convergence peaks with $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The left upper panel includes both maximum and minimum peaks, and the left lower panel shows the distribution function for maximum peaks only. The right panel is the zoom-in version of the left lower panel. Figure 5: Same as Figure 4 but for the case of MRLens with $\alpha_{0}=0.01$. For the Gaussian smoothing results in Figure 4, the black, red dashed, red solid, blue dashed, and blue solid lines are for the results of noise free peaks, pure noise peaks with $n_{g}=30\hbox{ arcmin}^{-2}$, noisy convergence peaks with $n_{g}=30\hbox{ arcmin}^{-2}$, pure noise peaks with $n_{g}=100\hbox{ arcmin}^{-2}$, noisy convergence peaks with $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. We can see that in the Gaussian smoothing cases, the noise peaks dominate over the real peaks at $\kappa<3\sigma_{0}$. At larger $\kappa>3\sigma_{0}$, real peaks can be detected with high efficiencies. Comparing the blue solid line with the red solid line, we see that by reducing the noise level from $\sigma_{0}\sim 0.015$ to $\sigma_{0}\sim 0.008$, we effectively reduce the number of noise peaks with $\kappa>0.025$, and thus increase the real peak detection efficiencies significantly. In Figure 5 for the MRLens results, the line styles are the same as those in Figure 4. Different from that in the Gaussian smoothing cases, here the noise peaks (red and blue dashed lines) contribute little to the total number of peaks with $\kappa>0.02$ in comparison with the real peaks (black solid line). However, the suppression process in the MRLens treatment mistakenly removes a large number of real peaks with $\kappa<0.1$. Thus we expect a high efficiency but a low completeness in weak-lensing peak detections after MRLens filtering. Reducing the original noise level by increasing $n_{g}$ form $30\hbox{ arcmin}^{-2}$ to $100\hbox{ arcmin}^{-2}$ leads to a less suppression effect. Therefore more peaks with $\kappa<0.1$ are kept and the completeness of peak detections increases considerably. In the next subsection, we investigate and compare explicitly the efficiency and completeness of weak-lensing cluster detections in the two smoothing treatments. ### 4.3 Efficiency and completeness of weak-lensing cluster detection The existence of noise from intrinsic ellipticities of source galaxies results false peaks in convergence maps, and thus lowers considerably the efficiency of weak-lensing cluster detections. Increasing the detection threshold can increase the efficiency, however at the expense of completeness. In this section, we compare the weak-lensing cluster detection with Gaussian smoothing and with the MRLens, respectively. Following Hamana et al. (2004), we define the efficiency $f_{e}$ and completeness $f_{c}$ of cluster detection with respect to the number of clusters (dark matter halos) above a certain mass threshold. Specifically, we have $f_{e}=\frac{N_{iii}}{N_{i}},$ (31) $f_{c}=\frac{N_{iii}}{N_{ii}},$ (32) where $N_{i}$ denotes the number of convergence peaks with their heights above a detection threshold, $N_{ii}$ represents the number of dark matter halo with mass above a certain mass threshold, and $N_{iii}$ is the number of peaks that have correspondences with dark matter halos among $N_{ii}$. A peak is defined to be associated with its nearest dark matter halo if the location of the peak is within a radius of $12$ pixels (corresponds to $2.11\hbox{ arcmin}$) around the halo. If there are two or more peaks associated with a same halo, the highest peak is defined to have the correspondence with the halo. Figure 6: The efficiency $f_{e}$ and completeness $f_{c}$ as functions of the peak detection threshold $\kappa$. The left and right panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The upper two panels are for the Gaussian smoothing and the lower two panels are for MRLens. The lines with symbols are for the completeness, and the lines without symbols are for the efficiency. The red, green and blue lines are for the results of halos with $M>5\times 10^{13}M_{\odot}$, $M>1\times 10^{14}M_{\odot}$, and $M>2\times 10^{14}M_{\odot}$, respectively. Figure 6 shows the results of $f_{e}$ and $f_{c}$ for Gaussian smoothing (upper panels) and MRLens (lower panels). The left panels are for $n_{g}=30\hbox{ arcmin}^{-2}$, and the right panels are for $n_{g}=100\hbox{ arcmin}^{-2}$. In each panel, the red, green and blue lines are for halos with mass $M>5\times 10^{13}M_{\odot}$, $M>1\times 10^{14}M_{\odot}$, and $M>2\times 10^{14}M_{\odot}$, respectively. The lines with and without symbols are, respectively, for the results of completeness and efficiency. The horizontal axis in each panel is the peak detection threshold $\kappa$. We first analyze the Gaussian smoothing cases. As we discuss previously, such a smoothing process reserves more or less all the real peaks with scales above the smoothing scale. At mean time, the number of noise peaks is large at $\kappa<3\sigma_{0}$. Thus a high completeness and a low efficiency are expected when the peak detection threshold is low. For $n_{g}=30\hbox{ arcmin}^{-2}$ (upper left), we have $\sigma_{0}\sim 0.015$. At the detection threshold $\kappa=0.02\sim 1.3\sigma_{0}$, we have the completeness $f_{c}\sim 50\%,65\%$ and $70\%$ for $M>5\times 10^{13}\hbox{ M}_{\odot}$, $1\times 10^{14}\hbox{ M}_{\odot}$ and $2\times 10^{14}\hbox{ M}_{\odot}$, respectively. The corresponding efficiencies are $30\%,10\%$ and $2\%$. When the detection threshold $\kappa>3\sigma_{0}$, the number of noise peaks drops significantly, leading to a large increase in the detection efficiency. On the other hand, a considerable fraction of halos are missed due to the high detection threshold, resulting a decrease in the completeness. Specifically, at $\kappa=0.045\sim 3\sigma_{0}$, the completeness $f_{c}\sim 20\%,35\%$ and $60\%$, and the efficiency $f_{e}\sim 50\%,25\%$ and $10\%$, for $M>5\times 10^{13}\hbox{ M}_{\odot}$, $1\times 10^{14}\hbox{ M}_{\odot}$ and $2\times 10^{14}\hbox{ M}_{\odot}$, respectively. With the increase of $n_{g}$ to $n_{g}=100\hbox{ arcmin}^{-2}$ (upper right), the noise level $\sigma_{0}$ decreases by a factor of $\sqrt{100/30}$ to $\sigma_{0}\sim 0.008$. Thus $3\sigma_{0}$ corresponds to $\kappa\sim 0.025$. At this detection threshold, the number of noise peaks is smaller and correspondingly the efficiency is higher than those with $n_{g}=30\hbox{ arcmin}^{-2}$. On the other hand, the number of real peaks does not change much as the noise level decreases, and thus the completeness is similar to that of $n_{g}=30\hbox{ arcmin}^{-2}$. Quantitatively, at the threshold $\kappa=0.025$, the efficiency $f_{e}\sim 50\%,20\%$ and $8\%$, in comparison with $f_{e}\sim 35\%,12\%$ and $3\%$ in the case of $n_{g}=30\hbox{ arcmin}^{-2}$, for $M>5\times 10^{13}\hbox{ M}_{\odot}$, $1\times 10^{14}\hbox{ M}_{\odot}$ and $2\times 10^{14}\hbox{ M}_{\odot}$, respectively. For the completeness, we have $f_{c}\sim 30\%,50\%$ and $80\%$ for $n_{g}=100\hbox{ arcmin}^{-2}$. For $n_{g}=30\hbox{ arcmin}^{-2}$, $f_{c}\sim 45\%,60\%$ and $70\%$. While being similar, $f_{c}$ decreases somewhat for $M>5\times 10^{13}\hbox{ M}_{\odot}$ and $1\times 10^{14}\hbox{ M}_{\odot}$ with the decrease of noise level. This is in accordance with the analyses of Fan et al. (2010) where they find that the existence of noise generates a systematic shift for the real peaks toward higher amplitudes. The shift depends on the density profile of dark matter halos associated with the real peaks, and can be as high as $\sim 1\sigma_{0}$ for NFW halos with low concentrations. In terms of $\kappa$ values, the shift is larger for larger $\sigma_{0}$. Thus, in the case of $n_{g}=30\hbox{ arcmin}^{-2}$, the relatively large $\sigma_{0}$ leads to a large shift of the real peak heights and consequently a larger number of real peaks above the detection threshold than that in the case of $n_{g}=100\hbox{ arcmin}^{-2}$. For MRLens, with $n_{g}=30\hbox{ arcmin}^{-2}$ (lower left), the completeness of the weak-lensing cluster detection is very low, and $f_{c}\sim 10\%,20\%$ and $40\%$ at the threshold $\kappa=0.02$, in comparison with $f_{c}\sim 50\%,65\%$ and $70\%$ in the corresponding Gaussian smoothing case. This is because the suppression of the wavelet coefficients aiming to reduce noise removes a large fraction of real peaks in the range of $\kappa<0.1$ as seen from Figure 5. The total number of peaks in $3\times 3\hbox{ deg}^{2}$ with $\kappa\geq 0.02$ is only $\sim 53$, while the total number of halos in the area with $M>5\times 10^{13}\hbox{ M}_{\odot}$ is $\sim 530$. Thus although the efficiency in MRLens here is rather high ($\sim 80\%$ for $M>5\times 10^{13}\hbox{ M}_{\odot}$), the very few number of detected halos makes the MRLens method be disadvantageous in comparing with that of simple Gaussian smoothing method. For $n_{g}=100\hbox{ arcmin}^{-2}$, the noise level is lower and thus the removal effect is less significant than the case of $n_{g}=30\hbox{ arcmin}^{-2}$. Consequently, the completeness increases considerably with $f_{c}\sim 20\%,40\%$ and $65\%$. Meanwhile, the efficiency decreases somewhat. In this low noise case, the differences between the MRLens and Gaussian smoothing in terms of the completeness and efficiency are less than those of high noise case. But still, the completeness is lower for MRLens, especially considering relatively low mass halos with $M>5\times 10^{13}\hbox{ M}_{\odot}$. To further demonstrate the differences between the Gaussian smoothing and the MRLens, in Figure 7, we show the peak-halo correspondences explicitly in $z-M$ plane for one of our $3\times 3\hbox{ deg}^{2}$ simulation maps, where $M$ is the halo mass in unit of $10^{13}M_{\odot}$ and $z$ is the halo redshift from simulations. The halos are the ones located in the solid angle of $3\times 3\hbox{ deg}^{2}$ in the considered direction and their redshift and mass are taken directly from the halo catalogs constructed by White and Vale (2004). The upper and lower panels are for the Gaussian smoothing and the MRLens, respectively. The left and right panels correspond to $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. In each panel, the ‘+’ symbols denote the dark matter halos identified in simulations with $M\geq 5\times 10^{13}M_{\odot}$ and in the redshift range of $0\leq z\leq 2$. There are very few halos extending to redshift beyond $z=2$. The green squares represent those halos that have corresponding convergence peaks with $\kappa\geq 0.02$. The differences between the two filtering methods are strikingly seen. For MRLens with $n_{g}=30\hbox{ arcmin}^{-2}$ (lower left), a majority of halos with $M<10^{14}\hbox{ M}_{\odot}$ or with $z>0.8$ are missed in weak-lensing detections, consistent with its extremely low completeness shown in Figure 6. Lowering the noise level by increasing $n_{g}$ to $n_{g}=100\hbox{ arcmin}^{-2}$ increases the number of halos with associated peaks by nearly a factor of $2$ (lower right). But the number is still much less than that in the Gaussian smoothing case. Therefore in studies aiming to detect a large number of clusters from blind surveys and subsequent cosmological applications, the Gaussian smoothing method is clearly much better than the MRLens. In addition, the noise field after a Gaussian smoothing with $\theta_{G}\sim 1\hbox{ arcmin}$ is approximately Gaussian in statistics, and thus its effects on weak-lensing cluster detection can be modeled much easier than the case of MRLens where the left-over noise is statistically highly non-Gaussian (Fan et al. 2010). Figure 7: Peak-halo correspondences in $z-M$ plane. The upper and lower panels are for the Gaussian smoothing and MRLens, respectively. The left and right panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. In each panel, the ‘+’ symbols show all the halos with $M\geq 5\times 10^{13}M_{\odot}$ and the squares show the halos with corresponding convergence peaks with $\kappa\geq 0.02$. In MRLens, the $\alpha_{0}$ parameter plays a crucial role in classifying significant and non-significant wavelet coefficients. A larger $\alpha_{0}$ leads to a larger fraction of significant coefficients, and thus a less suppression effect in MRLens restoration. To see if the problem of low completeness in MRLens cluster detection can be largely improved by increasing $\alpha_{0}$, we analyze the $\alpha_{0}$ dependence for the completeness $f_{c}$ as well as for the efficiency $f_{e}$. The results are shown in Figure 8. The upper and lower panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$, respectively. The red, green and blue lines are for $M\geq 5\times 10^{13}M_{\odot}$, $1\times 10^{14}M_{\odot}$, and $2\times 10^{14}M_{\odot}$, respectively. The peak detection threshold is set to be $\kappa=0.02$. We can see that both $f_{c}$ and $f_{e}$ are not very sensitive to $\alpha_{0}$. Increasing $\alpha_{0}$ from $0.01$ to $0.1$ improves the completeness only by $\Delta f_{c}\sim 10\%$ for both $n_{g}=30\hbox{ arcmin}^{-2}$, and $n_{g}=100\hbox{ arcmin}^{-2}$. Meanwhile, the efficiency decreases by $\Delta f_{e}\sim 10\%-20\%$. Therefore increasing $\alpha_{0}$ cannot overcome the shortcoming of MRLens considerably. Comparing to the Gaussian smoothing cases, the completeness is still low for MRLens weak-lensing cluster detection even with $\alpha_{0}=0.1$. We then conclude that in probing cosmologies with weak-lensing cluster abundance analyses, in which a large sample of clusters is needed, the Gaussian smoothing method performs much better than the MRLens method. To overcome the relatively low efficiency for low peaks in the Gaussian smoothing treatment, a detection threshold $\kappa>3\sigma_{0}$ is normally set. In Fan et al. (2010), the noise effects on convergence peak statistics can be accurately modeled for the Gaussian smoothing method. Therefore it is potentially possible to even include peaks with $\kappa<3\sigma_{0}$ in the abundance analyses, which can increase the number of detected clusters greatly so that to strengthen the derived cosmological constraints on different parameters. This will be explored further in our future studies. Figure 8: The $\alpha_{0}$ dependence of the completeness $f_{c}$ and the efficiency $f_{e}$. The upper and lower panels are for $n_{g}=30\hbox{ arcmin}^{-2}$ and $n_{g}=100\hbox{ arcmin}^{-2}$ respectively. The peak detection threshold is set to be $\kappa=0.02$. The red, green and blue lines are for the results of halos with $M>5\times 10^{13}M_{\odot}$, $M>1\times 10^{14}M_{\odot}$, and $M>2\times 10^{14}M_{\odot}$, respectively. The lines with and without symbols are for the completeness $f_{c}$ and the efficiency $f_{e}$, respectively. ## 5 Summary Constructing cluster samples through their weak-lensing effects has been an important aspect of weak-lensing studies. Their statistical abundance contains valuable cosmological information. Observations have shown the feasibility in detecting clusters with weak-lensing effects (e.g., Wittman et al. 2006; Dietrich et al. 2007; Gavazzi & Soucail 2007; Schirmer et al. 2007; Hamana et al. 2009). In conjunction with optical observations, the detailed analyses on the completeness and efficiency of weak-lensing selected cluster samples also become possible (e.g., Geller et al. 2010). It is noted, however, the efficiency and completeness depend on the method applied to reconstruct the convergence field from shear measurements. Different methods can result residual noise with different statistical properties, and can also change the weak-lensing signals differently. In order to extract cosmological information from observations, it is therefore crucial to understand how a particular reconstruction method affects the results in detail. In this paper, we systematically compare the Gaussian smoothing method and the MRLens treatment to suppress noise from intrinsic ellipticities in convergence maps. We concentrate on convergence peak statistics. It is found that while the MRLens method can remove noise very effectively, it mistakenly removes a large fraction of real peaks associated with clusters of galaxies. For $n_{g}=30\hbox{ arcmin}^{-2}$, the number of peaks with $\kappa\geq 0.02$ after MRLens filtering is only $\sim 50$ in an area of $3\times 3\hbox{ deg}^{2}$ in comparison with $\sim 530$ for the number of halos of $M>5\times 10^{13}\hbox{ M}_{\odot}$. On the other hand, for the Gaussian smoothing treatment, the number of detected clusters is $\sim 260$. Even with the detection threshold $\kappa=3\sigma_{0}\sim 0.045$, which is normally set in the Gaussian smoothing treatment to reduce the number of noise peaks in the peak catalog and thus to increase the cluster detection efficiency, the number of detected clusters is $\sim 100$, twice as many as that in the MRLens filtering with the threshold $\kappa=0.02$. As the accuracy of statistical abundance analyses depends crucially on the number of detected clusters, the Gaussian smoothing method is therefore strongly favored to detect clusters as many as possible. Furthermore, the Gaussian smoothing leads to a noise field which is approximately Gaussian in statistics, while the residual noise from MRLens filtering is highly non-Gaussian. Therefore the noise effects can be modeled more straightforwardly for the Gaussian smoothing case than that of MRLens (e.g., van Waerbeke 2000; Fan 2007; Fan et al. 2010). The recent studies of Fan et al. (2010) on the weak-lensing peak statistics with noise included provide an analytical model for the efficiency of peak detections in the Gaussian smoothing case. Thus it is possible for us to include peaks with $\kappa<3\sigma_{0}$ in the analyses. Then the number of detected clusters can increase considerably, which in turn can lead to a significant improvement in the cosmological constraints derived from weak-lensing cluster statistics. ###### Acknowledgements. This research is supported in part by the NSFC of China under grants 10373001, 10533010 and 10773001, and the 973 program No.2007CB815401. 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arxiv-papers
2010-12-29T09:23:43
2024-09-04T02:49:16.049113
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yangxiu Jiao, Huanyuan Shan, Zuhui Fan", "submitter": "HuanYuan Shan", "url": "https://arxiv.org/abs/1012.5894" }
1101.0019
Current Address: ]Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, 13560-970, Brazil Primary Address: ]Department of Physics, Umeå University, SE-901 87, Umeå, Sweden Current Address: ]School of Physics, Monash University, 3800, Australia # Superflow in a toroidal Bose-Einstein condensate: an atom circuit with a tunable weak link A. Ramanathan∗ Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD, 20899, USA K. C. Wright kcwright@nist.gov Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD, 20899, USA S. R. Muniz [ M. Zelan [ W. T. Hill III Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD, 20899, USA C. J. Lobb Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD, 20899, USA K. Helmerson [ W. D. Phillips Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD, 20899, USA G. K. Campbell Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD, 20899, USA ###### Abstract We have created a long-lived ($\approx$ 40 s) persistent current in a toroidal Bose-Einstein condensate held in an all-optical trap. A repulsive optical barrier across one side of the torus creates a tunable weak link in the condensate circuit, which can affect the current around the loop. Superflow stops abruptly at a barrier strength such that the local flow velocity at the barrier exceeds a critical velocity. The measured critical velocity is consistent with dissipation due to the creation of vortex-antivortex pairs. This system is the first realization of an elementary closed-loop atom circuit. ###### pacs: 03.75.Lm, 03.75.Kk, 67.85.De Quantum fluids can exhibit properties such as long-range coherence and superfluidity that make them useful for constructing sensors and other devices. For example, superconducting quantum interference devices (SQUIDs) are sensitive magnetic field detectors Clarke and Braginski (2004), and superfluid He circuits have been used to detect rotation Simmonds et al. (2001); Hoskinson et al. (2001). Ultracold atomic-gas analogs of electronic devices and circuits, or “atomtronics” have been proposed including diodes and transistors atomtronics . Of particular interest is the realization of an atomic-gas SQUID analog. SQUID circuits have been realized with either tunnel or weak link junctions Clarke and Braginski (2004); Likharev (1979); Davis and Packard (2002). In atomic Bose-Einstein condensates, Josephson junctions have been demonstrated only between adjacent wells Albiez et al. (2005); Levy et al. (2007). Here we present the first implementation of a non-trivial, closed- loop atom circuit, and show that it is possible to control the current at the single-quantum level by changing the strength of a weak link. This is an essential step toward realizing an atomic SQUID analog. Superfluids flow without dissipation if the flow velocity is below a threshold determined by the lowest energy excitations allowed for the system Landau (1941). In a homogeneous condensate the lowest energy excitations are phonons end , and the Landau critical velocity is the speed of sound Pitaevskii and Stringari (2003). Real systems are finite and therefore inhomogeneous; consequently the lowest energy excitations can be vortex-like Barenghi et al. (2001), and dissipation can occur at velocities well below the sound speed Feynman (1955). Dissipation involving vortex-like excitations has been previously observed in experiments with liquid He Avenel and Varoquaux (1985); Amar et al. (1992), superconductors Huebener (2001), and in a simply-connected condensate Neely et al. (2010). Figure 1: Experimental configuration. (a) Schematic of the toroidal optical dipole trap formed at the intersection of two red-detuned beams: a horizontal “sheet” beam, and a vertical Laguerre-Gaussian beam (LG${}_{0}^{1}$) with a ring-shaped intensity maximum. A pulsed pair of Raman beams (large downward arrows) co-propagating with the LG trapping beam creates circulation in the condensate. (b) Energy level diagram for the Raman transition: $|F\\!=\\!1,m_{F}\\!=\\!-1\rangle\\!\rightarrow\\!|F\\!=\\!1,m_{F}\\!=\\!0\rangle$. One Raman beam carries $\hbar$ orbital angular momentum per photon (LG${}_{0}^{1}$), the other carries none (Gaussian); the condensate is transferred to a quantized ($l$=1) circulating state. (c) False-color absorption image showing the normalized column density of a condensate in the trap, viewed from above. Arrows: Raman beam polarizations and magnetic bias. The critical velocity in simply-connected condensates has been measured previously by moving a defect created by a localized optical potential Onofrio et al. (2000); Engels and Atherton (2007); Neely et al. (2010). When the velocity of the defect was high enough, excitations and heating were observed. In contrast to this earlier work, we create a quantized, persistent flow around a multiply-connected (toroidal) condensate, and study the decay of that flow in the presence of a stationary barrier, as a function of barrier height and condensate atom number. In previous experiments Ryu et al. (2007), we created persistent currents in a harmonic magnetic potential pierced by a repulsive optical potential. Relative drift between these potentials limited the flow lifetime to $\approx$ 10 s. This motivated the construction of an all-optical trap which supports persistent currents for up to 40 s, and allows us to carefully study the stability of superflow. To create a toroidal condensate, $3^{2}S_{1/2}\,|F\\!=\\!1,m_{F}\\!=\\!-1\rangle\ $ 23Na atoms are cooled almost to degeneracy in a magnetic trap and then transferred into an optical dipole trap created by the intersection of red-detuned (1030 nm) “sheet” and “ring” beams (Fig. 1a). The horizontal sheet beam has a vertical (horizontal) $1/e^{2}$ half-width of $\approx$ 9 $\mu$m ($\approx$ 400 $\mu$m), and provides vertical confinement. The vertical ring beam is Laguerre-Gaussian (LG${}^{1}_{0}$), and confines the condensate to its $\approx$ 20 $\mu$m radial intensity peak, generating a toroidal potential minimum. With the atoms in the optical trap, the beam intensities are ramped down to force evaporative cooling. At the end of the ramp, the trap depth is $\approx$ 700 nK, with trap frequencies $\omega_{z}/2\pi=550$ Hz (vertical) and $\omega_{r}/2\pi=110$ Hz (radial). This produces a toroidal condensate of up to 3$\times$105 atoms with a chemical potential $\mu_{0}$ of up to $h\cdot 1200$ Hz, and temperature $<$ 10 nK (no discernible non-condensed fraction). The azimuthal variation of the potential minimum is less than $h\cdot 100$ Hz, as shown by the smooth condensate density profile in Fig. 1c. The condensate is initially nonrotating ywas . Superfluid circulation around any closed path must be quantized, such that the wave function has a $2\pi l$ phase winding ($l\\!\in\\!\mathbb{Z}$). We create circulation by transferring quantized angular momentum from optical fields during a Raman process Wright et al. (2008). The co-propagating Raman beams, detuned 2.3 GHz below the D2 transitions, are in two-photon resonance with the $|1,-1\rangle\\!\rightarrow\\!|1,\;0\rangle\ $ transition (Fig. 1b). They have orthogonal linear polarizations, parallel and perpendicular to the horizontal magnetic bias field (Fig. 1c). The nonlinear Zeeman shift from the 0.5 mT field applied during the interaction prevents coupling to $|1,1\rangle$. The angular momentum change of the condensate is determined by the spatial mode of the Raman beams. With one beam Gaussian, and the other in an LG${}^{1}_{0}$ spatial mode carrying $\hbar$ orbital angular momentum per photon, the Raman process coherently transfers the condensate to the $l$=1 circulating state Andersen et al. (2006); Wright et al. (2008); Ryu et al. (2007). With good mode matching and an optimized Raman $\pi$-pulse ($\approx$ 100 $\mu$s), we achieve a minimum transfer efficiency of 90%, with only a few percent atom loss due to spontaneous scattering. Residual atoms in $|1,-1\rangle$ after the Raman pulse are quickly removed from the trap by transferring them to $|2,-2\rangle$ with a microwave pulse, then ejecting them from the trap with resonant imaging light (see below). Circulation is detected by releasing the condensate from the trap and imaging the density distribution after several milliseconds time-of-flight (TOF) Madison et al. (2000). If the condensate is not rotating, the central hole closes after a short time. When a rotating condensate is released, the angular velocity of the flow prevents complete closure. The persistence of a central hole after sufficiently long TOF is the signature of circulation in the ring (see Fig 2(b) insets). The apparent size of the hole at a given time after release is related to the the azimuthal flow velocity prior to release and the velocity of the mean-field-driven inward expansion. For a rotating condensate released directly from our narrow annular trap, the hole size is below our imaging resolution for experimentally accessible TOFs ($<$ 15 ms). To make the signature of circulation visible earlier, we first adiabatically reduce the ring beam intensity by 90% over 100 ms, then release the condensate suddenly ($<$ 1 $\mu$s). We use this procedure, followed by 6 ms TOF, to detect circulation. Figure 2: (a) Flow survival as a function of chemical potential, $\mu_{0}$, for two barrier heights: $\beta/h$ = 650 Hz (upper, blue), and $\beta/h$ = 780 Hz (lower, red). Presence or absence of flow for a single condensate is shown by closed circles. Open circles are the average of data within the bins (vertical lines), representing the flow survival probability (P${}_{\textrm{flow}}$) of each bin. A critical chemical potential $\mu_{c}$ for stable flow is found from a sigmoidal fit (solid lines) to the data for each $\beta$. Inset: In situ absorption image of a condensate near $\mu_{c}$ ($\mu_{0}/h$ = 870 Hz, $\beta/h$ = 650 Hz). (b) Values of $\mu_{c}$ at different $\beta$, determined by fits as described in (a). The open circles correspond to the data in (a). The vertical error bars reflect the width of the sigmoidal fit, $\pm 2\mu_{w}$. The horizontal error bars are the 20 Hz uncertainty in calibrating $\beta$. The dotted line is a linear fit to the data, with slope 1.6(2). Insets: typical TOF absorption images showing the presence (top left) and absence (bottom right) of circulation. The Raman beams used to create circulation also cause small-amplitude oscillations in the radial density profile, due to small dipole forces and atom loss. These oscillations have no observable impact on the stability of the circulation, and damp out after $<0.5$ s. We add a wait time $\geq$ 3 s after the Raman transfer to ensure complete damping. The circulation is extremely robust, and continues until losses due to background collisions reduce $\mu_{0}$ to the level of the nonuniformities in the trap. For a 30 s vacuum-limited $1/e$ condensate lifetime, $\mu_{0}$ remains high enough for flow to survive up to 40 s. After the $\geq$ 3 s wait time, we insert a barrier into the path of the flow and study the stability of the circulation. The repulsive barrier is created with a blue-detuned (532 nm) laser beam focused to an elliptical spot. The major axis (15 $\mu$m $1/e^{2}$ radius) is aligned in the radial direction of the toroid, and the minor axis (4.3 $\mu$m $1/e^{2}$ radius) is parallel to the flow, and exceeds the bulk condensate healing length ($\xi=\hbar/\sqrt{2m\mu}$ $<$ 1 $\mu$m). The barrier depletes the local density of the condensate, $n$, as seen in the inset in Fig. 2(a). The reduction in density increases the local flow velocity (roughly $v\\!\propto\\!1/n$). Lowering the density also lowers the local interaction energy, $\mu_{l}\\!\propto\\!n$, decreasing the local sound speed. To study flow stability, we ramp up the barrier intensity over 100 ms to a chosen barrier height $\beta$, hold for 2 s, then ramp back down in another 100 ms. The presence or absence of flow is then detected in TOF as described above. This procedure is repeated many times for the same $\beta$, varying the total number of atoms (by varying the initial condensate number and/or the wait time) until the range of atom number is well-sampled. We then change $\beta$ and repeat the procedure. If the barrier is not applied, the flow always survives, so we can attribute the decay of the flow to the effect of the barrier. Separate measurements indicate that the flow decays in $<$ 100 ms. The analysis of flow stability depends on in situ observations of the condensate density profile in the presence of the barrier, and from TOF images after the barrier has been removed. From TOF images we determine whether the flow survived [insets Fig. 2(b)], and measure the condensate atom number, $N$. For an annular condensate with a Thomas-Fermi profile, the chemical potential $\mu_{0}=\hbar\bar{\omega}\sqrt{\pi/2\cdot(Na_{s}/R)}$ where $\bar{\omega}\equiv\sqrt{\omega_{z}\omega_{r}}$, $a_{s}$ is the s-wave scattering length, and $R$ is the radius of the ring. This calculation does not include small corrections ($\approx 6\%$) due to the azimuthal nonuniformity of the potential minimum and displacement of atoms from the barrier region, corrections which are less than the systematic uncertainty in determining $\mu_{0}$ ($\approx 10\%$). We calibrate $\beta$ by taking in situ images of the condensate and measuring the reduction in column density at the location of the barrier (see Fig. 2a inset). Due to the high optical depth (up to 10), we use a partial transfer imaging technique Freilich et al. (2010); Ramanathan et al. , in which a precise fraction (ranging from 15-40%) of the atoms is transferred to the $|2,-1\rangle$ state using a microwave pulse, then resonantly imaged on the $S_{1/2}\,F\\!=\\!2\rightarrow P_{3/2}\,F\\!=\\!3$ transition. The local interaction energy $\mu_{l}$ can be found from the measured column density $\tilde{n}$. For data where $\mu_{l}\\!<\\!\hbar\omega_{z}$, we assume the axial density profile is that of the harmonic oscillator ground state, with $\mu_{l}\\!=\\![8\pi\cdot(\hbar\omega_{z})(\hbar^{2}a_{s}^{2}\tilde{n}^{2}/m)]^{1/2}$. For data where $\mu_{l}>\hbar\omega_{z}$, we assume a Thomas-Fermi profile, with $\mu_{l}\\!=\\![9\pi^{2}/2\cdot(\hbar\omega_{z})^{2}(\hbar^{2}a_{s}^{2}\tilde{n}^{2}/m)]^{1/3}$. When measuring the column density at the barrier, we correct for loss of contrast due to the imaging resolution, which reduces the apparent depth of the density depletion by $\approx$ 15%. We take $\beta$ to be $\mu_{0}-\mu_{l}$. Figure 3: Critical flow velocity, $v_{c}$, above which the circulation becomes unstable, for each of the barrier heights in Fig 2(b). The value of $v_{c}$ is shown normalized to the effective sound speed at the barrier ($c_{\mathrm{eff}}$), and is plotted as a function of $c_{\mathrm{eff}}$. The horizontal error bars are the estimated uncertainty in $c_{\mathrm{eff}}$. The vertical error bars are the combined experimental uncertainty in $v_{c}$ and $c_{\mathrm{eff}}$. The measured ratio $v_{c}/c_{\mathrm{eff}}\approx 0.6$, and is independent of $c_{\mathrm{eff}}$ to within the experimental uncertainty. Vortex-like excitations are expected to occur in this system at and above a velocity $v_{F}$, where $v_{F}<c_{\mathrm{eff}}$ Feynman (1955). The gray band indicates estimated upper/lower bounds (see text) on $v_{F}/c_{\mathrm{eff}}$, using Feynman’s approximate expression for $v_{F}$. The solid circles in Fig. 2(a) show the flow survival or decay for single experimental runs, plotted against $\mu_{0}$ for that run. The open circles are the average of the solid circles within the bins shown. The upper plot (blue) is for $\beta/h$ = 650 Hz (upper); the lower (red) shows $\beta/h$ = 780 Hz. At low $\mu_{0}$, the flow is arrested by the barrier. At high $\mu_{0}$ the flow survival probability becomes unity. In between, the survival probability increases from zero to one over a narrow critical region. We characterize this critical region for each $\beta$ by fitting a sigmoidal function $P(\mu_{0})=1/(1+e^{(\mu_{c}-\mu_{0})/\mu_{w}})$ to each unbinned data set, where the parameters $\mu_{c}$ and $\mu_{w}$ are the critical chemical potential and the sigmoidal width respectively unc . The observed width is consistent with observed shot-to-shot variations in the trapping potential. Figure 2(b) shows the values of $\mu_{c}$ extracted from the fits of the data for seven different $\beta$. Over this range, $\mu_{c}$ increases approximately linearly with $\beta$, with a slope greater than unity. The functional dependence and slope are determined in a non-trivial way by trap geometry and the condition of quantized circulation around the ring. The experimental results are consistent with expectations for our geometry. The physics behind Fig. 2 is more apparent when the data is recast in terms of flow velocity and sound speed at the barrier. The barrier thickness is greater than $\xi$, so we expect the flow to become unstable when the velocity in the barrier region exceeds some local critical velocity $v_{c}$. The flow velocity at the barrier cannot be determined just from $\mu_{0}$ and $\beta$. The requirements of quantized circulation (global), and flow conservation (local), make it necessary to self-consistently calculate the velocity distribution around the entire ring. We do this by integrating the in situ column density radially to make a 1D approximation of the density profile, then solving for the velocity distribution of an $l=1$ circulation state. The critical velocity is determined by the lowest energy excitations allowed for the system Landau (1941). For phonon-like excitations in the ring, that velocity should be approximately the local sound speed in the barrier region Watanabe et al. (2009). We make an initial estimate for the critical velocity from the local interaction energy at the peak of the barrier, $c_{l}=\sqrt{\mu_{l}/m}$. However, the inhomogeneous (nearly parabolic) radial density profile lowers the effective sound speed to $c_{\mathrm{eff}}=c_{l}/\sqrt{2}$ for waves traveling azimuthally along the annulus Stringari (1998). Figure 3 shows the observed critical velocity normalized to $c_{\mathrm{eff}}$, as a function of $c_{\mathrm{eff}}$. As seen in previous work with finite inhomogeneous atomic condensates Onofrio et al. (2000); Engels and Atherton (2007); Neely et al. (2010), the observed critical velocity is less than the sound speed. For all tested values of $\beta$, $v_{c}/c_{\mathrm{eff}}\approx 0.6$ and is independent of $c_{\mathrm{eff}}$ to within the experimental uncertainty. In this experiment, flow is confined to a narrow, flattened channel, raising the possibility that vortex-like excitations are responsible for the observed critical velocity. Numerical simulations Piazza et al. (2009) with a model condensate similar to ours, but in an $l=8$ circulation state, showed vortices traversing the barrier region when the barrier was raised above a critical level. This suggests that for our $l=1$ circulation state, a similar decay mechanism could be at work. For vortex-like excitations in our quasi-2D geometry, the (Feynman) critical velocity $v_{F}$ can be estimated from energetic arguments Feynman (1955) to be $v_{F}=(\hbar/md)\ln(d/a)$, where $d$ is the channel width, and $a$ is the vortex core size. We take $d$ to be the Thomas-Fermi width, and $a$ the healing length, both calculated for the barrier region. Both $d$ and $a$ depend on $c_{\mathrm{eff}}$ via the interaction energy $\mu_{l}$. The grey band in Fig. 3 is an estimate of the probable value of $v_{F}/c_{\mathrm{eff}}$ with $c_{l}\geq c_{\mathrm{eff}}\geq c_{l}/\sqrt{2}$. While this calculation is in surprisingly good agreement with our data, a more complete model including geometric factors is needed to accurately calculate the energy of a vortex- antivortex pair in the barrier region. We have presented the first realization of a closed atomtronic circuit, demonstrating precise control both in inducing and arresting superfluid flow. We have clearly identified the critical velocity where flow stops, and our observations are in agreement with theoretical predictions in which vortex- antivortex excitations are the decay mechanism for the system. In future work, we plan to investigate the role of barrier geometry, condensate temperature, and dimensionality in determining the critical velocity and decay mode. In addition, rotating a barrier around the ring (oscillating it azimuthally) would be analogous to magnetically biasing (driving an AC current in) a SQUID. The present work constitutes a significant step toward realizing such an atomic SQUID analog. The authors thank L. Mathey for helpful discussions, and R. B. Blakestad for comments on the manuscript. This work was partially supported by ONR, the ARO atomtronics MURI, and the NSF PFC at JQI. ## References * Clarke and Braginski (2004) J. Clarke and A. I. Braginski, _The SQUID Handbook_ , vol. 1,2 (Wiley-VCH, Weinheim, 2004). * Simmonds et al. (2001) R. W. Simmonds et al., Nature 412, 55 (2001). * Hoskinson et al. (2001) E. Hoskinson, Y. Sato, R. Packard, Phys. Rev. B 74, 100509 (2006). * (4) B. T. Seaman et al., Phys. Rev. A 75, 023615 (2007). R. A. Pepino et al., Phys. Rev. Lett. 103, 140405 (2009). A. Ruschhaupt, and J. G. Muga, Phys. Rev. A 70, 061604 (2004). J. J. Thorn et al., Phys. Rev. Lett. 100, 240407 (2008). J. A. Stickney, D. Z. Anderson, and A. A. Zozulya, Phys. Rev. A 75, 013608 (2007). * Likharev (1979) K. K. Likharev, Rev. Mod. Phys. 51, 101 (1979). * Davis and Packard (2002) J. C. Davis and, R. E. Packard, Rev. Mod. Phys. 74, 741 (2001). * Albiez et al. (2005) M. Albiez et al., Phys. Rev. Lett. 95, 010402 (2005). * Levy et al. (2007) S. Levy et al., Nature 449, 579 (2007). * Landau (1941) L. D. Landau, J. Phys. (USSR) 5, 71 (1941). * (10) This assumes that the spatial scale of any perturbing potential is much less than the healing length. * Pitaevskii and Stringari (2003) L. P. Pitaevskii and S. Stringari, _Bose-Einstein Condensation_ (Clarendon, Oxford, 2003). * Barenghi et al. (2001) C. F. Barenghi, R. J. Donnelly, , and W. F. Vinen, eds., _Quantized Vortex Dynamics and Superfluid Turbulence_ (Springer-Verlag, Berlin, 2001). * Feynman (1955) R. Feynman, Prog. Low Temp. Phys. 1, 17 (1955). * Avenel and Varoquaux (1985) O. Avenel and E. Varoquaux, Phys. Rev. Lett. 55, 2704 (1985). * Amar et al. (1992) A. Amar et al., Phys. Rev. Lett. 68, 2624 (1992). * Huebener (2001) R. P. Huebener, _Magnetic Flux Structures in Superconductors_ (Springer, 2001). * Neely et al. (2010) T. W. Neely et al., Phys. Rev. Lett. 104, 160401 (2010). * Onofrio et al. (2000) R. Onofrio et al., Phys. Rev. Lett. 85, 2228 (2000). * Engels and Atherton (2007) P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007). * Ryu et al. (2007) C. Ryu et al., Phys. Rev. Lett. 99, 260401 (2007). * (21) We apply a strong barrier during evaporation, removing it adiabatically when well below the condensation temperature, several seconds before we create circulation. * Wright et al. (2008) K. C. Wright, L. S. Leslie, and N. P. Bigelow, Phys. Rev. A 77, 041601 (2008). * Andersen et al. (2006) M. F. Andersen et al., Phys. Rev. Lett. 97, 170406 (2006). * Madison et al. (2000) K. W. Madison et al., Phys. Rev. Lett. 84, 806 (2000). * Freilich et al. (2010) D. V. Freilich et al., Science 329, 1182 (2010). * (26) A. Ramanathan et al., to be published. * (27) Uncertainties herein are the uncorrelated combination of 1$\sigma$ statistical and systematic uncertainties unless stated otherwise. * Watanabe et al. (2009) G. Watanabe et al., Phys. Rev. A 80, 053602 (2009). * Stringari (1998) S. Stringari, Phys. Rev. A 58, 2385 (1998). * Piazza et al. (2009) F. Piazza, L. A. Collins, and A. Smerzi, Phys. Rev. A 80, 021601 (2009).
arxiv-papers
2010-12-29T22:43:04
2024-09-04T02:49:16.064742
{ "license": "Public Domain", "authors": "A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill III, C.\n J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell", "submitter": "Kevin Wright", "url": "https://arxiv.org/abs/1101.0019" }
1101.0082
# Probabilistic Dynamic Logic of Phenomena and Cognition ††thanks: Evgenii Vityaev is with the Department of Mathematical Logic, Sobolev Institute of Mathematics of the Russian Academy of Sciences and with the Department of Discrete mathematics and Informatics of the Novosibirsk State University, 630090, Novosibirsk, Russia, email: vityaev@math.nsc.ru ††thanks: Boris Kovalerchuk is with the Department of Computer Science, Central Washington University, Ellensburg, WA 98926-7520, e-mail: borisk@cwu.edu ††thanks: Leonid Perlovsky is with the Harvard University and the Air Force Research Laboratory, Sensors Directorate, Hanscom AFB, leonid@seas.harvard.edu ††thanks: Stanislav Smerdov, Novosibirsk State University, Sobolev Institute of Mathematics of the Russian Academy of Sciences, 630090, Novosibirsk, Russia, email: netid@ya.ru Evgenii Vityaev, Boris Kovalerchuk, Leonid Perlovsky, Stanislav Smerdov ###### Abstract The purpose of this paper is to develop further the main concepts of Phenomena Dynamic Logic (P-DL) and Cognitive Dynamic Logic (C-DL), presented in the previous paper. The specific character of these logics is in matching vagueness or fuzziness of similarity measures to the uncertainty of models. These logics are based on the following fundamental notions: _generality relation, uncertainty relation, simplicity relation, similarity maximization problem with empirical content and enhancement (learning) operator_. We develop these notions in terms of logic and probability and developed a Probabilistic Dynamic Logic of Phenomena and Cognition (P-DL-PC) that relates to the scope of probabilistic models of brain. In our research the effectiveness of suggested formalization is demonstrated by approximation of the expert model of breast cancer diagnostic decisions. The P-DL-PC logic was previously successfully applied to solving many practical tasks and also for modelling of some cognitive processes. ## I Introduction In the paper [1] there was introduced a Phenomena Dynamic Logic (P-DL) and Cognitive Dynamic Logic (C-DL) as a generalization of the Dynamic Logic and Neural Modelling Fields theory (NMF) introduced in the previous papers [2, 3]. Logics P-DL, C-DL provide the most general description of Dynamic Logic in the following fundamental notions _generality relation, uncertainty relation, simplicity relation, similarity maximization problem with empirical content and enhancement (learning) operator_. This generalization provide interpretation of P-DL, C-DL logics in the frame of other approaches. In this paper we interpret logics P-DL, C-DL in terms of logic and probability: uncertainty we interpret as probability, while the process of learning as a semantic probabilistic inference [4, 9, 6, 5]. We also interpret mentioned fundamental notions. The resulting Probabilistic Dynamic Logic of Phenomena and Cognition (P-DL-PC) belong to the scope of the probabilistic models of brain [19, 20]. Thus, through logics P-DL, C-DL we extend the interpretation of Dynamic Logic and Neural Modelling Fields theory to probabilistic models of brain. The P-DL-PC logic as probabilistic model of brain was previously applied to modelling of some cognitive process [7, 8, 9, 21]. The effectiveness of P-DL-PC logic demonstrated in this paper by approximation of the expert model of breast cancer diagnostic decisions. ## II Universal productions. Data for prediction In our study learning models will be generated as sets of _universal productions_ (_u-productions_), which are introduced in this section. Note that every set of _universal formulas_ is logically equivalent to a certain set of u-productions. Consider a fixed first-order language $\mathfrak{L}$ in a countable signature. Hereafter denote $\mathbf{A}_{\mathfrak{L}}$ the set of all atoms; $\mathbf{L}_{\mathfrak{L}}$ – the set of all literals; $\mathbf{S}_{\mathfrak{L}}^{0}$ – the set of ground sentences. The set of ground atoms and the set of ground literals are denoted $\mathbf{A}_{\mathfrak{L}}^{0}\rightleftharpoons\mathbf{A}_{\mathfrak{L}}\cap\mathbf{S}_{\mathfrak{L}}^{0}$ and $\mathbf{L}_{\mathfrak{L}}^{0}\rightleftharpoons\mathbf{L}_{\mathfrak{L}}\cap\mathbf{S}_{\mathfrak{L}}^{0}$ correspondingly. Following examples of atoms and literals are given in the section VIII for the task of approximation of the expert model of breast cancer diagnostic decisions: ‘number of calcifications per $cm^{3}$ less than 20‘, ‘volume of calcifications in $cm^{3}$ not less or equal to 5‘, ‘total number of calcifications more than 30 and etc. Let $\Theta$ be the set of all substitutions and $\Theta^{0}\subseteq\Theta$ the set of ground substitutions, that are mappings variables to ground terms. All necessary notions from model theory and logic programming are elementary and can be easily found in books [12], [13, 14]. ###### Definition. A record of the type $\mathrm{R}\leftrightharpoons\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n}\right),$ where $\mathrm{A}_{1},\cdots\mathrm{A}_{m},\mathrm{B}_{1},\cdots,\mathrm{B}_{n}$ are literals, and $\tilde{\forall}$ stands for a bloc of quantifiers over all free variables of the formulae in brackets (universal closure), is called a _u- production_. _A variant of u-production $\mathrm{R}$_ is $\mathrm{R}\theta\rightleftharpoons\tilde{\forall}\left(\mathrm{A}_{1}\theta\wedge\cdots\mathrm{A}_{m}\theta\Leftarrow\mathrm{B}_{1}\theta\wedge\cdots\wedge\mathrm{B}_{n}\theta\right),$ where $\theta$ is an arbitrary one-to-one correspondence over the set of variables. Let ${\tt Prod}$ be the set of all u-productions. For example in section X presented the following u-production that was discovered by the learning model: > IF TOTAL number of calcifications is more than 30, and VOLUME is more than 5 > $cm^{3}$, and DENSITY of calcifications is moderate, > THEN Malignant. Let $\mathtt{Fact}_{v}\subset\mathbf{A}_{\mathfrak{L}}$ be a set of atoms from A that are valid for verification in algebraic system ${\mathfrak{B}}$ appearing in practice. Our aim is to investigate as much “extra” facts about $\mathfrak{L}$ as possible, i.e., to predict or explain them. A natural assumption is that we can verify (falsify) each element of $\mathtt{Fact}_{\rm o}\leftrightharpoons\left\\{{\rm A}\theta\mid\theta\in\Theta^{0},~{}{\rm A}\in\mathtt{Fact}_{v}\right\\}.$ Certainly we may postulate our ability to check any literal of $\mathtt{Fact}_{v}^{\ast}\leftrightharpoons\mathtt{Fact}_{v}\cup\left\\{\neg{\rm A}\mid{\rm A}\in\mathtt{Fact}_{v}\right\\}$. For the rest of the literals (and their conjunctions) the machinery of _probabilistic prediction_ will be defined later on. Note that ${\tt Fact}_{\rm o}^{\ast}={\tt Fact}_{\rm o}\cup{\tt Fact}_{\rm o}^{\neg}$, where ${\tt Fact}_{\rm o}^{\neg}\rightleftharpoons\left\\{\neg\mathrm{A}\mid\mathrm{A}\in{\tt Fact}_{\rm o}\right\\}$ is _the complete set of alternatives_ allowing a real test. The _data_ are defined as a maximal (logically) consistent subset of the complete set of alternatives, i.e., being given a mapping $\zeta_{\mathfrak{B}}:{\tt Fact}_{\rm o}\mapsto\left\\{{\bot,\top}\right\\}$ (here _$\bot$ – “false”, $\top$ – “true”_) we conclude that ${\tt Data}\left[\mathfrak{B}\right]\rightleftharpoons\left\\{{{\rm A}\mid{\rm A}\in{\tt Fact}_{\rm o}~{}\mbox{{and}}~{}\zeta_{\mathfrak{B}}\left({\rm A}\right)=\top}\right\\}\cup$ $\left\\{{\neg{\rm A}\mid{\rm A}\in{\tt Fact}_{\rm o}~{}\mbox{{and}}~{}\zeta_{\mathfrak{B}}\left({\rm A}\right)=\bot}\right\\}.$ ## III Generality relation between theories The idea of a _generality relation_ between theories can be viewed, for example, as a reduction of the set of properties predicted by the use of these theories. A more general theory (potentially) predicts a greater number of formal features. We start with a generality relation between one-element specifications, i.e., between u-productions. ###### Definition. For two productions $\mathrm{R}_{1}\equiv\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m_{1}}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n_{1}}\right)$ and $\mathrm{R}_{2}\equiv\tilde{\forall}\left(\mathrm{C}_{1}\wedge\cdots\wedge\mathrm{C}_{m_{2}}\Leftarrow\mathrm{D}_{1}\wedge\cdots\wedge\mathrm{D}_{n_{2}}\right)$ a relation $\mathrm{R}_{1}\succ\mathrm{R}_{2}$ (“more general than”) takes place if and only if there exists $\theta\in\Theta$ such that $\left\\{{\mathrm{B}_{1}\theta,\cdots,\mathrm{B}_{n_{1}}\theta}\right\\}\subseteq\left\\{{\mathrm{D}_{1},\cdots,\mathrm{D}_{n_{2}}}\right\\}$, $\left\\{{\mathrm{A}_{1}\theta,\cdots,\mathrm{A}_{m_{1}}\theta}\right\\}\supseteq\left\\{{\mathrm{C}_{1},\cdots,\mathrm{C}_{m_{2}}}\right\\}$, and $n_{1}\leqslant n_{2}$, $m_{1}\geqslant m_{2}$, $\not\vdash{\rm R}_{1}\equiv{\rm R}_{2}$. The inclusion of the sets of premises designates that the more general u-production is, then the wider its field of application. The inverse inclusion (for conclusions) says that $\mathrm{R}_{1}$ predicts a greater number of properties using a smaller premise. Let $S\subseteq{\tt Prod}$. Denote ${\tt Fact}\left[S;\mathfrak{B}\right]$ the set of all $\mathrm{A}\in\mathbf{L}_{\mathfrak{L}}^{0}$ such that for some ${\rm R}\in S$ and $\theta\in\Theta^{0}$, $\mathrm{R}\theta\equiv\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\Leftarrow\mathrm{B}_{1}\wedge\cdots\wedge\mathrm{B}_{n}\right)$, holds $\left\\{\mathrm{B}_{1},\cdots,\mathrm{B}_{n}\right\\}\subseteq{\tt Data}\left[\mathfrak{B}\right]$ and $\mathrm{A}\in\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}$. Thus, ${\tt Fact}\left[S;\mathfrak{B}\right]$ is the set of ground literals predicted according to available data (about the model $\mathfrak{B}$) together with u-productions in $S$. In the sequel let $\succcurlyeq$ be a reflexive closure of $\succ$. One should pay attention to the fact: $\mathrm{R}_{1}\succcurlyeq\mathrm{R}_{2}$ entails that ${\tt Fact}\left[\left\\{\mathrm{R}_{1}\right\\};\mathfrak{B}\right]$ contains ${\tt Fact}\left[\left\\{\mathrm{R}_{2}\right\\};\mathfrak{B}\right]$. Thereafter it isn’t difficult to extend the domain of our generality relation to subsets of ${\tt Prod}$. ###### Definition. Let $S,S^{\prime}\subseteq{\tt Prod}$, and for any ${\rm R}^{\prime}\in S^{\prime}$ we find ${\rm R}\in S$ such that $\mathrm{R}\succcurlyeq\mathrm{R}^{\prime}$. In this case we say ‘$S$ is not less general than $S^{\prime}$’ ($S\vartriangleright S^{\prime}$). It’s straightforward to notice that ${\tt Fact}\left[S^{\prime};\mathfrak{B}\right]\subseteq{\tt Fact}\left[S;\mathfrak{B}\right]$ for $S$ and $S^{\prime}$ from the definition above. Remark that $S$ may include u-productions apart from those, which are generalizations of elements of $S^{\prime}$. ## IV Probability/degree of belief The topic of distributing probability over formulas of propositional logic (as well as over ground statements in a first order language) being widely discussed in a literature and meets Kolmogorov’s understanding of probability measure [11]. The following definition is given on the basis of analysis cited in [10]. ###### Definition. A probability over $F\subseteq{\mathbf{S}}_{\mathfrak{L}}^{0}$ closed with respect to $\wedge$, $\vee$ and $\neg$, is a function $\mu:F\mapsto\left[{0,1}\right]$ satisfying the following conditions: 1. 1. if $\vdash\Phi$ (“$\Phi$ is a tautology”), then $\mu\left(\Phi\right)=1$; 2. 2. if $\vdash\neg\left({\Phi\wedge\Psi}\right)$, then $\mu\left({\Phi\vee\Psi}\right)=\mu\left(\Phi\right)+\mu\left(\Psi\right)$. For any ground instance of a u-production its probability is defined as conditional, i.e., $\mu\left({\rm A}_{1}\wedge\cdots{\rm A}_{m}\Leftarrow{\rm B}_{1}\wedge\cdots\wedge{\rm B}_{n}\right)=$ $=\mu\left({\rm A}_{1}\wedge\cdots{\rm A}_{m}\mid{\rm B}_{1}\wedge\cdots\wedge{\rm B}_{n}\right)=\frac{\mu\left({{\rm A}_{1}\wedge\cdots{\rm A}_{m}\wedge{\rm B}_{1}\wedge\cdots\wedge{\rm B}_{n}}\right)}{\mu\left({{\rm B}_{1}\wedge\cdots\wedge{\rm B}_{n}}\right)}$ Let ${\rm R}\in{\tt Prod}$. Denote as ${\tt Sub}\left[{\rm R}\right]^{\mu}$ those substitutions $\theta\in\Theta^{0}$, for which the premise of u-production ${\rm R}\theta$ has a non-zero probability. ${\tt Prod}^{\mu}\rightleftharpoons\left\\{{\rm R}\in{\tt Prod}\mid{\tt Sub}\left[{\rm R}\right]^{\mu}\neq\varnothing\right\\}$; $\underline{\mu}\left({\rm R}\right)\rightleftharpoons{\rm inf}\left\\{\mu\left({{\rm R}\theta}\right)\mid\theta\in{\tt Sub}\left[{\rm R}\right]^{\mu}\right\\}$, where ${\rm R}\in{\tt Prod}^{\mu}$. A value of conditional probability serves to characterize our _degree of belief_ (and responsible for an _uncertainty relation_) in reliability of different causal connections included in temporary specification. Note that two productions are not necessary comparable with respect to generality relation $\succcurlyeq$; moreover, their premisses may not be contained in the complete set of alternatives (and so these productions will be not valid for a direct check in a real structure $\mathfrak{B}$). ## V Simplicity of probabilistic theories Adding comparison of lower probabilistic estimations to the definition of generality relation we obtain the following definition. ###### Definition. Let $S,S^{\prime}\subseteq{\tt Prod}^{\mu}$. We say that $S$ is _more $\mu$-general than_ $S^{\prime}$ iff for every $\mathrm{C}^{\prime}\in S^{\prime}$ there exists $\mathrm{C}\in S$ such that $\mathrm{C}\succcurlyeq\mathrm{C}^{\prime}$ and $\underline{\mu}(\mathrm{C})\geqslant\underline{\mu}(\mathrm{C}^{\prime})$, and in at least one of the cases the strong relation $\succ$ takes place. Hence, $\mu$-generalization allows us to define a more general set $S$ in such a way that the lower estimations of probabilities is not declined. When our belief to the elements of $S$ is no less than that of $S^{\prime}$, then we have a _simplicity relation_ – the set $S$ is simpler than $S^{\prime}$ in order to describe/predict the properties. ## VI Similarity measure with the empirical content By elaboration of u-productions we mean the gain of its conditional probability. ###### Definition. A relation ${\rm R}_{1}\sqsubset{\rm R}_{2}$ (‘probabilistic inference’) for ${\rm R}_{1},{\rm R}_{2}\in{\tt Prod}^{\mu}$ means that ${\rm R}_{1}\succ{\rm R}_{2}$ and $\underline{\mu}\left({{\rm R}_{1}}\right)<\underline{\mu}\left({{\rm R}_{2}}\right)$. ###### Definition. Let $\pi$ be some requirements to be applied to elements of ${\tt Prod}^{\mu}$, i.e. $\pi:{\tt Prod}^{\mu}\mapsto\left\\{\bot,\top\right\\}$ (value is equal to $\bot$, if u-production satisfies $\pi$, and $\top$ – otherwise); $\mathsf{\Pi}\leftrightharpoons\left\\{{\rm R}\in{\tt Prod}^{\mu}\mid\pi\left({\rm R}\right)=\top\right\\}$. We say that $\mathrm{R}_{2}\in\mathsf{\Pi}$ is a _minimal follower of $\mathrm{R}_{1}\in{\tt Prod}^{\mu}$ relative to $\sqsubset$ in $\mathsf{\Pi}$_ (denoted as $\mathrm{R}_{1}\sqsubset_{\pi}\mathrm{R}_{2}$), iff $\mathrm{R}_{1}\sqsubset\mathrm{R}_{2}$ and there is no intermediate u-production $\mathrm{R}_{3/2}\in\Pi$ such that $\mathrm{R}_{1}\sqsubset\mathrm{R}_{3/2}\sqsubset\mathrm{R}_{2}$. In the prediction of a literal ${\rm H}$ the _similarity measure_ for u-productions, which are valid for verification and applicable to the goal ${\rm H}$, is equal to conditional probability $\underline{\mu}\left(\cdot\right)$. Thus we deal with a uniform measure of similarity. ## VII Learning operator ###### Definition. A production $\mathrm{R}\equiv\tilde{\forall}\left(\mathrm{A}_{1}\wedge\cdots\wedge\mathrm{A}_{m}\leftarrow\mathrm{B}_{1}\wedge\dots\wedge\mathrm{B}_{n}\right)$ is called a _maximal specific u-production (ums-production) for prediction of a conjunction $\mathrm{H}\equiv\left(\mathrm{H}_{1}\wedge\cdots\wedge\mathrm{H}_{k}\right)$,_ where $\left\\{\mathrm{H}_{1},\cdots,\mathrm{H}_{k}\right\\}\subset\mathbf{L}_{\mathfrak{L}}$ and $m\leqslant k$, iff the following conditions are satisfied: 1. 1. there is a substitution $\theta$ (not necessary ground) such that $\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}\subseteq\left\\{\mathrm{H}_{1}\theta,\cdots,\mathrm{H}_{k}\theta\right\\}$, $\left\\{{\mathrm{B}_{1},\dots,\mathrm{B}_{n}}\right\\}\subseteq\left\\{\mathrm{B}\theta\mid\mathrm{B}\in\mathtt{Fact}_{v}^{\ast}\right\\}$; 2. 2. if $\mathrm{D}\in\left\\{\mathrm{A}_{1},\cdots,\mathrm{A}_{m}\right\\}$ and $\theta_{\rm o}\in{\tt Sub}\left[{\rm R}\right]^{\mu}$, then $\mu\left({\mathrm{A}_{1}\theta_{\rm o}\wedge\cdots\wedge\mathrm{A}_{m}}\theta_{\rm o}\right)<\\\ \mu\left({\mathrm{A}_{1}\theta_{\rm o}\wedge\cdots\wedge\mathrm{A}_{m}}\theta_{\rm o}\mid{\mathrm{B}_{1}\theta_{\rm o}\wedge\cdots\wedge\mathrm{B}_{n}}\theta_{\rm o}\right)$ and $\mu\left(\mathrm{D}\theta_{\rm o}\right)<\mu\left(\mathrm{D}\theta_{\rm o}\mid{\mathrm{B}_{1}\theta_{\rm o}\wedge\cdots\wedge\mathrm{B}_{n}}\theta_{\rm o}\right)$; 3. 3. there is no ${\rm R^{\prime}}\in\mathtt{Prod}^{\mu}$, for which points (1–2) are hold along with ${\rm R}\sqsubset{\rm R^{\prime}}$; 4. 4. the u-production ${\rm R}$ can’t be generalized up to some ${\rm R^{\prime}}\in\mathtt{Prod}^{\mu}$ satisfying all the previous points (1–3) without decreasing its estimation $\underline{\mu}\left(\cdot\right)$. The conditions above (for corresponding ums-productions) are denoted as ‘point.i’, $1\leqslant i\leqslant 4$. ###### Remark. Though condition point.4 emphasizes the nature of definition, but it isn’t necessary for indication. Indeed, if $\mathrm{R}$ may be generalized up to $\mathrm{R}^{\prime}$ under preserving point.1–3, then $\underline{\mu}\left(\mathrm{R}\right)\leqslant\underline{\mu}\left(\mathrm{R}^{\prime}\right)$ (otherwise we get ${\rm R^{\prime}}\sqsubset{\rm R}$ – that contradicts point.3 for $\mathrm{R}$. Let $\pi\left({\rm R}\right)=\top$ be fulfilled for ${\rm R}\in\mathtt{Prod}^{\mu}$ iff conditions points.1–2 are satisfied for ${\rm R}$ and ${\rm H}$ (the last one is fixed from this moment); denote $\mathsf{\Pi}\leftrightharpoons\pi^{-1}\left(\top\right)$. Define the _probabilistic fix-point operator_ $\mathrm{T}_{\pi}:2^{\mathtt{Prod}^{\mu}}\mapsto 2^{\mathtt{Prod}^{\mu}}$ as follows: for a set $S\subseteq\mathtt{Prod}^{\mu}$ it produces $S^{\prime}\leftrightharpoons\left\\{\mathrm{R}^{\prime}\mid\mathrm{R}\sqsubset_{\pi}\mathrm{R}^{\prime}\ \mbox{\emph{for some}}\ \mathrm{R}\in S\right\\}\cup$ $\cup\left\\{\mathrm{R}\mid\mathrm{R}\in S\cap\mathsf{\Pi}\ \mbox{\emph{and there is no}}\ \mathrm{R}^{\prime}\ \mbox{\emph{such that}}\ \mathrm{R}\sqsubset_{\pi}\mathrm{R}^{\prime}\right\\}$. Therefore the operator $\mathrm{T}_{\pi}:S\mapsto S^{\prime}$ possess important properties: 1. 1. the set $S^{\prime}$ is always more precise than $S$ (relative to $\succcurlyeq$); 2. 2. the conditional probabilities $\underline{\mu}\left(\cdot\right)$ increase during the conversion to more particular cases (and so fuzziness decreases); 3. 3. the similarity measure with the empirical content becomes greater for at least one u-production (in $S$) when the operator converts $S$ to $S^{\prime}$ (if not $S=S^{\prime}$, of course); As a result the operator $\mathrm{T}_{\pi}$ is the _enhancement, or learning, operator_ in the sense of [1]. ###### Definition. A fix-point (f.p., for short) $S$ of $\mathrm{T}_{\pi}$ is _optimal_ iff there is no other f.p. $S^{\prime}$ of considered operator, which is more $\mu$-general than $S$. ###### Statement. A subset $S\subseteq\mathtt{Prod}^{\mu}$ is a fix-point of the operator $\mathrm{T}_{\pi}$ iff every element of $S$ satisfies points.1–3 for $\mathrm{H}$. ###### Corrolary. A subset $S\subseteq\mathtt{Prod}^{\mu}$ is an optimal fix-point of the operator $\mathrm{T}_{\pi}$ iff every element of $S$ is a ums-production for prediction of $\mathrm{H}$. Ums-productions may be viewed as a result of performing generalized scheme of the _semantic probabilistic inference_ [4, 5], which is realized by the fix- point operator described above. The program system ‘Discovery’ (see [16, 17, 9, 21]) was developed: it carries out the propositional version of the probabilistic fix-point (learning) operator and was successfully applied to solving many practical tasks [21]. ## VIII Extraction of the expert model of breast cancer diagnostic decisions We applied our method to approximation of the expert model of breast cancer diagnostic decisions that was obtained from the radiologist J.Ruiz [17]. At first we extract this model from the expert by the special procedure using monotone boolean functions [17] and then apply the program system ‘Discovery’ [16] to approximate this model. ### VIII-A Hierarchical Approach At first we ask an expert to describe particular cases using the binary features. Then we ask a radiologist to evaluate a particular cases, when features take on specific values. A typical query will have the following format: ”If feature 1 has value $v_{1}$, feature 2 has value $v_{2}$, …, feature n has value $v_{n}$, then is a case suspicious of cancer or not?” Each set of values ($v_{1},v_{2},...,v_{n}$) represent a possible clinical case. It is practically impossible to ask a radiologist to generate diagnosis for thousands of possible cases. A hierarchical approach combined with the use of the property of monotonicity makes the problem manageable. We construct a hierarchy of medically interpretable features from a very generalized level to a less generalized level. This hierarchy follows from the definition of the 11 medically oriented binary attributes. The medical expert indicate that the original 11 binary attributes $w_{1},w_{2},w_{3},y_{1},y_{2},y_{3},y_{4},y_{5},x_{3},x_{4},x_{5}$ could be organized in terms of a hierarchy with development of two new generalized attributes $x_{1}$, depending on attributes $w_{1},w_{2},w_{3}$, and $x_{2}$, depending on attributes $y_{1},y_{2},y_{3},y_{4},y_{5}$. A new generalized feature, $x_{1}$ – ‘Amount and volume of calcifications’ with grades (0 - ‘benign’ and 1 - ‘cancer’) was introduced based on features: $w_{1}$ – number of calcifications/cm3, $w_{2}$ – volume of calcification, cm3 and $w_{3}$ – total number of calcifications. We view $x_{1}$ as a function $g(w_{1},w_{2},w_{3})$ to be identified. Similarly a new feature $x_{2}$ – ‘Shape and density of calcification’ with grades: (1) for ‘cancer’ and (0)-‘benign’ generalizes features: $y_{1}$ – ‘irregularity in shape of individual calcifications’ $y_{2}$ – ‘variation in shape of calcifications’ $y_{3}$ – ‘variation in size of calcifications’ $y_{4}$ – ‘variation in density of calcifications’ $y_{5}$ – ‘density of calcifications’. We view $x_{2}$ as a function $x_{2}=h(y_{1},y_{2},y_{3},y_{4},y_{5})$ to be identified for cancer diagnosis. As result we have a decomposition of our task as follows: $f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)=$ $f\left(g\left(w_{1},w_{2},w_{3}\right),h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right),x_{3},x_{4},x_{5}\right).$ ### VIII-B Monotonicity Giving the above definitions we can represent clinical cases in terms of binary vectors with five generalized features as: $(x_{1},x_{2},x_{3},x_{4},x_{5})$. Let us consider two clinical cases that are represented by the two binary sequences: (10110) and (10100). If radiologist correctly diagnose (10100) as cancer, then, by utilizing the property of monotonicity, we can also conclude that the clinical case (10110) should also be cancer. Medical expert agreed with presupposition about monotonicity of the functions $f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)$ and $h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$. Let us describe the interview with an expert using minimal sequence of questions to completely infer a diagnostic function using monotonicity. This sequence is based on fundamental Hansel lemma [15]. We omit a detailed description of the specific mathematical steps. They can be found in [18]. Table 1 illustrates this. ### VIII-C Expert model extraction Columns 2 and 3 present values of above defined functions $f$ and $h$. We omit a restoration of function $g\left(w_{1},w_{2},w_{3}\right)$ because few questions are needed to restore this function. All 32 possible cases with five binary features $\langle x_{1},x_{2},x_{3},x_{4},x_{5}\rangle$ are presented in column 1 in table 1. They are grouped and the groups are called Hansel chains [17]. The sequence of chains begins with the shortest chain 1 – $(01100)<(11100)$ for five binary features. Then largest chain 10 consists of 6 ordered cases: $(00000)<(00001)<(00011)<(00111)<(01111)<(11111)$. The chains are numbered there from 1 to 10 and each case has its number in the chain, e.g., 1.2 means the second case in the first chain. Asterisks in columns 2 and 3 mark answers obtained from an expert, e.g., 1* for case (01100) in column 3 means that the expert answered ‘yes’. The answers for some other chains in column 3 are automatically obtained using monotonicity. The value f(01100) = 1 for case 1.1 is extended for cases 1.2, 6.3. and 7.3 in this way. Similarly values of the monotone Boolean functions h are computed using the table 1. The attributes in the sequence (10010) are interpreted as $y_{1},y_{2},y_{3},y_{4},y_{5}$ for the function h instead of $x_{1},x_{2},x_{3},x_{4},x_{5}$. The Hansel chains are the same if the number of attributes is the same five in this case. Column 5 and 6 list cases for extending functions’ values without asking an expert. Column 5 is for extending functions’ values from 1 to 1 and column 6 is for extending them from 0 to 0. If an expert gave an answer opposite (f(01100) = 0) to that presented in table 1 for function $f$ in the case 1.1, then this 0 value could be extended in column 2 for cases 7.1 (00100) and 8.1 (01000). These cases are listed in column 5 for case (01100). There is no need to ask an expert about cases 7.1 (00100) and 8.1 (01000). Monotonicity provides the answer. The negative answer f(01100) = 0 can not be extended for f(11100). An expert should be queried regarding f(11100). If his/her answer is negative f(11100) = 0 then this value can be extended for cases 5.1. and 3.1 listed in column 5 for case 1.2. Relying on monotonicity, the value of f for them will also be 0. The total number of cases with asterisk (*) in columns 2 and 3 are equal to 13 and 12. These numbers show that 13 questions are needed to restore the function $f\left(x_{1},x_{2},x_{3},x_{4},x_{5}\right)$ and 12 questions are needed to restore the function $h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$. This is only 37.5% of 32 possible questions. The full number of questions for the expert without monotonicity and hierarchy is $2^{11}=2048$. ## IX Approximation of the expert model by learning operator For the Approximation of the expert model we used the program system ‘Discovery’ [16], that realizes the propositional case of the probabilistic fix-point learning operator. We discovered several dozens diagnostic rules that were statistically significant on the 0.01, 0.05 and 0.1 levels of (F-criterion). Rules were extracted using 156 cases (73 malignant, 77 benign, 2 highly suspicious and 4 with mixed diagnosis). In the Round-Robin test our rules diagnosed 134 cases and refused to diagnose 22 cases. The total accuracy of diagnosis is 86%. Incorrect diagnoses were obtained in 19 cases (14% of diagnosed cases). The false-negative rate was 5.2% (7 malignant cases were diagnosed as benign) and the false-positive rate was 8.9% (12 benign cases were diagnosed as malignant). Some of the rules are shown in table 2. This table presents examples of discovered rules with their statistical significance. In this table: * • ‘NUM’ – number of calcifications per $cm^{3}$; * • ‘VOL’ – volume in $cm^{3}$; * • ‘TOT’ – total number of calcifications; * • ‘DEN’ – density of calcifications; * • ‘VAR’ – variation in shape of calcifications; * • ‘SIZE’ – variation in size of calcifications; * • ‘IRR’ – irregularity in shape of calcifications; * • ‘SHAPE’ – shape of calcifications. We studied three levels of similarity measure: 0.7, 0.85 and 0.95. A higher level of conditional probability decreases the number of rules and diagnosed patients, but increases accuracy of diagnosis. Table 1. Dynamic sequence of questions to expert --- 1 | 2 | 3 | 4 | 5 | 6 | 7 Number | $f$ | $h$ | Monotonic extrapolation | Chain | Case | Diagnose | Form and V | $1\mapsto 1$ | $0\mapsto 0$ | | $(01100)$ | 1* | 1* | 1.2, 6.3, 7.3 | 7.1, 8.1 | Chain 1 | 1.1 $(11100)$ | 1 | 1 | 6.4, 7.4 | 5.1, 3.1 | | 1.2 $(01010)$ | 0* | 1* | 2.2, 6.3, 8.3 | 6.1, 8.1 | Chain 2 | 2.1 $(11010)$ | 1* | 1 | 6.4, 8.4 | 3.1, 6.1 | | 2.2 $(11000)$ | 1* | 1* | 3.2 | 8.1, 9.1 | Chain 3 | 3.1 $(11001)$ | 1 | 1 | 7.4, 8.4 | 8.2, 9.2 | | 3.2 $(10010)$ | 0* | 1* | 4.2, 9.3 | 6.1, 9.1 | Chain 4 | 4.1 $(10110)$ | 1* | 1 | 6.4, 9.4 | 6.2, 5.1 | | 4.2 $(10100)$ | 1* | 1* | 5.2 | 7.1, 9.1 | Chain 5 | 5.1 $(10101)$ | 1 | 1 | 7.4, 9.4 | 7.2, 9.2 | | 5.2 $(00010)$ | 0 | 0* | 6.2, 10.3 | 10.1 | Chain 6 | 6.1 $(00110)$ | 1* | 0* | 6.3, 10.4 | 7.1 | | 6.2 $(01110)$ | 1 | 1 | 6.4, 10.5 | | | 6.3 $(11110)$ | 1 | 1 | 10.6 | | | 6.4 $(00100)$ | 1* | 0* | 7.2, 10.4 | 10.1 | Chain 7 | 7.1 $(00101)$ | 1 | 0* | 7.3, 10.4 | 10.2 | | 7.2 $(01101)$ | 1 | 1* | 7.4, 10.5 | 8.2, 10.2 | | 7.3 $(11101)$ | 1 | 1 | 5.6 | | | 7.4 $(01000)$ | 0 | 1* | 8.2 | 10.1 | Chain 8 | 8.1 $(01001)$ | 1* | 1 | 8.3 | 10.2 | | 8.2 $(01011)$ | 1 | 1 | 8.4 | 10.3 | | 8.3 $(11011)$ | 1 | 1 | 10.6 | 9.3 | | 8.4 $(10000)$ | 0 | 1* | 9.2 | 10.1 | Chain 9 | 9.1 $(10001)$ | 1* | 1 | 9.3 | 10.2 | | 9.2 $(10011)$ | 1 | 1 | 9.4 | 10.3 | | 9.3 $(10111)$ | 1 | 1 | 10.6 | 10.4 | | 9.4 $(00000)$ | 0 | 0 | 10.2 | | Chain 10 | 10.1 $(00001)$ | 0* | 0 | 10.3 | | | 10.2 $(00011)$ | 1* | 0 | 10.4 | | | 10.3 $(00111)$ | 1 | 1* | 10.5 | | | 10.4 $(01111)$ | 1 | 1 | 10.6 | | | 10.5 $(11111)$ | 1 | 1 | | | | 10.6 Questions | 13 | 12 | | | | Table 2. Examples of discovered diagnostic rules --- Diagnosis | $f$-criteria | Value. $f$-criteria | Precision rule | | 0.01 | 0.05 | 0.1 | on control If $10<{\rm NUM}<20$ | NUM | 0.0029 | + | + | + | 93.3% and ${\rm VOL}>5$ | VOL | 0.0040 | + | + | + | then malignant | | | | | | If ${\rm TOT}>30$ | TOT | 0.0229 | - | + | + | 100.0% and ${\rm VOL}>5$ | VOL | 0.0124 | - | + | + | and DEN is moderate | DEN | 0.0325 | - | + | + | then malignant | | | | | | If VAR is marked | VAR | 0.0044 | + | + | + | 100.0% and $10<{\rm NUM}<20$ | NUM | 0.0039 | + | + | + | and IRR is moderate | IRR | 0.0254 | - | + | + | then malignant | | | | | | If SIZE is moderate | SIZE | 0.0150 | - | + | + | 92.86% and SHAPE is mild | SHAPE | 0.0114 | - | + | + | and IRR is mild | IRR | 0.0878 | - | - | + | then benign | | | | | | Results for them are marked as Discovery1, Discovery2 and Discovery3. We extracted 44 statistically significant diagnostic rules for 0.05 level of F -criterion with a conditional probability no less than 0.75 (Discovery1). There were 30 rules with a conditional probability no less than 0.85 (Discovery2) and 18 rules with a conditional probability no less than 0.95 (Discovery3). The most reliable 30 rules delivered a total accuracy of 90%, and the 18 most reliable rules performed with 96.6% accuracy with only 3 false positive cases (3.4%). ## X Decision rule (model) extracted from the expert through monotone Boolean functions We obtained Boolean expressions for function $h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)$ (‘shape and density of calcification’) from the information depicted in table 1 with the following steps: 1. - Find all the maximal lower units for all chains as elementary conjunctions; 2. - Take the disjunction of obtained conjunctions; 3. - Exclude the redundant terms (conjunctions) from the end formula. Using 1 and 3 columns we have $x_{2}=h\left(y_{1},y_{2},y_{3},y_{4},y_{5}\right)=y_{2}y_{3}\vee y_{2}y_{4}\vee y_{1}y_{2}\vee y_{1}y_{4}\vee y_{1}y_{3}\vee$ $\vee y_{2}y_{3}y_{5}\vee y_{2}\vee y_{1}\vee y_{3}y_{4}y_{5}\equiv y_{2}\vee y_{1}\vee y_{3}y_{4}y_{5}.$ Function $g\left(w_{1},w_{2},w_{3}\right)=w_{2}\vee w_{1}w_{3}$ we may obtain by direct $2^{3}=8$ questions for the expert. Using 1 and 2 columns we have $f\left(\overline{x}\right)=x_{2}x_{3}\vee x_{1}x_{2}x_{4}\vee x_{1}x_{2}\vee x_{1}x_{3}x_{4}\vee x_{1}x_{3}\vee x_{3}x_{4}\vee x_{3}$ $\vee x_{2}x_{5}\vee x_{1}x_{5}\vee x_{4}x_{5}\equiv x_{1}x_{2}\vee x_{3}\vee\left(x_{2}x_{1}x_{4}\right)x_{5}\equiv$ $\left(w_{2}\vee w_{1}w_{3}\right)\left(y_{1}\vee y_{2}\vee y_{3}y_{4}y_{5}\right)\vee x_{3}\vee$ $\vee\left(y_{1}\vee y_{2}\vee y_{3}y_{4}y_{5}\right)\left(w_{2}\vee w_{1}w_{3}\right)x_{4}x_{5}.$ ## XI Comparison of the expert model with its approximation by learning operator For compare rules discovered by the learning operator (Discovery system) with the expert model we asked the expert to evaluate this rules. Below we present some rules, discovered by Discovery system, and radiologists comments regarding these rules as approximation of his model. > IF TOTAL number of calcifications is more than 30, and VOLUME is more than 5 > $cm^{3}$, and DENSITY of calcifications is moderate, > THEN Malignant. $f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases – $100\%$. Radiologist’s comment - this rule might have promise, but I would consider it risky. > IF VARIATION in shape of calcifications is marked, and NUMBER of > calcifications is between 10 and 20, and IRREGULARITY in shape of > calcifications is moderate, > THEN Malignant. $f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases – $100\%$. Radiologist’s comment - I would trust this rule. > IF variation in SIZE of calcifications is moderate, and variation in SHAPE > of calcifications is mild, and IRREGULARITY in shape of calcifications is > mild, THEN Benign. $f$-criterion significant for $0.05$. Accuracy of diagnosis for test cases – $92.86\%$. Radiologist’s comment - I would trust this rule. ## Acknowledgment This work partially supported by the Russian Science Foundation grant 08-07-00272a and Integration projects of the Siberian Division of the Russian Academy of science grants 47, 111, 119. ## References * [1] Kovalerchuk B. Ya., Perlovsky L. I. _Dynamic logic of phenomena and cognition_. IJCNN, 2008, pp. 3530–3537. * [2] Perlovsky L. I. _Toward physics of the mind: concepts, emotions, consciousness, and symbols_ // Physics of Life Reviews, 3, 2006, pp. 23–55. * [3] Perlovsky L. I. _Neural Networks, Fuzzy Models and Dynamic Logic_ // R. Kohler and A. Mehler, eds., Aspects of Automatic Text Analysis (Festschrift in Honor of Burghard Rieger), Springer, Germany, 2007, pp. 363-386. * [4] Vityaev E. E. The logic of prediction // Mathematical Logic in Asia 2005, Proceedings of the 9th Asian Logic Conference, eds. Goncharov S.S., Downey R. and Ono H., August 16–19, Novosibirsk, Russia, World Scientific Publisher, 2006, pp. 263–276. * [5] Smerdov S. O., Vityaev E. E. Probability, logic & learning synthesis: formalizing prediction concept // Siberian Electronic Mathematical Reports, vol. 9, 2009, pp. 340–365., in russian, english abstract. * [6] Vityaev E. E., Smerdov S. O. New definition of prediction without logical inference // Proceedings of the IASTED international conference on Computational Intelligence (CI 2009), ed. Kovalerchuk B. Ya., August 17 -19, Honolulu, Hawaii, USA, pp. 48–54. * [7] Evgenii Vityaev, Principals of brain activity, supported the functional systems theory by P.K. Anokhin and emotional theory by P.V. Simonov, Neiroifformatics, v.3, N1, 2008, pp. 25-78 * [8] Akexander Demin, Evgenii Vityaev, Logical model of adaptive control system. Neiroifformatics, v.3, N1, 2008, pp. 79-107 * [9] Evgenii Vityaev. Knowledge discovery. Computational cognition. Cognitive processes modeling. Novosibirsk State University, Novosibirsk, 2006. pp.293. * [10] Halpern J. Y. An analysis of first-order logics of probability. In: Artificial Intelligence, 46, 1990, pp. 311–350. * [11] Shiryaev A. N. Probability. Springer, 1995. * [12] Keisler H. J., Chang C. C. Model theory. Elsevier, 1990. * [13] Maltsev A. I. Algebraic systems. Springer-Verlag, 1973. * [14] Lloyd J.W. Foundations of logic programming. Springer-Verlag, 1987. * [15] Hansel G. Sur le nombre des fonctions Boolenes monotones den variables // C. R. Acad. Sci. Paris, vol. 262, 20, 1966, pp. 1088–1090. * [16] Kovalerchuk B. Ya., Vityaev E. E. Data mining in finance: advances in relational and hybrid methods. Kluwer Academic Publisher, 2000. * [17] Kovalerchuk B. Ya., Vityaev E. E., Ruiz J. F. Consistent and complete data and “expert” mining in medicine // Medical data mining and knowledge discovery, Springer, 2001, pp. 238–280. * [18] Kovalerchuk B, Talianski V. Comparison of empirical and computed fuzzy values of conjunction. Fuzzy Sets and Systems 46: 49-53, 1992. * [19] The Probabilistic Mind. Prospects for Bayesian cognitive sciense // Eds. Nick Chater, Mike Oaksford, Oxfor University Press, 2008, pp.536 * [20] Probabilistic models of cognition // Special issue of the journal Trends in cognitive science, v.10, Issue 7, 2006, pp. 287-344 * [21] Scientific Discovery website. http://math.nsc.ru/AP/ScientificDiscovery
arxiv-papers
2010-12-30T13:15:15
2024-09-04T02:49:16.074972
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Evgenii Vityaev, Boris Kovalerchuk, Leonid Perlovsky, Stanislav\n Smerdov", "submitter": "Evgenii Vityaev", "url": "https://arxiv.org/abs/1101.0082" }
1101.0232
# Phase qubits fabricated with trilayer junctions M. Weides1,2, R. C. Bialczak1, M. Lenander1, E. Lucero1, Matteo Mariantoni1, M. Neeley1,3, A. D. O’Connell1, D. Sank1, H. Wang1,4, J. Wenner1, T. Yamamoto1,5, Y. Yin1, A. N. Cleland1, and J. Martinis1 1Department of Physics, University of California, Santa Barbara, CA 93106, USA 2Present address: National Institute of Standards and Technology, Boulder, CO 80305, USA 3Present address: Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA 02420, USA 4Present address: Department of Physics, Zhejiang University, Zhejiang 310027, China 5Present address: Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan martin.weides@nist.gov martinis@physics.ucsb.edu , ###### Abstract We have developed a novel Josephson junction geometry with minimal volume of lossy isolation dielectric, being suitable for higher quality trilayer junctions implemented in qubits. The junctions are based on in-situ deposited trilayers with thermal tunnel oxide, have micron-sized areas and a low subgap current. In qubit spectroscopy only a few avoided level crossings are observed, and the measured relaxation time of $T_{1}\approx 400\;\rm{nsec}$ is in good agreement with the usual phase qubit decay time, indicating low loss due to the additional isolation dielectric. ###### pacs: 74.50.+r, 85.25.Cp, 85.25.-j ## 1 Introduction The energy relaxation time $T_{1}$ of superconducting qubits is affected by dielectric loss, nonequilibrium quasiparticles [1], and charge or bias noise, and varies between a few nano- to several microseconds, depending on qubit type, material, and device layout. Superconducting qubits are commonly based on $\mathrm{Al}$ thin films, and their central element, the non-linear inductor given by a Josephson tunnel junction (JJ), is formed either by overlap [2] or window-type geometries [3]. Qubit spectroscopy reveals coupling to stochastically distributed two-level systems (TLSs) in the tunnel oxide [4, 5, 6, 7, 8] which provide a channel for qubit decoherence. While the physical nature of TLSs is still under debate, their number was shown to decrease with junction size and their density with higher atomic coordination number of the tunnel oxide [3, 9]. The number of coherent oscillations in the qubit is limited by, among other decoherence mechanisms such as nonequilibrium quasiparticles, the _effective_ dielectric loss tangent $\tan\delta_{\rm{eff}}$ [9]. The overlap geometry provides JJs with amorphous barriers with no need for isolation dielectrics, being itself a source for additional TLSs and dielectric losses. The window geometry is used for higher quality, e.g. epitaxial, trilayer JJs with in-situ grown barriers. Besides complex fabrication, they have the drawback of requiring additional isolation dielectrics [10]. The importance of keeping the total dielectric volume in qubits small to reduce the additional loss was shown in Ref. [9]. In this paper we give an overview of our standard technology for junction fabrication, and present an alternative junction based on sputtered trilayer stacks, which provide an intrinsically cleaner tunnel oxide and is well suited for micron-sized trilayer qubit junctions. The so-called _side-wall passivated JJs_ provide contact to the top electrode without adding too much lossy dielectric to the circuitry, which would negatively affect the loss tangent. The trilayer isolation is achieved via an electrolytic process. These novel JJs were realized in a flux-biased phase qubit and characterized by i) current transport measurements on reference junctions and ii) spectroscopy and time- domain measurements of the qubit. By systematically replacing only the Josephson junction, being central to any superconducting qubit, we aim to analyze the loss contributions of this specific element, and, ideally, develop low-loss Josephson junctions for superconducting qubits and improve our qubit performance. We found performance comparable to the current generation of overlap phase qubits. ## 2 Novel geometry Figure 1: (Color online) Schematics of the a) overlap JJ and b) side-wall passivated JJ, offering minimal volume of passivation region. Left (right) part: before (after) the top-layer deposition. After the edge etch in the trilayer stack, the side-wall oxide is grown by anodic oxidation. The trilayer JJ has in-situ grown tunnel oxides to avoid sources of residual impurities. Patterning of the top wiring and etching below the tunnel barrier yields the tunnel junction. Figure 1 depicts the patterning process for our standard overlap (a) and trilayer junctions (b). Our standard process has an oxide layer grown on an ion mill cleaned aluminum edge, which was previously chlorine etched. The top wiring is then etched back below the oxide layer using argon with $\sim 10\%$ chlorine mixture. For the trilayer process, the in-situ sputtered $\mathrm{Al}$-$\mathrm{Al}\mathrm{O}_{x}$-$\mathrm{Al}$ trilayer has a thermally grown tunnel oxide barrier, formed for 10 min at $140\>\rm{mTorr}$ at room temperature. After deposition of the trilayer stack an edge is etched. The bottom electrode of the trilayer stack is isolated from the top electrode wiring by a self-aligned nanometer thin dielectric layer, grown for $\mathrm{Al}$ (or other suitable electrode metals such as $\mathrm{Nb}$) by anodic oxidation [11]. The metallic aluminium serves as partly submerged anode in a liquid electrolytic mixture of $156\;\rm{g}$ ammonium pentaborate, $1120\;\rm{ml}$ ethylene glycol and $760\;\rm{ml}$ $\mathrm{H}_{2}\mathrm{O}$ at room temperature. A gold-covered metal served as cathode and the electric contact was made outside the electrolyte to the anode. By protecting parts of the aluminum electrode with photoresist only a well-defined area was oxidized by passing a constant current through the Al film and converting the metallic surface to its oxide form. The oxide thickness can be controlled by the voltage drop across the electrolyte. After a light ion clean and top wiring deposition the resist is patterned to define the junction area. Finally, the trilayer is etched below the tunnel barrier, yielding Josephson junctions with planar tunnel barrier and isolation dielectric on just one side of the tunnel area. For $\mathrm{Nb}$ junctions a similar patterning process, without minimizing the dielectric loss contribution, was developed using anodic $\mathrm{Nb}$ oxide and covered by $\mathrm{Si}\mathrm{O}_{2}$ [12]. The in- situ grown tunnel oxide avoids sources of residual impurities such as hydrogen, hydroxide or carbon at the interface vicinity, which may remain even after ion-milling in our standard process. These trilayer junctions are fully compatible with our standard process using overlap patterning and no junction side-wall. ### 2.1 Transport Transport measurements on a $\sim 3\>\rm{\mu m^{2}}$ reference junction at $100\;\rm{mK}$ are shown in Fig. 2. The critical current $I_{c}$ is $1.80\>\rm{\mu A}$, with normal resistance $R_{n}=150\rm{\Omega}$ yielding $I_{c}R_{n}=270\rm{\mu V}$, close to the calculated Ambegaokar-Baratoff value of $I_{c}R_{n}=298\rm{\mu V}$ for the measured superconducting gap of $190\;\rm{\mu V}$. The back bending of the voltage close to the gap voltage is attributed to self-heating inside the junction. The retrapping current of $\approx 0.01\cdot I_{c}$ indicates a very small subgap current. The current transport is consistent with tunneling, and we can exclude transport via metallic pinholes, located in the $\sim 5\;\rm{nm}$ thin side-wall dielectric. As a further check, the $I_{c}(T)$ dependence is as expected, see inset in Fig. 2, with a critical temperature $T_{c}$ of $1.2\;\rm{K}$. Figure 2: Current-voltage-characteristic at $100\;\rm{mK}$ and $I_{c}(T)$ dependence (lower inset) of a $3\;\rm{\mu m^{2}}$ side-wall passivated trilayer junction. Top inset: dielectric circuit elements of the junction. The tunnel oxide capacitance $C_{\rm{t}}$ is connected in parallel with the capacitor formed by the side-wall oxide $C_{\rm{sw}}$. ## 3 Measurement The qubit is a flux-biased phase qubit that is coupled via a tunable mutual inductance to the readout-SQUID [13]. The total qubit capacitance $C_{total}$, see upper inset of Fig. 3, is given by the tunnel oxide $C_{\rm{t}}$, the anodic side-wall oxide $C_{\rm{sw}}$ and shunt capacitor $C_{\rm{s}}\approx 1250\;\rm{fF}$ dielectric, provided by a parallel plate capacitor with relative permittivity $\epsilon^{\prime}\simeq 11.8$ made from hydrogenated amorphous silicon (a-Si:H). The measurement process follows the standard phase qubit characterization [14]. ### 3.1 Spectroscopy When operated as a qubit, spectroscopy over a range of more than $2.5\;\rm{GHz}$ revealed clean qubit resonance spectra with just two avoided level crossings of $40$–$50\;\rm{MHz}$ coupling strength (at $6.96$ and $7.32\>\rm{GHz}$, as shown in Fig. 3). The excitation pulse length is $1\;\rm{\mu sec}$, and the qubit linewidth is about $3\>\rm{MHz}$ in the weak power limit. The qubit visibility, measured in a separate experiment, is about $86\%$, which is in the range we found for our standard phase qubits. Qualitatively, the TLS number and coupling strength per qubit is lower than in other trilayer systems [3], that have larger tunnel areas. The TLS density per qubit has roughly the same order of magnitude as in conventional overlap qubits with similar tunnel area dimensions [2]. ### 3.2 Relaxation Figure 3: (Color online) 2D spectroscopy of side-wall passivated trilayer qubit at $25\;\rm{mK}$. Two avoided level crossings due to qubit-TLS coupling are observed at $6.96$ and $7.32\>\rm{GHz}$ (arrows). Top inset: Dielectric circuit schematics of the qubit. Bottom inset: Qubit relaxation measurement. Qubit relaxation measurements via $\pi$ pulse excitation and time-varied delay before readout pulse were obtained when operated outside the avoided level structures. We measured a relaxation time $T_{1}$ of about $400\;\rm{nsec}$, as shown in the lower inset of Fig. 3. This relaxation time is similar to that observed in the overlap qubits, which consistently have $300\textrm{--}500\>\rm{nsec}$ for $\sim 2$-$4\;\rm{\mu m^{2}}$ JJ size. Apart from the change to trilayer junctions, no modification from the previous design was made. ## 4 Loss estimation dielectric elements | | capacitance | loss $\tan\delta_{i}$ | $\frac{C_{i}}{C_{total}}\tan\delta_{i}$ | ---|---|---|---|---|--- | | $[\rm{fF}$] | | | shunt capacitor a-Si:H | $C_{\rm{s}}$ | 1250 | $2\cdot 10^{-5}$ | $1.83\cdot 10^{-5}$ | [15] anodic side-wall oxide | $C_{\rm{sw}}$ | 3.2 | $<1.6\cdot 10^{-3}$ | $<3.7\cdot 10^{-6}$ | [9] tunnel barrier | $C_{\rm{t}}$ | 116 | $<1.6\cdot 10^{-3}$ | $<1.36\cdot 10^{-4}$ | [9] _measured_ $\tan\delta_{m}$ | | | | $6.6\cdot 10^{-5}$ | Table 1: Dielectric parameters for anodic oxide, shunt capacitance, and tunnel barrier. $\tan\delta$ is given for low temperature and low power at microwave frequencies. The capacitance for the tunnel oxide $\mathrm{Al}\mathrm{O}_{x}$ is taken and corrected for $\mathrm{Al}$ electrodes from Ref. [16] for the dimensions given in the text. For qubits the loss tangent is calculated away from TLS resonances, as the losses in small size anodic side-wall oxide and tunnel barrier are smaller than the bulk value considered for the specific loss contribution $\frac{C_{i}}{C_{total}}\tan\delta_{i}$. The measured loss $\tan\delta_{m}$ is a factor 2-3 smaller than $\tan\delta_{\rm{eff}}$, the weighted sum of all specific loss contributions. We estimate the additional dielectric loss due to the sidewall oxide. The _effective_ loss tangent of a parallel combination of capacitors is given by $\tan{\delta_{\rm{eff}}}=\frac{\epsilon^{\prime\prime}_{\rm{eff}}}{\epsilon^{\prime}_{\rm{eff}}}=\frac{\sum\limits_{i}\epsilon^{\prime\prime}_{i}\frac{A_{i}}{d_{i}}}{\sum\limits_{i}\epsilon^{\prime}_{i}\frac{A_{i}}{d_{i}}}=\frac{\sum\limits_{i}C_{i}\tan\delta_{i}}{\sum\limits_{i}C_{i}}$ with $\epsilon^{\prime}_{i}$ and $\epsilon^{\prime\prime}_{i}$ being the real and imaginary part of the individual permittivity for capacitor $i$ with area $A_{i}$ and dielectric thickness $d_{i}$. Now, we discuss the individual loss contributions for all dielectrics. We design the qubit so that the dominant capacitance comes from the shunt capacitor made from a-Si:H, which has a relatively low loss tangent of $2\cdot 10^{-5}$. Including the non-negligible capacitance of the tunnel junction, this gives an effective loss tangent to the qubit of $1.83\cdot 10^{-5}$. Because the junction capacitance is about 10% of the shunting capacitance, the effective junction loss tangent is 10 times less than the loss tangent of the junction oxide. We statistically avoid the effects of two-level systems by purposely choosing to bias the devices away from the deleterious resonances. The loss tangent of the junction is smaller than the value for bulk aluminum oxide, approximately $1.6\cdot 10^{-3}$, and probably smaller than $5\cdot 10^{-5}$ since long energy decay times ($500\;\rm{nsec}$) have been observed for an unshunted junction when operated away from resonances [9]. The anodic side-wall oxide contributes a small capacitance of about 3.2 fF, which can be calculated assuming a parallel plate geometry. Here, we use the dielectric constant $\epsilon^{\prime}=9$ for aluminum oxide, assume an area given by $2\,\mu$m, the width of the overlap, multiplied by $0.1\mu$m the thickness of the base layer, and estimate the thickness of the oxide $\simeq 5\,$nm as determined by the anodic process [11]. The anodic oxide is assumed to have a bulk loss tangent of $1.6\cdot 10^{-3}$ [15], which gives a net qubit loss contribution of $3.7\cdot 10^{-6}$, about 5 times lower than for the a-Si:H capacitor. Note that we expect the loss from this capacitance to be even lower because of statistical avoidance of the TLS loss [9]. The small volume of the capacitor, equivalent to a $\sim 0.5\,\mu\textrm{m}^{2}$ volume tunnel junction, implies that most biases do not put the qubit on resonance with two-level systems in the anodic oxide. ## 5 Qubit lifetime and effective loss tangent From the measured energy decay time $T_{1}=400\;\rm{nsec}$ , we determine the loss tangent of the qubit to be $\delta_{m}=(T_{1}\;\omega_{10})^{-1}\approx 6.6\cdot 10^{-5}$, using a qubit frequency $\omega_{10}/2\pi=6\;\rm{GHz}$. This is 3-5 times larger than our estimation of our dielectric losses, as shown in Table 1. We believe the qubit dissipation mechanism comes from some other energy loss sources as well, such as non-equilibrium quasiparticles [1]. ## 6 Conclusion In conclusion, we have shown that the use of a anodic oxide, self-aligned to the junction edge, does not degrade the coherence of present phase qubits [14]. We found performance comparable to the current generation of overlap phase qubits. The new junction geometry may provide a method to integrate submicron sized, superior quality junctions (lower TLS densities) grown, for example, by MBE epitaxy to eliminate the need for shunt dielectrics. Also, our nanometer-thin, three dimensional-conform anodic passivation layer can be replaced by a self- aligned isolation dielectric at the side-wall, which could be used for all types of trilayer stacks. Devices were made at the UCSB Nanofabrication Facility, a part of the NSF- funded National Nanotechnology Infrastructure Network. The authors would like to thank D. Pappas for stimulating discussions. This work was supported by IARPA under grant W911NF-04-1-0204. M.W. acknowledges support from AvH foundation and M.M. from an Elings Postdoctoral Fellowship. ## References * [1] John M. Martinis, M. Ansmann, and J. Aumentado. Energy decay in superconducting josephson-junction qubits from nonequilibrium quasiparticle excitations. Phys. Rev. Lett., 103(9):097002, Aug 2009. * [2] Matthias Steffen, M. Ansmann, R. McDermott, N. Katz, Radoslaw C. Bialczak, Erik Lucero, Matthew Neeley, E. M. Weig, A. N. Cleland, and John M. Martinis. State tomography of capacitively shunted phase qubits with high fidelity. Phys. Rev. Lett., 97(5):050502, Aug 2006. * [3] Jeffrey S Kline, Haohua Wang, Seongshik Oh, John M Martinis, and David P Pappas. Josephson phase qubit circuit for the evaluation of advanced tunnel barrier materials. Supercond. Sci. Technol., 22(1):015004, 2009. * [4] R. W. Simmonds, K. M. Lang, D. A. Hite, S. Nam, D. P. Pappas, and John M. Martinis. Decoherence in josephson phase qubits from junction resonators. Phys. Rev. Lett., 93(7):077003, Aug 2004. * [5] A. Lupaşcu, P. Bertet, E. F. C. Driessen, C. J. P. M. Harmans, and J. E. Mooij. One- and two-photon spectroscopy of a flux qubit coupled to a microscopic defect. Phys. Rev. B, 80(17):172506, Nov 2009. * [6] Jürgen Lisenfeld, Clemens Müller, Jared H. Cole, Pavel Bushev, Alexander Lukashenko, Alexander Shnirman, and Alexey V. Ustinov. Rabi spectroscopy of a qubit-fluctuator system. Phys. Rev. B, 81(10):100511, Mar 2010. * [7] P. Bushev, C. Müller, J. Lisenfeld, J. H. Cole, A. Lukashenko, A. Shnirman, and A. V. Ustinov. Multiphoton spectroscopy of a hybrid quantum system. Phys. Rev. B, 82(13):134530, Oct 2010. * [8] F. Deppe, M. Mariantoni, E. P. Menzel, S. Saito, K. Kakuyanagi, H. Tanaka, T. Meno, K. Semba, H. Takayanagi, and R. Gross. Phase coherent dynamics of a superconducting flux qubit with capacitive bias readout. Phys. Rev. B, 76(21):214503, Dec 2007. * [9] John M. Martinis, K. B. Cooper, R. McDermott, Matthias Steffen, Markus Ansmann, K. D. Osborn, K. Cicak, Seongshik Oh, D. P. Pappas, R. W. Simmonds, and Clare C. Yu. Decoherence in josephson qubits from dielectric loss. Phys. Rev. Lett., 95(21):210503, Nov 2005. * [10] A. Barone and G. Paterno. Physics and Applications of the Josephson Effect. John Wiley & Sons, 1982. * [11] H. Kroger, L. N. Smith, and D. W. Jillie. Selective niobium anodization process for fabricating josephson tunnel-junctions. Appl. Phys. Lett., 39:280, 1981. * [12] F. Müller, H. Schulze, R. Behr, J. Kohlmann, and J. Niemeyer. The Nb-Al technology at PTB: a common base for different types of Josephson voltage standards. Physica C, 354:66, 2001. * [13] Matthew Neeley, M. Ansmann, Radoslaw C. Bialczak, M. Hofheinz, N. Katz, Erik Lucero, A. O’Connell, H. Wang, A. N. Cleland, and John M. Martinis. Transformed dissipation in superconducting quantum circuits. Phys. Rev. B, 77(18):180508, May 2008. * [14] John M. Martinis. Superconducting phase qubits. Quantum Inf. Process., 8:81, 2009. * [15] Aaron D. O’Connell, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, Erik Lucero, C. McKenney, M. Neeley, H. Wang, E. M. Weig, A. N. Cleland, and J. M. Martinis. Microwave dielectric loss at single photon energies and millikelvin temperatures. Appl. Phys. Lett., 92(11):112903, 2008. * [16] H. S. J. van der Zant, R. A. M. Receveur, T. P. Orlando, and A. W. Kleinsasser. One-dimensional parallel josephson-junction arrays as a tool for diagnostics. Appl. Phys. Lett., 65(16):2102–2104, 1994.
arxiv-papers
2010-12-31T11:54:34
2024-09-04T02:49:16.087858
{ "license": "Public Domain", "authors": "M. Weides, R. C. Bialczak, M. Lenander, E. Lucero, Matteo Mariantoni,\n M. Neeley, A. D. O'Connell, D. Sank, H. Wang, J. Wenner, T. Yamamoto, Y. Yin,\n A. N. Cleland, and J. Martinis", "submitter": "Martin Weides", "url": "https://arxiv.org/abs/1101.0232" }
1101.0470
SNSN-323-63 $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ from B-factories and Tevatron Gerald Eigen representing the BABAR collaboration 111Work supported by the Norwegian Research Council. Department of Physics and Technology University of Bergen, 5007 Bergen, NORWAY > BABAR and Belle measurements of branching fractions, rate asymmetries and > angular observables in the decay modes $B\rightarrow > K^{(*)}\ell^{+}\ell^{-}$ are reviewed and new results from CDF on > $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ branching fractions and angular > observables are discussed. A first search for $B^{+}\rightarrow > K^{+}\tau^{+}\tau^{-}$ is presented. > PRESENTED AT > > > > > CKM workshop 2010 > Warwick, UK, September 06–10, 2010 ## 1 Introduction The decays $b\rightarrow s\ell^{+}\ell^{-}$, where $\ell^{+}\ell^{-}$ is an $e^{+}e^{-},\mu^{+}\mu^{-}$ or $\tau^{+}\tau^{-}$ pair, are flavor-changing neutral current (FCNC) processes, which are forbidden in the Standard Model (SM) at tree level but are allowed to proceed via electroweak loops and weak box diagrams. An effective Hamiltonian is used to calculate decay amplitudes [1], which depend on three effective Wilson coefficients, $C_{7}^{eff}$, $C_{9}^{eff}$, and $C_{10}^{eff}$. The first is extracted from the $B\rightarrow X_{s}\gamma$ branching fraction, the latter two respectively represent the vector and axial vector part of the weak penguin and box diagrams. New Physics effects involve new loops that interfere with the SM processes modifying the measured values of $C_{7}^{eff}$, $C_{9}^{eff}$, and $C_{10}^{eff}$ with respect to the SM predictions [2]. In addition, scalar and pseudoscalar processes may contribute that introduce new Wilson coefficients $C_{s}$ and $C_{p}$ that are forbidden in the SM. Thus, it is important to measure many observables in order to overconstrain the complex Wilson coefficients [3]. These electroweak penguin modes contribute in probing New Physics at a scale of a few TeV [4]. In this review, we focus on exclusive decays presenting results from BABAR , Belle and CDF. The data samples are based on luminosities of $349~{}\rm fb^{-1}$, $605~{}\rm fb^{-1}$ and $4.4~{}\rm fb^{-1}$ corresponding to 384 million $B\overline{B}$ events, 656 Million $B\overline{B}$ events and $2\times 10^{10}~{}b\overline{b}$ events, respectively. ## 2 Selection of $B\rightarrow K^{(*)}e^{+}e^{-}$ and $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ Events BABAR and Belle fully reconstruct ten $B\rightarrow K^{(*)}e^{+}e^{-}$ and $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ final states, in which a $K^{+},K^{0}_{S},K^{+}\pi^{-},K^{+}\pi^{0}$ or $K^{0}_{S}\pi^{+}$ recoils against the lepton pair***Charge conjugation is implied unless otherwise stated., while CDF reconstructs $K^{+}\mu^{+}\mu^{-}$ and $K^{+}\pi^{-}\mu^{+}\mu^{-}$ final states. BABAR (Belle) selects lepton candidates with momenta $p_{e}>0.3(0.4)~{}\rm GeV/c$ and $p_{\mu}>0.7(0.7)~{}\rm GeV/c$. BABAR and Belle require good particle identification (PID) for $e,\mu,K$, and $\pi$, and select $K^{0}_{S}$ in the $\pi^{+}\pi^{-}$ channel. CDF requires muons with $p_{T}(\mu)>0.4~{}\rm GeV/c$, kaons and pions with $p_{T}(K,\pi)>\rm 1~{}GeV/c$ and $B$-mesons with $p_{T}(B)>6~{}\rm GeV/c$. Both, muons and hadrons must have good PID and the muon pair must originate from a secondary vertex. All three experiments suppress combinatorial $B\overline{B}$ and $q\overline{q}$ continuum backgrounds ($q=u,d,s,c)$. Here, the leptons dominantly originate from semileptonic $b$ and $c$ decays. BABAR trains neural networks (NN) using event shape variables, vertex information, missing energy, and lepton separation near the interaction region (IR) optimized in each mode and each $q^{2}$ bin†††This is the squared momentum transfer into the dilepton system.. Belle trains a Fisher discriminant using event shape variables, missing mass, B flavor tagging, and lepton separation in z near the IR. CDF trains NNs using vertex information, the angle between the signed vertex displacement with respect to the B momentum, and the $\mu$ separation. BABAR and Belle select signal candidates using the beam-energy substituted mass $m_{ES}=\sqrt{E^{*2}_{beam}-p^{*2}_{B}}$ and the energy difference $\Delta E=E^{*}_{B}-E^{*}_{beam}$, where $E^{*}_{beam},E^{*}_{B}$ and $p^{*}_{B}$ are the beam energy, B-meson energy and B-meson momentum in the $\mathchar 28935\relax(4S)$ center-of-mass frame, respectively. BABAR extracts the signal yield from a one-dimensional unbinned extended maximum log-likelihood fit in $m_{ES}$, while Belle performs a one (two) dimensional unbinned extended maximum log-likelihood fit in $m_{ES}$ (and $m_{K\pi}$) for $K^{(*)}\ell^{+}\ell^{-}$ modes. CDF selects signal candidates from an unbinned maximum log-likelihood fit in the B invariant-mass distribution. All experiments reject events in the $J/\psi$ and $\psi(2S)$ mass regions and require that $K\mu$ and $K\pi\mu$ masses are not consistent with a $D$ mass to reject background from $B\rightarrow DX$ decays. The rejected charmonium events are used as control samples for various cross checks. ## 3 Results for $B\rightarrow K^{(*)}e^{+}e^{-}$ and $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$ Modes Figure 1 (left) shows total branching fractions for $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ ($e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ modes combined) [6, 7, 8] and $B\rightarrow X_{s}\ell^{+}\ell^{-}$[9, 7] in comparison to the SM predictions [5]. The individual exclusive measurements are summarized in Table 1. The Belle inclusive measurement is a recent update based on a luminosity of $\rm 605~{}fb^{-1}$, yielding ${\cal B}(B\rightarrow X_{s}\ell^{+}\ell^{-})=3.33\pm 0.8^{+0.19}_{-0.24})\times 10^{-6}$ [10]. The partial branching fractions measured in the three experiments are also consistent with the SM predictions. Experiment | Mode | ${\cal B}~{}[10^{-6)}]$ | ${\cal A}_{CP}$ | ${\cal R}_{K^{(*)}}$ ---|---|---|---|--- BABAR [6] | $K\ell^{+}\ell^{-}$ | $0.394^{+0.073}_{-0.069}\pm 0.02$ | $-0.18^{+0.18}_{-0.18}\pm 0.01$ | $0.96^{+0.44}_{-0.34}\pm 0.05$ BABAR [6] | $K^{*}\ell^{+}\ell^{-}$ | $1.11^{+0.19}_{-0.18}\pm 0.07$ | $-0.01^{+0.16}_{-0.15}\pm 0.01$ | $1.10^{+0.42}_{-0.32}\pm 0.07$ Belle [7] | $K\ell^{+}\ell^{-}$ | $0.48^{+0.05}_{-0.04}\pm 0.03$ | $0.04\pm 0.1\pm 0.02$ | $1.03\pm 0.19\pm 0.06$ Belle [7] | $K^{*}\ell^{+}\ell^{-}$ | $1.07^{+0.11}_{-0.10}\pm 0.09$ | $-0.10\pm 0.1\pm 0.01$ | $0.83\pm 0.17\pm 0.8$ CDF [8] | $K\mu^{+}\mu^{-}$ | $0.38^{+0.05}_{-0.05}\pm 0.03$ | | CDF [8] | $K^{*}\mu^{+}\mu^{-}$ | $1.06^{+0.14}_{-0.14}\pm 0.09$ | | Table 1: Branching fractions, $C\\!P$ asymmetries and lepton flavor ratios for $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ modes in the entire $q^{2}$ region from BABAR, Belle, and CDF. Uncertainties are statistical and systematic, respectively. Figure 1: (Left) Total branching fractions measurements of $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ and $B\rightarrow X_{s}\ell^{+}\ell^{-}$ modes from BABAR (red dots), Belle (blue triangles) and CDF (magenta squares) in comparison to the SM prediction (grey-shaded region). For BABAR and Belle, $\ell^{+}\ell^{-}$ is a combination of $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$ modes, for CDF it is $\mu^{+}\mu^{-}$. (Right) Isospin asymmetry measurements for $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ versus $q^{2}$ from BABAR (black squares, blue dots) and Belle (red triangles, green triangles). Rate asymmetries are more precisely measured than branching fractions, since many uncertainties cancel [11]. The isospin asymmetry [12] ${\cal A}_{I}(q^{2})=\frac{d{\cal B}(B^{0}\rightarrow K^{(*)0}\ell^{+}\ell^{-})/dq^{2}-(\tau_{B^{0}}/\tau_{B^{+}})d{\cal B}(B^{+}\rightarrow K^{(*)+}\ell^{+}\ell^{-})/dq^{2}}{d{\cal B}(B^{0}\rightarrow K^{(*)0}\ell^{+}\ell^{-})/dq^{2}+(\tau_{B^{0}}/\tau_{B^{+}})d{\cal B}(B^{+}\rightarrow K^{(*)+}\ell^{+}\ell^{-})/dq^{2}},$ (1) corrected for the different $B^{0}$ and $B^{+}$ lifetimes ($\tau_{B^{0}}/\tau_{B^{+}}$), is expected to be small in the SM ($d{\cal A}_{I}(q^{2})/dq^{2}$ is $-0.01$ for $q^{2}=2.7-6~{}\rm GeV^{2}/c^{4}$ after dropping from $\simeq 0.075$ at $q^{2}=0.1~{}\rm GeV^{2}/c^{4}$ and crossing zero near $q^{2}=1.7~{}\rm GeV^{2}/c^{4}$) [12]. Figure 1 (right) shows the BABAR and Belle ${\cal A}_{I}$ measurements for different $q^{2}$ regions. The $q^{2}$ integrated isospin asymmetry and ${\cal A}_{I}$ for $q^{2}$ values above the $J/\psi$ are consistent with the SM prediction. Below the $J/\psi$, however, BABAR observes a negative ${\cal A}_{I}$ that deviates significantly from the SM prediction ($3.9\sigma$ from ${\cal A}_{I}=0$) . For models in which the sign in $C^{eff}_{7}$ is flipped with respect to the value in the SM, a small negative ${\cal A}_{I}$ is expected [12, 13], but it is too small to explain the BABAR measurement. For low $q^{2}$, the Belle results are consistent with both BABAR and the SM. In the SM, the direct $C\\!P$ asymmetry ${\cal A}_{CP}=\frac{{\cal B}(\overline{B}\rightarrow K^{(*)}\ell^{+}\ell^{-})-{\cal B}(B\rightarrow K^{(*)}\ell^{+}\ell^{-})}{{\cal B}(\overline{B}\rightarrow K^{(*)}\ell^{+}\ell^{-})+{\cal B}(B\rightarrow K^{(*)+}\ell^{+}\ell^{-})}.$ (2) is expected to be ${\cal O}(10^{-3})$, and new physics at the electroweak scale may provide significant enhancements [14]. BABAR performs a simultaneous fit to $B^{+}\rightarrow K^{+}\ell^{+}\ell^{-}$ and $B\rightarrow K^{*}\ell^{+}\ell^{-}$ modes. The results summarized in Table 1 together with Belle’s measurements are consistent with the SM expectations. In the SM, the lepton flavor ratios ${\cal R}_{K}={\cal B}(B\rightarrow K\mu^{+}\mu^{-})/{\cal B}(B\rightarrow Ke^{+}e^{-})$ and ${\cal R}_{K^{*}}={\cal B}(B\rightarrow K^{*}\mu^{+}\mu^{-})/{\cal B}(B\rightarrow K^{*}e^{+}e^{-})$ integrated over all $q^{2}$ are predicted to be one and 0.75, respectively. The theoretical uncertainties are just a few percent. For example, in two-Higgs-doublet models the presence of a SUSY Higgs might give $\sim 10\%$ corrections to ${\cal R}_{K^{(*)}}$ for large $\tan\beta$ [13].The BABAR and Belle measurements summarized in Table 1 are consistent with the SM expectations. The $B\rightarrow K^{*}\ell^{+}\ell^{-}$ angular distribution depends on three angles: $\theta_{K}$, the angle between the K momentum and the B momentum in the $K^{*}$ rest frame, $\theta_{\ell}$, the angle between the $\ell^{+}(\ell^{-})$ momentum and the $B(\overline{B})$ momentum in the $\ell^{+}\ell^{-}$ rest frame, and $\phi$, the angle between the two decay planes. The angular distribution involves 12 $q^{2}$-dependent coefficients $J_{i}$ [15, 16] that can be extracted from a full angular fit in individual bins of $q^{2}$. Since large data samples are necessary for this study, BABAR , Belle and CDF have analyzed only the one-dimensional angular distributions $\displaystyle W(\cos\theta_{K})=\frac{3}{2}{\cal F}_{L}\cos^{2}\theta_{K}+\frac{3}{4}(1-{\cal F}_{L})\sin^{2}\theta_{K},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ (3) $\displaystyle W(\cos\theta_{\ell})=\frac{3}{4}{\cal F}_{L}\sin^{2}\theta_{\ell}+\frac{3}{8}(1-{\cal F}_{L})(1+\cos^{2}\theta_{\ell})+{\cal A}_{FB}\cos\theta_{\ell},$ (4) where ${\cal F}_{L}$ is the $K^{*}$ longitudinal polarization and ${\cal A}_{FB}$ is the lepton forward-backward asymmetry. While Belle and CDF measure ${\cal F}_{L}$ and ${\cal A}_{FB}$ in six $q^{2}$ bins, BABAR measured ${\cal F}_{L}$ and ${\cal A}_{FB}$ in two $q^{2}$ bins due to the limited data sample. An update with the full BABAR data set in six $q^{2}$ bins is in progress. The measured $m_{ES}$ and angular distributions are fitted with signal, combinatorial background and peaking background components. After determining the signal yield from the $m_{ES}$ spectrum, ${\cal F}_{L}$ is extracted from a fit to the $\cos\theta_{K}$ distribution for fixed signal yield. Finally, ${\cal A}_{FB}$ is extracted from the $\cos\theta_{\ell}$ distribution for fixed signal yield and fixed ${\cal F}_{L}$. Figure 2 shows the BABAR, Belle, and CDF results for ${\cal F}_{L}$ (left) and ${\cal A}_{FB}$ (right) in comparison to the SM prediction (lower red curve) [18] and for flipped-sign $C^{eff}_{7}$ models (upper blue curve) [20, 23]. In the SM, ${\cal A}_{FB}$ is negative for small $q^{2}$, crosses zero at $q^{2}_{0}=(4.2\pm 0.6)~{}\rm GeV^{2}/c^{4}$ and is positive for large $q^{2}$, while for flipped-sign $C^{eff}_{7}$ models ${\cal A}_{FB}$ is positive for all $q^{2}$. Table 2 summarized the ${\cal F}_{L}$ and ${\cal A}_{FB}$ measurements from $B\rightarrow K^{*}\ell^{+}\ell^{-}$ in the low $q^{2}$ region in comparison to the SM prediction. For ${\cal F}_{L}$, the three measurements are consistent with each other and the SM prediction. For ${\cal A}_{FB}$, the three measurements are in good agreement. Though they are in better agreement with the flipped-sign $C^{eff}_{7}$ model, they are consistent with the SM prediction. For $B\rightarrow K\ell^{+}\ell^{-}$, ${\cal A}_{FB}$ is consistent with zero as expected in the SM. Experiment | $q^{2}$ bin $\rm[GeV^{2}/c^{4}]$ | ${\cal F}_{L}$ | ${\cal A}_{FB}$ ---|---|---|--- BABAR [17] | 0.1-6.25 | $0.35\pm 0.16\pm 0.04$ | $0.24^{+0.18}_{-0.23}\pm 0.05$ Belle [7] | 1-6 | $0.67\pm 0.23\pm 0.04$ | $0.26^{+0.27}_{-0.30}\pm 0.07$ CDF [8] | 1-6 | $0.5^{+0.27}_{-0.30}\pm 0.04$ | $0.43^{+0.36}_{-0.37}\pm 0.06$ SM [24] | 1-6 | $0.73^{+0.13}_{-0.23}$ | $-0.05^{+0.03}_{-0.04}$ Table 2: BABAR, Belle, and CDF measurements of ${\cal F}_{L}$ and ${\cal A}_{FB}$ from $B\rightarrow K^{*}\ell^{+}\ell^{-}$ modes in the low $q^{2}$ region. Figure 2: (left) Measurements of ${\cal F}_{L}$ and (right) Measurements of ${\cal A}_{FB}$ in $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ modes by BABAR (black squares), Belle ( brown dots) and CDF (green triangles). The SM prediction (flipped-sign $C^{eff}_{7}$ model) is shown by the upper red (lower blue) curve for ${\cal F}_{L}$ and the lower red (upper blue) curve for ${\cal A}_{FB}$. ## 4 Search for $B^{+}\rightarrow K^{+}\tau^{+}\tau^{-}$ In the SM, the $q^{2}$ dependence of the $B\rightarrow X_{s}\tau^{+}\tau^{-}$ decay rate has a shape similar to that of $B\rightarrow X_{s}\mu^{+}\mu^{-}$ in the high $q^{2}$ region. The $B^{+}\rightarrow K^{+}\tau^{+}\tau^{-}$ branching fraction is predicted to be $\sim 2\times 10^{-7}$ in the SM, which is $50-60\%$ of the total inclusive branching fraction [21]. Enhancements are predicted in models beyond the SM. In the next-to-minimal supersymmetric models (NMSSM), for example, the rate may be enhanced by the squared tau-to- muon mass ratio $(m_{\tau}/m_{\mu})^{2}\sim 280$. Since signal final states contain 2-4$\nu$, a different analysis strategy is needed here to control backgrounds. BABAR has performed the first search for $B^{+}\rightarrow K^{+}\tau^{+}\tau^{-}$ using an integrated luminosity of $423~{}\rm fb^{-1}$ which corresponds to 465 $B\overline{B}$ events. The recoiling (”tag”) $B$ is reconstructed in many hadronic final states, $B^{-}\rightarrow D^{(*)0,+}X$, where $X$ represents up to six hadrons ($\pi^{\pm},\pi^{0},K^{\pm},K^{0}_{S}$). Using $m_{ES}$ and $\Delta E$ the tag is selected with an efficiency of $\sim 0.2\%$. The single-prong $\tau$ decays $\tau\rightarrow e\nu\overline{\nu},\tau\rightarrow\mu\nu\overline{\nu}$ and $\tau\rightarrow\pi\nu$ are selected as signal modes. Thus, signal candidates are required to have only three charged particles of which one is an identified kaon with charge opposite to the tag $B$ and $0.44<p_{K}<1.4~{}\rm GeV/c$ in the center-of-mass frame. The two remaining particles must have opposite charge, be consistent with the signal $\tau$ decays, have $p<1.59~{}\rm GeV/c$ and a mass $M_{pair}<2.89~{}\rm GeV/c^{2}$. Further requirements are $q^{2}=(\vec{p}_{\mathchar 28935\relax(4S)}-\vec{p}_{tag}-\vec{p}_{K})^{2}/c^{2}>14.23~{}\rm GeV^{2}/c^{4}$, a missing energy ($i.e.$ the energy carried off by neutrinos estimated as the difference between $\mathchar 28935\relax(4S)$ energy and that of all observed particles) of $1.39<E_{miss}<3.38~{}\rm GeV$, and neutral energy deposited in the electromagnetic calorimeter $E_{extra}<0.74~{}\rm GeV$. Continuum background is suppressed by $|\cos\theta_{T}|<0.8$, where $\theta_{T}$ is the opening angle between the thrust axis of the tag and that of the rest of the event. The largest remaining background originates from $B^{+}\rightarrow D^{0}X^{+}$, which is suppressed by combining the signal $K^{+}$ with the $\tau$ daughter of opposite charge assigned the $\pi$ mass hypothesis and requiring a mass $M_{K\pi}>1.96~{}\rm GeV/c^{2}$. BABAR observes 47 events with an expected background of $64.7\pm 7.3$ events. Including systematic uncertainties a branching fraction upper limit of ${\cal B}(B\rightarrow K^{+}\tau^{+}\tau^{-})<3.3\times 10^{-3}$ is set at $90\%$ confidence level (CL). ## 5 Conclusion BABAR and Belle have measured branching fractions, rate asymmetries and angular observables in $B\rightarrow K^{(*)}\ell^{+}\ell^{-}$ final states. Recently, CDF contributed new measurements on branching fractions and angular observables in $B\rightarrow K^{(*)}\mu^{+}\mu^{-}$. Except for the isospin asymmetry at low values of $q^{2}$ all other measurements are consistent with the SM, though ${\cal F}_{L}$ and ${\cal A}_{FB}$ agree also with the flipped- sign $C^{eff}_{7}$ model. BABAR has performed the first search for $B^{+}\rightarrow K^{+}\tau^{+}\tau^{-}$ setting a branching fraction upper limit of ${\cal B}(B^{+}\rightarrow K^{+}\tau^{+}\tau^{-})<3.3\times 10^{-3}$ at $90\%~{}CL$. Although all experiments are expected to update results with the final data sets, significant improvement in precision will come from LHCb and the Super B-factories. In these new experiments, sufficiently large data samples will be collected to measure the full angular distribution from which the 12 observables $J_{i}$ [15] can be measured with high precision in different bins of $q^{2}$. In turn, the Wilson coefficients can be determined with high precision to reveal small discrepancies with respect to the SM predictions [3, 23]. ACKNOWLEDGEMENTS I would like to thank my BABAR colleague K. Flood for useful discussions. This work has been supported by the Norwegian Research Council. ## References * [1] G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996); C. Bobeth, M. Misiak and J. Urban, Nucl. Phys. B574, 291 (2000); H.H Asatryan $et~{}al.$, Phys. Rev. D65, 034009 (2002); Phys. Lett. B507, 162, (2001); G. Hiller and F.Krüger, Phys.Rev. D69, 074020 (2004); M. Beneke, Th. Feldmann, and D. Seidel; Nucl. Phys.B612, 25 (2001);M. Beneke, Th. Feldmann, and D. Seidel; Eur.Phys.J. C41, 173 (2005). * [2] G. Burdman, Phys. Rev. D52, 6400 (1995); J. L. Hewett and J. D. Wells, Phys. Rev. D55, 5549 (1997); W. J. Li, Y. B. Dai and C. S. Huang, Eur. Phys. J. C40, 565 (2005); Y. G. Xu, R. M. Wang and Y. D. Yang, Phys. Rev. D74, 114019 (2006); P. Colangelo et al., Phys. Rev. D73, 115006 (2006); C.-H. Chen and C.Q. Geng, Phys. Rev. D 66 094018 (2002);C. Bobeth et al, Phys. Rev. D64 074014 (2001). * [3] K.S.M. Lee et al., Phys. Rev. D75, 034016 (2007). * [4] G. Isidori, Y. Nir, G. Prerez, arXiv:1002.0900 (2010). * [5] A. Ali, E. Lunghi, C. Greub and G. Hiller, Phys. Rev. D 66, 034002 (2002). * [6] B. Aubert et al. (BABAR collaboration), Phys. Rev. Lett.102, 091803 (2009). * [7] J.T. Wei et al. (Belle collaboration), Phys. Rev. Lett.103, 171801 (2009). * [8] T. Aaltonen et al. (CDF collaboration), CDF note 10047 (2010). * [9] B. Aubert et al. (BABAR collaboration), Phys. Rev. Lett.93, 081862 (2004). * [10] C.C.Chiang (Belle collaboration), talk at ICHEP10 (2010). * [11] F. Krüger, L. M. Sehgal, N. Sinha and R. Sinha, Phys. Rev. D61, 114028 (2000), [Erratum-ibid. D63, 019901 (2001)]. * [12] T. Feldmann and J. Matias, JHEP 0301, 074 (2003). * [13] Q. S. Yan, C. S. Huang, W. Liao and S. H. Zhu, Phys. Rev. D 62, 094023 (2000). * [14] C. Bobeth, G. Hiller and G. Piranishvili, JHEP 0807, 106 (2008). * [15] F. Krüger et al., Phys. Rev. D61, 114028 (2000); Erratum-ibid D63, 019901 (2001). * [16] C.S. Kim et al., Phys. Rev. D 62, 034013 (2000). * [17] B. Aubert et al. (BABAR collaboration), Phys. Rev. D79, 031102 (2009). * [18] G. Buchalla $et~{}al.$, Phys. Rev. D63, 014015 (2001). * [19] G. Buchalla $et~{}al.$, Phys. Rev. D63, 014015 (2001). * [20] A. Hovhannisyan, W. S. Hou and N. Mahajan, Phys. Rev. D 77, 014016 (2008). * [21] J.L. Hewett, Phys. Rev. D53, 4964 (1995). * [22] K. Flood, talk at the Int. Conf. on HEP, Paris July 22-28 (2010). * [23] F. Krüger and J. Matias, Phys. Rev. D71, 094009 (2005). * [24] C. Bobeth, G. Hiller and D. van Dyk, JHEP 1007, 098 (2010).
arxiv-papers
2011-01-03T08:44:20
2024-09-04T02:49:16.098438
{ "license": "Public Domain", "authors": "Gerald Eigen", "submitter": "Gerald Eigen", "url": "https://arxiv.org/abs/1101.0470" }
1101.0649
aainstitutetext: Department of Physics, Kobe University, 1-1 Rokkodai-Cho, Nada-Ku, Kobe 657-8501, JAPAN # Higgs production and decay processes via loop diagrams in various 6D Universal Extra Dimension Models at LHC Kenji Nishiwaki nishiwaki@stu.kobe-u.ac.jp ###### Abstract We calculate loop-induced Higgs production and decay processes which are relevant for the LHC in various six-dimensional Universal Extra Dimension models. More concretely, we focus on the Higgs production through gluon fusion and the Higgs decay into two photons induced by loop diagrams. They are one- loop leading processes and the contribution of Kaluza-Klein particles is considered to be significant. These processes are divergent in six dimensions. Therefore, we employ a momentum cutoff, whose size is fixed from the validity of perturbative calculation through naive dimensional analysis. In these six- dimensional Universal Extra Dimension models, the Higgs production cross section through gluon fusion is highly enhanced and the Higgs decay width into two photons is suppressed. In particular in the case of the compactification on Projective Sphere, these effects are remarkable. The deviation of the $h^{(0)}\rightarrow 2\gamma$ signal from the prediction of the Standard model is much greater than that in the case of the five-dimensional minimal UED model. We also consider threshold corrections in the two processes and these effect are noteworthy even when we take a higher cutoff and/or a heavy KK scale. Comparing our calculation to the recent LHC results which were published at the Lepton-Photon 2011 and at the December of 2011 is performed briefly. ###### Keywords: Universal Extra Dimension model, Collider Physics ††arxiv: 1101.0649 [hep-ph]††dedication: KOBE-TH-10-04 ## 1 Introduction After a long shutdown, the LHC (Large Hadron Collider) restarted and new era of particle physics comes. Stimulated by the advent of two renowned works ArkaniHamed:1998rs ; Randall:1999ee , phenomenology in extra dimension has been well studied. Universal Extra Dimension (UED) is one of the interesting possibility along this direction and has been studied very well. In this model, all the fields describing particles of the Standard Model (SM) propagate in the bulk space.111 This possibility is first considered within string theory context Antoniadis:1990ew . The minimal UED (mUED) model is constructed with one extra spacial dimension of orbifold $S^{1}/Z_{2}$ Appelquist:2000nn . This orbifold imposes the identification between the extra spacial coordinate $y$ and $-y$ and there are two fixed points at $y=0,\pi R$, where $R$ is the radius of $S^{1}$. Due to this identification four- dimensional (4D) chiral fermions describing the SM fermions appear. One of the interesting points of UED model is that the constraints from the current experiments are very loose. The Kaluza-Klein (KK) mass scale ${M_{\text{KK}}}$, which is defined by the inverse of the compactification radius $R$, is constrained Appelquist:2000nn ; Agashe:2001ra ; Agashe:2001xt ; Appelquist:2001jz ; Appelquist:2002wb ; Oliver:2002up ; Chakraverty:2002qk ; Buras:2002ej ; Colangelo:2006vm ; Gogoladze:2006br in the mUED case. In UED model, the zero mode profile takes constant value and the overlap integral between zero modes and KK modes does not generate large deviation from the SM result. Therefore we can take the lower KK mass scale than in the other types of extra dimensional models. In addition, the existence of dark matter candidate is naturally explained by the KK parity, which is the remnant of the translational invariance along the extra spacial direction. The particle cosmology in the five-dimensional (5D) UED models has been studied strenuously Cheng:2002ej ; Servant:2002aq ; Kakizaki:2005en ; Matsumoto:2005uh ; Burnell:2005hm ; Kakizaki:2006dz ; Kong:2005hn ; Matsumoto:2007dp ; Kakizaki:2005uy ; Belanger:2010yx ; Hisano:2010yh . The collider signature of the 5D UED models is similar to the one of the supersymmetric theory with neutralino dark matter Cheng:2002ab . The discrimination between these models is also well studied Datta:2005zs ; Matsumoto:2009tb . And another thing, UED models with two spacial dimensions have been studied energetically. Six-dimensional (6D) UED models have remarkable theoretical properties, for example, prediction of the number of matter generations imposed by the condition of (global) anomaly cancellation Dobrescu:2001ae , ensuring proton stability Appelquist:2001mj , generating electroweak symmetry breaking ArkaniHamed:2000hv ; Hashimoto:2003ve ; Hashimoto:2004xz . These topics drive us into considering such a class of models. In phenomenological point of view, there are also interesting aspects in 6D UED model. In 6D case, the KK mass spectrum is not equally-spaced, up to radiative corrections Cheng:2002iz ; Ponton:2005kx . And a 4D new scalar particle named “spinless adjoint” emerges in the model corresponding to a 6D gauge boson. These are un- eaten physical scalars associated to the 4D vector components of the 6D gauge bosons. These two points exert considerable influence on collider physics and particle cosmology Burdman:2006gy ; Dobrescu:2007xf ; Dobrescu:2007ec ; Freitas:2007rh ; Freitas:2008vh ; Ghosh:2008dp ; Bertone:2009cb ; Blennow:2009ag . These studies are executed on the 6D UED model based on two- torus $T^{2}$ Dobrescu:2004zi ; Burdman:2005sr ; Cacciapaglia:2009pa . It is noted that recently the UED models based on two-sphere $S^{2}$ are proposed and these models have interesting properties Maru:2009wu ; Dohi:2010vc . In the $S^{2}$-based models, the KK mass spectrum is totally different from that of the ordinary $T^{2}$-based models and we consider that this difference would have an impact on collider and cosmological phenomenology.222In 5D case, there are also many approaches of considering the non-minimal UED models Flacke:2008ne ; Park:2009cs ; Csaki:2010az ; Haba:2009uu ; Haba:2009pb ; Haba:2009wa ; Haba:2010xz ; Nishiwaki:2010te . In this paper, we focus on the Higgs boson production and decay sequences through one-loop leading processes expected to occur at the LHC. In one-loop leading processes, the contribution of KK particles is considered to be significant. More concretely, we consider the Higgs production by gluon fusion and the Higgs decay to two photons. The former process is very important because it is the dominant Higgs production process at the LHC. The latter process becomes important in the case when Higgs boson mass is about from $120$ GeV to $150$ GeV. Actually, the ATLAS and CMS experiments at the CERN LHC have presented their latest results for the $\simeq 2\,\text{fb}^{-1}$ of data at the center of mass energy 7 TeV at the Lepton-Photon 2011, Mumbai, India, 22-27 August 2011 and the Higgs decay to two photons process plays a significant role at this range ATLAS-CONF-2011-135 ; CMS_PAS_HIG-11-022 .333 During revising this paper, both the ATLAS and CMS have published the new results, which claim that there is a peak around 125 GeV ATLAS:2012ae ; Chatrchyan:2012tx . In the SM, the branching ratio of the decay into two photons is too small, but the signal of this process is very clear at the LHC experiments. Using the result of the above two processes, we can perform a crude estimation of the difference of the number of the decay events to two photons from the SM expectation value. By naive power counting argument, the production and the decay processes are known to be divergent logarithmically. We adopt the regularization scheme by use of KK momentum cutoff, which is determined by naive dimensional analysis. We also consider threshold corrections in the two processes and these effect are noteworthy, especially when we choose low cutoff scale in 6D UED. As the end of the introduction, we show the organization of this paper briefly. In Section 2, we give a brief review of 6D UED model on $T^{2}/Z_{4}$, which is one of the $T^{2}$-based 6D UED model and has been studied well. In Section 3, we calculate the rate of the Higgs production process through gluon fusion and the Higgs decay process to two photons via loop diagrams in the 6D UED model on $T^{2}/Z_{4}$. These results can be applied for the $S^{2}$-based 6D UED cases with some modifications. In Section 4, we get an overview of gauge theory on $S^{2}$ and give a brief review of the two types of $S^{2}$-based 6D UED models. In Section 5, we estimate the maximal cutoff scale, where the validity of perturbation will break down. In Section 6, we estimate the deviation of the rate of the Higgs production and decay processes and evaluate the difference of the event number from the SM results with/without threshold corrections. Section 7 is devoted to summary and discussions. ## 2 Universal Extra Dimension on $T^{2}/Z_{4}$ We give a brief review of UED model on $T^{2}/Z_{4}$. A detailed construction of the minimal 5D UED based on $S^{1}/Z_{2}$ is studied in Appelquist:2000nn . We consider a gauge theory on six-dimensional spacetime $M^{4}\times T^{2}/Z_{4}$, which is a direct product of the four-dimensional Minkowski spacetime $M^{4}$ and two-torus $T^{2}$ with $Z_{4}$ orbifolding. We use the coordinate of six-dimensional spacetime defined by $x^{M}=(x^{\mu},y,z)$ and the mostly-minus metric convention $\eta_{MN}=\text{diag}(1,-1,-1,-1,-1,-1)$.444Latin indices ($M,N$) run for $0,1,2,3,y,z$ and Greek indices ($\mu,\nu$) run for $0,1,2,3$. The representation of Clifford algebra which we adopt is $\Gamma^{\mu}=\gamma^{\mu}\otimes I_{2}=\begin{bmatrix}\gamma^{\mu}&0\\\ 0&\gamma^{\mu}\end{bmatrix}\ ,\ \Gamma^{y}=\gamma^{5}\otimes i\sigma_{1}=\begin{bmatrix}0&i\gamma^{5}\\\ i\gamma^{5}&0\end{bmatrix}\ ,\ \Gamma^{z}=\gamma^{5}\otimes i\sigma_{2}=\begin{bmatrix}0&\gamma^{5}\\\ -\gamma^{5}&0\end{bmatrix},$ (1) where $\gamma^{5}$ is 4D chirality operator and $\sigma_{i}\ (i=1,2,3)$ are Pauli matrices. To obtain 4D Weyl fermion from 6D Weyl fermion, we choose the type of orbifold as $Z_{4}$, not as $Z_{2}$ in 5D case Dobrescu:2004zi ; Burdman:2005sr . $Z_{4}$ symmetry is realized as the rotation on the $y-z$ plane by an angle $\frac{\pi}{2}$ on $T^{2}$. This means a bulk scalar field $\Phi(x;y,z)$ obeys the following relation: $\Phi_{t}(x,-z,y)=t\Phi_{t}(x,y,z).$ (2) $t$ is $Z_{4}$ parity which takes the possible values $t=\pm 1,\pm i$ and all the fields are classified according to their parity. Following the general prescription Georgi:2000ks , mode functions of $T^{2}/Z_{4}$ $f^{(m,n)}_{t}(y,z)$ are obtained as follows:555For simplicity, we drop the overall $-i$ factor for $t=\pm i$ cases. $\displaystyle f^{(m,n)}_{t}(y,z)=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2\pi R}\frac{1}{\sqrt{1+3\delta_{m,0}\delta_{n,0}}}\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny- mz}{R}\right)\Big{]}&\text{for}\ t=1,\\\ \displaystyle\frac{1}{2\pi R}\Big{[}\cos\left(\frac{my+nz}{R}\right)-\cos\left(\frac{ny- mz}{R}\right)\Big{]}&\text{for}\ t=-1,\\\ \displaystyle\frac{1}{2\pi R}\Big{[}\sin\left(\frac{my+nz}{R}\right)-i\sin\left(\frac{ny- mz}{R}\right)\Big{]}&\text{for}\ t=i,\\\ \displaystyle\frac{1}{2\pi R}\Big{[}\sin\left(\frac{my+nz}{R}\right)+i\sin\left(\frac{ny- mz}{R}\right)\Big{]}&\text{for}\ t=-i,\end{array}\right.$ (3) where $m$ and $n$ are $y$ and $z$ directional KK numbers, respectively and take the values $m\geq 1,n\geq 0$ or $m=n=0$ (only for $t=1$).666The complex factor $i$ in $f_{t=\pm i}^{(m,n)}$ generates CP violating interactions after KK expansion in KK sector Lim:2009pj . And realizing cancellation of 6D gravitational and SU(2)L global anomalies requires the choice of 6D chiralities, for example, as follows Dobrescu:2001ae : $(\mathcal{Q}_{-},\mathcal{U}_{+},\mathcal{D}_{+},\mathcal{L}_{-},\mathcal{E}_{+},\mathcal{N}_{+}),$ (4) whose zero modes form single generation of the standard model; $\mathcal{Q}_{-}^{(0)}=(u,d)_{L},\ \mathcal{U}_{+}^{(0)}=u_{R},\ \mathcal{D}_{+}^{(0)}=d_{R},\ \mathcal{L}_{-}^{(0)}=(\nu,l)_{L},\ \mathcal{E}_{+}^{(0)}=l_{R},\ \mathcal{N}_{+}^{(0)}=\nu_{R}$. The $\pm$ suffixes represent 6D chirality of each field and 6D chirality operator is defined as $\Gamma_{7}=\gamma^{5}\otimes\sigma_{3}.$ (5) Using 6D chiral projective operator $\Gamma_{\pm}\equiv\frac{1}{2}(1\pm\Gamma_{7})$, 6D Weyl fermions $\Psi_{\pm}$ are described as follows; $\Psi_{+}=\begin{pmatrix}\psi_{+R}\\\ \psi_{+L}\end{pmatrix},\quad\Psi_{-}=\begin{pmatrix}\psi_{-L}\\\ \psi_{-R}\end{pmatrix},$ (6) where $\psi_{L(R)}$ is a left(right)- handed 4D Weyl fermion. We can take the boundary condition of 6D fermion $\Psi_{6}=(\psi,\Psi)^{\mathrm{T}}$ ($\mathrm{T}:$ transpose) as in Scrucca:2003ut : $\displaystyle\Psi_{6}(x,-z,y)$ $\displaystyle=(i)^{\frac{1}{2}+r}\left(\frac{1+\Gamma^{y}\Gamma^{z}}{\sqrt{2}}\right)P\Psi_{6}(x,y,z)$ $\displaystyle{\Longleftrightarrow}\begin{pmatrix}\psi\\\ \Psi\end{pmatrix}(x,-z,y)$ $\displaystyle=\begin{pmatrix}i^{r}&0\\\ 0&i^{r+1}\end{pmatrix}P\begin{pmatrix}\psi\\\ \Psi\end{pmatrix}(x,y,z).$ (7) $r$ is $Z_{4}$ twist factor which can takes the values ($r=0,1,2,3$) and $P$ is group twist matrix for fundamental representation with the $Z_{4}$ identification $(P^{4}=1)$, which we discuss soon later. When we choose the values of $r$ as $0$ or $3$, zero mode sectors of $\Psi_{\pm}$ become 4D chiral. Next, we go on to the gauge sector. The boundary condition of this part is as in Scrucca:2003ut : $A_{\mu}(x,i\omega)=PA_{\mu}(x,\omega)P^{-1},\quad A_{\omega}(x,i\omega)=(-i)PA_{\omega}(x,\omega)P^{-1}.$ (8) Here we define a complexified coordinate and a vector field component for clarity as $\omega\equiv\frac{y+iz}{\sqrt{2}},\quad A_{\omega}\equiv\frac{A_{y}-iA_{z}}{\sqrt{2}}.$ (9) In UED model, we do not break the gauge symmetry by boundary condition. Then the matrix $P$ is selected as $P=\mathbf{1}$. This means that none of the fields belonging to $A_{\omega}$ (or $A_{\bar{\omega}}$) takes zero mode, which is an would-be exotic SM particle. Finally we discuss the 6D scalar $\Phi$. The boundary condition for this field is very simple: $\Phi(x,i\omega)=P\Phi(x,\omega).$ (10) Choosing $P=\mathbf{1}$, $\Phi$’s zero mode remains and can be identified as the SM Higgs field. From above discussion, we can form the zero mode sector just as the SM one. We write down the part of the 6D UED Lagrangian which is requisite for our calculation. The 6D action takes the form as follows: $\displaystyle S$ $\displaystyle=\int_{0}^{2\pi R}dy{\int_{0}^{2\pi R}}dz\int d^{4}x\Bigg{\\{}-\frac{1}{2}\sum_{i=1}^{3}\mathrm{Tr}\big{[}F_{MN}^{(i)}F^{(i)MN}\big{]}$ $\displaystyle\qquad{+(D^{M}H)^{\dagger}(D_{M}H)+\bigg{[}\mu^{2}|H|^{2}-\frac{\lambda_{6}^{(H)}}{4}|H|^{4}\bigg{]}}$ $\displaystyle\qquad+i\bar{\mathcal{Q}}_{3-}\Gamma^{M}D_{M}\mathcal{Q}_{3-}+i\bar{\mathcal{U}}_{3+}\Gamma^{M}D_{M}\mathcal{U}_{3+}-\bigg{[}\lambda_{6}^{(t)}\bar{\mathcal{Q}}_{3-}(i\sigma_{2}H^{\ast})\mathcal{U}_{3+}+\mathrm{h.c.}\bigg{]}\Bigg{\\}}.$ (11) $F^{(i)}_{MN}$ are the field strengths of gauge fields, where $F^{(i)}_{MN}=\partial_{M}A_{N}^{(i)}-\partial_{N}A_{M}^{(i)}-ig_{6}^{(i)}[A_{M}^{(i)},A_{N}^{(i)}]$, and the gauge groups are those for $U(1)_{Y}\ (i=1),SU(2)_{L}\ (i=2)$ and $SU(3)_{C}\ (i=3)$ in the SM. The covariant derivatives $D_{M}$ are expressed in our convention as $D_{M}=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M},$ (12) where $g_{6}^{(i)}$ are the six-dimensional gauge couplings and $T^{(i)a}$ are the group generators of each corresponding gauge group. $H$ is the Higgs doublet, and $\mu$, $\lambda_{6}^{(H)}$ and $\lambda_{6}^{(t)}$ are the usual Higgs mass, Higgs self coupling and Yukawa coupling of the top quark in 6D theory, respectively.777All the six-dimensional couplings are dimensionful. After the KK expansion, corresponding 4D couplings become dimensionless as they should be so. $\mathcal{Q}_{3-}$ is the quark doublet in third generation and $\mathcal{U}_{3+}$ is the top quark singlet. We are ready to derive the four-dimensional effective action by expanding all the 6D fields by use of Eq. (3). The concrete forms of KK expansion are as follows: $\displaystyle A^{(i)}_{\mu}(x;y,z)$ $\displaystyle=\frac{1}{2\pi R}\bigg{\\{}A^{(i)(0)}_{\mu}(x)$ $\displaystyle\qquad{+{\sum_{m\geq 1,n\geq 0}}A_{\mu}^{(i)(m,n)}(x)\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny- mz}{R}\right)\Big{]}\bigg{\\}}},$ (13) $\displaystyle A^{(i)}_{\omega}(x;y,z)$ $\displaystyle=\frac{1}{2\pi R}\bigg{\\{}{\sum_{m\geq 1,n\geq 0}}A_{\omega}^{(i)(m,n)}(x)\Big{[}\sin\left(\frac{my+nz}{R}\right)+i\sin\left(\frac{ny- mz}{R}\right)\Big{]}\bigg{\\}},$ (14) $\displaystyle H(x;y,z)$ $\displaystyle=\frac{1}{2\pi R}\bigg{\\{}H^{(0)}(x)+{\sum_{m\geq 1,n\geq 0}}H^{(m,n)}(x)\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny- mz}{R}\right)\Big{]}\bigg{\\}},$ (15) $\displaystyle\mathcal{Q}_{3-}(x;y,z)$ $\displaystyle=\frac{1}{2\pi R}\begin{pmatrix}\displaystyle Q_{3L}^{(0)}(x)+{\sum_{m\geq 1,n\geq 0}}Q_{3L}^{(m,n)}\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny- mz}{R}\right)\Big{]}\\\ \qquad\displaystyle{\sum_{m\geq 1,n\geq 0}}Q_{3R}^{(m,n)}\Big{[}\sin\left(\frac{my+nz}{R}\right)-i\sin\left(\frac{ny- mz}{R}\right)\Big{]}\end{pmatrix},$ (16) $\displaystyle\mathcal{U}_{3+}(x;y,z)$ $\displaystyle=\frac{1}{2\pi R}\begin{pmatrix}\displaystyle t_{R}^{(0)}(x)+{\sum_{m\geq 1,n\geq 0}}t_{R}^{(m,n)}\Big{[}\cos\left(\frac{my+nz}{R}\right)+\cos\left(\frac{ny- mz}{R}\right)\Big{]}\\\ \qquad\displaystyle{\sum_{m\geq 1,n\geq 0}}t_{L}^{(m,n)}\Big{[}\sin\left(\frac{my+nz}{R}\right)-i\sin\left(\frac{ny- mz}{R}\right)\Big{]}\end{pmatrix}.$ (17) In the fermionic part, we choose all the twist factors as $r=0$. Now we can find the SM fields $A^{(0)(i)}_{\mu},H^{(0)},Q^{(0)}_{3L}\big{(}=\big{(}t^{(0)}_{L},b^{(0)}_{L}\big{)}^{\mathrm{T}}\big{)}$ and $t^{(0)}_{R}$ in the zero mode sectors. Here we focus on the 5D Higgs doublet in terms of 4D component fields: $H^{(0)}=\begin{pmatrix}\phi^{+(0)}\\\ \frac{1}{\sqrt{2}}\big{(}v+h^{(0)}+i\chi^{(0)}\big{)}\end{pmatrix},\quad H^{(m,n)}=\begin{pmatrix}\phi^{+(m,n)}\\\ \frac{1}{\sqrt{2}}\big{(}h^{(m,n)}+i\chi^{(m,n)}\big{)}\end{pmatrix}.$ (18) At the zero mode part, $v$ and $h^{(0)}$ are the ordinary four-dimensional Higgs Vacuum Expectation Value (VEV) and the usual SM physical Higgs field. $\phi^{+(0)}$ is the would-be Nambu-Goldstone boson of $W_{\mu}^{+(0)}$ and generate the longitudinal d.o.f. for $W^{+(0)}_{\mu}$ and $\chi^{(0)}$ is for $Z^{(0)}_{\mu}$. Subsequently, we take notice of the (4D) scalar KK excitation modes. In addition to the Higgs KK excitation modes ${\\{}h^{(m,n)},\phi^{+(m,n)},\chi^{(m,n)}{\\}}$, there are other excitation modes closely related to the (zero mode) massive gauge bosons, which are $y$ and $z$ components of 6D gauge fields. Throughout this paper, we use information about W boson zero mode and its KK particles and their associative particles, which are zero and KK modes of $\phi^{{+}}$, $W^{{+y}}$ and $W^{{+z}}$. In what follows, we discuss only the free Lagrangian with respect to the non-zero KK modes of W boson and their associative particles since the zero mode part is the same with the SM one. From Eq. (11), we can read off the free Lagrangian part $S^{W}|_{{\text{free}}}$ as $\displaystyle S^{W}|_{{\text{free}}}=\int d^{4}x{\sum_{m\geq 1,n\geq 0}}\Bigg{\\{}-\frac{1}{2}\Big{[}F_{\mu\nu}^{W(m,n)}F^{W(m,n)\mu\nu}\Big{]}_{{\text{quad}}}$ $\displaystyle\ +\frac{1}{2}\Big{[}(\partial_{\mu}\phi^{+(m,n)})(\partial^{\mu}\phi^{-(m,n)})+(\partial_{\mu}W^{+(m,n)y})(\partial^{\mu}W^{-(m,n)y})+(\partial_{\mu}W^{+(m,n)z})(\partial^{\mu}W^{-(m,n)z})\Big{]}$ $\displaystyle\ +\big{(}m_{W}^{2}+m_{(m,n)}^{2}\big{)}W_{\mu}^{+(m,n)}W^{\mu-(m,n)}-m_{(n)}^{2}W^{+(m,n)y}W^{-(m,n)y}$ $\displaystyle\ -m_{(m)}^{2}W^{+(m,n)z}W^{-(m,n)z}+m_{(m)}m_{(n)}\Big{[}W^{+(m,n)y}W^{-(m,n)z}+W^{-(m,n)y}W^{+(m,n)z}\Big{]}$ $\displaystyle\ -m_{(m,n)}^{2}\phi^{+(m,n)}\phi^{-(m,n)}-im_{W}\phi^{-(m,n)}\Big{[}m_{(m)}W^{+(m,n)y}+m_{(n)}W^{+(m,n)z}\Big{]}$ $\displaystyle\ +im_{W}\phi^{+(m,n)}\Big{[}m_{(m)}W^{-(m,n)y}+m_{(n)}W^{-(m,n)z}\Big{]}$ $\displaystyle\ -m_{W}^{2}\Big{[}W^{+(m,n)y}W^{-(m,n)y}+W^{+(m,n)z}W^{-(m,n)z}\Big{]}$ $\displaystyle\ -im_{W}\Big{[}(\partial^{\mu}\phi^{-(m,n)})W_{\mu}^{+(m,n)}-(\partial^{\mu}\phi^{+(m,n)})W_{\mu}^{-(m,n)}\Big{]}$ $\displaystyle\ -\Big{[}m_{(m)}(\partial^{\mu}W^{+(m,n)y})+m_{(n)}(\partial^{\mu}W^{+(m,n)z})\Big{]}W^{-(m,n)}_{\mu}$ $\displaystyle\ -\Big{[}m_{(m)}(\partial^{\mu}W^{-(m,n)y})+m_{(n)}(\partial^{\mu}W^{-(m,n)z})\Big{]}W^{+(m,n)}_{\mu}\Bigg{\\}},$ (19) where $\Big{[}F_{\mu\nu}^{W(m,n)}F^{W(m,n)\mu\nu}\Big{]}_{{\text{quad}}}=\big{(}\partial^{\mu}W^{+(m,n)\nu}-\partial^{\nu}W^{+(m,n)\mu}\big{)}\big{(}\partial_{\mu}W^{-(m,n)}_{\nu}-\partial_{\nu}W^{-(m,n)}_{\mu}\big{)}$ is the KK W-boson’s kinetic term, $m_{W}$ is the W-boson mass; $m_{(m)}=\frac{m}{R}$ and $m_{(m,n)}^{2}=m_{(m)}^{2}+m_{(n)}^{2}$ are describing the KK masses. Here we adopt the following type of gauge-fixing term about W boson to eliminate cross terms in Eq. (19) as $\displaystyle S^{W}_{{\text{gf}}}$ $\displaystyle=-\frac{1}{\xi}\int_{0}^{2\pi R}dy{\int_{0}^{2\pi R}}dz\int d^{4}x\Big{[}\partial_{\mu}W^{+\mu}+\xi\big{(}\partial_{y}W^{+y}+\partial_{z}W^{+z}-im_{W}\phi^{+}\big{)}\Big{]}$ $\displaystyle\qquad\times\Big{[}\partial_{\mu}W^{-\mu}+\xi\big{(}\partial_{y}W^{-y}+\partial_{z}W^{-z}+im_{W}\phi^{-}\big{)}\Big{]}.$ (20) From Eq. (19), the mass of the field $W^{+(m,n)}_{\mu}$ is determined as $m_{W,(m,n)}^{2}=m_{W}^{2}+m_{(m,n)}^{2}$. Meanwhile, we have to diagonalize the scalar mass terms about $\phi^{+(m,n)}$, $W^{+(m,n)y}$ and $W^{+(m,n)z}$ to execute perturbative calculations. When we focus on this part $S^{W}_{\text{scalar-mass}}$ out of $S^{W}+S^{W}_{{\text{gf}}}$, $S^{W}_{\text{scalar-mass}}=-\int d^{4}x{\sum_{m\geq 1,n\geq 0}}\Big{(}W^{+(m,n)y},W^{+(m,n)z},\phi^{+(m,n)}\Big{)}\mathcal{M}_{(m,n)}\begin{pmatrix}W^{-(m,n)y}\\\ W^{-(m,n)z}\\\ \phi^{-(m,n)}\end{pmatrix},$ (21) $\mathcal{M}_{(m,n)}=\begin{bmatrix}m_{W}^{2}+\xi m_{(m)}^{2}+m_{(n)}^{2}&&-(1-\xi)m_{(m)}m_{(n)}&&-i(1+\xi)m_{W}m_{(m)}\\\ -(1-\xi)m_{(m)}m_{(n)}&&m_{W}^{2}+m_{(m)}^{2}+\xi m_{(n)}^{2}&&-i(1+\xi)m_{W}m_{(n)}\\\ +i(1+\xi)m_{W}m_{(m)}&&+i(1+\xi)m_{W}m_{(n)}&&\xi m_{W}^{2}+m_{(m)}^{2}+m_{(n)}^{2}\end{bmatrix}.$ (22) By using those mass eigenstates ${\\{}G^{+(m,n)},a^{+(m,n)},H^{+(m,n)}{\\}}$, we can diagonalize the matrix $\mathcal{M}_{(m,n)}$ to the following form: $\begin{pmatrix}G^{\pm(m,n)}\\\ a^{\pm(m,n)}\\\ H^{\pm(m,n)}\end{pmatrix}=N_{(m,n)}^{\pm}\begin{pmatrix}W^{\pm(m,n)y}\\\ W^{\pm(m,n)z}\\\ \phi^{\pm(m,n)}\end{pmatrix},$ (23) $N_{(m,n)}^{\pm}=\frac{1}{m_{W,(m,n)}m_{(m,n)}}\begin{bmatrix}m_{(m)}m_{(m,n)}&&m_{(n)}m_{(m,n)}&&\mp im_{W}m_{(m,n)}\\\ \mp im_{W}m_{(m)}&&\mp im_{W}m_{(n)}&&m_{(m,n)}^{2}\\\ -m_{(n)}m_{W,(m,n)}&&+m_{(m)}m_{W,(m,n)}&&0\end{bmatrix},$ (24) $N^{-}_{(m,n)}\mathcal{M}_{(m,n)}\big{(}N^{-}_{(m,n)}\big{)}^{\dagger}=\text{diag}\big{(}\xi m_{W,(m,n)}^{2}\ ,\ m_{W,(m,n)}^{2}\ ,\ m_{W,(m,n)}^{2}\big{)}.$ (25) This result means that $G^{+(m,n)}$ is the would-be Nambu-Goldstone boson of $W^{+(m,n)}_{\mu}$ and the others $a^{+(m,n)},H^{+(m,n)}$ are physical 4D scalars. It is noted that ${H}^{+(m,n)}$ is called “spinless adjoint” because ${H}^{+(m,n)}$ is constructed only by extra spacial components of the 6D gauge boson $W^{+(m,n)}_{\mu}$. They contribute to $h^{(0)}\rightarrow 2\gamma$ Higgs decay process via loop diagrams. Next, we derive the mass eigenstates of fermions. Just like the case mentioned above, we again consider the KK part only. The kinetic terms are diagonal, and therefore there is no need to discuss the part. The mass term of $(m,n)$-th KK mode fermions arising from Eq. (11) is $\Big{(}\bar{t}^{(m,n)}_{R}\ ,\ \bar{Q}^{(m,n)}_{tR}\Big{)}\begin{pmatrix}-m_{(m)}+im_{(n)}&m_{t}\\\ m_{t}&m_{(m)}+im_{(n)}\end{pmatrix}\begin{pmatrix}t_{L}^{(m,n)}\\\ Q_{tL}^{(m,n)}\end{pmatrix}+\text{h.c.},$ (26) where $m_{t}$ is the zero mode top quark mass and $Q_{t}^{(m,n)}$ is the upper component of the SU(2) doublet $Q_{3}^{(m,n)}$. By using the following unitary transformation including chiral rotation, we can derive the ordinary diagonalized Dirac mass term as follows: $\begin{pmatrix}t^{(m,n)}\\\ Q_{t}^{(m,n)}\end{pmatrix}=\begin{pmatrix}e^{\frac{i}{2}\gamma^{5}\varphi_{(m,n)}}&0\\\ 0&e^{-\frac{i}{2}\gamma^{5}\varphi_{(m,n)}}\end{pmatrix}\begin{pmatrix}-\cos{\alpha_{(m,n)}}\gamma^{5}&\sin{\alpha_{(m,n)}}\\\ \sin{\alpha_{(m,n)}}\gamma^{5}&\cos{\alpha_{(m,n)}}\end{pmatrix}\begin{pmatrix}t^{{}^{\prime}(m,n)}\\\ Q_{t}^{{}^{\prime}(m,n)}\end{pmatrix},$ (27) where $t^{{}^{\prime}(m,n)}$ and $Q_{t}^{{}^{\prime}(m,n)}$ are mass eigenstates of their corresponding fields with degenerate $(m,n)$-th level masses; $m_{t,(m,n)}^{2}=m_{t}^{2}+m_{(m,n)}^{2}$. The mixing angles $\varphi_{(m,n)}$ and $\alpha_{(m,n)}$ are determined as $\tan{\varphi_{(m,n)}}=-\frac{m_{(n)}}{m_{(m)}},\quad\cos{{2}\alpha_{(m,n)}}=\frac{m_{(m,n)}}{m_{t,(m,n)}},$ (28) from the condition to obtain the ordinary diagonalized Dirac mass matrix. Now we are ready to calculate the rates of Higgs processes at the LHC. Some requisite interactions in this paper are discussed at the next section. ## 3 Calculation of one loop Higgs production and decay processes We calculate some virtual effects of KK particle via loop diagrams in the Higgs production process through gluon ($g$) fusion $2g\rightarrow h^{(0)}$ and the Higgs decay process to two photon ($\gamma$) $h^{(0)}\rightarrow 2\gamma$. Those processes are 1-loop leading and it is expected that the effects of massive KK particles are significant. In addition, there is another 1-loop leading Higgs decay process to photon and Z-boson ($Z$) $h^{(0)}\rightarrow\gamma Z$, which we do not discuss in this paper. Before the concrete discussion about interactions, we have to understand the general structure of interactions which is needed for our study. In the scope of this paper, all external particles are SM particles, which are described by zero modes. This means the effective couplings which we use are obtained by the following type of integrals concerning mode functions $f_{t}^{(m,n)}$, $\displaystyle\int_{0}^{2\pi R}dy{\int_{0}^{2\pi R}}dz\left\\{f_{t_{i}}^{(0,0)}f_{t_{j}}^{(m,n)}f_{t_{k}}^{(m^{\prime},n^{\prime})}\right\\}\ [\text{3-point}],$ (29) $\displaystyle\int_{0}^{2\pi R}dy{\int_{0}^{2\pi R}}dz\left\\{f_{t_{l}}^{(0,0)}f_{t_{i}}^{(0,0)}f_{t_{j}}^{(m,n)}f_{t_{k}}^{(m^{\prime},n^{\prime})}\right\\}\ [\text{4-point}],$ (30) where the Latin indices $i,j,k,l$ indicate types of the particles. The $Z_{4}$-parities $t_{l},t_{i}$ are determined as $t_{l}=t_{i}=1$ and the condition $(t_{j})^{\ast}=t_{k}$ is required from $Z_{4}$ invariance of the action in Eq. (11). Because of orthonormality of mode functions, we know that the integrals are non-vanishing only when $(m,n)=(m^{\prime},n^{\prime})$ and the integrals can be reduced to the ones for the zero modes alone. In other words, the value of the vertex containing KK modes, which are described by the above integrals, is exactly the same with the value of the corresponding vertex for the zero mode alone in the basis of gauge eigenstates. We give a comment on the Higgs mass $m_{h}$ and the lowest KK mass ${M_{\text{KK}}}$, which is defined as $1/R$ on the geometry of $T^{2}$. In UED model, those two parameters are free, which means they are not determined by the theory, but there are some constraints on these parameters. From the result of LEP2 experiment, $m_{h}$ is bounded from below as $m_{h}>114\ \text{GeV}$. And recently another bound is announced from the LHC experiments ATLAS-CONF-2011-135 ; CMS_PAS_HIG-11-022 ; ATLAS:2012ae ; Chatrchyan:2012tx . We discuss this point in Section 6. We ignore the graviton contributions. In any 6D UED model, 6D Planck scale $M_{\ast}$ is related to 4D Planck scale $M_{\text{pl}}$ through a KK mass scale ${M_{\text{KK}}}$ as follows: $M_{\ast}^{2}\sim{M_{\text{KK}}}M_{\text{pl}}.$ (31) $M_{\text{pl}}$ is approximately $10^{18}$ GeV and we are interested in the case ${M_{\text{KK}}}\sim\mathcal{O}(1)$ TeV. Then the magnitude of $M_{\ast}$ is estimated easily as $\sim 10^{10}$ GeV and gravitons are still weekly coupled to other fields. ### 3.1 $2g\rightarrow h^{(0)}$ process This gluon fusion process gets contribution only from the fermion triangle loops at 1-loop level. The SM contribution is calculated in Eq. Georgi:1977gs ; Rizzo:1979mf . In UED model, the intermediate fermions are not only SM ones (zero modes) but also their KK excitations. Studies of the production process for the case of 5D minimal UED Petriello:2002uu and 6D $S^{2}/Z_{2}$ UED Maru:2009cu are made. We consider only contributions from the top quark and its KK states. The reason why we ignore other types of quarks and its KK modes is that the coupling of fermions to the Higgs is proportional to each zero mode quark mass, and thereby those effects are negligible in our analysis. In terms of the fermion mass eigenstates, the interactions of KK quarks are $\displaystyle S^{t}_{{\text{int}}}$ $\displaystyle=\int d^{4}x{\sum_{m\geq 1,n\geq 0}}$ $\displaystyle\times\Bigg{\\{}\Big{(}\bar{t}^{{}^{\prime}(m,n)}\ ,\ \bar{Q}^{{}^{\prime}(m,n)}_{t}\Big{)}\Bigg{[}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}g^{(3)}\gamma^{\mu}g_{\mu}$ $\displaystyle\qquad-\frac{m_{t}}{v}h^{(0)}\begin{pmatrix}\sin{2\alpha_{(m,n)}}&\cos{2\alpha_{(m,n)}}\gamma^{5}\\\ -\cos{2\alpha_{(m,n)}}\gamma^{5}&\sin{2\alpha_{(m,n)}}\end{pmatrix}\Bigg{]}\begin{pmatrix}t^{{}^{\prime}(m,n)}\\\ Q_{t}^{{}^{\prime}(m,n)}\end{pmatrix}\Bigg{\\}}.$ (32) The production cross section of $2g\rightarrow h^{(0)}$ process is given as follows: ${\sigma_{2g\rightarrow h^{(0)}}=\frac{\sqrt{2}\pi G_{F}}{64}\left(\frac{\alpha_{s}}{\pi}\right)^{2}|F_{\text{{gluonfusion}}}|^{2},}$ (33) where $G_{F}$ is the Fermi constant and $\alpha_{s}$ is the QCD coupling. $F_{\text{{gluonfusion}}}$ is the loop function, which consists of the SM top quark effect $F_{t}^{{\text{SM}}}$, the KK top quark effect $F_{t}^{{\text{KK}}}$ and the threshold correction $F_{\text{gluonfusion}}^{\text{TC}}$. Then we can write $F_{\text{{gluonfusion}}}=F_{t}^{{\text{SM}}}+F_{t}^{{\text{KK}}}{+F_{\text{gluonfusion}}^{\text{TC}}}$ and $F_{t}^{{\text{SM}}}$ is given in Ref. Rizzo:1979mf in our notation as $F_{t}^{\text{{SM}}}={-2}\lambda(m_{t}^{2}){+}\lambda(m_{t}^{2})(1-4\lambda(m_{t}^{2}))J\left(\lambda(m_{t}^{2})\right),$ (34) where $\lambda(m^{2})$ and the loop function $J(\lambda)$ are defined as $\displaystyle{\lambda(m^{2})}$ $\displaystyle{=m^{2}/m_{h}^{2},}$ (35) $\displaystyle{J(\lambda)}$ $\displaystyle{=\int_{0}^{1}{dx\over x}\ln\left[{x(x-1)\over\lambda}+1-i\epsilon\right]}$ $\displaystyle{=\begin{cases}\displaystyle-2\left[\arcsin{1\over\sqrt{4\lambda}}\right]^{2}&\text{(for $\lambda\geq{1\over 4}$)},\\\ \displaystyle{1\over 2}\left[\ln{1+\sqrt{1-4\lambda}\over 1-\sqrt{1-4\lambda}}-i\pi\right]^{2}&\text{(for $\lambda<{1\over 4}$)},\end{cases}}$ (36) respectively.888Our loop function $J$ is related to the three-point scalar Passarino-Veltman function $C_{0}$ in Refs. Passarino:1978jh ; Denner:1991kt as $J=m_{h}^{2}C_{0}$. The KK top quark coupling to the gluon and the zero mode Higgs is shown in Eq. (32). After some calculation, we can get the form of $F_{t}^{{\text{KK}}}$ in 6D UED model, where the concrete form is $\displaystyle F_{t}^{{\text{KK}}}$ $\displaystyle=2\sum_{m\geq 1,n\geq 0}\left(\frac{m_{t}}{m_{t,(m,n)}}\right)^{2}$ $\displaystyle\quad\times\left\\{{-2}\lambda(m_{t,(m,n)}^{2}){+}\lambda(m_{t,(m,n)}^{2})(1-4\lambda(m_{t,(m,n)}^{2}))J\left(\lambda(m_{t,(m,n)}^{2})\right)\right\\}.$ (37) It is noted that $F_{t}^{\text{SM}}$ and $F_{t}^{\text{KK}}$ contain the $(-1)$ factor due to fermionic loop. Our result is directly related to the minimal 5D case in Ref. Petriello:2002uu . The reason is that the only difference between 5D and 6D case is the KK top quark mass spectrum and the structure of the Feynman diagrams itself describing this process is completely the same. The $m^{2}$ in ${\lambda(={m^{2}}/{m_{h}^{2}})}$ indicates the intermediate mass scale propagating in the loops and we consider the situation that KK scale $m_{(m,n)}$ is much greater than the Higgs scale $m_{h}$. It is noted that we only focus on the light Higgs possibility; $120\,\text{GeV}\lesssim m_{h}\lesssim 150\,\text{GeV}$ and thereby we have to only consider the ${\lambda}\geq 1/4$ case. Finally the contribution from threshold correction is obtained as $F_{\text{gluonfusion}}^{\text{TC}}=\left[\left(\frac{\alpha_{s}}{\pi}\right)\frac{1}{v}\right]^{-1}C^{\prime}_{hgg},$ (38) where $C^{\prime}_{hgg}$ is a dimensionful coefficient describing the threshold correction and is related to the dimensionless constant in a part of the Lagrangian $C_{hgg}$ with the UED cutoff scale ${\Lambda_{\text{UED}}}$ as $C^{\prime}_{hgg}=\frac{C_{hgg}\left(v\over\sqrt{2}\right)}{{\Lambda_{\text{UED}}}^{2}}.$ (39) We see the details in Appendix B. From naive power counting, this result is logarithmically divergent. The reason is the following. Higgs decay through gluon fusion is described with dimension-six operator in four-dimensional picture after KK reduction. In UED model, there is no shift symmetry alleviating divergence, then this process obeys the above simple estimation.999In 5D UED, we can calculate this process without cutoff dependence. Therefore, we introduce a cutoff scale ${\Lambda_{\text{UED}}}$ to regularize the $F_{t}^{\text{KK}}$ in Eq. (37). We estimate an upper bound of ${\Lambda_{\text{UED}}}$ by use of Naive Dimensional Analysis (NDA) technique Appelquist:2000nn in Section 5. ### 3.2 $h^{(0)}\rightarrow 2\gamma$ process Now we turn to the Higgs decay process $h^{(0)}\rightarrow 2\gamma$, which is the experimentally favorable at the LHC with Higgs mass region $120\,\text{GeV}\lesssim m_{h}\lesssim 150\,\text{GeV}$. The Feynman diagrams describing $h^{(0)}\rightarrow 2\gamma$ process due to the contribution of W boson and its associated particles are shown in Fig. 1, 2, 3, and 4. $\omega_{W}^{(m,n)}$ and $\bar{\omega}_{W}^{(m,n)}$ indicate $(m,n)$-th ghost and anti-ghost modes originated from $W^{(m,n)}_{\mu}$ boson, respectively. We also need to consider a flipped $(\mu\leftrightarrow\nu)$ one for each diagram if it exists. It is noted that there are another triangle loop diagrams contributing to this process, whose intermediate particles are the top quark and its KK states. But we can take these effects into account by use of the previous result in Eq. (37) with some modifications. The decay width can be written as ${\Gamma_{h^{(0)}\rightarrow 2\gamma}=\frac{\sqrt{2}G_{F}}{16\pi}\left(\frac{\alpha_{\text{EM}}}{\pi}\right)^{2}m_{h}^{3}|F_{\text{decay}}|^{2},}$ (40) where $\alpha_{\text{EM}}$ is the electromagnetic coupling strength. In this process, the function describing loop effects $F_{\text{decay}}$ is written by $F_{\text{decay}}=F_{W}+3Q_{t}^{2}\left({F_{t}^{\text{SM}}+F_{t}^{\text{KK}}}\right){+F^{\text{TC}}_{\text{decay}}},$ (41) where the first term represents the effect of W boson and its associated particles, and the second term represents that of the top quark and its KK states.1010103 is color factor and $Q_{t}$ is the electromagnetic charge of the top quark $(=\frac{2}{3})$. The third term describes the threshold correction. The SM result for $F_{t}^{{\text{SM}}}$ is previously obtained in Eq. (34) and the concrete form of $F_{W}^{{\text{SM}}}$ is derived in Ellis:1975ap as ${F_{W}^{{\text{SM}}}=\frac{1}{2}+3\lambda{(m_{W}^{2})}-3\lambda{(m_{W}^{2})}(1-2\lambda{(m_{W}^{2})})J\left(\lambda(m_{W}^{2})\right)},$ (42) where $J$ is given in Eq. (36). We set $F_{W}=F_{W}^{{\text{SM}}}+F^{{\text{KK}}}_{W}$, where $F_{t}^{{\text{KK}}}$ has been already discussed and $F_{W}^{{\text{KK}}}$ represents the contribution of KK W boson and its associated KK particles. And we decompose $F_{W}^{{\text{KK}}}$ into four pieces as $F_{W}^{{\text{KK}}}=F^{{\text{KK}}}_{\text{gauge}}+F^{{\text{KK}}}_{\text{NG}}+F^{{\text{KK}}}_{\text{scalar1}}+F^{{\text{KK}}}_{\text{scalar2}},$ (43) where each term $F_{W}^{{\text{KK}}}$ indicates the loop effects coming from gauge, would-be NG boson, scalar particles, respectively and corresponding Feynman diagrams are found in Fig. 1, 2, 3, and 4, respectively. The four sets of diagrams are $U(1)_{EM}$ gauge invariant and we can check this fact by use of Ward identity. Figure 1: Feynman diagrams of 4D gauge sector describing $h^{(0)}\rightarrow 2\gamma$ process. Figure 2: Feynman diagrams of 4D would-be NG boson sector describing $h^{(0)}\rightarrow 2\gamma$ process. Figure 3: Feynman diagrams of 4D scalar sector describing $h^{(0)}\rightarrow 2\gamma$ process. Figure 4: Feynman diagrams of 4D scalar (“spinless adjoint”) sector describing $h^{(0)}\rightarrow 2\gamma$ process. After some tedious but straightforward calculation, we can get the result as follows:111111In Appendix A, we write down some Feynman rules to calculate this process. $\displaystyle F^{{\text{KK}}}_{\text{gauge}}$ $\displaystyle=\sum_{m\geq 1,n\geq 0}\left\\{3\lambda(m_{W}^{2})+2\lambda(m_{W}^{2})\big{(}3\lambda(m_{W,(m,n)}^{2})-2\big{)}J\left(\lambda(m_{W,(m,n)}^{2})\right)\right\\},$ (44) $\displaystyle F^{{\text{KK}}}_{\text{NG}}$ $\displaystyle=\sum_{m\geq 1,n\geq 0}\left(\frac{1}{2}\frac{m_{h}^{2}}{m_{W,(m,n)}^{2}}\right)\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(m,n)}^{2})J\left(\lambda(m_{W,(m,n)}^{2})\right)\right\\},$ (45) $\displaystyle F^{{\text{KK}}}_{\text{scalar1}}$ $\displaystyle=\sum_{m\geq 1,n\geq 0}\left(\frac{1}{2}\frac{1}{m_{W,(m,n)}^{2}}\right)\left[\frac{m_{h}^{2}}{m_{W}^{2}}m_{(m,n)}^{2}+{2}m_{W,(m,n)}^{2}\right]$ $\displaystyle\quad\times\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(m,n)}^{2})J\left(\lambda(m_{W,(m,n)}^{2})\right)\right\\},$ (46) $\displaystyle F^{{\text{KK}}}_{\text{scalar2}}$ $\displaystyle=\sum_{m\geq 1,n\geq 0}\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(m,n)}^{2})J\left(\lambda(m_{W,(m,n)}^{2})\right)\right\\},$ (47) By adding up Eqs. (44)-(47), the concrete form of $F_{W}^{{\text{KK}}}$ is given as $\displaystyle F_{W}^{{\text{KK}}}$ $\displaystyle=\sum_{m\geq 1,n\geq 0}\Big{\\{}\frac{1}{2}+5\lambda(m_{W}^{2})-\Big{[}\lambda(m_{W}^{2})(4-10\lambda(m_{W,(m,n)}^{2}))$ $\displaystyle\qquad\qquad-\lambda(m_{W,(m,n)}^{2})\Big{]}J\left(\lambda(m_{W,(m,n)}^{2})\right)\Big{\\}},$ (48) where we use the relation $m_{W,{(m,n)}}^{2}=m_{W}^{2}+m_{(m,n)}^{2}$. This loop-induced process is also described by dimension-six operator in 4D point of view and we have to introduce the cutoff scale ${\Lambda_{\text{UED}}}$ to regularize the summations. The concrete form of the third term in Eq. (41), which originates form threshold correction, is as follows: $F^{\text{TC}}_{\text{decay}}=\left[\left(\frac{\alpha_{\text{EM}}}{\pi}\right)\frac{2}{v}\right]^{-1}C^{\prime}_{h\gamma\gamma},$ (49) where $C^{\prime}_{h\gamma\gamma}$ is a dimensionful coefficient describing the threshold correction and is related to the dimensionless constant in a part of the Lagrangian $C_{h\gamma\gamma}$ with the UED cutoff scale ${\Lambda_{\text{UED}}}$ as $C^{\prime}_{h\gamma\gamma}=\frac{C_{h\gamma\gamma}\left(v\over\sqrt{2}\right)}{{\Lambda_{\text{UED}}}^{2}}.$ (50) We also see the details in Appendix B. ## 4 Universal Extra Dimension Models based on $S^{2}$ Recently, Universal Extra Dimension Models based on $S^{2}$ are proposed in Refs. Maru:2009wu ; Dohi:2010vc . After an overview of gauge theory on $S^{2}$, we give a brief review of these models. ### 4.1 Gauge Theory on $S^{2}$ We consider a gauge theory on six-dimensional spacetime $M^{4}\times S^{2}$, which is a direct product of the four-dimensional Minkowski spacetime $M^{4}$ and two-sphere $S^{2}$. We use the coordinate of six-dimensional spacetime defined by $x^{M}=(x^{\mu},\theta,\phi)$. $\theta\ (\phi)$ is zenith (azimuthal) angle of $S^{2}$, respectively and we use the same coordinate conventions as in Section 2. The metric ansatz of $M^{4}\times S^{2}$ is $g_{MN}=\text{diag}(1,-1,-1,-1,-R^{2},-R^{2}\sin^{2}{\theta})$ (51) and we also need to introduce the vielbein $e_{M}^{\ \ \underline{N}}=\text{diag}(1,1,1,1,R,R\sin{\theta})$ to describe tangent space which fermions live in. In this tangent space, the coordinate is expressed with barred letters and we choose the same representation of Clifford algebra as in Eq. (1). $S^{2}$ has a positive curvature and then a radius of $S^{2}$ described by $R$ only can take an infinite value by the consistency with the 6D Einstein equation. To stabilize the system, we introduce a $U(1)_{X}$ gauge field which has a monopole-like configuration in classical level $X^{c}_{M}$ RandjbarDaemi:1982hi . This configuration is defined as follows: $[X^{c}_{\phi}(x^{\mu},\theta,\phi)]^{{N}\atop{S}}={n\over 2g^{(X)}_{6}}(\cos{\theta}\mp 1),\quad(\text{other components})=0,$ (52) where $g^{(X)}_{6}$ is a $U(1)_{X}$ gauge coupling and $n$ is a monopole index. The superscript $N\atop S$ indicates that the field is given in north (involving the $\theta=0$ point) and south (involving the $\theta=\pi$ point) patches, respectively and we use this notation throughout the rest of this paper. The gauge transformation from the north to the south patch is given by $[X_{M}(x^{\mu},\theta,\phi)]^{S}=[X_{M}(x^{\mu},\theta,\phi)]^{N}+\frac{1}{g_{6}^{(X)}}\partial_{M}\alpha(x^{\mu},\theta,\phi){,}$ (53) where the function $\alpha(x^{\mu},\theta,\phi)=n\phi$. Because of the monopole-like configuration, the radius of $S^{2}$ is stabilized spontaneously as $R^{2}=\left({n\over 2g_{6}^{(X)}M_{\ast}^{2}}\right)^{2}.$ (54) Every 6D field $\Phi$ on $S^{2}$ is KK expanded by use of the spin-weighted spherical harmonics ${}_{s}Y_{jm}(\theta,\phi)$ as follows:121212Newman- Penrose edth formalism Newman:1966ub is useful for description of spin weighted spherical harmonics. $\Phi(x,\theta,\phi)^{N\atop S}=\sum_{j=|s|}^{\infty}\sum_{m=-j}^{j}\varphi^{(j,m)}(x)f_{\Phi}^{(j,m)}(\theta,\phi)^{N\atop S},\quad f_{\Phi}^{(j,m)}(\theta,\phi)^{N\atop S}:={{}_{s}Y_{jm}(\theta,\phi)e^{\pm is\phi}\over R},$ (55) where $s$ is the spin weight of the field $\Phi$. The spin-weighted spherical harmonics ${}_{s}Y_{jm}(\theta,\phi)$ matches the orthonormal condition as ${\int_{0}^{2\pi}d\phi\int_{-1}^{1}d\cos{\theta}\overline{{}_{s}Y_{jm}(\theta,\phi)}{}_{s}Y_{j^{\prime}m^{\prime}}(\theta,\phi)=\delta_{jj^{\prime}}\delta_{mm^{\prime}}.}$ (56) A spin weight of a fermion is closely related to its $U(1)_{X}$ charge. When we assign $U(1)_{X}$ charges of 6D Weyl fermions $\Psi_{\pm}$ as $q_{\Psi_{\pm}}$, the corresponding spin weights of 4D Weyl fermions $\\{\psi_{+{R\atop L}},\psi_{-{R\atop L}}\\}$ are given as follows in our convention: $s_{+{R\atop L}}=-\left({nq_{\Psi_{+}}\mp 1\over 2}\right),\quad s_{-{R\atop L}}=-\left({nq_{\Psi_{-}}\pm 1\over 2}\right).$ (57) We can find the fact that if a 6D Weyl fermion takes the $s=0$ spin weight, one zero mode $(j=0)$ appears as a 4D Weyl fermion with no KK mass. This means we can get the SM fermions without orbifolding in the case of $S^{2}$. When we take the values as $(s_{+R},s_{+L},s_{-R},s_{-L})=(0,-1,-1,0)$, we can create the same situation as in $T^{2}/Z_{4}$ which we discussed before. A spin weight of a 4D vector component of a 6D gauge boson is $s=0$ and then there is a zero mode which we can assign as a SM gauge boson. However, extra dimensional components of 6D gauge boson are expanded by the $|s|=1$ spin- weighted spherical harmonics. This is because these parts are closely related to $S^{2}$ structure. Concretely speaking, the combinations of components $A_{\pm}=\frac{1}{\sqrt{2}}(A_{\underline{\theta}}\pm iA_{\underline{\phi}})$ are KK expanded with $s=\pm 1$ spin weighted spherical harmonics, respectively, where $A_{\underline{M}}$ is a gauge field on tangent space defined as $A_{\underline{M}}=e_{\underline{M}}^{\ \ N}A_{N}$. Then there is no zero mode in these parts. After the introduction of gauge fixing term concerning a gauge field $A_{M}$, whose concrete form is $-\frac{1}{\xi}\text{tr}\left(\eta^{\mu\nu}\partial_{\mu}A_{\nu}-\frac{\xi}{R^{2}\sin{\theta}}\partial_{\theta}\sin{\theta}A_{\theta}-\frac{\xi}{R^{2}\sin^{2}{\theta}}A_{\phi}\right)^{2}\quad(\xi:\text{gauge fixing parameter}),$ (58) the mass eigenstates are obtained as follows: $\begin{pmatrix}A_{\underline{\theta}}\\\ A_{\underline{\phi}}\end{pmatrix}=\begin{pmatrix}\partial_{\theta}&-\csc{\theta}\partial_{\phi}\\\ \partial_{\theta}&+\csc{\theta}\partial_{\phi}\end{pmatrix}\begin{pmatrix}\phi_{1}^{(A)}\\\ \phi_{2}^{(A)}\end{pmatrix}.$ (59) $\phi_{1}^{(A)}$ and $\phi_{2}^{(A)}$ are 4D physical scalar field and unphysical would-be Nambu-Goldstone mode, respectively. A 6D scalar field can take a nonzero spin weight through the interaction with the $U(1)_{X}$ gauge boson. But we would like to regard the zero mode of a 6D scalar field as the SM Higgs, then the value of the spin weight must be $s=0$. In our configuration, any $(j,m)$-th KK mode has the KK mass, $m_{(j,m)}^{2}=\frac{j(j+1)}{R^{2}}.$ (60) An important point is that the form of the above KK mass is independent of the index of $m$. This means there are $2j+1$ degenerated modes for each $j$. It is noted that each KK mode summation over $j$ begins from one. In contrast to the $T^{2}$ case, the value of the first KK mass is represented as $M_{\text{KK}}=\sqrt{2}/R$. We can construct an Universal Extra Dimension model on $S^{2}$ along the direction which we have discussed. But there are two problems in this model. One is absence of KK parity. In usual UED models based on orbifold, there are fixed points of orbifold discrete symmetry and KK parity is realized as a remnant of extra spatial symmetry, which is an invariance of system in exchange of fixed points. It ensures the existence of dark matter candidate in these models. But the geometry of $S^{2}$ do not have fixed point and thereby the UED on $S^{2}$ cannot possess KK parity. The other is more serious. As we discussed before, a 4D vector component of a 6D gauge boson has zero mode in $S^{2}$. In case of the $U(1)_{X}$ gauge boson, which has the monopole-like configuration, this is true. We should notice that the gauge coupling of an extra massless gauge boson is severely constrained to be $g^{(X)}\lesssim 10^{-23}$ by a torsion balance experiment Smith:1999cr . $g^{(X)}$ is the 4D effective coupling of the 6D $U(1)_{X}$ gauge coupling $g^{(X)}_{6}$ and is described as $g^{(X)}={{g^{(X)}_{6}}/{\sqrt{4\pi R^{2}}}}$. By use of (54), we can estimate the value of $g^{(X)}$ in the UED model on $S^{2}$ as $g^{(X)}{\simeq}\frac{n{M_{\text{KK}}}}{M_{\text{pl}}}{{.}}$ (61) In the viewpoint of our phenomenological motivation, ${M_{\text{KK}}}$ must be $\sim\mathcal{O}(1)$ TeV. In such a situation, $g^{(X)}$ becomes $\sim 10^{-15}\cdot n$ and its value is far from the experimental bound. Since monopole charge $n$ only can take integer value, we cannot resolve this pathology by tuning of the parameter $n$. Fortunately, we can solve these problems by some modifications in the $S^{2}$ geometry. In the rest of this section, we follow some essential points of these ideas. ### 4.2 UED on $S^{2}/Z_{2}$ Following Ref. Maru:2009wu , we take a $Z_{2}$ orbifold on the geometry of $S^{2}$. On this orbifold, the point ${(\theta,\phi)}$ is identified with $(\pi-\theta,-\phi)$. The 6D action is as follows: $\displaystyle S$ $\displaystyle=\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\int d^{4}x\sqrt{-g}\Bigg{\\{}-\frac{1}{2}\sum_{i=1}^{3}\mathrm{g}^{MN}g^{KL}\text{Tr}\big{[}F_{MK}^{(i)}F^{(i)}_{NL}\big{]}-\frac{1}{4}g^{MN}g^{KL}\big{[}F_{MK}^{(X)}F^{(X)}_{NL}\big{]}$ $\displaystyle\qquad+g^{MN}(D_{M}H)^{\dagger}(D_{N}H)+\bigg{[}\mu^{2}|H|^{2}-\frac{\lambda_{6}^{(H)}}{4}|H|^{4}\bigg{]}$ $\displaystyle\qquad+i\bar{\mathcal{Q}}_{3-}\Gamma^{M}D_{M}\mathcal{Q}_{3-}+i\bar{\mathcal{U}}_{3+}\Gamma^{M}D_{M}\mathcal{U}_{3+}-\bigg{[}\lambda_{6}^{(t)}\bar{\mathcal{Q}}_{3-}(i\sigma_{2}H^{\ast})\mathcal{U}_{3+}+\mathrm{h.c.}\bigg{]}\Bigg{\\}}$ (62) where $\sqrt{-g}=R^{2}\sin{\theta}$. In this model, the form of 6D action and matter content are almost the same with these of the $T^{2}/Z_{4}$ except the existence of the $U(1)_{X}$ gauge field and $F_{MN}^{(X)}$ has the classical part arising from the monopole-like configuration as $F_{\theta\phi}=-\frac{n}{2g_{6}^{(X)}}\sin{\theta},\quad(\text{other components})=0.$ (63) The covariant derivative of the Higgs is given in an ordinary form as $D_{M}=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M},$ (64) and the covariant derivatives of fermions are obtained as follows: $D_{M}=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M}-ig_{6}^{(X)}q_{\Psi}(X^{c}_{M}+X_{M})+\Omega_{M}.$ (65) $q_{\Psi}$ is a $U(1)_{X}$ charge of a fermion and $X^{c}_{M}$ is the monopole-like classical configuration in Eq. (52). The other additional term $\Omega_{M}$ is the spin connection in $S^{2}$, whose concrete form is $(\Omega_{\phi})^{N\atop S}=\frac{i}{2}(\cos{\theta}\mp 1)\begin{pmatrix}1_{4}&0\\\ 0&-1_{4}\end{pmatrix},\quad(\text{other components})=0,$ (66) where $1_{4}$ is a four-by-four unit matrix. We can easily construct mode functions of $S^{2}/Z_{2}$ $f_{s,t}^{(j,m)}(\theta,\phi)$ with spin weight $s$ in both north and south patches following the general prescription in Ref. Georgi:2000ks as follows: $f_{s,t}^{(j,m)}(\theta,\phi)^{N\atop S}=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{2R}\left[{}_{s}Y_{jm}(\theta,\phi)+(-1)^{j-s}{}_{s}Y_{j-m}(\theta,\phi)\right]e^{\pm is\phi}&\text{for}\ t=+1\\\ \displaystyle\frac{1}{2R}\left[{}_{s}Y_{jm}(\theta,\phi)-(-1)^{j-s}{}_{s}Y_{j-m}(\theta,\phi)\right]e^{\pm is\phi}&\text{for}\ t=-1\end{array}\right.,$ (67) where $t=\pm 1$ is the $Z_{2}$ parity. These mode functions have the property that $f_{s,t=\pm 1}^{(j,m)}(\pi-\theta,-\phi)^{N\atop S}=\pm f_{s,t=\pm 1}^{(j,m)}(\theta,\phi)^{S\atop N}$. To realize the $Z_{2}$ symmetry, we identify a field at $(\theta,\phi)$ in the north patch with the same field at $(\pi-\theta,-\phi)$ in the south patch. The conditions are as follows: $\displaystyle H(x,\pi-\theta,-\phi)^{N\atop S}$ $\displaystyle=+H(x,\theta,\phi)^{S\atop N},$ (68) $\displaystyle\\{A^{(i)}_{\mu},X_{\mu}\\}(x,\pi-\theta,-\phi)^{N\atop S}$ $\displaystyle=+\\{A^{(i)}_{\mu},X_{\mu}\\}(x,\theta,\phi)^{S\atop N},$ (69) $\displaystyle\\{A^{(i)}_{\theta,\phi},X_{\theta,\phi}\\}(x,\pi-\theta,-\phi)^{N\atop S}$ $\displaystyle=-\\{A^{(i)}_{\theta,\phi},X_{\theta,\phi}\\}(x,\theta,\phi)^{S\atop N},$ (70) $\displaystyle\\{\mathcal{Q}_{3-},\mathcal{U}_{3+}\\}(x,\pi-\theta,-\phi)^{N\atop S}$ $\displaystyle=+i\Gamma^{\underline{y}}\Gamma^{\underline{z}}\\{\mathcal{Q}_{3-},\mathcal{U}_{3+}\\}(x,\theta,\phi)^{S\atop N},$ (71) where we take the choice that all gauge twist matrices are trivial ($P={\bf 1}$). And we define the transformation of 6D Weyl fermion $\Psi_{\pm}$ from the north to the south patch as $\Psi^{S}_{\pm}(x,\theta,\phi)=\exp(iq_{\Psi_{\pm}}\alpha+2\phi\Sigma^{\underline{y}\underline{z}})\Psi^{N}_{\pm}(x,\theta,\phi){,}$ (72) where $\alpha$ is the $U(1)_{X}$ gauge transformation function in Eq. (53) and $\Sigma^{\underline{y}\underline{z}}$ is the $(\underline{y},\underline{z})$ component of the local Lorentz generator of a 6D Weyl fermion.131313In our notation, $\Sigma^{\underline{y}\underline{z}}=\frac{-i}{2}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$. The Higgs does not transform along the patches because the Higgs does not have spin and interaction with the $U(1)_{X}$ gauge field. By use of the above facts and some specific information of this model,141414We can find the details in Ref. Dohi:2010vc . we can show that the action in Eq. (62) is equal at both the north and the south patches. Combining this result with Eqs. (68)-(71), it is clear that the $Z_{2}$ symmetry is entailed on the action in Eq. (62). The specific forms of each KK expansion are as follows: $\displaystyle\\{A^{(i)}_{\mu},X_{\mu}\\}(x,\theta,\phi)^{N\atop S}$ $\displaystyle=\frac{1}{\sqrt{4\pi}R}\\{A^{(i)(0)}_{\mu},X^{(0)}_{\mu}\\}(x)$ $\displaystyle\quad+\sum_{j=1}^{\infty}\sum_{m=0}^{j}\\{A^{(i)(j,m)}_{\mu},X^{(j,m)}_{\mu}\\}(x)\cdot(\sqrt{2}(i)^{j+m})f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop S},$ (73) $\displaystyle\\{A^{(i)}_{\pm},X_{\pm}\\}(x,\theta,\phi)^{N\atop S}$ $\displaystyle=\sum_{j=1}^{\infty}\sum_{m=0}^{j}\\{A^{(i)(j,m)}_{\pm},X^{(j,m)}_{\pm}\\}(x)\cdot(\sqrt{2}(i)^{j+m+1})f_{s=\pm 1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop S},$ (74) $\displaystyle H(x,\theta,\phi)^{N\atop S}$ $\displaystyle=\frac{1}{\sqrt{4\pi}R}H^{(0)}(x)+\sum_{j=1}^{\infty}\sum_{m=0}^{j}H^{(j,m)}(x)\cdot\sqrt{2}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop S},$ (75) $\displaystyle\mathcal{Q}_{3-}(x,\theta,\phi)^{N\atop S}$ $\displaystyle=\begin{pmatrix}\displaystyle\frac{1}{\sqrt{4\pi}R}Q_{3L}^{(0)}(x)+\sum_{j=1}^{\infty}\sum_{m=0}^{j}Q_{3L}^{(j,m)}(x)\cdot\sqrt{2}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop S}\\\ \displaystyle\sum_{j=1}^{\infty}\sum_{m=0}^{j}Q_{3R}^{(j,m)}(x)\cdot\sqrt{2}f_{s=-1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop S}\end{pmatrix},$ (76) $\displaystyle\mathcal{U}_{3+}(x,\theta,\phi)^{N\atop S}$ $\displaystyle=\begin{pmatrix}\displaystyle\frac{1}{\sqrt{4\pi}R}t_{R}^{(0)}(x)+\sum_{j=1}^{\infty}\sum_{m=0}^{j}t_{R}^{(j,m)}(x)\cdot\sqrt{2}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop S}\\\ \displaystyle\sum_{j=1}^{\infty}\sum_{m=0}^{j}t_{L}^{(j,m)}(x)\cdot\sqrt{2}f_{s=-1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop S}\end{pmatrix}.$ (77) Here we introduce suitable normalization factor $(\sqrt{2})$ in each KK modes and some phase factors ($(i)^{j+m},(i)^{j+m+1}$) in Eqs. (73,74) to ensure the reality of these fields. The range of the summation over $m$ shrinks from $[-j,j]$ to $[0,j]$ after the $Z_{2}$ identification. This system has two fixed points of the $Z_{2}$ symmetry at $(\theta,\phi)=(\frac{\pi}{2},0),(\frac{\pi}{2},\pi)$ and under the transformation of $(\theta,\phi)\rightarrow(\theta,\phi+\pi)$, mode functions behave as $\displaystyle f_{s=0,t=+1}^{(j,m)}(\theta,\phi+\pi)^{N\atop S}$ $\displaystyle=(-1)^{m}f_{s=0,t=+1}^{(j,m)}(\theta,\phi)^{N\atop S},$ $\displaystyle f_{s=\pm 1,t=-1}^{(j,m)}(\theta,\phi+\pi)^{N\atop S}$ $\displaystyle=-(-1)^{m}f_{s=\pm 1,t=-1}^{(j,m)}(\theta,\phi)^{N\atop S}.$ (78) Thereby after the fields redefinition as $\displaystyle\\{A^{(i)(j,m)}_{\pm},X^{(j,m)}_{\pm},Q^{(j,m)}_{3R},t^{(j,m)}_{L}\\}\rightarrow(-1)\\{A^{(i)(j,m)}_{\pm},X^{(j,m)}_{\pm},Q^{(j,m)}_{3R},t^{(j,m)}_{L}\\},$ (79) we can find that each KK field has KK parity $(-1)^{m}$, whose origin is considered to be a remnant of KK angular momentum conservation. We focus on the $m=0$ modes of each $j$ level. When we see the concrete forms of mode functions in $m=0$, which are $\displaystyle f_{s=0,t=+1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$ $\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{0}Y_{j0}(\theta,\phi),$ (80) $\displaystyle f_{s=+1,t=-1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$ $\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{1}Y_{j0}(\theta,\phi)e^{\pm i\phi},$ (81) $\displaystyle f_{s=-1,t=-1}^{(j,m=0)}(\theta,\phi)^{N\atop S}$ $\displaystyle=\frac{1}{2R}(1+(-1)^{j})\cdot{}_{-1}Y_{j0}(\theta,\phi)e^{\mp i\phi},$ (82) we find that $m=0$ modes appear only in the case of even $j$. Then the degeneracy of KK masses is $\begin{array}[]{cl}j+1&\text{for}\quad j=\text{even},\\\ j&\text{for}\quad j=\text{odd},\end{array}$ (83) since $m$ runs from $0$ to $j$. These results play an essential role at the Higgs production and decay processes via loop diagrams. After the $Z_{2}$ identification, the massless zero mode of $U(1)_{X}$ gauge boson survives. In this model, it is assumed that the $U(1)_{X}$ symmetry is anomalous and it is broken at the quantum level Scrucca:2003ra . Therefore gauge bosons should be heavy and decoupled from the low energy physics. ### 4.3 UED on Projective Sphere We can also construct a UED model based on a non-orbifolding idea in Ref. Dohi:2010vc .151515 In Dohi:2010vc the terminology “real projective plane” is employed for the compactified space, the sphere with its antipodal points being identified. The projective sphere $(PS)$ is a sphere $S^{2}$ with its antipodal points identified by $(\theta,\phi)\sim(\pi-\theta,\phi+\pi)$. In the UED model based on ${PS}$, the 6D action takes a different form from that of ordinary 6D UED model. It is written as follows: $\displaystyle S$ $\displaystyle=\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\int d^{4}x\sqrt{-g}\Bigg{\\{}-\frac{1}{2}\sum_{i=1}^{3}\mathrm{g}^{MN}g^{KL}\text{Tr}\big{[}F_{MK}^{(i)}F^{(i)}_{NL}\big{]}-\frac{1}{4}g^{MN}g^{KL}\big{[}F_{MK}^{(X)}F^{(X)}_{NL}\big{]}$ $\displaystyle\qquad+g^{MN}(D_{M}H)^{\dagger}(D_{N}H)+\bigg{[}\mu^{2}|H|^{2}-\frac{\lambda_{6}^{(H)}}{4}|H|^{4}\bigg{]}$ $\displaystyle\qquad+\frac{1}{2}\bigg{[}i\bar{\mathcal{Q}}_{3-}\Gamma^{M}D_{M}\mathcal{Q}_{3-}+i\bar{\mathcal{Q}}_{3+}\Gamma^{M}D_{M}\mathcal{Q}_{3+}\bigg{]}+\frac{1}{2}\bigg{[}i\bar{\mathcal{U}}_{3+}\Gamma^{M}D_{M}\mathcal{U}_{3+}+i\bar{\mathcal{U}}_{3-}\Gamma^{M}D_{M}\mathcal{U}_{3-}\bigg{]}$ $\displaystyle\qquad-\frac{1}{2}\bigg{[}\lambda_{6}^{(t)}\Big{(}\bar{\mathcal{Q}}_{3-}(i\sigma_{2}H^{\ast})\mathcal{U}_{3+}+\bar{\mathcal{Q}}_{3+}(i\sigma_{2}H^{\ast})^{\text{T}}\mathcal{U}_{3-}\Big{)}+\mathrm{h.c.}\bigg{]}\Bigg{\\}}.$ (84) Here the $``1/2"$ factors are introduced for a later convenience. Like the $S^{2}/Z_{2}$ case, $F_{MN}^{(X)}$ has the classical part. A new feature of this model is that we introduce “mirror” 6D Weyl fermions $\\{\mathcal{Q}_{3+},\mathcal{U}_{3-}\\}$, which have opposite 6D chirality and opposite SM and $U(1)_{X}$ charges when compared with the fields $\\{\mathcal{Q}_{3-},\mathcal{U}_{3+}\\}$, respectively. And the covariant derivatives in this model are given as $\displaystyle D_{M}$ $\displaystyle=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M},$ $\displaystyle\text{for}\quad H,$ (85) $\displaystyle D_{M}$ $\displaystyle=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}T^{(i)a}A^{(i)a}_{M}-ig_{6}^{(X)}q_{\Psi}(X^{c}_{M}+X_{M})+\Omega_{M},$ $\displaystyle\text{for}\quad\mathcal{Q}_{3-},\mathcal{U}_{3+},$ (86) $\displaystyle D_{M}$ $\displaystyle=\partial_{M}-i\sum_{i=1}^{3}g_{6}^{(i)}\big{[}-T^{(i)a}\big{]}^{\text{T}}A^{(i)a}_{M}-ig_{6}^{(X)}q_{\Psi}(X^{c}_{M}+X_{M})+\Omega_{M},$ $\displaystyle\text{for}\quad\mathcal{Q}_{3+},\mathcal{U}_{3-}.$ (87) The covariant derivative of the Higgs is the same with that in the $S^{2}/Z_{2}$ case, but there is a difference between fermions and these “mirror” fermions. We discuss these points shortly below. ${PS}$ is a non-orientable manifold and has no fixed point. Therefore, we cannot perform identification like the $S^{2}/Z_{2}$ case. We focus on the 6D $P$ and $CP$ transformations, which are defined as $[\text{6D}\ P]=\left\\{\begin{array}[]{lcr}A_{\mu}(x,\theta,\phi)&\rightarrow&A_{\mu}(x,\pi-\theta,\phi+\pi),\\\ A_{\theta}(x,\theta,\phi)&\rightarrow&-A_{\theta}(x,\pi-\theta,\phi+\pi),\\\ A_{\phi}(x,\theta,\phi)&\rightarrow&A_{\phi}(x,\pi-\theta,\phi+\pi),\\\ \Psi(x,\theta,\phi)&\rightarrow&P\Psi(x,\pi-\theta,\phi+\pi),\\\ H(x,\theta,\phi)&\rightarrow&H(x,\pi-\theta,\phi+\pi),\end{array}\right.$ (88) $[\text{6D}\ CP]=\left\\{\begin{array}[]{lcr}A_{\mu}(x,\theta,\phi)&\rightarrow&A_{\mu}^{C}(x,\pi-\theta,\phi+\pi),\\\ A_{\theta}(x,\theta,\phi)&\rightarrow&-A_{\theta}^{C}(x,\pi-\theta,\phi+\pi),\\\ A_{\phi}(x,\theta,\phi)&\rightarrow&A_{\phi}^{C}(x,\pi-\theta,\phi+\pi),\\\ \Psi(x,\theta,\phi)&\rightarrow&P\Psi^{C}(x,\pi-\theta,\phi+\pi),\\\ H(x,\theta,\phi)&\rightarrow&H^{\ast}(x,\pi-\theta,\phi+\pi).\end{array}\right.$ (89) Like before, we consider $\Psi$ is a 6D fermion and the concrete shapes of 6D $C$ and $P$ transformations are $A_{M}^{C}=-A_{M}^{\text{T}}=-A_{M}^{\ast},\quad\Psi^{C}=\Gamma^{\underline{2}}\Gamma^{\underline{y}}\Psi^{\ast},\quad P=\Gamma^{\underline{y}}.$ (90) It must be noted that the monopole-like configuration of the $U(1)_{X}$ gauge boson in Eq. (52) behaves under the antipodal identification as $\\{X^{c}_{\phi}\\}^{N\atop S}(x,\pi-\theta,\phi+\pi)=-\\{X^{c}_{\phi}\\}(x,\theta,\phi)^{S\atop N}=\\{(X^{c}_{\phi})^{C}\\}(x,\theta,\phi)^{S\atop N}.$ (91) We use a property of $U(1)$ gauge field $(X_{M}^{\text{T}}=X_{M})$. We can notice that the monopole-like configuration is invariant under the 6D $CP$ transformation and transition between patches. Then we consider the identification of the $U(1)_{X}$ gauge field as 161616We pay attention the fact that identification conditions of classical field $(X_{\phi}^{c})$ and quantum field $(X_{\phi})$ must be the same. $\left\\{\begin{array}[]{lcc}X_{\mu}(x,\pi-\theta,\phi+\pi)^{N\atop S}&=&X_{\mu}^{C}(x,\theta,\phi)^{S\atop N},\\\ X_{\theta}(x,\pi-\theta,\phi+\pi)^{N\atop S}&=&-X_{\theta}^{C}(x,\theta,\phi)^{S\atop N},\\\ \\{X_{\phi}^{c},X_{\phi}\\}(x,\pi-\theta,\phi+\pi)^{N\atop S}&=&\\{(X_{\phi}^{c})^{C},X_{\phi}^{C}\\}(x,\theta,\phi)^{S\atop N}.\end{array}\right.$ (92) These conditions ensure the monopole-like configuration after the antipodal identification and projected out the non-desirable $U(1)_{X}$ zero mode. It is clearly understood by the additional minus factor coming from the 6D CP transformation of gauge field in Eq. (90). In contrast, since we want the zero modes which describe the SM gauge bosons in UED model construction, identification of $A^{(i)}_{M}$ should be done by another condition. We adopt the 6D $P$ transformation and those identifications are written as $\left\\{\begin{array}[]{lcc}A^{(i)}_{\mu}(x,\pi-\theta,\phi+\pi)^{N\atop S}&=&A^{(i)}_{\mu}(x,\theta,\phi)^{S\atop N},\\\ A^{(i)}_{\theta}(x,\pi-\theta,\phi+\pi)^{N\atop S}&=&-A^{(i)}_{\theta}(x,\theta,\phi)^{S\atop N},\\\ A^{(i)}_{\phi}(x,\pi-\theta,\phi+\pi)^{N\atop S}&=&A^{(i)}_{\phi}(x,\theta,\phi)^{S\atop N},\end{array}\right.$ (93) where it is evident that $A_{\mu}^{(i)}$’s zero mode survives. We also identify the Higgs with the 6D $P$ transformation to obtain its zero mode as $H(x,\pi-\theta,\phi+\pi)^{N\atop S}=H(x,\theta,\phi)^{S\atop N}.$ (94) Finally, we discuss the identification of 6D Weyl fermions. Since 6D Weyl fermions have $U(1)_{X}$ charge and interact with the $U(1)_{X}$ gauge boson, they should be identified by the 6D $CP$ transformation. But if we do not consider the “mirror” fermions, a fundamental problem arises. The 6D $P$ transformation of fermion changes the 6D chirality like the ordinary 4D transformation. However, the 6D $C$ transformation of fermion does not change the 6D chirality unlike the ordinary 4D case. This means 6D chirality flips under the 6D $CP$ transformation and we should introduce the “mirror” fermions with opposite 6D chirality and opposite SM and $U(1)_{X}$ charges to perform identification. The specific forms are as follows: $\\{\mathcal{Q}_{3+},\mathcal{U}_{3-}\\}(x,\pi-\theta,\phi+\pi)^{N\atop S}=P\\{\mathcal{Q}_{3-}^{C},\mathcal{U}_{3+}^{C}\\}(x,\theta,\phi)^{S\atop N}=\Gamma^{\underline{2}}\\{\mathcal{Q}_{3-}^{\ast},\mathcal{U}_{3+}^{\ast}\\}(x,\theta,\phi)^{S\atop N}.$ (95) And we determine the forms of the covariant derivatives in Eqs. (86,87) on the criterion of invariance of the action under the 6D $CP$ transformation in advance. Using the identification conditions in Eqs. (92)-(95), we can see that the “mirror” fermions vanish from the action in Eq. (84) after the identifications as $\displaystyle S$ $\displaystyle\longrightarrow\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\int d^{4}x\sqrt{-g}\Bigg{\\{}-\frac{1}{2}\sum_{i=1}^{3}\mathrm{g}^{MN}g^{KL}\text{Tr}\big{[}F_{MK}^{(i)}F^{(i)}_{NL}\big{]}-\frac{1}{4}g^{MN}g^{KL}\big{[}F_{MK}^{(X)}F^{(X)}_{NL}\big{]}$ $\displaystyle\qquad+g^{MN}(D_{M}H)^{\dagger}(D_{N}H)+\bigg{[}\mu^{2}|H|^{2}-\frac{\lambda_{6}^{(H)}}{4}|H|^{4}\bigg{]}$ $\displaystyle\qquad+\bigg{[}i\bar{\mathcal{Q}}_{3-}\Gamma^{M}D_{M}\mathcal{Q}_{3-}\bigg{]}+\bigg{[}i\bar{\mathcal{U}}_{3+}\Gamma^{M}D_{M}\mathcal{U}_{3+}\bigg{]}-\bigg{[}\lambda_{6}^{(t)}\Big{(}\bar{\mathcal{Q}}_{3-}(i\sigma_{2}H^{\ast})\mathcal{U}_{3+}\Big{)}+\mathrm{h.c.}\bigg{]}\Bigg{\\}},$ (96) and we obtain a usual type of UED model action. Next we discuss the mass spectrum of the UED model on ${PS}$. Roughly speaking, about a half of modes are projected out. First, we focus on the $U(1)_{X}$ gauge boson. By use of properties of spin weighted spherical harmonics, we can conclude that its identification conditions in terms of 4D KK fields are as follows: $\displaystyle X_{\mu}^{(j,m)}(x)$ $\displaystyle=(-1)^{j}(X^{(j,m)}_{\mu})^{\text{c}}(x)=(-1)^{j+1}(X^{(j,m)}_{\mu})(x),$ (97) $\displaystyle X_{\pm}^{(j,m)}(x)$ $\displaystyle=(-1)^{j+1}(X^{(j,m)}_{\mp})^{\text{c}}(x),$ (98) $\displaystyle\phi^{(X)(j,m)}_{1}(x)$ $\displaystyle=(-1)^{j+1}(\phi^{(X)(j,m)}_{1})^{\text{c}}(x)=(-1)^{j}(\phi^{(X)(j,m)}_{1})(x),$ (99) $\displaystyle\phi^{(X)(j,m)}_{2}(x)$ $\displaystyle=(-1)^{j}(\phi^{(X)(j,m)}_{2})^{\text{c}}(x)=(-1)^{j+1}(\phi^{(X)(j,m)}_{2})(x),$ (100) where the superscript ${}^{\text{c}}$ means 4D charge conjugation and has the property that $(X_{M}^{(j,m)})^{\text{c}}(x)=-(X_{M}^{(j,m)})^{\text{T}}$. $\phi_{1,2}^{(X)}$ are a 4D physical scalar field and an unphysical would-be Nambu-Goldstone mode of $U(1)_{X}$ gauge field, respectively in Eq. (59). In Eq. (97), it is clear that its unwanted zero mode is projected out correctly. In ${PS}$ case, the range of the summation over $m$ does not shrink under the identification and is still $[-j,j]$. This means that degeneracy of KK masses is $2j+1$ in this model. But from Eqs. (97)-(100), we can find that the even $j$ modes of both $X_{\mu}^{(j,m)}$ and $\phi^{(X)(j,m)}_{2}$ and the odd $j$ modes of $\phi^{(X)(j,m)}_{1}$ are projected out. The structure of these mass spectrums is one of the most characteristic feature in the UED model on ${PS}$ and influences the rates of the Higgs production and decay processes via loop diagrams. Next, we go on to the gauge bosons $A_{M}^{(i)}$ and the Higgs $H$. These field are identified by the 6D $P$ transformation and its identification conditions in terms of 4D KK fields are as follows: $\displaystyle A_{\mu}^{(i)(j,m)}(x)$ $\displaystyle=(-1)^{j}(A^{(i)(j,m)}_{\mu})(x),$ (101) $\displaystyle A_{\pm}^{(i)(j,m)}(x)$ $\displaystyle=(-1)^{j+1}({A}^{(i)(j,m)}_{\mp})(x),$ (102) $\displaystyle\phi^{(i)(j,m)}_{1}(x)$ $\displaystyle=(-1)^{j+1}(\phi^{(i)(j,m)}_{1})(x),$ (103) $\displaystyle\phi^{(i)(j,m)}_{2}(x)$ $\displaystyle=(-1)^{j}(\phi^{(i)(j,m)}_{2})(x).$ (104) $H^{(j,m)}(x)=(-1)^{j}H^{(j,m)}(x).$ (105) From Eqs. (101)-(105), it is obvious that the even $j$ modes of $\phi^{(i)(j,m)}_{1}$ and the odd $j$ modes of $A_{\mu}^{(i)(j,m)},\phi^{(i)(j,m)}_{2}$ and $H^{(j,m)}$ are projected out. As a previous argument, the zero modes of $A_{\mu}^{(i)(j,m)}$ do not vanish. Finally, 6D Weyl fermion must be discussed. It is important that the “mirror” fermions are completely projected out from the action in Eq. (96) after the antipodal identification. This is interpreted that all modes of the “mirror” fermions $\\{\mathcal{Q}_{3+},\mathcal{U}_{3-}\\}$ are erased and no mode of $\\{\mathcal{Q}_{3-},\mathcal{U}_{3+}\\}$ is projected out. We comment on the dark matter candidate briefly. In this model, there is no KK-parity because of lack of fixed points. But alternatively, the conservation of KK angular momentum exists and it implies that the lightest KK particle is stable. ## 5 Naive Dimensional Analysis In 6D UED models, since the gluon fusion Higgs production and Higgs decay to two photons processes are logarithmically divergent, we must consider upper limit of the summations of KK number in such models. We review Naive Dimensional Analysis (NDA) in these 6D models briefly. Following the concept of NDA, a loop expansion parameter $\epsilon$ in D-dimensional SU(N) gauge theory at a scale $\mu$ is obtained as $\displaystyle\epsilon{(\mu)}$ $\displaystyle=\frac{1}{2}\frac{2\pi^{D/2}}{(2\pi)^{D}\Gamma(D/2)}N_{g}\,g_{Di}^{2}(\mu)\,{\Lambda_{\text{UED}}}^{D-4},$ (106) where $N_{g}$ is a group index, $g_{Di}$ is a gauge coupling in $D$-dimensions and ${\Lambda_{\text{UED}}}$ is a cutoff scale. The index $i$ is introduced to express the type of gauge interaction and the remaining part is originated from D-dimensional momentum loop integral. We should mention that $g_{Di}({\mu})$, which is the effective running coupling, has energy dependency and obeys power-of-two law scaling. When we consider a 6D theory $(D=6)$ with two compact spacial directions, an effective 4D gauge coupling $g_{i}$ emerges after KK decomposition and it is described with the volume of two extra dimensions $V_{2}$ as $g_{i}={g_{6i}}/{\sqrt{V_{2}}}$. The cutoff scale $\mu$ is the scale where the perturbation breaks down $\epsilon({\Lambda_{\text{UED}}})\sim 1$. It is obvious that the upper bound of ${\Lambda_{\text{UED}}}$ depends on the value of $V_{2}$, whose value is $(2\pi R)^{2}$ in $T^{2}$ and $4\pi R^{2}$ in $S^{2}$, where $R$ is the radius of $T^{2}$ or $S^{2}$. Next we would like to focus on the behavior of running of the 4D effective gauge coupling strength $\alpha_{i}({\Lambda_{\text{UED}}})$ along the energy. We consider the following renormalization group equation: $\displaystyle{\alpha_{4i}^{{-1}}(\mu)=\alpha_{4i}^{{-1}}(m_{Z})-\frac{\textsf{b}_{i}^{\text{SM}}}{2\pi}\ln{\mu\over m_{Z}}+2C\,{\textsf{b}_{i}^{\text{6D}}\over 2\pi}\ln{\mu\over M_{\text{KK}}}-C\,\frac{\textsf{b}_{i}^{\text{6D}}}{2\pi}\left[\left(\frac{\mu}{M_{\text{KK}}}\right)^{2}-1\right],}$ (107) where $C$ represents $\pi/2\,(1)$ in the case of $T^{2}$ ($S^{2}$) geometry. We note that the coefficient of the quadratic term for $T^{2}$ coincides with that in Refs. Dienes:1998vh ; Dienes:1998vg obtained from a different regularization scheme. The value of $C$ differs due to the structure of the background geometry.171717Readers who are interested in the details see Appendix in Ref. Nishiwaki:2011gm . In Eq. (107), we take a scheme of approximation; masses of particles are almost degenerated in each KK level regardless of type of the fields, the effect of KK particles appears after the reference energy $\mu$ exceeds the value of $M_{\text{KK}}$.181818 When we consider PS model with non-orientable manifold, there are differences in KK spectrum of gauge and Higgs fields compared to that of the other “ordinary” UED models as we discussed before. We ignore the effect coming from this in our analysis. The coefficients are summarized in Table. 1.191919 Note that we do not employ the GUT normalization for the $U(1)_{Y}$ coupling and the beta function. gauge group | SM contribution ($\textsf{b}^{\text{SM}}_{i}$) | KK contribution ($\textsf{b}^{\text{6D}}_{i}$) ---|---|--- $SU(3)_{C}$ | $\displaystyle-7$ | $-2$ $SU(2)_{W}$ | $\displaystyle-{19}/{6}$ | $\displaystyle{3}/{2}$ $U(1)_{Y}$ | $\displaystyle{41/6}$ | $\displaystyle{27}/{2}$ Table 1: Coefficients of renormalization group equation in Eq. (107). Considering only the quadratic term, Eq. (107) reads $\displaystyle\alpha_{4i}^{-1}(\Lambda_{\text{UED}})\sim\alpha_{4i}^{-1}({m_{Z}})-\frac{C\textsf{b}^{\text{6D}}_{i}}{{2}\pi}\frac{\Lambda_{\text{UED}}^{2}}{M_{\text{KK}}^{2}}.$ (108) From Eq. (108) and $\epsilon({\Lambda_{\text{UED}}})\sim 1$, we get $\displaystyle\Lambda_{\text{UED}}^{2}\sim{{4\pi M_{\text{KK}}^{2}\over C\left(N_{g}+2\textsf{b}_{i}^{\text{6D}}\right)\alpha_{4i}(m_{Z})},}$ (109) In the above analysis, we take values of $N_{g}$ as $3$, $2$ and $1$ in each case of $SU(3)_{C}$, $SU(2)_{W}$ and $U(1)_{Y}$, respectively, and adopt some latest data announced by Particle Data Group (PDG) as $\left\\{\begin{array}[]{rcl}\alpha_{U(1)_{Y}}(m_{Z})^{-1}\big{|}_{\text{MS}}&=&97.99,\\\ \alpha_{SU(2)_{W}}(m_{Z})^{-1}\big{|}_{\text{MS}}&=&29.46,\\\ \alpha_{SU(3)_{C}}(m_{Z})^{-1}\big{|}_{\text{MS}}&=&8.445,\\\ m_{Z}&=&91.18\,\text{[GeV]}.\end{array}\right.$ (110) We do not consider “TeV-scale gauge coupling unification” in this paper. In the both $T^{2}$ and $S^{2}$ cases, the most stringent bounds come from the $U(1)_{Y}$ cutoff scales, which restrict the effective range of the perturbation the most severely. Therefore we can conclude that the “cutoff” scales are as follows: $\displaystyle{\Lambda_{\text{UED}}}$ $\displaystyle\lesssim{{5.3}\,M_{\text{KK}}},$ $\displaystyle\text{for $T^{2}$-case}\ (V_{2}=(2\pi R)^{2},\,M_{\text{KK}}=1/R),$ (111) $\displaystyle{\Lambda_{\text{UED}}}$ $\displaystyle\lesssim{{6.6}\,M_{\text{KK}}},$ $\displaystyle\text{for $S^{2}$-case}\ (V_{2}=4\pi R^{2},\,M_{\text{KK}}=\sqrt{2}/R).$ (112) We truncate the KK mode summations up to these upper bounds in each case to regularize the process. Before going on to the concrete calculation, we have to declare our choice of the UED cutoff scales. We choose three patterns in $T^{2}$ and $S^{2}$ cases separately and the concrete forms are summarized in Table 2. We also list up the value of the QCD and electromagnetic coupling strengths ${\\{}\alpha_{s},\alpha_{\text{EM}}{\\}}$ at the cutoff scales by use of Eq. (108) in Table 3. It is noted that the values derived form Eq. (108) do not depend on the value of the KK mass scale $M_{\text{KK}}$ up to our approximation in Eq. (108). The electromagnetic coupling strength is defined by using $\alpha_{SU(2)_{W}}$ and $\alpha_{U(1)_{Y}}$ as $\displaystyle\alpha_{\text{EM}}(\mu)^{-1}=\alpha_{SU(2)_{W}}(\mu)^{-1}+\alpha_{U(1)_{Y}}(\mu)^{-1}.$ (113) | $T^{2}$-based | $S^{2}$-based ---|---|--- | high | low | high | low KK index | $m^{2}+n^{2}\leq 30$ | $m^{2}+n^{2}\leq 10$ | $j(j+1)\leq 100$ | $j(j+1)\leq 30$ UV cutoff | $\Lambda_{\text{UED}}\sim 5{M_{\text{KK}}}$ | $\Lambda_{\text{UED}}\sim 3{M_{\text{KK}}}$ | $\Lambda_{\text{UED}}\sim 7{M_{\text{KK}}}$ | $\Lambda_{\text{UED}}\sim 4{M_{\text{KK}}}$ Table 2: Two choices of high and low upper bounds for KK indices and for the corresponding UV cutoff scale. | $T^{2}$-based | $S^{2}$-based ---|---|--- | high | low | high | low $\alpha_{s}(\Lambda_{\text{UED}})^{{-1}}$ | $20.9$ | $12.9$ | $24.0$ | $13.5$ $\alpha_{\text{EM}}(\Lambda_{\text{UED}})^{{-1}}$ | $33.7$ | $93.7$ | $10.5$ | $89.3$ Table 3: The value of the QCD and electromagnetic coupling strengths at the cutoff scales. ## 6 The deviation of the rates of Higgs production and its decay from the standard model predictions ### 6.1 Formulation of calculation From the discussions which we have done, we evaluate the ratio (fractional deviation) of the Higgs production cross section through gluon fusion and the Higgs decay width into two photons to the SM ones in the three types of 6D UED models, which are denoted by $\mathcal{R}_{2g\rightarrow h^{(0)}}$ and $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$, respectively. These ratio are represented as $\mathcal{R}_{2g\rightarrow h^{(0)}}\equiv\frac{\sigma(2g\rightarrow h^{(0)};\ \text{UED})}{\sigma(2g\rightarrow h^{(0)};\ \text{SM})}=\left(1+{F_{t}^{{\text{KK}}}{+F_{\text{gluonfusion}}^{\text{TC}}}\over F_{t}^{{\text{SM}}}}\right)^{2},$ (114) $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}\equiv\frac{\Gamma(h^{(0)}\rightarrow 2\gamma;\ \text{UED})}{\Gamma(h^{(0)}\rightarrow 2\gamma;\ \text{SM})}=\left(1+{F_{W}^{{\text{KK}}}+3Q_{t}^{2}F_{t}^{{\text{KK}}}{+F_{\text{decay}}^{\text{TC}}}\over F_{W}^{{\text{SM}}}+3Q_{t}^{2}F_{t}^{{\text{SM}}}}\right)^{2}.$ (115) We have obtained $F_{W}^{{\text{KK}}},F_{t}^{{\text{KK}}},{F_{\text{gluonfusion}}^{\text{TC}}}$ and ${F_{\text{decay}}^{\text{TC}}}$ in Section 3 in the case of $T^{2}/Z_{4}$ by 1-loop calculation and we can apply these results for the $S^{2}/Z_{2}$ and the ${PS}$ cases with some modifications. It is important that the $U(1)_{X}$ gauge boson does not contribute to either the production process and the decay process at the 1-loop level. Therefore no new type of diagram appears and only difference appears in the KK mass spectrum and the multiplicity of each KK mode. Once the $Z_{2}$ orbifolding or the antipodal identification is understood, the structure of KK mass spectrum itself is the same as the case of $S^{2}$ up to degeneracy. We summarize the information which is needed for the estimation in Table 4. type of field | $S^{2}/Z_{2}$ case | ${PS}$ case ---|---|--- fermion | $\begin{array}[]{cl}j+1&\text{for}\ j=\text{even}\\\ j&\text{for}\ j=\text{odd}\end{array}$ | $\begin{array}[]{cl}2j+1&\text{for}\ j=\text{even}\\\ 2j+1&\text{for}\ j=\text{odd}\end{array}$ “mirror” fermion | N/A | $\begin{array}[]{cl}0&\text{for}\ j=\text{even}\\\ 0&\text{for}\ j=\text{odd}\end{array}$ | gauge boson & would-be NG boson --- & scalar(Higgs) $\begin{array}[]{cl}j+1&\text{for}\ j=\text{even}\\\ j&\text{for}\ j=\text{odd}\end{array}$ | $\begin{array}[]{cl}2j+1&\text{for}\ j=\text{even}\\\ 0&\text{for}\ j=\text{odd}\end{array}$ scalar(“spinless adjoint”) | $\begin{array}[]{cl}j+1&\text{for}\ j=\text{even}\\\ j&\text{for}\ j=\text{odd}\end{array}$ | $\begin{array}[]{cl}0&\text{for}\ j=\text{even}\\\ 2j+1&\text{for}\ j=\text{odd}\end{array}$ Table 4: Multiplicities of fields at j level in $S^{2}$-based UED models. In the $S^{2}/Z_{2}$ case, the KK state multiplicity is the same irrespective of the type of field. With the modification ${\sum_{m\geq 1,n\geq 0}\rightarrow\sum_{j\geq 1}}\ n_{{S^{2}/Z_{2}}}(j),\quad m_{(m,n)}\rightarrow m_{(j,m)},$ (116) where $n_{S^{2}/Z_{2}}(j)$ shows the multiplicity of each level of KK modes ($j+1$ for $j=\text{even}$ or $j$ for $j=\text{odd}$) and $m_{(j,m)}$ is the KK mass on $S^{2}$ in Eq. (60), we can obtain the results as follows: $\displaystyle F_{t}^{{\text{KK}}}$ $\displaystyle=2\sum_{j\geq 1}n_{S^{2}/Z_{2}}(j)\left(\frac{m_{t}}{m_{t,(j,m)}}\right)^{2}$ $\displaystyle\quad\times\left\\{{-2}\lambda(m_{t,(j,m)}^{2}){+}\lambda(m_{t,(j,m)}^{2})(1-4\lambda(m_{t,(j,m)}^{2}))J\left(\lambda(m_{t,(j,m)}^{2})\right)\right\\},$ (117) $\displaystyle F_{W}^{{\text{KK}}}$ $\displaystyle=\sum_{j\geq 1}n_{S^{2}/Z_{2}}(j)\Big{\\{}\frac{1}{2}+5\lambda(m_{W}^{2})-\Big{[}\lambda(m_{W}^{2})(4-10\lambda(m_{W,(j,m)}^{2}))$ $\displaystyle\qquad\qquad-\lambda(m_{W,(j,m)}^{2})\Big{]}J\left(\lambda(m_{W,(j,m)}^{2})\right)\Big{\\}}{,}$ (118) where we use $J(m^{2})$ in Eq. (36). In ${PS}$ case, we should pay attention to the KK state multiplicity of each type of field. There is no contribution from the “mirror” fermions. The concrete forms are as follows: $\displaystyle{F_{t}^{\text{KK}}}$ $\displaystyle=2\sum_{j\geq 1}(2j+1)\left(\frac{m_{t}}{m_{t,(j,m)}}\right)^{2}$ $\displaystyle\quad\times\left\\{{-2}\lambda(m_{t,(j,m)}^{2}){+}\lambda(m_{t,(j,m)}^{2})(1-4\lambda(m_{t,(j,m)}^{2}))J\left(\lambda(m_{t,(j,m)}^{2})\right)\right\\},$ (119) $\displaystyle F^{{\text{KK}}}_{\text{gauge}}$ $\displaystyle={\sum_{j\geq 1}n_{PS\text{even}}(j)\left\\{3\lambda(m_{W}^{2})+2\lambda(m_{W}^{2})\big{(}3\lambda(m_{W,(j,m)}^{2})-2\big{)}J\left(\lambda(m_{W,(j,m)}^{2})\right)\right\\}},$ (120) $\displaystyle F^{{\text{KK}}}_{\text{NG}}$ $\displaystyle={\sum_{j\geq 1}n_{PS\text{even}}(j)\left(\frac{1}{2}\frac{m_{h}^{2}}{m_{W,(j,m)}^{2}}\right)\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(j,m)}^{2})J\left(\lambda(m_{W,(j,m)}^{2})\right)\right\\}},$ (121) $\displaystyle F^{{\text{KK}}}_{\text{scalar1}}$ $\displaystyle=\sum_{j\geq 1}n_{PS\text{even}}(j)\Big{(}\frac{1}{2}\frac{1}{m_{W,(j,m)}^{2}}\Big{)}\Big{[}\frac{m_{h}^{2}}{m_{W}^{2}}m_{(j,m)}^{2}+{2}m_{W,(j,m)}^{2}\Big{]}$ $\displaystyle\quad\times\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(j,m)}^{2})J\left(\lambda(m_{W,(j,m)}^{2})\right)\right\\},$ (122) $\displaystyle F^{{\text{KK}}}_{\text{scalar2}}$ $\displaystyle={\sum_{j\geq 1}n_{PS\text{odd}}(j)\lambda(m_{W}^{2})\left\\{1+2\lambda(m_{W,(j,m)}^{2})J\left(\lambda(m_{W,(j,m)}^{2})\right)\right\\}},$ (123) where $n_{PS\text{even}}(j)$: $2j+1$ for $j=$even, $0$ for $j=$odd and $n_{PS\text{odd}}(j)$: $0$ for $j=$even, $2j+1$ for $j=$odd. We have already discussed the cutoff scale in both the $T^{2}$ and $S^{2}$ cases concretely in Section 5 and we are ready to estimate the ratio in the various 6D UED models. ### 6.2 Results without threshold corrections The numerical results of the ratios of the production cross section via gluon fusion to the standard model prediction $\mathcal{R}_{2g\rightarrow h^{(0)}}$ are given as functions of the first KK mass scale $(M_{\text{KK}})$ in a unit of GeV in Fig. 5. In this paper, we consider two possibilities of Higgs mass; $m_{h}=120\,\text{GeV}$ and $m_{h}=145\,\text{GeV}$ and take the KK mass range between $600\,\text{GeV}$ and $2000\,\text{GeV}$. We use the values of the W boson mass $m_{W}$ and the top quark mass $m_{t}$, which are $m_{W}=80.3\,\text{GeV},m_{t}=173\,\text{GeV}$. From top to bottom, the green, blue, red curves represent the results of ${PS}$, $S^{2}/Z_{2}$, $T^{2}/Z_{4}$ with $m_{h}=120\,\,\text{GeV}$, providing no threshold correction, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. The left, right are these with the high, low cutoff choices, respectively. It is noted that the $\mathcal{R}_{2g\rightarrow h^{(0)}}=1$ shows the SM predictions and there are few differences between $m_{h}=120\,\text{GeV}$ case and $m_{h}=145\,\text{GeV}$ case in all the models. Contrast to the case of little Higgs Han:2003gf ; Dib:2003zj ; Chen:2006cs or gauge-Higgs unification Falkowski:2007hz ; Maru:2007xn ; Maru:2008cu , the contribution from KK fermions is constructive and the results of UED cases are enhanced compared to the SM prediction. These results are naturally understood because the number of the intermediate particles are much greater than these of the SM.202020These results are consistent with the results in Ref. Maru:2009cu . It is expected that 6D UED models predict a significant collider signature in the Higgs production at the LHC, especially in the ${PS}$ case. The origin of the remarkable enhancement in the ${PS}$ case is that numerous fermions contribute to the production process in each KK level. Besides, we can find the fact that when we choose the higher cutoff, the larger number of KK top modes propagate in the triangle loop and therefore the deviation from the SM gets significant. This tendency do not depend on the type of the 6D UED models. Figure 5: These plots represent the ratios of the Higgs boson production cross sections via gluon fusion to the SM prediction $\mathcal{R}_{2g\rightarrow h^{(0)}}$ in 6D UED on ${PS}$(green), $S^{2}/Z_{2}$(blue), $T^{2}/Z_{4}$(red) with $m_{h}=120\,\text{GeV}$ providing no threshold correction from top to bottom. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. The left, right are these with the high, low cutoff choices, respectively. The numerical results of the ratios of the rate of Higgs decay into two photons $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ are also given as functions of the first KK mass scale ($M_{\text{KK}}$) in a unit of GeV in Fig. 6. From bottom to top, the green, blue, red curves represent the results of ${PS}$, $S^{2}/Z_{2}$, $T^{2}/Z_{4}$ with $m_{h}=120\,\,\text{GeV}$, providing no threshold correction, respevtively. Each black dashed line located above the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. The left, right are these with the high, low cutoff choices, respectively. Differently from the production, the ratios are suppressed because the contributions from quarks and gauge bosons are destructive each other. The reason of the large reduction in the ${PS}$ case is understood as the results of the enormous effects of KK top quarks, which we discussed before. In any type of 6D UED models, this ratio takes the lower value than 5D mUED one. We can find some differences between $m_{h}=120\,\text{GeV}$ and $m_{h}=145\,\text{GeV}$ in each case of 6D UED model, which are sizable in particular at the KK mass range between $600\,\text{GeV}$ and $1200\,\text{GeV}$. Figure 6: These plots represent the ratios of the Higgs boson decay width to two photons to the SM prediction $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ in 6D UED on ${PS}$(green), $S^{2}/Z_{2}$(blue), $T^{2}/Z_{4}$(red) with $m_{h}=120\,\text{GeV}$ providing no threshold correction from bottom to top. Each black dashed line located above the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. The left, right are these with the high, low cutoff choices, respectively. We now define a value defined as $\displaystyle\Delta\equiv\mathcal{R}_{2g\rightarrow h^{(0)}}\times\mathcal{R}_{h^{(0)}\rightarrow 2\gamma},$ (124) which shows the “total ratio” of the deviation of the $h^{(0)}\rightarrow 2\gamma$ signals coming from the $2g\rightarrow h^{(0)}$ Higgs production. At the LHC, the Higgs production process through gluon fusion is dominant and the value of $\Delta$ is considered to be an appropriate approximation of the $h^{(0)}\rightarrow 2\gamma$ signal deviation coming from all the Higgs production processes. Of course the numerical results of $\Delta$ are given as functions of the first KK mass scale ($M_{\text{KK}}$) in a unit of GeV and are shown in Fig. 7. It should be noted that in 6D UED models, the collider signal deviations from the SM in the $h^{(0)}\rightarrow 2\gamma$ process take the greater values than in the 5D mUED case. When we take the reference value as ${M_{\text{KK}}}=800$ GeV in the $m_{h}=120$ GeV and each high cutoff case, approximately $40\%(T^{2}/Z_{4})$, $60\%(S^{2}/Z_{2})$, $110\%({PS})$ enhancements from the SM expectation value can be seen. It should be mentioned that the shapes of $\Delta$ in each model do not have large dependence on the value of the UED cutoff $\Lambda_{\text{UED}}$. This reason can be considered that the behavior of the ratios of the gluon fusion Higgs production and the Higgs decay to two photons is opposite when we change the value of the cutoff and a large part of the distinctions due to the value of the cutoff are cancelled out. This property is accidental but do not depend on the type of the background geometry and thereby we consider that this is one of the interesting aspects of 6D UED model. The difference between the above results and the SM expectation value is significant and we hope that this could be tested at the LHC experiments in the near future. Finally, we comment on the up-to-date collider experimental results at the LHC. The ATLAS group announced their results, which conclude the upper limit of the cross section of the $h^{(0)}\rightarrow 2\gamma$ process in the form of the ratio to the SM result $(\sigma/\sigma_{\text{SM}})$ based on the $1.7\,\text{fb}^{-1}$ data within the $95\%$ confidence level in the August of 2011. According to ATLAS-CONF-2011-135 , the value of the upper bound of $(\sigma/\sigma_{\text{SM}})$ is about $3.5$ $(5.0)$ at the point of $m_{h}=120\,\text{GeV}$ ($m_{h}=145\,\text{GeV}$). The CMS group also announced their results, which says that the value of the upper bound of $(\sigma/\sigma_{\text{SM}})$ is about $3.5$ $(4.0)$ at the point of $m_{h}=120\,\text{GeV}$ ($m_{h}=145\,\text{GeV}$) CMS_PAS_HIG-11-022 . And at the December of 2011, the new results have been published by both the ALTAS and CMS. The ATLAS claims that there is an excess of events close to 126 GeV with a 3.6 $\sigma$ confidence ATLAS:2012ae . On the other hand, the excess also have been observed by the CMS, but the location of the peak is 124 GeV with a 3.1 $\sigma$ confidence Chatrchyan:2012tx . It is noted that both results are these before taking looking-elsewhere effect. The allowed region of the SM Higgs becomes highly constrained as $115.5\,\,\text{GeV}<m_{h}<127\,\,\text{GeV}$ except the unexplored high mass region $m_{h}>600\,\,\text{GeV}$. We do not execute detailed analysis in this paper on this topics but we can conclude from our result in Fig. 7 that the $T^{2}/Z_{4}$, $S^{2}/Z_{2}$, and $PS$ 6D UED with $m_{h}=120\,\,\text{GeV}$ still survive only in the KK mass region above ${M_{\text{KK}}=600,750,1150\,\text{GeV}}$, respectively, when we consider the high cutoffs, judging from the constraint on the value of $\sigma/\sigma_{\text{SM}}$ in the December’s ALTAS and CMS results, whose maximum value is roughly 1.6.212121 We note that the newer CMS diphoton data set includes vector boson fusion (VBF) events that occurs at the tree level in the SM. The VBF Higgs production process is not significantly enhanced by the UED loop effects. It is obvious that the possibility of $m_{h}=145\,\,\text{GeV}$ is discarded in all the 6D UED models since the signals are expected to be greater than these in the SM. Figure 7: These plots represent the values of $\Delta$ (total ratio) in 6D UED on ${PS}$(green), $S^{2}/Z_{2}$(blue), $T^{2}/Z_{4}$(red) with $m_{h}=120\,\text{GeV}$ providing each high cutoff case and no threshold correction from top to bottom. Each black dashed line located above the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. The left, right are these with the high, low cutoff choices, respectively. ### 6.3 Results with threshold corrections When we switch on the threshold corrections accompanying the processes of $2g\rightarrow h^{(0)}$ and $h^{(0)}\rightarrow 2\gamma$, the shapes of the ratios ${\\{}\mathcal{R}_{2g\rightarrow h^{(0)}},\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}{\\}}$ and the total ratios $\Delta$ are modified forcefully. There are two dimensionless new parameters in Eqs. (39,50) $C_{hgg},C_{h\gamma\gamma}$, which describe the threshold correction in the process of $2g\rightarrow h^{(0)}$ or $h^{(0)}\rightarrow 2\gamma$, respectively. In this paper, we only consider some extremal choices of $C_{hgg},C_{h\gamma\gamma}$ as $\displaystyle C_{hgg}=0,\pm 1,\quad C_{h\gamma\gamma}=0,\pm 1.$ (125) We mention that the plus/minus sign of $C_{hgg},C_{h\gamma\gamma}$ determines the direction of interference effects to the UED part. We show the results of our numerical calculations in Figs. 8-16. We write down our convention about the color/shape of curves in Figs. 14,15,16 (total ratios) in Table 5. In the range of our approximation, the values of $\alpha_{s}^{-1}(m_{h})$ and $\alpha_{\text{EM}}^{-1}(m_{h})$ only appears in the terms describing the threshold corrections in Eqs. (39,50) and we adopt these values at the Z boson mass pole as $\alpha_{s}^{-1}(m_{Z})=8.44,\alpha_{\text{EM}}^{-1}(m_{Z})=127$ with ignoring the renormalization group effects between $m_{Z}$ and $m_{h}$ $(=120\ \text{or}\ 145\,\,\text{GeV})$. We make several comments in order. * • In the gluon fusion process in each model, due to the $(-1)$ factor which originates from Fermi statistics in $F_{t}^{\text{SM}}$, the interference term between the threshold correction and the UED effect is destructive (constructive) in case of $C_{hgg}=+1\ (-1)$, respectively. It is considered that the degree of a threshold correction is inversely proportional to the value of a cutoff. When we look at the PS case with its high cutoff choice in Fig. 10, we notice that the threshold correction works a little compared to the cases of the $T^{2}/Z_{4}$ or $S^{2}/Z_{2}$. We can find some differences between the cases of $PS$ and $S^{2}/Z_{2}$ with a same cutoff value, which stem from the differences in the corresponding UED contributions. We mention that the threshold correction is still observable in the many cases with $M_{\text{KK}}=2\,\text{TeV}$. By contraries, in all the low cutoff cases in Figs. 8,9,10, the threshold correction plays a very important role. Here we mention that the cases with $m_{h}=145\,\text{GeV}$ are almost identical with these with $m_{h}=120\,\text{GeV}$. * • In the Higgs decay to two photons in each model, unlike with the previous gluon fusion case, the interference term between the threshold correction and the UED effect is constructive (destructive) in case of $C_{h\gamma\gamma}=+1\ (-1)$, respectively. In this case, the degree of the effect which only comes from the threshold correction is also smaller than the others. It is an interesting point that in some cases with $C_{g\gamma\gamma}=+1$, the value of the ratio $(\mathcal{R}_{h^{(0)}\rightarrow 2\gamma})$ exceeds one, which we never find in the no-threshold-correction cases in the range of the parameter region of $M_{\text{KK}}$ which we consider. Another remarkable point compared to the gluon fusion, the cases with $m_{h}=145\,\text{GeV}$ are not identical with these with $m_{h}=120\,\text{GeV}$ but this difference is still not so significant since the other effects (cutoff scale, threshold correction and so on) are more effective. Of course, in all the low cutoff cases in Figs. 11,12,13, the threshold correction works very well. We mention that we can find the $10\sim 20\,\%$ deviations from the no-threshold-correction cases even with $M_{\text{KK}}=2\,\text{TeV}$ in all the 6D UED models. * • After combining the above two results, we can estimate the total ratio $\Delta$ in each 6D UED model with the extremal threshold corrections. In this analysis, we only consider the $m_{h}=120\,\text{GeV}$ cases. There are nine curves in each graph and our convention about the color/shape of curves is summarized in Table 5. Here we would like to only focus on a few important topics. Firstly, we can find the tendency that every orange line $(C_{hgg}=-1,C_{h\gamma\gamma}=+1)$ is located at the top of each graph and any cyan one $(C_{hgg}=+1,C_{h\gamma\gamma}=-1)$ is located at the bottom of each graph. The reason is that the two threshold corrections function toward maximally enhancing (suppressing) the process in the former (latter) case. Secondly, the deviation from the no-threshold-correction case (black dot- dashed curve) is noteworthy, in particular, in the $M_{\text{KK}}$ range below $1\,\text{TeV}$. All the results tend to converge with the no-threshold- correction curve proportional to the value of $M_{\text{KK}}$ and it is notable even around $M_{\text{KK}}=2\,\text{TeV}$ in many choices of $C_{hgg}$ and $C_{h\gamma\gamma}$ because the tens of percents of the deviations still remain. Thirdly, in the low cutoff cases, the interference effects dominate the whole process and the predictions about two photon signals via the gluon fusion Higgs production possibly become extraordinary. Finally we comment on the constraint from the LHC results briefly. We also do not execute detailed analysis in this case but we can conclude that some cases which predict too great value of the total ratio are already excluded. On the other hand, the total ratio can be suppressed in some choices of the parameters describing the threshold corrections, and in this case the possibility with $m_{h}=145\,\,\text{GeV}$ is not totally rejected. At the end of this section, we give comments for more precise analysis. We need to take into account the correction from QCD (parton distribution function and K-factor) Rai:2005vy . However the KK contributions would receive almost the same QCD corrections as in the case of the SM and this deviation from the SM result would not be large. Actually, to get the ratios of event rates in 6D UED models to that in the SM, the partial decay width in $\Delta$ should be replaced by corresponding branching ratios. We, however, expect that the effects of heavy KK particles to the leading decay processes at the tree level are small because of decoupling. Thus $\Delta$ is expected to be enough for crude estimation. But there are two other one-loop leading decay processes of $h^{(0)}\rightarrow 2g$ and $h^{(0)}\rightarrow\gamma Z$, which may possibly give considerable contribution to the Branching Ratio. These points are beyond the scope of this paper and are left for future work Nishiwaki:2011gk . ## 7 Summary In this paper, we have discussed the main Higgs production process through gluon fusion and the important one-loop leading decay channel to two photons at the LHC in various 6D UED models. The Higgs production cross sections in 6D UED models are much enhanced than the prediction of the SM or the 5D mUED. In contrast, the decay width in 6D UED models are decreased because of the destructive contribution between quarks and gauge bosons. In both cases, the results of ${PS}$ model are significant. This is because numerous fermions contribute to the process in each level of KK modes. We also have discussed the threshold corrections in the processes and their effects become significant even when we take the higher cutoff and/or a heavy KK scale. Some parameter region are already excluded by the current LHC experimental results obviously. By use of the data announced by the ALTAS and CMS experiments in the December of 2011, we can estimate the lower bounds of the KK scale as $M_{\text{KK}}=600\,(T^{2}/Z_{4}),750\,(S^{2}/Z_{2}),1150\,(PS)\,\text{GeV}$ when we consider the high cutoffs with no threshold correction. These results are modified by the threshold corrections substantially. The SM with the $145\,\,\text{GeV}$ Higgs boson is rejected but 6D UED with the Higgs mass parameter is still survived in some ranges of the parameters describing the threshold corrections. Our results are affected by ultraviolet physics because the calculation has logarithmic cutoff scale dependence. There seem to be some ambiguities coming from this fact. We expect that our prediction would be verified by forthcoming LHC experimental results. Detailed analysis of the final states in the single Higgs production at the LHC is important for discriminating UED from the other models. The collider physics and particle cosmology of $S^{2}$-based 6D UED models are unexplored and we would like to pursue these topics in future work Nishiwaki:2011gm ; Nishiwaki:2011gk . Acknowledgments We are most grateful to C. S. Lim and Kin-ya Oda for valuable comments and discussions. In particular C. S. Lim red the manuscript carefully and gave us very useful comments. And we also thank Nobuhito Maru, Naoya Okuda, Makoto Sakamoto, Joe Sato, Takashi Shimomura, Ryoutaro Watanabe and Masato Yamanaka for fruitful discussions. Yasuhiro Okada and Hideo Ito suggest the recent Tevatron experimental results to us. We express our appreciation to them very much. We again appreciate C. S. Lim and Kin-ya Oda for advising me in the revision. Finally we appreciate the referee for giving a lot of useful comments. Figure 8: These plots represent the values of the ratios $\mathcal{R}_{2g\rightarrow h^{(0)}}$ in 6D UED on $T^{2}/Z_{4}$ with/without threshold correction. The red, blue, green curves show these with $C_{hgg}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the QCD coupling strength as that at the Z boson mass scale. Figure 9: These plots represent the values of the ratios $\mathcal{R}_{2g\rightarrow h^{(0)}}$ in 6D UED on $S^{2}/Z_{2}$ with/without threshold correction. The red, blue, green curves show these with $C_{hgg}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the QCD coupling strength as that at the Z boson mass scale. Figure 10: These plots represent the values of the ratios $\mathcal{R}_{2g\rightarrow h^{(0)}}$ in 6D UED on $PS$ with/without threshold correction. The red, blue, green curves show these with $C_{hgg}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the QCD coupling strength as that at the Z boson mass scale. Figure 11: These plots represent the values of the ratios $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ in 6D UED on $T^{2}/Z_{4}$ with/without threshold correction. The red, blue, green curves show these with $C_{h\gamma\gamma}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the electromagnetic coupling strength as that at the Z boson mass scale. Figure 12: These plots represent the values of the ratios $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ in 6D UED on $S^{2}/Z_{2}$ with/without threshold correction. The red, blue, green curves show these with $C_{h\gamma\gamma}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the electromagnetic coupling strength as that at the Z boson mass scale. Figure 13: These plots represent the values of the ratios $\mathcal{R}_{h^{(0)}\rightarrow 2\gamma}$ in 6D UED on $PS$ with/without threshold correction. The red, blue, green curves show these with $C_{h\gamma\gamma}=0,+1,-1$, respectively. Each black dashed line near the lines for $m_{h}=120\,\text{GeV}$ corresponds to that with $m_{h}=145\,\text{GeV}$. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the value of the electromagnetic coupling strength as that at the Z boson mass scale. value of $C_{hgg}$ | value of $C_{h\gamma\gamma}$ | color/shape of curve ---|---|--- $C_{hgg}=0$ | $C_{h\gamma\gamma}=0$ | black, dot-dashed $C_{hgg}=+1$ | $C_{h\gamma\gamma}=0$ | red $C_{hgg}=0$ | $C_{h\gamma\gamma}=+1$ | blue $C_{hgg}=-1$ | $C_{h\gamma\gamma}=0$ | green $C_{hgg}=0$ | $C_{h\gamma\gamma}=-1$ | magenta $C_{hgg}=+1$ | $C_{h\gamma\gamma}=+1$ | yellow, dotted $C_{hgg}=-1$ | $C_{h\gamma\gamma}=+1$ | orange, dotted $C_{hgg}=+1$ | $C_{h\gamma\gamma}=-1$ | cyan, dotted $C_{hgg}=-1$ | $C_{h\gamma\gamma}=-1$ | brown, dotted Table 5: Our convention about the color/shape of curves in Figs. 14,15,16 (total ratios). Figure 14: These plots represent the values of the total ratios $\Delta$ in 6D UED on $T^{2}/Z_{4}$ with/without threshold corrections with $m_{h}=120\,\text{GeV}$. The color/shape convention is summarized in Table 5. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the values of the QCD and electromagnetic coupling strengths as that at the Z boson mass scale. Figure 15: These plots represent the values of the total ratios $\Delta$ in 6D UED on $S^{2}/Z_{2}$ with/without threshold corrections with $m_{h}=120\,\text{GeV}$. The color/shape convention is summarized in Table 5. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the values of the QCD and electromagnetic coupling strengths as that at the Z boson mass scale. Figure 16: These plots represent the values of the total ratios $\Delta$ in 6D UED on $PS$ with/without threshold corrections with $m_{h}=120\,\text{GeV}$. The color/shape convention is summarized in Table 5. In the left and right plots, which correspond to the high and low cutoff cases, respectively, we take the values of the QCD and electromagnetic coupling strengths as that at the Z boson mass scale. ## Appendix A Feynman Rules containing scalar particle In this appendix, we list the Feynman rules containing scalar particle in the ’t Hooft-Feynman gauge. We omit the rules containing no scalar particle, which are the same with the corresponding rules of the SM for the zero modes alone. In the vertices all momenta $(k_{1},k_{2})$ and directions of propagation are considered as incoming. $g^{(2)}$ and $e$ are $SU(2)_{L}$ the 4D gauge coupling and the 4D elementary electric charge, respectively. $\displaystyle=-i(k_{1}-k_{2})_{\mu}\mathcal{F}$, $\begin{array}[]{|c|c|c||c|}\hline\cr A_{\mu}&B&C&\mathcal{F}\\\ \hline\cr W^{+(m,n)}_{\mu}&G^{-(m,n)}&h^{(0)}&\frac{g^{(2)}}{2}\frac{-im_{W}}{m_{W,(m,n)}}\\\ \hline\cr W^{+(m,n)}_{\mu}&a^{-(m,n)}&h^{(0)}&\frac{g^{(2)}}{2}\frac{m_{(m,n)}}{m_{W,(m,n)}}\\\ \hline\cr W^{-(m,n)}_{\mu}&G^{+(m,n)}&h^{(0)}&\frac{g^{(2)}}{2}\frac{-im_{W}}{m_{W,(m,n)}}\\\ \hline\cr W^{-(m,n)}_{\mu}&a^{+(m,n)}&h^{(0)}&\frac{g^{(2)}}{2}\frac{-m_{(m,n)}}{m_{W,(m,n)}}\\\ \hline\cr A^{(0)}_{\mu}&G^{+(m,n)}&G^{-(m,n)}&e\\\ \hline\cr A^{(0)}_{\mu}&a^{+(m,n)}&a^{-(m,n)}&e\\\ \hline\cr A^{(0)}_{\mu}&H^{+(m,n)}&H^{-(m,n)}&e\\\ \hline\cr\end{array}$ $\displaystyle=i\eta_{\mu\nu}\mathcal{F}$, $\begin{array}[]{|c|c|c||c|}\hline\cr A&B_{\mu}&C_{\nu}&\mathcal{F}\\\ \hline\cr h^{(0)}&W^{+(m,n)}_{\mu}&W^{-(m,n)}_{\nu}&m_{W}g^{(2)}\\\ \hline\cr G^{+(m,n)}&W^{-(m,n)}_{\mu}&A^{(0)}_{\nu}&-iem_{W,(m,n)}\\\ \hline\cr G^{-(m,n)}&W^{+(m,n)}_{\mu}&A^{(0)}_{\nu}&iem_{W,(m,n)}\\\ \hline\cr\end{array}$ $\displaystyle=-im_{W}g^{(2)}\mathcal{F}$, $\begin{array}[]{|c|c|c||c|}\hline\cr A&B&C&\mathcal{F}\\\ \hline\cr h^{(0)}&G^{+(m,n)}&G^{-(m,n)}&\frac{m_{h}^{2}}{2m_{W,(m,n)}^{2}}\\\ \hline\cr h^{(0)}&G^{+(m,n)}&a^{-(m,n)}&i\frac{m_{(m,n)}}{2m_{W,(m,n)}^{2}}\left(\frac{m_{h}^{2}}{m_{W}}-\frac{m_{W,(m,n)}^{2}}{m_{W}}\right)\\\ \hline\cr h^{(0)}&a^{+(m,n)}&G^{-(m,n)}&i\frac{m_{(m,n)}}{2m_{W,(m,n)}^{2}}\left(-\frac{m_{h}^{2}}{m_{W}}+\frac{m_{W,(m,n)}^{2}}{m_{W}}\right)\\\ \hline\cr h^{(0)}&a^{+(m,n)}&a^{-(m,n)}&\frac{1}{2m_{W,(m,n)}^{2}}\left(\frac{m_{h}^{2}}{m_{W}^{2}}m_{(m,n)}^{2}+2{m_{W,(m,n)}^{2}}\right)\\\ \hline\cr h^{(0)}&H^{+(m,n)}&H^{-(m,n)}&1\\\ \hline\cr\end{array}$ $\displaystyle=i\eta_{\mu\nu}\mathcal{F}$, $\begin{array}[]{|c|c|c|c||c|}\hline\cr A_{\mu}&B_{\nu}&C&D&\mathcal{F}\\\ \hline\cr W_{\mu}^{+(m,n)}&A_{\nu}^{(0)}&G^{-(m,n)}&h^{(0)}&\frac{eg^{(2)}}{2}\left(\frac{im_{W}}{m_{W,(m,n)}}\right)\\\ \hline\cr W_{\mu}^{-(m,n)}&A_{\nu}^{(0)}&G^{+(m,n)}&h^{(0)}&\frac{eg^{(2)}}{2}\left(\frac{-im_{W}}{m_{W,(m,n)}}\right)\\\ \hline\cr W_{\mu}^{+(m,n)}&A_{\nu}^{(0)}&a^{-(m,n)}&h^{(0)}&\frac{eg^{(2)}}{2}\left(\frac{-m_{(m,n)}}{m_{W,(m,n)}}\right)\\\ \hline\cr W_{\mu}^{-(m,n)}&A_{\nu}^{(0)}&a^{+(m,n)}&h^{(0)}&\frac{eg^{(2)}}{2}\left(\frac{-m_{(m,n)}}{m_{W,(m,n)}}\right)\\\ \hline\cr A_{\mu}^{(0)}&A_{\nu}^{(0)}&G^{+(m,n)}&G^{-(m,n)}&2e^{2}\\\ \hline\cr A_{\mu}^{(0)}&A_{\nu}^{(0)}&a^{+(m,n)}&a^{-(m,n)}&2e^{2}\\\ \hline\cr A_{\mu}^{(0)}&A_{\nu}^{(0)}&H^{+(m,n)}&H^{-(m,n)}&2e^{2}\\\ \hline\cr\end{array}$ ## Appendix B Detail on threshold correction In this Appendix, we explain the concrete forms of threshold corrections in the gluon fusion $(2g\rightarrow h^{(0)})$ and the Higgs decay to two photons $(h^{(0)}\rightarrow 2\gamma)$. The parts of the Lagrangian describing the former ($\mathcal{L}_{hgg}$) and the latter ($\mathcal{L}_{h\gamma\gamma}$) processes are defined as $\displaystyle\mathcal{L}_{hgg}$ $\displaystyle=-\frac{1}{4}\frac{C_{hgg}}{{\Lambda_{\text{UED}}}^{2}}V_{2}F_{MN}^{[\text{QCD}]}F^{[\text{QCD}]MN}H^{\dagger}H,$ (126) $\displaystyle\mathcal{L}_{h\gamma\gamma}$ $\displaystyle=-\frac{1}{4}\frac{C_{h\gamma\gamma}}{{\Lambda_{\text{UED}}}^{2}}V_{2}F_{MN}^{[\text{QED}]}F^{[\text{QED}]MN}H^{\dagger}H,$ (127) where $C_{hgg}$ and $C_{h\gamma\gamma}$ are dimensionless coefficients characterizing the processes, $\Lambda_{\text{UED}}$ is 6D UED cutoff, $V_{2}$ is the volume of the two extra dimensions, $H$ is the 6D Higgs doublet, and $F_{MN}^{[\text{QCD}]}$ ($F_{MN}^{[\text{QED}]}$) is the 6D field strength of gluon (photon), respectively. It is an important thing that the Higgs doublet should be introduced in these effective operators in a bilinear form of $H^{\dagger}H$ because the electroweak symmetry breaking (EWSB) is realized by the usual Higgs mechanism in 6D UED models. After EWSB and KK reduction, the Higgs doublet can acquire the VEV as $\langle H\rangle=\left(0,v\right)^{\text{T}}/\sqrt{2V_{2}}$, where $v\simeq 246\,\text{GeV}$, and we would like to focus on the parts, which are $\displaystyle\mathcal{L}_{hgg}$ $\displaystyle\supset-\frac{v/\sqrt{2}}{4}\frac{C_{hgg}}{{\Lambda_{\text{UED}}}^{2}}F_{\mu\nu}^{(0)[\text{QCD}]}F^{(0)[\text{QCD}]\mu\nu}h^{(0)},$ (128) $\displaystyle\mathcal{L}_{h\gamma\gamma}$ $\displaystyle\supset-\frac{v/\sqrt{2}}{4}\frac{C_{h\gamma\gamma}}{{\Lambda_{\text{UED}}}^{2}}F_{\mu\nu}^{(0)[\text{QED}]}F^{(0)[\text{QED}]\mu\nu}h^{(0)}.$ (129) The superscript “$(0)$” means that the fields are zero modes and we take integration toward the two extra spacial directions in Eqs. 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arxiv-papers
2011-01-04T04:02:16
2024-09-04T02:49:16.109296
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Kenji Nishiwaki", "submitter": "Kenji Nishiwaki", "url": "https://arxiv.org/abs/1101.0649" }
1101.0711
# Optical Intraday Variability Studies of Ten Low Energy Peaked Blazars Bindu Rani1, Alok C. Gupta1, U. C. Joshi2, S. Ganesh2, Paul J. Wiita3 1Aryabhatta Research Institute of Observational Sciences (ARIES), Manora Peak, Nainital – 263129, India 2Physical Research Laboratory, Navrangpura, Ahmedabad-380 009, India 3Department of Physics, The College of New Jersey, P.O. Box 7718, Ewing, NJ 08628, USA E-mail: bindu@aries.res.in (Accepted ……. Received ……; in original form ……) ###### Abstract We have carried out optical (R band) intraday variability (IDV) monitoring of a sample of ten bright low energy peaked blazars (LBLs). Forty photometric observations, of an average of $\sim 4$ hours each, were made between 2008 September and 2009 June using two telescopes in India. Measurements with good signal to noise ratios were typically obtained within 1–3 minutes, allowing the detection of weak, fast variations using N-star differential photometry. We employed both structure function and discrete correlation function analysis methods to estimate any dominant timescales of variability and found that in most of the cases any such timescales were longer than the duration of the observation. The calculated duty cycle of IDV in LBLs during our observing run is $\sim$52$\%$, which is low compared to many earlier studies; however, the relatively short periods for which each source was observed can probably explain this difference. We briefly discuss possible emission mechanisms for the observed variability. ###### keywords: galaxies: active — galaxies: BL Lacs — galaxies: photometry ††pagerange: Optical Intraday Variability Studies of Ten Low Energy Peaked Blazars–9††pubyear: 2010 ## 1 Introduction Blazars are a subclass of radio-loud AGNs characterized by strong and rapid flux variability across the entire EM spectrum and strong polarization from radio to optical wavelengths. Microvariability, or intraday variability (IDV) is commonly observed across much of the electromagnetic (EM) spectrum of AGNs but is particularly common in blazars. A change of flux of $\sim$1–15% within a few minutes to hours reflect extreme physical conditions embedded in small sub-parsec scales. According to the usually accepted orientation based unified model of radio-loud AGNs, blazar jets usually make an angle $\leq 10^{\circ}$ to our line-of-sight (Urry & Padovani, 1995). The Doppler boosting of the jet emission means that most of what we see from blazars arises in those jets. These jet dominated AGNs provide a natural laboratory to study the mechanisms of energy extraction from the vicinity of central supermassive black holes, the physical properties of jets and perhaps also accretion disks. The radiation of blazars across the whole EM spectrum is predominantly non- thermal. At lower frequencies (through the UV or X-ray bands) the mechanism is almost certainly synchrotron emission while at higher frequencies it is very probably dominated by Inverse-Compton (IC) emission (Sikora & Madejski, 2001; Krawczynski, 2004). The spectral energy distributions (SEDs) of blazars have a double-peaked structure (Fossati et al., 1998; Ghisellini et al., 1997). Based on the location of the peak of their SEDs, blazars are often sub-classified into the low energy peaked blazars (LBLs) and high energy peaked blazars (HBLs); the first component peaks in the near-infrared (NIR)/optical in case of LBLs and in the UV/X-rays in HBLs, while the second component usually peaks at GeV energies in LBLs and at TeV energies in HBLs. The study of variability is one of the most powerful tools for revealing the nature of blazars and probing the various processes occurring in them. Based on their different timescales, the variability of blazars can be broadly divided into three classes, IDV, short-term variability (STV), and long-term variability (LTV). Variations in the flux of source up to a few tenths of magnitude over a time scale of a day or less is known as IDV (Wagner & Witzel, 1995) or microvariability, or intra-night optical variability. Variations of days to a few months are often considered to be STV, while those from several months to many years are usually called LTV (e.g., Gupta et al., 2004); both of these classes of variations for blazars typically exceed $\sim$ 1 magnitude and can exceed even 5 magnitudes. Over the last two decades, the optical variability of blazars has been extensively studied on diverse timescales (e.g., Heidt & Wagner, 1996; Sillanpää et al., 1996a, b; Gupta et al., 2004, 2008a, 2008b, 2008c, 2009; Ciprini et al., 2003, 2007, and references therein). There are several theoretical models that might be able to explain the observed variability over wide time-scales for all bands, with the leading contenders all variants of models based upon shocks propagating down relativistic jets (Marscher & Gear, 1985; Qian et al., 1991; Hughes et al., 1991; Marscher et al., 1992; Wagner & Witzel, 1995). Some of the variability may arise from helical structures, precession or other geometrical effects occurring within the jets (e.g., Camenzind & Krockenberger, 1992; Gopal- Krishna & Wiita, 1992) and some of the radio variability is due to extrinsic propagative effects (Rickett et al., 2001). Hot spots or other disturbances in or above accretion disks surrounding the black holes at the centres of AGNs (e.g., Mangalam & Wiita, 1993; Chakrabarti & Wiita, 1993) are likely to play a key role in the variability of non-blazar AGNs and might provide seed fluctuations that could be advected into a rotating blazar jet and then be Doppler amplified. Despite the large amount of information we have about blazars that is very briefly summarized above, we still lack sufficient understanding of basic parameters of the emission regions, such as jet composition, a quantitative assessment of beaming parameters, or the processes leading to the origin of shocks in the jets. These physical quantities are obviously important in understanding the physics of jets and their emission regions and additional IDV studies leading to statistically valid pictures of many blazars can help constrain them. In this paper we present the results of an extensive IDV studies of a sample of ten LBLs including six BL Lacs and four FSRQs. The work presented here is focused on intraday variations in the R passband magnitudes of these sources, which were the brightest blazars visible from ARIES, Nainital, India and PRL, Mount Abu India. We also compare our observational results to some of those presented in the literature. The paper is structured as follows. In Section 2, we present the observations and data reduction procedure. Section 3 provides our analysis and results. We present our discussion and conclusions in Section 4. ## 2 Observations and data reduction We have carried out optical R band photometric observations of ten LBLs from September 2008 to June 2009 using two telescopes in India. The details of the telescopes and instruments used are given in Table 1 and the observation details are in Table 2. For doing image processing, or data pre-processing, we generated a master bias frame for each observing night by taking the median of all bias frames; the master bias frame was subtracted from all flat and source image frames taken on that night. Then the master flat in each passband was generated by median combine of flat frames in that passband. Finally, the normalized master flat in each passband was generated. As usual, each source image frame was divided by the normalized master flat in the respective passband to remove pixel-to- pixel inhomogeneities (flat fielding). Finally cosmic ray removal was done from all source image frames. This pre-processing of the data was done by using standard routines in the Image Reduction and Analysis Facility111IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. (IRAF) software. Our data analysis, or processing of the data, utilizes Dominion Astronomical Observatory Photometry (DAOPHOT II) software to perform aperture photometry (Stetson, 1987, 1992). We carried out aperture photometry with four different aperture radii, 1$\times$FWHM, 2$\times$FWHM, 3$\times$FWHM and 4$\times$FWHM. On comparing the results, we observed that aperture radii = 3$\times$FWHM provided the best S/N ratio, and we have adopted that in this work. For all of the ten blazars, we observed more than three local standard stars. The magnitudes of the standard stars we used in the fields of our sources are given in Table 3. The multiple comparison stars were used to check that the usual standard stars were non-variable. We have used two non-varying standard stars from each blazar field and plotted their differential instrumental magnitudes in Figs. 1$-$9\. Finally, for the calibration of blazar data, we have used the one standard star that has a colour close to the blazar from those two standards stars. The calibrated light curves of the blazars are plotted in the same panel with the differential instrumental magnitudes of two standard stars. ## 3 Analysis and Results ### 3.1 Variability Parameters We checked for the presence of microvariability both by using the $F$-test (de Diego, 2010) and the variability detection parameter, $C,$ for the sake of comparison with earlier papers, which nearly always used that approach. For two sample variances, say s${}^{2}_{Q}$, for the quasar differential light curves and s${}^{2}_{stS}$, for that of the standard star $F=\frac{s^{2}_{Q}}{s^{2}_{stS}}.$ (1) The $F$-statistic is compared with a critical value corresponding to the significance level set for the test. We have used the inbuilt F test code available in R222R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org.. A $p$-value of $\leq$ 0.01 ($\geq$99$\%$ significance level) is adopted for our variability detection criterion. The variability detection parameter is defined as Romero et al. (e.g., 1999)) the average of $C1$and $C2$ where $\displaystyle C1=\frac{\sigma(BL-starA)}{\sigma(starA-starB)}\hskip 14.22636pt\rm{and}\hskip 14.22636ptC2=\frac{\sigma(BL-starB)}{\sigma(starA- starB)}.$ (2) Here (BL $-$ starA) and (BL $-$ starB) are the differential instrumental magnitudes of the blazar and standard star A and the blazar and standard star B, respectively, while $\sigma$(BL$-$starA) and $\sigma$(BL$-$starB) are the observational scatters of the differential instrumental magnitude of the blazar and star A and the blazar and star B, respectively. If $C\geq 2.57$, a conservative confidence level of a variability detection is $>99$%, and we consider this to be a positive detection of a variation using this criterion. As noted by de Diego (2010) this C-statistic does not behave as a proper statistic should, but as it has been used in most of the IDV studies in literature, we also employed this test. The percentage variation in the intraday light curves of LBLs is calculated by using the variability amplitude parameter, $A$, introduced by Heidt & Wagner (1996), defined as $\displaystyle A=100\times\sqrt{{(A_{max}-A_{min}})^{2}-2\sigma^{2}}(\%),$ (3) where $A_{max}$ and $A_{min}$ are the maximum and minimum magnitudes in the calibrated light curves of the blazar and $\sigma$ is the average measurement error of the blazar light curve. The calculated $F$-statistics, $C$-“statistics” and variability amplitude parameter, $A$, values are listed in Table 4. ### 3.2 Structure Function The first order structure function (SF) is a very useful tool to search for periodicities and timescales of variability in time series data trains (Simonetti et al., 1985). Here we give only a very brief introduction to the method; for details refer to Rani et al. (2009). The first order SF for a data train, $a$, is defined as $\displaystyle D^{1}_{a}(k)={\frac{1}{N^{1}_{a}(k)}}{\sum_{i=1}^{N}}w(i)w(i+k){[a(i+k)-a(i)]}^{2},$ (4) where $k$ is the time lag, ${N^{1}_{a}(k)}=\sum w(i)w(i+k)$, and the weighting factor, $w(i)$ is 1 if a measurement exists for the $i^{th}$ interval, and 0 otherwise. The behaviour of the first order SF can be simply summarized. The SF curves for AGN usually at first rise with time lag. Following this rising portion, the SF will then fall into one of the following classes: (i) if no plateau exists, any time scale of variability exceeds the length of the data train; (ii) if there are one or more plateaus, each one indicates a possible time scale of variability; and (iii) if that plateau is followed by a dip in the SF, the lag corresponding to the minimum of that dip indicates a possible periodic cycle (unless such a dip is seen at a lag close to the maximum length of the data train, when it is probably an artifact). However, (iv) uncorrelated data produce a white noise behaviour, characterized by a constant slope (Ciprini et al., 2003). We have carried out the SF analysis of all of those LCs which satisfy the variability detection criteria. Recently, some weaknesses of the SF method, including spurious indications of timescales and periodicities have been discussed by Emmanoulopoulos et al. (2010), so we cross check the SF results by using the discrete correlation function (DCF) method. The timescales of variability calculated using the SF analysis are listed in Table 4. ### 3.3 Discrete Correlation Functions The Discrete Correlation function (DCF) method was first introduced by Edelson & Krolik (1988) and it was later generalized to provide better error estimates (Hufnagel & Bregman, 1992). Here we give only a brief introduction to the method; for details refer to Hovatta et al. (2007), Rani et al. (2009), and references therein. The first step is to calculate the unbinned correlation (UDCF) using the given time series through (Hovatta et al., 2007) $UDCF_{ij}={\frac{(a(i)-\bar{a})(b(j)-\bar{b})}{\sqrt{\sigma_{a}^{2}\sigma_{b}^{2}}}}.$ (5) Here $a(i)$ and $b(j)$ are the individual points in two time series $a$ and $b$, respectively, $\bar{a}$ and $\bar{b}$ are respectively the means of the time series, and $\sigma_{a}^{2}$ and $\sigma_{b}^{2}$ are their variances. The correlation function is binned after calculation of the UDCF. The DCF method does not automatically define a bin size, so several values need to be tried. If the bin size is too big, useful information is lost, but if the bin size is too small, a spurious correlation can be found. Taking $\tau$ as the centre of a time bin and $n$ as the number of points in each bin, the DCF is found from the UDCF via $DCF(\tau)={\frac{1}{n}}\sum~{}UDCF_{ij}(\tau).$ (6) The error for each bin can be calculated using $\sigma_{\mathrm{d}ef}(\tau)={\frac{1}{n-1}}\Bigl{\\{}\sum~{}\bigl{[}UDCF_{ij}-DCF(\tau)\bigr{]}^{2}\Bigr{\\}}^{0.5}.$ (7) A DCF analysis is frequently used for finding the correlation and possible lags between multi-frequency AGN data where different data trains are used in the calculation (e.g., Villata et al., 2004; Raiteri et al., 2003; Hovatta et al., 2007, and references therein). When the same data train is used, so $a$=$b$, there is obviously a peak at zero DCF, indicating that there is no time lag between the two, but any other strong peaks in the DCF give indications of variability timescales. The calculated $t_{v}$ values using DCF analyses are listed in Table 4. ### 3.4 Intraday Variability (IDV) of Individual Blazars The R filter light curves (LCs) of the blazars in which significant variability has been detected, along with their corresponding SF and DCF analysis curves, are displayed in Figures 1$-$7; the remaining non-variable LCs of the blazars are plotted along with the corresponding curves of the standard stars used for comparison in Figures 8 and 9. The complete observing log for the blazars in given in Table 3. The values of $A$, $C$, $F$-test along with any estimated timescales of variability found using SF and DCF analysis methods on the individual blazar LCs are listed in Table 4. A detailed multiband optical short term variability (STV) study of the fluxes and colours of all of these blazars over the same time period of observation is reported in Rani et al. (2010b). There we showed that the colour versus brightness correlations seen in these sources support the hypothesis that BL Lacs tends to be bluer with increase in brightness while FSRQs shows the opposite trend. We now report some key individual results for each of our sources, placed in the context of earlier work. 3C 66A: This is a low energy peaked blazar (LBL) at redshift $z=0.444$ (Lanzetta et al., 1993) and belongs to the class of BL Lac objects. Since its optical identification (Wills & Wills, 1974), the source has been regularly monitored at many observable frequencies, although less regularly at radio frequencies (Aller et al., 1992; Takalo et al., 1996). Fan & Lin (1999, 2000) have studied the long-term optical and IR variability of the source and reported a variation of $\leq$1.5 mag at time scales of $\sim$1 week to several years at those two frequencies. Böttcher et al. (2005) reported a large microvariability of $\sim$0.2 mag within 6 hours; they also reported several major outbursts in the source separated by $\sim$50 days and argued that the outbursts seem to have a quasi-periodic behaviour. At the end of 2007 the source was found to be in an optically active phase, which triggered a new Optical-IR-Radio Whole Earth Blazar Telescope (WEBT) campaign on the source (Böttcher et al., 2009). Our optical IDV observations of the source 3C 66A comprise a total of seven LCs, spanning a time period between October 2008 and January 2009. During this period a change of $\sim$1 magnitude in brightness of the source is seen (Rani et al., 2010b). The source showed significant microvariability only on October 22 and 26, 2008 (Fig. 1). There is a continuous fading trend of $\sim$0.08 and $\sim$0.06 magnitude, respectively on those two nights. The SF and DCF analysis of the LCs revealed that any timescale of variability in this source at those epochs is greater than the lengths of our observations. AO 0235+164: The blazar AO 0235+164 at $z=0.94$ (Nilsson et al., 1996) was classified as a BL Lac object by Spinrad & Smith (1975). Over the past few decades this blazar has been seen to be highly variable over all timescales and at all frequencies (Ghosh & Soundararajaperumal, 1995; Heidt & Wagner, 1996; Raiteri et al., 2001) and a very high fractional polarization of $\sim$40$\%$ has been reported in the source both at visible and IR frequencies (Impey et al., 1982). By analysing 25 years (1975$-$2000) of optical and radio data, Raiteri et al. (2001) argued that the source seemed to have an optical outburst period of $\sim$5.70 years but the expected outburst in 2004 was not detected by a 2003–2005 multiwavelength WEBT observing campaign (Raiteri et al., 2006a). A more detailed long term optical data analysis suggested a possible outburst period of $\sim$8 years in this source (Raiteri et al., 2006b) and this period was supported by subsequent observations Gupta et al. (2008c). Strong IDV flux variations of 9.5$\%$ and 13.7$\%$ during two nights were observed by Gupta et al. (2008c). Recently, Rani et al. (2009) reported the possible presence of nearly periodic fluctuations, with a timescale of $\sim$17 days, in a 12.7 year long X-ray light curve of AO 0235$+$164 obtained by the All Sky Monitor (ASM) instrument on the Rossi X-ray Timing Explorer (RXTE) satellite. We observed three IDV LCs of the source AO 0235+164 between October 2008 and January 2009. The brightness of the source decayed by $\sim$2.2 magnitude during this period (Rani et al., 2010b). We found a significant flux variation in two out of three LCs. A continuous fading trend of $\sim$0.13 magnitude and both brightening and decaying of $\sim$0.04 magnitude were observed on October 20 (Fig. 1) and October 23, 2008 (Fig. 2), respectively. A possible timescale of variability is $\sim$5.2 hr (from the SF analysis) for the LC observed on 20 October, while any such timescale exceeds the length of our observation on 23 October. However this putative timescale is not supported by the DCF analysis. PKS 0420$-$014: The blazar PKS 0420$-$014 is classified as a FSRQ and has a redshift of 0.915. It has been observed in optical bands since 1969. Several papers have reported multiple optically active and bright phases of the source and perhaps regular major flaring cycles (e.g., Villata et al., 1997; Webb et al., 1998; Raiteri et al., 1998, and references therein). Webb et al. (1998) reported that there were increases of $\sim$ 2$-$3 magnitudes during the active phases of this blazar during their observations that stretched from December 1969 to January 1986. Clements et al. (1995) have reported variations of $\Delta$mag $\cong$ 2.8 mag with a time scale of $\sim$22 years. We found a variation of $>$0.1 magnitude within 2 hrs in the brief optical LCs of the source on both of the days of observation in October and December 2008. During this period the source brightened by a factor of $\sim$0.7 magnitude (Rani et al., 2010b). The nominal timescale of variability (from the peak of the SF) is 0.12 hrs for the LC observed on December 26 and multiple dips might indicate a possible periodicity around 0.18 hrs, which is weakly supported by the DCF curve. The other LC is irregular and shows no hint of a timescale (Fig. 2). S5 0716$+$714: The blazar S5 0716$+$714 is classified as a BL Lac object. Nilsson et al. (2008) made a recent claim of redshift determination of the source to be $z=0.31\pm 0.08$. This source has been extensively studied at all observable wavelengths from radio to $\gamma$-rays on diverse time scales (Wagner et al., 1990; Heidt & Wagner, 1996; Villata et al., 2000; Raiteri et al., 2003; Montagni et al., 2006; Foschini et al., 2006; Ostorero et al., 2006; Gupta et al., 2008a, c). This source is one of the brightest BL Lacs in optical bands with an IDV duty cycle of nearly 1. Unsurprisingly, it has been the subject of several optical monitoring campaigns on IDV timescales (Heidt & Wagner, 1996; Montagni et al., 2006; Gupta et al., 2008c). This source has shown five major optical outbursts (Gupta et al., 2008c) at intervals of $\sim$3.0$\pm$0.3 years. High optical polarizations of $\sim$ 20$\%$ \- 29$\%$ has also been observed in the source (Takalo et al., 1994; Fan et al., 1997). Gupta et al. (2009) reported good evidence for nearly periodic oscillations in a few of the intraday optical light curves of the source observed by Montagni et al. (2006). Good evidence of presence of a $\sim$15 minute periodic oscillation at optical frequencies has been reported by Rani et al. (2010a). Our optical IDV observations of S5 0716$+$714 spans a time period from October 2008 to January 2009. The source brightened by a factor of $\sim$2 magnitude during this period (Rani et al., 2010b). We found significant microvariability of $\sim$0.1 magnitude in three out of four LCs of the source. The LCs observed on December 24, 2008 and January 03, 2009 show continuous fading trends trend of the order of $\sim$0.1 magnitude, though the former is abrupt while the latter is gradual. Both fading and brightening and fading trends of $\sim$0.05 magnitude were observed over just a few minutes on December 23, 2008. The calculated possible variability timescales are listed in Table 4, but the lack of agreement between the SF and DCF possibilities leads us to discount their reality. PKS 0735$+$178: The blazar PKS 0735$+$718 has been classified as a BL Lac object (Carswell et al., 1974). Papers concerning its redshift determination (Carswell et al., 1974; Falomo & Ulrich, 2000) had set a lower limit of $z>0.4$ and $z=0.424$ was reported for the source using a HST snapshot image (Sbarufatti et al., 2005). Since its optical identification, the source has been extensively observed across the whole electromagnetic spectrum (Teräsranta et al., 2004; Gu et al., 2006; Gupta et al., 2008c; Ciprini et al., 2007). A periodicity of $\sim$14 years has been suggested to be present in the source using a century long, but still sparse, optical light curve (Fan et al., 1997). Optical variability on IDV and STV timescales has been observed for 0735$+$178 (Xie et al., 1992; Massaro et al., 1995; Fan et al., 1997; Zhang et al., 2004; Ciprini et al., 2007; Gupta et al., 2008c). A significant amount of polarization ($\sim$ 1$\%$ to 30$\%$) has been observed in the source both at optical and IR bands (Mead et al., 1990; Takalo, 1991; Takalo et al., 1992b; Valtaoja et al., 1991a, 1993; Tommasi et al., 2001). Our IDV observations of the source PKS 0735$+$718 comprise four LCs taken between December 2008 and January 2009\. The brightness of the source changes by $\sim$0.6 magnitude during this period (Rani et al., 2010b). We found significant microvariations of an order of $\sim$0.1 magnitude in three out of four observed LCs of the source (Fig. 4). The only conceivable hints of timescales for variability from the SFs are $\sim$2.3 hrs for the LC observed on December 28, 2008 and $\sim$0.58 hrs for January 04, 2009. However, since both of these peaks are close to the total lengths of the observations they are not likely to be real, nor supported by DCF results. OJ 287: The blazar OJ 287 at $z=0.306$ is one of the most extensively observed and best studied BL Lac objects. It is also among the very few AGN’s for which more than a century of optical observations are available (Sillanpää et al., 1996a, b; Fan et al., 1998; Abraham, 2000; Gupta et al., 2008c; Fan et al., 2009). Using the binary black hole model (Sillanpää et al., 1988) for the long-term optical light curve of the source, an outburst with a predicted $\sim$12 year period was detected in the source by the OJ-94 programme (Sillanpää et al., 1996a; Valtonen et al., 2008). A very high optical polarization and variability in the degree and angle of polarization has been also reported for OJ 287 (Efimov et al., 2002). The observational properties of the source from radio to X-ray energy bands have been reviewed by Takalo et al. (1994). Recently, Fan et al. (2009), reported large variations in the source of $\Delta$V = 1.96 mag, $\Delta$R = 2.36 mag, and $\Delta$I = 1.95 mag during their observations spanning 2002 to 2007. Our observations of OJ 287 span a period from December 2008 to January 2009, and during this time the brightness of the source changed significantly by $\sim$1 magnitude (Rani et al., 2010b). The source showed significant microvariations only in one out of three observed nightly LCs during which the brightness of the source continuously faded by $\sim$0.08 magnitude within 2 hrs (Fig. 5). The SF and DCF analysis showed that any timescale of variability is longer than the timescale of observation. 3C 273: The FSRQ 3C 273 was the first quasar discovered and has a redshift of 0.158 (Schmidt, 1963). It is categorized as a LBL (Nieppola et al., 2006) and its spectral energy distribution, correlations among different energy band flares and the approaching jet orientation have been extensively studied at all wavelengths. There are many papers covering the observational properties of the source in the optical band (Angione, 1971; Sitko et al., 1982; Corso et al., 1985, 1986; Moles et al., 1986; Hamuy & Maza, 1987; Sillanpää et al., 1991; Valtaoja et al., 1991b; Takalo et al., 1992a, b; Elvis et al., 1994; Lichti et al., 1995; Ghosh et al., 2000). An analysis of the optical light curve of 3C 273 spanning over 100 years can be interpreted to suggest a LTV timescale of $\sim$13.5 years (Fan et al., 2001). Recently, the short-term optical variability and colour index properties of the source have been studied by Dai et al. (2009). Our IDV observations of the source 3C 273 span the period from December 2008 to April 2009 during which a total of eight LCs were obtained. There is no change in overall flux of the source during this period (Rani et al., 2010b). This source showed microvariations (of $\sim$0.05 magnitude) in only two out of eight observed LCs (Fig. 5). On the night of 19 April 2009 there is a hint of a timescale of variability of $\sim$6 hrs from both the SF and DCF approaches. PKS 1510$-$089: The blazar PKS 1510$-$089 is classified as a FSRQ and has $z=0.361$. It also belongs to the category of highly polarized quasars. Significant optical flux variations in the source were first reported by Lu (1972) over a time span of $\sim$5 years. The historical light curve of the source shows a large variation of $\Delta$B = 5.4 mag during an outburst in 1948 after which it faded by $\sim$2.2 mag within 9 days (Liller & Liller, 1975). Strong variations on IDV time scales also have been reported for PKS 1510$-$089; e.g., $\Delta$R = 0.65 mag within 13 min. (Xie et al., 2001), $\Delta$R = 2.0 mag in 42 min. (Dai et al., 2001), $\Delta$V = 1.68 mag in 60 min. (Xie et al., 2002a). In the optical light curves of this source, a few deep minima have been observed on different days (Xie et al., 2001; Dai et al., 2001; Xie et al., 2002b), that nominally correspond to a time scale of $\sim$42 min, though no more than 3 such dips were ever seen in a single night. Nonetheless, an eclipsing binary black hole model was actually proposed to explain the occurrences of these minima (Wu et al., 2005). Other observations by this group have yielded a claim of another possible time scale between minima of $\sim$89 min (Xie et al., 2004). We carried out IDV observations of the source from April to June 2009. There is a large change in the brightness of the source during this period , with $\Delta$Rmag = 1.5 (Rani et al., 2010b). We found significant microvariations of $\sim$0.05–0.08 magnitude in three out of four LCs (Fig. 6). We found that any timescale of variability is larger than the timescale of observations, except perhaps for the LC observed on April 19, 2009 for which it is formally $\sim$0.6 hrs from the SF curve and $\sim$00.5 hr from the DCF; however, this LC has too few points to allow the production of a crisp SF or DCF, so this evidence is very weak. BL Lac: The object BL Lac at $z=0.069$ (Miller et al., 1978) is the archetype of its class. Observations over the past few decades have showed that its optical and radio emissions are highly variable and polarized and the polarization at those widely separated frequencies is found to be strongly correlated (Sitko et al., 1985). It is among the very few sources for which more than 100 years of optical data is available in the literature (Shen, 1970; Webb et al., 1988; Fan et al., 1998). An optical variation of $\Delta$B = 5.3 mag and a possible periodicity of $\sim$14 years has been reported for BL Lac by Fan et al. (1998). Very recently, Nieppola et al. (2009) have studied the long term variability of the source at radio frequencies and generalized the shock model that can explain it. Our IDV observations of the source BL Lac were made between September 2008 and June 2009. The brightness of the source faded by $\sim$1.6 magnitude during this period (Rani et al., 2010b). BL Lac faded by $\sim$0.05 magnitude within 1 hr of observation on September 04, 2008 which had a nominal variability timescale of $\sim$2.5 hrs according to both the SF and DCF plots (Fig. 7). A continuous rising trend of $\sim$0.1 magnitude in the LC of the source was observed on June 21, 2009, and the calculated timescale of variability for the LC that night exceeds the time period of the observations. 3C 454.3: A FSRQ at a redshift of 0.859, 3C 454.3 is among the most intense and variable sources. The source has been detected in the flaring state in July 2007 and July 2008 at $\gamma$-ray frequencies and those flares have been found to be well correlated with optical and longer wavelength flares (Ghisellini et al., 2007; Raiteri et al., 2008; Villata et al., 2007). The long term observational properties of the source at optical and radio frequencies have been well studied through multiwavelength campaigns (e.g., Villata et al., 2006, 2007). The IDV of the source was recently studied by Gupta et al. (2008c). They have reported that the amplitude varied by $\sim$5 - 17$\%$ during their observing span. An extraordinary flaring activity above 100 MeV has been reported in the source in December 2009 (Striani et al., 2010). We observed two IDV LCs of the source on October 24 and 28, 2008. During the period the brightness of the source increases by $\sim$0.4 magnitude (Rani et al., 2010b). The source showed significant microvariations of $\sim$0.13 magnitude on October 24, 2008 (Fig. 7) while no significant variations has been detected for the LC observed on October 28, 2008. The SF and DCF analysis revealed that any timescale of variability exceeds the length of the observation on the first of those nights. ## 4 Discussion and Conclusions We have carried out optical R band IDV observations of ten LBLs spanning over a time period of 2008 September to 2009 June. The sources PKS 0420$-$014 and PKS 1510$-$089 were in faint states; AO 0235$+$164, BL Lac and 3C 454.3 were possibly in post-outburst states; S5 0716$+$714 and 3C 66A were in pre- outburst states, while PKS 0735$+$178 and OJ 287 were in some intermediate states during the observing run (Rani et al., 2010b). In our search of microvariations in ten LBLs we found significant IDV in 21 out of 40 observed LCs; so the calculated duty cycle of IDV in LBLs during our observing run is $\sim$52$\%$. We performed the SF and DCF analysis to calculate the nominal timescales of variability; however we found that in most of the cases this timescale of variability is longer than the length of observations and in a substantial majority of the cases where the SF indicated a possible timescale the DCF did not support it. The blazar emission mechanism in the outburst state is quantitatively understood by relativistic shocks propagating through a relativistic jet of plasma. In general, blazar emission in the outburst state is non-thermal Doppler-boosted emission from jets enhanced by that arising from shocks in the flows. (Blandford & Rees, 1978; Marscher & Gear, 1985; Marscher et al., 1992). The other models of AGNs that can explain the IDV in any type of AGN are optical flares, disturbances or hot spots on the accretion disk surrounding the black hole of the AGN (e.g., Mangalam & Wiita, 1993, and references therein). Models based on the instabilities on the accretion disk could convinancibly yield blazar IDV only when the blazar is in the very low state. When a blazar is in the low state, any contribution from the jets, if at all present, is very weak. So, we consider that the observed IDV in the sources AO 0235$+$164, BL Lac, 3C 454.3, S5 0716$+$714, 3C 66A, 0735$+$178 and OJ 287 is almost certainly related to a shock propagating through a relativistic jet (Blandford & Konigl, 1979; Marscher & Gear, 1985). Turbulence behind a shock propagating down such a jet is a very feasible way to explain the observed IDV (Marscher, 1996). Since the blazars PKS 0420$-$014 and PKS 1510$-$089 were observed in relatively faint states, there is a chance that the observed optical IDV in these source may be because of hot spots or any other enhanced emission on the accretion disk (Mangalam & Wiita, 1993). In one source, 3C 273, we are unable to classify the state of the source since this blazar has been in an essentially steady state for last six years (Dai et al., 2009; Rani et al., 2010b). This source showed significant microvariations in two out of eight observed LCs and on both of days of observation the brightness of the source follows both rising and fading trends of $\sim$0.05 magnitude. Whatever may be the mechanism responsible for the origin of of microvariability in this source, it does not seem to be strong enough to introduce day-to-day variations in the flux of the source. It is worth noting that an extrinsic mechanism can also be responsible for some of the observed IDV in blazars. Extrinsic IDV could be caused by refractive interstellar scintillation, which is only relevant in low-frequency radio observations, or microlensing, which is achromatic. We note that the blazar AO 0235$+$164 has two foreground galaxies at $z=0.524$ and $z=0.851$ (Cohen et al., 1987; Nilsson et al., 1996; Webb et al., 2000). The flux of this source is contaminated and absorbed by foreground galaxies, the stars of which can act as gravitational microlenses. Thus the observed optical IDV in AO 0235+164 could arise, at least partially, from gravitational microlensing. Using two separated telescopes to simultaneously observe 0716$+$714 Pollock, Webb & Azarnia (2007) showed that instrumental and atmospheric effects cannot account for the microvariations they measured for that blazar. We found significant IDV in 21 out of 40 observed LCs of ten LBLs; so the calculated duty cycle of IDV in LBLs during our observing run is $\sim$52$\%$. The average duration of our observation was 3.7 hrs per LC. The SF and DCF curves revealed that in $\sim$60$\%$ cases any timescale of variability is longer than the timescale of observations. In quite a few cases there were hints of timescales in the data from the SF plots, but only a few of those hints were supported by the DCF plots. Extensive IDV studies of different subclasses of AGNs revealed that the occurrence of IDV in blazars observed on a timescale of $<$6 hrs is $\sim$60$-$65$\%$ and if the blazar is observed more than 6 hrs then the possibility of IDV detection is 80$-$85$\%$ (Carini, 1990; Gupta & Joshi, 2005, and references therein). If we consider observations over days to months, i.e., at STV timescales then the observed duty cycle of variations is $>$92$\%$ (e.g. Rani et al., 2010b) and at LTV timescales it is almost 100$\%$, which confirms that the probability of detection of variability in blazars rises with the duration of the observations. 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(2004) Zhang, X., Zhang, L., Zhao, G., Xie, Z., Wu, L., & Zheng, Y. 2004, , 128, 1929 Table 1: Properties of Telescopes and Instruments Site: | ARIES Nainital | PRL Mount Abu ---|---|--- Telescope: | 1.04-m RC Cassegrain | 1.20 m Cassegrain CCD model: | Wright 2K CCD | Andor EMCCD Chip size: | $2048\times 2048$ pixels | $2048\times 2048$ pixels Pixel size: | $24\times 24$ $\mu$m | $13\times 13$ $\mu$m Scale: | 0.37″/pixel | 0.17″/pixel Field: | $13\arcmin\times 13\arcmin$ | $3\arcmin\times 3\arcmin$ Gain: | 10 $e^{-}$/ADU | 5 $e^{-}$/ADU Read Out Noise: | 5.3 $e^{-}$ rms | 4.9 $e^{-}$ rms Binning used: | $2\times 2$ | $2\times 2$ Typical seeing : | 1″to 2.8″ | 1″to 2.6″ Table 2: Observation Log Blazar Name | Date of Observation | Telescope | Filter | Data Points | Duration (hrs) ---|---|---|---|---|--- 3C 66A | 2008 Oct 22 | A | R | 68 | 3.86 | 2008 Oct 26 | A | R | 72 | 3.50 | 2008 Dec 23 | B | R | 90 | 1.45 | 2008 Dec 24 | B | R | 113 | 1.85 | 2008 Dec 27 | B | R | 417 | 3.46 | 2008 Dec 28 | B | R | 203 | 2.23 | 2009 Jan 03 | B | R | 320 | 3.54 AO 0235$+$164 | 2008 Oct 20 | A | R | 98 | 6.30 | 2008 Oct 23 | A | R | 45 | 2.60 | 2008 Dec 26 | B | R | 248 | 1.23 PKS 0420$-$014 | 2008 Oct 23 | A | R | 18 | 1.94 | 2008 Dec 26 | B | R | 29 | 0.73 S5 0716+714 | 2008 Oct 24 | A | R | 25 | 1.35 | 2008 Dec 23 | B | R | 114 | 0.40 | 2008 Dec 24 | B | R | 292 | 1.62 | 2009 Jan 03 | B | R | 2685 | 3.73 PKS 0735$+$178 | 2008 Dec 23 | B | R | 90 | 0.63 | 2008 Dec 28 | B | R | 300 | 3.20 | 2009 Jan 04 | B | R | 50 | 1.02 | 2009 Jan 20 | A | R | 67 | 3.98 OJ 287 | 2008 Dec 26 | B | R | 70 | 0.57 | 2009 Jan 20 | A | R | 60 | 3.87 | 2009 Jan 22 | A | R | 70 | 3.35 3C 273 | 2008 Dec 23 | B | R | 130 | 0.72 | 2009 Jan 22 | A | R | 90 | 3.88 | 2009 Feb 25 | A | R | 101 | 3.61 | 2009 Mar 24 | A | R | 36 | 1.38 | 2009 Apr 01 | A | R | 110 | 5.60 | 2009 Apr 18 | A | R | 91 | 3.54 | 2009 Apr 19 | A | R | 195 | 7.42 | 2009 Apr 27 | A | R | 66 | 3.31 PKS 1510$-$089 | 2009 Apr 17 | A | R | 77 | 3.82 | 2009 Apr 19 | A | R | 22 | 1.09 | 2009 Apr 27 | A | R | 64 | 4.47 | 2009 June 21 | A | R | 68 | 4.41 BL Lac | 2008 Sep 04 | A | R | 83 | 5.10 | 2008 Oct 26 | A | R | 73 | 4.25 | 2009 June 21 | A | R | 62 | 2.98 3C 454.3 | 2008 Oct 24 | A | R | 55 | 3.01 | 2008 Oct 28 | A | R | 65 | 4.25 A : 1.04 m Sampuranand Telescope, ARIES, Nainital, India B : 1.20 m Telescope, PRL, Mount Abu, India Table 3: Standard Stars in the Blazar Fields Source | Standard | R magnitude | Refrencesa ---|---|---|--- Name | star | (error) | 3C 66A | 1 | 13.36(0.01) | 5 | 2 | 14.28(0.04) | 5 | 3 | 15.46(0.12) | 5 | 4 | 12.70(0.04) | 6 | 5 | 13.62(0.05) | 6 AO 0235+164 | 1 | 12.69(0.02) | 1 | 2 | 12.23(0.02) | 1 | 3 | 12.48(0.03) | 1 | 6 | 13.64(0.04) | 6 | 8 | 15.79(0.10) | 1 | C1 | 14.23(0.05) | 6 PKS 0420$-$014 | 1 | 12.09(0.03) | 4 | 2 | 12.80(0.02) | 5 | 3 | 12.89(0.01) | 5 | 4 | 14.47(0.01) | 5 | 5 | 14.37(0.03) | 4 | 6 | 14.70(0.03) | 4 | 7 | 14.91(0.03) | 4 | 8 | 15.46(0.03) | 4 | 9 | 15.58(0.04) | 4 S5 0716+714 | 1 | 10.63(0.01) | 2 | 2 | 11.12(0.01) | 2, 3 | 3 | 12.06(0.01) | 2, 3 | 4 | 12.89(0.01) | 2 | 5 | 13.18(0.01) | 2, 3 | 6 | 13.26(0.01) | 2, 3 | 7 | 13.32(0.01) | 2 | 8 | 13.79(0.02) | 2 PKS 0735+178 | A | 13.14(0.05) | 1 | C | 13.87(0.06) | 1 | D | 15.45(0.06) | 1 OJ 287 | 2 | 12.46(0.05) | 1 | 4 | 13.72(0.06) | 1 | 10 | 14.26(0.06) | 1 | 11 | 14.67(0.07) | 1 3C 273 | C | 11.30(0.04) | 1 | D | 12.31(0.04) | 1 | E | 12.27(0.05) | 1 | G | 13.16(0.05) | 1 PKS 1510$-$089 | 1 | 11.23(0.03) | 5 | 2 | 12.95(0.03) | 5 | 3 | 13.98(0.09) | 5 | 4 | 14.34(0.05) | 5 | 5 | 14.35(0.05) | 4 | 6 | 14.61(0.02) | 4 BL Lac | B | 11.93(0.05) | 1 | C | 13.69(0.03) | 1 | H | 13.60(0.03) | 1 | K | 14.88(0.05) | 1 3C 454.3 | A | 15.32(0.09) | 6, 9 | B | 14.73(0.05) | 6, 9 | C | 13.98(0.02) | 9, 10 | D | 13.22(0.01) | 9, 10 | E | 14.92(0.08) | 6, 9 | F | 14.83(0.03) | 4, 9 | G | 14.83(0.02) | 4, 9 | H | 13.10(0.04) | 6, 9 | C1 | 15.27(0.06) | 6 a1\. (Smith et al., 1985); 2. (Villata et al., 1998); 3. (Ghisellini et al., 1997); 4. (Raiteri et al., 1998); 5. (Smith & Balonek, 1998); 6\. (Fiorucci & Tosti, 1996); 7. Craine E.R.:Handbook of Quasistellar and BL Lacertae Objects, (Angione, 1971); 8\. http://www.lsw.uni- heidelberg.de/projects/extragalactic/charts/2251+158.html Table 4: IDV Results Blazar Name | Date of | C - Test | F - Test | V | A ($\%$) | tv ---|---|---|---|---|---|--- | Observation | value | F-value | p-value | | | SF (hrs) | DCF (hrs) 3C 66A | 2008 Oct 22 | 3.59 | 3.65 | 3.1e-7 | V | 8.1 | $>$3.86 | $>$3.86 | 2008 Oct 26 | 2.65 | 7.03 | 1.9e-14 | V | 5.7 | $>$3.50 | $>$3.50 | 2008 Dec 23 | 0.78 | 0.41 | 0.06 | NV | | | | 2008 Dec 24 | 3.31 | 0.61 | 0.02 | NV | | | | 2008 Dec 27 | 0.88 | 0.58 | 0.26 | NV | | | | 2008 Dec 28 | 0.75 | 0.37 | 0.3 | NV | | | | 2009 Jan 03 | 0.83 | 0.63 | 0.35 | NV | | | AO 0235$+$164 | 2008 Oct 20 | 8.32 | 76.84 | 2.2e-16 | V | 13.4 | 5.20? | $>$6.30 | 2008 Oct 23 | 3.23 | 10.53 | 1.4e-14 | V | 4.2 | $>$2.60 | $>$2.60 | 2008 Dec 26 | 1.09 | 1.43 | 0.04 | NV | | | PKS 0420$-$014 | 2008 Oct 23 | 5.44 | 31.44 | 6.6e-15 | V | 14.4 | | | 2008 Dec 26 | 5.42 | 31.24 | 6.1e-15 | V | 12.3 | 0.12,0.18 | 0.18 S5 0716$+$714 | 2008 Oct 24 | 0.92 | 1.57 | 0.27 | NV | | | | 2008 Dec 23 | 3.47 | 7.77 | 2.2e-16 | V | 9.1 | 0.0014 | 0.0028 | 2008 Dec 24 | 4.51 | 3.07 | 2.2e-16 | V | 14.6 | $>$1.62 | $>$1.62 | 2009 Jan 03 | 3.71 | 1.26 | 7.7e-7 | V | 31.6 | 3.28 | $>$3.73 PKS 0735$+$178 | 2008 Dec 23 | 3.14 | 9.49 | 2.2e-16 | V | 11.1 | | | 2008 Dec 28 | 4.22 | 18.36 | 2.2e-16 | V | 19.3 | 2.90 | 2.30 | 2009 Jan 04 | 2.31 | 4.64 | 3.1e-7 | V | 9.8 | 0.58 | 0.60 | 2009 Jan 20 | 1.30 | 1.99 | 0.055 | NV | | | OJ 287 | 2008 Dec 26 | 1.11 | 1.48 | 0.10 | NV | | | | 2009 Jan 20 | 0.88 | 1.20 | 0.47 | NV | | | | 2009 Jan 22 | 3.21 | 11.13 | 2.2e-16 | V | 7.9 | $>$3.35 | $>$3.35 3C 273 | 2008 Dec 23 | 0.91 | 0.73 | 0.08 | NV | | | | 2009 Jan 22 | 0.81 | 0.65 | 0.05 | NV | | | | 2009 Feb 25 | 0.75 | 0.49 | 0.06 | NV | | | | 2009 Mar 24 | 0.76 | 0.50 | 0.05 | NV | | | | 2009 Apr 01 | 2.59 | 2.58 | 1.2e-6 | V | 5.2 | $>$5.2 | $>$5.60 | 2009 Apr 18 | 1.59 | 0.25 | 0.11 | NV | | | | 2009 Apr 19 | 3.74 | 2.22 | 4.7e-8 | V | 4.5 | 6.0 | 6.0 | 2009 Apr 27 | 0.86 | 0.84 | 0.49 | NV | | | PKS 1510$-$089 | 2009 Apr 17 | 2.67 | 2.31 | 0.0003 | V | 9.3 | $>$3.82 | $>$3.82 | 2009 Apr 19 | 2.72 | 7.11 | 3.3e-5 | V | 7.6 | 0.60 | 0.50 | 2009 Apr 27 | 1.27 | 1.54 | 0.09 | NV | | | | 2009 June 21 | 4.46 | 2.65 | 0.0001 | V | 8.3 | $>$4.41 | $>$4.41 BL Lac | 2008 Sep 04 | 2.73 | 5.71 | 1.0e-13 | V | 8.7 | 2.5 | 2.5 | 2008 Oct 26 | 1.05 | 1.56 | 0.06 | NV | | | | 2009 June 21 | 2.87 | 7.34 | 4.9e-13 | V | 12.9 | $>$2.98 | $>$2.98 3C 454.3 | 2008 Oct 24 | 5.14 | 22.95 | 2.2e-16 | V | 16.3 | $>$3.01 | $>$3.01 | 2008 Oct 28 | 1.42 | 2.09 | 0.01 | NV | | | Figure 1: R band optical IDV LCs of the blazars 3C 66A and AO 0235$+$164 and their respective SFs and DCFs. Figure 2: R band optical IDV LCs of the blazars AO 0235$+$164 and PKS 0420$-$014 and their respective SFs and DCFs. Figure 3: R band optical IDV LCs of the blazar S5 0716$+$714 and their respective SFs and DCFs. Figure 4: R band optical IDV LCs of the blazar PKS 0735$+$178 and their respective SFs and DCFs. Figure 5: R band optical IDV LCs of the blazars 3C 273 and OJ 287 and their respective SFs and DCFs. Figure 6: R band optical IDV LCs of blazar PKS 1510$-$089 and their respective SFs and DCFs. Figure 7: R band optical IDV LC of the blazars BL Lac and 3C 454.3 and their respective SFs and DCFs. Figure 8: The R filter LCs of several blazars during which no significant variability has been detected, plotted along with the differential magnitudes of standard stars. Figure 9: The R filter LCs of several blazars during which no significant variability has been detected, plotted along with the differential magnitudes of standard stars.
arxiv-papers
2011-01-04T11:57:23
2024-09-04T02:49:16.125128
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Bindu Rani (1), Alok C. Gupta (1), U. C. Joshi (2), S. Ganesh (2),\n Paul J. Wiita (3), ((1) Aryabhatta Research Institute of Observational\n Sciences (ARIES), India, (2) Physical Research Laboratory, Navrangpura,\n India, (3) Department of Physics, The College of New Jersey)", "submitter": "Bindu Rani Ms.", "url": "https://arxiv.org/abs/1101.0711" }
1101.0811
# Measuring the Spins of Accreting Black Holes Jeffrey E. McClintock11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Ramesh Narayan11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Shane W. Davis22affiliation: Canadian Institute for Theoretical Astrophysics, Toronto, ON M5S3H4, Canada , Lijun Gou11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Akshay Kulkarni11affiliation: Harvard- Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Jerome A. Orosz33affiliation: Department of Astronomy, San Diego State University, 5500 Companile Drive, San Diego, CA 92182 , Robert F. Penna11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 , Ronald A. Remillard44affiliation: MIT Kavli Institute for Astrophysics and Space Research, MIT, 70 Vassar Street, Cambridge, MA 02139 , James F. Steiner11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 ###### Abstract A typical galaxy is thought to contain tens of millions of stellar-mass black holes, the collapsed remnants of once massive stars, and a single nuclear supermassive black hole. Both classes of black holes accrete gas from their environments. The accreting gas forms a flattened orbiting structure known as an accretion disk. During the past several years, it has become possible to obtain measurements of the spins of the two classes of black holes by modeling the X-ray emission from their accretion disks. Two methods are employed, both of which depend upon identifying the inner radius of the accretion disk with the innermost stable circular orbit (ISCO), whose radius depends only on the mass and spin of the black hole. In the Fe K$\alpha$ method, which applies to both classes of black holes, one models the profile of the relativistically- broadened iron line with a special focus on the gravitationally redshifted red wing of the line. In the continuum-fitting method, which has so far only been applied to stellar-mass black holes, one models the thermal X-ray continuum spectrum of the accretion disk. We discuss both methods, with a strong emphasis on the continuum-fitting method and its application to stellar-mass black holes. Spin results for eight stellar-mass black holes are summarized. These data are used to argue that the high spins of at least some of these black holes are natal, and that the presence or absence of relativistic jets in accreting black holes is not entirely determined by the spin of the black hole. ## 1 Introduction Our focus throughout is on the two principal methods for measuring the spins of accreting black holes: modeling the thermal continuum X-ray spectrum and modeling the profile of the relativistically-broadened Fe K$\alpha$ line. The continuum-fitting (CF) method, has thus far only been applied to stellar-mass black holes in X-ray binaries, whereas the Fe K$\alpha$ method has been applied to both stellar-mass ($M\sim 10$ $M_{\odot}$) and supermassive ($M\sim 10^{6}-10^{10}$ $M_{\odot}$) black holes1111 $M_{\odot}$ = 1 solar mass = $2.0\times 10^{33}$ g.. This paper is chiefly focused on measuring the spins of stellar-mass black holes via the CF method because we have been deeply engaged in this work during the past several years. In Section 6, we secondarily discuss the Fe K$\alpha$ method, which is very important because it is the primary approach to measuring the spins of supermassive black holes. We note that spin estimates have also been obtained by modeling the high- frequency X-ray oscillations (100–450 Hz) observed for several stellar-mass black holes (Wagoner et al. 2001; Török et al. 2005). At present, this method is not providing dependable results because the correct model of these oscillations is not known. X-ray polarimetry is another potential avenue for measuring spin (Dovčiak et al. 2008; Li et al. 2009; Schmoll et al. 2009), which may be realized soon with the 2014 launch of the Gravity and Extreme Magnetism Small Explorer (Swank et al. 2008). Meanwhile, several other methods of measuring spin have been proposed or applied (e.g., Takahashi 2004; Barai et al. 2004; Huang et al. 2007; Suleimanov et al. 2008; Shcherbakov & Huang 2011). ## 2 Stellar-Mass Black Holes in X-ray Binaries Observations in 1972 of the X-ray binary Cygnus X-1 provided the first strong evidence that black holes exist. Today, a total of 23 such X-ray binary systems are known that contain a compact object too massive to be a neutron star or a degenerate star of any kind (i.e., $M>3$ $M_{\odot}$; Özel et al. 2010). These compact objects, which have typical masses of $10$$M_{\odot}$, are referred to as black holes. Their host systems are mass-exchange binaries containing a nondegenerate star that supplies gas to the black hole via a stellar wind or via Roche-lobe overflow in a stream that emanates from the inner Lagrangian point. The mass-donor star in the Roche-lobe overflow systems is typically a low mass ($M\sim 1$ $M_{\odot}$) sun-like star, and the X-ray source is transient, alternating between yearlong outbursts ($L_{\rm max}\sim L_{\rm Edd}=1.3{\times}10^{39}{M/10M_{\odot}}$ erg s-1) and years or decades of quiescence ($L\sim 10^{-7}L_{\rm Edd}$)222$L_{\rm Edd}$ is the critical Eddington luminosity above which radiation pressure exceeds gravity.. The wind-fed X-ray sources, on the other hand, are fueled by massive hot stars ($M\gtrsim 10M_{\odot}$) and are persistently luminous. A schematic sketch to scale of 21 of these systems is shown in Figure 1: The four at top are persistent systems and the 17 at bottom are the transients. For a review of the properties of black hole binaries, see Remillard & McClintock (2006). ## 3 The Continuum-Fitting Method A definite prediction of relativity theory is the existence of an innermost stable circular orbit (ISCO) for a particle orbiting a black hole. This inherently relativistic effect has a major impact on the structure of an accretion disk. At radii $R\geq R_{\rm ISCO}$ (the radius of the ISCO), accreting gas moves on nearly circular orbits and slowly spirals in toward the black hole. At the ISCO, however, the dynamics change suddenly and the gas, finding no more stable circular orbits, plunges into the hole. In the continuum-fitting (CF) method, one identifies the inner edge of the accretion disk with the ISCO (see Secs. 4 and 5 for supporting evidence) and estimates $R_{\rm ISCO}$ by fitting the X-ray continuum spectrum. Since the dimensionless radius $r_{\rm ISCO}\equiv R_{\rm ISCO}/(GM/c^{2})$ is solely a monotonic function of the black hole spin parameter $a_{*}$ (Fig. 2)333We express black hole spin in the customary way as the dimensionless quantity $a_{*}\equiv cJ/GM^{2}$ with $|a_{*}|\leq 1$, where $M$ and $J$ are respectively the black hole mass and angular momentum., knowing its value allows one immediately to infer the value of $a_{*}$. We note that the truncation of the disk at the ISCO is also a crucial assumption of the Fe K$\alpha$ method of measuring spin (Reynolds & Fabian 2008). Before describing the CF methodology, we stress that for this technique to succeed it is essential to have accurate measurements of the distance $D$ to the source, the inclination $i$ of the accretion disk, and the mass $M$ of the black hole (for reasons discussed below). The methodologies for measuring $D$, $i$ and $M$ are firmly established. Therefore, rather than digressing to discuss how these measurements are made, we refer the interested reader to some recent papers on the subject (Orosz et al. 2007, 2009, 2011; Cantrell et al. 2010). The gaseous matter flowing from the companion star to the black hole has appreciable angular momentum as a consequence of the binary orbital motion. As the gas flows, viscous forces cause it to spread out into an orbiting structure known as an accretion disk. The gas flowing into the outer disk spirals slowly inward on Keplerian orbits on a time scale of weeks, reaching a typical temperature near the ISCO of $kT\sim 1$ keV. Because the accretion disk is of fundamental importance to the measurement of black hole spin, we now describe in some detail the thin-disk model we employ. The model we use is that described in Novikov & Thorne (1973, hereafter NT), which is a relativisitic generalization of a Newtonian model developed by Shakura & Sunyaev (1973). The NT model describes an axisymmetric radiatively- efficient accretion flow which, for a given black hole mass $M$, mass accretion rate $\dot{M}$ and black hole spin parameter $a_{*}$, gives a precise prediction for the local radiative flux $F(R)$ emitted at each radius $R$ of the disk. Moreover, the accreting gas is optically thick and the emission is thermal and blackbody-like, making it straightforward to compute the spectrum of the emission. Most importantly, the inner edge of the disk is located at the ISCO. Therefore, from the measurement of $R_{\rm ISCO}$, and if we know the mass $M$ of the black hole, we can immediately obtain $a_{*}$ (Fig. 2). This is the principle behind the CF method of estimating black hole spin, which was first described by Zhang et al. (1997). Before discussing how to measure $R_{\rm ISCO}$ of a disk, we remind the reader how one measures the radius $R_{*}$ of a star. Given the distance $D$ to the star, the radiation flux $F_{\rm obs}$ received from the star, and the temperature $T$ of its continuum radiation, the luminosity of the star is given by $L_{*}=4\pi D^{2}F_{\rm obs}=4\pi R_{*}^{2}\sigma T^{4},$ (1) where $\sigma$ is the Stefan-Boltzmann constant. From $F_{\rm obs}$ and $T$, we immediately obtain the quantity $\pi(R_{*}/D)^{2}$, which is the solid angle subtended by the star. From this and the distance $D$, we immediately obtain $R_{*}$. For accurate results we must allow for limb darkening and other non-blackbody effects in the stellar emission by computing a stellar atmosphere model, but this is a detail. The same principle applies to an accretion disk, but with some differences. First, since the flux $F(R)$ emitted locally by the disk varies with radius $R$, the radiation temperature $T(R)$ also varies with $R$. But the precise variation is known (if we assume the NT disk model), so it is easily incorporated into the model. Second, since the bulk of the emission is from the very inner regions of the disk, the effective area of the radiating surface is directly proportional to the square of the disk inner radius, $A_{\rm eff}=CR_{\rm ISCO}^{2}$, where the constant $C$ is known. Third, the observed flux $F_{\rm obs}$ depends not only on the luminosity and the distance, but also on the inclination $i$ of the disk to the line of sight. Allowing for these differences, one can write a relation for the disk problem similar in spirit to eq. (1), i.e., given $F_{\rm obs}$ and a characteristic $T$ (from X-ray observations), one obtains the solid angle subtended by the ISCO: $\pi\cos i\,(R_{\rm ISCO}/D)^{2}$. If we further know $i$ and $D$, we obtain $R_{\rm ISCO}$; and if we also know $M$, we obtain $a_{*}$ (Fig. 2). This is the basic idea of the CF method. There are three main issues that must be dealt with before applying the method: (1) One must carefully trace rays from the disk to the observer in the Kerr metric of the rotating black hole in order to compute accurately the observed flux and spectrum. To this end, we have developed an accretion disk model called kerrbb444This model name and those that follow designate publicly-available programs that comprise a suite of X-ray data analysis software known as XSPEC (Arnaud 1996; http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/index.html). (Li et al. 2005) for fitting X-ray data. kerrbb assumes the NT model of the disk and carries out all the necessary ray-tracing to relate disk properties to observables. (2) One must have an accurate model for computing the spectral hardening factor $f=T/T_{\rm eff}$, where $T$ is the temperature of the radiation at a given radius and $T_{\rm eff}$ is the effective temperature at the same radius defined by $F(R)=\sigma T_{\rm eff}^{4}(R)$. This correction for non-blackbody effects is important at the high temperatures typically found in black hole disks. To carry out this correction we use the advanced disk atmosphere models of Davis et al. (2005) and Davis & Hubeny (2006). (3) Most importantly, the inner accretion disk must be well described by the standard geometrically- thin, optically-thick NT disk model that we employ. To ensure this, we restrict our attention strictly to observations with a strong thermal component (Steiner et al. 2009a) and with disk luminosities below 30% of the Eddington limit (McClintock et al. 2006). In the two sections that follow, we present observational and theoretical evidence that at these luminosities the NT model is quite accurate. For a full description of the mechanics of the CF method we refer the reader to Section 4 in McClintock et al. (2006). In brief, we fit the broadband X-ray continuum spectrum in conjunction with other components as needed, principally a Compton tail component that in more recent work is described using an empirical model of Comptonization called simpl (Steiner et al. 2009b). The accretion-disk component, which is key for the CF method, is modeled using kerrbb (Li et al. 2005), which includes all relativistic effects within the context of the NT model, and also incorporates the advanced treatment of spectral hardening mentioned above. It furthermore includes self-irradiation of the disk (“returning radiation”) and the effects of limb darkening. The key fit parameters returned are the black hole spin parameter $a_{*}$ and the mass accretion rate $\dot{M}$. ## 4 Truncation of the Accretion Disk at the ISCO: Observational Evidence A crucial assumption that underlies both the CF and Fe K$\alpha$ approaches to measuring spin is that the accretion disk is quite sharply truncated at the ISCO. This assumption, which is fundamental to the NT model, is clearly valid if one considers only geodesic forces in the midplane. However, there are strong magnetohydrodynamic (MHD) forces acting in black hole accretion disks, and it is therefore unclear a priori that the disk terminates sharply at the ISCO. In this section, we present the observational evidence that there exists a fixed inner-disk radius in black hole binaries and, in the following section, we discuss the theoretical evidence for identifying this radius with $r_{\rm ISCO}$. There is a long history of evidence suggesting that fitting the X-ray continuum is a promising approach to measuring black hole spin. This history begins in the mid-1980s with the application of a simple nonrelativistic multicolor disk model (Mitsuda et al. 1984; Makishima et al. 1986), now known as diskbb, which returns the color temperature $T_{\rm in}$ at the inner-disk radius $R_{\rm in}$. In their review paper on black hole binaries, Tanaka & Lewin (1995) summarize examples of the steady decay (by factors of 10–100) of the thermal flux of transient sources during which $R_{\rm in}$ remains quite constant (see their Fig. 3.14). They remark that the constancy of $R_{\rm in}$ suggests that this fit parameter is related to the radius of the ISCO. Zhang et al. (1997) then outlined how, using a relativistic disk model and corrections for the effects of radiative transfer, the fixed inner disk radius provides an observational basis to infer black hole spin. More recently, the evidence for a constant inner radius in the thermal state has been presented for a number of sources via plots showing that the bolometric luminosity of the thermal component is approximately proportional to $T_{\rm in}^{4}$ (e.g., Kubota et al. 2001; Kubota & Makishima 2004; Gierliński & Done 2004; Abe et al. 2005; McClintock et al. 2009; Dunn et al. 2010). A recent study of the persistent source LMC X-3 presents the most compelling evidence to date for a constant inner-disk radius (Steiner et al. 2010a). We analyzed many spectra collected during eight X-ray missions that span 26 years. As shown in Figure 3, for a selected sample of hundreds of spectra obtained using the Rossi X-ray Timing Explorer (RXTE), we find that to within $\approx 2$ percent the inner radius of the accretion disk is constant over time and unaffected by the gross variability of the source (top panel). Meanwhile, even considering an ensemble of eight X-ray missions, we find consistent values of the radius to within $\approx 4$ percent. These results provide compelling evidence for the existence of a fixed inner-disk radius and establish a firm foundation for the measurement of black hole spin. The only reasonable inference is that this radius is closely associated with the radius of the ISCO, as we show to be the case in the following section. ## 5 Truncation of the Accretion Disk at the ISCO: Theoretical Evidence The relativistic NT model on which the CF method is currently built assumes, as does its predecessor the Newtonian model of Shakura & Sunyaev (1973), that the accretion disk under consideration is geometrically thin. That is, the model assumes that the vertical thickness $H$ at any radius $R$ satisfies $H\ll R$. Assuming in addition that the disk is axisymmetric and in steady state, the model derives a number of relations which follow directly from basic conservation laws. One of the powerful results of this analysis is a formula for the disk flux profile $F(R)$ which depends only on the mass $M$ and spin $a_{*}$ of the black hole and the mass accretion rate $\dot{M}$, but is independent of messy details such as the viscosity of the accreting gas. It is the existence of this robust result for $F(R)$ that enables the CF method to work so well. There is, however, one unproven assumption in the NT model which is incorporated via a boundary condition: The model assumes that the shear stress (which drives the accretion at larger radii) vanishes at the ISCO. This “zero- torque” assumption is intuitively reasonable (since the gas switches to a rapidly plunging state once it crosses the ISCO, why should there be a stress at the transition radius?), but as NT themselves realized, it is ultimately an assumption. Paczyński (2000) and Afshordi & Paczyński (2003) argued that deviations from the NT model decrease monotonically with decreasing disk thickness and that thin disks with $H/R\ll 1$ should be very well described by the model. Their argument, which was based on a hydrodynamical description of the disk, was confirmed by detailed calculations carried out by Shafee et al. (2008b). However, this leaves open the question of whether magnetized disks might deviate substantially from the NT model. In their paper, NT explicitly mention that magnetized disks could very well violate the zero-torque boundary condition. Arguments have been advanced to suggest that a magnetized accreting gas will indeed have a non-zero shear stress at the ISCO (Krolik 1999; Gammie 1999), and that furthermore this stress could be so large that it may completely invalidate the NT model even in very thin disks. This is clearly an important question that strikes at the heart of the CF method. A number of recent studies of magnetized disks using three-dimensional general relativistic magnetohydrodynamic (GRMHD) simulations have explored this question (Shafee et al. 2008a; Reynolds & Fabian 2008; Noble & Krolik 2009; Noble et al. 2010; Penna et al. 2010). The conclusion of these authors is that the shear stress and the luminosity of the simulated disks do differ from the NT model, but perhaps not by a large amount. Figure 4, taken from Kulkarni et al. (2010), shows the disk luminosity distribution $dL/d\ln R=4\pi R^{2}F(R)$ derived from a set of four GRMHD thin- disk models simulated by Penna et al. (2010). These models have thicknesses $H/R\sim 0.045-0.08$ (see Penna et al. and Kulkarni et al. for a precise definition of $H$, which varies directly with luminosity). As is clear from the figure, the numerical models follow the NT model reasonably well, although they do deviate from it. Two kinds of deviation are seen. First, the numerical models produce some radiation inside the ISCO, whereas the NT model predicts no radiation there. Second, the peak of the emission in the simulated disks is shifted inward relative to the peak in the NT model. Both of these effects cause the disk to appear to have a smaller ISCO radius. This in turn means that, if one were to infer the black hole spin by fitting this luminosity distribution (or the corresponding spectrum) using the NT model, one would infer an erroneously large value for the spin. This systematic error arises because the NT model is not a perfect description of the simulated disk. To the extent that the simulated disk is a closer match to a real accretion disk than is the NT model, this allows us to estimate the corresponding systematic error in our measurements of spin. How serious is this systematic error? We answer this in three parts (see Kulkarni et al. 2010, for details). 1\. Each of the models shown in Figure 4 causes an error in the spin estimate that is smallest for a low disk inclination (face-on disk; see Fig. 1) and largest for a high disk inclination ($i=75^{\circ}$). For the latter (most unfavorable) case, the four models, which correspond to true spin values of $a_{*}=0$, 0.7, 0.9 and 0.98, give via the CF method spin values of 0.17, 0.83, 0.936 and 0.991, respectively. 2\. A similar exercise can be carried out for disks with other thicknesses. It is found that the error in the spin estimate is larger for thicker disks and smaller for thinner disks. Very roughly, the error appears to scale as $H/R$. Thus, the conclusion of Paczyński (2000), Afshordi & Paczyński (2003) and Shafee et al. (2008b), that deviations from the NT model vanish in the limit of vanishingly small disk thickness, appears to be valid also for MHD disks. 3\. The particular models shown in Figure 4 correspond to disk luminosities in the range $L/L_{\rm Edd}\sim 0.4-0.8$, based on their $H/R$. (It is difficult to be very quantitative since the mapping between luminosity and $H/R$ is not known precisely.) Since the CF method is applied only to observations at $L/L_{\rm Edd}<0.3$, the systematic error due to inaccuracies in the theoretical model could be up to a factor of 2 smaller than the errors quoted in point 1 above. Although the above results are based on numerical simulations that do not necessarily mimic real disks perfectly, we believe they still provide an estimate of the likely magnitude of the systematic error. The key point is that the level of systematic error we find is not serious at the current time. The observational errors considered in the following section are significantly larger in all cases. Stating this differently, while magnetized disks do behave as if their inner edges are shifted inward relative to the position of the ISCO (Fig. 4), the effect is quantitatively not serious for the disk luminosities (or disk thicknesses) at which the CF method is applied. ## 6 Results of Continuum Fitting The spin results obtained to date for eight stellar-mass black holes are summarized in Table 1. The spins we find cover the full allowable range of prograde spins (Fig. 2) from $a_{*}\approx 0$ (Schwarzschild) to $a_{*}\approx 1$ (extreme Kerr). Interestingly, the spin values that have been obtained (Table 1), while spanning the full range of prograde spins, are all in the canonical physical range, namely $|a_{*}|\leq 1$. This is an important result and not at all a foregone conclusion: Given the hard external constraints on the dynamical model parameters ($D$, $i$ and $M$; Section 3), it is entirely possible that a black hole will be found that implies a spin beyond the reach of our current model, i.e., $|a_{*}|>1$. Such a result could simply be caused by large systematic errors in $D$, $i$ and $M$. Or, more interestingly, it could falsify our model, or even possibly point to new physics. This is why we are excited about improving our measurements of $D$, $i$ and $M$ for the near- extreme Kerr hole GRS 1915+105 (see Sec. 6.2). Table 1: Spin Results to Date for Eight Black HolesaaErrors are quoted at the 68% level of confidence. | Source | Spin $a_{*}$ | Reference ---|---|---|--- 1 | GRS 1915+105 | $>0.98$ | McClintock et al. 2006 2 | LMC X–1 | $0.92_{-0.07}^{+0.05}$ | Gou et al. 2009 4 | M33 X–7 | $0.84\pm 0.05$ | Liu et al. 2008, 2010 3 | 4U 1543–47 | $0.80\pm 0.05$ | Shafee et al. 2006 5 | GRO J1655–40 | $0.70\pm 0.05$ | Shafee et al. 2006 6 | XTE J1550–564 | $0.34_{-0.28}^{+0.20}$ | Steiner et al. 2010b 7 | LMC X–3 | $<0.3$bbProvisional result pending improved measurements of $M$ and $i$. | Davis et al. 2006 8 | A0620–00 | $0.12\pm 0.18$ | Gou et al. 2010 Error estimates are primitive for the first four spin results published in 2006 (Table 1). In recent work on the other four black holes, the principal sources of observational error, as well as the uncertainties in the key model parameters (e.g., the viscosity parameter), have been treated in detail. In particular, in our most recent paper on XTE J1550–564, we exhaustively explored many different sources of error (see Table 3 and Appendix A in Steiner et al. 2010b). The upshot of the work to date is that in every case the uncertainty in $a_{*}$ is completely dominated by the errors in the three key dynamical parameters that we input when fitting the X-ray spectral data. As discussed in Section 3, these parameters are the distance $D$, the black hole mass $M$, and the inclination of the inner disk $i$ (which we assume is aligned with the orbital angular momentum vector of the binary; Li et al. 2009). In order to determine the error in $a_{*}$ due to the combined uncertainties in $D$, $M$ and $i$, we perform Monte Carlo simulations assuming that these parameters are normally and independently distributed (e.g., see Gou et al. 2009). Note that the errors introduced by our use of the NT model (Sec. 5), which are not considered here, are significantly smaller than the observational errors (see Table 7 and Sec. 4 in Kulkarni et al. 2010). We now discuss the results for four black holes in some detail. We first consider the persistent source M33 X–7. We then turn to GRS 1915+105, the prototype microquasar (Mirabel & Rodríguez 1994), which hosts a near-extreme Kerr hole. Finally, we consider the microquasars A0620–00 and XTE J1550–564, contrasting their behavior with that of GRS 1915+105. ### 6.1 M33 X-7: The First Eclipsing Black Hole A long observation of the galaxy M33 with the Chandra X-ray Observatory led to the discovery of a black hole that is eclipsed by its companion star (Pietsch et al. 2006). We made a detailed follow-up dynamical study of the optical counterpart of M33 X-7, the first such study of a black hole binary beyond the environs of the Milky Way. We determined a precise mass for the black hole, $M=15.65\pm 1.45$ $M_{\odot}$ (Orosz et al. 2007), as well as the mass of its exceptional companion star ($\approx 70$ $M_{\odot}$). As we discuss in Orosz et al., it is difficult to understand the origin of this system – a massive black hole in a 3.5-day orbit, separated by only 42 solar radii from its supergiant companion (see Fig. 1). Recently, a consistent evolutionary model has been proposed that accounts for all the key properties of the system (Valsecchi et al. 2010). It assumes that M33 X-7 started as a primary of $\sim 95$ $M_{\odot}$ and a secondary of $\sim 30$ $M_{\odot}$, with an orbital period that is close to its present 3.5-day value. Using as input our precise values for the black hole mass and orbital inclination angle, and the well-established distance of M33, we fitted 15 Chandra and XMM-Newton X-ray spectra and obtained a precise value for the spin of the black hole primary, $a_{*}=0.84\pm 0.05$ (Liu et al. 2008, 2010). Remarkably, given that an (uncharged) astrophysical black hole is described by just its mass and spin, this result yields a complete description of an asteroid-size object at a distance of 2.74 million light-years (840 kpc5551 parsec (pc) = 3.26 light-years.). What is the origin of the spin of M33 X-7? Was the black hole born with its present spin, or was it torqued up gradually via the accretion flow supplied by its companion? In order to achieve a spin of $a_{*}=0.84$ via disk accretion, an initially non-spinning black hole must accrete $5.7M_{\odot}$ from its donor star (King & Kolb 1999) in becoming the $M=15.65M_{\odot}$ black hole that we observe today. However, to transfer this much mass even in the case of Eddington-limited accretion requires $>17$ million years666$\dot{M}_{\rm Edd}\equiv L_{\rm Edd}/{\eta}c^{2}$, where $L_{\rm Edd}=1.3{\times}10^{39}{M/10M_{\odot}}$ erg s-1 and the efficiency $\eta$ increases from 5.7% to 13.3% as $a_{*}$ increases from 0 to 0.84 (Shapiro & Teukolsky 1983)., whereas the massive companion star, and hence its host system, can not be older than about 2–3 million years (Orosz et al. 2007). Thus, it appears that the spin of M33 X-7 must be chiefly natal – i.e., the event horizon trapped much of the angular momentum of the collapsing stellar core – a conclusion that has been reached for two other stellar-mass black holes (McClintock et al. 2006; Shafee et al. 2006). (However, see Moreno Méndez et al. 2008 on hypercritical accretion). ### 6.2 GRS 1915+105: A Near-Extreme Kerr Black Hole GRS 1915+105 has unique and striking properties that sharply distinguish it from the 50 or so known binaries that are believed to contain a stellar-mass black hole (McClintock et al. 2006; Özel et al. 2010). It is the most reliable source of relativistic radio jets in the Milky Way and is the prototype of the microquasars (Mirabel & Rodríguez 1994). It frequently displays extraordinary X-ray variability that is not mimicked by any other black hole system. The properties of its high-frequency X-ray oscillations are equally extraordinary (Remillard & McClintock 2006). Among the 17 transient black hole systems, GRS 1915+105 is unique in having remained active for more than a decade since its discovery in 1992. The system has an orbital period of 30.8 days, and it is the largest of the black hole binary systems (Fig. 1). The pc-scale radio jets of GRS 1915+105, with apparent velocities greater than the speed of light (superluminal motion), are the analogue of the kpc-scale jets that have long been observed for quasars. The source episodically ejects material at relativistic speeds, which can easily be tracked for weeks at centimeter wavelengths as clouds of plasma moving outward on the plane of the sky (Mirabel & Rodríguez 1994; Fender et al. 1999). Based on a kinematic model, the jet velocity for a plausible distance of $D\sim 11$ kpc is $v_{\rm J}/c>0.9$, and the inclination of the jet from our line of sight is $i_{\rm J}\approx 65^{\circ}$. Based on the analysis of X-ray spectral data for reasonable estimates of $D$, $M$ and $i$, we discovered that GRS 1915+105 contains a near-extreme Kerr hole with $a_{*}>0.98$ (McClintock et al. 2006). However, the current estimates of both $D$ and $M$ are poor: The distance is uncertain by a factor of $\approx 2$ (Fig. 5$b$), and the mass is uncertain by $\approx 30$% ($M=14.4\pm 4.4$ $M_{\odot}$; Greiner et al. 2001; Harlaftis & Greiner 2004). Remarkably, the extraordinarily high spin of GRS 1915+105 and other properties of the source allow one – for this source only – to place tight constraints on the allowable values of $M$ and $D$, which must lie within the triangular region shown in Figure 5$a$. Our spin model constrains the black hole’s mass and distance to lie to the right of the slanted line (99% confidence level) because the model implies values of $a_{*}>1$ to the left of the line. (Our model is only valid for $a_{*}<0.999$). Distances $>12$ kpc are ruled out by the kinematic model of the radio jets (Fender et al. 1999). The lower bound on $M$ is an estimate and is based on work in progress: We (Danny Steeghs et al.) are in the act of obtaining near-infrared spectroscopic data at ESO’s VLT Observatory that we fully expect will improve the measurement of $M$ by at least a factor of two. When this result is in hand, we will have $M$ and $D$ constrained to lie within the small shaded triangle, thereby constraining the distance to lie within the range 9.5–12 kpc. As noted above, the distance is also highly uncertain. Furthermore, all measurements to date are model-dependent estimates with large, systematic uncertainties that are difficult to assess (Fig. 5$b$). With Mark Reid, we are in the act of obtaining a model-independent trigonometric distance with an uncertainty of 10% via a parallax measurement (the gold-standard method in astronomy) using the Very Long Baseline Array (VLBA), a worldwide array of radio telescopes. We have made successful observations at four epochs in 2008–2010 and anticipate that observations at several additional epochs will be required to reach our goal. Two hypothetical and possible outcomes of these VLBA observations are indicated in Figure 5$a$: VLBA2 would confirm our spin model and VLBA1 would rule it out. ### 6.3 A0620–00 and XTE J1550–564: Two Schwarzschild-Like Black Holes The host systems of these two black holes are quite small (Fig. 1). The optical companion in A0620–00, which has an orbital period of only 0.3 days, is a star somewhat cooler than the Sun with about half its size and mass. During its yearlong X-ray outburst in 1975–1976, this nearby transient ($D\approx 1$ kpc) became the brightest celestial X-ray source ever observed (apart from the Sun). For several days, the flux at Earth from this source was greater than the combined flux of all of the hundreds of other X-ray binaries in our galaxy. During this period, A0620–00 was also a bright transient radio source, which was observed with the early radio telescopes of the day. A reanalysis of these data by Kuulkers et al. (1999) indicates that multiple jet ejections occurred. The authors find that, like GRS 1915+105, the radio source was extended on parsec scales, and they infer a relativistic expansion velocity. The cool companion star in XTE J1550–564 also has a mass about half that of the Sun, although its radius is about twice as great (Fig. 1); the orbital period of the system is 1.5 days (Orosz et al. 2011). During its principal 1998–1999 outburst cycle, this transient source produced one of the most remarkable X-ray flare events ever observed for a black hole binary. For $\approx 1$ day, the source intensity rose fourfold and the flux in the dominant nonthermal component of emission rose by the same factor. Then, just as quickly, the source intensity declined to its pre-outburst level (Sobczak et al. 2000). Four days later, small-scale superluminal radio jets were observed (Hannikainen et al. 2009); their separation angle and relative velocity link them to the impulsive X-ray flare. The subsequent detection of pc-scale radio jets in 2000 led to the discovery of relativistic X-ray jets (Corbel et al. 2002). All of the available evidence strongly indicates that these pc-scale X-ray and radio jets are associated with the powerful X-ray flare. Using our recently-determined estimates of $D$, $M$ and $i$ for A0620–00 and XTE J1550–564 (Cantrell et al. 2010; Orosz et al. 2011), we fitted the X-ray spectra of these black holes and determined their spins (Table 1). Figure 6 shows a pair of fitted spectra for the latter source. The spectrum in the left panel is completely dominated by the thermal component and is therefore ideal for the determination of spin. Now, however, using our improved methodologies (Steiner et al. 2009a, 2010b), we are able to obtain useful and consistent values of spin as well for spectra that have a strong Compton component of emission, like the one shown in the right panel of Figure 6. The origin of this component is widely attributed to Compton upscattering of the soft disk photons by coronal electrons (see Sec. 7). The CF spins of both A0620–00 and XTE J1550-564 are quite low: $a_{*}\approx 0.1$ and $a_{*}\approx 0.3$, respectively. The corresponding nominal radii of their ISCOs are $5.7M$ and $5.0M$, which differ only modestly from the Schwarzschild value of $6M$. The low spins of these two microquasars challenge the long-standing and widely-held belief that there is a strong connection between black hole spin and relativistic jets (Blandford & Znajek 1977, hereafter BZ). If relativistic jets are powered by black hole spin, then theory predicts that jet power will increase dramatically with increasing $a_{*}$ (Tchekhovskoy et al. 2010). For low spins, the black hole contributes very little power; in fact, for $a_{*}<0.4$, the accretion disk apparently provides more power than the black hole (McKinney 2005). Given the low spins of XTE J1550–564 and A0620–00, it would appear that their episodic jets are driven largely by their accretion disks. One well-known candidate mechanism is the centrifugally driven outflow of matter from a disk described by Blandford & Payne (1982, hereafter BP). A useful comparison of the operational regimes of BP and BZ is given by Garofalo et al. (2010). They show that BP is always viable, but that BZ is a more likely mechanism for the most rapidly rotating sources, such as the extreme-Kerr black hole GRS 1915+105 (see Sec. 6.2). In closing, we note that a statistical study by Fender et al. (2010), which is based on data of uneven quality, found no evidence that black hole spin powers jets. ## 7 The Fe K$\alpha$ Reflection Method In the Fe K$\alpha$ method, one determines $r_{\rm ISCO}$ by modeling the profile of the broad and skewed iron line, which is formed in the inner disk by Doppler effects, light bending, relativistic beaming, and gravitational redshift (Fabian et al. 2000; Reynolds & Nowak 2003; Miller 2007). Of central importance is the effect of the gravitational redshift on the red wing of the line. This wing extends to very low energies for a rapidly rotating black hole ($a_{*}\sim 1$) because in this case gas can orbit near the event horizon, deep in the potential well of the black hole. Relative to the CF method, measuring the extent of the red wing in order to infer $a_{*}$ is hindered by the relative faintness of the signal. However, the Fe K$\alpha$ method has the virtues that it is independent of $M$ and $D$, while the blue wing of the line even allows an estimate if $i$. As noted in Section 1, this method, while applicable to both classes of black holes, is presently the only viable approach to measuring the spins of supermassive black holes. For stellar-mass black holes, in addition to the thermal disk component of emission (which is central to the CF method), a higher-energy power-law component of emission is always observed (e.g., see Fig. 6). For supermassive black holes in active galactic nuclei (AGN), this power-law component is dominant, and it is thought to be produced by inverse Compton scattering of soft thermal photons in a hot ($kT\sim 100$ keV) corona (Reynolds & Nowak 2003; Done et al. 2007). Meanwhile, the disks of both stellar-mass and supermassive black holes are too cool ($kT\sim 1$ and 0.01 keV, respectively) to produce the observed Fe K$\alpha$ emission line. Rather, this dominant line, and a host of other lines, are generated via X-ray fluorescence as a result of irradiation of the disk by the hard, coronal power-law component. The complex spectrum so generated is referred to as a “reflection spectrum.” In order to determine the spin using the Fe-line method, one must model the reflection spectrum in detail (Ross & Fabian 2005, 2007). The spins of several stellar-mass black holes have been measured using the Fe K$\alpha$ method. An early suggestion of high spins for two black holes was made by Miller et al. (2002, 2004) and preliminary results for a total of eight stellar-mass black holes are given in Miller et al. (2009). Other important papers on the spins of stellar-mass black holes include Reis et al. (2008, 2009). For a review, see Miller (2007). Very recently, we teamed up with Fe K$\alpha$ experts to measure the spin of XTE J1550–564 (Steiner et al. 2010b). The spin estimate obtained using the Fe K$\alpha$ method is $a_{*}=0.55_{-0.22}^{+0.15}$, which is quite consistent with the CF value (see Table 1). The spins of several supermassive black holes have been reported, which range from $a_{*}\approx 0.6$ to $>0.98$ (Brenneman & Reynolds 2006; Fabian et al. 2009; Schmoll et al. 2009; Miniutti et al. 2009). By far, the most well studied of these is the Seyfert 1 galaxy MCG–6–30–15 (for background, see Reynolds & Nowak 2003). The 6.4 keV Fe K$\alpha$ line of this AGN is extremely broad and skewed. Brenneman & Reynolds (2006) and Miniutti et al. (2009) show that the red wing extends downward to below 4 keV and conclude that $a_{*}>0.98$. ## 8 Conclusion We have discussed the only two established classes of black holes, stellar- mass and supermassive777There is evidence for a class of intermediate-mass ($100-10^{5}$) black holes (Miller & Colbert 2004). However, to date there existence remains uncertain because no direct and confirming measurement of mass has been made., and the two principal approaches to measuring their spins, the continuum-fitting and Fe-K$\alpha$ methods. Spin measurements for eight stellar-mass black holes are presented, and these data are used to argue that the high spin of M33 X-7 is natal, and that at least some relativistic black-hole jets are powered by their accretion disks, not the spin of the black hole. Two aspects of this work excite us greatly. First, by measuring a black hole’s spin, after earlier measuring its mass, we are able to completely characterize the intrinsic properties of each of the black holes we study. The No Hair Theorem states that a macroscopic black hole, regardless of how massive it may be, is described by just two parameters: $M$ and $a_{*}$888In principle it could also have an electric charge, but astrophysical black holes are unlikely to have enough charge to be dynamically important.. But is the No Hair Theorem really true? The only way we will answer this question is by first measuring $M$ and $a_{*}$ for a good sample of black holes, and then testing whether the Kerr metric corresponding to these values of $M$ and $a_{*}$ is completely consistent with all observables that are sensitive to the space-time near the black hole. In a sense, we have attempted the first test of the No Hair Theorem by measuring the spin of XTE J1550-564 by two independent methods, the continuum fitting method and the Fe K$\alpha$ method (Sec. 7), and finding agreement. However, the errors in the two measurements are currently rather large, and we do not yet understand all systematic sources of error, so it would be premature to claim success. But this example provides a taste of how astrophysics can contribute to deep questions in physics. The other aspect that excites us is all the areas of astrophysics that our work ties to, e.g., the connections that are beginning to be made between measurements of spin and the phenomenology and theory of relativistic jets (Sec. 6.3), and the processes that lead to black hole formation (Sec. 6.1). We hope to see spin data used to help constrain models of gamma-ray bursts, black hole formation, black-hole binary evolution, high- and low-frequency X-ray oscillations, black hole X-ray states and state transitions, models of X-ray coronae, etc. These two strong motivations stimulate us to continue firming up the measurements of black hole spin, with the goal of amassing a good sample of a total of 12–15 measurements during the next several years. We conclude by noting that it is reasonable to expect LIGO, LISA and other gravitational-wave observatories to provide us with intimate knowledge concerning black holes. However, these breakthroughs are still some years in the future whereas astrophysical techniques are providing information on black holes today. Also, gravitational-wave facilities are unlikely to help us understand MHD accretion flows in strong fields, or the origin of relativistic jets, or the formation of relativistically-broadened Fe lines and high- frequency quasi-periodic oscillations, etc., phenomena that are now routinely observed for black holes. In short, observations of accreting black holes show us uniquely how a black hole interacts with its environment. There is no straight path to unlocking the mysteries of black holes, probing the extreme physical conditions they generate, and understanding their importance to astrophysics and cosmology. It behooves us to explore widely because it is the synergistic exploration of all paths that will enlighten us. Therefore, it is important to maintain balance between gravitational-wave and electromagnetic studies of black holes. J.E.M. acknowledges support from NASA grant NNX08AJ55G and the Smithsonian Endowment Funds. 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(2001) Wagoner, R. V., Silbergleit, A. S., & Ortega-Rodríguez, M. 2001, ApJ, 559, L25 * Zhang et al. (1997) Zhang, S. N., Cui, W., & Chen, W. 1997, ApJ, 482, L155 Figure 1: Scale drawings of 21 black hole binaries. The size of the Sun and the Sun-Mercury distance (0.4 AU) are indicated at the top. The systems range in size from the giant GRS 1915+105 with an orbital period of 30.8 days to tiny XTE J1118+480 with an orbital period of 0.2 days. The shapes of the tidally distorted stars are accurately rendered, and the black hole is located at the center of the accretion disk (see key in inset). The inclination of the binary to our line of sight is indicated by the tilt of the accretion disk; an inclination angle of $i=0^{\circ}$ corresponds to a system whose accretion disk lies in the plane of the sky and is viewed face on (e.g., $i=21^{\circ}$ for 4U 1543–47 and $i=75^{\circ}$ for SAX J1819.3-2525). Figure 2: Radius of the ISCO in units of $GM/c^{2}$ versus the black hole spin parameter. Negative values of $a_{*}$ correspond to retrograde motion, with the black hole spinning in the opposite sense of the disk. Stellar black holes are expected to have prograde spins ($a_{*}>0$) as a consequence of their formation in a binary system, whereas the spins of supermassive black holes, which are conditioned by galaxy merger events, may be either prograde or retrograde (e.g., Garofalo et al. 2010). Figure 3: $(top)$ Accretion-disk luminosity in Eddington-scaled units (for $M=10$ $M_{\odot}$) versus time for all the 766 spectra considered in the study of LMC X-3 by Steiner et al. (2010a). The downward arrows show RXTE data which are off scale. Data in the unshaded region satisfy our thin-disk selection criterion $L/L_{\rm Edd}<0.3$ (Sec. 3). The dotted line indicates the lower luminosity threshold (5% $L/L_{\rm Edd}$) set to avoid confusion with strongly Comptonized data. $(bottom)$ Fitted values of the inner disk radius $r_{\rm in}\equiv R_{\rm in}/(GM/c^{2})$ are shown for thin-disk data in the top panel that meet the selection criteria of the study (a total of 411 spectra). Despite large variations in luminosity, $r_{\rm in}$ remains constant to within a few percent over time. The median value for just the 391 selected RXTE spectra is shown as a red dashed line. Figure 4: Luminosity profiles from GRMHD simulations (solid lines) compared with those from the Novikov & Thorne model (dashed lines) for $a_{*}=0$, $0.7$, $0.9$ and $0.98$ (bottom to top). The ISCO is located at the radius where the NT disk luminosity goes to zero. Figure 5: ($a$) Allowed values of black hole mass and distance for GRS 1915+105 fall within the shaded triangular region (see text). ($b$) Six estimates of the distance to GRS 1915+105 are shown. They range from below 7 to above 12 kpc. We are working toward a 10% trigonometric distance. Two hypothetical and possible outcomes of our VLBA observations, labeled VLBA1 and VLBA2, are indicated at the top of panel $a$. For references on distance estimates, see Figure 18 in McClintock et al. (2006). Figure 6: Model fits for a pair of spectra of XTE J1550–564. $(left)$ A spectrum with a strongly dominant thermal component; shown are the data, the fit to the data and the fitted thermal component. $(right)$ A strongly Comptonized spectrum. Note the intensity of the power-law component relative to its intensity in the left panel. For details, see Figure 4 and text in Steiner et al. (2010b).
arxiv-papers
2011-01-04T21:00:02
2024-09-04T02:49:16.137963
{ "license": "Public Domain", "authors": "Jeffrey E. McClintock, Ramesh Narayan, Shane W. Davis, Lijun Gou,\n Akshay Kulkarni, Jerome A. Orosz, Robert F. Penna, Ronald A. Remillard, James\n F. Steiner", "submitter": "Jeffrey McClintock", "url": "https://arxiv.org/abs/1101.0811" }
1101.0827
# O Algoritmo Usado No Programa de Criptografia PASME Péricles Lopes Machado Laboratório de Análises Numéricas em Eletromagnetismo (LANE), Universidade Federal do Pará, caixa postal 8619, CEP 66073-900, Brasil; e-mail: pericles.machado@itec.ufpa.br ###### Abstract Neste trabalho será apresentado o principal algoritmo de criptografia da ferramenta PASME, a qual permite encriptação e ocultamento de informações em diversos tipos de arquivos. O algoritmo utiliza o fato da fatoração de números grandes ser um problema difícil do ponto de vista computacional, efetuando assim, os principais passos da encriptação. ###### Index Terms: Criptografia, Teoria dos números, Teoria da informação This work will present the main encryption algorithm of the PASME tool, PASME allows encrypt and hide information in various types of files. The algorithm uses the fact that factoring large numbers is a difficult issue in terms of computational performing to make the main steps of the encryption. ## I Introdução A ideia fundamental de qualquer algoritmo de criptografia é modificar a representação de uma informação para garantir proteção contra acesso indevidos. No decorrer dos anos, muitos algoritmos de criptografia foram desenvolvidos. Um dos mais antigos realiza uma permutação no alfabeto que contém todos os símbolos da mensagem que será encriptada. Contudo, este algoritmo apresenta grande vulnerabilidade a uma análise da frequência de ocorrência de determinados símbolos, principalmente quando aplicados à textos escritos. Outro método clássico, usado em mensagens binárias, consiste em inverter certos bits e armazenar a posição dos bits que foram invertidos em outra palavra, a folha-chave, com o mesmo tamanho da mensagem que foi encriptada. Um problema desse método é que o tamanho da folha-chave pode ser muito grande, inviabilizando o processo de encriptação. Muitos algoritmos modernos utilizam estratégias envolvendo teoria dos números através da utilização de problemas que atualmente são intratáveis do ponto de vista computacional. Um exemplo clássico desta classe de algoritmo é o RSA [2] [3]. A ideia do presente trabalho é utilizar a intratabilidade da fatoração de inteiros grandes para realizar os passos-chave de sua encriptação. Nas próximas seções, serão descritos os passos realizados pelo algoritmo de encriptação PASME, além de serem comentados alguns detalhes de sua implementação [1]. ## II Algumas funções fundamentais ### II-A A função $\mp$ (inflar) A ideia fundamental do algoritmo PASME é a mudança na base de representação de um número. Mudar a base de representação de um número inteiro $n=a_{0}a_{1}a_{2}...a_{k}$ para a base $b$ consiste em realizar a operação descrita na equação 1 $T(n,b)=a_{0}b^{k}+a_{1}b^{k-1}+...+a_{k}b^{0}$ (1) A função $\mp$ descrita em 2 é uma mudança de base onde a cada digito é adicionado um ”lixo”. $\mp(n,b,v)=(a_{0}+c_{0})b^{1}+(a_{1}+c_{1})b^{2}+...+(a_{k}+c_{k})b^{k+1}$ (2) Onde $c_{i}$ é descrito na equação 5. $\displaystyle c_{i}=\left\\{\begin{array}[]{clcr}\triangleright(v)&,se&i=0\\\ \triangleright(c_{i-1})&,se&i>0\end{array}\right.$ (5) Nas equações 2 e 5, $\triangleright(x)$ é o próximo primo depois de $x$, $v$ é um inteiro qualquer, $n$ é a informação representada na forma de um inteiro, $a_{k}$ é um digito de $n$ na base original e $b$ é a base alvo. ### II-B A função $\pm$ (sujar) $\pm$ é semelhante a função $\mp$, só que o ”lixo” $v$ usado é o mesmo em todos dígitos, conforme pode ser visto na equação 6. $\pm(n,b,v)=(a_{0}+v)b^{0}+(a_{1}+v)b^{1}+...+(a_{k}+v)b^{k}$ (6) ## III O algoritmo de encriptação PASME A seguir, serão descritos os procedimentos para encriptar ou desencriptar uma mensagem usando o algoritmo PASME. O algoritmo $PASME(n,key)$ encripta uma mensagem $n$ usando a frase-chave $key$. ### III-A Encriptando uma mensagem O processo de encriptação inicia com a geração de 7 números aleatórios (de preferência, grandes) $r_{i},i=1...7$. Em seguida, são definidos 4 números $K_{i}=\triangleright(r_{i})$, para $i=1...5$ e $i\neq 3$, $K_{3}=\triangleright(K_{5}+d_{max}+r_{3}+1)$, $d_{max}$ é o maior digito da base em que a informação originalmente está representada. Para continuar o processo de encriptação, uma frase-chave $key$ tem de ser fornecida. Usando-se a frase-chave, são gerados os números $W=\mp(key,K_{3},K_{2})+K_{1}$, $Q=\triangleright(\pm(n,K_{3},K_{5})+r_{7})$, $P=WQ+K_{4}$, e $X=\pm(n,K_{3},K_{5})$ xor $Q$. $X$ é a mensagem $n$ encriptada. As informações divulgadas são os números $K_{i}(i=1...5)$, $P$ e $X$. ### III-B Desencriptando uma mensagem Para desencriptar uma mensagem, é preciso que sejam fornecidos os números $K_{i}(i=1...5)$, $P$ e $X$, além da frase-chave $key$. O primeiro passo da desencriptação é a validação da chave, para realizar essa operação, gera-se o número $W^{\prime}=\mp(key,K_{3},K_{2})+K_{1}$ e é verificado se $P$ mod $W^{\prime}=K_{4}$. Efetuada a validação, pode-se recuperar $Q=(P-K_{4})/W^{\prime}$ . Com $Q$ recuperado, a mensagem $n$ ocultada em $X$ poderá ser revelada. Para revelar a mensagem $n$, gera-se o número $Y=X$ xor $Q$ e o procedimento descrito a seguir tem de ser efetuado. 1. 1. $X^{\prime}=\emptyset$, $X^{\prime}$ é uma palavra vazia 2. 2. Enquanto Y $\neq$ 0: 1. (a) $a\leftarrow Y$ mod $K_{3}$ 2. (b) $Y\leftarrow Y-a$ 3. (c) $Y\leftarrow Y/K_{3}$, efetua-se a divisão inteira de $Y$ por $K_{3}$. 4. (d) $a\leftarrow a-K_{5}$ 5. (e) $X^{\prime}\leftarrow X^{\prime}\oplus a$, $\oplus$ é a operação de concatenação, ou seja, a união de duas palavras (por exemplo,$33\oplus 5=335$). 3. 3. $X^{\prime}$ é a mensagem desencriptada ## IV Comentários sobre a implementação de PASME disponível em [1] Em [1] está disponível uma implementação do algoritmo de criptografia descrito na seção III. Essa implementação utiliza a biblioteca GMP [4] para realizar as operações envolvendo inteiros presentes no algoritmo PASME. Como a ferramenta [1] permite encriptar arquivos com tamanho variáveis, usar o algoritmo PASME nem sempre é uma boa escolha, já que dependendo do tamanho da mensagem o tempo de execução pode ser alto. Então, por questões de eficiência, a implementação [1] utiliza o processo de encriptação de dois passos descrito a seguir para encriptar uma mensagem $n$. 1. 1. Gera-se uma folha-chave $fc$ com um tamanho de $L(fc)$ bytes. 2. 2. Cria-se aleatoriamente uma frase-chave $key$ com $L(key)$ bytes de tamanho. 3. 3. Utiliza-se o algoritmo descrito em III para encriptar a folha-chave $fc$. 4. 4. Quebra-se a mensagem $n$ em $L(n)$ bytes, 5. 5. $i\leftarrow 0$ 6. 6. $k\leftarrow 0$ 7. 7. $X\leftarrow\emptyset$ 8. 8. Enquanto $i\leq L(n)$: 1. (a) $X\leftarrow X\oplus(n_{i}$ xor $fc_{k})$, $n_{i}$ é i-ésimo byte da mensagem $n$ e $fc_{k}$ é o k-ésimo byte da folha-chave $fc$. 2. (b) $i\leftarrow i+1$ 3. (c) $k\leftarrow(k+1)$ mod $L(fc)$ Para desencriptar, o passo (1) do algoritmo anterior não é executado, no passo (2) é fornecido a frase-chave que ”abre” a mensagem e no passo (3) é chamado o algoritmo de desencriptação descrito em III. Na implementação [1], cada simbolo (digito num número) tem 1 byte (8 bits) de comprimento. A implementação [1] armazena em um arquivo-alvo as informações públicas geradas pelo algoritmo III e a mensagem $X$ gerada pelo procedimento anterior. Para ocultar informações em arquivos, [1] primeiramente verifica o tamanho, em bytes, da informação que será ocultada. Após isso, a informação é concatenada ao arquivo e, por fim, concatena-se o tamanho da informação (em [1], um inteiro com 4 bytes de comprimento). O procedimento para recuperar a informação é semelhante, só que, primeiramente, recupera-se o tamanho $L$ (em [1], os 4 últimos bytes do arquivo) da informação que está oculta, depois recua-se $L-4$ bytes a partir do fim do arquivo, no caso de [1], e armazena-se os $L$ bytes seguintes em um arquivo-alvo. A interface gráfica da implementação [1] foi criada utilizando-se o framework QT4 [5]. ## V Conclusões Este trabalho apresentou um algoritmo de encriptação que usa o fato da mesma informação ter significados distintos dependendo da base em que está representada e de, atualmente, certos problemas em teoria dos números serem intratáveis. Tal algoritmo faz parte da ferramenta PASME que permite a encriptação e ocultamento da informação em arquivos nos mais diversos formatos. ## VI Agradecimentos O autor agradece a Diego Aranha por apontar uma falha no algoritmo inicial, a João Augusto Palmitesta Neto por sugestões e testes na implemtentação [1] do algoritmo e Fabio Lobato por revisar o artigo. ## References * [1] “Projeto pasme,” _http://sourceforge.net/projects/pasme/_. * [2] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, _Algoritmos_. Editora Campus, 2002\. * [3] L. Lovász, J. Pelikán, and K. Vesztergombi, _Matemática Discreta_. Sociedade Brasileira de Matemática, 2003. * [4] “The gnu multiple precision arithmetic library,” _http://gmplib.org/_. * [5] “qt - cross platform and ui framework,” _http://qt.nokia.com/_.
arxiv-papers
2011-01-04T21:35:59
2024-09-04T02:49:16.147871
{ "license": "Public Domain", "authors": "P\\'ericles Lopes Machado", "submitter": "P\\'ericles Lopes Machado Machado", "url": "https://arxiv.org/abs/1101.0827" }
1101.0869
# A New Variation of Hat Guessing Games Tengyu Ma Xiaoming Sun Huacheng Yu Institute for Theoretical Computer Science Tsinghua University, Beijing, China Email: matengyu1989@gmail.comEmail: xiaomings@tsinghua.edu.cnEmail: yuhch123@gmail.com ###### Abstract Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of $n$ players is assigned a hat from a given color set. Simultaneously, each player tries to guess the color of his/her own hat by looking at colors of hats worn by other players. In this paper, we consider a new variation of this game, in which we require at least $k$ correct guesses and no wrong guess for the players to win the game, but they can choose to “pass”. A strategy is called perfect if it can achieve the simple upper bound $\frac{n}{n+k}$ of the winning probability. We present sufficient and necessary condition on the parameters $n$ and $k$ for the existence of perfect strategy in the hat guessing games. In fact for any fixed parameter $k$, the existence of perfect strategy can be determined for every sufficiently large $n$. In our construction we introduce a new notion: $(d_{1},d_{2})$-regular partition of the boolean hypercube, which is worth to study in its own right. For example, it is related to the $k$-dominating set of the hypercube. It also might be interesting in coding theory. The existence of $(d_{1},d_{2})$-regular partition is explored in the paper and the existence of perfect $k$-dominating set follows as a corollary. Keywords: Hat guessing game; perfect strategy; hypercube; k-dominating set; perfect code ## 1 Introduction Several different hat guessing games have been studied in recent years [1, 2, 3, 4, 5, 6, 7]. In this paper we investigate a variation where players can either give a guess or pass. It was first proposed by Todd Ebert in [3]. In a standard setting there are $n$ players sitting around a table, who are allowed to coordinate a strategy before the game begins. Each player is assigned a hat whose color is chosen randomly and independently with probability $1/2$ from two possible colors, red and blue. Each player is allowed to see all the hats but his own. Simultaneously, each player guesses its own hat color or passes, according to their pre-coordinated strategy. If at least one player guesses correctly and no player guesses wrong, the players win the game. Their goal is to design a strategy to maximum their winning probability. By a simple counting argument there is an upper bound of the maximum winning probability, $n/(n+1)$. It is known that this upper bound can be achieved if and only if $n$ has the form $2^{t}-1$ [3]. It turns out that the existence of such perfect strategy that achieves the upper bound corresponds to the existence of perfect 1-bit error-correcting code in $\\{0,1\\}^{n}$. In this paper, we present a natural generalization of Ebert’s hat guessing problem: The setting is the same as in the original problem, every player can see all other hats except his own, and is allowed to guess or pass. However, the requirement for them to win the game is generalized to be that at least $k$ players from them should guess correctly, and no player guesses wrong ($1\leq k\leq n$). Note that when $k=1$, it is exactly the original problem. We denote by $P_{n,k}$ the maximum winning probability of players. Similarly to the $k=1$ case, $P_{n,k}$ has a simple upper bound $P_{n,k}\leq\frac{n}{n+k}$. We call a pair $(n,k)$ perfect if this upper bound can be achieved, i.e. $P_{n,k}=\frac{n}{n+k}$. There is a simple necessary condition for a pair $(n,k)$ to be perfect, and our main result states that this condition is almost sufficient: ###### Theorem 1. For any $d,k,s\in\mathbb{N}$ with $s\geq 2\lceil\lg k\rceil$, $(d(2^{s}-k),dk)$ is perfect, in particular, $(2^{s}-k,k)$ is perfect. There exists pair $(n,k)$ with the necessary condition but not perfect, see the remark in Section 4. Here is the outline of the proof: first we give a general characterization of the winner probability $P_{n,k}$ by using the size of the minimum $k$-dominating set of the hypercube. Then we convert the condition of $(n,k)$ perfect to some kind of regular partition of the hypercube (see the definition in Section 2). Our main contribution is that we present a strong sufficient condition for the existence of such partition, which nearly matches the necessary condition. Then we can transform it into a perfect hat guessing strategy. As a corollary of Theorem 1, we also give asymptotic characterization of the value $P_{n,k}$. For example, we show that for any fixed $k$, the maximum winning probability approaches 1 as $n$ tends to the infinity. Related work: Feige [4] considered some variations including the discarded hat version and the everywhere balanced version. Lenstra and Seroussi [6] studied the case that $n$ is not of form $2^{m}-1$, they also considered the case with multiple colors. In [2], Butler, Hajiaghayi, Kleinberg and Leighton considered the worst case of hat placement with sight graph $G$, in which they need to minimize the maximum wrong guesses over all hat placements. In [5] Feige studied the case that each player can see only some of other players’ hats with respect to the sight graph $G$. In [7], Peterson and Stinson investigated the case that each player can see hats in front of him and they guess one by one. Very recently, Buhler, Butler, Graham and Tressler [1] studied the case that every player needs to guess and the players win the game if either exactly $k_{1}$ or $k_{2}$ players guess correctly, they showed that the simple necessary condition is also sufficient in this game. The rest of the paper is organized as follows: Section 2 describes the definitions, notations and models used in the paper. Then, Section 3 presents the result of the existence of $(d_{1},d_{2})$-regular partition of hypercube while Section 4 shows the main result of the hat guessing game. Finally, we conclude the paper in Section 5 with some open problems. ## 2 Preliminaries We use $Q_{n}$ to denote the the $n$ dimension boolean hypercube $\\{0,1\\}^{n}$. Two nodes are adjacent on $Q_{n}$ if they differ by only one bit. We encode the blue and red color by 0 and 1. Thus the placement of hats on the $n$ players’ heads can be represented as a node of $Q_{n}$. For any $x\in Q_{n}$, $x^{(i)}$ is used to indicate the string obtained by flipping the $i^{th}$ bit of $x$. Throughout the paper, all the operations are over $\mathbb{F}_{2}$. We will clarify explicitly if ambiguity appears. Here is the model of the hat guessing game we consider in this paper: The number of players is denoted by $n$ and players are denoted by $p_{1},\ldots,p_{n}$. The colors of players’ hats will be denoted to be $h_{1},\dots,h_{n}$, which are randomly assigned from $\\{0,1\\}$ with equal probability. $h=(h_{1},\ldots,h_{n})$. Let $h_{-i}\in Q_{n-1}$ denote the tuple of colors $(h_{1},\dots,h_{i-1},h_{i+1},\dots,h_{n})$ that player $p_{i}$ sees on the others’ heads. The strategy of player $p_{i}$ is a function $s_{i}:Q_{n-1}\rightarrow\\{0,1,\bot\\}$, which maps the tuple of colors $h_{-i}$ to $p_{i}$’s answer, where $\bot$ represents $p_{i}$ answers “pass” (if some player answers pass, his answer is neither correct nor wrong). A strategy $\mathcal{S}$ is a collection of $n$ functions $(s_{1},\ldots,s_{n})$. The players win the game if at least $k$ of them guess correctly and no one guesses wrong. We use $P_{n,k}$ to denote the maximum winning probability of the players. The following two definitions are very useful in characterization $P_{n,k}$: ###### Definition 2. A subset $D\subseteq V$ is called a $k$-dominating set of graph $G=(V,E)$ if for every vertex $v\in V\setminus D$, $v$ has at least $k$ neighbors in $D$. ###### Definition 3. A partition $(V_{1},V_{2})$ of hypercube $Q_{n}$ is called a $(d_{1},d_{2})$-regular partition if each node in $V_{1}$ has exactly $d_{1}$ neighbors in $V_{2}$, and each node in $V_{2}$ has exactly $d_{2}$ neighbors in $V_{1}$. For example, consider the following partition $(V_{1},V_{2})$ of $Q_{3}$: $V_{1}=\\{000,111\\}$, and $V_{2}=Q_{3}\setminus V_{1}$. For each vertex in $V_{1}$, there are $3$ neighbors in $V_{2}$, and for each vertex in $V_{2}$, there is exactly one neighbor in $V_{1}$. Thus $(V_{1},V_{2})$ forms a $(3,1)$-regular partition of $Q_{3}$. ## 3 ($d_{1},d_{2}$)-Regular Partition of $Q_{n}$ In this section we study the existence of $(d_{1},d_{2})$-regular partition of $Q_{n}$. ###### Proposition 4. Suppose $d_{1},d_{2}\leq n$, if there exists a $(d_{1},d_{2})$-regular partition of hypercube $Q_{n}$, then the parameters $d_{1},d_{2},n$ should satisfy $d_{1}+d_{2}=\gcd(d_{1},d_{2})2^{s}$ for some $s\leq n$. ###### Proof. Suppose the partition is $(V_{1},V_{2})$, we count the total number of vertices $\left|V_{1}\right|+\left|V_{2}\right|=2^{n},$ and the number of edges between two parts $d_{1}\left|V_{1}\right|=d_{2}\left|V_{2}\right|.$ By solving the equations, we obtain $\left|V_{1}\right|=\frac{d_{2}}{d_{1}+d_{2}}2^{n},\ \ \left|V_{2}\right|=\frac{d_{1}}{d_{1}+d_{2}}2^{n}.$ Both $\left|V_{1}\right|$ and $\left|V_{2}\right|$ should be integers, therefore $d_{1}+d_{2}=\gcd(d_{1},d_{2})2^{s}$ holds, since $\gcd(d_{1},d_{1}+d_{2})=\gcd(d_{2},d_{1}+d_{2})=\gcd(d_{1},d_{2})$. ∎ ###### Proposition 5. If there exists a $(d_{1},d_{2})$-regular partition of hypercube $Q_{n}$, then there exists a $(d_{1},d_{2})$-regular partition of $Q_{m}$ for every $m\geq n$. ###### Proof. It suffices to show that the statement holds when $m=n+1$, since the desired result follows by induction. $Q_{n+1}$ can be treated as the union of two copies of $Q_{n}$ (for example partition according to the last bit), i.e. $Q_{n+1}=Q_{n}^{(1)}\cup Q_{n}^{(2)}$. Suppose $(V_{1},V_{2})$ is a $(d_{1},d_{2})$-regular partition of $Q_{n}^{(1)}$. We can duplicate the partition $(V_{1},V_{2})$ to get another partition $(V_{1}^{\prime},V_{2}^{\prime})$ of $Q_{n}^{(2)}$. Then we can see that $(V_{1}\cup V_{1}^{\prime},V_{2}\cup V_{2}^{\prime})$ forms a partition of $Q_{n+1}$, in which each node has an edge to its duplicate through the last dimension. Observe that each node in $V_{1}$ ($V_{1}^{\prime}$) still has $d_{1}$ neighbors in $V_{2}$ ($V_{2}^{\prime}$) and same for $V_{2}$ ($V_{2}^{\prime}$), and the new edges introduced by the new dimension are among $V_{1}$ and $V_{1}^{\prime}$, or $V_{2}$ and $V_{2}^{\prime}$, which does not contribute to the edges between two parts of the partition. Therefore we constructed a $(d_{1},d_{2})$-regular partition of $Q_{n+1}$. ∎ ###### Proposition 6. If there exists a $(d_{1},d_{2})$-regular partition of $Q_{n}$, then there exists $(td_{1},td_{2})$-regular partition of $Q_{tn}$, for any positive integer $t$. ###### Proof. Suppose $(V_{1},V_{2})$ is a $(d_{1},d_{2})$-regular partition of $Q_{n}$. Let $x=x_{1}x_{2}\cdots x_{nt}$ be a node in $Q_{nt}$. We can divide $x$ into $n$ sections of length $t$, and denote the sum of $i^{th}$ section by $w_{i}$, i.e. $w_{i}(x)=\sum_{j=ti-t+1}^{ti}x_{j},\ \ (1\leq i\leq n).$ Let $R(x)=w_{1}(x)w_{2}(x)\ldots w_{n}(x)\in Q_{n}$. Define $V_{i}^{\prime}=\\{x\in Q_{nt}|R(x)\in V_{i}\\},\ (i=1,2).$ We claim that $(V_{1}^{\prime},V_{2}^{\prime})$ is a $(td_{1},td_{2})$-regular partition of $Q_{nt}$. This is because for any vertex $x$ in $V_{1}^{\prime}$, $R(x)$ is in $V_{1}$. So $R(x)$ has $d_{1}$ neighbors in $V_{2}$, and each of which corresponds $t$ neighbors of $x$ in $V_{2}^{\prime}$, thus in total $td_{1}$ neighbors in $V_{2}^{\prime}$. It is the same for vertices in $V_{2}^{\prime}$. ∎ By Proposition 4-6 we only need to consider the existence of $(d_{1},d_{2})$-regular partition of $Q_{n}$ where $\gcd(d_{1},d_{2})=1$ and $d_{1}+d_{2}=2^{s}$ (where $s\leq n$), or equivalently, the existence of $(d,2^{s}-d)$-regular partition of $Q_{n}$, where $s\leq n$ and $d$ is odd. The following Lemma from [1] showed that when $n=2^{s}-1$ such regular partition always exists. ###### Lemma 7. [1] There exists a $(t,2^{s}-t)$-regular partition of $Q_{2^{s}-1}$, for any integer $s,t$ with $0<t<2^{s}$. The following theorem shows how to construct the $(t,2^{s}-t)$-regular partition for $n=2^{s}-r$ (where $r\leq t$). ###### Theorem 8. Suppose there exists a $(t,2^{s}-t)$-regular partition for $Q_{2^{s}-r}$ and $t>r$, then there exists a $(t,2^{s}-t)$-regular partition for $Q_{2^{s+1}-\min\\{t,2r\\}}$. ###### Proof. For convenience, let $m=2^{s}-r$, and $l=2r-\min\\{t,2r\\}(\geq 0)$. Observe that $2^{s+1}-\min\\{t,2r\\}=2m+l$, and if $t\geq 2r$ then $l=0$. Suppose that $(V_{1},V_{2})$ is a $(t,2^{s}-t)$-regular partition for $Q_{m}$. We want to construct a $(t,2^{s+1}-t)$-regular partition for $Q_{2m+l}$. The basic idea of the construction is as follows: We start from set $V_{2}$. We construct a collection of linear equation systems, each of which corresponds to a node in $V_{2}$. The variables of the linear systems are the $(2m+l)$ bits of node $x\in Q_{2m+l}$. Let $V_{2}^{\prime}$ be the union of solutions of these linear equation systems, and $V_{1}^{\prime}$ be the complement of $V_{2}^{\prime}$. Then $(V_{1}^{\prime},V_{2}^{\prime})$ is the $(t,2^{s+1}-t)$-regular partition as we desired. Here is the construction. Since $(V_{1},V_{2})$ is a $(t,2^{s}-t)$ regular partition for $Q_{m}$, the subgraph induced by $V_{2}$ of $Q_{m}$ is a $(t-r)$-regular graph, i.e. for every node $p\in V_{2}$, there are $(t-r)$ neighbors of $p$ in $V_{2}$. For each $p\in V_{2}$, arbitrarily choose a subset $N(p)\subseteq V_{2}$ of neighbors of node $p$ with size $\left|N(p)\right|=r-l$. (here $r-l=r-(2r-\min\\{t,2r\\})=\min\\{t,2r\\}-r$, so $r-l\leq t-r$, and $r-l>0$ since $t>r$) Now for each node $p=(p_{1},\dots,p_{m})\in V_{2}$, we construct a linear equation system as follows: $\begin{cases}x_{1}+x_{2}=p_{1},\\\ x_{3}+x_{4}=p_{2},\\\ \ \ldots\ \ \ldots\ \ \ldots,\\\ x_{2m-1}+x_{2m}=p_{m},\\\ \sum_{j=1}^{m}x_{2j-1}+\sum_{j\in N(p)}x_{2j}+\sum_{1\leq j\leq l}x_{2m+j}=0.\\\ \end{cases}$ (1) Note that in the last equation the last term $\sum_{1\leq j\leq l}x_{2m+j}$ vanishes if $l=0$. Denote by $S(p)\subseteq Q_{2m+l}$ the solutions of this linear system. For convenience, let $f:Q_{2m+l}\rightarrow Q_{m}$ be the operator such that $f(x_{1},\ldots,x_{2m+l})=(x_{1}+x_{2},x_{3}+x_{4},\dots,x_{2m-1}+x_{2m}).$ Then in the linear system (1) the first $m$ equations is nothing but $f(x)=p$. Let $V_{2}^{\prime}=\cup_{p\in V_{2}}S(p)$, and $V_{1}^{\prime}=Q_{2m+l}\setminus V_{2}^{\prime}$ be its complement. We claim that $(V_{1}^{\prime},V_{2}^{\prime})$ is a $(t,2^{s+1}-t)$-regular partition of $Q_{2m+l}$. To begin with, observe the following two facts. Observation 1 For every $x\in V_{2}^{\prime}$, we have $f(x)\in V_{2}$. It can be seen clearly from the first $m$ equations in each equation system. Observation 2 If $k\leq 2m$, then $f(x^{(2k)})=f(x^{(2k-1)})=(f(x))^{(k)}$. If $k>2m$, $f(x^{(k)})=f(x)$. Recall the $x^{(i)}$ is the node obtained by flipping the $i^{th}$ bit of $x$. The observation can be seen from the definition of $f(x)$. For any node $x\in V_{1}^{\prime}$, we show that there are $t$ different ways of flipping a bit of $x$ so that we can get a node in $V_{2}^{\prime}$. There are two possible cases: Case 1: $f(x)\not\in V_{2}$. In this case if we flip the $i^{th}$ bit of $x$ for some $i>2m$, then from Observation 2, $f(x^{(i)})=f(x)$, so $f(x^{(i)})$ will remain not in $V_{2}$, and therefore $x^{(i)}$ will not be in $V_{2}^{\prime}$, by Observation 1. So we can only flip the bit in $(x_{1},\ldots,x_{2m})$. Suppose by flipping the $i^{th}$ bit of $x$ we get $x^{(i)}\in V_{2}^{\prime}$ ($i\in[2m]$), from the definition of $V_{2}^{\prime}$ we have : $f(x^{(i)})\in V_{2}$, and $x^{(i)}$ satisfies the last equation in the equation systems corresponding to $f(x^{(i)})$: $\sum_{j=1}^{m}x_{2j-1}+\sum_{j\in N(f(x^{(i)}))}x_{2j}+\sum_{1\leq j\leq l}x_{2m+j}=0.$ (2) Since $f(x)\notin V_{2}$ and $(V_{1},V_{2})$ is a $(t,2^{s}-t)$-regular partition of $Q_{m}$, so there are exactly $t$ neighbors of $f(x)$ in $V_{2}$, which implies there are $t$ bits of $f(x)$ by flipping which we can get a neighbor of $f(x)$ in $V_{2}$. Let $\\{j_{1},\ldots,j_{t}\\}\subseteq[m]$ be these bits, i.e. $f(x)^{(j_{1})},\ldots,f(x)^{(j_{t})}\in V_{2}$, by Observation 2, $f(x^{(2j_{k}-1)})=f(x^{(2j_{k})})=f(x)^{(j_{k})}\in V_{2},\ \ (k=1,\ldots,t).$ But exactly one of $\\{x^{(2j_{k}-1)},x^{(2j_{k})}\\}$ satisfies the equation (2) (here we use the fact $f(x)\notin V_{2}$, note that $j_{k}\not\in N(f(x)^{(j_{k})})$. Thus totally, there are $t$ possible $i$ such that $x^{(i)}\in V_{2}^{\prime}$. Case 2: $f(x)\in V_{2}$. Since $x\notin V_{2}^{\prime}$, the last linear equation must be violated, i.e. $\sum_{j=1}^{m}x_{2j-1}+\sum_{j\in N(f(x^{(i)}))}x_{2j}+\sum_{1\leq j\leq l}x_{2m+j}=1.$ (3) We further consider three cases here: flip a bit in $\\{x_{1},\ldots,x_{2m}\\}\setminus\\{x_{2j},x_{2j-1}:j\in N(f(x))\\}$; flip a bit in $\\{x_{2j},x_{2j-1}:j\in N(f(x))\\}$; flip a bit in $\\{x_{2m+1},\ldots,x_{2m+l}\\}$: a) if $i\in[m]$ , $i\notin N(f(x))$, and $f(x)^{(i)}\in V_{2}$. Since $f(x)\in V_{2}$, $(V_{1},V_{2})$ is a $(t,2^{s}-t)$-regular partition of $Q_{m}$, there are $m-(2^{s}-t)-\left|N(f(x))\right|=(2^{s}-r)-(2^{s}-t)-(r-l)=t-2r+l$ such index $i$, and $x^{(2i-1)}$ is the exactly the one in $\\{x^{(2i-1)},x^{(2i)}\\}$ which is in $V_{2}^{\prime}$. (determined by Equation 3). Thus in this case there are $(t-2r+l)$ neighbors of $x$ in $V_{2}^{\prime}$. b) if $i\in N(f(x))$, then both of $x^{(2i-1)},x^{(2i)}$ are in $V_{2}^{\prime}$, there are $2\cdot\left|N(f(x))\right|=2(r-l)$ such neighbors. c) if $i>2m$, then every $x^{(i)}$ is in $V_{2}^{\prime}$, ($i=2m+1,\ldots,2m+l$), there are $l$ such neighbors. Hence, totally $x$ has $(t-2r+l)+2(r-l)+l=t$ neighbors in $V_{2}^{\prime}$. The rest thing is to show that every node $x\in V_{2}^{\prime}$ has $(2^{s+1}-t)$ neighbors in $V_{1}^{\prime}$. The proof is similar to the proof of Case 2 above, we consider three cases: a) If $i\in[m]$, $i\not\in N(f(x))$, and $f(x)^{(i)}\in V_{2}$. Then exactly one of $x^{(2k-1)},x^{(2k)}$ in $V_{2}^{\prime}$, thus there are $m-(2^{s}-t)-\left|N(f(x))\right|=2^{s}-r-(2^{s}-t)-(r-l)=t-2r+l$ such neighbors of $x$ in $V_{2}^{\prime}$. b) If $i\in N(f(x))$ both $x^{(2i-1)},x^{(2i)}$ are not in $V_{2}^{\prime}$. c) If $i>2m$, then every $x^{(i)}$ is not in $V_{2}^{\prime}$. Hence totally, $x$ has $(t-2r+l)$ neighbors in $V_{2}^{\prime}$, and therefore $(2m+l)-(t-2r+l)=2^{s+1}-t$ neighbors in $V_{1}^{\prime}$. Hence we prove that $(V_{1}^{\prime},V_{2}^{\prime})$ is indeed a $(t,2^{s+1}-t)$-regular partition of $Q_{2^{s+1}-\min\\{t,2r\\}}$. ∎ ###### Theorem 9. For any odd number $t$ and any $c\leq t$, when $s\geq\lceil\lg t\rceil+\lceil\lg c\rceil$, there exists a $(t,2^{s}-t)$-regular partition of $Q_{2^{s}-c}$. ###### Proof. Let $s_{0}=\lceil\lg t\rceil$. By Lemma 7, there exists a $(t,2^{s_{0}}-t)$-regular partition of $Q_{2^{s_{0}}-1}$. By repeatedly using Theorem 8, we obtain that there exists $(t,2^{s_{0}+1}-t)$-regular partition of $Q_{2^{s_{0}+1}-2}$, $(t,2^{s_{0}+2}-t)$-regular partition of $Q_{2^{s_{0}+2}-2^{2}}$, etc., $(t,2^{s_{0}+\lceil\lg{c}\rceil-1}-t)$-regular partition of $Q_{2^{s_{0}+\lceil\lg{c}\rceil-1}-2^{\lceil\lg{c}\rceil-1}}$, and $(t,2^{s_{0}+\lceil\lg{c}\rceil}-t)$-regular partition of $Q_{2^{s_{0}+\lceil\lg{c}\rceil}-c}$. By using Proposition 5, we get that there exists a $(t,2^{s}-t)$-regular partition of $Q_{2^{s}-c}$, for any $s\geq s_{0}+\lceil\lg{c}\rceil=\lceil\lg t\rceil+\lceil\lg{c}\rceil$. ∎ Combining Proposition 6 and Theorem 9, we have the following corollary. ###### Corollary 10. Suppose $d_{1}=dt,d_{2}=d(2^{s}-t),n=d(2^{s}-c)$, where $d,t,s$ are positive integers with $0<t<2^{s}$, $c\leq t$ and $s\geq\lceil\lg c\rceil+\lceil\lg t\rceil$, then there exists a $(d_{1},d_{2})$-regular partition for $Q_{n}$. ## 4 The Maximum Winning Probability $P_{n,k}$ The following lemma characterizes the relationship between the maximum winner probability $P_{n,k}$ and the minimum $k$-dominating set of $Q_{n}$. The same result was showed in [5] for $k=1$. ###### Lemma 11. Suppose $D$ is a $k$-dominating set of $Q_{n}$ with minimum number of vertices. Then $P_{n,k}=1-\frac{|D|}{2^{n}}.$ ###### Proof. Given a $k$-dominating set $D$ of $Q_{n}$, the following strategy will have winning probability at least $1-\frac{\left|D\right|}{2^{n}}$: For any certain placement of hats, each player can see all hats but his own, so player $p_{i}$ knows that current placement $h$ is one of two adjacent nodes $\\{x,x^{(i)}\\}$ of $Q_{n}$. If $x\in D$ (or $x^{(i)}\in D$), he guesses that the current placement is $x^{(i)}$ (or $x$), otherwise he passes. We claim that by using this strategy, players win the game when the placement is a node which is not in $D$. Observe that since $D$ is a $k$-dominating set, for any node $y\notin D$, $y$ has $l$ neighbors $y^{(i_{1})},y^{(i_{2})},\ldots,y^{(i_{l})}$ that are in $D$, where $l\geq k$. According to the strategy desribed, players $p_{i_{1}},\ldots,p_{i_{l}}$ would guess correctly and all other players will pass. This shows the winning probability is at least $1-\frac{\left|D\right|}{2^{n}}$. Next we show that $P_{n,k}\leq 1-\frac{\left|D\right|}{2^{n}}$. Suppose we have a strategy with winning probability $P_{n,k}$. We prove that there exists a $k$-dominating set $D_{0}$, such that $|D_{0}|=2^{n}(1-P_{n,k})$. The construction is straightforward: Let $D_{0}=\\{h\in Q_{n}:h\textrm{ is not a winning placement}\\}$. Thus $|D_{0}|=N(1-P_{n,k})$. For every winning placement $h\notin D_{0}$, suppose players $p_{i_{1}},\ldots,p_{i_{l}}$ will guess correctly ($l\geq k$), consider the placement $h^{(i_{1})}$, which differs from $h$ only at player $p_{i_{1}}$’s hat, so player $p_{i_{1}}$ will guess incorrectly in this case, thus $h^{(i_{1})}\in D_{0}$. Similarly $h^{(i_{2})},\ldots,h^{(i_{l})}\in D_{0}$, therefore $D_{0}$ is a $k$-dominating set. We have $|D|\leq|D_{0}|=2^{n}(1-P_{n,k}),$ which implies $P_{n,k}\leq 1-\frac{\left|D\right|}{2^{n}}.$ Combining these two results, we have $P_{n,k}=1-\frac{\left|D\right|}{2^{n}}$ as desired. ∎ ###### Proposition 12. The following properties hold: 1. (a) If $n_{1}<n_{2}$ then $P_{n_{1},k}\leq P_{n_{2},k}$. 2. (b) $(n,k)$ is perfect iff there exists a $(k,n)$-regular partition of $Q_{n}$. 3. (c) For any $t\in\mathbb{N}$, $P_{nt,kt}\geq P_{n,k}$. As a consequence, if $(n,k)$ is perfect, $(nt,kt)$ is perfect. ###### Proof. For part (a), suppose that $D$ is a minimum $k$-dominating set of $Q_{n_{1}}$. We make $2^{n_{2}-n_{1}}$ copies of $Q_{n_{1}}$, and by combining them we get a $Q_{n_{2}}$, which has dominating set of size $2^{n_{2}-n_{1}}|D|$. By Lemma 11, $P_{n_{2},k}\geq 1-\frac{2^{n_{2}-n_{1}}|D|}{2^{n_{2}}}=P_{n_{1},k}$. For part (b), suppose $(U,V)$ is a $(k,n)$-regular partition of $Q_{n}$, note that $V$ is a $k$-dominating set of $Q_{n}$ and $|V|=\frac{k}{n+k}\cdot 2^{n}$, thus $V$ is a minimum $k$-dominating set of $Q_{n}$. We have that $P_{n,k}=1-\frac{|V|}{2^{n}}=\frac{n}{n+k}$, which implies that $(n,k)$ is perfect. On the other hand, if $(n,k)$ is perfect, suppose $D$ is the minimum $k$-dominating set, we have $\left|D\right|=\frac{k}{n+k}\cdot 2^{n}$. It can be observed that $\left(Q_{n}\setminus D,D\right)$ is a $(k,n)$-regular partition of $Q_{n}$. For part (c), since $\frac{n}{n+k}=\frac{nt}{nt+kt}$, once $P_{nt,kt}\geq P_{n,k}$ holds, it’s an immediate consequence that the perfectness of $(n,k)$ implies the perfectness of $(nk,nt)$. Suppose for $n$ players, we have a strategy $\mathcal{S}$ with probability of winning $P_{n,k}$. For $nt$ players, we divide them into $n$ groups, each of which has $t$ players. Each placement $h=(h_{1},h_{2},\ldots,h_{nt})$ of $nt$ players can be mapped to a placement $P(h)$ of $n$ players in the following way: for Group $i$, suppose the sum of colors in the group is $w_{i}$, i.e. $w_{i}(h)=\sum_{j=ti-t+1}^{ti}h_{j},\ \ (1\leq i\leq n).$ Let $P(h)=(w_{1}(h),w_{2}(h),\ldots,w_{n}(h))$ be a placement of $n$ players. Each player in Group $i$ knows the color of all players in $P(h)$ other than Player $i$, thus he uses Player $i$’s strategy $s_{i}$ in $\mathcal{S}$ to guess the sum of colors in Group $i$ or passes. Moreover once he knows the sum, his color can be uniquely determined. Note that the players in Group $i$ would guess correctly or incorrectly or pass, if and only if Player $i$ in the $n$-player-game would do. Since the hat placement is uniformly at random, the probability of winning using this strategy is at least $P_{n,k}$, thus $P_{nt,kt}\geq P_{n,k}$. ∎ Now we can prove our main theorem: ###### Theorem 1. For any $d,k,s\in\mathbb{N}$ with $s\geq 2\lceil\lg k\rceil$, $(d(2^{s}-k),dk)$ is perfect, in particular, $(2^{s}-k,k)$ is perfect. ###### Proof. It’s an immediate corollary of part (b) of Proposition 12 and Theorem 9. ∎ Remark: By Proposition 4 and Proposition 12(b) there is a simple necessary condition for $(n,k)$ to be perfect, $n+k=\gcd(n,k)2^{t}$. Theorem 1 indicates that when $n+k=\gcd(n,k)2^{t}$ and $n$ is sufficiently large, $(n,k)$ is perfect. The necessary condition and sufficient condition nearly match in the sense that for each $k$, there’s only a few $n$ that we don’t know whether $(n,k)$ is perfect. Moreover, the following proposition shows that the simple necessary condition can’t be sufficient. The first counterexample is $(5,3)$, it is not perfect while it satisfies the simple necessary condition. But $(13,3)$ is perfect by Theorem 1 and more generally for all $s\geq 4$, $(2^{s}-3,3)$ is perfect. We verified by computer program that $(2^{4}-5,5)=(11,5)$ is not perfect, while by our main theorem $(2^{6}-5,5)=(59,5)$ is perfect. But we still don’t know whether the case between them, $(2^{5}-5,5)=(27,5)$, is perfect. ###### Proposition 13. $(n,k)$ is not perfect unless $2k+1\leq n$ when $n\geq 2$ and $k<n$. ###### Proof. Suppose $(n,k)$ is perfect. According to part (b) of Proposition 12, we can find $(U,V)$, a $(k,n)$-regular partition of $Q_{n}$. Suppose $x$ is some node in $U$, and $y$ is some neighbor of $x$ which is also in $U$, $y$ has $k$ neighbors in $V$. They all differ from $x$ at exactly $2$ bits and one of them is what $y$ differs from $x$ at, i.e. each of them “dominates” $2$ neighbors of $x$, one of them is $y$. So $x$ has totally $k+1$ neighbors “dominated” by $k$ of nodes in $V$. Since all nodes in $V$ are pairwise nonadjacent, these $k+1$ nodes must be in $U$. Now we have $k+1$ neighbors of $x$ are in $U$ and $k$ neighbors are in $V$, it has totally $n$ neighbors. We must have $2k+1\leq n$. ∎ For each odd number $k$, let $s(k)$ be the smallest number such that $(2^{s(k)}-k,k)$ is perfect. We know that $s(k)\in[\lceil\lg{k}\rceil,2\lceil\lg{k}\rceil]$. The following proposition indicates that all $s\geq s(k)$, $(2^{s}-k,k)$ is also perfect. ###### Proposition 14. If $(2^{s}-k,k)$ is perfect, $(2^{s+1}-k,k)$ is perfect. ###### Proof. If $(2^{s}-k,k)$ is perfect, by Proposition 12(b) there is a $(k,2^{s}-k)$-regular partition of $Q_{2^{s}-k}$. Thus by Proposition 5, we have a $(k,2^{s}-k)$-regular partition of $Q_{2^{s}-k+1}$. Combine this partition and Theorem 8, we get a $(k,2^{s+1}-k)$-regular partition of $Q_{2^{s+1}-k}$. Therefore $(2^{s+1}-k,k)$ is perfect. ∎ Using Theorem 1 we can give a general lower bound for the winning probability $P_{n,k}$. Recall that there’s upper bound $P_{n,k}\leq 1-\frac{k}{n+k}$. ###### Lemma 15. $P_{n,k}>1-\frac{2k}{n+k}$, when $n\geq 2^{2\lceil\lg k\rceil}-k$. ###### Proof. Let $n^{\prime}$ be the largest integer of form $2^{t}-k$ which is no more than $n$. By Theorem 1, $(n^{\prime},k)$ is perfect, i.e. $P_{n^{\prime},k}=1-\frac{k}{n^{\prime}+k}$. By part (a) of Proposition 12, $P_{n,k}\geq P_{n^{\prime},k}$. On the other hand we have $n+k<2^{t+1}$, so we have $P_{n,k}\geq 1-\frac{k}{n^{\prime}+k}=1-\frac{2k}{2^{t+1}}>1-\frac{2k}{n+k}.$ ∎ ###### Corollary 16. For any integer $k>0$, $\lim_{n\rightarrow\infty}P_{n,k}=1$. ## 5 Conclusion In this paper we investigated the existence of regular partition for boolean hypercube, and its applications in finding perfect strategies of a new hat guessing games. We showed a sufficient condition for $(n,k)$ to be perfect, which nearly matches the necessary condition. Several problems remain open: for example, determine the minimum value of $s(k)$ such that $(2^{s(k)}-k,k)$ is perfect, and determine the exact value of $P_{n,k}$. It is also very interesting to consider the case when there are more than two colors in the game. ## References * [1] Joe Buhler, Steve Butler, Ron Graham, and Eric Tressler. Hypercube orientations with only two in-degrees. http://arxiv.org/abs/1007.2311, 2010. * [2] Steve Butler, Mohammad T. Hajiaghayi, Robert D. Kleinberg, and Tom Leighton. Hat guessing games. SIAM J. Discrete Math., 22(2):592–605, 2008. * [3] Todd T. Ebert. Applications of recursive operators to randomness and complexity. PhD thesis, University of California at Santa Barbara, 1998. * [4] Uriel Feige. You can leave your hat on (if you guess its color). Technical Report MCS04-03, Computer Science and Applied Mathematics, The Weizmann Institute of Science, 2004. * [5] Uriel Feige. On optimal strategies for a hat game on graphs. SIAM Journal of Discrete Mathematics, 2010. * [6] Hendrik W. Lenstra and Gadiel Seroussi. On hats and other covers. http://arxiv.org/abs/cs/0509045, 2005. * [7] Maura B . Peterson and Douglas R. Stinson. Yet another hatt game. the electronic journal of combinatorics, 17(1), 2010.
arxiv-papers
2011-01-05T02:13:50
2024-09-04T02:49:16.154026
{ "license": "Public Domain", "authors": "Tengyu Ma and Xiaoming Sun and Huacheng Yu", "submitter": "Xiaoming Sun", "url": "https://arxiv.org/abs/1101.0869" }
1101.1136
Marginal Likelihood Estimation via Arrogance Sampling By Benedict Escoto ###### Abstract This paper describes a method for estimating the marginal likelihood or Bayes factors of Bayesian models using non-parametric importance sampling (“arrogance sampling”). This method can also be used to compute the normalizing constant of probability distributions. Because the required inputs are samples from the distribution to be normalized and the scaled density at those samples, this method may be a convenient replacement for the harmonic mean estimator. The method has been implemented in the open source R package margLikArrogance. ## 1 Introduction When a Bayesian evaluates two competing models or theories, $T_{1}$ and $T_{2}$, having observed a vector of observations $\boldsymbol{x}$, Bayes’ Theorem determines the posterior ratio of the models’ probabilities: $\frac{p(T_{1}|\boldsymbol{x})}{p(T_{2}|\boldsymbol{x})}=\frac{p(\boldsymbol{x}|T_{1})}{p(\boldsymbol{x}|T_{2})}\frac{p(T_{1})}{p(T_{2})}.$ (1) The quantity $\frac{p(\boldsymbol{x}|T_{1})}{p(\boldsymbol{x}|T_{2})}$ is called a _Bayes factor_ and the quantities $p(\boldsymbol{x}|T_{1})$ and $p(\boldsymbol{x}|T_{2})$ are called the theories’ _marginal likelihoods_. The types of Bayesian models considered in this paper have a fixed finite number of parameters, each with their own probability function. If $\boldsymbol{\theta}$ are parameters for a model $T$, then $p(\boldsymbol{x}|T)=\int p(\boldsymbol{x}|\boldsymbol{\theta},T)p(\boldsymbol{\theta}|T)\,d\boldsymbol{\theta}=\int p(\boldsymbol{x}\wedge\boldsymbol{\theta}|T)\,d\boldsymbol{\theta}$ (2) Unfortunately, this integral is difficult to compute in practice. The purpose of this paper is to describe one method for estimating it. Evaluating integral (2) is sometimes called the problem of computing normalizing constants. The following formula shows how $p(\boldsymbol{x}|T)$ is a normalizing constant. $p(\boldsymbol{\theta}|\boldsymbol{x},T)=\frac{p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)}{p(\boldsymbol{x}|T)}$ (3) Thus the marginal likelihood $p(\boldsymbol{x}|T)$ is also the normalizing constant of the posterior parameter distribution $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ assuming we are given the density $p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ which is often easy to compute in Bayesian models. Furthermore, Bayesian statisticians typically produce samples from the posterior parameter distribution $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ even when not concerned with theory choice. In these case, computing the marginal likelihood is equivalent to computing the normalizing constant of a distribution from which samples and the scaled density at these samples are available. The method described in this paper takes this approach. ## 2 Review of Literature Given how basic (1) is, it is perhaps surprising that there is no easy and definitive way of applying it, even for simple models. Furthermore, as the dimensionality and complexity of probability distributions increase, the difficulty of approximation also increases. The following three techniques for computing bayes factors or marginal likelihoods are important but will not be mentioned further here. 1. 1. Analytic asymptotic approximations such as Laplace’s method, see for instance Kass and Raftery (1995), 2. 2. Bridge sampling/path sampling/thermodynamic integration (Gelman and Meng, 1998), and 3. 3. Chib’s MCMC approximation (Chib, 1995; Chib and Jeliazkov, 2005). Kass and Raftery (1995) is a popular overview of the earlier literature on Bayes factor computation. All these methods can be very successful in the right circumstances, and can often handle problems too complex for the method described here. However, the method of this paper may still be useful due to its convenience. The rest of section 2 describes three approaches that are relevant to this paper. ### 2.1 Importance Sampling Importance sampling is a technique for reducing the variance of monte carlo integration. This section will note some general facts; see Owen and Zhou (1998) for more information. Suppose we are trying to compute the (possibly multidimensional) integral $I$ of a well-behaved function $f(\boldsymbol{\theta})$. Then $I=\int f(\boldsymbol{\theta})\,d\boldsymbol{\theta}=\int\frac{f(\boldsymbol{\theta})}{g(\boldsymbol{\theta})}g(\boldsymbol{\theta})\,d(\boldsymbol{\theta})$ so if $g(\boldsymbol{\theta})$ is a probability density function and $\boldsymbol{\theta}_{i}$ are independent samples from it, then $I=\mbox{E}_{g}[f(\boldsymbol{\theta})/g(\boldsymbol{\theta})]\approx\frac{1}{n}\sum_{i=1}^{n}\frac{f(\boldsymbol{\theta}_{i})}{g(\boldsymbol{\theta}_{i})}=I_{n}.$ (4) $I_{n}$ is an unbiased approximation to $I$ and by the central limit theorem will tend to a normal distribution. It has variance $\mbox{Var}[I_{n}]=\frac{1}{n}\int\left(\frac{f(\boldsymbol{\theta})}{g(\boldsymbol{\theta})}-I\right)^{2}g(\boldsymbol{\theta})\,d\boldsymbol{\theta}=\frac{1}{n}\int\frac{(f(\boldsymbol{\theta})-Ig(\boldsymbol{\theta}))^{2}}{g(\boldsymbol{\theta})}\,d\boldsymbol{\theta}$ (5) Sometimes $f$ is called the _target_ and $g$ is called the _proposal_ distribution. Assuming that $f$ is non-negative, then minimum variance (of $0!$) is achieved when $g=f/I$—in other words when $g$ is just the normalized version of $f$. This cannot be done in practice because normalizing $f$ requires knowing the quantity $I$ that we wanted to approximate; however (5) is still important because it means that the more similar the proposal is to the target, the better our estimator $I_{n}$ becomes. In particular, $f$ must go to 0 faster than $g$ or the estimator will have infinite variance. To summarize this section: 1. 1. Importance sampling is a monte carlo integration technique which evaluates the target using samples from a proposal distribution. 2. 2. The estimator is unbiased, normally distributed, and its variance (if not 0 or infinity) decreases as $O(n^{-1})$ (using big-$O$ notation). 3. 3. The closer the proposal is to the target, the better the estimator. The proposal also needs to have longer tails than the target. ### 2.2 Nonparametric Importance Sampling A difficulty with importance sampling is that it is often difficult to choose a proposal distribution $g$. Not enough is known about $f$ to choose an optimal distribution, and if a bad distribution is chosen the result can have large or even infinite variance. One approach to the selection of proposal $g$ is to use non-parametric techniques to build $g$ from samples of $f$. I call this class of techniques self-importance sampling, or arrogance sampling for short, because they attempt to sample $f$ from itself without using any external information. (And also isn’t it a bit arrogant to try to evaluate a complex, multidimensional integral using only the values at a few points?) The method of this paper falls into this class and particularly deserves the name because the target and proposal (when they are both non-zero) have exactly the same values up to a multiplicative constant. Two papers which apply nonparametric importance sampling to the problem of marginal likelihood computation (or computation of normalizing constants) are Zhang (1996) and Neddermeyer (2009). Although both authors apply their methods to more general situations, here I will use the framework suggested by (3) and assume that we can compute $p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ for arbitrary $\boldsymbol{\theta}$ and also that we can sample from the posterior parameter distribution $p(\boldsymbol{\theta}|\boldsymbol{x},T)$. The goal is to estimate the normalizing constant, the marginal likelihood $p(\boldsymbol{x}|T)$. Zhang’s approach is to build the proposal $g$ using traditional kernel density estimation. $m$ samples are first drawn from $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and used to construct $g$. Then $n$ samples are drawn from $g$ and used to evaluate $p(\boldsymbol{x}|T)$ as in traditional importance sampling. This approach is quite intuitive because kernel estimation is a popular way of approximating an unknown function. Zhang proves that the variance of his estimator decreases as $O(m^{\frac{-4}{4+d}}n^{-1})$ where $d$ is the dimensionality of $\boldsymbol{\theta}$, compared to $O(n^{-1})$ for standard (parametric) importance sampling. There were, however, a few issues with Zhang’s method: 1. 1. A kernel density estimate is equal to 0 at points far from the points the kernel estimator was built on. This is a problem because importance sampling requires the proposal to have longer tails than the target. This fact forces Zhang to make the restrictive assumption that $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ has compact support. 2. 2. It is hard to compute the optimal kernel bandwidth. Zhang recommends using a plug-in estimator because the function $p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ is available, which is unusual for kernel estimation problems. Still, bandwidth selection appears to require significant additional analysis. 3. 3. Finally, although the variance may decrease as $O(m^{\frac{-4}{4+d}}n^{-1})$ as $m$ increases, the difficulty of computing $g(\boldsymbol{\theta})$ also increases with $m$, because it requires searching through the $m$ basis points to find all the points close to $\boldsymbol{\theta}$. In multiple dimensions, this problem is not trivial and may outweigh the $O(m^{\frac{-4}{4+d}})$ speedup (in the worst case, practical evaluation of $g(\boldsymbol{\theta})$ at a single point may be $O(m)$). See Zlochin and Baram (2002) for some discussion of these issues. Neddermeyer (2009) uses a similar approach to Zhang and also achieves a variance of $O(m^{\frac{-4}{4+d}}n^{-1})$. It improves on Zhang’s approach in two ways relevant to this paper: 1. 1. The support of $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ is not required to be compact. 2. 2. Instead of using kernel density estimators, linear blend frequency polynomials (LBFPs) are used instead. LBFPs are basically histograms whose density is interpolated between adjacent bins. As a result, the computation of $g(\boldsymbol{\theta})$ requires only finding which bin $\boldsymbol{\theta}$ is in, and looking up the histogram value at that and adjacent bins ($2^{d}$ bins in total). As we will see in section 3, the arrogance sampling described in this paper is similar to the methods of Zhang and Neddermeyer. ### 2.3 Harmonic Mean Estimator The harmonic mean estimator is a simple and notorious method for calculating marginal likelihoods. It is a kind of importance sampling, except the proposal $g$ is actually the distribution $p(\boldsymbol{\theta}|\boldsymbol{x},T)=p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)/p(\boldsymbol{x}|T)$ to be normalized and the target $f$ is the known distribution $p(\boldsymbol{\theta}|T)$. Then if $\boldsymbol{\theta}_{i}$ are samples from $p(\boldsymbol{\theta}|x,T)$, we apparently have $1\approx\frac{1}{n}\sum_{i=1}^{n}\frac{p(\boldsymbol{\theta}_{i}|T)}{p(\boldsymbol{\theta}_{i}|\boldsymbol{x},T)}=\frac{1}{n}\sum_{i=1}^{n}\frac{p(\boldsymbol{\theta}_{i}|T)}{p(\boldsymbol{x}|\boldsymbol{\theta}_{i},T)p(\boldsymbol{\theta}_{i}|T)/p(\boldsymbol{x}|T)}=\frac{1}{n}\sum_{i=1}^{n}\frac{1}{p(\boldsymbol{x}|\boldsymbol{\theta}_{i},T)/p(\boldsymbol{x}|T)}$ hence $p(\boldsymbol{x}|T)\stackrel{{\scriptstyle?}}{{\approx}}\left(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{p(\boldsymbol{x}|\boldsymbol{\theta}_{i},T)}\right)^{-1}$ (6) Two advantages of the harmonic mean estimator are that it is simple to compute and only depends on samples from $p(\boldsymbol{\theta}|x,T)$ and the likelihood $p(\boldsymbol{x}|\boldsymbol{\theta},T)$ at those samples. The main drawback of the harmonic mean estimator is that it doesn’t work—as mentioned earlier the importance sampling proposal distribution needs to have longer tails than the target. In this case the target $p(\boldsymbol{\theta}|T)$ typically has longer tails than the proposal $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and thus (6) has infinite variance. Despite not working, the harmonic mean estimator continues to be popular (Neal, 2008). ## 3 Description of Technique This paper’s arrogance sampling technique is a simple method that applies the nonparametric importance techniques of Zhang and Neddermeyer in an attempt to develop a method almost as convenient as the harmonic mean estimator. The only required inputs are samples $\boldsymbol{\theta}_{i}$ from $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and the values $p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)=p(x|\boldsymbol{\theta}_{i},T)p(\boldsymbol{\theta}_{i}|T)$. This is similar to the harmonic mean estimator, but perhaps slightly less convenient because $p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)$ is required instead of $p(\boldsymbol{x}|\boldsymbol{\theta}_{i},T)$. There are two basic steps: 1. 1. Take $m$ samples from $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and using modified histogram density estimation, construct probability density function $f(\boldsymbol{\theta})$. 2. 2. With $n$ more samples from $p(\boldsymbol{\theta}|\boldsymbol{x},T)$, estimate $1/p(\boldsymbol{x}|T)$ via importance sampling with target $f$ and proposal $p(\boldsymbol{\theta}|\boldsymbol{x},T)$. These steps are described in more detail below. ### 3.1 Construction of the Histogram Of the $N$ total samples $\boldsymbol{\theta}_{i}$ from $p(\boldsymbol{\theta}|\boldsymbol{x},T)$, the first $m$ will be used to make a histogram. The optimal choice of $m$ will be discussed below, but in practice this seems difficult to determine. An arbitrary rule of $\mbox{min}(0.2N,2\sqrt{N})$ can be used in practice. With a traditional histogram, the only available information is the location of the sampled points. In this case we also know the (scaled) heights $p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ at each sampled point. We can use this extra information to improve the fit. Our “arrogant” histogram $f$ is constructed the same as a regular histogram, except the bin heights are not determined by the number of points in each bin, but rather by the minimum density over all points in the bin. If a bin contains no sampled points, then $f(\boldsymbol{\theta})=0$ for $\boldsymbol{\theta}$ in that bin. Then $f$ is normalized so that $\int f(\boldsymbol{\theta})\,d\boldsymbol{\theta}=1$. To determine our bin width, we can simply and somewhat arbitrarily set our bin width $h$ so that the histogram is positive for 50% of the sampled points from the distribution $p(\boldsymbol{\theta}|\boldsymbol{x},T)$. To approximate $h$, we can use a small number of samples (say, 40) from $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ and set $h$ so that $f(\boldsymbol{\theta})>0$ for exactly half of these samples. Figure 1 compares the traditional and new histograms for a one dimensional normal distribution based on 50 samples. The green rug lines indicate the $50$ sampled points which are the same for all. The arrogant histogram’s bin width is chosen as above. The traditional histogram’s optimal bin width was determined by Scott’s rule to minimize mean squared error. As the figure shows, the modified histogram is much smoother for a given bin width, so a smaller bin width can be used. On the other hand, $f$ will either equal 0 or have about twice the original density at each point, while the traditional histogram’s density is numerically close to the original density. Figure 1: Histogram Comparison ### 3.2 Importance Sampling The remaining $n=N-m-40$ sampled points can be used for importance sampling. Using equation (4) with histogram $f$ as our target and $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ as the proposal, we have $1\approx I_{n}=\frac{1}{n}\sum_{i=1}^{n}\frac{f(\boldsymbol{\theta}_{i})}{p(\boldsymbol{\theta}_{i}|\boldsymbol{x},T)}=\frac{1}{n}\sum_{i=1}^{n}\frac{f(\boldsymbol{\theta}_{i})}{p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)/p(\boldsymbol{x}|T)}$ hence $p(\boldsymbol{x}|T)\approx p(\boldsymbol{x}|T)/I_{n}=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{f(\boldsymbol{\theta}_{i})}{p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)}\right)^{-1}=A_{n}$ (7) To underscore the self-important/arrogant nature of this approximation $A_{n}$, we can rewrite (7) as $p(\boldsymbol{x}|T)\approx H\left(\frac{1}{n}\sum_{i=1}^{n}\frac{\mbox{min}\\{p(\boldsymbol{\theta}_{j}\wedge\boldsymbol{x}|T):\boldsymbol{\theta}_{j}\mbox{ and }\boldsymbol{\theta}_{j}\mbox{ are in the same bin}\\}}{p(\boldsymbol{\theta}_{i}\wedge\boldsymbol{x}|T)}\right)^{-1}$ where $H$ is the histogram normalizing constant. This equation shows that all the values in the numerator and the denominator of our importance sampling are from the same distribution $p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$. Note that the histogram $f$ is the target of the importance sampling and $p(\boldsymbol{\theta}\wedge\boldsymbol{x}|T)$ is the proposal. This is backwards from the usual scheme where the unknown distribution is the target and the known distribution is the proposal. Instead here the unknown distribution is the proposal, as in the harmonic mean estimator (see Robert and Wraith (2009) for another example of this.) As in section 2.1, our approximation of $p(\boldsymbol{x}|T)^{-1}$ tends to a normal distribution as $n\to\infty$ by the central limit theorem. This fact can be used to estimate a confidence interval around $p(\boldsymbol{x}|T)$. ## 4 Validity of Method This section will investigate the performance of the method. First, note that this method is just an implementation of importance sampling, so $A_{n}^{-1}$ should converge to $p(\boldsymbol{x}|T)^{-1}$ with finite variance as long as the proposal density $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ exists and is finite and positive on the compact region where the target histogram density is positive. To calculate the speed of convergence we will use equation (5) where $f$ is the histogram, $g(\boldsymbol{\theta})=p(\boldsymbol{\theta}|\boldsymbol{x},T)$, and $I=1$ because the histogram has been normalized. Unless otherwise noted, we will assume below that $g:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is finite, twice differentiable and positive, and that $\int\frac{\lVert\nabla\cdot g(\boldsymbol{\theta})\rVert^{2}}{g(\boldsymbol{\theta})}d\boldsymbol{\theta}$ is finite. ### 4.1 Histogram Bin Width One important issue will be how quickly the $d$-dimensional histogram’s selected bin width $h$ goes to 0 as the number of samples $m\rightarrow\infty$. This section will only offer an intuitive argument. For any $m$, the histogram will enclose about the same probability ($\frac{1}{2}$) and will have about the same average density in a fixed region. Each bin has volume $h^{d}$, so if $l$ is the number of bins then $lh^{d}=O(1)$ and $h\propto l^{-d}$. Furthermore, the distribution of the sampled points converges to the actual distribution $g(\boldsymbol{\theta})$. If $m>O(l)$, an unbounded number of sampled points would end up in each bin. If $m<O(l)$, then some bins would have no points in them. Neither of these is possible because exactly one sampled point is necessary to establish each bin. Thus $m\propto l$ and $h\propto m^{-d}$. ### 4.2 Conditional Variance Before estimating the convergence rate of $A_{n}$ we will prove something about the conditional variance of importance sampling. Let $A=\\{\boldsymbol{\theta}:f(\boldsymbol{\theta})>0\\}$, $\mathbf{1}_{A}$ be the characteristic function of $A$, and $q=\int_{A}g(\boldsymbol{\theta})\,d\boldsymbol{\theta}$. Define $g_{A}(\boldsymbol{\theta})=\left\\{\begin{array}[]{cl}g(\boldsymbol{\theta})/q&\mbox{ if }\boldsymbol{\theta}\in A\\\ 0&\mbox{ otherwise }\end{array}\right.$ Then $g_{A}$ is the density of $g$ conditional on $f>0$. Define $\mbox{Var}_{A}$ and $\mbox{E}_{A}$ to mean the variance and expectation conditional on $f(\boldsymbol{\theta})>0$. Thus $\displaystyle\mbox{Var}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))$ $\displaystyle=$ $\displaystyle\mbox{Var}(\mbox{E}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta})|\mathbf{1}_{A}))+\mbox{E}(\mbox{Var}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta})|\mathbf{1}_{A}))$ $\displaystyle=$ $\displaystyle\mbox{Var}\left(\begin{array}[]{cl}\mbox{E}_{A}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))&\mbox{ if }\boldsymbol{\theta}\in A\\\ 0&\mbox{ otherwise}\\\ \end{array}\right)$ $\displaystyle+\,\mbox{E}\left(\begin{array}[]{cl}\mbox{Var}_{A}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))&\mbox{ if }\boldsymbol{\theta}\in A\\\ 0&\mbox{ otherwise}\\\ \end{array}\right)$ $\displaystyle=$ $\displaystyle\mbox{Var}\left(\begin{array}[]{cl}1/q&\mbox{ if }\boldsymbol{\theta}\in A\\\ 0&\mbox{ otherwise}\\\ \end{array}\right)+q\mbox{Var}_{A}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))$ $\displaystyle=$ $\displaystyle(1/q)^{2}q(1-q)+\frac{1}{q}\mbox{Var}_{A}(f(\boldsymbol{\theta})/qg(\boldsymbol{\theta}))$ $\displaystyle=$ $\displaystyle\frac{1-q}{q}+\frac{1}{q}\mbox{Var}_{A}(f(\boldsymbol{\theta})/g_{A}(\boldsymbol{\theta}))$ We will assume below that $q=\frac{1}{2}$, so that $\mbox{Var}(f(\boldsymbol{\theta})/g(\boldsymbol{\theta}))=1+2\mbox{Var}_{A}(f(\boldsymbol{\theta})/g_{A}(\boldsymbol{\theta}))$ (11) ### 4.3 Importance Sampling Convergence With $f$, $g$, and $A$ as defined above, $f$ and $g_{A}$ have the same domain. Assuming errors in estimating $q$ and normalization errors are of a lesser order of magnitude, we can treat the histogram heights as being sampled from $g_{A}$. Suppose the histogram has $l$ bins $\\{B_{j}\\}$, each with width $h$ and based around the points $g_{A}(\boldsymbol{\theta}_{j})$. Then by equation (5), $\displaystyle\mbox{Var}_{A}(f(\boldsymbol{\theta})/g_{A}(\boldsymbol{\theta}))$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{l}\int_{B_{j}}\frac{(f(\boldsymbol{\theta})-g_{A}(\boldsymbol{\theta}))^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{l}\int_{B_{j}}\frac{(g_{A}(\boldsymbol{\theta})+\nabla g_{A}(\boldsymbol{\theta})\cdot(\boldsymbol{\theta}_{j}-\boldsymbol{\theta})+O((\boldsymbol{\theta}_{j}-\boldsymbol{\theta})^{2})-g_{A}(\boldsymbol{\theta}))^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{l}\int_{B_{j}}\frac{(\nabla g_{A}(\boldsymbol{\theta})\cdot(\boldsymbol{\theta}_{j}-\boldsymbol{\theta}))^{2}+O((\boldsymbol{\theta}_{j}-\boldsymbol{\theta})^{3})}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$ $\displaystyle\leq$ $\displaystyle\sum_{j=1}^{l}\int_{B_{j}}\frac{\lVert\nabla\cdot g_{A}(\boldsymbol{\theta})\rVert^{2}h^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$ $\displaystyle=$ $\displaystyle h^{2}\int\frac{\lVert\nabla\cdot g_{A}(\boldsymbol{\theta})\rVert^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$ Because $h\propto m^{-d}$ where $d$ is the number of dimensions, and $m$ is the number of samples used to make the histogram, $\mbox{Var}_{A}(f(\boldsymbol{\theta})/g_{A}(\boldsymbol{\theta}))\leq Cm^{-2/d}\\\ $ where $C\propto\int\frac{\lVert\nabla\cdot g_{A}(\boldsymbol{\theta})\rVert^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$. Putting this together with (11), we get $\mbox{Var}(I_{n})=\mbox{Var}(p(\boldsymbol{x}|T)/A_{n})=n^{-1}(1+O(Cm^{-2/d}))$ (12) ## 5 Implementation Issues ### 5.1 Speed of Convergence The variance of $n^{-1}(1+O(Cm^{-2/d}))$ given by (12) is asymptotically equal to $n^{-1}$, which is the typical importance sampling rate. In practice however, the asymptotic results cannot distinguish useful from impractical estimators. If $Cm^{-2/d}$ is small and $\mbox{Var}(p(\boldsymbol{x}|T)/A_{n})\approx n^{-1}$, then $p(\boldsymbol{x}|T)$ can be approximated in only 1000 samples to about $6\%=\frac{1.96}{\sqrt{1000}}$ with 95% confidence. For many theory choice purposes, this is quite sufficient. Thus in typical problem cases the factor of $Cm^{-2/d}$ will be very significant. If $Cm^{-2/d}\gg 1$, then the convergence rate may in practice be similar to $n^{-1}m^{-2/d}$. Compare this to the rate of $n^{-1}m^{-4/(4+d)}$ for the methods proposed by Zhang and Neddermeyer. This method also uses simple histograms, instead of a more sophisticated density estimation method (Zhang uses kernel estimation, Neddermeyer uses linear blend frequency polynomials). Although simple histograms converge slower for large $d$ as shown above, they are much faster to compute for large $d$. Neddermeyer’s LBFP algorithm is quite efficient compared to Zhang’s, but its running time is $O(2^{d}d^{2}n^{\frac{d+5}{d+4}})$. $d$ is a constant for any fixed problem, but if, say, $d=10$, then the dimensionality constant multiplies the running time by $2^{10}10^{2}\approx 10^{5}$. By contrast, this paper’s method takes only $O(dm\mbox{log}(m))$ time to construct the initial histogram, and an additional $O(dn\mbox{log}(m))$ time to do the importance sampling. The main reason for the difference is that querying a simple histogram can be done in $\mbox{log}(m)$ time by computing the bin coordinates and looking up the bin’s height in a tree structure. However, querying a LBFP requires blending all nearby bins and is thus exponential in $d$. ### 5.2 When $g=0$ Our discussion assumed that $g(\boldsymbol{\theta})=p(\boldsymbol{\theta}|\boldsymbol{x},T)$ was always positive. If $g$ goes to 0 where the histogram is positive, the variance of $A_{n}^{-1}$ will be infinite. However, this paper’s method can still be used if $g(\boldsymbol{\theta})$ is 0 over some well-defined area. For instance, suppose one dimension $\theta_{k}$ of $p(\boldsymbol{\theta}|T)$ is defined by a gamma distribution, so that $p(\theta_{k}|T)=0$ if and only if $\theta_{k}\leq 0$. Then we can ensure the variance is not infinite by checking that the histogram is only defined where $\theta_{k}>\epsilon>0$ for some fixed $\epsilon$. The margLikArrogance package contains a simple mechanism to do this. The user may specify a range along each dimension of $\boldsymbol{\theta}$ where it is known that $g>0$. If the histogram is non-zero outside of this range, the method aborts with an error. Note that the variance of the estimator increases with $\int\frac{\lVert\nabla\cdot g_{A}(\boldsymbol{\theta})\rVert^{2}}{g_{A}(\boldsymbol{\theta})}d\boldsymbol{\theta}$. In practice the estimator will work well only when $g$ doesn’t go to 0 too quickly where the histogram is positive. In these cases the histogram will be defined well away from any region where $g=0$ and infinite variance won’t be an issue even if $g=0$ somewhere. ### 5.3 Bin Shape Cubic histogram bins were used above—their widths were fixed at $h$ in each dimension. Although the asymptotic results aren’t affected by the shape of each bin, for usable convergence rates the bins’ dimensions need to compatible with the shape of the high probability region of $p(\boldsymbol{\theta}|\boldsymbol{x},T)$. Unfortunately, it is difficult to determine the best bin shapes. The margLikArrogance package contains a simple workaround: by default the distribution is first scaled so that the sampled standard deviation along each dimension is constant. This is equivalent to setting each bin’s width by dimension in proportion to that dimension’s standard deviation. If this simple rule of thumb is insufficient, the user can scale the sampled values of $p(\boldsymbol{\theta}|\boldsymbol{x},T)$ manually (and make the corresponding adjustment to the estimate $A_{n}$). ## 6 Conclusion This paper has described an “arrogance sampling” technique for computing the marginal likelihood or Bayes factor of a Bayesian model. It involves using samples from the model’s posterior parameter distribution along with the scaled values of the distribution’s density at those points. These samples are divided into two main groups: $m$ samples are used to build a histogram; $n$ are used to importance sample the histogram using the posterior parameter distribution as the proposal. This method is simple to implement and runs quickly in $O(d(m+n)\mbox{log}(m))$ time. Its asymptotic convergence rate, $n^{-1}(1+O(Cm^{-2/d}))$, is not remarkable, but in practice convergence is fast for many problems. Because the required inputs are similar to those of the harmonic mean estimator, it may be a convenient replacement for it. ## 7 References 1. 1. S. Chib. “Marginal Likelihood from the Gibbs Output” _Journal of the American Statistical Association_. Vol 90, No 432. (1995) 2. 2. S. Chib and I. Jeliazkov. “Accept-reject Metropolis-Hastings sampling and marginal likelihood estimation” _Statistica Neerlandica_. Vol 59, No 1. (2005) 3. 3. A. Gelman and X. Meng. “Simulating Normalizing Constants: From Importance Sampling to Bridge Sampling to Path Sampling” _Statistical Science_. Vol 13, No 2. (1998) 4. 4. R. Kass and A. Raftery. “Bayes Factors” _Journal of the American Statistical Association_. Vol 90, No 430. (1995) 5. 5. R. Neal. “The Harmonic Mean of the Likelihood: Worst Monte Carlo Method Ever”. Blog post, http://radfordneal.wordpress.com/2008/08/17/the-harmonic-mean-of- the-likelihood-worst-monte-carlo-method-ever/. (2008) 6. 6. J. Neddermeyer. “Computationally Efficient Nonparametric Importance Sampling” _Journal of the American Statistical Association_. Vol 104, No 486. (2009) arXiv:0805.3591v2 7. 7. A. Owen and Y. Zhou. “Safe and effective importance sampling” _Journal of the American Statistical Association_. Vol 95, No 449. (2000) 8. 8. C. Robert and D. Wraith. “Computational methods for Bayesian model choice” arXiv:0907.5123v1 9. 9. P. Zhang. “Nonparametric Importance Sampling” _Journal of the American Statistical Association_. Vol 91, No 435. (1996) 10. 10. M. Zlochin and Y. Baram. “Efficient Nonparametric Importance Sampling for Bayesian Inference” _Proceedings of the 2002 International Joint Conference on Neural Networks_ 2498–2502. (2002)
arxiv-papers
2011-01-06T04:30:27
2024-09-04T02:49:16.168435
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Benedict Escoto", "submitter": "Benedict Escoto", "url": "https://arxiv.org/abs/1101.1136" }
1101.1193
Dedicated to Alexei Sisakian co-author $\&$ friend On the connection between quantum and classical descriptions J.Manjavidze Joint Institute for Nuclear Research (VBLHEP & LNP, Dubna, Russia) $\&$ Andronikashvili Institute of Physics (LThP, Tbilisi State University, Georgia), Tel: (09621) 6 35 17, Fax: (09621) 6 66 66, E-mail: joseph@jinr.ru ###### Abstract The review paper presents generalization of d’Alembert’s variational principle: the dynamics of a quantum system for an external observer is defined by the exact equilibrium of all acting in the system forces, including the random quantum force $\hbar j$, $\forall\hbar$. Spatial attention is dedicated to the systems with (hidden) symmetries. It is shown how the symmetry reduces the number of quantum degrees of freedom down to the independent ones. The sin-Gordon model is considered as an example of such field theory with symmetry. It is shown why the particles $S$-matrix is trivial in that model. ###### Contents 1. 1 Introduction 2. 2 Simplest examples 1. 2.1 Introduction 2. 2.2 The generalized stationary-phase method 3. 2.3 Complex trajectories 4. 2.4 Conclusions 3. 3 Path integrals on Dirac measure 1. 3.1 Introduction 2. 3.2 Canonical transformation 3. 3.3 Selection rule 4. 3.4 Coordinate transformation 5. 3.5 Conclusions 4. 4 Reduction of quantum degrees of freedom 1. 4.1 Introduction 2. 4.2 Unitary definition of the path-integral measure 3. 4.3 Perturbation theory on the cotangent manifold 4. 4.4 Cancelation of angular perturbations 5. 4.5 Conclusions 5. 5 Example: H-atom 1. 5.1 Introduction 2. 5.2 Mapping 3. 5.3 Reduction 4. 5.4 Perturbations 5. 5.5 Conclusions 6. 6 Example: sin-Gordon model 1. 6.1 Introduction 2. 6.2 Reduction procedure 3. 6.3 Quantum corrections 7. 7 Summary 8. 8 Conclusion Preface Present paper is the review of the works which was done after my first paper [1]. I returned from time to time to the idea [1] that it seems interesting to embed the total probability conservation condition into the quantum field theory formalism and discuss it with Alexei Sissakian during our team-work. It seems that this suggestion is unnecessary noting that the $S$-matrix is the unitary operator and it is not evident why this attempt can give something new. But it turns out that exist the correspondence among quantum theory and classics which is independent from the value of quantum corrections. Besides this new quantum field theory is free from divergences and the value of quantum corrections ingenuously depend on the topology of classical field. All that is new from the point of view of ordinary theory and at last Alexei Sissakian propose to write on paper all result in details. Present introductory paper devoted to simplest examples and more interesting field theory models will be published later. ## 1 Introduction The basis of the method of calculations is the following [1]. The $S$-matrix unitarity condition, $S^{\dagger}S=1$, in terms of amplitudes, $S=1+iA$, looks as follows: $iA^{\dagger}A=(A-A^{\dagger}).$ (1\. 1) The nonlinearity of this equality points on existence of the cancelations mechanism (of the real part of amplitude) which reduces quadratic form down the linear one. Our purpose is to show how this reduction removes the ”unwanted” contributions111This means that the theory must be formulated directly in terms of probability. But notice that it is the frequently used method of particle physics. For example, one must integrate over unobserved final state in the inclusive approach to the multiple production phenomena. Another example: describing the very high multiplicity (VHM) processes the number of produced particles $n$ must be considered as the dynamical parameter. In the frame of $S$-matrix thermodynamics, where the ”rough” description of final state is used, one must also integrate over final particles momenta. In all cases one must consider quantities $\sim|A|^{2}$ directly, where $A$ is the corresponding amplitude.. One may consider the simplest vacuum-into-vacuum transition probability, $|Z|^{2}$, as the main quantity, where $Z$ is the functional integral over fields, $Z=\int D\varphi~{}e^{iS(\varphi)},~{}D\varphi=\prod_{x}d\varphi(x).$ (1\. 2) One may include into the action, $S$, also the linear over field $\varphi$ term, $\int dxJ(x)\varphi(x)$ (1\. 3) to describe production of particles. We will assume on the early stages that $J=0$. Then the vacuum-into-vacuum transition probability $|Z|^{2}=\int D\varphi_{+}D\varphi^{*}_{-}~{}e^{iS(\varphi_{+})-iS^{*}(\varphi_{-})},$ (1\. 4) where $\varphi_{+}$ and $\varphi_{-}$ are completely independent fields. It can be shown that Eq. (1\. 1) means that a reduced form must also exist [1]: $|Z|^{2}=\lim_{j=e=0}e^{i\hat{\mathbb{K}}(j,e)}\int DMe^{iU(\varphi,e)},$ (1\. 5) where $\hat{\mathbb{K}}=\hat{\mathbb{K}}(j,e)$ is a definite differential operator over $j(x)$ and $e(x)$, see the examples (2\. 42), (6\. 7). The expansion of $\exp\\{i\hat{\mathbb{K}}\\}$ generates perturbation series. The functional $U(\varphi,e)$ introduces interaction among quantum degrees of freedom and the integral measure is $\delta$-functional: $DM=\prod_{x}d\varphi(x)\delta\left(\frac{\delta S(\varphi)}{\delta\varphi(x)}+\hbar j(x)\right).$ (1\. 6) Sometimes the $\delta$-like measure [2] is called in mathematical literature as the ”Dirac measure”. It follows from (1\. 6) that — the quantum system for an externa observer looks like classical which is excited by the external random force $\hbar j,~{}\forall\hbar$. The established generalized correspondence principle222This formulation of the principle was offered by A.Sisakian. is the main consequence of Eq. (1\. 1). Therefore the complete set of acceptable field states for external observer333Since the ”probability” is considered is known having (1\. 6). It is important that the restricted problem is considered. We will calculate the imaginary part of amplitude believing that it will be sufficient for us. In this case the unmeasurable phase of amplitude stay undefined444 Therewith why must the calculations of unnecessary, i.e. unmeasurable, phase be performed? Just in this sense the unitarity condition (1\. 1) is a necessary one. It says that the real part is the ”unwanted” part of the amplitude.. A main mathematical problem in the searching representation (1\. 5) is to find the way how to find the imaginary part from the modulo square of amplitude. To be more precise, we will find the imaginary part as a result of cancelation of ”unwanted” contribution in the modulo square of amplitude. The $\delta$-function (1\. 6) solves the problem of definition of contributions into the path integral but can not solve the problem completely since the action of operator $\hat{\mathbb{K}}$ remains unknown. It must be noted that $\exp\\{i\hat{\mathbb{K}}\\}$ generates the asymptotic series ordinary in quantum theories [3] and it seems that $\delta$-like measure gives nothing new555Looking at the approach from the point of view of the stationary phase methods. In other words, one can think that the present approach gives nothing new to the Bohr’s correspondence principle.. But this is not entirely so. I would like draw attention to the appearance of source of quantum excitations $\hbar j$ in the r.h.s. of classical Lagrange equation, i.e. the changes of l.h.s. in equation of motion leads to the change of $j$. It is crucially important that (1\. 6) is rightful independently from the value of $\hbar$. The theory defined on the Dirac measure (1\. 6) for this reason has quite unexpected properties, e.g. allows to perform transformation of the path integral variables. So, it will be shown that in theories with symmetry the reduced form of representation (1\. 5) exists: $|Z|^{2}=\lim_{j=e=0}e^{i\hat{\mathbb{K}}(j,e)}\int DM(j)e^{iU(\varphi_{c},e)},$ (1\. 7) where $\hat{\mathbb{K}}$ is again the perturbations generating operator and $U$ introduces interactions. Note that $\hat{\mathbb{K}}$ and $U$ in (1\. 7) depends on the sets $\\{j_{\xi_{k}},j_{\eta_{k}}\\}$, $\\{e_{\xi_{k}},e_{\xi_{k}}\\}$ of new variables. One must take this auxiliary variables equal to zero at the very end of calculations. At the same time the transformed measure $DM$ is again $\delta$-like: $DM=\prod_{k}\prod_{t}d\xi_{k}(t)d\eta_{k}(t)\times$ $\times\delta\left(\dot{\xi}_{k}(t)-\frac{\delta h}{\delta\eta_{k}(t)}-j_{\xi_{k}}(t)\right)\delta\left(\dot{\eta}_{k}(t)+\frac{\delta h}{\delta\xi_{k}(t)}+j_{\eta_{k}}(t)\right),$ (1\. 8) where $t$ is the time variable and $h=h(\eta)$ is the transformed Hamiltonian: $h(\eta)=H(\varphi_{c}),$ (1\. 9) where $\varphi_{c}=\varphi_{c}({\mathbf{x}};\xi,\eta)$ is given solution of Lagrange equation at $j=0$. The formulae (1\. 8) is the main result. Therefore, as it follows from it the problem of the quantum field theory with symmetry is reduced down to quantum mechanical one, with potential defined by $\varphi_{c}$. (A) The Dirac measure (1\. 6) prescribes that $|Z|^{2}$ is defined by the $sum$ of $strict$ solutions of equation of motion: $\frac{\delta S(\varphi)}{\delta\varphi(x)}=\hbar j(x),$ (1\. 10) in vicinity of $j=0$, i.e. by definition Eq. (1\. 10) must be solved expanding the solution over $j$ 666It should be noted that it may be that the limit $j=0$ is absent. For example it may happen if the system is unstable against symmetry breaking. This important possibility will not be considered in present paper.. Following the ordinary rule we obviously leave the contribution which ensures the minimal vacuum energy. On the other hand, having theory on Dirac measure, which calls for the complete set of contributions, we have offered another selection rule in our dynamic theory of $S$-matrix. Namely, we simply propose777This selection rule is used widely in classical mechanics, see e.g. formulation of Kolmogorov-Arnold-Mozer (KAM) theorem [4]. that — the largest terms in the sum over solutions of (1\. 10) are significant from the physics point of view. To be more precise, this selection rule means that if $G$ is the symmetry of action and $TG^{*}$ is the symmetry of the extremum of the action, then in the situation of a general position only the trajectories with the highest dimension factor group, $(G/TG^{*})$, are sufficient. It will be seen that this kind of definition of the ”ground state” extracts the maximally ”feeling” symmetry contributions since other ones will be realized on a zero measure, or, more precisely, only maximal symmetry breaking field configurations, $\varphi_{c}$, are most probable. We will call such solution of the problem the field theory with symmetry. It is the main formal distinction of present approach. It is important here that the zero width of $\delta$-function excludes the interference among contributions from various trajectories. Therefore the formalism naturally takes into account the orthogonality of Hilbert spaces built on various trajectories. This is achieved through the special boundary conditions in the frame of which the total action of the product $Z\cdot Z^{*}=<in|out>\cdot<out|in>$ always describes a closed path, i.e. the necessary for d’Alembert variational principle time reversible motion. It points to the necessity to be careful with boundary conditions in a considered formalism888 The necessity to count all possible boundary conditions of a given problem was mention to author by L.Lipatov.. (B) The Dowker’s theorem [5] insists that the semiclassical approximation to be exact for path integrals on the simple Lie group manifolds. For this reason one can expect that the quantum-mechanical problems, as well as the field- theoretical ones, may be at least transparent to the symmetry manifolds. However we know how to construct correctly the path integral formalism only in the restricted case of canonical variables [6]. At first sight the path integrals in terms of generalized coordinates can be defined through the corresponding transformation. But there is an opinion that it is impossible to perform the transformation of path-integral variables: the naive transformation of coordinates give wrong results because of their stochastic nature in quantum theories999One can find corresponding examples in [6, 7]. The mostly popular method of transformation of the path-integral variables is a ”time-sliced” method [8], which induces corrections to the interaction Lagrangian proportional at least to $\hbar^{2}$ [9], i.e. the problem of transformation is of a quantum nature. For this reason the usage of the ”time- sliced” method in general case is cumbersome, see also [10].. That is why such general principle as the conservation of total probability (1\. 1) should play important role. Indeed, it is evident that $\delta$-like Dirac measure (1\. 6) allows to perform arbitrary transformation [1] just as in the classical mechanics. Therefore, the theory on Dirac measure straight away leads to the new for quantum field theory selection rule and latter one gives the theory with symmetry. All this is attained by transition to the appropriate variables, $(\xi,\eta)\in W$ in our notations. The last circumstance means that we go away from ordinary spectral analysis of quantum fluctuations to the description of the classical trajectories topology conserving deformations, since $\varphi_{c}=\varphi_{c}({\mathbf{x}};\xi,\eta)$ is given, of symmetry manifold, $W$ 101010It will be seen from our selection rule that the measure on which particles mechanics is realized is equal to zero in the field theories with symmetry.. It must be underlined that our method of transformations is rightful for arbitrary case, i.e. not only for simple Lie group manifolds, where the semiclassical approximation is exact. Next, the dimensions of initial phase space of field and of the transformed space of independent degrees of freedom, i.e. of the symmetry manifold, will not coincide. That means that the mapping to the independent degrees of freedom, $(\xi,\eta)$, will be singular. For this reason the transformation $\varphi_{c}:\varphi\rightarrow(\xi,\eta)$ will be irreversible and the notion of particle should be considered as the wrong idea of quantum field theory with symmetry 111111Considering gluon production in the frame of Yang-Mills field theory with symmetry the conclusion that gluons can not be created should be confirmed by direct calculations, taking into account also the quark fields. That was mentioned to the author by P.Culish and will be shown in later publications. It is noticeable that the mapping in quantum mechanics is not singular and for this reason both representations before and after transformation have the equal status.. (C) It will be shown that the result of action of the operator $\exp\\{i\hat{\mathbb{K}}\\}$ for transformed theories may be expressed as the sum of contributions on all boundaries $\partial W$: $|Z|^{2}=|Z|^{2}_{sc}+\sum_{k}\int d\xi_{k}(0)\frac{\partial}{\partial\xi_{k}(0)}C_{\xi}+\sum_{k}\int d\eta_{k}(0)\frac{\partial}{\partial\eta_{k}(0)}C_{\eta}$ (1\. 11) where the first term presents a semiclassical contribution and $C_{\xi}$, $C_{\eta}$ contains quantum corrections. This result shows that the quantum corrections greatly depend on the topology of classical trajectory. This important observation solves a number of problems. For instance, it is known that the Coulomb trajectory is closed because of Bargman-Fock symmetry, independent from the initial conditions. For this reason the corrections on $\partial W$ of Coulomb problem are canceled and the H-atom problem is pure semiclassical. We will find the same for sine-Gordon model [11] as the consequence of mapping on the Arnold’s hypertorus [12]. It is extremely important to keep in mind that the symmetry constraints can not be taken into account perturbatively over the interaction constant, $g$. Indeed, we will see below that the expansion in $polinomial$ theories with symmetry is performed in terms of the inverse interaction constant, $1/g$. It points to the absence of the weak-coupling limit in such theories. In the end our present aim is — to find representation (1\. 5); — to investigate the main properties of theory defined on the Dirac measure (1\. 6); — to investigate the structure of perturbation theory generated by operator $\hat{\mathbb{K}}$ on the measure (1\. 8); — to find particles production probabilities for theories with symmetry. I understand that the perturbations scheme in terms of new variables, especially in theories with symmetry, is outside the habitual one121212See [13, 14, 15] and for this reason the approach will be describe in more details, giving step-by-step the properties of a new quantization scheme by appropriate examples. I think that such non-formal scheme of the description is much more transparent, although the text may contain reiterations with the used method of description far from completeness. ## 2 Simplest examples ### 2.1 Introduction It it has mentioned above a technical aspect of our idea is the suggestion to calculate the probability without the intermediate step of calculations of the amplitudes. In present Section we restrict ourselves to the simplest problem - to the motion of a particle in a potential $V(x)$. Let the amplitude $A(x_{2},T;x_{1},0)$ describes the motion of the particle from the point $x_{1}$ to the point $x_{2}$ during the time $T$. Using the spectral representation: $A(x_{2},T;x_{1},0)=\sum_{n}\psi_{n}(x_{2})\psi_{n}^{*}(x_{1})e^{iE_{n}T},$ (2\. 1) we have for probability: $W(x_{2},T;x_{1},0)=\sum_{n_{1},n_{2}}\psi_{n_{1}}(x_{2})\psi_{n_{1}}^{*}(x_{1})\psi_{n_{2}}^{*}(x_{2})\psi_{n_{2}}(x_{1})e^{i(E_{n_{1}}-E_{n_{2}})T}.$ (2\. 2) Taking into account the ortho-normalizability condition: $\int dx\psi_{n}(x)\psi_{m}^{*}(x)=\delta_{n,m},$ (2\. 3) the total probability: $\int dx_{2}dx_{1}W(x_{2},T;x_{1},0)=\sum_{n}\delta_{n,n}=\Omega$ (2\. 4) is the time independent quantity which coincides with the number of existing physics states. Therefore, the amplitude (2\. 1) is time dependent, but the total probability (2\. 4) is not. This means that the time is the unwanted parameter from the point of view of experiment described by the probability (2\. 4). Notice also the role of boundary condition (2\. 3). The quantity (2\. 4) is of no interest to experiment. Much more interesting the probability $\rho(E)$, where $E$ is the energy experimentally measured. The Fourier transform of $A(x_{2},T;x_{1},0)$ with respect to $T$ $a(x_{2},x_{1};E)=\sum_{n}\frac{\psi_{n}(x_{2})\psi_{n}^{*}(x_{1})}{E-(E_{n}+i\varepsilon)}$ (2\. 5) leads to the probability $\omega(x_{2},x_{1};E)=|a(x_{2},x_{1};E)|^{2}=\sum_{n_{1},n_{2}}\frac{\psi_{n_{1}}(x_{2})\psi_{n_{1}}^{*}(x_{1})}{E-(E_{n_{1}}+i\varepsilon)}\frac{\psi_{n_{2}}^{*}(x_{2})\psi_{n_{2}}(x_{1})}{E-(E_{n_{2}}-i\varepsilon)}$ (2\. 6) and the total probability: $\rho(E)=\int dx_{1}dx_{2}\omega(x_{2},x_{1};E)=\sum_{n}\left|\frac{1}{E-E_{n}-i\varepsilon}\right|^{2}=$ $=\frac{1}{\varepsilon}\sum_{n}{\rm Im}\frac{1}{E-E_{n}-i\varepsilon}=\frac{\pi}{\varepsilon}\sum_{n}\delta(E-E_{n}).$ (2\. 7) The total probability $\rho(E)$ again coincides with number of existing states but for all that it is seen that the unphysical, i.e. needless, states from the point of view of measurement with $E\neq E_{n}$ was canseled131313Such contributions enter into the real part of $a(x_{2},x_{1};E)$.. Let us use now the proper-time representation: $a(x_{1},x_{2};E)=\sum_{n}\Psi_{n}(x_{1})\Psi^{*}_{n}(x_{2})i\int^{\infty}_{0}dTe^{i(E-E_{n}+i\varepsilon)T}$ (2\. 8) to see the integral form of cancelation of unwanted contributions and insert it into definition of total probability ($\varepsilon\rightarrow+0$): $\rho(E)=\int dx_{1}dx_{2}|a(x_{1},x_{2};E)|^{2}=\sum_{n}\int^{\infty}_{0}dT_{+}dT_{-}e^{-(T_{+}+T_{-})\varepsilon}e^{i(E-E_{n})(T_{+}-T_{-})}.$ (2\. 9) We will introduce new time variables instead of $T_{\pm}$: $T_{\pm}=T\pm\tau,$ (2\. 10) where, as it follows from Jacobian of transformation, $|\tau|\leq T,~{}0\leq T\leq\infty$. But we can put $|\tau|\leq\infty$ since $T\sim 1/\varepsilon\rightarrow\infty$ is essential in integral over $T$. As a result, $\rho(E)=4\pi\sum_{n}\int^{\infty}_{0}dTe^{-2\varepsilon T}\int^{+\infty}_{-\infty}\frac{d\tau}{\pi}e^{2i(E-E_{n})\tau}=\frac{\pi}{\varepsilon}\sum_{n}\delta(E-E_{n}).$ (2\. 11) In the last integral all contributions with $E\neq E_{n}$ has been canceled and only the acceptable from physics point of view contributions with $E=E_{n}$ has survived. This peculiarity of considered interference phenomena which is the consequence of unitarity condition, i.e. its ability to extract only physics states, would have the significant applications. Note also that the product of amplitudes $a\cdot a^{*}$ was ”linearized” after introduction of ”virtual” time $\tau=(T_{+}-T_{-})/2$, i.e. after transformation (2\. 10) we start calculation of the imaginary part. The meaning of such variables will be discussed also in Sec.2.2. ### 2.2 The generalized stationary-phase method 1. 0-dimensional model Let us practise considering the ”$0$-dimensional” integral: $A=\int^{+\infty}_{-\infty}\frac{dx}{(2\pi)^{1/2}}e^{i(\frac{1}{2}ax^{2}+\frac{1}{3}bx^{3})},$ (2\. 12) with ${\rm Im}a\rightarrow+0$ and $b>0$. This example is useful since it allows to illustrate practically all technical tricks of the approach. We want to compute the ”probability” $R=|A|^{2}=\int^{+\infty}_{-\infty}\frac{dx_{+}dx_{-}}{2\pi}e^{i(\frac{1}{2}ax_{+}^{2}+\frac{1}{3}bx_{+}^{3})-i(\frac{1}{2}a^{*}x_{-}^{2}+\frac{1}{3}bx_{-}^{3})}.$ (2\. 13) New variables: $x_{\pm}=x\pm e$ (2\. 14) will be introduced to find out the cancelation phenomenon. In result: $R=\int^{+\infty}_{-\infty}\frac{dxde}{\pi}e^{-2(x^{2}+e^{2}){\rm Im}a}e^{2i({\rm Re}a\;x+2bx^{2})e}e^{2i\frac{b}{3}e^{3}},$ (2\. 15) where the prescription that ${\rm Im}a\rightarrow+0$ has been used. Note that integrations are performed along the real axis. We will compute the integral over $e$ perturbatively. For this purpose the transformation: $F(e)=\lim_{j=e^{\prime}=0}e^{\frac{1}{2i}\hat{j}\hat{e}^{\prime}}e^{2ije}F(e^{\prime}),$ (2\. 16) which is valid for any differentiable function, will be used. In (2\. 16) two auxiliary variables $j$ and $e^{\prime}$ has been introduced and the ”hat” symbol means the differential over corresponding quantity: $\hat{j}=\frac{\partial}{\partial j},\;\;\;\hat{e^{\prime}}=\frac{\partial}{\partial e^{\prime}}.$ (2\. 17) The auxiliary variables must be taken equal to zero at the very end of calculations. Choosing $\ln F(e)=-2e^{2}{\rm Im}a+2i\frac{b}{3}e^{3}$ (2\. 18) we will find: $R=\lim_{j=e=0}e^{\frac{1}{2i}\hat{j}\hat{e}}\int^{+\infty}_{-\infty}dxe^{-2(x^{2}+e^{2}){\rm Im}a}e^{2i\frac{b}{3}e^{3}}\delta({\rm Re}a~{}x+bx^{2}+j).$ (2\. 19) Therefore, the destructive interference among two exponents in the product $a\cdot a^{*}$ unambiguously determines both integrals, over $x$ and over $e$. The integral over difference $e=(x_{+}-x_{-})/2$ gives $\delta$-function and then this $\delta$-function defines the contributions in the last integral over $x=(x_{+}+x_{-})/2$. Following the definition of $\delta$-function only a strict solutions of equation ${\rm Re}a\;x+bx^{2}+j=0$ (2\. 20) gives the contribution into $R$. But one can note that this is not the complete solution of the problem: the expansion of operator exponent $\exp\\{\frac{1}{2i}\hat{j}\hat{e}\\}$ generates the asymptotic series. Note also that it is impossible to remove the source, $j$, dependence (only harmonic case, $b=0$, is free from $j$). The equation (2\. 20) at $j=0$ has the solutions, at $x_{1}=0$ and at $x_{2}=-a/b$. Performing trivial transformation $e\rightarrow ie$, $\hat{e}\rightarrow-i\hat{e}$ of auxiliary variable we find at the limit ${\rm Im}a=0$ that the contribution from $x_{1}$ extremum (minimum) has the expression141414The contribution of $x_{2}$ leads to divergent series.: $R=\frac{1}{a}e^{-\frac{1}{2}\hat{j}\hat{e}}(1-4bj/a^{2})^{-1/2}e^{2\frac{b}{3}e^{3}}$ (2\. 21) and the expansion of an operator exponent gives the asymptotic series: $R=\frac{1}{a}\sum^{\infty}_{n=0}(-1)^{n}\frac{(6n-1)!!}{n!}\left(\frac{2b^{4}}{3a^{6}}\right)^{n},\;\;\;\;(-1)!!=0!!=1.$ (2\. 22) This series is convergent in Borel’s sense. Therefore the described destructive interference has not an action upon the value of perturbation series convergence radii. Let us calculate now $R$ using stationary phase method. The contribution from the minimum $x_{1}$ gives $({\rm Im}a=0)$: $A=e^{-i\hat{j}\hat{x}}e^{-\frac{i}{2a}j^{2}}e^{i\frac{b}{3}x^{3}}({i}/{a})^{1/2}.$ (2\. 23) The corresponding “probability” is $R=\frac{1}{a}e^{-i(\hat{j}_{+}\hat{x}_{+}-\hat{j}_{-}\hat{x}_{-})}e^{-\frac{i}{2a}(j_{+}^{2}-j_{-}^{2})}e^{i\frac{b}{3}(x_{+}^{3}-x_{-}^{3})}.$ (2\. 24) Introducing new auxiliary variables: $j_{\pm}=j\pm j_{1},\;\;\;\;x_{\pm}=x\pm e$ (2\. 25) and, correspondingly, $\hat{j}_{\pm}=(\hat{j}\pm\hat{j}_{1})/2,\;\;\;\;\hat{x}_{\pm}=(\hat{x}\pm\hat{e})/2$ (2\. 26) we find from (2\. 24): $R=\frac{1}{a}e^{-\frac{1}{2}\hat{j}\hat{e}}e^{2\frac{b}{3}e^{3}}e^{\frac{2b}{a^{2}}ej^{2}}$ (2\. 27) This expression does not coincide with (2\. 21) but it leads to the same asymptotic series (2\. 22). We may conclude that both considered methods of calculation of $R$ are equivalent since the Borel’s regularization scheme of asymptotic series gives a unique result. The difference between this two methods of calculation is in different organization of perturbations. So, if $F(e)$, instead of (2\. 18), is chosen in the form: $\ln F(e)=-2e^{2}{\rm Im}a+2i\frac{b}{3}e^{3}+2ibx^{2}e,$ (2\. 28) we may find (2\. 27) straightforwardly. Therefore, our method has the freedom in choice of (quantum) source $j$151515This freedom was mentioned firstly by A.Ushveridze.. Indeed, the transition from perturbation theory with Eq.(2\. 18) to the theory with Eq.(2\. 28) formally looks like following transformation of the argument of $\delta$-function: $\delta(ax+bx^{2}+j)=\lim_{e^{\prime}=j^{\prime}=0}e^{-i\hat{j}^{\prime}\hat{e}^{\prime}}e^{i(bx^{2}+j)e^{\prime}}\delta(ax+j^{\prime}).$ (2\. 29) Here the transformation (2\. 16) of the Fourier image of $\delta$-function was used. Inserting Eq.(2\. 29) into (2\. 19) we easily find (2\. 27). During analytic calculations it will be useful to have a corresponding quantum sources of the new dynamical variables. Formally this will be done using transformation (2\. 29). Note that this transformation will not lead to changing of the Borel’s regularization procedure. 2\. 1-dimensional model Let us calculate now the probability using the path-integral definition of amplitudes [1]. Calculating the quantity $|A|^{2}=\rm<in|out><in|out>^{*}=<in|out><out|in>,$ (2\. 30) the converging and diverging waves in the product $A\cdot A^{*}$ interfere in such a way that the continuum of contributions cancel each other. Indeed, the amplitude $A(x_{2},T;x_{1},0)=\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{C_{T}}e^{-iS_{T}(x)},~{}Dx=\prod_{t=0}^{T}\frac{dx(t)}{(2\pi)^{1/2}},$ (2\. 31) where the action $S_{T}$ is given by the expression: $S_{T}(x)=\int^{T}_{0}dt\left(\frac{1}{2}~{}\dot{x}^{2}-v(x)\right)$ (2\. 32) and $C_{T}$ is the standard normalization coefficient: $C_{T}=\int^{x(T)=x_{2}}_{x(0)=x_{1}}Dxe^{\frac{i}{2}\int^{T}_{0}dt~{}\dot{x}^{2}}$ (2\. 33) Let us calculate the quantity $R(x_{2},T;x_{1},0)=\int^{x_{\pm}(T)=x_{2}}_{x_{\pm}(0)=x_{1}}\frac{Dx_{+}}{C_{T}}\frac{Dx_{-}}{C_{T}^{*}}e^{-iS_{T}(x_{+})+iS_{T}(x_{-})}$ (2\. 34) We assume for simplicity that the integration in (2\. 31) is performed over real trajectories. Later a general case of complex trajectories will be considered. The convergence of functional integral at that is not important. One may restrict the range of integration for better confidence, or introduce into the Lagrangian $i\varepsilon$ term, and later remove the restriction in the expression (2\. 40). It is interesting that the interference phenomena naturally regularize divergent integrals of (2\. 31) type, accumulating divergence into $\delta$-function. In order to take into account explicitly the interference between contributions of the trajectories $x_{+}(t)$ and $x_{-}(t)$ we shall go over from the integration over two independent trajectories $x_{+}$ and $x_{-}$ to the pair $(x,e)$: $x_{\pm}(t)=x(t)\pm e(t).$ (2\. 35) It must be stressed that the transformation (2\. 35) is linear and for this reason may be done in the path integral. Substituting (2\. 35) into (2\. 34) the argument of the exponent takes the form $S_{T}(x+e)-S_{T}(x-e)=2\int_{0}^{T}dte(\ddot{x}+v^{\prime}(x))-U_{T}(x,e),$ (2\. 36) where $U_{T}(x,e)$ is the remainder of the expansion in powers of $e(t)$ ($U_{T}=O(e^{3})$). Note that in (2\. 36) we have discarded the ”surface” term $\int_{0}^{T}dt\partial_{t}(e\dot{x})=e(T)\dot{x}(T)-e(0)\dot{x}(0)=0,$ (2\. 37) since the boundary points of the trajectories $x_{+}(0)=x_{-}(0)=x_{1}$ and $x_{+}(T)=x_{-}(T)=x_{2}$ are not varied, i.e. $e(0)=e(T)=0.$ (2\. 38) Next, $Dx_{+}Dx_{-}=JDxDe=2\pi J\prod_{t=0}^{T}dx(t)\prod_{t\neq 0,T}\frac{de(t)}{2\pi},$ (2\. 39) where $J$ is an unimportant Jacobian of the transformation. As a result of the replacement (2\. 35) we have $R(x_{2},T;x_{1},0)=2\pi J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\int^{e(T)=0}_{e(0)=0}De~{}e^{2i\int_{0}^{T}dte(\ddot{x}+v^{\prime}(x))+U_{T}(x,e)}.$ (2\. 40) One can make use of the formulae $e^{iU_{T}(x,e)}=e^{\hat{\mathbb{K}}(e^{\prime},j)}e^{iU_{T}(x,e^{\prime})}e^{-2i\int_{0}^{T}e(t)j(t)dt},$ (2\. 41) where we have introduced the operator $\hat{\mathbb{K}}(e,j)=\lim_{e=j=0}\exp\left\\{-\frac{1}{2i}\int_{0}^{T}\frac{\delta}{\delta j(t)}\frac{\delta}{\delta e(t)}\right\\},$ (2\. 42) after which from (2\. 40) we have found that $R(x_{2},T;x_{1},0)=2\pi Je^{\hat{\mathbb{K}}(e^{\prime},j)}\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}e^{iU_{T}(x,e^{\prime})}\times$ $\times\int^{e(T)=0}_{e(0)=0}De~{}\exp\left\\{2i\int_{0}^{T}dt(\ddot{x}+v^{\prime}(x)-j)e\right\\}=$ $=2\pi Je^{\hat{\mathbb{K}}(e,j)}\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}e^{iU_{T}(x,e)}\prod_{t\neq 0,T}\delta(\ddot{x}+v^{\prime}(x)-j),$ (2\. 43) where the functional $\delta$-function $\prod_{t\neq 0,T}\delta(\ddot{x}+v^{\prime}(x)-j)=\int^{e(T)=0}_{e(0)=0}De~{}\exp\left\\{2i\int_{0}^{T}dt(\ddot{x}+v^{\prime}(x)-j)e\right\\}$ (2\. 44) has arisen as a result of total reduction of unnecessary contributions from the point of view of equation of motion $\ddot{x}(t)+V^{\prime}(x)=j(t).$ (2\. 45) The operator (2\. 42) is Gaussian so that the system is perturbed by the random force $j(t)$. If $x(t)$ is the ”true” trajectory and the virtual deviation is $e(t)$ then the quantity $e(\ddot{x}+v^{\prime}(x)-j)$ coincides with the virtual work. It must be equal to zero in classical mechanics since only the time reversible motion is considered. In result we came to equation of motion since $e$ is arbitrary in classics. The difference $S_{T}(x_{+})-S_{T}(x_{-})$ in (2\. 34) with boundary conditions (2\. 38) coincides with the action of reversible motion. Upon the substitution (2\. 35) we have identified the mean trajectory, $x(t)$, and the deviation from it, $e(t)$. One must integrate over $e(t)$ in quantum case, in contrast to classical one. In result the measure of the remaining path integral over mean trajectory $x(t)$ takes the Dirac $\delta$-function form which unambiguously chooses the ”true” trajectory. In other words, the proposed definition of the measure of the path integral is generalization of classical d’Alambert’s principle on the quantum case. The theory in the frame of this principle can take into account any external perturbations, $j(t)$ in our case, if the time reversibility of motion is conserved. In quantum case the reversibility is established through the boundary conditions (2\. 38). Next, one may generalize the approach adding also the probe force which can lead to dynamical symmetry breaking [16]161616It is important that if the expectation value of the probe force is not equal to zero then the symmetry is broken. This important possibility will not be considered in present work.. In the semiclassical approximation $\hat{\mathbb{K}}(e,j)=1$ and taking the limit $e=j=0$ we find that $R(x_{2},T;x_{1},0)=2\pi J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\prod_{t\neq 0,T}\delta(\ddot{x}+v^{\prime}(x)),$ (2\. 46) Let the solution of the homogeneous equation $\ddot{x}+v^{\prime}(x)=0$ (2\. 47) be $x_{c}(t)$, with $x_{c}(0)=x_{1}$ and $x_{c}(T)=x_{2}$. Then $R(x_{2},T;x_{1},0)=2\pi J\int^{x(T)=x_{2}}_{x(0)=x_{1}}\frac{Dx}{|C_{T}|^{2}}\prod_{t\neq 0,T}\delta(\ddot{x}+v^{\prime\prime}(x_{c})x),$ (2\. 48) The remaining integral is calculated by the standard methods171717Here it is more convenient to represent (2\. 48) as a production of two Gauss integrals; later on more effective method of calculation of the functional determinant will be offered.. As a result we find $R(x_{2},T;x_{1},0)=\frac{1}{2\pi}\left|\frac{\partial^{2}S_{T}(x_{c})}{\partial x_{c}(0)\partial x_{c}(T)}\right|_{x_{c}(0)=x_{1},x_{c}(T)=x_{2}}.$ (2\. 49) Next, let us recall that the full derivative of the classical action is $dS=p_{2}dx_{2}-p_{1}dx_{1},$ (2\. 50) where $p_{2}$ and $p_{1}$ are, respectively, the final and initial momentum. Noting this definition, $\left|\frac{\partial^{2}S_{T}}{\partial x_{1}\partial x_{2}}\right|dx_{2}=dp_{1},$ (2\. 51) and in result we find that $\int dx_{1}dx_{2}R(x_{2},T;x_{1},0)=\int\frac{dx_{1}dp_{1}}{2\pi}=\Omega^{2},$ (2\. 52) which coincides with (2\. 4), i.e. it agree with conservation of total probability since (2\. 52) again coincides with the total number of physical states. Deriving (2\. 52) we somewhat simplify the problem considering a unique solution of Eq.(2\. 47). A more complicate and important examples will be considered in the next Sections. ### 2.3 Complex trajectories Let us consider the one dimensional motion with fixed energy $E$ on the complex trajectory181818The necessity to extend the formalism on the case of complex trajectories was mention to the author by A.Slavnov.. The corresponding amplitude has the form: $A(x_{1},x_{2};E)=i\int^{\infty}_{0}dTe^{iET}\int_{x_{1}=x(0)}^{x_{2}=x(T)}D_{C_{+}}xe^{iS_{C_{+}}(x)},$ (2\. 53) where the action $S_{C_{+}}(x)=\int_{C_{+}}dt(\frac{1}{2}\dot{x}^{2}-v(x))$ (2\. 54) and the measure $D_{C_{+}}x=\prod_{t\in C_{+}}\frac{dx(t)}{(2\pi)^{1/2}}$ (2\. 55) are defined on the shifted in the upper half time plane Mills’ contour $C_{+}=C_{+}(T)$ [17]: $t\rightarrow t+i\varepsilon,\;\;\;\varepsilon\rightarrow+0,\;\;\;0\leq t\leq T.$ (2\. 56) Therefore, we will consider integration over real functions of complex variables: $x^{*}(t)=x(t^{*}).$ (2\. 57) It must be underlined also that the boundary conditions in (2\. 53) have the classical meaning, i.e. they do not vary, and $x_{1}$, $x_{2}$ are the real quantities. The probability looks as follows: $R(E)=\int^{\infty}_{0}e^{iE(T_{+}-T_{-})}\int^{x_{\pm}(T_{\pm})=x_{2}}_{x_{\pm}(0)=x_{1}}D_{C_{+}}x_{+}D_{C_{-}}x_{-}\times$ $\times e^{iS_{C_{+}(T_{+})}(x_{+})-iS_{C_{-}(T_{-})}(x_{-})},$ (2\. 58) where $C_{-}(T)=C^{*}_{+}(T)$ is the time contour in the lower half of complex time plane. New time variables $T_{\pm}=T\pm\tau$ (2\. 59) will be used. Considering ${\rm Im}E\rightarrow+0$ we can consider $T$ and $\tau$ as the independent variables: $0\leq T\leq\infty,\;\;\;-\infty\leq\tau\leq\infty.$ (2\. 60) We will apply the boundary conditions, see (2\. 58): $x_{1}=x_{+}(0)=x_{-}(0),~{}~{}x_{2}=x_{+}(T_{+})=x_{-}(T_{-}).$ (2\. 61) Inserting (2\. 59) one can find in zero order over $\tau$ from (2\. 61) that $x_{+}(0)=x_{-}(0),~{}~{}x_{+}(T)=x_{-}(T),$ (2\. 62) Now we will introduce also the mean trajectory $x(t)=(x_{+}(t)+x_{-}(t))/2$ and the deviation $e(t)$ from $x(t)$: $x_{\pm}(t)=x(t)\pm e(t).$ (2\. 63) We have consider $e(t)$ and $\tau$ as the virtual quantities. The integrals over $e$ and $\tau$ will be calculated perturbatively. In zero order over $e$ and $\tau$, i.e. in the semiclassical approximation, $x$ is the classical path and $T$ is the total time of classical motion. Note that one can do surely the linear transformations (2\. 63) in the path integrals. The higher terms over $\tau$ put a unphysical constrains on the trajectory $x(t)$: $\frac{d^{(2n+1)}x(T)}{dT^{(2n+1)}}=0,~{}n=0,1,2,...,$ since $e(t)$ must be arbitrary. Therefore, to avoid this constraints and since the boundaries have classical unvaried meaning we will use the minimal boundary conditions: $e(0)=e(T)=0,$ (2\. 64) which ensures the time reversibility. Note that it is sufficient to have (2\. 64) if the integrals over $e(t)$ are calculated perturbatively. At the same time $x(0)=x_{1},~{}x(T)=x_{2}.$ (2\. 65) Let us extract now the linear over $e$ and $\tau$ terms from the closed-path action: $S_{C_{+}(T_{+})}(x_{+})-S_{C_{-}(T_{-})}(x_{-})=$ $=-2\tau H_{T}(x)-\int_{C^{(+)}(T)}dte(\ddot{x}+v^{\prime}(x))-\tilde{H}_{T}(x;\tau)-U_{T}(x,e),$ (2\. 66) where $C^{(+)}(T)=C_{+}(T)+C_{-}(T)$ (2\. 67) is the total-time path, $H_{T}$ is the Hamiltonian: $2H_{T}(x)=-\frac{\partial}{\partial T}(S_{C_{+}(T)}(x)+S_{C_{-}(T)}(x)),$ (2\. 68) and $-\tilde{H}_{T}(x;\tau)=S_{C_{+}(T+\tau)}(x)-S_{C_{-}(T-\tau)}(x)+2\tau H_{T}(x),$ (2\. 69) $-U_{T}(x,e)=S_{C_{+}(T)}(x+e)-S_{C_{-}(T)}(x-e)+\int_{C^{(+)}}dte(\ddot{x}+v^{\prime}(x))$ (2\. 70) are the remainder terms, where $v^{\prime}(x)=\partial v(x)/\partial x$. Deriving the decomposition (2\. 66) the definition $C_{-}(T)=C^{*}_{+}(T)$ (2\. 71) and the boundary conditions (2\. 64) was used. One can find the compact form of expansion of $e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)}$ over $\tau$ and $e$ using formulae (2\. 16): $\exp\\{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)\\}=\exp\left\\{\frac{1}{2i}\hat{\omega}\hat{\tau}^{\prime}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}^{\prime}(t)\right\\}\times$ $\times\exp\left\\{2i\omega\tau+i\int_{C^{(+)}(T)}dtj(t)e(t)\right\\}\exp\\{-i\tilde{H}_{T}(x;\tau^{\prime})-iU_{T}(x,e^{\prime})\\}.$ (2\. 72) At the end of calculations the auxiliary variables $(\omega,\tau^{\prime},j,e^{\prime})$ should be taken equal to zero. Using (2\. 66) and (2\. 72) we find from (2\. 58) that $R(E)=2\pi\int^{\infty}_{0}dT\exp\left\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)\right\\}\times$ $\times\int Dx\exp\\{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)\\}\delta(E+\omega- H_{T}(x))\prod_{C^{(+)}}\delta(\ddot{x}+v^{\prime}(x)-j).$ (2\. 73) The expansion over the differential operators: $\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)=\frac{1}{2i}\left(\frac{\partial}{\partial\omega}\frac{\partial}{\partial\tau}+{\rm Re}\int_{C+}dt\frac{\delta}{\delta j(t)}\frac{\delta}{\delta e(t)}\right)$ (2\. 74) will generate the perturbation series. We propose that it is summable in Borel sense. The first $\delta$-function in (5\. 33) fixes the conservation of energy: $E+\omega=H_{T}(x)$ (2\. 75) where $E$ is the observed energy, $H_{T}(x)$ is the energy at the mean trajectory at the time moment $T$ and $\omega$ is the energy of quantum fluctuations. The second $\delta$-function191919Following shorthand entry of $\delta$-function of the complex argument: $\prod_{C^{(+)}}\delta(f(t))=\prod_{C_{+}}\delta(f(t))\prod_{C_{-}}\delta(f(t))=\prod_{C_{+}}\delta({\rm Re}f(t)+i{\rm Im}f(t))\delta({\rm Re}f(t)-i{\rm Im}f(t))=\prod_{C_{+}}\delta({\rm Re}f(t))\cdot\\\ \delta({\rm Im}f(t))$ will be useful during calculations. The condition (2\. 57) is important here. The inessential constant can be canceled by normalization. So, in the result of analytical continuation of $C_{\pm}$ on the real axis the product of two $\delta$-functions reduces to single one since $\delta^{2}({\rm Re}f(x))=\delta(0)\delta({\rm Re}f(x))=\delta(0)\delta(f(x))$ and $\delta(0)$ must be canceled by normalization. Offered abbreviated notation will allow to consider $\delta$-function on the complex time contour as the ordinary one. $\prod_{t\in C^{(+)}}\delta(\ddot{x}+v^{\prime}(x)-j)=(2\pi)^{2}\int\prod_{t\in C^{(+)}}\frac{de(t)}{\pi}\delta(e(0))\delta(e(T))\times$ $\times e^{-2i{\rm Re}\int_{C_{+}}dte(\ddot{x}+v^{\prime}(x)-j)}=\prod_{t\in C_{+}(T)}\delta({\rm Re}(\ddot{x}+v^{\prime}(x)-j))\delta({\rm Im}(\ddot{x}+v^{\prime}(x)-j))$ (2\. 76) fixes the function $x(t)$ of complex argument on $C^{(+)}$ completely by the equation $\ddot{x}+v^{\prime}(x)=j.$ (2\. 77) The physics meaning of $\delta$-function (2\. 76) was discussed in Sec.2.3 noting that the unitarity condition of quantum theories plays the same role as the d’Alambert’s variational principle in classical mechanics. In (2\. 77) $j(t)$ describes the external quantum force. The solution $x_{j}(t)$ of this equation will be found expanding it over $j(t)$: $x_{j}(t)=x_{c}(t)+\int dt_{1}G(t,t_{1})j(t_{1})+...$ (2\. 78) This is sufficient since $j(t)$ is the auxiliary variable202020See also footnote 15.. In this decomposition $x_{c}(t)$ is the strict solution of unperturbed equation: $\ddot{x}+v^{\prime}(x)=0$ (2\. 79) Note that the functional $\delta$-function in (2\. 76) does not contain the end-point values of $x(t)$, at $t=0$ and $t=T$. This means that if we integrate over $x_{1}$ and $x_{2}$ then the initial conditions to the Eq.(2\. 79) are not fixed and the integration over them must be performed. Inserting (2\. 78) into (2\. 77) we find the equation for Green function: $(\partial^{2}+v^{\prime\prime}(x_{c}))_{t}G(t,t^{\prime};x_{c})=\delta(t-t^{\prime}).$ (2\. 80) It is too hard to find the exact solution of this equation if $x_{c}(t)$ is the nontrivial function of $t$. We will see that the canonical transformation to the (action-angle)-type variables can help to avoid this problem, see following Section. ### 2.4 Conclusions 1\. The path integral must be defined on the Mills time contour. This condition will be important in the field theories with high space-time symmetries (such as the Yang-Mills type theory) since it seems that for such theories with symmetry one can not perform surely the analytic continuation over time variable212121The fact that a theory must satisfy certain conditions upon analytic continuation over time variable is clear from [18].. 2\. The quantization can be performed without transition to the canonical formalism, using only the Lagrange one which is more natural for relativistic field theories. 3\. Only the exact solutions of the equation of motion must be taken into account defining the contributions into the functional integral. ## 3 Path integrals on Dirac measure ### 3.1 Introduction In present Section we will offer two methods which may simplify calculation of path integrals on Dirac measure. They are based on the possibility to perform transformation of the path-integral variables. We will consider two examples. In the first example the transformation to the (action,angle)-type variables will be considered. This example shows how much the calculations of path integrals may be simplified. In the second part of present Section the coordinate transformation will be described. For the sake of definiteness the transformation to cylindrical coordinates will be considered. ### 3.2 Canonical transformation Let us introduce the first-order formalism. We will insert in (2\. 73) $1=\int Dp\prod_{t}\delta(p-\dot{x}).$ (3\. 1) As a result, $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+{\rm Re}\int_{C_{+}(T)}dt\hat{j}(t)\hat{e}(t))}\int DxDpe^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)}\times$ $\times\delta(E+\omega- H_{T}(x))\prod_{t}\delta\left(\dot{x}-\frac{\partial H_{j}}{\partial p}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial x}\right),$ (3\. 2) where $H_{j}=\frac{1}{2}p^{2}+v(x)-jx$ (3\. 3) may be considered as the total Hamiltonian which is time dependent through $j(t)$. Notice that in present simplest case $x$ and $p$ are independent parameters and therefore (3\. 3) define the Hamiltonian. Instead of pare $(x(t),p(t))$ we introduce new pare $(\theta(t),h(t))$ inserting in (3\. 2) $1=\int\prod_{t}d\theta dh\delta\left(h-\frac{1}{2}p^{2}-v(x)\right)\delta\left(\theta-\int^{x}dx(2(h-v(x)))^{-1/2}\right).$ (3\. 4) Note that the integral measures in (3\. 2) and (3\. 4) are both $\delta$-like, i.e. have the equal power. It allows to change the order of integration and firstly integrate over $(x,p)$. We find that $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+{\rm Re}\int_{C_{+}(T)}dt\hat{j}(t)\hat{e}(t))}\int D\theta Dhe^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e)}\times$ $\times\delta(E+\omega-h(T))\prod_{t}\delta\left(\dot{\theta}-\frac{\partial H_{c}}{\partial h}\right)\delta\left(\dot{h}+\frac{\partial H_{c}}{\partial\theta}\right),$ (3\. 5) where $H_{c}=h-jx_{c}(h,\theta)$ (3\. 6) is the transformed Hamiltonian and $x_{c}(\theta,h)$ is the given solution of algebraic equation: $\theta=\int^{x}dx(2(h-v(x)))^{-1/2},$ (3\. 7) i.e. $x_{c}$ is the classical trajectory parametrized in terms of $h(t)$ and $\theta(t)$. As it follows from (3\. 5) new variables, $h(t)$ and $\theta(t)$, are subjected to the action of quantum force $j(t)$ and the topology of classical trajectory $x_{c}$ remains unchanged. So, instead of Eq.(2\. 77) we must solve the equations: $\dot{h}=j\frac{\partial x_{c}}{\partial\theta},\;\;\;\;\;\dot{\theta}=1-j\frac{\partial x_{c}}{\partial h},$ (3\. 8) which have a simpler structure. Expanding the solutions over $j$ we will find the infinite set of recursive equations. This is the important peculiarity of used quantization scheme. Note now that $j\partial x_{c}/\partial\theta$ and $j\partial x_{c}/\partial h$ in the r.h.s. can be considered as the new sources. We will use this property of Eqs.(3\. 8) and introduce in the perturbation theory new ”renormalized” sources: $j_{h}=j\frac{\partial x_{c}}{\partial\theta},\;\;\;\;\;j_{\theta}=j\frac{\partial x_{c}}{\partial h},$ (3\. 9) i.e. $j_{\xi}$ and $j_{\eta}$ are the forces on the cotangent bundle. We will use transformation (2\. 29): $\prod_{t}\delta(\dot{h}-j\frac{\partial x_{c}}{\partial\theta})=e^{\frac{1}{2i}{\rm Re}\int_{C_{+}}dt\hat{j}_{h}(t)\hat{e}_{h}(t)}e^{2i{\rm Re}\int_{C_{+}}e_{h}j\frac{\partial x_{c}}{\partial\theta}}\prod_{t}\delta(\dot{h}-j_{h})$ (3\. 10) and $\prod_{t}\delta(\dot{\theta}-1+j\frac{\partial x_{c}}{\partial h})=e^{\frac{1}{2i}{\rm Re}\int_{C_{+}}dt\hat{j}_{\theta}(t)\hat{e}_{\theta}(t)}e^{2i{\rm Re}\int_{C_{+}}e_{\theta}j\frac{\partial x_{c}}{\partial h}}\prod_{t}\delta(\dot{\theta}-1-j_{\theta})$ (3\. 11) to introduce them. The re-scaling of source $j$ lead to the re-scaling of auxiliary field $e$. In the new perturbation theory we will have two sources $j_{h}$, $j_{\theta}$ and two auxiliary fields $e_{h}$, $e_{\theta}$. Notice that the momentum $p$ never arose. Inserting (3\. 10), (3\. 11) into (3\. 5) we find: $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}-i\int_{C^{(+)}}dt(\hat{j}_{h}(t)\hat{e}_{h}(t)+\hat{j}_{\theta}(t)\hat{e}_{\theta}(t)))}\times$ $\times\int DhD\theta e^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\times$ $\times\delta(E+\omega-h(T))\prod_{t}\delta(\dot{\theta}-1-j_{\theta})\delta(\dot{h}-j_{h}),$ (3\. 12) where $e_{c}=e_{h}\frac{\partial x_{c}}{\partial\theta}-e_{\theta}\frac{\partial x_{c}}{\partial h}$ (3\. 13) carry the simplectic structure of Hamilton equations of motion and the ”hat” symbol means differential operator over corresponding quantity. At the very end one should take all auxiliary variables, $(e_{h},j_{h},e_{\theta},j_{\theta})$, equal to zero. Hiding the $x_{c}(t)$ dependence into $e_{c}$ we solve the problem of the functional determinants, see (3\. 12), and simplify the Hamilton equations of motion as much as possible: $\dot{h}(t)=j_{h}(t),\;\;\;\;\;\dot{\theta}(t)=1+j_{\theta}(t)$ (3\. 14) We will use the boundary conditions $h(0)=h_{0},\;\;\;\;\theta(0)=\theta_{0},$ (3\. 15) as the extension of boundary conditions in (2\. 58). This lead to the following Green function of transformed perturbation theory: $g(t-t^{\prime})=\Theta(t-t^{\prime}),$ (3\. 16) with the properties of projection operator: $\displaystyle\int dtdt^{\prime}g^{2}(t-t^{\prime})=\int dtdt^{\prime}g(t-t^{\prime}),$ $\displaystyle\int dtdt^{\prime}g(t-t^{\prime})g(t^{\prime}-t)=0$ (3\. 17) and, at the same time, we will assume that $g(0)=1.$ (3\. 18) It is important to note that ${\rm Im}g(t)$ is regular on the real time axis. This is the very simplification of the perturbation theory since it eliminates the doubling of degrees of freedom. One may use here the analytical continuation to the real time axis. In result, shifting $C_{+}$ and $C_{-}$ contours on the real time axis we find: $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}(\hat{\omega}\hat{\tau}+\int^{\infty}_{0}dt_{1}dt_{2}\Theta(t_{1}-t_{2})(\hat{e}_{h}(t_{1})\hat{h}(t_{2})+\hat{e}_{\theta}(t_{1})\hat{\theta}(t_{2})))}\times$ $\times\int dh_{0}d\theta_{0}e^{-i\tilde{H}_{T}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega- h_{0}+h(T)),$ (3\. 19) where the solutions of eqs.(3\. 14) was used. In this expression $x_{c}(t)=x_{c}(h_{0}-h(t),t+\theta_{0}-\theta(t))$ and $(h(t),e_{h}(t),\theta(t),e_{\theta}(t))$ are the auxiliary fields. At the very end one must take them equal to zero. ### 3.3 Selection rule Let us consider the theory with Lagrangian $L(x)=\frac{1}{2}\dot{x}^{2}-\frac{1}{2}\omega^{2}x^{2}-\frac{g}{4}x^{4}.$ (3\. 20) The Dirac measure gives the equation (of motion): $\ddot{x}+\omega^{2}x+gx^{3}=j.$ (3\. 21) It has two solutions: $x_{1}(t)=x_{c}(t)+O(j),~{}~{}x_{2}(t)=O(j).$ (3\. 22) For this reason $R(E)=R_{1}(E;x_{1})+R_{2}(E;x_{2})$ (3\. 23) and which one defines $R(E)$ is a question. Following to our selection rule just $R_{1}$. This will be shown. Let us return now to the example with Lagrangian (3\. 20). In the semiclassical approximation $R_{1}(E;x_{1})=\int_{0}^{\infty}dT\int_{0}^{\infty}dh_{0}\int_{-\infty}^{+\infty}d\theta_{0}e^{-iU_{T}(x_{c},0)}\delta(E-h_{0}).$ (3\. 24) Therefore, $R_{1}(E;x_{1})\sim\int_{-\infty}^{+\infty}d\theta_{0}\equiv\Omega,$ (3\. 25) i.e. it is proportional to the volume of group of time translations. At the same time $R_{2}(E;x_{2})=O(1)$ (3\. 26) in the semiclassical approximation. Therefore, $R=R_{1}(1+O(1/\Omega)).$ (3\. 27) This result explains the source of chosen selection rule. ### 3.4 Coordinate transformation In this section the coordinate transformation of two dimensional quantum mechanical model with potential $v=v((x^{2}_{1}+x^{2}_{2})^{1/2})$ (3\. 28) will be considered. Repeating calculations of previous sections, $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{\vec{j}}(t)\hat{\vec{e}}(t)}\int D^{(2)}M(x)e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e)},$ (3\. 29) where the $\delta$-like Dirac measure $D^{(2)}M(x)=\delta(E+\omega- H_{T}(x))\prod_{t}d^{2}x(t)\delta^{(2)}(\ddot{x}+v^{\prime}(x)-j).$ (3\. 30) In the classical mechanics the problem with potential (3\. 28) is solved in the cylindrical coordinates: $x_{1}=r\cos\phi,\;\;\;\;\;x_{2}=r\sin\phi.$ (3\. 31) We insert into (3\. 29) $1=\int DrD\phi\prod_{t}\delta(r-(x^{2}_{1}+x^{2}_{2})^{1/2})\delta(\phi-\arctan\frac{x_{2}}{x_{1}}).$ (3\. 32) to perform the transformation. Note that the transformation (3\. 31) is not canonical. In result we will find a new measure: $D^{(2)}M(r,\phi)=\delta(E+\omega-H_{T}(x))\prod_{t}drd\phi J(r,\phi),$ (3\. 33) where the Jacobian of transformation $J(r,\phi)=\int\prod d^{2}x\delta^{(2)}(\ddot{x}+v^{\prime}(x)-j)\delta(\phi-\arctan\frac{x_{2}}{x_{1}})\delta(r-(x^{2}_{1}+x^{2}_{2})^{1/2})$ (3\. 34) is the product of two $\delta$-functions: $J(r,\phi)=\prod_{t}r^{2}(t)\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})\delta(\partial_{t}(\dot{\phi}r^{2})-rj_{\phi}),$ (3\. 35) where $v^{\prime}(r)=\partial v(r)/\partial r$ and $j_{r}=j_{1}\cos\phi+j_{2}\sin\phi,\;\;\;\;j_{\phi}=-j_{1}\sin\phi+j_{2}\cos\phi$ (3\. 36) are the components of $\vec{j}$ in the cylindrical coordinates. It is useful to organize the perturbation theory in terms of $j_{r}$ and $j_{\phi}$. For this purpose following transformation of arguments of $\delta$-functions will be used: $\prod_{t}\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})=e^{-i\int_{C^{(+)}}dt\hat{j}^{\prime}_{r}\hat{e}_{r}}e^{i\int_{C^{(+)}}dtj_{r}e_{r}}\prod_{t}\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j^{\prime}_{r})$ (3\. 37) and $\prod_{t}\delta(\partial_{t}(\dot{\phi}r^{2})-rj_{\phi})=e^{-i\int_{C^{(+)}}dt\hat{j}^{\prime}_{\phi}\hat{e}_{\phi}}e^{i\int_{C^{(+)}}dtj_{\phi}re_{\phi}}\prod_{t}r(t)\delta(\partial_{t}(\dot{\phi}r^{2})-j^{\prime}_{\phi}).$ (3\. 38) Here $j_{r}$ and $j_{\phi}$ was defined in (3\. 36). In result, we get to the path integral formalism written in terms of cylindrical coordinates. This is a very simplification which will help to solve a lot of mechanical problems. One can note that in result of mapping our problem reduced to the description of quantum fluctuations of the surface of cylinder: $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}_{r}(t)\hat{e}_{r}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$ $\times\int D^{(2)}M(r,\phi)e^{-i\tilde{H}_{T}(x;\tau)-iU_{T}(x,e_{C})},$ (3\. 39) where $\displaystyle D^{(2)}M(r,\phi)=\delta(E+\omega- H_{T}(r,\phi))\prod_{t}r^{2}(t)dr(t)d\phi(t)\times$ $\displaystyle\times\delta(\ddot{r}-\dot{\phi}^{2}r+v^{\prime}(r)-j_{r})\delta(\partial_{t}(\dot{\phi}r^{2})-j_{\phi})$ (3\. 40) and $e_{C,1}=e_{r}\cos\phi- re_{\phi}\sin\phi,\;\;\;\;e_{C,2}=e_{r}\sin\phi+re_{\phi}\cos\phi.$ (3\. 41) This is the final result. The transformation looks quite classically but (3\. 39) can not be deduced from naive coordinate transformation of initial path integral for amplitude. Inserting $1=\int DpDl\prod_{t}\delta(p-\dot{r})\delta(l-\dot{\phi}r^{2})$ (3\. 42) into (3\. 39) we can introduce the motion in the phase space with Hamiltonian $H_{j}=\frac{1}{2}p^{2}+\frac{l^{2}}{2r^{2}}+v(r)-j_{r}r-j_{\phi}\phi.$ (3\. 43) The Dirac’s measure becomes four dimensional: $D^{(4)}M(r,\phi,p,l)=\delta(E+\omega- H_{T}(r,\phi,p,l))\prod_{t}dr(t)d\phi(t)dp(t)dl(t)\times$ $\times\delta\left(\dot{r}-\frac{\partial H_{j}}{\partial p}\right)\delta\left(\dot{\phi}-\frac{\partial H_{j}}{\partial l}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial r}\right)\delta\left(\dot{l}+\frac{\partial H_{j}}{\partial\phi}\right)$ (3\. 44) Note absence of the coefficient $r^{2}$ in this expression. This is the result of special choice of transformation (3\. 38). Since the Hamilton’s group manifolds are more rich then Lagrange ones the measure (3\. 44) can be considered as the starting point of farther transformations. One must to note that the $(action,angle)$ variables are mostly useful [12]. Note also that to avoid the technical problems with equations of motion and with functional determinants it is useful to linearize the argument of $\delta$-functions in (3\. 44) hiding nonlinear terms in the corresponding auxiliary variables $e_{c}$. ### 3.5 Conclusions 1\. Our perturbation theory describes the quantum fluctuations of the parameters $(h,\theta)$ of classical trajectory $x_{c}$. It is more complicated than canonical one, over an interaction constant [19], since demands investigation of analytic properties of $4N$-dimensional integrals, where $2N$ is the phase space dimension. Indeed, in the considered case with $N=1$ the perturbations generating operator, $\hat{\mathbb{K}}$, see (3\. 12), contain derivatives over four auxiliary parameters, $(j_{h},e_{h},j_{\theta},e_{\theta})$. Our transformed theory describes the ”direct” deformations of classical trajectory $x_{c}=x_{c}(h,\theta)$, i.e. just $h$ and $\theta$ are the objects of quantization in the considered example. In another words, the quantum deformations of the invariant hypersurface, $(h,\theta)$, is described in the new quantum theory. This possibility is the consequence of $\delta$-likeness of measure, i.e. it based on the conservation of total probability. Dirac measure allows to perform classical transformations of the measure and to use high resources of classical mechanics. For example, the interesting possibility may arise in connection with Kolmogorov-Arnold-Mozer (KAM) theorem [4]: the system which is not strictly integrable can show the stable motion peculiar to integrable systems. This is the argument in favor of the idea that there may be another, non-topological, mechanism of suppression of the quantum excitations. 2\. One can note that the transformed perturbation theory describes only the retarded quantum fluctuations, see definition of Green function (3\. 16). This feature of the theory can lead to the imaginary time irreversibility of quantum processes and it must be explained. The starting expression (2\. 58) describes the reversible in time motion since total action $S_{C_{+}(T_{+})}(x_{+})-S_{C_{-}(T_{-})}(x_{-})$ is time reversible. But the unitarity condition forced us to consider the interference picture between expanding and converging waves. This is fixed by the boundary conditions $e(0)=e(T)=0$. The quantum theory remain time reversible up to canonical transformation to the invariant hypersurface of the constant energy. The causal Green function $G(t,t^{\prime})$ , see (2\. 80), is able to describe both advanced and retarded perturbations and the theory contains the doubling of degrees of freedom. It means that the theory ”keeps in mind” the time reversibility. But after the canonical transformation, using above mentioned boundary conditions, and continuing the theory to the real time, the quantum perturbations were transferred on the inner degrees of freedom of classical trajectory. In result the memory of doubling of the degrees of freedom was disappeared and the theory becomes ”time irreversible”. The key step in this calculations was an extraction of the classical trajectory $x_{c}$ which can not be defined without definition of boundary conditions. Just $x_{c}$ introduces the direction of motion and the order of quantum perturbations of trajectories inner degrees of freedom play no role, i.e. the mechanical motion is time reversible while the corrections to energy of trajectory, $h$, and to the phase, $\theta$, can not be time reversible. Therefore, the considered irreversibility of the quantum mechanics in terms of $(h,\theta)$ seems to be imaginary. ## 4 Reduction of quantum degrees of freedom ### 4.1 Introduction It will be shown in this Section that the quantum fluctuations of angular variables may be removed if the classical motion is periodic. This cancelation mechanism can be used for path-integral explanation of integrability of the quantum-mechanical problems, for example of H-atom problem where the classical trajectories is closed independently from the initial conditions222222The approach may be extended on the case of rigid rotator problem [20]. Last one is isomorphic to the Pocshle-Teller problem [21]. The main result of present Section is based on the statement that the topology properties of classical trajectory takes special significance232323Since the action of perturbations generating operator of transformedf theory, $\hat{\mathbb{K}}$, maps quantum corrections on the boundaries of cotangent foliation, $\partial W$, see (4\. 41).. Our technical problem consist in necessity to extract the quantum angular degrees of freedom. For this purpose we will define path integral in the phase space of action-angle variables. For simplicity the effect of cancelations we will demonstrate on the one-dimensional $\lambda x^{4}$ model. In the following subsection the brief description of unitary definition of the path- integral measure will be given. The perturbation theory in terms of action- angle variables will be contracted in Sec.4.3 (the scheme of transformed perturbation theory was given firstly in [1]). In Sec.4.4 the cancelation mechanism will be demonstrated. ### 4.2 Unitary definition of the path-integral measure We will calculate the probability $R(E)=\int dx_{1}dx_{2}|A(x_{1},x_{2};E)|^{2},$ (4\. 1) to introduce the unitary definition of path-integral measure [1]. Here $A(x_{1},x_{2};E)=i\int^{\infty}_{0}dTe^{iET}\int_{x(0)=x_{1}}^{x(T)=x_{2}}Dxe^{iS_{C_{+}(T)}(x)}$ (4\. 2) is the amplitude of the particle with energy $E$ moving from $x_{1}$ to $x_{2}$. The action $S_{C_{+}(T)}(x)=\int_{C_{+}(T)}dt(\frac{1}{2}\dot{x}^{2}-\frac{\omega_{0}^{2}}{2}x^{2}-\frac{\lambda}{4}x^{4})$ (4\. 3) is defined on the Mills’ contour [17]: $C_{\pm}(T):t\rightarrow t\pm i\epsilon,\;\;\;\epsilon\rightarrow+0,\;\;\;0\leq t\leq T.$ (4\. 4) So, we will omit the calculation of the amplitude. Inserting (4\. 2) into (4\. 1) we find, see previous Section, that $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}\int Dxe^{-i\tilde{H}(x;\tau)-iU_{T}(x,e)}\times$ $\times\delta(E+\omega- H_{T}(x))\prod_{t}\delta(\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}-j).$ (4\. 5) The ”hat” symbol means differentiation over corresponding auxiliary quantity. For instance, $\hat{\omega}\equiv\frac{\partial}{\partial\omega},~{}~{}~{}\hat{j}(t)=\frac{\delta}{\delta j(t)}.$ (4\. 6) It will be assumed that $\hat{j}(t\in C_{\pm})j(t^{\prime}\in C_{\pm})=\delta(t-t^{\prime}),$ $\hat{j}(t\in C_{\pm})j(t^{\prime}\in C_{\mp})=0.$ (4\. 7) The time integral over contour $C^{(\pm)}(T)$ means that $\int_{C^{(\pm)}(T)}=\int_{C_{+}(T)}\pm\int_{C_{-}(T)}.$ (4\. 8) At the end of calculations the limit $(\omega,\tau,j,e)=0$ must be calculated. The explicit form of $\tilde{H}(x;\tau)$ $U_{T}(x,e)$ will be given later; $H_{T}(x)$ is the Hamiltonian at the time moment $t=T$. The functional $\delta$-function unambiguously determines the contributions in the path integral. For this purpose we must find the strict solution $x_{j}(t)$ of the equation of motion: $\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}-j=0,$ (4\. 9) expanding it over $j$. In zero order over $j$ we have the classical trajectory $x_{c}$ which is defined by the equation of motion: $\ddot{x}+\omega_{0}^{2}x+\lambda x^{3}=0.$ (4\. 10) This equation is equivalent to the following one: $t+\theta_{0}=\int^{x}dx\\{2(h_{0}-\omega_{0}^{2}x^{2}-\lambda x^{4})\\}^{-1/2}.$ (4\. 11) The solution of this equation is the periodic elliptic function. Here $(h_{0},\theta_{0})$ are the constants of integration of Eq.(4\. 10), i.e. $(h_{0},\theta_{0})$ are the coordinates of point on the surface defined by elliptic function. The integration over $(h_{0},\theta_{0})$ is assumed since the integration over all trajectories in (4\. 2) must be performed, i.e. $(h_{0},\theta_{0})$ takes on all values available by elliptic function. Let $W$ be the corresponding manyfold. One can say therefore that classical trajectory belongs $W$ completely. The mapping of our problem on the action-angle phase space will be performed using representation (4\. 5) [22]. Using the obvious definition of the action: $I=\frac{1}{2\pi}\oint\\{2(h-\omega_{0}^{2}x^{2}-\lambda x^{4})\\}^{1/2},$ (4\. 12) and of the angle $\phi=\frac{\partial h}{\partial I}\int^{x_{c}}\\{2(h-\omega_{0}^{2}x^{2}-\lambda x^{4})\\}^{-1/2}$ (4\. 13) variables [12] we easily find from (4\. 5) that $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}\int DID\phi e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e)}\times$ $\times\delta(E+\omega-h_{T}(I))\prod_{t}\delta(\dot{I}-j\frac{\partial x_{c}}{\partial\phi})\delta(\dot{\phi}-\Omega(I)+j\frac{\partial x_{c}}{\partial I}),$ (4\. 14) where $x_{c}=x_{c}(I,\phi)$ is the solution of Eq.(4\. 13) with $h=h(I)$ as the solution of Eq.(4\. 12) and the frequency $\Omega(I)=\frac{\partial h}{\partial I}.$ (4\. 15) Representation (4\. 14) is not the full solution of our problem: the action and angle variables are still interdependent since they both are exited by the same source $j(t)$. This reflects the Lagrange nature of the path-integral description of phase-space motion. The true Hamilton’s description must contain independent quantum sources of action and angle variables. ### 4.3 Perturbation theory on the cotangent manifold The structure of source terms, $j\partial x_{c}/\partial\phi$ and $j\partial x_{c}/\partial I$, show that the source of quantum fluctuations is the classical trajectories perturbation and $j$ is the auxiliary variable. It allows to regroup the perturbation series in a following manner. Let us consider the action of the perturbation-generating operators on $\delta$-functions: $e^{-i\int_{C^{(+)}(T)}dt\hat{j}(t)\hat{e}(t)}e^{-iU_{T}(x,e)}\prod_{t}\delta\left(\dot{I}+j\frac{\partial x_{c}}{\partial\phi}\right)\delta\left(\dot{\phi}-\Omega(I)-j\frac{\partial x_{c}}{\partial I}\right)=$ $=\int D_{C^{(+)}}e_{I}D_{C^{(+)}}e_{\phi}e^{i\int_{C^{(+)}}dt(e_{I}\dot{I}+e_{\phi}(\dot{\phi}-\Omega(I)))}e^{-iU_{T}(x,e_{c})},$ (4\. 16) where $e_{c}(e_{I},e_{\phi})=e_{I}\frac{\partial x_{c}}{\partial\phi}-e_{\phi}\frac{\partial x_{c}}{\partial I}.$ (4\. 17) The integrals over $(e_{I},e_{\phi})$ will be calculated perturbatively: $e^{-iU_{T}(x,e_{c})}=\sum^{\infty}_{n_{I},n_{\phi}=0}\frac{1}{n_{I}!n_{\phi}!}\int\prod^{n_{I}}_{k=1}(dt_{k}e_{I}(t_{k}))\prod^{n_{\phi}}_{k=1}(dt^{\prime}_{k}e_{\phi}(t^{\prime}_{k}))\times$ $\times P_{n_{I},n_{\phi}}(x_{c},t_{1},...,t_{n_{I}},t^{\prime}_{1},...,t_{n_{\phi}}),$ (4\. 18) where $P_{n_{I},n_{\phi}}(x_{c},t_{1},...,t_{n_{I}},t^{\prime}_{1},...,t_{n_{\phi}})=\prod^{n_{I}}_{k=1}\hat{e}^{\prime}_{I}(t_{k})\prod^{n_{\phi}}_{k=1}\hat{e}^{\prime}_{\phi}(t^{\prime}_{k})e^{-iU_{T}(x,e^{\prime}_{c})},$ (4\. 19) where $e^{\prime}_{c}\equiv e_{c}(e^{\prime}_{I},e^{\prime}_{\phi})$ and the derivatives in (4\. 19) are calculated at $e^{\prime}_{I}=0$, $e^{\prime}_{\phi}=0$. At the same time, $\prod^{n_{I}}_{k=1}e_{I}(t_{k})\prod^{n_{\phi}}_{k=1}e_{\phi}(t^{\prime}_{k})=\prod^{n_{I}}_{k=1}(i\hat{j}_{I}(t_{k}))\prod^{n_{\phi}}_{k=1}(i\hat{j}_{\phi}(t^{\prime}_{k}))e^{-i\int_{C^{(+)}}dt(j_{I}(t)e_{I}(t)+j_{\phi}(t)e_{\phi}(t))}.$ (4\. 20) The limit $(j_{I},j_{\phi})=0$ is assumed. Inserting (4\. 19), (4\. 20) into (4\. 16) we will find new representation for $R(E)$: $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}_{I}(t)\hat{e}_{I}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$ $\times\int DID\phi e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\times$ $\times\delta(E+\omega- h_{T}(I))\prod_{t}\delta(\dot{I}-j_{I})\delta(\dot{\phi}-\Omega(I)-j_{\phi}),$ (4\. 21) in which the action and the angle are the decoupled degrees of freedom. Solving the canonical equations of motion: $\dot{I}=j_{I},\;\;\;\dot{\phi}=\Omega(I)+j_{\phi}$ (4\. 22) the boundary conditions: $I_{j}(0)=I_{0},\;\;\;\phi_{j}(0)=\phi_{0}$ (4\. 23) will be used. This will lead to the following Green function: $g(t-t^{\prime})=\Theta(t-t^{\prime}),$ (4\. 24) with boundary condition: $\Theta(0)=1$. The solutions of eqs.(4\. 22) have the form: $\displaystyle I_{j}(t)=I_{0}+\int dt^{\prime}g(t-t^{\prime})j_{I}(t^{\prime})\equiv I_{0}+I^{\prime}(t),$ $\displaystyle\phi_{j}(t)=\phi_{0}+\tilde{\Omega}(I_{j})t+\int dt^{\prime}g(t-t^{\prime})j_{\phi}(t^{\prime})\equiv\phi_{0}+\tilde{\Omega}(I_{0}+I^{\prime})t+\phi^{\prime}(t),$ (4\. 25) where $\tilde{\Omega}(I_{j})=\frac{1}{t}\int dt^{\prime}g(t-t^{\prime})\Omega(I_{0}+I^{\prime}(t^{\prime})).$ (4\. 26) Inserting (4\. 25) into (4\. 21) we find: $R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dt(\hat{j}{{}_{I}}(t)\hat{e}_{I}(t)+\hat{j}_{\phi}(t)\hat{e}_{\phi}(t))}\times$ $\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega- h_{T}(I_{j})),$ (4\. 27) where $x_{c}=x_{c}(I_{j},\phi_{j})=x_{c}(I_{0}+I(t),\phi_{0}+\tilde{\Omega}(I_{0}+I)t+\phi(t))$ (4\. 28) and $e_{c}$ was defined in (4\. 17). Note that the measure of the integrals over $(I_{0},\phi_{0})$ was defined without of the Faddeev-Popov’s ansatz and there is not any “hosts” since the Jacobian of transformation is equal to one. We can extract the Green function into the perturbation-generating operator using the equalities: $\hat{j}_{I}(t)=\int dt^{\prime}g(t-t^{\prime})\hat{I}(t),\hat{j}_{\phi}=\int dt^{\prime}g(t-t^{\prime})\hat{\phi}(t),$ (4\. 29) which evidently follows from (4\. 25). In result, $R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{C^{(+)}(T)}dtdt^{\prime}g(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$ $\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega- h_{T}(I_{0}+I)),$ (4\. 30) where $x_{c}$ was defined in (4\. 28). We can define the formalism without doubling of the degrees of freedom. One can use fact that the action of perturbation-generating operators and the analytical continuation to the real times are commuting operations. This can be seen easily using the definition (4\. 7). In result the expression: $R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}\hat{\omega}\hat{\tau}-i\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$ $\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}d\phi_{0}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}\delta(E+\omega- h_{T}(I_{0}+I(T)),$ (4\. 31) where $\tilde{H}_{T}(x_{c};\tau)=2\sum^{\infty}_{n=1}\frac{\tau^{2n+1}}{(2n+1)!}\frac{d^{2n}}{dT^{2n}}h(I_{0}+I(T))$ (4\. 32) and $-U_{T}(x_{c},e_{c})=S(x_{c}+e_{c})-S(x_{c}-e_{c})-2\int_{0}^{T}dte_{c}\frac{\delta S(x_{c})}{\delta x_{c}}$ (4\. 33) defines quantum theory on the cotangent manifold $W$. Now we can use the last $\delta$-function: $R(E)=2\pi\int^{\infty}_{0}dTe^{\\{\frac{1}{2i}(\hat{\omega}\hat{\tau}+\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))\\}}\times$ $\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E+\omega)}e^{-i\tilde{H}(x_{c};\tau)-iU_{T}(x_{c},e_{c})}.$ (4\. 34) Here $x_{c}(t)=x_{c}(I_{0}(E+\omega)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$ (4\. 35) Eq.(4\. 34) contains unnecessary contributions: the action of the operator $\int^{T}_{0}dtdt^{\prime}\Theta(t-t^{\prime})\hat{e}_{I}(t)\hat{I}(t^{\prime})$ (4\. 36) on $\tilde{H}_{T}$, defined in (4\. 32), leads to the time integrals with zero integration range: $\int^{T}_{0}dt\Theta(T-t)\Theta(t-T)=0.$ Using this fact, $\displaystyle R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})+\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime}))}\times$ $\displaystyle\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}e^{-iU_{T}(x_{c},e_{c})},$ (4\. 37) where $x_{c}(t)=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$ (4\. 38) is the periodic function: $x_{c}(I_{0}(E)+I(t)-I(T),(\phi_{0}+2\pi)+\tilde{\Omega}t+\phi(t))=$ $=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t+\phi(t)).$ (4\. 39) Now we can consider the cancelation of angular perturbations. ### 4.4 Cancelation of angular perturbations 1\. Simples example Introducing the perturbation-generating operator into the integral over $\phi_{0}$: $\displaystyle R(E)=2\pi\int^{\infty}_{0}dTe^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{I}(t)\hat{e}_{I}(t^{\prime})}\times$ $\displaystyle\times\int^{\infty}_{0}dI_{0}\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime})}e^{-iU_{T}(x_{c},e_{c})},$ (4\. 40) the mechanism of cancelations of the angular perturbations becomes evident. One can formulate the statement: (i) if $e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)\hat{\phi}(t)\hat{e}_{\phi}(t^{\prime})}e^{-iU_{T}(x_{c},e_{c})}=e^{-iU_{T}(x_{c},e_{c})}|_{e_{\phi}=\phi=0}+dF(\phi_{0})/d\phi_{0},$ (4\. 41) and (ii) if $F(\phi_{0}+2\pi)=F(\phi_{0}),$ (4\. 42) then: $R(E)=2\pi\int^{2\pi}_{0}\frac{d\phi_{0}}{\Omega(E)}\int^{\infty}_{0}dTdI_{0}e^{\frac{1}{2i}\int_{0}^{T}dtdt^{\prime}\Theta(t^{\prime}-t)(\hat{I}(t)\hat{e}_{I}(t^{\prime})}$ $\times e^{S(x_{c}+e\partial x_{c}/\partial\phi_{0})-S(x_{c}-e\partial x_{c}/\partial\phi_{0})},$ (4\. 43) i.e. we find the expression in which the angular corrections was canceled. In this case the problem becomes semiclassical over the angular degrees of freedom. For the $(\lambda x^{4})_{1}$-model $S(x_{c}+e\partial x_{c}/\partial\phi_{0})-S(x_{c}-e\partial x_{c}/\partial\phi_{0})=S_{0}(x_{c})-2\lambda\int^{T}_{0}dtx_{c}(t)\\{e\partial x_{c}/\partial\phi_{0}\\}^{3},$ (4\. 44) where [1] $S_{0}(x_{c})=\oint_{T}dt\left(\frac{1}{2}\dot{x}_{c}^{2}-\frac{\omega_{0}^{2}}{2}x_{c}^{2}-\frac{\lambda}{4}x_{c}^{4}\right)$ (4\. 45) is the closed time-path action and $x_{c}(t)=x_{c}(I_{0}(E)+I(t)-I(T),\phi_{0}+\tilde{\Omega}t).$ (4\. 46) Here $I(t)$ and $I(T)$ are the auxiliary variables. The condition (4\. 42) requires that the classical trajectory $x_{c}$ with all derivatives over $I_{0}$, $\phi_{0}$ is the periodic function. In the considered case of $(\lambda x^{4})_{1}$-model $x_{c}$ is periodic function with period $1/\Omega$, see (4\. 39). Therefore, we can concentrate the attention on the condition (4\. 41) only. Expanding $F(\phi_{0})$ over $\lambda$: $F(\phi_{0})=\lambda F_{1}(\phi_{0})+\lambda^{2}F_{2}(\phi_{0})+...$ (4\. 47) we find that $\frac{d}{d\phi_{0}}F_{1}(\phi_{0})=$ $=\int^{T}_{0}\prod^{3}_{k=1}dt^{\prime}_{k}\hat{\phi}(t^{\prime}_{k})\left(\left(-\frac{6}{(2i)^{3}}\right)\int^{T}_{0}dt\prod^{3}_{k=1}\Theta(t-t^{\prime}_{k})x_{c}(t)(\partial x_{c}/\partial I_{0})^{3}e^{iS_{0}(x_{c})}_{k}\right)=$ $=\int^{T}_{0}dt^{\prime}\hat{\phi}(t^{\prime})B_{1}(\phi),$ (4\. 48) where $B_{1}(\phi)=\left\\{-\frac{6}{(2i)^{3}}\int^{T}_{0}dt\Theta(t-t^{\prime})\right.\times$ $\times\left.\prod^{2}_{k=1}(\Theta(t-t^{\prime}_{k})\hat{\phi}(t^{\prime}_{k}))x_{c}(t)(\partial x_{c}/\partial I_{0})^{3}e^{iS_{0}(x_{c})}\right\\}$ (4\. 49) This example shows that the sum over all powers of $\lambda$ can be written in the form: $\frac{d}{d\phi_{0}}F(\phi_{0})=\int^{T}_{0}dt^{\prime}\hat{\phi}(t^{\prime})B(\phi),$ (4\. 50) where, using the definition (4\. 35), $B(\phi)=\int^{T}_{0}dt\tilde{B}(\phi_{0}+\phi(t)).$ (4\. 51) Therefore, $\hat{\phi}(t^{\prime})B(\phi)=\frac{d}{d\phi_{0}}\int^{T}_{0}dt\delta(t-t^{\prime})\tilde{B}(\phi_{0}+\phi(t))$ (4\. 52) coincides with the total derivative over initial phase $\phi_{0}$, and $F(\phi_{0})=\tilde{B}(\phi_{0}+\phi(t))|_{\phi=0}.$ (4\. 53) This result ends the prove of (4\. 41). 2\. General case Now we will offer following important statement: — each order of perturbation theory in the invariant subspace can be represented as the sum of total derivative over the subspace coordinate. This statement directly follows from structure of perturbations generating operator $\hat{\mathbb{K}}$ and the assumption (3\. 18). It explains the statement, offered in Preface. Let us remind that integration with last $\delta$-function gives the result of action of operator $\hat{\mathbb{K}}$ written in the form: $R(E)=2\pi\int_{0}^{\infty}dT\int_{0}^{2\pi}\frac{d\varphi_{0}}{\Omega(E)}:e^{-iU_{(}x_{c},\hat{e}/2i)}:,$ (4\. 54) where the colons mean normal product, $\hat{e}=\hat{j}_{\varphi}\frac{\partial x_{c}}{\partial I}-\hat{j}_{I}\frac{\partial x_{c}}{\partial\varphi},$ (4\. 55) and by definition $U_{T}$ is the odd over $\hat{e}_{c}$ functional: $U_{T}(x_{c},e_{c})=2\int_{0}^{T}\sum_{n=1}(\hat{e}_{c}(t)/2i)^{2n+1}u_{n}(x_{c}),$ (4\. 56) where $u_{n}$ is the function of only $x_{c}$ at the time $t$. Inserting (4\. 55) one can write: $:e^{-iU_{(}x_{c},\hat{e}/2i)}:=\prod_{n=1}^{\infty}\prod_{k=0}^{2n+1}:e^{-iU_{k,n}(j,x_{c})}:,$ (4\. 57) where $U_{k,n}(j,x_{c})=\int_{0}^{T}dt(\hat{j}_{\varphi}(t))^{2n-k+1}(\hat{j}_{I}(t))^{k}b_{k,n}(x_{c}(t))$ (4\. 58) and the explicit form of $b_{k,n}(x_{c})$ is not important. Using the evident definition: $\hat{j}_{X}=\int_{0}^{T}dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime}),~{}~{}X=\varphi,I,$ it is easy to find that $j_{X}(t_{1})b_{k,n}(x_{c}(t_{2}))=\Theta(t_{1}-t_{2})\partial b_{k,n}(x_{c}(t_{2}))/\partial X_{0},$ since $x_{c}=x_{c}(X+X_{0})$, or shortly: $j_{1}b_{2}=\Theta_{12}\partial_{X_{0}}b_{2}=\partial_{X_{0}}(\Theta_{12}b_{2})$ (4\. 59) since the indexes $(k,n)$ are not important. Let us start consideration from the first term with $k=0$. In this case we describe only the angular fluctuations. Noting that $\partial_{X_{0}}$ and $\hat{j}$ commute we can consider the lowest order over $\hat{j}$. The typical term looks as follows (omitting the index $X_{0}$): $\hat{j}_{1}\hat{j}_{2}\cdots\hat{j}_{m}b_{1}b_{2}\cdots b_{m}.$ It is sufficient to show that this expression is the total derivative over $X_{0}$. Case $m=1$. In this approximation we have, see (4\. 59): $\hat{j}_{1}b_{1}=\Theta_{11}\partial_{0}b_{1}\neq 0.$ (4\. 60) Here (3\. 18) was used. Case $m=2$. This order is less trivial: $\hat{j}_{1}\hat{j}_{2}b_{1}b_{2}=\Theta_{21}b_{1}^{2}b_{2}+b_{1}^{1}b_{2}^{1}+\Theta_{12}b_{1}b_{2}^{2},$ (4\. 61) where $b_{i}^{n}\equiv\partial^{n}b_{i}.$ (4\. 62) At first glance (4\. 61) is not the total derivative. But inserting $1=\Theta_{12}+\Theta_{21}$ we can symmetrize it: $\hat{j}_{1}\hat{j}_{2}b_{1}b_{2}=\Theta_{21}(b_{1}^{2}b_{2}+b_{1}^{1}b_{2}^{1})+\Theta_{12}(b_{1}b_{2}^{2}+b_{1}^{1}b_{2}^{1})=$ $=\partial_{0}(\Theta_{21}b_{1}^{1}b_{2}+\Theta_{12}b_{1}b_{2}^{1})\equiv$ $\equiv\partial_{0}(b_{1}^{1}\rightarrow b_{2}+b_{2}^{1}\rightarrow b_{1})$ (4\. 63) since the explicit form of the function is not important. Therefore, the second order term can be also reduced to the total derivative. Notice that (4\. 63) shows time reversibility. Case $m=3$. In this order one can find that $\hat{j}_{1}\hat{j}_{2}\hat{j}_{3}b_{1}b_{2}b_{3}=\partial_{0}\left\\{\sum_{i\neq j\neq k=1}^{3}(i^{2}\rightarrow j\rightarrow k+i^{1}\rightarrow j^{1}\rightarrow k)\right\\}$ (4\. 64) The $m$-th order contribution is also total derivative: $\hat{j}_{1}\hat{j}_{2}\cdots\hat{j}_{m}b_{1}b_{2}\cdots b_{m}=\partial_{0}\\{\sum_{i_{1}\neq i_{2}\neq i_{3}\neq\cdots\neq i_{m}=1}^{m}(i_{1}^{m}\rightarrow i_{2}\rightarrow i_{3}\rightarrow\cdots\rightarrow i_{m}+$ $+i_{1}^{m-1}\rightarrow i_{2}^{1}\rightarrow i_{3}\rightarrow\cdots\rightarrow i_{m}+i_{1}^{m-2}\rightarrow i_{2}^{1}\rightarrow i_{3}^{1}\rightarrow\cdots\rightarrow i_{m}+\cdots$ $\cdots+i_{1}^{1}\rightarrow i_{2}^{1}\rightarrow i_{3}^{1}\rightarrow\cdots\rightarrow i_{m-1}^{1}\rightarrow i_{m})\\}$ (4\. 65) Let us consider now the case with $k\neq 0$. The typical term looks as follows: $\hat{j}_{1}^{1}\hat{j}_{2}^{1}\cdots\hat{j}_{l}^{1}\hat{j}_{l+1}^{2}\hat{j}_{l+2}^{2}\cdots\hat{j}_{m}^{2}b_{1}b_{2}\cdots b_{m},~{}0<l<m,$ (4\. 66) where, for instance $\hat{j}_{k}^{1}\equiv\hat{j}_{I}(t_{k}),~{}~{}\hat{j}_{k}^{2}\equiv\hat{j}_{\varphi}(t_{k})$ (4\. 67) and $\hat{j}_{1}^{i}b_{2}=\Theta_{12}\partial_{0}^{i}b_{2}.$ (4\. 68) Case $m=2,l=1$.In this case: $\hat{j}_{1}^{1}\hat{j}_{2}^{2}b_{1}b_{2}=\Theta_{21}(b_{2}\partial_{0}^{1}\partial_{0}^{2}b_{1}+(\partial_{0}^{2}b_{2})(\partial_{0}^{1}\partial_{0}^{2}b_{1}))+\Theta_{12}(b_{1}\partial_{0}^{1}\partial_{0}^{2}b_{2}+(\partial_{0}^{2}b_{2})(\partial_{0}^{1}\partial_{0}^{2}b_{1}))=$ $=\partial_{0}^{1}(\Theta_{21}b_{2}\partial_{0}^{2}b_{1}+\Theta_{12}b_{1}\partial_{0}^{2}b_{2})+\partial_{0}^{2}(\Theta_{21}b_{2}\partial_{0}^{1}b_{1}+\Theta_{12}b_{1}\partial_{0}^{1}b_{2}).$ (4\. 69) Therefore we have the total-derivative structure yet. This property is conserved in arbitrary order over $m$ and $l$ since the time-ordered structure does not depends from upper index of $\hat{j}$, see (4\. 68). One can conclude that the contribution are defined by topology properties of classical trajectory $x_{c}$. We will see that this important property of perturbation theory remains unchanged also for field theories with symmetry. ### 4.5 Conclusions 1\. It was shown that the real-time quantum problem can be semiclassical over the part of the degrees of freedom and quantum over another ones. Following to the result of this Section one may introduce the (probably naive) interpretation of the quantum systems integrability (we suppose that the classical system is integrable and can be mapped on the compact hypersurface in the phase space [12]): the quantum system is strictly integrable in result of cancelation of all quantum degrees of freedom. The mechanism of cancelation of the quantum corrections is varied from case to case. For some problems (as the rigid rotator, or the Pocshle-Teller) the cancelation of angular degrees of the freedom is enough since they carry only the angular ones. In an another case (as in the Coulomb problem, or in the one-dimensional models) the problem may be partly integrable since the quantum fluctuations of action degrees of freedom just survive. Theirs absence in the Coulomb problem needs special discussion (one must take into account the dynamical (hidden) symmetry of Coulomb problem [23]). The transformation to the action-angle variables maps the $N$-dimensional Lagrange problem on the $2N$-dimensional phase-space torus. If the winding number on this hypertorus is a constant (i.e. the topological charge is conserved) one can expect the same cancelations. This is important for the field-theoretical problems (for instance, for sin-Gordon model [24]). 2\. In the classical mechanics following approximated method of calculations is used [12]. The canonical equations of motion: $\dot{I}=a(I,\phi),~{}~{}\dot{\phi}=b(I,\phi)$ (4\. 70) are changed on the averaged equations: $\dot{J}=\frac{1}{2\pi}\int^{2\pi}_{0}d\phi a(J,\phi),~{}~{}\dot{\phi}=b(J,\phi),$ (4\. 71) It is possible if the oscillations can be extracted from the systematic evolution of the degrees of freedoms. In our case $a(I,\phi)=j\partial x_{c}/\partial\phi,~{}~{}b(I,\phi)=\Omega(I)-j\partial x_{c}/\partial I.$ (4\. 72) Inserting this definitions into (4\. 71) we find evidently wrong result since in this approximation the problem looks like pure semiclassical for the case of periodic motion: $\dot{J}=0,~{}~{}\dot{\phi}=\Omega(J).$ (4\. 73) The result of this Section was used here. This shows that the procedure of extraction of the oscillations from the systematic evolution is not trivial and this method should be used carefully in the quantum theories. (This approximation of dynamics is ”good” on the time intervals $\sim 1/|a|$ [12].) ## 5 Example: H-atom ### 5.1 Introduction The mapping $J:T\rightarrow W,$ (5\. 1) where $T$ is the $2N$-dimensional phase space and $W$ is a linear space solves the mechanical problem iff $J=\otimes^{N}_{1}J_{i},$ (5\. 2) where $J_{i}$ are the first integrals in involution, see e.g. [12]242424The formalism of reduction (5\. 1) in classical mechanics is described also in [25]. The aim of this Section is to adopt this procedure for H-atom. The mapping (5\. 1) introduces integral $manifold$ $J_{\omega}=J^{-1}(\omega)$ in such a way that the $classical$ phase space flaw belongs to $J_{\omega}$ $completely$. We wish quantize the $J_{\omega}$ manifold instead of flow in $T$ noting that the quantum trajectory also should belong to $J_{\omega}$ completely. This important conclusion was demonstrated in previous Section by transformation of the path-integral measure to the canonical variables $(\xi,\eta)$. New perturbation theory is extremely simple since $W$ is the linear space. The ”direct” mapping (5\. 1) used in [26] assumes that $J$ is known. But it seems inconvenient having in mind the general problem of nonlinear waves quantization, when the number of degrees of freedom $N=\infty$, or if the transformation is not canonical. We will consider by this reason the ”inverse” approach assuming that just the classical flow is known. Then, since the flow belongs to $J_{\omega}$ completely [26], we would be able to find the quantum motion in $W$. It is the main technical result illustrated in this Section. The manifold $J_{\omega}$ is invariant relatively to some subgroup $G_{\omega}$ [27] in accordance to topological class of classical flaw. This introduces the $J_{\omega}$ classification and summation over all (homotopic) classes should be performed. Note, the classes are separated by the boundary bifurcation lines in $W$ [27]. If the quantum perturbations switched on adiabatically then the homotopic group should stay unbroken. It is the ordinary statement for quantum mechanics, but, generally speaking, this is not true for field theories. We will calculate the bound state energies in the Coulomb potential252525 We will restrict ourselves by the plane problem. Corresponding phase space $T=(p,l,r,\varphi)$ is 4-dimensional.. This popular problem was considered by many authors, using various methods, see e.g. [23]. The path-integral solution of this problem was offered firstly in [28]. The classical flaw of this problem can be parameterized by the angular momentum $l$, corresponding angle $\varphi$ and by the normalized on total Hamiltonian Runge-Lentz vector length $n$. So, we will consider the mapping ($p$ is the conjugate to $r$ radial momentum in the cylindrical coordinates): $J_{l,n}:(p,l,r,\varphi)\rightarrow(l,n,\varphi)$ (5\. 3) to construct the perturbation theory in the $W=(l,n,\varphi)$ space. I.e. $W$ is not considered as the cotangent foliation on $T$. The mapping (5\. 3) assumes additional reduction of the four-dimensional incident phase space up to three-dimensional linear subspace262626$W$ would not have the simplectic structure. Actually in considered case $W=R+TW$, where $R$ is the zero-modes space and $TW$ is the simplectic subspace.. Just this reduction phenomena leads to corresponding stability of $n$ concerning quantum perturbations and will allow to solve our H-atom problem completely272727 In other words, we would demonstrate that the hidden Bargman-Fock [23] $O(4)$ symmetry is stay unbroken concerning quantum perturbations.. In Subsec. 5.2 we will show how the mapping (5\. 3) can be performed for path- integral differential measure. In Subsec. 5.3 the consequence of reduction will be derived and in Subsec. 5.4 the perturbation theory in the $W$ space will be analyzed. The calculations are based on the formalism offered in previous Sections. ### 5.2 Mapping We will calculate the integral [26]: $\rho(E)=\int^{\infty}_{0}dTe^{-i{\hat{\mathbb{K}}}(j,e)}\int DM(p,l,r,\varphi)e^{-iU(r,e)},$ (5\. 4) where $\rho(E)$ is the $probability$ to find a particle with energy $E$, i.e. we should find [22] that normalized on the zero-modes volume $\rho(E)=\pi\sum_{n}\delta(E-E_{n}),$ (5\. 5) where $E_{n}$ are the bound states energies. For $H$-atom problem $E_{n}\leq 0$. This condition will define considered homotopy class. Expansion over operator ${\hat{\mathbb{K}}}(j,e)=\frac{1}{2}\int^{T}_{0}dt(\hat{j}_{r}\hat{e}_{r}+\hat{j}_{\varphi}\hat{e}_{\varphi}),~{}~{}~{}\hat{X}(t)\equiv\delta/\delta X(t),$ (5\. 6) generates the perturbation series. It will be seen that in our case we may omit the question of perturbation theories convergence. The differential measure $DM(p,l,r,\varphi)=\delta(E-H_{0})\prod_{t}dr(t)dp(t)dl(t)d\varphi(t)\times$ $\delta\left(\dot{r}-\frac{\partial H_{j}}{\partial p}\right)\delta\left(\dot{p}+\frac{\partial H_{j}}{\partial r}\right)\delta\left(\dot{\varphi}-\frac{\partial H_{j}}{\partial l}\right)\delta\left(\dot{l}+\frac{\partial H_{j}}{\partial\varphi}\right),$ (5\. 7) with total Hamiltonian ($H_{0}=H_{j}|_{j=0}$) $H_{j}=\frac{1}{2}p^{2}-\frac{l^{2}}{2r^{2}}-\frac{1}{r}-j_{r}r-j_{\varphi}\varphi$ (5\. 8) allows perform arbitrary transformation of variables because of its $\delta$-likeness. Notice that $H_{j}$ contains only the ”Lagrange forces” $j_{r}$ and $j_{\varphi}$. The functional $U(r,e)=-s_{0}(r)+$ $+\int^{T}_{0}dt\left[\frac{1}{((r+e_{r})^{2}+r^{2}e_{\varphi}^{2})^{1/2}}-\frac{1}{((r-e_{r})^{2}+r^{2}e_{\varphi}^{2})^{1/2}}+2\frac{e_{r}}{r}\right]$ (5\. 9) describes the interaction between various quantum modes and $s_{0}(r)$ defines the non-integrable phase factor [22]. The quantization of this factor determines the bound state energy. Such factor will appear if the phase of amplitude can not be fixed 282828As, for instance, in the Aharonov-Bohm case.. Note that the Hamiltonian (5\. 8) contains the energy of radial $j_{r}r$ and angular $j_{\varphi}\varphi$ excitation independently. Let us introduce the functional $\Delta=\int\prod_{t}d^{2}\xi d^{2}\eta\times$ $\times\delta(r(t)-r_{c}(\xi,\eta))\delta(p(t)-p_{c}(\xi,\eta))\delta(l(t)-l_{c}(\xi,\eta))\delta(\varphi(t)-\varphi_{c}(\xi,\eta))$ (5\. 10) which is defined by given functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$. If given functions $(\xi,\eta)$ zeroes argument of $\delta$-functions in (5\. 10) then it is assumed that the functional determinant $\displaystyle\Delta_{c}=\int\prod_{t}d^{2}\bar{\xi}d^{2}\bar{\eta}\delta\left(\frac{\partial r_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial r_{c}}{\partial\eta}\cdot\bar{\eta}\right)\delta\left(\frac{\partial p_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial p_{c}}{\partial\eta}\cdot\bar{\eta}\right)\times$ $\displaystyle\times\delta\left(\frac{\partial\varphi_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial\varphi_{c}}{\partial\eta}\cdot\bar{\eta}\right)\delta\left(\frac{\partial l_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial l_{c}}{\partial\eta}\cdot\bar{\eta}\right)\neq 0.$ (5\. 11) Note that this is the condition only for $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$. To perform the mapping we will insert $1=\Delta/\Delta_{c}$ (5\. 12) into (5\. 4) and integrate over $r(t)$, $p(t)$, $\varphi(t)$ and $l(t)$. In result we find the measure: $\displaystyle DM(\xi,\eta)=\frac{1}{\Delta_{c}}\delta(E-H_{0})\prod_{t}d^{2}\xi d^{2}\eta\delta\left(\dot{r_{c}}-\frac{\partial H_{j}}{\partial p_{c}}\right)\times$ $\displaystyle\times\delta\left(\dot{p_{c}}+\frac{\partial H_{j}}{\partial r_{c}}\right)\delta\left(\dot{\varphi_{c}}-\frac{\partial H_{j}}{\partial l_{c}}\right)\delta\left(\dot{l_{c}}+\frac{\partial H_{j}}{\partial\varphi_{c}}\right),$ (5\. 13) Note that the functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$ must obey only one condition (5\. 11). A simple algebra gives: $\displaystyle DM(\xi,\eta)=\frac{\delta(E-H_{0})}{\Delta_{c}}\prod_{t}d^{2}\xi d^{2}\eta\int\prod_{t}d^{2}\bar{\xi}d^{2}\bar{\eta}$ $\displaystyle\times\delta^{2}\left(\bar{\xi}-\left(\dot{\xi}-\frac{\partial h_{j}}{\partial\eta}\right)\right)\delta^{2}\left(\bar{\eta}-\left(\dot{\eta}+\frac{\partial h_{j}}{\partial\xi}\right)\right)$ $\displaystyle\times\delta\left(\frac{\partial r_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial r_{c}}{\partial\eta}\cdot\bar{\eta}+\\{r_{c},h_{j}\\}-\frac{\partial H_{j}}{\partial p_{c}}\right)$ $\displaystyle\times\delta\left(\frac{\partial p_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial p_{c}}{\partial\eta}\cdot\bar{\eta}+\\{p_{c},h_{j}\\}+\frac{\partial H_{j}}{\partial r_{c}}\right)$ $\displaystyle\times\delta\left(\frac{\partial\varphi_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial\varphi_{c}}{\partial\eta}\cdot\bar{\eta}+\\{\varphi_{c},h_{j}\\}-\frac{\partial H_{j}}{\partial l_{c}}\right)$ $\displaystyle\times\delta\left(\frac{\partial l_{c}}{\partial\xi}\cdot\bar{\xi}+\frac{\partial l_{c}}{\partial\eta}\cdot\bar{\eta}+\\{l_{c},h_{j}\\}+\frac{\partial H_{j}}{\partial\varphi_{c}}\right).$ (5\. 14) The Poisson notation: $\\{X,h_{j}\\}=\frac{\partial X}{\partial\xi}\frac{\partial h_{j}}{\partial\eta}-\frac{\partial X}{\partial\eta}\frac{\partial h_{j}}{\partial\xi}$ was introduced in (5\. 14). Next, the ”auxiliary” quantity $h_{j}$ have been introduced by following equalities: $\displaystyle\\{r_{c},h_{j}\\}-\frac{\partial H_{j}}{\partial p_{c}}=0,~{}\\{p_{c},h_{j}\\}+\frac{\partial H_{j}}{\partial r_{c}}=0,$ $\displaystyle\\{\varphi_{c},h_{j}\\}-\frac{\partial H_{j}}{\partial l_{c}}=0,~{}\\{l_{c},h_{j}\\}+\frac{\partial H_{j}}{\partial\varphi_{c}}=0.$ (5\. 15) Then the functional determinant $\Delta_{c}$ is canceled and $DM(\xi,\eta)=\delta(E-H_{0})\prod_{t}d^{2}\xi d^{2}\eta\delta^{2}(\dot{\xi}-\frac{\partial h_{j}}{\partial\eta})\delta^{2}(\dot{\eta}+\frac{\partial h_{j}}{\partial\xi}),$ (5\. 16) It is the desired result of transformation of the measure for given generating functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$. In this case the ”Hamiltonian” $h_{j}(\xi,\eta)$ is defined by four equations (5\. 15). But there is another possibility. Let us assume that $h_{j}(\xi,\eta)=H_{j}(r_{c},p_{c},\varphi_{c},l_{c})$ (5\. 17) and the functions $(r_{c},p_{c},\varphi_{c},l_{c})(\xi,\eta)$ are unknown. Then eqs.(5\. 15) are the equations for this functions. It is not hard to see that the eqs.(5\. 15) simultaneously with equations fixed by $\delta$-functions in (5\. 16) are equivalent of incident equations if the equality (5\. 17) is hold. Indeed, for example, $\dot{r}_{c}=\frac{\partial r_{c}}{\partial\xi}\cdot\dot{\xi}+\frac{\partial r_{c}}{\partial\eta}\cdot\dot{\eta}=\\{r_{c},h_{j}\\}=\frac{\partial H_{j}}{\partial p_{c}},$ (5\. 18) where (5\. 16) and (5\. 15) was used successively. So, incident dynamical problem was divided on two parts. First one defines the trajectory in the $W$ space through eqs.(5\. 15). Second one defines the dynamics, i.e. the time dependence, through the equations fixed by $\delta$-functions in the measure (5\. 16). Therefore, we should consider $r_{c},~{}p_{c},~{}\varphi_{c},~{}l_{c}$ as the solutions in the $\xi,~{}\eta$ parametrization. The desired parametrization of classical orbits has the form (one can find it in arbitrary textbook of classical mechanics): $r_{c}=\frac{\eta_{1}^{2}(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}}{(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}+\eta_{2}\cos\xi_{1}},~{}p_{c}=\frac{\eta_{2}\sin\xi_{1}}{\eta_{1}(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}},~{}\varphi_{c}=\xi_{1},~{}l_{c}=\eta_{1},$ (5\. 19) i.e. $r_{c}$ and $p_{c}$ are $\xi_{2}$ independent. At the same time, $h_{j}=\frac{1}{2(\eta_{1}^{2}+\eta_{2}^{2})^{1/2}}-j_{r}r_{c}-j_{\varphi}\xi_{1}\equiv h(\eta)-j_{r}r_{c}-j_{\varphi}\xi_{1}.$ (5\. 20) Noting that the derivatives of $h_{j}$ over $\xi_{2}$ are equal to zero292929To have the condition (5\. 11) we should assume that $\partial r_{c}/\partial\xi_{2}\sim\epsilon\neq 0$. We put $\epsilon=0$ completing the transformation. we find that $DM(\xi,\eta)=\delta(E-h(T))\prod_{t}d^{2}\xi d^{2}\eta\delta\left(\dot{\xi}_{1}-\omega_{1}+j_{r}\frac{r_{c}}{\partial\eta_{1}}\right)$ $\times\delta\left(\dot{\xi}_{2}-\omega_{2}+j_{r}\frac{r_{c}}{\partial\eta_{2}}\right)\delta\left(\dot{\eta}_{1}-j_{r}\frac{\partial r_{c}}{\partial\xi_{1}}-j_{\varphi}\right)\delta(\dot{\eta}_{2}),$ (5\. 21) where $\omega_{i}=\partial h/\partial\eta_{i}$ (5\. 22) are the conserved in classical limit $j_{r}=j_{\varphi}=0$ ”velocities” in the $W$ space. ### 5.3 Reduction We see from (5\. 21) that the length of Runge-Lentz vector is not perturbated by the quantum forces $j_{r}$ and $j_{\varphi}$. To investigate the consequence of this fact it is useful to project this forces on the axis of $W$ space. This means splitting of $j_{r},~{}j_{\varphi}$ on $j_{\xi},~{}j_{\eta}$. The equality $\prod_{t}\delta\left(\dot{\xi}_{1}-\omega_{1}+j_{r}\frac{r_{c}}{\partial\eta_{1}}\right)=e^{\frac{1}{2i}\int^{T}_{0}dt\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}}e^{2i\int^{T}_{0}dtj_{r}e_{\xi_{1}}\partial r_{c}/\partial\eta_{1}}\prod_{t}\delta(\dot{\xi}_{1}-\omega_{1}+j_{\xi_{1}})$ becomes evident if the Fourier representation of $\delta$-function is used (see also [26]). The same transformation of arguments of other $\delta$-functions in (5\. 21) can be applied. Then, noting that the last $\delta$-function in (5\. 21) is source-free, we find the same representation as (5\. 4) with $\hat{\mathbb{K}}(j,e)=\int^{T}_{0}dt(\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}+\hat{j}_{\xi_{2}}\hat{e}_{\xi_{2}}+\hat{j}_{\eta_{1}}\hat{e}_{\eta_{1}}),$ (5\. 23) where the operators $\hat{j}$ are defined by the equality: $\hat{j}_{X}(t)=\int^{T}_{0}dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime})$ (5\. 24) and $\Theta(t-t^{\prime})$ is the Green function of our perturbation theory [26]. We should change also $e_{r}\rightarrow e_{c}=e_{\eta_{1}}\frac{\partial r_{c}}{\partial\xi_{1}}-e_{\xi_{1}}\frac{\partial r_{c}}{\partial\eta_{1}}-e_{\xi_{2}}\frac{\partial r_{c}}{\partial\eta_{2}},~{}~{}e_{\varphi}\rightarrow e_{\xi_{1}}$ (5\. 25) in the Eq.(5\. 9). The differential measure takes the simplest form: $DM(\xi,\eta)=\delta(E-h(T))\prod_{t}d^{2}\xi d^{2}\eta\delta(\dot{\xi}_{1}-\omega_{1}-j_{\xi_{1}})\delta(\dot{\xi}_{2}-\omega_{2}-j_{\xi_{2}})$ $\times\delta(\dot{\eta}_{1}-j_{\eta_{1}})\delta(\dot{\eta}_{2}).$ (5\. 26) Note now that the $\xi,\eta$ variables are contained in $r_{c}$ only: $r_{c}=r_{c}(\xi_{1},\eta_{1},\eta_{2}).$ This means that the action of the operator $\hat{j}_{\xi_{2}}$ gives identical to zero contributions into perturbation theory series. And, since $\hat{e}_{\xi_{2}}$ and $\hat{j}_{\xi_{2}}$ are conjugate operators, see (5\. 23), we can put $j_{\xi_{2}}=e_{\xi_{2}}=0.$ This conclusion ends the reduction: $\hat{\mathbb{K}}(j,e)=\int^{T}_{0}dt(\hat{j}_{\xi_{1}}\hat{e}_{\xi_{1}}+\hat{j}_{\eta_{1}}\hat{e}_{\eta_{1}}),$ (5\. 27) $e_{c}=e_{\eta_{1}}\frac{\partial r_{c}}{\partial\xi_{1}}-e_{\xi_{1}}\frac{\partial r_{c}}{\partial\eta_{1}}.$ (5\. 28) The measure has the form: $DM(\xi,\eta)=\delta(E-h(T))d\xi_{2}(0)d\eta_{2}(0)\prod_{t}d\xi_{1}d\eta_{1}\delta(\dot{\xi}_{1}-\omega_{1}-j_{\xi_{1}})\delta(\dot{\eta}_{1}-j_{\eta_{1}})$ (5\. 29) since $V=V(r_{c},e_{c},\xi_{1})$ is $\xi_{2}$ independent and $\int\prod_{t}dX(t)\delta(\dot{X})=\int dX(0).$ ### 5.4 Perturbations One can see from (5\. 29) that the reduction can not solve the H-atom problem completely: there are nontrivial corrections to the orbital degrees of freedom $\xi_{1},\eta_{1}$. By this reason we should consider the expansion over $\hat{\mathbb{K}}$. Using last $\delta$-functions in (5\. 29) we find, see also [26] (normalizing $\rho(E)$ on the integral over $\xi_{2}(0)\eta_{2}(0)$): $\rho(E)=\int^{\infty}_{0}dTe^{-i\hat{\mathbb{K}}(j,e)}\int dMe^{-iU(r_{c},e)},$ (5\. 30) where $dM=\frac{d\xi_{1}d\eta_{1}}{\omega_{2}(E)}.$ (5\. 31) The operator $\hat{\mathbb{K}}(j,e)$ was defined in (5\. 27) and $U(r_{c},e_{c})=-s_{0}(r)+$ $+\int^{T}_{0}dt[\frac{1}{((r_{c}+e_{c})^{2}+r_{c}^{2}e_{\xi_{1}}^{2})^{1/2}}-\frac{1}{((r_{c}-e_{c})^{2}+r_{c}^{2}e_{\xi_{1}}^{2})^{1/2}}+2\frac{e_{c}}{r_{c}}]$ (5\. 32) with $e_{c},~{}e_{\xi_{1}}$ was defined in (5\. 28), (5\. 25) and $r_{c}(t)=r_{c}(\eta_{1}+\eta(t),\bar{\eta}_{2}(E,T),\xi_{1}+\omega_{1}(t)+\xi(t)),~{}~{}E\equiv h(\eta_{1}+\eta(T),\bar{\eta}_{2}),$ (5\. 33) where $\bar{\eta}_{2}(E,T)$ is the solution of equation $E=h$. The integration range over $\xi_{1}$ and $\eta_{1}$ is as follows: $0\leq\xi_{1}\leq 2\pi,~{}~{}-\infty\leq\eta_{1}\leq+\infty.$ (5\. 34) First inequality defines the principal domain of the angular variable $\varphi$ and second ones take into account the clockwise and anticlockwise motions of particle on the Kepler orbits. We can write: $\rho(E)=\int^{\infty}_{0}dT\int dM:e^{-iV(r_{c},\hat{e})}:$ (5\. 35) since the operator $\ln\hat{\mathbb{K}}$ is linear over $\hat{e}_{\xi_{1}},\hat{e}_{\eta_{1}}$. The colons means ”normal product” with differential operators staying to the left of functions and $U(r_{c},\hat{e})$ is the functional of operators: $2i\hat{e}_{c}=\hat{j}_{\eta_{1}}\frac{\partial r_{c}}{\partial\xi_{1}}-\hat{j}_{\xi_{1}}\frac{\partial r_{c}}{\partial\eta_{1}},~{}~{}2i\hat{e}_{\xi_{1}}=\hat{j}_{\xi_{1}}.$ (5\. 36) Expanding $U(r_{c},\hat{e})$ over $\hat{e}_{c}$ and $\hat{e}_{\eta_{1}}$ we find: $U(r_{c},\hat{e})=-s_{0}(r_{c})+2\sum_{n+m\geq 1}C_{n,m}\int^{T}_{0}dt{\hat{e}_{c}^{2n+1}\hat{e}_{\eta_{1}}^{m}}\frac{1}{r_{c}^{2n+2}},$ (5\. 37) where $C_{n,m}$ are the numerical constants. We see that the interaction part presents expansion over $1/r_{c}$ and, therefore, the expansion over $U$ generates an expansion over $1/r_{c}$. In result, see Sec.4.5, $\rho(E)=\int^{\infty}_{0}dT\int dM\\{e^{is_{0}(r_{c})}+B_{\xi_{1}}(\xi_{1},\eta_{1})+B_{\eta_{1}}(\xi_{1},\eta_{1})\\}.$ (5\. 38) The first term is the pure semiclassical contribution and last ones are the quantum corrections. The functionals $B$ are the total derivatives: $B_{\xi_{1}}(\xi_{1},\eta_{1})=\frac{\partial}{\partial\xi_{1}}b_{\xi_{1}}(\xi_{1},\eta_{1}),~{}~{}B_{\eta_{1}}(\xi_{1},\eta_{1})=\frac{\partial}{\partial\eta_{1}}b_{\eta_{1}}(\xi_{1},\eta_{1}).$ (5\. 39) This means that the mean value of quantum corrections in the $\xi_{1}$ direction are equal to zero: $\int^{2\pi}_{0}d\xi_{1}\frac{\partial}{\partial\xi_{1}}b_{\xi_{1}}(\xi_{1},\eta_{1})=0$ (5\. 40) since $r_{c}$ is the closed trajectory independently from initial conditions, see (5\. 19). In the $\eta_{1}$ direction the motion is classical: $\int^{+\infty}_{-\infty}d\eta_{1}\frac{\partial}{\partial\eta_{1}}b_{\eta_{1}}(\xi_{1},\eta_{1})=0$ (5\. 41) since (i) $b_{\eta_{1}}$ is the series over $1/r_{c}^{2}$ and (ii) $r_{c}\rightarrow\infty$ when $|\eta_{1}|\rightarrow\infty$. Therefore, $\rho(E)=\int^{\infty}_{0}dT\int dMe^{is_{0}(r_{c})}.$ (5\. 42) This is the desired result. Noting that $s_{0}(r_{c})=kS_{1}(E),~{}~{}k=\pm 1,\pm 2,...,$ where $S_{1}(E)$ is the action over one classical period $T_{1}$: $\frac{\partial S_{1}(E)}{\partial E}=T_{1}(E),$ and using the identity [22]: $\sum^{+\infty}_{-\infty}e^{inS_{1}(E)}=2\pi\sum^{+\infty}_{-\infty}\delta(S_{1}(E)-2\pi n),$ we find: $\rho(E)=\pi\Omega\sum_{n}\delta(E+1/2n^{2})$ (5\. 43) where $\Omega$ is the zero-modes volume. ### 5.5 Conclusions The demonstrated above mechanism of reduction is universal: one can introduce from the very beginning the arbitrary number of coordinates $(\xi,\eta)$. But later on the formalism automatically, through dependence of classical trajectory on coordinates of $W$, will extract the necessary set of variables $(\xi,\eta)$. At the same time $\dim(\xi,\eta)=\dim W$ and the integrals over other ones will give the volume $V_{0}=\int\prod d\xi(0)d\eta(0),$ see (5\. 29) where $\dim V_{0}=2$. Notice that appearance of the ”0-dimensional” integral measure $d\xi_{2}(0)d\eta_{2}(0)$ in (5\. 29) reflects the hidden $O(4)$ symmetry of H-atom problem [23]. Therefore, following our selection rule, we must consider in a first place the classical trajectory which leads to the maximal value of $\dim V_{0}$, i.e. we must consider the contributions with maximal number of zero modes. ## 6 Example: sin-Gordon model ### 6.1 Introduction First of all we will describe ”canonical” transformation in the path-integral formalism. The method of canonical transformations in spite of its expected effectiveness is unpopular in quantum theories since on this way exist the problem: it is necessary to find the transformation from Lagrangian to Hamiltonian descriptions. This transition in general is very difficult if $\varphi(x)$ and $\dot{\varphi}(x)=p(x)$ are not the independent quantities [13]. But we may use following trick. We start from the simplest verse of the canonical formalism introducing the ”first-order” description303030In other words, we will still stay in the frame of Lagrangian formalism. and after transformation come to independent canonically conjugate pares, $(\xi,\eta)$, i.e. come to Hamiltonian description. It is evident that in general the transformation $\varphi_{c}:(\varphi,p)\rightarrow(\xi,\eta)$ will not be canonical. The formalism of present Section is the same as in the H-atom problem but there is some distinction. We will continue in this Section description of influence of the phase-space structure on the result of quantum-mechanical measurements started in previous Sections. Now we will calculate the expectation value of the ”order parameter” (mass-shell particles production vertex) $\Gamma(q;u)$ [29]: $\rho(q)=<|\Gamma(q;u)|^{2}>_{u},$ where $q$ is the mass-shell ($q^{2}=m^{2}$) particles momentum and $<>_{u}$ means averaging over the field $u(x,t)$. Just the procedure of averaging would be the object of our interest considering the quantum Hamiltonian system with symmetry $G$. By definition, $\rho$ is the $probability$ to find one mass- shell particle. Certainly, $\rho(q)=0$ on the sourceless vacuum but, generally speaking, $\rho(q)\neq 0$ in a field with nonzero energy density. Calculations will be illustrated by the integrable (1+1)-dimensional model with non-polynomial Lagrangian $L=\frac{1}{2}(\partial_{\mu}u)^{2}+\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda u)-1],$ (6\. 1) We will consider following formulation of the problem. Formally nothing prevents to linearize partly our problem considering the Lagrangian $L=\frac{1}{2}[(\partial_{\mu}u)^{2}-m_{h}^{2}u^{2}]+\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda u)-1+\frac{\lambda^{2}}{2}u^{2}]\equiv L_{0}(u)-v(u)$ (6\. 2) to describe creation (and absorption) of the mass $m_{h}$ particles. Then the last term in (6\. 2), $v(u)=-\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda u)-1+\frac{\lambda^{2}}{2}u^{2}],$ (6\. 3) describes interactions. The corresponding to this theory order parameter is $\Gamma(q;u)=\int dxdte^{iqx}(\partial^{2}+m_{h}^{2})u(x,t),~{}~{}~{}q^{2}=m_{h}^{2}.$ (6\. 4) It will be shown by explicit calculations that $\rho(q)=0$ (6\. 5) as the consequence of unbroken $\tilde{sl}(2,C)$ Kac-Moody algebra on which the solitons of theory (6\. 1) live313131Trivialness of soliton $S$-matrix was shown in [30], see e.g. [31] and references cited therein323232 It may be useful at this point to compare our approach with ordinary thermodynamics of ferromagnetic. The external magnetic field is $\sim<{\mu}>$, where the order parameter $<\mu>$ is the mean value of the spin, and the phase transition means that $<\mu>\neq 0$, i.e. $<\mu>=0$ means that corresponding symmetry stay unbroken. We will suppose that the mean value of $|\Gamma(q,u)|^{2}$, which is the function of external fields parameter $q$, play the same role for field theories with symmetry, i.e. $<|\Gamma(q,u)|^{2}>_{u}=0$ means that corresponding symmetry stay unbroken. Therefore in our approach only the ”external” display of symmetry can be described.. The solution (6\. 5) seems interesting since it can be interpreted as the explicit demonstration of field $u(x,t)$ confinement. The main purpose of this paper is to investigate how the solution (6\. 5) appears. We will be able to find exact equality (6\. 5) since the model (6\. 1) possess infinite number of integrals of motion. It is well known that each integral of motion in involution allows to shrink a number of phase space $\bar{\gamma}$ variables on two units, see e.g. [12]. Resulting phase space $\gamma$ is called as the reduced phase space [25]. The summation over all reduced phase space topological classes [27] is assumed. By this way the field-theoretical problem will reduced to the quantum- mechanical one. We would consider $\eta$ as the ”particles” generalized momentum and would introduce $\xi$ as the conjugate to $\eta$ coordinate of soliton. The $2N$-dimensional phase space (cotangent manifold) $\gamma_{N}$ with local coordinates $(\xi,\eta)$ on it has natural simplectic structure, and $DM(\gamma_{N})=D^{N}M(\xi,\eta)$ in practical calculations (see Subsec.6.2). The summation over $N$ is assumed. The quantum corrections to semiclassical approximation of transformed theory are simply calculable since $\eta$ are conserved in the classical limit. This is the particularity of solitons dynamics (solitons momenta is the conserved quantities). One can consider the developed in this paper formalism as the path-integral version of nonlinear waves (solitons in our case) quantum theory (the canonical quantization of sin-Gordon model in the soliton sector was described also in [14].) In Subsec.6.3 we will demonstrate Eq.(6\. 5). It will be shown that this solution is consequence of the previously developed proposition (we would justify it in Subsec.6.2) that the semiclassical approximation is exact for sin-Gordon model [11]. The semiclassical approximation in the $\gamma_{N}$ phase space will be considered in Subsec.6.2. We would not use the complicated algebra to show the reduction procedure explicitly noting that all solutions of model (6\. 1) are known [24]. Then, using the $\delta$-likeness of measure $DM(\tilde{\gamma})$, we will find in Subsec.6.2 $DM(\gamma_{N})$ considering the mapping as an ordinary transformation to useful variables333333We will apply inverse reduction procedure. Let $G$ be a group of canonical transformations acting on the simplectic manifold $\tilde{\gamma}$ and let $\bar{G}$ be the Lie algebra of $G$ with $G^{*}$ dual of it. Then the momentum [32] mapping $J:~{}\tilde{\gamma}\rightarrow G^{*}$ introduces the integrals of motion which reduces the $\tilde{\gamma}$ manifold. Noting that the set of levels $J^{-1}({\eta})$ (solution of equations $J(\pi)=\eta$, $\pi\in\tilde{\gamma}$) is a manifold then $\gamma_{\eta}=J^{-1}({\eta})/\bar{G}_{\eta}$ is the reduced phase space, where $\bar{G}_{\eta}$ is the co-adjoint isotropy subgroup of $G$. Therefore, the differential measure $dM=dM(\eta,\gamma_{\eta})$ for reduced phase space. For integrable mechanical systems (infinite dimensional as well, see e.g. [24]) $\gamma_{\eta}$ shrinks to the point and in this case $dM=dM(\eta)$ is the measure of momentum manifold. Just this simplest case would be considered working with Lagrangian (6\. 1) and more general and interesting case with measure $DM=DM(\eta,\gamma_{\eta})$, $\gamma_{\eta}\neq\emptyset$, will be considered later. So, the reduction procedure of our Hamiltonian system with symmetry $G$ looks like canonical transformation [31]. This problem is nontrivial since, generally speaking, $\dim\tilde{\gamma}$ and $\dim\gamma$ are not the same for model (6\. 1).. Corresponding perturbation theory, see Subsec.6.3, in the momentum space $J$ was described in [26]. In Subsec.6.2 the path-integral definition of $\rho(q)$ will be given. We would conclude (this is the main result) that a theory in the ”nonlinear waves” sector may be nontrivial ($\rho\neq 0$) iff the manifold $\gamma$ is not compact. ### 6.2 Reduction procedure 6.2.1. Introduction into formalism. Our aim is to calculate the integral: $\rho(q)=e^{-i\hat{\mathbb{K}}(j,e)}\int DM(u,p)|\Gamma(q;u)|^{2}e^{iS_{O}(u)-iU(u,e)},$ (6\. 6) where $\Gamma(q;u)$ was defined in (6\. 4). In this expression the expansion over operator $\hat{\mathbb{K}}(j,e)={\rm Re}\int_{C_{+}}dxdt\frac{\delta}{\delta j(x,t)}\frac{\delta}{\delta e(x,t)}\equiv{\rm Re}\int_{C_{+}}dxdt\hat{j}(x,t)\hat{e}(x,t)$ (6\. 7) generates the perturbation theory series. We will assume that this series exist. The functionals $U(u,e)$ and $S_{O}(u)$ are defined by the equalities: $\displaystyle V(u+e)-V(u-e)=U(u,e)+\int dxdte(x,t)v^{\prime}(u),$ $\displaystyle S_{0}(u+e)-S_{0}(u-e)=S_{O}(u)+\int dxdte(x,t)(\partial^{2}+m_{h}^{2})u(x,t).$ (6\. 8) The action $S_{0}(u)$ corresponds to the free part of Lagrangian (6\. 1) and $V(u)$ describes interactions. The quantity $S_{O}(u)$ is not equal to zero since the soliton configurations have nontrivial topological charge (see also [1]). All time integrals in this expressions were defined on the Mills time contour [17]: $2{\rm Re}\int_{C_{+}}=\int_{C_{+}}+\int_{C_{-}}$ and $C_{\pm}:t\rightarrow t\pm i\epsilon,~{}~{}~{}\epsilon\rightarrow+0,~{}~{}~{}-\infty\leq t\leq+\infty,$ to avoid the possible light-cone singularities of the perturbation theory. The variational derivatives in (6\. 7) are defined by the following way: $\frac{\delta u(x,t\in C_{i})}{\delta u(x^{\prime},t^{\prime}\in C_{j})}=\delta_{ij}\delta(x-x^{\prime})\delta(t-t^{\prime}),~{}~{}~{}i,j=+,-.$ The auxiliary variables $(j,e)$ must be taken equal to zero at the very end of calculations. Considering the first order formalism with new coordinates $(u,p)$ the measure $DM(u,p)$ has the form: $DM(u,p)=\prod_{x,t}du(x,t)dp(x,t)\delta\left(\dot{u}-\frac{\delta H_{j}(u,p)}{\delta p}\right)\delta\left(\dot{p}+\frac{\delta H_{j}(u,p)}{\delta u}\right)$ (6\. 9) with the total ”Hamiltonian” $H_{j}(u,p)=\int dx\left\\{\frac{1}{2}p^{2}+\frac{1}{2}(\partial_{x}u)^{2}-\frac{m_{h}^{2}}{\lambda^{2}}[\cos(\lambda u)-1]-ju\right\\}.$ (6\. 10) The problem will be considered assuming that $u(x,t)$ belongs to Schwartz space: $u(x,t)|_{|x|=\infty}=0~{}({\rm{mod}}\frac{2\pi}{\lambda}).$ (6\. 11) This means that $u(x,t)$ tends to zero $(\rm{mod}\frac{2\pi}{\lambda})$ at $|x|\rightarrow\infty$ faster then any power of $1/|x|$. Note that $\dot{u}=p$, i.e. $u$ and $p$ are not the independent quantities. The measure (6\. 9) allows to perform arbitrary transformations. But, as was explained in Introduction, we will use the analog of canonical transformation which conserves the form of equations of motion. Hence, it is sufficient on this stage of calculations to know only the fact that this transformation exist [24]. One may propose that in result we should find for $N$-soliton topology: $D^{N}M(\xi,\eta)=\prod_{t}d^{N}\xi(t)d^{N}\eta(t)\delta^{(N)}\left(\dot{\xi}-\frac{\partial h_{j}(\xi,\eta)}{\partial\eta(t)}\right)\delta^{(N)}\left(\dot{\eta}+\frac{\partial h_{j}(\xi,\eta)}{\partial\xi(t)}\right),$ (6\. 12) where $h_{j}$ is the ”transformed Hamiltonian”: $h_{j}=h_{N}(\eta)-\int dxj(x,t)u_{N}(x;\xi,\eta)$ (6\. 13) and $u_{N}(x;\xi,\eta)$ is the $N$-soliton configuration the time dependence of which is parameterized by $(\xi,\eta)$. Therefore, the local coordinates $(\xi,\eta)$ are defined by the equations: $\dot{\xi}=\frac{\partial h_{j}}{\partial\eta},~{}~{}~{}\dot{\eta}=-\frac{\partial h_{j}}{\partial\xi},$ (6\. 14) where $h_{j}$ must obey the Poisson conditions343434See previous Section: $\\{u_{c}(x,t),h_{j}\\}=\frac{\delta H_{j}}{\delta p_{c}(x,t)},~{}~{}~{}\\{p_{c}(x,t),h_{j}\\}=-\frac{\delta H_{j}}{\delta u_{c}(x,t)}.$ (6\. 15) One can see choosing $h_{j}(\xi,\eta)=H_{j}(u_{c},p_{c})$ (6\. 16) that the initial equations have been restored: $\dot{u}_{c}=\frac{\partial u_{c}}{\partial\xi}\dot{\xi}+\frac{\partial u}{\partial\eta}\dot{\eta}=\\{u_{c},h_{j}\\}=\frac{\delta H_{j}}{\delta p_{c}}.$ The same we will have for $\dot{p}_{c}$. Therefore $(u_{c},p_{c})$ are solutions of equations of motion (6\. 14), if the equality (6\. 16) is hold. The field theory case in $(1+1)$-dimensional configuration space needs additional explanations. First of all, the analog of (5\. 10) must be introduced: $\Delta(u,p)=\int\prod_{t}d^{N}\xi(t)d^{N}\eta(t)\prod_{x,t}\delta(u(x,t)-u_{c}(x;\xi,\eta))\delta(p(x,t)-p_{c}(x;\xi,\eta))$ (6\. 17) if the $N$-soliton configuration is considered. Notice that the one- dimensional $\delta$-functions are introduced in (6\. 17) and $u_{c}$, $p_{c}$ are the functions of sets $(\xi,\eta)$, $\dim(\xi,\eta)=2N$. Introducing (6\. 17) we make the attempt to ”hide” the time dependence entirely into the set of $independent$ variables $(\xi,\eta)$. Comparing (6\. 9) and (6\. 12) one can note that $x$ dependence disappeared and the transformed measure depends on the number $N=1,2,...$ Therefore, occurs the reduction of the quantum degrees of freedom since the power of the coordinate set is continuum and the number of solitons $N$ is the countable set. This means that the proposed transformation to coordinates of solitons will be unavoidably singular. Notice then that the $x$ dependence of $\Delta(u,p)$ remain unimportant since last one always appear under the integrals over all $u(x,t)$ and $p(x,t)$. At the same time it is important that introduced in previous Section $\Delta_{c}$ disappeared in the final result, if the integral form of Poisson brackets (6\. 15) are hold353535 See the transformation (5\. 12), described in previous Section. For more confidence one can introduce the appropriate cells in the $x$ space [24].. One can try to propose also the local form of canonical commutators (6\. 15), if the definition (6\. 16) is hold. Indeed, one can find inserting (6\. 16) into (6\. 15) that: $\\{u_{c}(x,t),H_{j}(u_{c},p_{c})\\}=\frac{\delta H_{j}(u_{c},p_{c})}{\delta p_{c}(x,t)},~{}~{}~{}\\{p_{c}(x,t),H_{j}(u_{c},p_{c})\\}=-\frac{\delta H_{j}(u_{c},p_{c})}{\delta u_{c}(x,t)}.$ (6\. 18) This equalities must hold for arbitrary $j$. Making use the definition: $H_{j}(x_{c},p_{c})=\int dy\tilde{H}_{j}(x_{c},p_{c}),$ where $\tilde{H}_{j}$ is the Hamiltonian density, one can write from (6\. 18): $\int dy\\{u_{c}(x;\xi,\eta),u_{c}(y;\xi,\eta)\\}\frac{\delta\tilde{H}_{j}}{\delta u_{c}(y,t)}+$ $+\int dy(\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}-\delta(x-y))\frac{\delta\tilde{H}_{j}}{\delta p_{c}(y,t)}=0$ and $\int dy\\{p_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}\frac{\delta\tilde{H}_{j}}{\delta p_{c}(y,t)}-$ $-\int dy(\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}-\delta(x-y))\frac{\delta\tilde{H}_{j}}{\delta u_{c}(y,t)}=0.$ Then one can propose the solutions of these equations: $\\{u_{c}(x;\xi,\eta),u_{c}(y;\xi,\eta)\\}=\\{p_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}=0,$ $\\{u_{c}(x;\xi,\eta),p_{c}(y;\xi,\eta)\\}=\delta(x-y).$ (6\. 19) But it is interesting that the local commutators (6\. 19) are not satisfied363636That circumstances was mentioned firstly by V.Voronyuk.. One can see this inserting the soliton solution into (6\. 19). On the other hand the integral form (6\. 18) is satisfied. All this means that $u_{c}$ and $p_{c}$ are not the completely independent variables. It must be stressed that the local relations (6\. 19) are not the necessary conditions in our formalism. In our terms, the quantum force $j(x,t)$ excites the $(\xi,\eta)$ manifold only, leaving the topology of classical trajectory $(u,p)_{c}$ unchanged. We can use them immediately since the complete set of canonical coordinates $(\xi,\eta)$ of sin-Gordon model is known, see e.g. [24]. 6.2.3. Perturbation theory on the cotangent bundle. The classical Hamiltonian $h_{j}$ is the sum: $h_{j}(\eta)=\int dp\sigma(r)\sqrt{r^{2}+m_{h}^{2}}+\sum^{N}_{i=1}h(\eta_{i}),$ (6\. 20) where $\sigma(r)$ is the continuous spectrum and $h(\eta)$ is the soliton energy. Note absence of interaction energy among solitons. New degrees of freedom $(\xi,\eta)(t)$ must obey the equations (6\. 14): $\dot{\xi}_{i}=\Omega(\eta_{i})-\int dxj(x,t)\frac{\partial u_{N}(x;\xi,\eta)}{\partial\eta_{i}},~{}~{}~{}\Omega(\eta)\equiv\frac{\partial h(\eta)}{\partial\eta},$ $\dot{\eta}_{i}=\int dxj(x,t)\frac{\partial u_{N}(\xi,\eta)}{\partial\xi_{i}}.$ (6\. 21) Hence the sources of quantum perturbations are proportional to the time-local fluctuations of soliton configurations $\frac{\partial u_{N}(x;\xi,\eta)}{\partial\eta_{i}},~{}~{}~{}\frac{\partial u_{N}(x;\xi,\eta)}{\partial\xi_{i}}.$ One can split the Lagrange source onto ”Hamiltonian” ones: $j(x,t)\rightarrow(j_{\xi},j_{\eta}).$ This gives weight functional $U(u_{N};e_{\xi},e_{\eta})$ and operator $\hat{\mathbb{K}}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})$. In result: $\displaystyle\rho(q)=\sum_{N}e^{-i\hat{K}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})}\int D^{N}M(\xi,\eta)e^{iS_{O}(u_{N})}e^{-iU(u_{N};e_{\xi},e_{\eta})}\times$ $\displaystyle\times|\Gamma(q;u_{N})|^{2}$ (6\. 22) where, using vector notations, $\hat{\mathbb{K}}(e_{\xi},e_{\eta};j_{\xi},j_{\eta})=\frac{1}{2}\int dt\\{\hat{j}_{\xi}(t)\cdot\hat{e}_{\xi}(t)+\hat{j}_{\eta}(t)\cdot\hat{e}_{\eta}(t)\\}.$ (6\. 23) The measure takes the form: $D^{N}M(\xi,\eta)=\prod^{N}_{i=1}\prod_{t}d\xi_{i}(t)d\eta_{i}(t)\delta(\dot{\xi}_{i}-\Omega(\eta_{i})-j_{\xi,i}(t))\delta(\dot{\eta}_{i}-j_{\eta,i}(t))$ (6\. 24) The effective interaction potential $U(u_{N};e_{\xi},e_{\eta})=-\frac{2m^{2}}{\lambda^{2}}\int dxdt\sin\lambda u_{N}~{}(\sin\lambda e-\lambda e)$ (6\. 25) with $e(x,t)=e_{\xi}(t)\cdot\frac{\partial u_{N}(x;\xi,\eta)}{\partial\eta(t)}-e_{\eta}(t)\cdot\frac{\partial u_{N}(x;\xi,\eta)}{\partial\xi(t)}.$ (6\. 26) Performing the shifts: $\displaystyle\xi_{i}(t)\rightarrow\xi_{i}(t)+\int dt^{\prime}g(t-t^{\prime})j_{\xi,i}(t^{\prime})\equiv\xi_{i}(t)+\xi^{\prime}_{i}(t),$ $\displaystyle\eta_{i}(t)\rightarrow\eta_{i}(t)+\int dt^{\prime}g(t-t^{\prime})j_{\eta,i}(t^{\prime})\equiv\eta_{i}(t)+\eta^{\prime}_{i}(t),$ (6\. 27) we can move the Green function $g(t-t^{\prime})$ into the operator: $\hat{\mathbb{K}}(e_{\xi},e_{\eta};{\xi}^{\prime},{\eta^{\prime}})=\frac{1}{2}\int dtdt^{\prime}g(t-t^{\prime})\\{\hat{\xi}^{\prime}(t^{\prime})\cdot\hat{e}_{\xi}(t)+\hat{\eta}^{\prime}(t^{\prime})\cdot\hat{e}_{\eta}(t)\\}.$ (6\. 28) Notice that the Green function $g(t-t^{\prime})$ of eqs.(6\. 21) is again the step function: $g(t-t^{\prime})=\Theta(t-t^{\prime})$ (6\. 29) Its imaginary part is equal to zero for real times and this allows to shift $C_{\pm}$ to the real-time axis (see [26]). In result: $D^{N}M(\xi,\eta)=\prod^{N}_{i=1}\prod_{t}d\xi_{i}(t)d\eta_{i}(t)\delta(\dot{\xi}_{i}-\Omega(\eta+\eta^{\prime}))\delta(\dot{\eta}_{i})$ (6\. 30) with $u_{N}=u_{N}(x;\xi+\xi^{\prime},\eta+\eta^{\prime}).$ (6\. 31) The equations: $\dot{\xi}_{i}=\Omega(\eta_{i}+\eta^{\prime}_{i})$ (6\. 32) are trivially integrable. In quantum case $\eta^{\prime}_{i}\neq 0$ this equation describes the motion on nonhomogeneous and anisotropic manifold. So, the expansion over $(\hat{\xi^{\prime}},~{}\hat{e}_{\xi},~{}\hat{\eta}^{\prime},~{}\hat{e}_{\eta})$ generates the local in time deformations of $\gamma_{N}$ manifold, $(\xi,\eta)\in\gamma_{N}$ completely. The weight of this deformations is defined by $U(u_{N};e_{\xi},e_{\eta})$. Using the definition: $\int Dx\delta(\dot{x})=\int dx(0)=\int dx_{0}$ functional integrals are reduced to the ordinary integrals over initial data $(\xi,\eta)_{0}$. This integrals define the zero modes volume. ### 6.3 Quantum corrections The proof of (6\. 5) we would divide on two parts. First of all we would consider the semiclassical approximation (Sec.6.3.1) and in Sec.6.3.2. we will show that this approximation is exact. 6.1. Introduction and definitions. The $N$-soliton solution $u_{N}$ depends from $2N$ parameters. Half of them $N$ can be considered as the position of solitons and other $N$ as the solitons momentum. Generally at $|t|\rightarrow\infty$ the $u_{N}$ solution decomposed on the single solitons $u_{s}$ and on the double soliton bound states $u_{b}$ [24]: $u_{N}(x,t)=\sum^{n_{1}}_{j=1}u_{s,j}(x,t)+\sum^{n_{2}}_{k=1}u_{b,k}(x,t)+O(e^{-|t|})$ We will see later that main elements of our formalism are the one soliton $u_{s}$ and two-soliton bound state $u_{b}$ configurations. Its $(\xi,\eta)$ parameterizations, confirmed to eqs.(6\. 15), have the form: $u_{s}(x;\xi,\eta)=-\frac{4}{\lambda}\arctan\\{\exp(m_{h}x\cosh\beta\eta-\xi)\\},~{}~{}~{}\beta=\frac{\lambda^{2}}{8}$ (6\. 33) and $u_{b}(x;\xi,\eta)=-\frac{4}{\lambda}\arctan\\{\tan\frac{\beta\eta_{2}}{2}\frac{m_{h}x\sinh\frac{\beta\eta_{1}}{2}\cos\frac{\beta\eta_{2}}{2}-\xi_{2}}{m_{h}x\cosh\frac{\beta\eta_{1}}{2}\sin\frac{\beta\eta_{2}}{2}-\xi_{1}}\\}.$ (6\. 34) The $(\xi,\eta)$ parametrization of solitons individual energies $h(\eta)$ takes the form: $h_{s}(\eta)=\frac{m_{h}}{\beta}\cosh\beta\eta,~{}~{}~{}h_{b}(\eta)=\frac{2m_{h}}{\beta}\cosh\frac{\beta\eta_{1}}{2}\sin\frac{\beta\eta_{2}}{2}\geq 0.$ The bound-states energy $h_{b}$ depends from $\eta_{1}$ and $\eta_{2}$. First one defines inner motion of two bounded solitons and second one the bound states center of mass motion. Correspondingly we will call this parameters as the internal and external ones. Note that the inner motion is periodic, see (6\. 24). Performing last integration in (6\. 22) with measure (6\. 30) we find: $\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}e^{-i\hat{\mathbb{K}}}e^{iS_{O}(u_{N})}e^{-iU(u_{N};e_{\xi},e_{\eta})}|\Gamma(q;u_{N})|^{2}$ (6\. 35) where $u_{N}=u_{N}(\eta_{0}+\eta^{\prime},\xi_{0}+\Omega(t)+\xi^{\prime}).$ (6\. 36) and $\Omega(t)=\int dt^{\prime}\Theta(t-t^{\prime})\Omega(\eta_{0}+\eta^{\prime}(t^{\prime}))$ (6\. 37) In the semiclassical approximation $\xi^{\prime}=\eta^{\prime}=0$ we have: $u_{N}=u_{N}(x;\eta_{0},\xi_{0}+\Omega(\eta_{0})t).$ (6\. 38) Note now that if the surface term $\int\partial_{\mu}(e^{iqx}\partial^{\mu}u_{N})=0$ (6\. 39) then $\int d^{2}xe^{iqx}(\partial^{2}+m_{h}^{2})u_{N}(x,t)=-(q^{2}-m_{h}^{2})\int d^{2}xe^{iqx}u_{N}(x,t)=0$ (6\. 40) since $q^{2}$ belongs to mass shell by definition. The condition (6\. 39) is satisfied since $u_{N}$ belong to Schwartz space (the periodic boundary condition for $u(x,t)$ do not alter this conclusion). Therefore, in the semiclassical approximation (6\. 5) is hold. Expending the operator exponent in (6\. 35) we will find the expansion over $\rho_{n,m}(q)=\frac{(1/2i)^{n}}{n!}\frac{(1/2i)^{m}}{m!}\lim_{(\xi^{\prime},\eta^{\prime},e_{\xi},e_{\eta})=0}\sum_{N}\int d^{N}\xi_{0}d^{N}\eta_{0}\times$ $\times\int\prod^{n}_{i=1}\\{dt_{i}dt^{\prime}_{i}\theta(t_{i}-t^{\prime}_{i})\hat{\xi}^{\prime}(t^{\prime}_{i})$ $\times\int\prod^{m}_{i=1}\\{dt_{i}dt^{\prime}_{i}\theta(t_{i}-t^{\prime}_{i})\hat{\eta}^{\prime}(t^{\prime}_{i})\\}e^{iS_{O}(u_{N})}|\Gamma(q;u_{N})|^{2}$ $\times\\{\prod^{n}_{i=1}\hat{e}_{\xi}(t_{i})\prod^{m}_{j=1}\hat{e}_{\eta}(t_{j})e^{-iU(u_{N};e_{\xi},e_{\eta})}\\}|_{e=0},$ (6\. 41) where $U(u_{N};e_{\xi},e_{\eta})$ was defined in (6\. 25), (6\. 26). Notice that the action of operators $\hat{\xi}^{\prime}$, $\hat{\eta}^{\prime}$ create terms $\int d^{2}xe^{iqx}\theta(t-t^{\prime})(\partial^{2}+m^{2})u_{N}(x,t)\neq 0.$ (6\. 42) 6.2. Quantum corrections Now we will show that The semiclassical approximation is exact in the soliton sector of (6\. 1), (6\. 11) theory. The structure of the perturbation theory is readily seen in the ”normal- product” form: $\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}:e^{-iU(u_{N};\hat{j}/2i)}e^{iS_{O}(u_{N})}|\Gamma(q;u_{N})|^{2}:,$ (6\. 43) where $\hat{j}=\hat{j}_{\xi}\cdot\frac{\partial u_{N}}{\partial\eta}-\hat{j}_{\eta}\cdot\frac{\partial u_{N}}{\partial\xi}=\omega\hat{j}_{X}\frac{\partial u_{N}}{\partial X}$ (6\. 44) and $\hat{j}_{X}=\int dt^{\prime}\Theta(t-t^{\prime})\hat{X}(t^{\prime})$ (6\. 45) with $2N$-dimensional vector $X=(\xi,\eta)$. In Eq. (6\. 44) $\omega$ is the ordinary simplectic matrix. The colons in (6\. 43) mean that the operator $\hat{j}$ should stay to the left of all functions. The structure (6\. 44) shows that each order over $\hat{j}_{X_{i}}$ is proportional at least to the first order derivative of $u_{N}$ over conjugate to $X_{i}$ variable. The expansion of (6\. 43) over $\hat{j}_{X}$ can be written [26] in the form of total derivatives (omitting the semiclassical approximation): $\rho(q)=\sum_{N}\int\prod^{N}_{i=1}\\{d\xi_{0}d\eta_{0}\\}_{i}\left\\{\sum^{2n}_{i=1}\frac{\partial}{\partial X_{0i}}P_{X_{i}}(u_{N})\right\\},$ (6\. 46) where $P_{X_{i}}(u_{N})$ is the infinite sum of ”time-ordered” polynomials (see [26]) over $u_{N}$ and its derivatives. The explicit form of $P_{X_{i}}(u_{N})$ is complicated since the interaction potential is non- polynomial. But it is enough to know, see (6\. 44), that $P_{X_{i}}(u_{N})\sim\omega_{ij}\frac{\partial u_{N}}{\partial X_{0j}}.$ (6\. 47) Therefore, $\rho(q)=0$ (6\. 48) since (i) each term in (6\. 46) is the total derivative, (ii) we have (6\. 47) and (iii) $u_{N}$ belongs to Schwartz space. We can conclude that the equality (6\. 48) is hold since $\frac{\partial u_{N}}{\partial X_{0}}=0~{}~{}at~{}~{}X_{0}\in\partial W,$ (6\. 49) where $\partial W$ is the boundary of $W$. In our consideration we did not touch the continuous spectrum contributions. In considered approach this contributions are absent since they are realized on zero measure: theirs contributions are $\sim\\{volume~{}of~{}\gamma_{N}\\}^{-1}$. ## 7 Summary Let as summarize the general results of present and of the previous sections. 1\. The $m$\- into $n$-particles transition (non-normalized) $probability$ $R_{nm}$ would have on the Dirac measure the following symmetrical form: $\rho_{nm}(p_{1},...,p_{n},q_{1},...,q_{m})=<\prod^{m}_{k=1}|\Gamma(q_{k};u)|^{2}\prod^{n}_{k=1}|\Gamma(p_{k};u)|^{2}>_{u}=$ $=e^{-i\hat{K}(j,e)}\int DM(u)e^{iS_{O}(u)-iU(u,e)}\prod^{m}_{k=1}|\Gamma(q_{k};u)|^{2}\prod^{n}_{k=1}|\Gamma(p_{k};u)|^{2}\equiv$ $\equiv\hat{\cal O}(u)\prod^{m}_{k=1}|\Gamma(q_{k};u)|^{2}\prod^{n}_{k=1}|\Gamma(p_{k};u)|^{2}.$ (7\. 50) Here $p(q)$ are the in(out)-going particle momenta. It should be underlined that this representation is strict and is valid for arbitrary Lagrange theory of arbitrary dimensions. 2\. The operator $\hat{\cal O}$ contains three element. The Dirac measure $DM$, the functionals $S_{O}$, $U(x,e)$ and the operator $\hat{\mathbb{K}}(j,e)$. The expansion over the operator $\hat{\mathbb{K}}(j,e)=\frac{1}{2}{\rm Re}\int_{C_{+}}dxdt\frac{\delta}{\delta j(x,t)}\frac{\delta}{\delta e(x,t)}\equiv\frac{1}{2}{\rm Re}\int_{C_{+}}dxdt\hat{j}(x,t)\hat{e}(x,t)$ (7\. 51) generates the perturbation series. We will assume that this series exist (at least in Borel sense). 3\. The functionals $U(u,e)$ and $S_{O}(u)$ are defined by the equalities: $S_{O}(u)=(S_{0}(u+e)-S_{0}(u-e))+2{\rm Re}\int_{C_{+}}dxdte(x,t)(\partial^{2}+m^{2})u(x,t),$ (7\. 52) $U(u,e)=V(u+e)-V(u-e)-2{\rm Re}\int_{C_{+}}dxdte(x,t)v^{\prime}(u),$ (7\. 53) where $S_{0}(u)$ is the free part of the Lagrangian and $V(u)$ describes interactions. The quantity $S_{O}(u)$ is not equal to zero if $u$ have nontrivial topological charge. 4\. The measure $DM(u,p)$ has the Dirac form: $DM(u,p)=\prod_{x,t}du(x,t)dp(x,t)\delta\left(\dot{u}-\frac{\delta H_{j}(u,p)}{\delta p}\right)\delta\left(\dot{p}+\frac{\delta H_{j}(u,p)}{\delta u}\right)$ (7\. 54) with the total Hamiltonian $H_{j}(u,p)=\int dx\\{\frac{1}{2}p^{2}+\frac{1}{2}(\nabla u)^{2}+v(u)-ju\\}.$ (7\. 55) This last one includes the energy $ju$ of quantum fluctuations. 5\. Dirac measure contains following information: a. Only $strict$ solutions of equations $\dot{u}-\frac{\delta H_{j}(u,p)}{\delta p}=0,~{}\dot{p}+\frac{\delta H_{j}(u,p)}{\delta u}=0$ (7\. 56) with $j=0$ should be taken into account. This ”rigidness” of the formalism means the absence of pseudo-solutions (similar to multi-instanton, or multi- kink) contribution. b. $\rho_{nm}$ is described by the $sum$ of all solutions of Eq.(7\. 56), independently from their ”nearness” in the functional space; c. $\rho_{nm}$ did not contain the interference terms from various topologically nonequivalent contributions. This displays the orthogonality of corresponding Hilbert spaces; d. The measure (7\. 54) includes $j(x)$ as the external adiabatic source. Its fluctuation disturbs the solutions of Eq.(7\. 56) and vice versa since the measure (7\. 54) is strict; e. In the frame of the adiabatical condition, the field disturbed by $j(x)$ belongs to the same manifold (topology class) as the classical field defined by (7\. 56) [26]. f. The Dirac measure is derived for $real-time$ processes only, i.e. (7\. 54) is not valid for tunnelling ones. For this reason, the above conclusions should be taken carefully. g. It can be shown that theory on the measure (7\. 54) restores ordinary (canonical) perturbation theory. 6\. The parameter $\Gamma(q;u)$ plays the role of particle production vertex. It is connected directly with $external$ particle energy, momentum, spin, polarization, charge, etc., and is sensitive to the symmetry properties of the interacting fields system. For the sake of simplicity, $u(x)$ is the real scalar field. The generalization would be evident. As a consequence of (7\. 54), $\Gamma(q;u)$ is the function of the external particle momentum $q$ and is a $linear$ functional of $u(x)$: $\Gamma(q;u)=-\int dxe^{iqx}\frac{\delta S_{0}(u)}{\delta u(x)}=\int dxe^{iqx}(\partial^{2}+m^{2})u(x),~{}~{}q^{2}=m^{2},$ (7\. 57) for the mass $m$ field. This parameter presents the momentum distribution of the interacting field $u(x)$ on the remote hypersurface $\sigma_{\infty}$ if $u(x)$ is the regular function. Notice, the operator $(\partial^{2}+m^{2})$ cancels the mass-shell states of $u(x)$. The construction (7\. 57) means, because of the Klein-Gordon operator and since the external states being mass-shell by definition [33], the solution $\rho_{nm}=0$ is possible for a particular topology (compactness and analytic properties) of $quantum$ field $u(x)$. So, $\Gamma(q;u)$ carries the following remarkable properties: – it directly defines the observables, – it is defined by the topology of $u(x)$, – it is the linear functional of the actions symmetry group element $u(x)$. If (7\. 56) have nontrivial solution $u_{c}(x,t)$, then this ”extended objects” quantization problem arises. We solve it introducing convenient dynamical variables [34]. Then the measure (7\. 54) admits the transformation: $u_{c}:~{}(u,p)\rightarrow(\xi,\eta)\in W=G/G_{c}.$ (7\. 58) and the transformed measure has the form: $DM(u,p)=\prod_{x,t\it C}d\xi(t)d\eta(t)\delta\left(\dot{\xi}-\frac{\delta h_{j}(\xi,\eta)}{\delta\eta}\right)\delta\left(\dot{\eta}+\frac{\delta h_{j}(\xi,\eta)}{\delta\xi}\right),$ (7\. 59) where $h_{j}(\xi,\eta)=H_{j}(u_{c},p_{c})$ is the transformed Hamiltonian. It is evident that $(\xi,\eta)$ are parameters of integration of eqs.(7\. 56) and they form the factor space $W=G/G_{c}$. As a result of mapping of the perturbation generating operator $\hat{\mathbb{K}}$ on the manifold $W$ the equations of motion became linearized: $DM=\prod_{t}\delta\left(\dot{\xi}-\frac{\delta h(\eta)}{\delta\eta}-j_{\xi}\right)\delta\left(\dot{\eta}-j_{\eta}\right).$ (7\. 60) If Feynman’s $i\epsilon$-prescription is adopted, then the Green function of Eq.(7\. 60) $g(t-t^{\prime})=\Theta(t-t^{\prime})$ (7\. 61) with boundary property: $\Theta(0)=1.$ 7\. Expansion of $\exp\\{\hat{\mathbb{K}}(j,e)\\}$ gives the ”strong coupling’ perturbation series. Its analysis shows that the action of the integro- differential operator $\hat{\cal O}$ leads to the following representation: $\rho_{nm}(p,q)=\int_{W}\\{d\xi(0)\cdot\frac{\partial}{\partial\xi(0)}\rho^{\xi}_{nm}(p,q)+d\eta(0)\cdot\frac{\partial}{\partial\eta(0)}\rho^{\eta}_{nm}(p,q)\\}.$ (7\. 62) This means that the contributions into $R_{nm}(p,q)$ are accumulated strictly on the boundary, ”bifurcation manifold”, $\partial W$, i.e. depends directly on the topology property of $W$. 8\. It was shown that the MP is absent in the frame of Lagrangian (6\. 1). For this purpose one should modify the sin-Gordon Lagrangian adding for instance the term: $\frac{1}{2}(\partial\Phi)^{2}-\frac{1}{2}M^{2}\Phi^{2}-\frac{c}{3}u\Phi^{2}$ (7\. 63) to describe collision of ”external” field $\Phi$ on the solitons. This model allows to introduce the nontrivial probabilities $\rho(q_{1},q_{2},...)$ considering creation (and absorption) of the field $\Phi$. Note that field $u(x)$ is still ”confined” even with this adding. ## 8 Conclusion The final goal of present approach is to construct the workable at arbitrary distances, i.e. for arbitrary momenta of produced hadrons, $S$-matrix formalism for theories with (hidden) symmetry. But this aim remains unachieved in present paper. In subsequent papers more realistic field models in $4d$ Minkowski space-time metric will be described. But one should not consider the demonstrated examples of Yang-Mills $S$-matrix as the definite proves since I am note sure that the used $O(4)\times O(2)$ solution of Yang-Mills equation in the Minkowski in the situation of general position guarantee the largest contribution. Moreover, only the $SU(2)$ theory will be considered. Unfortunately we can not find in the frame of t’Hooft ansatz [35] the solution for larger $SU(N)$ group [36]. It will be to shown how one or another physical phenomena may be seen in the field theory with symmetry. Namely, — no plain waves production exist in theories with symmetry, i.e. for instance the gluons can not be seen in a free state since simply the last ones are absent in quantum theory of the symmetry manifolds, or, in other words, since the gluon states and the ”states” of the symmetry manifold belong to the orthogonal Hilbert spaces. The quark fields will not be included in this simplest example. But more realistic model with quarks shows that — inclusion of matter can not change previous conclusion that the gluons can not be created. In the other example we will show how the — binding potential may arise among quarks. Here the situation of general position selection rule will be extremely important: it will be used that the situation when $(q\bar{q})$ potential is independent from the scale of Yang-Mills fields is mostly probable. The quantum field theory with constraints will obey following important property: — the perturbation theory of quantum systems with symmetry may be free from any divergences, i.e. it $may$373737One can not be sure that the approach is universal, can be used, for instance, in quantum gravity case. be rightful at arbitrary distances, for VHM case as well. It is the evident consequence of lessening of the number of dynamical degrees of freedom because of symmetry constraints383838 And it is unnecessary to have in that case any new mechanism, such as the supersymmetry for example, to achieve the field theory without divergences. Possible scenario of such theory will be discussed later.. Exist also the intriguing question of asymptotic freedom. The point is that there is no running coupling constants in our strong coupling perturbation theory without divergences. On the other hand the asymptotic freedom is the experimental fact. We will show how — the effect of asymptotic freedom may arise in our quantum theory of the symmetry manifolds. The main question here is to find the experimentally observable corrections to the asymptotic freedom law. In summary, the aim of future publications would be the question: is the offered approach complete from physical point of view? It is important since offered quantization scheme in the situation of general position on Dirac measure must be true for arbitrary distances, since it is free from arbitrary scale parameters393939That is why I hope that it may give the predictions acceptable from physical point of view at arbitrary distances.. Acknowledgments First of all I am thankful to Alexei Sisakian for fruitful conversations during the work upon the topology conserving perturbation theories ideology. The offered text was arranged under last, before his sudden death, proposition to put in order my present-day understanding of the approach. I would like to note the significant role of E.Levin and L.Lipatov in realization of discussed formalism. I am grateful to V.Kadyshevski for interest to the discussed in the paper questions. Various parts of the approach were offered to auditory of many Institutes and Universities and I am grateful for theirs interest and comments. ## References * [1] J.Manjavidze, Sov.J.Nucl.Phys., 45 (1987) 442 * [2] M.V.Fedoryuk, Asymptotics: integrals and series (Nauka, Moscow, 1987) * [3] G.Parisi and Y.Wu, Ecientia Sinica, 24 (1981) 483, A.A.Migdal and T.A.Kozhamkulov, Yad. Phys., 39 (1984) 1596 [Sov. J. Nucl. Phys. 39 (1984) 1012], L.Lipatov * [4] A.N.Kolmogorov, DAN SSSR, 98 (1954) 527; V.I.Arnold, Izv. AN SSSR, 25 (1961) 21, V.I.Arnold, UMN, 18 (1963) 81; Yu.Mozer, Math. Phys., Bd.11a (1962) 1 * [5] J.S.Dowker, Ann. Phys.(NY), 62 (1971) 361 * [6] M.S.Marinov, Phys. Rep., 60 (1980) 1 * [7] S.F.Edvards and Y.Guliaev, Proc. Roy. Soc., A279 (1964) 229 * [8] R.P.Feynman and A.R.Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York, 1965) * [9] C.Grosche, Path Integrals, Hyperbolic Spaces, and Selberg Trace Formulae (World Scint., Singapore, New Jersey, london, Hong Kong, 1995) * [10] H.Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polimer Physics (World Scientific, Singapore, 1989) * [11] R.Dashen, B.Hasslacher and A.Neveu, Phys. Rev., D10 (1974) 4114 * [12] V.I.Arnold, Mathematical Methods of Classical Mechanics, (Springer Verlag, New York, 1978) * [13] P.A.M.Dirac, Lectures on quantum mechanics (Yeshiva Univ., New York, !964) * [14] V.E.Korepin and L.D.Faddeev, Sov. TMF, 25 (1975) 147 * [15] L.Faddeev and V.Korepin, Phys. Rep., 42C (1978) 3; J.Goldstone and R.Jackiw, Phys.Rev., D11 (1975) 1486; V.A.Rubakov, Classical and Gauge Fields (Editorial URSS, Moscow, 1999) * [16] S.Coleman, Whys in Subnuclear Physics, ed. by Zichichi, Ettore Majorana School, Erice, Italy (1976) * [17] R.Mills, Propagators of Many-Particles Systems (Gordon & Breach, 1969) * [18] K.Osterwalder and E.Seiler, Ann. Phys. (N.Y.) 110 (1978) 440 * [19] J.Manjavidze and A.Sissakian, Theor. Math. Phys., 130 (2002) 153 * [20] I.H.Duru, Phys. Rev., D30 (1984) 143 * [21] G.Pocshle and E.Teller, Zs. Phys. 83 (1933) 143 * [22] J.Manjavidze, J.Math.Phys. 41 (2000) 5710 * [23] V.Fock, Zs. Phys., 98 (1935) 145; V.Bargman, Zs. Phys., 99 (1935) 576; V.S.Popov, High Energy Physics and Elementary Particles Theory, (Naukova Dumka, Kiev, 1967) * [24] L.A.Takhtajan and L.D.Faddeev, Hamiltonian Approach in Solitons Theory (Moskow, Nauka, 1986) * [25] R.Abraham and J.E.Marsden, Foundations of Mechanics (Benjamin/ Cummings Publ. Comp., Reading, Mass., 1978) * [26] J.Manjavidze, Perturbation Theory on the Imvariant Subspace, hep-th/9801188 * [27] S.Smale, Inv.Math., 10:4 (1970) 305, ibid., 11:1 (1970) 45 * [28] I.H.Duru and H.Kleinert, Phys. Lett., 84B (1979) 185 * [29] J.Manjavidze and A.Sisakian, Phys. Rep., 346 (2001) 1 * [30] Zamolodchikov, A.B. and A.B.Zamolodchikov, Phys. Lett., 72B, 503 (1978) * [31] Solitons. (Ed. by R.K.Bullough and J.Caudry, Springer-Verlag, Berlin, Heidelberg, New York, 1980); T.D.Lee, Phys. Scr., 20 (1979) 440; R.Jakiw, Rev. Mod. Phys., 49 (1977) 681; J.Goldstone and R.Jackiw, Phys. Rev., D11, (1975) 1485; R.Rajaraman, Solitons and Instantons (North-Holland Publ. Comp., Amsterdam, New York, Oxford, 1982) * [32] J.M.Souriae, Structure des Systems Dynamiques (Dunod, Paris, 1970) * [33] L.Landau and R.Peierls, Zs.Phys., 69 (1931) 56 * [34] J.Manjavidze and A.Sissakian, J. Math. Phys. 42, (2001) 641 * [35] G.t’Hooft, ”Computation of the quantum effects due to a four-dimensional pseudoparticle”, Phys. Rev. D14 (1976) 3432 * [36] J.Manjavidze and V.Voronyuk, Phys.Part.Nucl.Lett. 3, (2006) 391
arxiv-papers
2011-01-06T11:31:31
2024-09-04T02:49:16.180121
{ "license": "Public Domain", "authors": "J. Manjavidze", "submitter": "Joseph Manjavidze", "url": "https://arxiv.org/abs/1101.1193" }
1101.1496
# On the k-nullity foliations in Finsler geometry and completeness B. Bidabad111Faculty of Mathematics, Amirkabir University of Technology, Tehran, Iran.(email:bidabad@aut.ac.ir) and M. Rafie-Rad222Faculty of Mathematics, Mazandaran University, Babolsar, Iran.(email: m.rafiei.rad@gmail.com) ###### Abstract Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant then the distribution is involutive and each maximal integral manifold is totally geodesic. Characterization of the $k$-nullity foliation is given, as well as some results concerning constancy of the flag curvature, and completeness of their integral manifolds, providing completeness of $(M,F)$. The introduced $k$-nullity space is a natural extension of nullity space in Riemannian geometry, introduced by S. S. Chern and N. H. Kuiper and enlarged to Finsler setting by H. Akbar-Zadeh and contains it as a special case. Keywords: Foliation, k-nullity, Finsler manifolds, Curvature operator. MSC: 2000 Mathematics subject Classification: 58B20, 53C60, 53C12. ## 1 Introduction Foliations of manifolds occur naturally in various geometric contexts. They arise in connections with some essential topics as vector fields without singularities, integrable $m$-dimensional distributions, submersions and fibrations, actions of Lie groups, direct constructions of foliations such as Hopf fibrations, Reeb foliations and finally they appear in the existence study of solution of certain differential equations. In the later case, S. Tanno in [15] applied the concept of the k-nullity spaces to achieve a complete proof for the famous Obata Theorem which is a subject of numerous rigidity results in Riemannian geometry. The nullity space of the Riemannian curvature tensor was first studied by S. S. Chern and N. H. Kuiper [3] in 1952. They have shown that, if the index of nullity, $\mu$, of a Riemannian manifold is locally constant, then the manifold admits a locally integrable $\mu$-dimensional distribution whose integral submanifolds are locally flat. O. Kowalski and M. Sekizawa have proved that vanishing of the index of nullity in some senses, results that the tangent sphere bundle is a space of negative scalar curvature [8]. The concept of nullity spaces are generalized to the ${\bf k}$-nullity spaces in Riemannian geometry in a number of works such as [4, 7] and [11]. In this work we answer to the following natural questions: Is there any extension for the concept of k-nullity space in Finsler geometry? Is its maximal integral manifold totally geodesic? And finally is its maximal integral manifold complete, provided that $(M,F)$ is complete? Fortunately, the answer to these questions is affirmative. More precisely, we obtain the following results. ###### Theorem 1.1. Let $(M,F)$ be a Finsler manifold for which the index of k-nullity $\mu_{\bf k}$ be constant on an open subset $U\subseteq M$. Then, the local ${\bf k}$-nullity distribution on $U$ is completely integrable. ###### Theorem 1.2. The ${\bf k}$-nullity space of a Finsler manifold $(M,F)$ at a point $x\in M$, coincides with the kernel of the related curvature operator of $\Omega$. D. Ferus has proved that the maximal integral manifolds of nullity foliation are totally geodesic [6]. This result has been extended to the Finsler case by H. Akbar-Zadeh [2]. Here, we prove the same result for k-nullity foliation in Finsler manifolds. ###### Theorem 1.3. Let $(M,F)$ be a Finsler manifold. If the ${\bf k}$-nullity space is locally constant on the open subset $U$ of $M$, then every $\bf k$\- nullity integral manifold $N$ in $U$ is an auto-parallel Finsler submanifold with non-negative constant flag curvature k. Moreover, $(N,\tilde{F})$ is a $P$-symmetric space. The completeness of the nullity foliations is studied by D. Ferus in [5]. The similar result is carried out for Finsler manifolds by H. Akbar-Zadeh [2] in 1972. ###### Theorem 1.4. Let $(M,F)$ be a complete Finsler manifold and $G$ an open subset of $M$ on which $\mu_{\bf k}$ is minimum. Then, every integral manifold of the k-nullity foliation in $G$ is a complete submanifold of $M$. It is worth mentioning that M. Sekizawa and S. Tachibana have studied $k^{th}$ nullity foliations as another generalization of Chern and Kuiper’s nullity in Riemannian geometry by considering $k^{th}$ consecutive derivative of the curvature tensor [13, 14]. ## 2 Preliminaries and terminologies. ### 2.1 Regular connections and Finsler manifolds. Let $M$ be a connected differentiable manifold of dimension $n$. We adopt here the notations and terminologies of [1]. Denote the bundle of tangent vectors of $M$ by $p:TM\longrightarrow M$, the fiber bundle of non-zero tangent vectors of $M$ by $\pi:TM_{0}\longrightarrow M$ and the pulled-back tangent bundle by $\pi^{*}TM\longrightarrow TM_{0}$. Any point of $TM_{0}$ is denoted by $z=(x,v)$, where $x=\pi z\in M$ and $v\in T_{\pi z}M$. By $TTM_{0}$ we denote the tangent bundle of $TM_{0}$ and by $\varrho$ the canonical linear mapping $\varrho:TTM_{0}\longrightarrow\pi^{*}TM,$ where $\varrho=\pi_{*}$. For all $z\in TM_{0}$, let ${\cal V}_{z}TM$ be the set of vertical vectors at $z$, that is, the set of vectors which are tangent to the fiber through $z$. Equivalently, ${\cal V}_{z}TM=\ker\pi_{*}$ where $\pi_{*}:TTM_{0}\longrightarrow TM$ is the linear tangent mapping. Let $\nabla$ be a linear connection on the vector bundle $\pi^{*}TM\longrightarrow TM_{0}$. We define a linear mapping $\mu:TTM_{0}\longrightarrow\pi^{*}TM,$ by $\mu(\hat{X})=\nabla_{\hat{X}}{\bf v}$ where $\hat{X}\in TTM_{0}$ and ${\bf v}$ is the canonical section of $\pi^{*}TM$. The connection $\nabla$ is said to be regular, if $\mu$ defines an isomorphism between ${\cal V}TM_{0}$ and $\pi^{*}TM$. In this case, there is the horizontal distribution ${\cal H}TM$ such that we have the Whitney sum: $TTM_{0}={\cal H}TM\oplus{\cal V}TM.$ This decomposition permits to write a vector $\hat{X}\in TTM_{0}$ into the form $\hat{X}=H\hat{X}+V\hat{X}$ uniquely. In the sequel we denote all vector fields on $TM_{0}$ by $\hat{X},\hat{Y}$, etc and the corresponding sections of $\pi^{*}TM$ by $X=\varrho(X)$, $Y=\varrho(Y)$, etc, respectively, unless otherwise specified. The structural equations of the regular connection $\nabla$ are given by: $\tau(\hat{X},\hat{Y})=\nabla_{\hat{X}}Y-\nabla_{\hat{Y}}X-\varrho[\hat{X},\hat{Y}],$ (1) $\Omega(\hat{X},\hat{Y})Z=\nabla_{\hat{X}}\nabla_{\hat{Y}}Z-\nabla_{\hat{Y}}\nabla_{\hat{X}}Z-\nabla_{[\hat{X},\hat{Y}]}Z,$ (2) where $X=\varrho(\hat{X})$, $Y=\varrho(\hat{Y})$, $Z=\varrho(\hat{Z})$ and $\hat{X}$, $\hat{Y}$ and $\hat{Y}$ are vector fields on $TM_{0}$. The tensors $\tau$ and $\Omega$ are called Torsion and Curvature tensors of $\nabla$, respectively. They determine two torsion tensors denoted here by $S$ and $T$ and three curvature tensors denoted by $R$, $P$ and $Q$ defined by: $S(X,Y)=\tau(H\hat{X},H\hat{Y}),\ \ \ T(\dot{X},Y)=\tau(V\hat{X},H\hat{Y}),$ $R(X,Y)=\Omega(H\hat{X},H\hat{Y}),\ \ P(X,\dot{Y})=\Omega(H\hat{X},V\hat{Y}),\ \ Q(\dot{X},\dot{Y})=\Omega(V\hat{X},V\hat{Y}),$ where $X=\varrho(\hat{X})$, $Y=\varrho(\hat{Y})$, $\dot{X}=\mu(\hat{X})$ and $\dot{Y}=\mu(\hat{Y})$. The tensors $R$, $P$ and $Q$ are called $hh-$, $hv-$ and $vv-$curvature tensors, respectively. Using the Jacobi identity for three vector fields ${\hat{X}}$, ${\hat{Y}}$ and ${\hat{Z}}$, one obtains the Bianchi identities for a regular connection $\nabla$ with curvature 2-forms $\Omega$, as follows: $\sigma\Omega({\hat{X}},{\hat{Y}})Z=\sigma\nabla_{{\hat{Z}}}\tau({\hat{X}},{\hat{Y}})+\sigma\tau({\hat{Z}},[{\hat{X}},{\hat{Y}}]),$ (3) $\sigma\nabla_{{\hat{Z}}}\Omega({\hat{X}},{\hat{Y}})+\sigma\Omega({\hat{Z}},[{\hat{X}},{\hat{Y}}])=0,$ (4) where, $\sigma$ denotes the circular permutation in the set $\\{{\hat{X}},{\hat{Y}},{\hat{Z}}\\}$. Let $(x^{i})$ be a local chart with the domain $U\subseteq M$ and $(x^{i},v^{i})$ the induced local coordinates on $\pi^{-1}(U)$ where ${\bf v}=v^{i}\frac{\partial}{\partial x^{i}}\in T_{\pi z}M$, where $i$ run over the range $1,2,...,n$. A Finsler structure $F$ is defined to be a function $F$ on $TM_{0}$ satisfying the following conditions: (1)$\ F>0\ \textrm{and}\ C^{\infty}\ \textrm{on}\ TM_{0}$, (2)$\ F(x,\lambda v)=\lambda F(x,v),$ for every $\lambda>0$ and (3)$\ \ g_{ij}(x,v)=\frac{1}{2}\frac{\partial^{2}F^{2}}{\partial v^{i}\partial v^{j}}$ is positive definite. The pair $(M,F)$ is called a Finsler manifold. There is a unique regular connection associated to $F$ such that: $\displaystyle\nabla_{\hat{Z}}g$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle S(X,Y)$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle g(\tau(V\hat{X},\hat{Y}),Z)$ $\displaystyle=$ $\displaystyle g(\tau(V\hat{X},\hat{Z}),Y),$ where $X=\varrho(\hat{X})$, $Y=\varrho(\hat{Y})$ and $Z=\varrho(\hat{Z})$ for all $\hat{X}$, $\hat{Y}$, $\hat{Z}\in TTM_{0}$. The regular connection $\nabla$ is called the Cartan connection. Given an induced natural coordinates on $\pi^{-1}(U)$, the coefficients of $\nabla$ can be written as follows: $\nabla_{\partial_{j}}\partial_{i}=\Gamma^{k}_{\ ij}\partial_{k},\ \ \nabla_{\overset{\bullet}{\partial}_{j}}\partial_{i}=C^{k}_{\ ij}\partial_{k},$ where, $\partial_{i}=\frac{\partial}{\partial x^{i}},\ \overset{\bullet}{\partial}_{i}=\frac{\partial}{\partial v^{i}}$ and $\Gamma^{k}_{\ ij}$ and $C^{k}_{\ ij}$ are smooth functions defined on $\pi^{-1}(U)$. One can observe that components of the second torsion tensor $T$ coincides with components of Cartan tensor $C$ in this coordinates, that is $T_{ijk}=\frac{1}{2}\overset{\bullet}{\partial}_{k}g_{ij}$, where, $T_{ijk}=g_{ir}T^{r}_{\ jk}$. It can be shown that the set $\\{\delta_{j}\\}$ defined by $\delta_{j}=\partial_{j}-\Gamma^{k}_{\ 0j}\overset{\bullet}{\partial}_{k}$ form a local frame field for the horizontal space ${\cal H}TM$. Assume that $\nabla_{\delta_{j}}\partial_{i}=\overset{*}{\Gamma}{{}^{k}_{\ ij}}\partial_{k}$. One can easily see that $\overset{*}{\Gamma}{{}^{k}_{\ ij}}$ is symmetric with respect to the indices $i$ and $j$. The curvature operator $\Omega(\hat{X},\hat{Y})$ of Cartan connection is anti-symmetric in the following sense $g(\Omega(\hat{X},\hat{Y})Z,W)=-g(\Omega(\hat{X},\hat{Y})W,Z),$ (5) where $\hat{X},\hat{Y}\in{\cal X}(TM_{0})$, $Z=\varrho(\hat{Z})$ and $W=\varrho(\hat{W})$. The $hv-$curvature tensor $P$ and the $vv-$ curvatures tensor $Q$ of the Cartan connection $\nabla$ are given respectively by $P^{i}_{\ jkl}=\nabla^{i}T_{jkl}-\nabla_{j}T^{i}_{\ kl}+T^{i}_{\ kr}\nabla_{0}T^{r}_{\ jl}-T^{r}_{\ kj}\nabla_{0}T^{i}_{\ rl},$ (6) $Q^{i}_{\ jkl}=T^{i}_{\ rl}T^{r}_{\ jk}-T^{i}_{\ rk}T^{r}_{\ jl}.$ (7) Among the Finsler manifolds, there are some classes determined by non- Riemannian quantities. One of them which is appeared in the present work is the $P$-symmetric Finsler manifolds required a kind of partial symmetry in the indices of $P$. This class of Finsler manifolds, was introduced by M. Matsumoto an H. Shimada in [9] and [10], and has been extensively studied by many authors. The curvature tensor $P^{i}_{\ jkl}$ can be decomposed into the sum of two symmetric and anti-symmetric tensors with respect to the indices $k$ and $l$, that is to say $P={{}^{s}P}+{{}^{a}P}$. By means of Eq.(6) the symmetric tensor ${{}^{s}P}$ can be written in the following form: ${{}^{s}P}^{i}_{\ jkl}=\nabla^{i}T_{jkl}+\frac{1}{2}\\{T^{i}_{\ kr}\nabla_{0}T^{r}_{\ jl}-T^{r}_{\ kj}\nabla_{0}T^{i}_{\ rl}+T^{i}_{\ lr}\nabla_{0}T^{r}_{\ jk}-T^{r}_{\ lj}\nabla_{0}T^{i}_{\ rk}\\}.$ (8) A Finsler manifold is said to be P-symmetric if $P(X,Y)=P(Y,X),$ $\forall X,Y\in\Gamma(\pi^{*}TM)$. $P$-symmetric spaces are closely related to the Finsler manifolds of isotropic sectional curvature. In this relation the following result is well-known. ###### Proposition 2.1. [9] A Finsler manifold is $P$-symmetric if and only if $\nabla_{\hat{\bf v}}Q=0$. Next we consider the Berwald connection $D$ which is not metric-compatible but a torsion free regular connection relative to $F$. There is the following relation between the connections $\nabla$ and $D$ $D_{H\hat{X}}Y=\nabla_{H\hat{X}}Y+(\nabla_{\hat{{\bf v}}}T)(X,Y),\ \ \ \ D_{V\hat{X}}Y=(V\hat{X}.Y^{i})\partial_{i}\,$ (9) where the vector field $\hat{{\bf v}}=v^{i}\delta_{i}$ is the canonical geodesic spray of $F$. If we assume $D_{\delta_{j}}\partial_{i}=G^{k}_{\ ij}\partial_{k},\ \ $ then Eqs.(9) can be written in the following local form: $G^{i}_{\ jk}=\overset{*}{\Gamma}{{}^{i}_{\ jk}}+\nabla_{0}T^{i}_{\ jk},\ \ \ D_{\overset{\bullet}{\partial}_{j}}\partial_{i}=0.$ It is clear from Eq.(9) that the connections $D$ and $\nabla$ associate to the same geodesic spray, since we have $\nabla_{\hat{X}}{\bf v}=D_{\hat{X}}{\bf v}$. The metric tensor $g$ related to the Finsler structure $F$ is parallel along any geodesic of Berwald connection, that is equivalent to $D_{\hat{\bf v}}g=0$. The Berwald connection $D$ admits the $hh-$ curvature tensors $H$ and the $hv-$ curvature tensors $G$ with the components $H^{i}_{\ jkl}$ and $G^{i}_{\ jkl}$. $G^{i}_{\ jkl}$ and $H^{i}_{\ jkl}$ can be determined by $G^{i}_{\ jkl}=\overset{\bullet}{\partial}_{l}\overset{\bullet}{\partial}_{k}\overset{\bullet}{\partial}_{j}G^{i}=\overset{\bullet}{\partial}_{l}G^{i}_{\ jk},$ $H^{i}_{\ jkl}=(\delta_{k}G^{i}_{\ jl}-G^{i}_{\ ljs}G^{s}_{\ k})-(\delta_{l}G^{i}_{\ jk}-G^{i}_{\ kjs}G^{s}_{\ l})+G^{i}_{\ rk}G^{i}_{\ jl}-G^{i}_{\ rl}G^{i}_{\ jk}.$ Let $z\in TM_{0}$ and ${\cal P}({\bf v},X)\subseteq T_{\pi z}M$ be a plane, generated by ${\bf v}$ and a linearly independent vector $X$ in $T_{\pi z}M$. The flag curvature at the point $z\in TM_{0}$ with respect to ${\cal P}({\bf v},X)$ is denoted by ${\bf K}(z,{\cal P}({\bf v},X))$ and is defined as follows: ${\bf K}(z,{\cal P}({\bf v},X))=\frac{g(R(X,{\bf v}){\bf v},X)}{g(X,X)F^{2}-g(X,{\bf v})^{2}},$ where, $R$ denotes the $hh-$curvature of Cartan connection [12]. Note that, the flag curvature ${\bf K}(z,{\cal P}({\bf v},X))$ does not depend on the choice of Berwald and Cartan connection, since, after a simple calculation. In fact, one can easily show that $H(X,{\bf v}){\bf v}=R(X,{\bf v}){\bf v}.$ (10) The Finsler manifold $(M,F)$ is said to be of scalar flag curvature at the point $z\in TM_{0}$ if ${\bf K}(z,P({\bf v},X))$ does not depend on the choice of the plane ${\cal P}({\bf v},X)$ and it is said to be of scalar flag curvature if it is of scalar flag curvature at all points $z\in TM_{0}$. In this case we have: $R(X,{\bf v}){\bf v}={\bf K}(z)\\{F^{2}X-g(X,{\bf v}){\bf v}\\},\ \ \ \forall X\in\Gamma(\pi^{*}TM).$ ### 2.2 Finsler submanifolds. Let $S$ be a k-dimensional embedded submanifold of the Finsler manifold $(M,F)$ defined by embedding ${\bf i}:S\longrightarrow M$. We identify a point $\tilde{x}\in S$ and a tangent vector $\widetilde{X}\in T_{\tilde{x}}S$ by by ${\bf i}(\tilde{x})$ and ${\bf i}_{*}\widetilde{X}$, respectively. Hence, $T_{\tilde{x}}S$ can be considered as a subspace of $T_{\tilde{x}}M$. The embedding ${\bf i}$ induces a map $\tilde{{\bf i}}={\bf i}_{*}:TS_{0}\longrightarrow TM_{0}$. If we identify a point $\widetilde{z}\in TS_{0}$ with its image $\tilde{{\bf i}}(\widetilde{z})$, then $TS_{0}$ can be considered as a sub-fiber bundle of $TM_{0}$. Restricting the map $\pi:TM_{0}\longrightarrow M$ to $TS_{0}$, we obtain the mapping $q:TS_{0}\longrightarrow M$. Denote by $\hat{T}S={\bf i}^{*}TM$, the pulled back bundle of $TM$. The Finsler metric $g$ on $M$ induces a Finsler metric on $S$ which is denoted by $\widetilde{g}$. Given any point $\widetilde{x}=q(\widetilde{z})\in S$, where $\widetilde{z}\in TS_{0}$, we denote by $N_{q(\widetilde{z})}$ the orthogonal complementary subspace of $T_{q(\widetilde{z})}M$ in $\hat{T}_{q(\widetilde{z})}S$. Therefore we have the Whitney sum $\hat{T}_{q(\tilde{z})}S=T_{q(\tilde{z})}S\oplus N_{q(\tilde{z})}.$ (11) The above decomposition defines the two projection maps ${\bf P}_{1}$ and ${\bf P}_{2}$ as follows: ${\bf P}_{1}:\hat{T}S\longrightarrow TS,$ ${\bf P}_{2}:\hat{T}S\longrightarrow N,$ where $N=\bigcup_{\tilde{z}\in TS_{0}}N_{q(\tilde{z})}$. We have $q^{*}\hat{T}S=q^{*}TS\oplus N$. $N$ is called the normal fiber bundle. We denote by $\rho$ the canonical linear mapping $TTS_{0}\longrightarrow q^{*}TS$, that is, $\rho=q_{*}$. Let $\widetilde{X}$ and $\widetilde{Y}$ be two vector fields on $TS_{0}$. Given $\tilde{z}\in TS_{0}$, $(\nabla_{\widetilde{X}}Y)_{\tilde{z}}$ belongs to $\hat{T}_{q(\tilde{z})}S$. Therefore, using the decomposition (11), we get $\nabla_{\widetilde{X}}Y=\widetilde{\nabla}_{\widetilde{X}}Y+\alpha(\widetilde{X},Y),$ (12) where, $\nabla$ is the Cartan connection, $Y=\rho(\widetilde{Y})$, $\widetilde{\nabla}_{\widetilde{X}}Y\in T_{q(\tilde{z})}S$ and $\alpha(\widetilde{X},Y)\in N_{q(\tilde{z})}$. $\alpha$ is called the second fundamental form of $S$. From Eq.(12), it follows that, $\widetilde{\nabla}$ is a covariant derivative in the vector bundle $q^{*}TS\longrightarrow TS_{0}$ and satisfies $\widetilde{\nabla}\widetilde{g}=0$. $\widetilde{\nabla}$ is called the tangential covariant derivation. $\alpha(\widetilde{X},\rho(\widetilde{Y}))$ is a bilinear form possessing its values in $N$. Let us denote by $\widetilde{\tau}$ the torsion tensor of $\widetilde{\nabla}$. Then, we have ${\bf P}_{1}\tau(\widetilde{X},\widetilde{Y})=\widetilde{\tau}(\widetilde{X},\widetilde{Y})=\widetilde{\nabla}_{\widetilde{X}}Y-\widetilde{\nabla}_{\widetilde{Y}}X-\rho[\widetilde{X},\widetilde{Y}],$ ${\bf P}_{2}\tau(\widetilde{X},\widetilde{Y})=\alpha(\widetilde{X},Y)-\alpha(\widetilde{Y},X),$ where $X=\rho(\widetilde{X})$ and $Y=\rho(\widetilde{Y})$. The submanifold $S$ is said to be totally geodesic at a point $\tilde{x}\in S$ if, for every tangent vector $\widetilde{X}\in T_{\tilde{x}}S$, the geodesic $\gamma(t)$ of $M$ in the direction of $\widetilde{X}$ lies in $S$ for small values of the parameter $t$. If $S$ is totally geodesic at every point of $S$, it is called a totally geodesic submanifold of $M$. ###### Theorem 2.2. [1] Let $S$ be a submanifold of the Finsler manifold $(M,F)$ with the second fundamental form $\alpha$. Then, $S$ is a totally geodesic submanifold if and only if $\alpha(\widetilde{X},{\bf v})=0$, for all $\widetilde{X}\in{\cal X}(TS_{0})$. The submanifold $S$ is also said to be an auto-parallel submanifold of $M$ if the second fundamental form $\alpha$ vanishes identically. Note that, in the Riemannian manifolds, the concepts of auto-parallel and totally geodesic submanifolds coincide. Clearly, every auto-parallel submanifold is also totally geodesic. Notice that, on an auto-parallel submanifold $S$, the induced connection $\widetilde{\nabla}$ coincides with the Cartan connection of the induced Finsler structure $\widetilde{F}=\tilde{{\bf i}}^{*}F$. ### 2.3 Nullity space of curvature operator in Finsler geometry. Let $(M,F)$ be a Finsler manifold and $\nabla$ the Cartan connection related to $F$. Given any point $z\in TM_{0}$, consider the subspace of ${\cal H}_{z}TM$ defined by: $N_{z}:=\\{\hat{X}\in{\cal H}_{z}TM|\ \ \Omega(\hat{X},\hat{Y})=0,\ \ \forall\hat{Y}\in{\cal H}_{z}TM\\},$ where, $\Omega$ is the curvature operator of $\nabla$. For any point $z\in TM_{0}$ where $\pi z=x$. The subspace ${\cal N}_{x}=\varrho(N_{z})\subset T_{x}M$ is linearly isomorphic to $N_{z}$. ${\cal N}_{x}$ is called the nullity space of the curvature operator on the Finsler manifold $(M,F)$ at the point $x\in M$, while ${\cal N}$ will denote the field of nullity spaces. Its orthogonal complementary space in $T_{x}M$ is called the co-nullity space at $x$ and is denoted by $\overset{\perp}{{\cal N}}_{x}$. Every element of ${\cal N}_{x}$ is called a nullity vector. The non-negative integer valued function $\mu_{0}:M\longrightarrow I\\!{N}$ defined by $\mu_{0}(p)=\dim{\cal N}_{p}$ is called the index of nullity and $\mu_{0}(p)$ is called the index of nullity at the point $p\in M$. Nullity space is called locally constant if given any $x\in M$, there is a neighborhood $U$ of $x$ such that the function $\mu_{0}$ is constant on $U$. In this case, the correspondence $x\in U\mapsto{\cal N}_{x}$ is a distribution called the nullity distribution on $U$. In the sequel we assume $0<\mu_{0}<n$ unless otherwise specified. Let $\ker_{x}\Omega$ be the kernel of the operator $\Omega$, that is $\ker_{x}\Omega=\\{Z\in T_{x}M|\ \Omega(\hat{X},\hat{Y})Z=0,\ \forall\hat{X},\hat{Y}\in{\cal H}_{z}TM\\}.$ (13) Akbar-Zadeh proved that, ${\cal N}_{x}=\ker_{x}\Omega$ and moreover, if the nullity space is locally constant on $U$, then the nullity distribution on $U$ is completely integrable. This is an extension of the similar result in Riemannian manifolds, established by Maltz [11] and Gray [7]. Akbar-Zadeh proved the following result: ###### Theorem 2.3. Let $(M,F)$ be a complete Finsler manifold and $G$ an open subset in $M$ on which $\mu_{0}$ is minimum. Then, every nullity manifold is a geodesically complete submanifold of $M$. ## 3 k-Nullity space of Cartan connection’s curvature operator. Let $(M,F)$ be an n-dimensional Finsler manifold endowed with the Cartan connection $\nabla$. The aim of this section is to associate to $(M,F)$ a k-nullity space of the Cartan connection’s curvature operator. We first introduce the concept of k-nullity space as a natural extension of nullity space in Finsler geometry containing nullity space as a special case ${\bf k}=0$. Furthermore, we study fundamental properties of k-nullity spaces. Given any non-negative real number k, we define the tensors $\eta^{\bf k}$ and $\bar{\Omega}$ as follows $\eta^{\bf k}(\hat{X},\hat{Y})Z={\bf k}\\{g(Y,Z)X-g(X,Z)Y\\}+{{}^{a}P}(X,\dot{Y})Z,$ $\bar{\Omega}(\hat{X},\hat{Y})Z=\Omega(\hat{X},\hat{Y})Z-\eta^{\bf k}(\hat{X},\hat{Y})Z,$ (14) where, $\hat{X},\hat{Y},\hat{Z}\in{\cal X}(TM_{0})$, $X=\varrho(\hat{X})$, $Y=\varrho(\hat{Y})$, $Z=\varrho(\hat{Z})$ and ${{}^{a}P}$ is the anti- symmetric part of $hv-$curvature tensor $P(X,Y)$. We refer to $\bar{\Omega}$ as the related curvature operator of $\Omega$. The local representation of $\bar{\Omega}(H\hat{X},H\hat{Y})$ is given by $\bar{\Omega}^{i}_{\ jkl}=R^{i}_{\ jkl}-{\bf k}\\{g_{jk}\delta^{i}_{\ l}-g_{jl}\delta^{i}_{\ k}\\},$ and we have from Eq.(14) $\bar{\Omega}(H\hat{X},V\hat{Y})={{}^{s}P}(X,\dot{Y}),$ (15) where, $\dot{Y}=\mu(\hat{Y})$. Notice that, Eq.(8) yields $\bar{\Omega}(H\hat{X},V\hat{Y}){\bf v}={{}^{s}P}(X,\dot{Y}){\bf v}=0,$ (16) where ${\bf v}$ is the canonical section of $\pi^{*}TM$ given by ${\bf v}=v^{i}{\partial}_{i}$. Given any point $z\in TM_{0}$, we define the subspace $N^{\bf k}_{z}$ of ${\cal H}_{z}TM$ by $N^{{\bf k}}_{z}:=\\{\hat{X}\in{\cal H}_{z}TM|\ \bar{\Omega}(\hat{X},\hat{Y})=0,\ \ \ \forall\hat{Y}\in{\cal H}_{z}TM\\}.$ For any point $z\in TM_{0}$ and $\pi z=x$, we consider the subspace ${\cal N}^{{\bf k}}_{x}=\varrho(N^{{\bf k}}_{z})\subset T_{x}M$. Clearly, the subspace ${\cal N}^{\bf k}_{x}=\varrho(N^{\bf k}_{z})\subset T_{x}M$ is linearly isomorphic to $N^{\bf k}_{z}$, since $\varrho$ is a linear isomorphism between ${\cal H}TM$ and $\pi^{*}TM$. Now, we are in position to define a non-Riemannian k-nullity space on Finsler manifolds. ###### Definition 3.1. Let $(M,F)$ be a Finsler manifold. ${\cal N}^{\bf k}_{x}$ is called the k-nullity space of the curvature operator on the Finsler manifold $(M,F)$ at the point $x\in M$, while ${\cal N}^{\bf k}$ will denote the field of k-nullity spaces. Its orthogonal complementary space in $T_{x}M$ is denoted by ${\scriptstyle\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}}$. Every element of ${\cal N}^{\bf k}_{x}$ is called a k-nullity vector. The non-negative integer valued function $\mu_{\bf k}:M\longrightarrow I\\!{N}$ defined by $\mu_{\bf k}(p)=\dim{\cal N}^{\bf k}_{p}$ is called the index of k-nullity at the point $p\in M$. k-nullity space is called locally constant if given any $x\in M$, there is a neighborhood $U$ of $x$ such that the function $\mu_{\bf k}$ is constant on $U$. In this case, the correspondence $x\in U\mapsto{\cal N}^{\bf k}_{x}$ is a distribution called the k-nullity distribution on $U$. The function $\mu_{\bf k}:M\longrightarrow I\\!\\!{N}$ is upper semi- continuous. In the sequel we assume that $0<\mu_{\bf k}<n$ unless otherwise specified. Observe that, the following relations hold for $\eta^{\bf k}$: $\sigma\eta^{\bf k}({\hat{X}},{\hat{Y}})Z=0,\ \ \ \nabla_{\hat{Z}}\eta^{\bf k}=0,\ \ \ \forall\hat{X},\hat{Y},\hat{Z}\in{\cal H}TM.$ (17) where, $\sigma$ is a circular permutation on the set $\\{\hat{X},\hat{Y},\hat{Z}\\}$. Thus, it is clear that we have: $\sigma\bar{\Omega}({\hat{X}},{\hat{Y}})Z=\sigma\Omega({\hat{X}},{\hat{Y}})Z,\ \ \ \forall\hat{X},\hat{Y},\hat{Z}\in{\cal H}TM.$ (18) The tensor $\bar{\Omega}$ has somehow the same algebraic properties as $\Omega$. The following properties of $\bar{\Omega}$ are easily verified: ###### Lemma 3.2. The following statements hold for $\bar{\Omega}$: $(1)\ \ \sigma\bar{\Omega}({\hat{X}},{\hat{Y}})Z=\sigma\nabla_{{\hat{Z}}}\tau({\hat{X}},{\hat{Y}})+\sigma\tau({\hat{Z}},[{\hat{X}},{\hat{Y}}]),$ $(2)\ \ \sigma\nabla_{{\hat{Z}}}\bar{\Omega}({\hat{X}},{\hat{Y}})+\sigma\bar{\Omega}({\hat{Z}},[{\hat{X}},{\hat{Y}}])=0,$ $(3)\ \ g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=-g(\bar{\Omega}(\hat{X},\hat{Y})W,Z)$, where, $\hat{X},\hat{Y},\hat{Z},\hat{W}\in{\cal H}TM$ and $\sigma$ is a circular permutation in the set $\\{\hat{X},\hat{Y},\hat{Z}\\}$. ###### Proof. The proof is a simple application of Bianchi identities, Eq.(5), Eq.(17) and Eq.(18). ∎ Proof of Theorem 1.1. Let $\hat{X}$, $\hat{Y}$ and $\hat{Z}$ be three horizontal vector fields on $TM_{0}$ such that $\hat{X},\hat{Y}\in N^{\bf k}_{z}$. Taking into account Eq.(16) and Eq.(10), by a straightforward computation, we have $\varrho[\hat{X},\hat{Y}]=[X,Y]_{\pi},$ $\mu([\hat{X},\hat{Y}])=-\Omega(\hat{X},\hat{Y}){\bf v}=-\eta^{\bf k}(\hat{X},\hat{Y}){\bf v},$ (19) $H[\hat{X},\hat{Y}]=[\hat{X},\hat{Y}]+\eta^{\bf k}(\hat{X},\hat{Y})v^{r}\overset{\bullet}{\partial}_{r}.$ (20) In this case, the relation (2) in Lemma 3.2 reduces to $\bar{\Omega}(\hat{X},[\hat{Y},\hat{Z}])+\bar{\Omega}(\hat{Y},[\hat{Z},\hat{X}])+\bar{\Omega}(\hat{Z},[\hat{X},\hat{Y}])=0$ The last equation can be written in the following form: $\bar{\Omega}(\hat{X},V[\hat{Y},\hat{Z}])+\bar{\Omega}(\hat{Y},V[\hat{Z},\hat{X}])+\bar{\Omega}(\hat{Z},[\hat{X},\hat{Y}])=0.$ (21) Following Eq.(15) and Eq.(19), first and second terms of Eq.(21) become: $\bar{\Omega}(\hat{X},V[\hat{Y},\hat{Z}])={{}^{s}P}(X,\mu[\hat{Y},\hat{Z}])=-{{}^{s}P}(X,\eta^{\bf k}(\hat{Y},\hat{Z}){\bf v})$ (22) $={\bf k}g(Z,{\bf v}){{}^{s}P}(X,Y)-{\bf k}g(Y,{\bf v}){{}^{s}P}(X,Z),$ $\bar{\Omega}(\hat{Y},V[\hat{Z},\hat{X}])={{}^{s}P}(Y,\mu[\hat{Z},\hat{X}])=-{{}^{s}P}(Y,\eta^{\bf k}(\hat{Z},\hat{X}){\bf v})$ (23) $={\bf k}g(X,{\bf v}){{}^{s}P}(Y,Z)-{\bf k}g(Z,{\bf v}){{}^{s}P}(Y,X).$ By means of Eq.(22) and Eq.(23) and the symmetry property ${{}^{s}P}(X,Y)={{}^{s}P}(Y,X)$, Eq.(21) can be written in the following form: $\bar{\Omega}(\hat{Z},[\hat{X},\hat{Y}]+\eta^{\bf k}(\hat{X},\hat{Y})v^{r}\overset{\bullet}{\partial}_{r})=0,$ Following Eq.(20), the last equation becomes: $\bar{\Omega}(\hat{Z},H[\hat{X},\hat{Y}])=0,\ \ \ \hat{Z}\in{\cal H}_{z}TM.$ Indeed $H[\hat{X},\hat{Y}]\in N^{\bf k}_{z}$ and $[X,Y]=\varrho[\hat{X},\hat{Y}]=\varrho(H[\hat{X},\hat{Y}])\in{\cal N}^{\bf k}_{x}$. Therefore, ${\bf k}$-nullity distribution is involutive or completely integrable.∎ Considering the kernel of the operator $\bar{\Omega}$ $\ker_{x}\bar{\Omega}=\\{Z\in T_{x}M|\ \bar{\Omega}(\hat{X},\hat{Y})Z=0,\ \ \hat{X},\hat{Y}\in{\cal H}_{z}TM\\},$ we shall show that ${\cal N}^{{\bf k}}_{x}=\ker_{x}\bar{\Omega}$. Proof of Theorem 1.2. Let $\hat{X}$, $\hat{Y}$ and $\hat{Z}$ be three horizontal vector fields on $TM_{0}$ such that $\hat{X},\hat{Y}\notin N^{\bf k}_{z}$ but $\hat{Z}\in N^{\bf k}_{z}$. In this case, the relation (1) in Lemma 3.2 reduces to $\bar{\Omega}(\hat{X},\hat{Y})Z=\tau(\hat{X},[\hat{Y},\hat{Z}])+\tau(\hat{Y},[\hat{Z},\hat{X}])+\tau(\hat{Z},[\hat{X},\hat{Y}])$ On the other hand, for every vector field $\hat{W}\in{\cal X}(TM_{0})$, we have: $g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=g(\tau(\hat{X},[\hat{Y},\hat{Z}]),W)+g(\tau(\hat{Y},[\hat{Z},\hat{X}]),W)+g(\tau(\hat{Z},[\hat{X},\hat{Y}]),W).$ (24) Considering Eq.(19), we have the following relations for the torsion tensor $\tau$. $\tau(\hat{X},[\hat{Y},\hat{Z}])=T(\hat{X},\mu[\hat{Y},\hat{Z}])={\bf k}g(Y,{\bf v})T(X,Z)-{\bf k}g(Z,{\bf v})T(X,Y),$ (25) $\tau(\hat{Y},[\hat{Z},\hat{X}])=T(\hat{Y},\mu[\hat{Z},\hat{X}])={\bf k}g(Z,{\bf v})T(Y,X)-{\bf k}g(X,{\bf v})T(Y,Z),$ (26) $\tau(\hat{Z},[\hat{X},\hat{Y}])=T(\hat{Z},\mu[\hat{X},\hat{Y}])={\bf k}g(X,{\bf v})T(Z,Y)-{\bf k}g(Y,{\bf v})T(Z,X).$ (27) Replacing Eqs.(25),(27) and (27), in Eq.(24) we obtain $g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=2{\bf k}\\{g(Y,{\bf v})g(T(X,Z),W)+g(Z,{\bf v})g(T(Y,X),W)$ $+g(X,{\bf v})g(T(Z,Y),W)\\}.$ As a consequence of the relation (3) in Lemma 3.2, the left hand side of the previous equation is anti-symmetric with respect to $W$ and $Z$. Thus, it follows that $2{\bf k}\\{g(Y,{\bf v})g(T(X,Z),W)+g(X,{\bf v})g(T(Z,Y),W)\\}=0.$ Since $W$ is arbitrarily chosen, we have the following relation: $g(Y,{\bf v})T(X,Z)+g(X,{\bf v})T(Z,Y)=0.$ (28) From Eq.(19) one can conclude that $\tau(\hat{Z},[\hat{X},\hat{Y}])=\tau(\hat{Z},V[\hat{X},\hat{Y}])=T(Z,\mu[\hat{X},\hat{Y}])$ $={\bf k}g(X,{\bf v})T(Z,Y)-{\bf k}g(Y,{\bf v})T(Z,X).$ By anti-symmetry property of the tensor $T$ and Eq.(28), we get $\tau(\hat{Z},V[\hat{X},\hat{Y}])={\bf k}g(X,{\bf v})T(Z,Y)+{\bf k}g(Y,{\bf v})T(X,Z)=0.$ Plugging Eqs.(25),(26) and (27) into Eq.(24) results $g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=g(\tau(\hat{X},[\hat{Y},\hat{Z}]),W)+g(\tau(\hat{Y},[\hat{Z},\hat{X}]),W)+g(\tau(\hat{Z},[\hat{X},\hat{Y}]),W)=0.$ Therefore, we have $g(\bar{\Omega}(\hat{X},\hat{Y})Z,W)=g(\tau(\hat{Z},[\hat{X},\hat{Y}]),W)=0.$ Finally, since $W$ is arbitrarily chosen, we obtain the following equation, $\bar{\Omega}(\hat{X},\hat{Y})Z=\tau(\hat{Z},[\hat{X},\hat{Y}])=T(Z,\mu[\hat{X},\hat{Y}])=0.$ (29) The last equation shows that $Z\in\ker_{x}\bar{\Omega}$, that is ${\cal N}^{\bf k}_{x}\subseteq\ker_{x}\bar{\Omega}$ and $\ker\bar{\Omega}^{\perp}\subseteq\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$. Now, let $W\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$ and $U\in{\cal N}^{\bf k}_{x}$, we have $g(\bar{\Omega}(\hat{X},\hat{Y})W,U)=-g(\bar{\Omega}(\hat{X},\hat{Y})U,W)=0.$ The previous equation shows that $\bar{\Omega}(\hat{X},\hat{Y})W\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$, that is $Im_{x}\bar{\Omega}\subseteq\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$. For every k-nullity vector $U\in{\cal N}^{\bf k}_{x}$, Eq.(19) yields $g(\mu([\hat{X},\hat{Y}])+\eta^{\bf k}(\hat{X},\hat{Y}){\bf v},U)=-g(\bar{\Omega}(\hat{X},\hat{Y}){\bf v},U)=g(\bar{\Omega}(\hat{X},\hat{Y})U,{\bf v})=0.$ By definition of $\eta^{\bf k}$ and the fact that $X,Y\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$, we obtain $g(\eta^{\bf k}(\hat{X},\hat{Y}){\bf v},U)=0$. Therefore, $\mu([\hat{X},\hat{Y}])\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}.$ (30) From which $g(\mu([\hat{X},\hat{Y}]),U)=0$. Consider the following homomorphism of vector spaces: $\Psi:\frac{T_{x}M}{\ker_{x}\bar{\Omega}}\cong Im_{x}\bar{\Omega}\longrightarrow\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}},$ defined by $W+\ker_{x}\bar{\Omega}\mapsto\bar{\Omega}(\hat{X},\hat{Y})W$. It is clear that $\Psi$ is one-to-one and thus it is onto and therefore, ${\scriptstyle\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}}=\ker_{x}{\scriptstyle\bar{\Omega}^{\perp}}$ and ${\cal N}^{\bf k}_{x}=\ker_{x}\bar{\Omega}$. This completes the proof of Theorem. ∎ ## 4 Auto-parallel k-nullity maximal integral manifold Proof of Theorem 1.3. The method used here is inspired by Akbar-Zadeh’s technic. Let $N$ be an integral manifold of k-nullity distribution in $U$. For all vector fields $\widetilde{X},\widetilde{W}\in{\cal X}(TN_{0})$ we have by means of Eq.(12) $\nabla_{\widetilde{W}}X=\widetilde{\nabla}_{\widetilde{W}}X+\alpha(\widetilde{W},X),$ (31) where, $\widetilde{\nabla}$ denotes the induced connection on $TN_{0}$, $X=\rho(\widetilde{X})$ and $\alpha(\widetilde{W},X)$ is the second fundamental form of $N$. Let $\widetilde{X},\widetilde{Y}\in{\cal H}TN$ such that $X,Y\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$ and $U\in{\cal N}^{\bf k}_{x}$. By means of Theorem 1.2, we have $\bar{\Omega}(\hat{X},\hat{Y})U=0$. Suppose that $\widetilde{Z}\in N^{\bf k}_{z}$. It follows immediately from Eq.(31) that the covariant derivative of $\bar{\Omega}$ along $\widetilde{Z}$ becomes $\displaystyle(\nabla_{\widetilde{Z}}\bar{\Omega}(\widetilde{X},\widetilde{Y}))U$ $\displaystyle=$ $\displaystyle\nabla_{\widetilde{Z}}\bar{\Omega}(\widetilde{X},\widetilde{Y})U-\bar{\Omega}(\widetilde{X},\widetilde{Y})\nabla_{\widetilde{Z}}U$ $\displaystyle=$ $\displaystyle-\bar{\Omega}(\widetilde{X},\widetilde{Y})\nabla_{\widetilde{Z}}U$ $\displaystyle=$ $\displaystyle-\bar{\Omega}(\widetilde{X},\widetilde{Y})(\widetilde{\nabla}_{\widetilde{Z}}U+\alpha(\widetilde{Z},U))$ $\displaystyle=$ $\displaystyle-\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U).$ Therefore, $(\nabla_{\widetilde{Z}}\bar{\Omega}(\widetilde{X},\widetilde{Y}))U+\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U)=0.$ Using the identity (2) in Lemma 3.2 and the above equation, we obtain $\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U)=\bar{\Omega}(\widetilde{Z},[\widetilde{X},\widetilde{Y}])U={{}^{s}P}(\widetilde{Z},\mu[\widetilde{X},\widetilde{Y}])U.$ (32) If we assume $\mu[\widetilde{X},\widetilde{Y}]=0$, then we have $\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U)=0$. On the other hand $\alpha(\widetilde{Z},U)\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$, it follows that $\alpha(\widetilde{Z},U)=0$. In this case, the integral manifold $N$ is an auto-parallel submanifold. Otherwise, assume that $\mu[\widetilde{X},\widetilde{Y}]\neq 0$. Consider a basis $\\{{\bf e}_{1},{\bf e}_{2},...,{\bf e}_{n}\\}$ for $T_{x}M$ such that, the first $r$ vectors form a basis for ${\cal N}^{\bf k}_{x}$ and the rest $(n-r)$ vectors is a basis for $\overset{\perp}{{\cal N}}{{}^{\bf k}_{x}}$. In virtue of Eq.(30), without loss of generality, one can assume that the vector $\mu[\widetilde{X},\widetilde{Y}]$ is an element of basis $\\{{\bf e}_{r-1},...,{\bf e}_{n}\\}$. In the sequel, assume that the following indices run over the indicated ranges $a,b=1,2,...,n,\ \ \ \ \alpha,\beta=1,2,...,r,\ \ \ \ i,j=r-1,...,n.$ (33) Eq.(29) states that, in this basis, we have $T_{a\alpha j}=0$ (34) Plugging it into Eq.(8), it results ${{}^{s}P}_{ia\beta j}=\nabla_{i}T_{a\beta j}+\frac{1}{2}\\{T_{i\beta r}\nabla_{0}T^{r}_{\ aj}-T^{r}_{\ \beta a}\nabla_{0}T_{irj}+T_{ijr}\nabla_{0}T^{r}_{\ a\beta}-T^{r}_{\ ja}\nabla_{0}T_{ir\beta}\\}=0.$ From the last equation, it results that ${{}^{s}P}(\widetilde{Z},\mu[\widetilde{X},\widetilde{Y}])U=0$. Eq.(32) implies that $\bar{\Omega}(\widetilde{X},\widetilde{Y})\alpha(\widetilde{Z},U)=0$, that is to say $\alpha(\widetilde{Z},U)\in\ker\bar{\Omega}={\cal N}^{\bf k}_{x}$. It follows $\alpha(\widetilde{Z},U)=0$ and $N$ is an auto-parallel submanifold. Denote the curvature 2-forms of $\widetilde{\nabla}$ by $\widetilde{\Omega}$. Since $N$ is an auto-parallel submanifold of $M$, its curvature tensors are given by $\widetilde{\Omega}(H\widetilde{X},H\widetilde{Y})Z=\Omega(H\widetilde{X},H\widetilde{Y})Z={\bf k}\\{\widetilde{g}(Y,Z)X-\widetilde{g}(X,Z)Y\\},$ $\widetilde{\Omega}(H\widetilde{X},V\widetilde{Y})Z=\Omega(H\widetilde{X},V\widetilde{Y})Z={{}^{s}P}(X,\dot{Y})Z,$ $\widetilde{\Omega}(V\widetilde{X},V\widetilde{Y})Z=\Omega(V\widetilde{X},V\widetilde{Y})Z=Q(\dot{X},\dot{Y})Z,$ where, $\widetilde{X},\widetilde{Y}\in{\cal X}(TN_{0})$ and $Z\in\Gamma(\pi^{*}TN)$. The above relation shows that, $N$ is a $P$-symmetric space. Indeed components of the $hh-$curvature $\widetilde{R}^{\alpha}_{\ \beta\gamma\theta}$ of $(N,\widetilde{F})$ are given by $\widetilde{R}^{\alpha}_{\ \beta\gamma\theta}={\bf k}\\{\widetilde{g}_{\beta\gamma}\delta^{\alpha}_{\ \theta}-\widetilde{g}_{\beta\theta}\delta^{\alpha}_{\ \gamma}\\},$ where, $\widetilde{g}$ denotes the induced metric on $(N,\widetilde{F})$. Following Eq.(10), we have $\widetilde{H}(X,{\bf v}){\bf v}=\widetilde{R}(X,{\bf v}){\bf v}={\bf k}\\{\widetilde{g}({\bf v},{\bf v})Y-\widetilde{g}(Y,{\bf v}){\bf v}\\},$ which shows that $N$ is of constant flag curvature ${\bf k}$. ∎ ### 4.1 Completeness of the k-nullity foliation. Proof of Theorem 1.4. Let $(M,F)$ be an $n$-dimensional Finsler manifold and $\gamma:[0,c)\longrightarrow N$ a geodesic on the integral manifold $N$ of the k-nullity foliation in $G$. We shall prove that, $\gamma$ can be extended to a geodesic $\tilde{\gamma}:[0,\infty)\longrightarrow N$ on $N$. We shall proceed the proof with the contrary assumption, by supposing that such geodesic does not exist. Following Theorem 1.3, every ${\bf k}$-nullity manifold is auto- parallel and hence it is totally geodesic. Therefore, $\gamma$ is a geodesic on $M$ and has an extension $\tilde{\gamma}:[0,\infty)\longrightarrow M$ such that $\gamma=\tilde{\gamma}\cap N$. It follows that, $p=\tilde{\gamma}(c)\notin G$. Suppose that $p_{0}=\gamma(0)=\tilde{\gamma}(0)$ and put $r_{0}=\mu_{\bf k}(p_{0})$, the dimension of the k-nullity space at $p_{0}$. The function $\mu_{\bf k}:M\longrightarrow M$ attains its minimum on $G$ and it results that $\mu_{\bf k}(p)>r_{0}$. Consider a basis ${\cal B}_{0}=\\{{\bf e}_{1},{\bf e}_{2},...,{\bf e}_{r_{0}},{\bf e}_{r_{0}+1},...,{\bf e}_{n}\\}$ for $T_{p_{{}_{0}}}M$ such that $\\{{\bf e}_{1},{\bf e}_{2},...,{\bf e}_{r_{0}}\\}$ is a basis for ${\cal N}^{\bf k}_{p_{{}_{0}}}$ and ${\bf e}_{1}$ is the tangent vector to $\gamma$ at the point $p_{0}=\gamma(0)$. Using the following system of differential equations $\frac{\nabla{\bf E}_{i}}{dt}=0,\ \ \ \ \ {\bf E}_{i}(0)={\bf e}_{i},$ where $i=1,2,...,n$, one can translate the basis ${\cal B}_{0}$ into the parallel frame ${\cal B}=\\{{\bf E}_{1},{\bf E}_{2},...,{\bf E}_{r_{0}},{\bf E}_{r_{0}+1},...,{\bf E}_{n}\\}$ along $\tilde{\gamma}$. There is a neighborhood $U$ of $p$ on $M$ such the subset $\\{{\bf E}_{1},{\bf E}_{2},...,{\bf E}_{r_{0}}\\}$ is a basis for the ${\bf k}$-nullity space at every point $\tilde{\gamma}(t)$ in $G\cap U$. Since $\mu_{\bf k}(p)>r_{0}$, there is a vector field ${\bf E}_{a}$ along $\tilde{\gamma}$, for a fixed number $a$ in the range $r_{0}+1,...,n$, such that for every $t\in[0,c)$, we have ${\bf E}_{a}(t)\in\overset{\perp}{{\cal N}}{{}^{\bf k}_{\gamma(t)}}$ and ${\bf E}_{a}(c)\in{\cal N}^{\bf k}_{p}$. Now, let $\hat{\tilde{\gamma}}=(\tilde{\gamma},\dot{\tilde{\gamma}})$ be the natural lift of $\tilde{\gamma}$ to $TM_{0}$ and $\hat{{\cal B}}=\\{\hat{{\bf E}}_{1},\hat{{\bf E}}_{2},...,\hat{{\bf E}}_{r_{0}},\hat{{\bf E}}_{r_{0}+1},...,\hat{{\bf E}}_{n}\\}$ the basis for ${\cal H}_{\hat{\tilde{\gamma}}(t)}TM$ such that $\varrho(\hat{{\bf E}}_{i})={\bf E}_{i}$. Assume that the coefficients $f_{ija}$ are defined as follows $\bar{\Omega}(\hat{{\bf E}}_{i},\hat{{\bf E}}_{j}){\bf E}_{a}=f_{ija}.$ (35) Using the relation (2) in Lemma 3.2, the Cartan horizontal derivative of both sides of Eq.(35) along $\hat{\tilde{\gamma}}$ in $\pi^{-1}(G\cap U)$ and using the fact that, $\bar{\Omega}(\hat{{\bf E}}_{1},V[\hat{{\bf E}}_{j},\hat{{\bf E}}_{i}])=0$, we obtain $f^{\prime}_{ija}+\bar{\Omega}(\hat{{\bf E}}_{j},[\hat{{\bf E}}_{i},\hat{{\bf E}}_{1}])+\bar{\Omega}(\hat{{\bf E}}_{i},[\hat{{\bf E}}_{1},\hat{{\bf E}}_{j}])=0,$ (36) where, $i,j=r_{0}+1,...,n$. Plugging $\hat{{\bf E}}_{j}$, $\hat{{\bf E}}_{i}$ and $\hat{{\bf E}}_{1}$ instead of $\hat{X}$, $\hat{Y}$ and $\hat{Z}$ into Eq.(22) and Eq.(23) respectively, we obtain $\bar{\Omega}(\hat{{\bf E}}_{j},V[\hat{{\bf E}}_{i},\hat{{\bf E}}_{1}])+\bar{\Omega}(\hat{{\bf E}}_{i},V[\hat{{\bf E}}_{1},\hat{{\bf E}}_{j}])=0.$ Therefore, Eq.(36) becomes $f^{\prime}_{ija}+\bar{\Omega}(\hat{{\bf E}}_{j},H[\hat{{\bf E}}_{i},\hat{{\bf E}}_{1}])+\bar{\Omega}(\hat{{\bf E}}_{i},H[\hat{{\bf E}}_{1},\hat{{\bf E}}_{j}])=0.$ (37) But, the horizontal part of $[\hat{{\bf E}}_{j},\hat{{\bf E}}_{1}]$ can be written in the basis $\hat{{\cal B}}$ in the form $H[\hat{{\bf E}}_{1},\hat{{\bf E}}_{j}]={\bf W}^{k}_{j}\hat{{\bf E}}_{k}+{\bf W}^{a}_{j}\hat{{\bf E}}_{a},$ for some functions ${\bf W}^{k}_{j}$ defined on $\hat{\tilde{\gamma}}$ in $\pi^{-1}(U)$, where the index $k$ runs over the range $1,...,\hat{a},...n$ and the hat over $a$ indicates that the index $a$ is omitted. Plugging the terms $H[\hat{{\bf E}}_{j},\hat{{\bf E}}_{1}]$ and $H[\hat{{\bf E}}_{1},\hat{{\bf E}}_{i}]$ into Eq.(37), we obtain the homogenous system of ODEs $f^{\prime}_{ija}+{\bf W}^{k}_{i}f_{jka}-{\bf W}^{k}_{j}f_{ika}=0.$ Since ${\bf E}_{a}$ is a ${\bf k}$-nullity vector field at $p$, by means of Eq.(35), we have clearly for the fixed index $a$, $f_{lma}(c)=0$, where, $l,m=r_{0}+1,...,n$. Solving the system of ODEs above with initial value $f_{lma}(c)=0$ implies that $f_{lma}\equiv 0$. Eq.(35), implies that, ${\bf E}_{a}$ is a ${\bf k}$-nullity vector at every point of $\tilde{\gamma}$ in $G\cap U$ and specially, it is a ${\bf k}$-nullity vector at every point of $\gamma$ in $G\cap U$. Obviously, this is merely a contradiction to the contrary hypothesis and $\gamma$ can be extended to a geodesic $\tilde{\gamma}:[0,\infty)\longrightarrow N$. ∎ We remark that, relaxing the constant ${\bf k}$ to be zero in the Eq.(14) leads to a notion of non-Riemannian nullity in Finsler geometry which is a special case of the nullity space in [2]. ## References * [1] Akbar-Zadeh, H.: Initiation to Global Finslerian Geometry, North Holland, 2006. * [2] Akbar-Zadeh, H.: Espaces de nullité de cértains opérateurs géométrie des sous-variétés, _C. R. Acad. Sci. Paris_ Sér. A-B, 274 (1972) ,A490–A493. * [3] Chern S. S. and Kuiper N. H., Some theorems on isometric imbedding of compact Riemann manifolds in Euclidean space, _Ann. of Math._ 56 (1952), 313-316. * [4] Clifton, Y. and Maltz, H.: The k-nullity space of the curvature operator, _Michigan Math. J._ 17 (1970). * [5] Ferus, D.: On the completeness of nullity foliation, _Michigan Math. J._ 18 (1971). * [6] Ferus, D.: Totally geodesic foliation, _Math. Ann._ 188 313-316, (1970). * [7] Gray, A.: Space of constancy of curvature operator, _Proc. Amer. Math. Soc._ 17 897-902, (1966). * [8] Kowalski, O. and Sekizawa, M.: On Tangent Sphere Bundles with small or Large Constant Radius, _Ann. Global Anal. Geom._ 18: 207-219, (2000). * [9] Matsumoto, M. and Shimada, H.: On Finsler spaces with the curvature tensors $P_{hijk}$ and $S_{hijk}$ satisfying special conditions, _Rep. on Math. Phys._ 12 1 ,77-87,(1977). * [10] Matsumoto, M.: Finsler spaces with the hv-curvature tensor $P_{hijk}$ of a special form, , _Rep. on Math. Phys._ 14 1 ,1-13, (1978). * [11] Maltz, R.: The nullity space of the curvature operator, _Thesis_ , University of California Los Angeles, (1965). * [12] Shen, Z.: Lectures on Finsler Geometry, _World Scientific_ , Singapore, 2001. * [13] Sekizawa, M.: Completeness of the $k$-th nullity foliations, _J. Diff. Geom._ 11 no.3 , 461-465, (1976). * [14] Tachibana, S. I. and Sekizawa, M.: On the $k$-th nullity space of the Riemannian curvature tensor, _T hoku Math. J._ 2 27, 25-30, (1975). * [15] Tanno, S.: Some differential equations on Riemannian manifolds, _J. Math. Soc. Japan._ 30, no. 3, (1978).
arxiv-papers
2011-01-07T18:50:06
2024-09-04T02:49:16.204734
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "B. Bidabad, M. Rafie-Rad", "submitter": "Mehdi Rafie-Rad", "url": "https://arxiv.org/abs/1101.1496" }
1101.1537
# Time-Optimal solutions of Parallel Navigation and Finsler geodesics M. Rafie-Rad111Email address:_m.rafiei.rad@gmail.com_ Department of Mathematics, Faculty of Sciences, Mazandaran University, Bablosar, Iran. ###### Abstract A geometric approach to kinematics in control theory is illustrated. A non- linear control system is derived for the problem and the Pontryagin maximum principle is used to find the time-optimal trajectories of the Parallel navigation. The time-optimal trajectories of the Parallel navigation are characterized through a geometric formulation. It is notable that the approach has the advantages using feedback.222 2000 Mathematics subject Classification: Primary 53C60; Secondary 53B40. Keywords: Finsler geometry, Parallel navigation, Kinematics, Optimal control, Pontryagin maximum principle. ## 1 Introduction The historical development of what became the Calculus of Variations is closely linked to certain minimization principles in the majority subjects in mechanics, namely, the principle of least distance, the principle of least time and ultimately, the principle of least action [7]. To understand solution of the well-known brachistochrone problem, (i.e finding a curve from point $A$ to point $B$ along which a free-sliding particle will descend more quickly than on any other $AB$-curve), we are led through Fermat’s principle of least time: light always takes a path that minimizes travel time. The Parallel navigation, or briefly P-navigation, is a quiet old problem and has been studied using several techniques from the viewpoints of kinematics and dynamics in optimal control theory [17]. The application of Finsler geometry in Physics, seismology and Biology is a subject of numerous papers such as [1], [2],[3], [5], [9], [13], [15], [18], etc. Let $O$ be the origin of an inertial reference frame of coordinates (FOC). The positions of $M$ and $T$ in this (FOC) are given by the vectors ${\bf r}_{M}=OM$ and ${\bf r}_{T}=OT$, respectively. In two-point guidance systems, the vector ${\bf r}={\bf r}_{T}-{\bf r}_{M}$ is conventionally called the range. Its time derivative $\dot{\bf r}=\dot{\bf r}_{T}-\dot{\bf r}_{M}={\bf v}_{T}-{\bf v}_{M}$ is the relative velocity between the two objects, and ${\bf v}_{T}$ and ${\bf v}_{M}$ are the velocities of $T$ and $M$, respectively. W always denote the vectors by bold face and their norms will be shown by the same normal letter. As an application, it is notable for mariners wishing to rendez-vous each other at sea. $M$ could be a boat and $T$, a tanker with fuel for it (or vice-versa). Or, back in history, $T$ could be a merchantman and $M$ a pirate ship. This rule assumes, of course, constant speeds. Thus, in most realistic cases, $v_{T}$ and $v_{M}$ are supposed to be constant. However, it is easy to extend the theory if they are not constant. The closing velocity, a term often used in the study of guidance, is simply ${\bf v}_{C}=-\dot{\bf r}$. Notice that, we wish to study the kinematics of P-navigation in a relative (FOC) rather than a absolute one, i.e., we shall seek the location of $M$ in a (FOC) attached to $T$. Thus, a trajectory in the relative (FOC) shows the situation as seen by an observer located at $T$. As the special cases, we assume that $M=R^{3}$ or $M=R^{2}$. In reality, the velocity ${\bf v}_{T}$ and ${\bf r}_{T}$ can be detected and reported at any ${\bf r}$ by a grounded radar. Suppose that $\delta({\bf r})$ be the angle between ${\bf v}_{M}$ and $MT$ and given any $\delta$, there is Finsler metric $F$ given by: $F({\bf r},{\bf v},\delta)=\frac{|{\bf v}|^{2}}{v_{M}\cos\delta|{\bf v}|-\langle{\bf v},{\bf v}_{T}\rangle},$ (1) where, $|.|$ denotes the Riemannian norm on $M$. A solution of the described P-navigation is a curve $({\bf r}(t),\delta(t))$ such that respects the required constraints on velocities. ###### Theorem 1.1 Given any solution $({\bf r},\delta)$ of parallel navigation, the curve ${\bf r}$ can be reparametrized so that it satisfies $F({\bf r}(t),{\bf v}(t),\delta(t))=1$. The indicatrix $S({\bf r},\delta)$ of the metric (1) is the set of unit tangent vectors ${\bf v}$ with respect to (1) which is defined by $S({\bf r},\delta)=\\{{\bf v}\in T_{\bf r}M\ |\ F({\bf r},{\bf v},\delta)=1\\}$. Following Theorem 1.1, at any time $t$ we have $\dot{\bf r}={\bf v}\in S({\bf r},\delta)$. Hence, at any time $t$, there is a unit vector $f({\bf r},\delta)\in S({\bf r},\delta)$ such that $\dot{\bf r}={\bf v}=f({\bf r},\delta)$. Control problems typically concern finding a (not necessarily unique) control law $\delta(.)$ , which transfers the system in finite time from a given initial state $x_{i}={\bf r}(0)$ , to a given final state $x_{f}={\bf r}(t_{f})$. This transition is to occur along an admissible path, i.e. ${\bf r}(.)$ and respects all kinematic constraints imposed on it. Let us consider it as $\dot{\bf r}=f({\bf r},\delta).$ (2) We further assume that $\delta(.)$ is admissible, i.e. is piecewise continuous and belongs to ${\cal U}$ , the admissible control space. Let there now be a rule which assigns a unique, real-valued number to each of these transfers. Such a rule can be viewed as the transition cost between $x_{i}$ and $x_{f}$ along an admissible path, completely specified by $\delta(.)$. The Optimal control concerns specifying this rule and thereby providing a systematic method for selecting the best , or optimal control law, according to some prescribed cost functional. One can find an analogue discussion in [5], to calculate the travel-time along the trajectories of the so called Pure pursuit navigation. Here, the P-navigation optimal control problem can be founded by the cost function $C({\bf r},\delta)=F({\bf r},\dot{\bf r},\delta)$ and has the following form $\textrm{minimize}\int_{0}^{t_{f}}C({\bf r},\delta)dt,$ (3) where, $t_{f}\in(0,\infty)$ is the final time which is going to be optimized. From everyday experience we know that collision courses need not be straight lines if $T$ changes its speed or direction; so what is exactly the collision course? It may be curved in some sense. One of our goal in this paper is to make known the best collision course. ###### Theorem 1.2 Given any time-optimal solution $({\bf r},\delta)$ of P-navigation, the curve ${\bf r}$ is a geodesic of the Finsler metric (1). The trajectory ${\bf r}_{M}$ can be obtained ${\bf r}_{M}={\bf r}_{T}-{\bf r}$ when ${\bf r}$ is known. One can freely consider ${\bf v}_{M}$ and ${\bf v}_{T}$ as vector fields alon ${\bf r}$. Now, let $\frac{\nabla}{\ dt}$ be the covariant derivative defined for any vector field $Y$ along ${\bf r}$ defined by $\frac{\nabla Y^{i}}{dt}:=\frac{dY^{i}}{dt}+G^{i}_{jk}({\bf r},\dot{\bf r},\delta)Y^{j}Y^{k},$ where, $G^{i}_{jk}$ are the connection coefficients of Berwald connection associated to the Finsler metric (1). As a result of Theorem 1.2, we can mention the following result: ###### Theorem 1.3 The time-optimal trajectory ${\bf r}_{M}$ of P-navigation satisfies the following second order ODE: $\ddot{\bf r}_{M}^{i}+G^{i}_{jk}({\bf r},{\bf v},\delta){\bf v}_{M}^{j}{\bf v}_{M}^{k}=\frac{\nabla{\bf v}_{T}^{i}}{dt},\ \ \ \ i=1,...,n.$ Our approach is closely related with subjects such as non-holonomic mechanics, sub-Finslerian geometries, see for a deeper sight [8] and [4]. One may find various techniques in missile guidance and control in [17]. ## 2 Preliminaries Let $M$ be a n-dimensional $C^{\infty}$ manifold. $T_{x}M$ denotes the tangent space of M at $x$. The tangent bundle of $M$ is the union of tangent spaces $TM:=\cup_{x\in M}T_{x}M$. We will denote the elements of $TM$ by $(x,y)$ where $y\in T_{x}M$. Let $TM_{0}=TM\setminus\\{0\\}.$ The natural projection $\pi:TM_{0}\rightarrow M$ is given by $\pi(x,y):=x$. A Finsler metric on $M$ is a function $F:TM\rightarrow[0,\infty)$ with the following properties; (i) $F$ is $C^{\infty}$ on $TM_{0}$, (ii) $F$ is positively 1-homogeneous on the fibers of tangent bundle $TM$, and (iii) the $y$-Hessian of $\frac{1}{2}F^{2}$ with elements $g_{ij}(x,y):=\frac{1}{2}[F^{2}(x,y)]_{y^{i}y^{j}}$ is positive definite on $TM_{0}$. The pair $(M,F)$ is then called a Finsler space. The Riemannian metrics are special Finsler metrics. Traditionally, a Riemannian metric is denoted by $a_{ij}(x)dx^{i}\otimes dx^{j}$. It is a family of inner products on tangent spaces. Let $\alpha(x,y):=\sqrt{g_{ij}(x)y^{i}y^{j}}$, ${\bf y}=y^{i}{{\partial}\over{\partial}x^{i}}|_{x}\in T_{x}M$. $\alpha$ is a family of Euclidean norms on tangent spaces. Throughout this paper, we also denote a Riemannian metric by $\alpha=\sqrt{a_{ij}(x)y^{i}y^{j}}$. An $(\alpha,\beta)$-metric is a scalar function on $TM$ defined by $F:=\Phi(\frac{\beta}{\alpha})\alpha$, where $\phi=\phi(s)$ is a $C^{\infty}$ on $(-b_{0},b_{0})$ with certain regularity. $\alpha=\sqrt{a_{ij}(x)y^{i}y^{j}}$ is a Riemannian metric and $\beta=b_{i}(x)y^{i}$ is a 1-form on a manifold $M$. One may find another important class of $(\alpha,\beta)$-metrics in [16]. The Randers and Matsumoto metrics are special $(\alpha,\beta)$-metrics defined by $\Phi=1+s$ and $\Phi=\frac{1}{1-s}$, respectively, i.e, $F=\alpha+\beta$ and $F=\frac{\alpha^{2}}{\alpha-\beta}$. Randers metrics were introduced by Randers in 1941 [13] in the context of general relativity. In [6], applying Fermat’s principle, the authors proved that the time-optimal solutions of the well-known Zermelo’s navigation-moving that is the motion of a vehicle equipped with an engine with a fixed power output in presence of a wind current-are actually the geodesics of a Randers metric. M. Matsumoto gave an exact formulation of a Finsler surface to measuring the time on the slope of a hill and introduced the Matsumoto metrics in [9], see also [15]. A Lagrangian on the manifold $M$ is a mapping $L:TM\longrightarrow R$ which is smooth on $TM_{0}$. A Lagrangian is said to be regular if it has non- degenerate $y$-Hessian on $TM_{0}$. Thus, given a Finsler metric $F$, the function $L=\frac{F^{2}}{2}$ is a regular Lagrangian. A large area of applicability of this geometry is suggested by the connections to Biology, Mechanics, and Physics and also by its general setting as a generalization of Finsler and Riemannian geometries [10]. For every smooth curve $c:[a,b]\longrightarrow R$, the extremal curves of the action integral given by $I(c)=\int_{a}^{b}L(c(t),\dot{c}(t))dt,$ (4) are characterized locally by the Euler-Lagrange equations given as follows: $\frac{d}{dt}\frac{\partial L}{\partial\dot{x}^{i}}-\frac{\partial L}{\partial x^{i}}=0,$ (5) where, $x^{i}(t)$ is a local coordinate expression of $c$. The extremal curves of the action integral (4) are usually called the geodesics of L. In [1] it is shown that the Lagrangian and Finslerian approaches are projectively the same. Given a Finsler manifold $(M,F)$, a globally defined vector field $G$ is induced by $F$ on $TM_{0}$, which in a standard coordinate $(x^{i},y^{i})$ for $TM_{0}$ is given by $G=y^{i}{{\partial}\over{\partial x^{i}}}-2G^{i}(x,y){{\partial}\over{\partial y^{i}}},$ where $G^{i}(x,y)$ are local functions on $TM_{0}$ satisfying $G^{i}(x,\lambda y)=\lambda^{2}G^{i}(x,y)\,\,\,,\lambda>0$, see [14]. G is called the associated spray to $(M,F)$. In local coordinates, a curve $c(t)$ is a geodesic of $F$ if and only if its coordinates $(c^{i}(t))$ satisfy $\ddot{c}^{i}+2G^{i}(c,\dot{c})=0$. ### 2.1 The kinematics of Parallel navigation We shall refer to the target as $T$ and to the pursuer as $M$ and their velocities as $v_{M}$ and $v_{T}$, respectively. To begin, we set up a coordinate system called reference frame of coordinates, in which the pursuer is initially located at the origin $O$. When considering planar motion we shall use Cartesian coordinates $(x,y)$ or $(x,z)$, and the angles will be positive if measured counterclockwise. The ray that starts at the pursuer $M$ and is directed at the target $T$ along the positive sense of ${\bf r}$ is called the line of sight (LOS). The parallel navigation geometrical rule,has been known since antiquity, mostly by mariners. According to this rule, the direction of the line of sight, $MT$, is kept constant relative to inertial space, i.e., the LOS is kept parallel to the initial LOS. In three-dimensional vector terminology, the rule is very concisely stated as ${\bf r}\times\dot{\bf r}=0$. Suppose that $\theta$ and $\lambda$ denote, respectively, the angles between ${\bf v}_{T}$ and ${\bf v}_{M}$ and, ${\bf v}_{M}$ and the horizontal axis (Figure 1). , Figure 1: The range ${\bf r}$, the velocity vectors ${\bf v}_{M}$ and ${\bf v}_{T}$. Let us put $r=|{\bf r}|$. The basic rule for moving of the pursuer is presented by the following two equations [17]: $\displaystyle\dot{r}$ $\displaystyle=$ $\displaystyle v_{T}\cos\theta- v_{M}\cos\delta,$ (6) $\displaystyle r\dot{\lambda}$ $\displaystyle=$ $\displaystyle v_{T}\sin\theta-v_{M}\sin\delta.$ (7) Notice that, in a planar framework, ${\bf v}_{M}$ , ${\bf v}_{T}$ and ${\bf r}$ being on the same (fixed) plane by definition, therefore, the parallel navigation geometrical rule can be restated as $\dot{\lambda}=0$. The requirement $\langle{\bf r},{\bf v}\rangle<0$ must be added in order to ensure that $M$ should approach $T$ not recede from it. In this case, we have $\dot{r}<0$, that is $v_{T}\cos\theta<v_{M}\cos\delta$. Let us denote the projection of any vector ${\bf v}_{T}$ on ${\bf v}$ by $Proj_{{\bf v}}{\bf v}_{T}$. A solution of the described P-navigation is a curve $({\bf r}(t),\delta(t))$ such that respects the equations (6) and (7). By the trajectory of P-navigation, we mean a curve ${\bf r}(t)$ such that $({\bf r}(t),\delta(t))$ is a solution, for some control $\delta$. Initiating the process, we have ${\bf r}(0)={\bf r}_{0}$ which shows that, $M$ stands at a point with distance $r_{0}$ from $T$. Through the performance, $r$ decreases by time and hence, $M$ approaches $T$. Therefore, ${\bf r}$ tends to the origin $O$ and $M$ hits $T$ when ${\bf r}(t_{f})=0$, (Figure 2). It follows that, P-navigation trajectories are characterized by a curve ${\bf r}$ joining $Q={\bf r}_{0}$ to the origin $O$ (Figure 3). It is of our interests to find the best $QO$-trajectory. More precisely, the problem is to find a curve from point $Q$ to point $O$ along which a particle will descend more quickly than on any other $QO$-curve of P-navigation. In this way, the problem somehow resembles to a brachistochrone problem. , Figure 2: Some possible ranges initiated at the point $Q$. , Figure 3: Schematic of exemplary collision courses for $M$. ## 3 The optimal control theory. A control system of ordinary differential equations is a family of differential equations in normal form $\frac{d{\bf r}^{i}}{dt}=f^{i}({\bf r},\delta)$, where ${\bf r}^{i}$ are called state variables, $t$ is the parameter of evolution (usually the time) and $\delta^{a}$ are the controls. Geometrically, it can be regarded as a fibred mapping $X:U\longrightarrow TM$, from a control fiber bundle $(U,\eta,M)$ over the state manifold $M$ to the tangent bundle $(TM,\pi,M)$, see [11]. Using local coordinates $({\bf r}^{i}),\ i=1,...,n$ in $M$, adapted coordinates $({\bf r}^{i},\delta^{a}),\ a=1,...,k$ in $U$, and natural coordinates $({\bf r}^{i},{\bf v}^{i})$ in $TM$, the coordinate expression for $X$ is $X({\bf r},\delta)=f^{i}({\bf r},\delta)\frac{\partial}{\partial{\bf r}^{i}}$ , or ${\bf v}^{i}=f^{i}({\bf r},\delta)$, the family of control equations. Admissible curves of the control system are curves $\gamma:I\subset R\longrightarrow U$ such that $(\eta o\gamma)^{c}=Xo\gamma$, where c denotes the natural lifting to $TM$ of a curve in $M$. Interested readers are advised to see [11] for getting familiar to the geometry of control systems. In Optimal Control Theory, a cost functional ${\cal C}(\gamma)=\int C({\bf r}(t),\delta(t))dt$ is given and the goal is to obtain admissible curves of the control system, satisfying some boundary conditions (e.g. $x_{i}={\bf r}(0)$, $x_{f}={\bf r}(t_{f})$) and minimizing the cost functional. It is therefore a Classical Variational problem with non- integrable constraints defined by the control equations. Pontryagin maximum principle [12] provides a set of necessary conditions for a solution $({\bf r}(t),\hat{\delta}(t))$ to be optimal; introducing a Hamiltonian function $\displaystyle H({\bf r},{\bf p},\delta)$ $\displaystyle:=$ $\displaystyle\langle{\bf p},X\rangle-C({\bf r},\delta)={\bf p}_{i}f^{i}({\bf r},\delta)-C({\bf r},\delta),$ $\displaystyle\hat{H}({\bf r},{\bf p})$ $\displaystyle:=$ $\displaystyle\underset{\delta}{\max}\ H({\bf r},{\bf p},\delta).$ where the variables $({\bf p}_{i})$ are momenta coordinates, the optimal curves $({\bf r}(t),\hat{\delta}(t))$ must satisfy the control system equations ${\bf v}^{i}=\frac{\partial\hat{H}}{\partial{\bf p}^{i}}=f^{i}({\bf r}(t),\hat{\delta}(t))$ and there must exist a solution curve for the adjoint differential equations $\frac{d{\bf p}_{i}}{dt}=-\frac{\partial\hat{H}}{\partial{\bf r}^{i}},$ Define the Lagrangian $L$ by $L({\bf r},{\bf v})={\bf p}_{i}{\bf v}^{i}-\hat{H}$. Observe that we have the following relations $\frac{d{\bf r}}{dt}=\frac{\partial\hat{H}}{\partial{\bf p}}={\bf v},\ \ \ \ \ \ \frac{d{\bf p}}{dt}=-\frac{\partial\hat{H}}{\partial{\bf r}}=\frac{\partial L}{\partial{\bf r}},\ \ \ \ \ \frac{\partial\hat{H}}{\partial{\bf v}}={\bf p}-\frac{\partial L}{\partial{\bf v}}=0.$ From the above equations, it results the well-known Euler-Lagrange for $L$ $\frac{d}{dt}\frac{\partial L}{\partial{\bf v}}-\frac{\partial L}{\partial{\bf r}}=0.$ ###### Proposition 3.1 [12] In order for $({\bf r}(t),\hat{\delta}(t))$ to be an optimal solution of (3), the following are necessary conditions: (a) There exists a solution curve for the adjoint differential equations $\frac{d{\bf p}_{i}}{dt}=-\frac{\partial\hat{H}}{\partial{\bf r}^{i}}.$ (b) $\hat{\delta}=\arg\ \underset{\delta}{\max}\ H({\bf r},{\bf p},\delta),\ \ \ \ \forall t\in[0,t_{f}]$. (c) $\hat{H}({\bf r},{\bf p})=0,\ \ \ \ \forall t\in[0,t_{f}]$. ## 4 Proof of Theorems. ### 4.1 Proof of Theorem 1.1 Let $({\bf r}(t),\delta(t))$ be a pair of the curve ${\bf r}$ and a function $\delta(t)$. We are going to show that, if $({\bf r}(t),\delta(t))$ be a solution of P-navigation, then ${\bf t}(t)$ must be reparametrized so that we we have $F({\bf r}(t),\dot{\bf r}(t),\delta(t))=1$. We notice that, in P-navigation, ${\bf r}$ and ${\bf v}$ are collinear and $\dot{r}<0$, hence we have $\dot{r}=\frac{\langle{\bf r},{\bf v}\rangle}{r}=\pm|Proj_{\bf r}{\bf v}|=\pm|Proj_{\bf v}{\bf v}|=-|{\bf v}|.$ Now, we summarize (6) in the following relation $|{\bf v}|=v_{M}\cos\delta-\frac{\langle{\bf v}_{T},{\bf v}\rangle}{|{\bf v}|}.\\\ $ After simplification, we obtain the following equation $F({\bf r},{\bf v},\delta)=\frac{|{\bf v}|^{2}}{v_{M}\cos\delta|{\bf v}|-\langle{\bf v}_{T},{\bf v}\rangle}=1.$ Q.E.D. ### 4.2 Proof of Theorem 1.2 Following Theorem 1.1, at any time $t$ we have $\dot{\bf r}={\bf v}\in S({\bf r},\delta)$. Hence, at any time $t$, there is a unit vector $X({\bf r},\delta)\in S({\bf r},\delta)$ such that $\dot{\bf r}={\bf v}=X({\bf r},\delta)$. Consider the unit canonical vector field $\ell({\bf r},\dot{\bf r},\delta)=\frac{\dot{\bf r}}{F({\bf r},\dot{\bf r},\delta)}$. We notice that, in P-navigation framework, we always assume that ${\bf r}$ and $\dot{\bf r}$ are collinear and hence, one can understand $\ell$ as a function of ${\bf r}$ and $\delta$, as well. It follows that, given any trajectory ${\bf r}$ of P-navigation, $X$ is given by $X({\bf r},\delta)=\ell({\bf r},\dot{\bf r},\delta)$. Therefore, it is clear that, $\displaystyle\langle{\bf p},X\rangle$ $\displaystyle=$ $\displaystyle p_{i}f^{i}({\bf r},\delta)=p_{i}\ell^{i}({\bf r},\dot{\bf r},\delta)=F({\bf r},\dot{\bf r},\delta),$ $\displaystyle\langle{\bf p},{\bf v}\rangle$ $\displaystyle=$ $\displaystyle p_{i}{\bf v}^{i}=F^{2}({\bf r},\dot{\bf r},\delta).$ Now, we return to the control system of P-navigation given by (2) with the cost functional $C({\bf r},\delta)=F({\bf r},\dot{\bf r},\delta)$. It is easy to verify that, $H=0$, $\hat{H}=0$ and one may consider $\hat{\delta}$ as any possible control law. The conditions of Proposition 3.1 holds as well and the Lagrangian $L_{\hat{\delta}}=\langle{\bf p},{\bf v}\rangle-\hat{H}$ is obtained as $L_{\hat{\delta}}({\bf r},\dot{\bf r})=F^{2}({\bf r},\dot{\bf r},\hat{\delta}).$ Therefore, based on Pontryagin maximum principle, the optimal trajectories ${\bf r}(t)$ are geodesics of the Lagrangian $L_{\hat{\delta}}$. Clearly, they are geodesics of the Finsler metric $F({\bf r},\dot{\bf r},\delta)$. Now, consider the control-parametric family of Finsler metrics defined by $F_{\delta}({\bf r},\dot{\bf r}):=F({\bf r},\dot{\bf r},\delta)$. Let ${\cal L}_{\delta}(\gamma)=\int_{0}^{t_{f}}F_{\delta}(\gamma,\dot{\gamma})dt$ be the length of any admissible curve $\gamma(t)$ on $(M,F_{\delta})$. A simple calculation gives the following inequality: $F_{0}({\bf r},\dot{\bf r})\leq F_{\delta}({\bf r},\dot{\bf r}),\ \ \ \textrm{for all possible controls}\ \delta.$ From that, it follows that the functional ${\cal L}_{\delta}(\gamma)$ takes its minimum at $\delta=0$, that is ${\cal L}_{0}(\gamma)\leq{\cal L}_{\delta}(\gamma),\ \ \ \textrm{for all possible controls}\ \delta.$ Therefore, to find a time-optimal solution, one should minimize the cost functional ${\cal C}(\gamma)=\int F_{0}(\gamma,\dot{\gamma})dt$ and this leads us to obtain it as a geodesic of $F_{0}$. Q.E.D. ###### Theorem 4.1 The time-optimal trajectory of P-navigation is a geodesic ${\bf r}(t)$ of the Finsler metric $F_{0}=\frac{|{\bf v}|^{2}}{v_{M}|{\bf v}|-\langle{\bf v}_{T},{\bf v}\rangle}$. However, given any control law, one may obtain a geodesic of the metric $F_{\delta}$ as the time-optimal trajectory. As a remark, we quote that the target $T$ may not be reachable by the control $\delta=0$. ###### Example 4.1 (Case of plane nonmaneuvering target.) The target $T$ is said to be nonmaneuvering if ${\bf a}_{T}=0$. In this case, $T$ moves on a straghit line at velocity $v_{T}$ in the direction with a constant angle $\theta_{0}$ if measured counterclockwise, see Figure 4. Let us suppose ${\bf v}_{T}(x^{1},x^{2})=v_{T}\\{\cos\theta_{0}\frac{\partial}{\partial x^{1}}+\sin\theta_{0}\frac{\partial}{\partial x^{2}}\\}$. Thus, from (7), it follows that $\delta=\sin^{-1}(\frac{\sin\theta_{0}}{K})$, where, $K$ is the velocity ratio $K=\frac{v_{M}}{v_{T}}$. Then, $\delta$ is a constant say $\delta_{0}$. Moreover, ${\bf v}_{T}$ is a parallel vector field and then $F_{\delta}$ is a Minkowski metric and is flat. Thus, it geodesics are straight lines. We obtain ${\bf r}(t)={\bf r}_{0}+t{\bf v}_{0}$. But, from (6), we have $|{\bf v}|=|{\bf v}_{0}|=v_{M}\cos\delta_{0}-v_{T}\cos\theta_{0}$. Intercept occur when we have ${\bf r}(t_{f})=0$, thus, the total flight time $t_{f}$ is obtained by $t_{f}=\frac{r_{0}}{v_{M}\cos\delta_{0}-v_{T}\cos\theta_{0}}=\frac{r_{0}}{v_{T}(K\cos\delta_{0}-\cos\theta_{0})}$ and the total range of $M$ equals $r_{0}$ which is the shortest curve joining ${\bf r}_{0}$ to the origin $O$. , Figure 4: Collision course for a target moving on a straight line at a direction with a constant angle $\theta_{0}$. ## References * [1] O. Amici, B. Casciaro, M. Hashiguchi, On Finsler metrics associated with a Lagrangian, Rep. Fac. Sci., Kagoshima Univ., (Math., Phys. and Chem), No. 20. p. 33-41, 1987. * [2] P. L. Antonelli, A. Bóna, M. Slawiński, Seismic rays as Finsler geodesics, Nonlinear Analysis: Real World Applications, 4 (2003) 711-722. * [3] P.L. Antonelli, R.S. Ingarden, M. Matsumoto, The Theory of Sprays and Finsler Spaces with Application in Physics and Biology, Kluwer Academic Publishers, Dordrecht, Boston, London, 1993. * [4] A. M. Bloch, J. Baillieul, P. Crouch, J. Marsden, Nonholonomic Mechanics and Control, Springer, (2003). * [5] B. Bidabad, M. Rafie-Rad, Pure pursuit navigation on Riemannian manifolds, Nonlinear Analysis: Real World Applications, 10 (2009), 1265-1269. * [6] D. Bao, C. Robles, Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom., 66 (2004) 377-435 * [7] C. Lanczos, The variational principles of mechanics, University of Toronto Press, Toronto, 1970 (1st ed. 1949). * [8] C. López, E. Martínez, Sub-Finslerian metric associated to an optimal control system, SIAM J. Control Optim. , to appear. * [9] M. Matsumoto, A slope of a hill is a Finsler surface with respect to a time measure, J. Math. Kyoto. Univ. 29 (1980) 17 25. * [10] R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications Vol. 59, Fundamental Theories of Physics Series, Kluwer Academic Publishers, Dordrecht, Boston, London, 1994\. * [11] H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer- Verlag, New York (1990). * [12] L. S. Pontryagin et al., The Mathematical Theory of Optimal Processes, InterScience Pub., New York (1962). * [13] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59 (1941) 195-199. * [14] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001. * [15] H. Shimada, S.V. Sabau, An introduction to Matsumoto metric, Nonlinear Anal. 63 (2005), 165-168. * [16] C. Shibata, On Finsler spaces with Kropina metric, Rep. Math. Phys. 13 (1978), 117-128. * [17] N. A. Shneydor, Missile Guidance and Pursuit: Kinematics, Dynamics and Control, Horwood Publishing Chichester, 1998\. * [18] T. Yajima, H. Nagahama, Zermelo’s condition and seismic ray path, Nonlinear Analysis: Real World Applications 8 (2007) 130-135.
arxiv-papers
2011-01-07T21:39:05
2024-09-04T02:49:16.214183
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. Rafie-Rad", "submitter": "Mehdi Rafie-Rad", "url": "https://arxiv.org/abs/1101.1537" }
1101.1546
arxiv-papers
2011-01-07T22:42:52
2024-09-04T02:49:16.221544
{ "license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/", "authors": "Therese A. Hart, Gabriel Khan, Mizan R. Khan", "submitter": "Mizan Khan", "url": "https://arxiv.org/abs/1101.1546" }
1101.1567
# Age and mass constraints for a young massive cluster in M31 based on spectral-energy-distribution fitting Jun Ma,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn 22affiliation: Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China Song Wang,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn 33affiliation: Graduate University, Chinese Academy of Sciences, Beijing 100039, P. R. China Zhenyu Wu,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn Zhou Fan,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn Yanbin Yang,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn Tianmeng Zhang,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn Jianghua Wu,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn Xu Zhou,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn Zhaoji Jiang,11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn and Jiansheng Chen11affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China; majun@nao.cas.cn ###### Abstract VDB0-B195D is a massive, blue star cluster in M31. It was observed as part of the Beijing-Arizona-Taiwan-Connecticut (BATC) Multicolor Sky Survey using 15 intermediate-band filters covering a wavelength range of 3000–10,000 Å. Based on aperture photometry, we obtain its spectral-energy distribution (SED) as defined by the 15 BATC filters. We apply previously established relations between the BATC intermediate-band and the Johnson-Cousins $UBVRI$ broad-band systems to convert our BATC photometry to the standard system. A detailed comparison shows that our newly derived $VRI$ magnitudes are fully consistent with previous results, while our new $B$ magnitude agrees to within $2\sigma$. In addition, we determine the cluster’s age and mass by comparing its SED (from 3000 to 20,000Å, comprising photometric data in the 15 BATC intermediate bands, optical broad-band $BVRI$, and 2MASS near-infrared $JHK_{\rm s}$ data) with theoretical stellar population synthesis models, resulting in age and mass determinations of $60.0\pm 8.0$ Myr and $(1.1-1.6)\times 10^{5}M_{\odot}$, respectively. This age and mass confirms previous suggestions that VDB0-B195D is a young massive cluster in M31. ###### Subject headings: galaxies: individual (M31) – galaxies: star clusters – galaxies: stellar content ††slugcomment: AJ, in press ## 1\. Introduction Young massive star clusters (YMCs) are among the main objects resulting from violent star-forming episodes triggered by galaxy collisions, mergers, and close encounters (see de Grijs & Parmentier, 2007, and references therein). They are also referred to as ‘young populous clusters,’ a term first coined by Hodge (1961), who used it to describe 23 clusters containing bright, blue stars in the Large Magellanic Cloud. In Hodge (1961), the ‘young’ aspect is demonstrated by the fact that all clusters have main sequences that extend to absolute magnitudes brighter than $M_{V}=0$, while ‘populous’ describes their richness (stellar membership). However, YMCs are also observed in quiescent galaxies (Larsen & Richtler, 1999) and in the disks of isolated spirals, although higher cluster-formation efficiencies are associated with environments exhibiting high star-formation rates (see Larsen, 2004; Cao & Wu, 2007, and references therein). It has become clear that, in many ways, YMCs resemble young versions of the old globular clusters (GCs) associated with all large galaxies (see Larsen et al., 2004, and references therein). YMCs are seemingly absent in the Milky Way; possibly the best example of a Galactic YMC is Westerlund 1, a heavily reddened cluster with an age and mass of 4–5 Myr (Crowther et al., 2006) and $M_{\rm cl}\sim 10^{5}~{}M_{\odot}$ (Clark et al., 2005), respectively. Since the pioneering work of Tinsley (1968, 1972) and Searle (1973), evolutionary population synthesis modeling has become a powerful tool to interpret integrated spectrophotometric observations of galaxies and their components, such as star clusters (e.g., Anders et al., 2004). The evolution of star clusters is usually modeled by means of the simple stellar population (SSP) approximation. An SSP is defined as a single generation of coeval stars formed from the same progenitor molecular cloud (thus implying a single metallicity), and governed by a given stellar initial mass function (IMF). Age and metallicity are two basic star cluster parameters. The most direct method to determine a cluster’s age is by employing main-sequence photometry, since the absolute magnitude of the main-sequence turnoff is predominantly affected by age (see Puzia et al., 2002, and references therein). However, until recently (cf. Perina et al., 2009), this method was only applied to the star clusters in the Milky Way and its satellites (e.g., Rich et al., 2001), although Brown et al. (2004) estimated the age of an M31 GC using extremely deep images observed with the Hubble Space Telescope (HST)’s Advanced Camera for Surveys. Generally, the ages of extragalactic star clusters are determined by comparing their observed spectral-energy distributions (SEDs) and/or spectroscopy with the predictions of SSP models (Williams & Hodge, 2001a, b; de Grijs et al., 2003a, b, c; Bik et al., 2003; Jiang et al., 2003; Beasley et al., 2004; Puzia et al., 2005; Ma et al., 2006; Fan et al., 2006; Ma et al., 2007, 2009; Caldwell et al., 2009; Wang et al., 2010). Nevertheless, SSP models assume that cluster IMFs are fully populated, i.e., that clusters contain infinite numbers of stars with a continuous distribution of stellar masses, and that all evolutionary stages are well sampled. Real clusters, however, contain a finite number of stars. Therefore, a disagreement between the observed cluster colors and theoretical colors derived from SSP models may become apparent (see Piskunov et al., 2009; Popescu & Hanson, 2010, and references therein). Other limitations inherent to SSP models arise from our poor understanding of some advanced stellar evolutionary stages, such as the supergiant and the asymptotic-giant-branch (AGB) phases (see Bruzual & Charlot, 2003, and references therein). Located at a distance of $785\pm 25$ kpc, corresponding to a distance modulus of $(m-M)_{0}=24.47\pm 0.07$ mag (McConnachie et al., 2005), M31 is the nearest and largest spiral galaxy in the Local Group of galaxies. It has been the subject of many GC studies and surveys, dating back to the early study of Hubble (1932). Based on previous publications (Hubble, 1932; Seyfert & Nassau, 1945; Hiltner, 1958; Mayall & Eggen, 1953; Kron & Mayall, 1960), Vetes̆nik (1962) compiled the first large M31 GC catalog, containing $UBV$ photometric data of approximately 300 GC candidates. Over the past decades, several major catalogs of M31 GCs and GC candidates have been published, including major efforts by the Bologna group (Battistini et al., 1980, 1987, 1993), Barmby et al. (2000), Galleti et al. (2004, 2005, 2006, 2007), Kim et al. (2007), Caldwell et al. (2009), and Peacock et al. (2010). Following on from the first extensive spectroscopic survey of M31 GCs by van den Bergh (1969), a significant number of authors (e.g., Huchra et al., 1982, 1991; Dubath & Grillmair, 1997; Federici et al., 1993; Jablonka et al., 1998; Barmby et al., 2000; Perrett et al., 2002; Galleti et al., 2006; Lee et al., 2008, and references therein) have studied their spatial, kinematic, and chemical (metallicity) properties. M31 is known to host a large number of young star clusters (e.g., Fusi Pecci et al., 2005; Caldwell et al., 2009; Wang et al., 2010, and references therein). Fusi Pecci et al. (2005) presented a comprehensive study of 67 very blue star clusters, which they referred to as ‘blue luminous compact clusters’ (BLCCs). Since they are quite bright ($-6.5\leq M_{V}\leq-10.0$ mag) and very young ($<2$ Gyr), BLCCs may be equivalent to YMCs (see for details Perina et al., 2009, 2010). To ascertain their properties, Perina et al. (2009, 2010) performed an imaging survey of 20 BLCCs in the disk of M31 using the HST’s Wide Field and Planetary Camera-2 (WFPC2). They obtained the reddening values, ages, and metallicities of their sample clusters by comparing the observed color-magnitude diagrams (CMDs) and luminosity functions with theoretical models. VDB0-B195D was first detected by van den Bergh (1969). Its color is extremely blue (e.g., $U-B=-0.48$ mag; van den Bergh, 1969) and it is very bright in blue bands (e.g., $U=14.66$ mag; van den Bergh, 1969). As a consequence, van den Bergh (1969) asserted that VDB0-B195D is the brightest open cluster in M31. He determined an integrated stellar spectral type equivalent to A0, which implies that the cluster contains massive stars. In addition, VDB0-B195D is particularly extended and most previous photometric studies did not include the full extent of the object’s light distribution (see for details Perina et al., 2009). We will provide an overview of previous studies that included the cluster in §2.1. It was observed as part of the galaxy calibration program of the Beijing-Arizona-Taiwan-Connecticut (BATC) Multicolor Sky Survey (e.g., Fan et al., 1996; Zheng et al., 1999) in 15 intermediate-band filters. Combined with photometry in optical broad-band $BVRI$ and near-infrared $JHK_{\rm s}$ filters from the Two Micron All Sky Survey (2MASS) taken from Perina et al. (2009), we obtained the SED of VDB0-B195D in 22 filters, covering the wavelengh range from 3000 to 20,000 Å. In this paper, we describe the details of the observations and our approach to the data reduction in §2. In §3, we determine the age and mass of VDB0-B195D by comparing observational SEDs with population synthesis models. We discuss the implications of our results and provide a summary in §4. ## 2\. Optical and near-infrared observations of the YMC VDB0-B195D ### 2.1. Historical overview VDB0-B195D was first given the designation ‘0’ (i.e., VDB0), the brightest open cluster in M31, in the catalog of van den Bergh (1969). Battistini et al. (1987) identified VDB0-B195D independently and called it B195D. In Battistini et al. (1987), B195D was given a low level of confidence (class D) of being a genuine cluster (classes A and B were assigned very high and high levels of confidence, respectively). It was only recently independently confirmed to be a single object. Caldwell et al. (2009) presented a new catalog containing 670 likely star clusters, stars, possible stars, and galaxies in the field of M31, all with updated high-quality coordinates accurate to $0.2^{\prime\prime}$, based on images from either the Local Group Galaxies Survey (LGGS) (Massey, 2006) or the Digitized Sky Survey (DSS). They use the designation VDB0-B195D, associated with $\rm\alpha_{0}=00^{\rm h}40^{\rm m}29^{\rm s}.43$ and $\rm\delta_{0}=+40^{\circ}36^{\prime}14^{\prime\prime}.8$ (J2000.0), which are the coordinates we adopt in this paper. Independently, Perina et al. (2009) studied the properties of VDB0-B195D in detail based on their HST/WFPC2 imaging survey of young massive GCs in M31. They initially selected VDB0-B195D as two YMCs in M31, but their WFPC2 images showed unequivocally that these two sample objects are, in fact, the same cluster. In addition, the HST images clearly confirmed that VDB0-B195D is a real cluster. However, it is difficult to establish whether it is more similar to ordinary open clusters, similar to those in the disk of the Milky Way, than to YMCs that may evolve to become disk GCs (see for details Perina et al., 2009). Spectral observations of VDB0-B195D were obtained by van den Bergh (1969)—yielding classification spectra and the object’s radial velocity—and Perrett et al. (2002), who used them for determination of its radial velocity and metallicity. ### 2.2. Archival images of the BATC Multicolor Sky Survey Observations of the YMC VDB0-B195D were obtained with the BATC 60/90cm Schmidt telescope located at the XingLong station of the National Astronomical Observatory of China (NAOC). This telescope is equipped with 15 intermediate- band filters covering the optical wavelength range from 3000 to 10,000 Å. The filter system was specifically designed to avoid contamination by the brightest and most variable night-sky emission lines. Descriptions of the BATC photometric system can be found in Fan et al. (1996). Before February 2006, a Ford Aerospace 2k$\times$2k thick CCD camera was installed, with a pixel size of 15 $\mu$m and a field of view of $58^{\prime}\times 58^{\prime}$, yielding a resolution of $1.7^{\prime\prime}$ pixel-1. Since February 2006, a new E2V 4k$\times$4k thinned CCD with a pixel size of 12 $\mu$m has been in operation, featuring a resolution of $1.3^{\prime\prime}$ pixel-1. The blue quantum efficiency of the new, thinned CCD is 92.2% at 4000 Å, which is much higher than for the old, thick device (see for details Fan et al., 2009). A field including VDB0-B195D in the $a$–$c$ filters was observed with the thinned CCD, and in $d$–$p$ bands with the thick CCD. Fig. 1 shows a finding chart of VDB0-B195D in the BATC $g$ band (centered at 5795 Å), obtained with the NAOC 60/90cm Schmidt telescope. We adopt an aperture with a radius of $15^{\prime\prime}$ (shown in Fig. 1) for the integrated photometry discussed in this paper. Figure 1.— Image of VDB0-B195D in the BATC $g$ band, obtained with the NAOC 60/90cm Schmidt telescope. VDB0-B195D is circled using an aperture with a radius of $15^{\prime\prime}$. The field of view of the image is $11^{\prime}\times 11^{\prime}$. The BATC survey team obtained 61 images of VDB0-B195D in 15 BATC filters between January 2004 and November 2006. Fan et al. (2009) performed the data reduction of these images, which formed part of their M31-7 field. Table 1 contains the observation log, including the BATC filter names, the central wavelength and bandwidth of each filter, the number of images observed through each filter, and the total observing time per filter. Multiple images through the same filter were combined to improve image quality (i.e., increase the signal-to-noise ratio and remove spurious signal). ### 2.3. Intermediate-band photometry of VDB0-B195D We determined the intermediate-band magnitudes of VDB0-B195D on the combined images using a standard aperture photometry approach, i.e., the phot routine in daophot (Stetson, 1987). Calibration of the magnitude zero level in the BATC photometric system is similar to that of the spectrophotometric AB magnitude system. For flux calibration, the Oke-Gunn (Oke & Gunn, 1983) primary flux standard stars HD 19445, HD 84937, BD +26∘2606, and BD +17∘4708 were observed during photometric nights (Yan et al., 2000). VDB0-B195D is located in the M31-7 field of Fan et al. (2009). The absolute flux of the M31-7 field was calibrated based on secondary standard transformations using the M31-1 field, which was calibrated, in turn, by the four Oke-Gunn primary flux standard stars by Jiang et al. (2003). Since VDB0-B195D is an extended object, an appropriate aperture size must be adopted to determine its total luminosity. The (radial) photometric asymptotic growth curve, in all BATC bands, flattens out at a radius of $\sim 15^{\prime\prime}$. Inspection ensured that this aperture is adequate for photometry, i.e., VDB0-B195D does not show any obvious signal beyond this radius. In addition, this aperture is nearly the same as that adopted by Perina et al. (2009) to determine the cluster’s photometry in the $BVRI$ bands, based on the M31 imaging survey of Massey (2006) (see §2.4 below). Therefore, we use an aperture with $r\approx 15^{\prime\prime}$ for integrated photometry, i.e., $r=9$ pixels for the 2k$\times$2k thick CCD camera, and $r=12$ pixels for the 4k$\times$4k thinned CCD camera. VDB0-B195D is projected onto the disk of M31, where the background is bright and fluctuates, potentially as a function of distance from the cluster center. To avoid contamination from background fluctuations, we adopted annuli for background subtraction spanning between 10 and 15 pixels for the 2k$\times$2k thick CCD camera, and from 13 to 20 pixels for the 4$\times$4k thinned CCD camera, both corresponding to $\sim 17$–26′′. While these annuli are spatially as close as possible to the region dominated by cluster light (so that any differences in background flux are minimized), they are wide enough to average out any expected background fluctuations. The calibrated photometry of VDB0-B195D in 15 filters is summarized in column (6) of Table 1, in conjunction with the $1\sigma$ magnitude uncertainties, which include uncertainties from the calibration errors of both the M31-1 field standard stars (see for details Fan et al., 2009; Jiang et al., 2003) and ‘the secondary standard stars’ in common between the M31-1 and M31-7 fields used for calculation of the mean magnitude offsets between the standard and instrumental magnitudes (see for details Fan et al., 2009), as well as those resulting from our daophot application. ### 2.4. Optical broad-band and near-infrared 2MASS photometry of VDB0-B195D Four independent sets of photometric data exist for VDB0-B195D. van den Bergh (1969) obtained $UBV$ photometry using observations of the 200-inch Hale telescope, Battistini et al. (1987) performed $UBVR$ photometry based on photographic plates observed with the 152 cm Ritchey-Chrétien $f$/8 telescope of the University of Bologna in Loiano, King & Lupton (1991) obtained $UBV$ photometry for VDB0-B195D using observations with the University of Hawaii’s 2.2 m telescope on Mauna Kea using the $f$/10 secondary and coronene-coated $584\times 416$ GEC CCD, and Sharov et al. (1995) performed $UBV$ photometry based on photo-electric observations with the 2.6 m Shain telescope of the Crimean Astrophysical Observatory. In addition, in the Revised Bologna Catalogue (RBC) of M31 GCs published by Galleti et al. (2004), the photometric data of VDB0-B195D in optical bands are based on Battistini et al. (1987) and Sharov et al. (1995), and transformed to the reference system of Barmby et al. (2000) by applying offsets derived from objects in common between the relevant catalog and the data set of Barmby et al. (2000). In the RBC, VDB0-B195D was regarded as two objects. We list these photometric data in Table 2 for comparison. Note that, in the latest RBC incarnation (version 3.5, updated on 27 March 2008), VDB0-B195D is included as a single object. Galleti et al. (2004) also determined 2MASS $JHK_{\rm s}$ photometric magnitudes for VDB0-B195D (transformed to the CIT photometric system; Elias et al., 1982, 1983), which we have included in Table 3. In addition, Perina et al. (2009) realized that VDB0-B195D is a particularly extended object and that it is possible that the photometry of Sharov et al. (1995) (compiled in the RBC) was obtained with apertures that were not large enough to include all of its flux. Therefore, they redetermined its photometric values in the $BVRI$ bands based on the M31 imaging survey of Massey (2006) using an aperture with $r=14.4^{\prime\prime}$, which are also listed in Table 3. From a comparison of the values in Tables 2 and 3, it is clear that the magnitudes of van den Bergh (1969) are brighter, while the results of the three other references are consistent. The magnitudes determined by Perina et al. (2009) are much brighter, however, because of their careful inclusion of all of the cluster’s flux. To compare our photometric results with previously published values, we transformed the magnitudes of VDB0-B195D in the BATC intermediate bands to broad-band $UBVRI$-equivalent photometry based on the relationships obtained by Zhou et al. (2003). These are also listed in Table 3, and the uncertainties include those originating from the transformation based on the relationships of Zhou et al. (2003) and their calibration errors (column 5 of their Table 3). In Fig. 2, we show the result of the comparison. In general, the other photometric data are fainter than ours and those of Perina et al. (2009). Fig. 2 and Table 3 show that our new $VRI$ magnitudes agree with the results of Perina et al. (2009), and that the $B$ magnitude obtained in this paper is 0.32 mag brighter than that of Perina et al. (2009). Considering the photometric errors of both Perina et al. (2009) and our current study, these two $B$-band photometric results are consistent within $2\sigma$. In addition, we should keep in mind that, although the $VRI$ magnitudes obtained in this paper are consistent with the results of Perina et al. (2009) within $1\sigma$, the disagreement in $B$ magnitudes at this level is understandable. This is caused by the fact that the original photometry in the present paper was obtained in the proprietary BATC filters and transformed to the $UBVRI$ system using transformation equations. Zhou et al. (2003) determined these conversions based on the broad-band $UBVRI$ magnitudes of 48 stars from Landolt (1983, 1992) and Galadí-Enríquez et al. (2000) in the Landolt SA95 field, and their photometric data in the 15 BATC intermediate- band filters. In addition, the central wavelengths and bandwidths of the BATC and $UBVRI$ systems differ. In fact, a similar significant disagreement of $B$-band photometric data for some M31 GCs was reported by Wang et al. (2010), citing similar arguments. Figure 2.— Comparison of photometric data from different sources with new determinations in this paper for VDB0-B195D. The data points shown as black dots are from Perina et al. (2009). ## 3\. Stellar population of VDB0-B195D ### 3.1. Stellar populations and synthetic photometry To determine the age and mass of VDB0-B195D, we compared its SED with theoretical stellar population synthesis models. The SED consists of photometric data in the 15 BATC intermediate bands obtained in this paper and optical broad-band $BVRI$ and 2MASS near-infrared $JHK_{s}$ data from Perina et al. (2009), listed in Table 3. We used the galev SSP models (e.g., Kurth et al., 1999; Schulz et al., 2002; Anders & Fritze-v. Alvensleben, 2003) for our comparisons. The galev SSPs are based on the Padova stellar isochrones, with the most recent versions using the updated Bertelli et al. (1994) isochrones (which include the thermally pulsing asymptotic giant-branch phase), and a Salpeter (1955) stellar IMF with lower- and upper-mass limits of 0.10 and between 50 and 70 $M_{\odot}$, respectively, depending on metallicity. The full set of models spans the wavelength range from 91Å to 160 $\mu$m. These models cover ages from $4\times 10^{6}$ to $1.6\times 10^{10}$ yr, with an age resolution of 4 Myr for ages up to 2.35 Gyr, and 20 Myr for greater ages. The galev SSP models include five initial metallicities, $Z=0.0004,0.004,0.008,0.02$ (solar metallicity), and 0.05. Since our observational data consist of integrated luminosities through the set of BATC filters, we convolved the galev SSP SEDs with the BATC intermediate-, optical broad-band $BVRI$, and 2MASS filter-response curves to obtain synthetic optical and near-infrared photometry for comparison. The synthetic $i^{\rm th}$ filter magnitude can be computed as $m=-2.5\log\frac{\int_{\nu}F_{\nu}\varphi_{i}(\nu){\rm d}\nu}{\int_{\nu}\varphi_{i}(\nu){\rm d}\nu}-48.60,$ (1) where $F_{\nu}$ is the theoretical SED and $\varphi_{i}$ the response curve of the $i^{\rm th}$ filter of the BATC, $BVRI$, and 2MASS photometric systems. Here, $F_{\nu}$ varies with age and metallicity. Since the observed magnitudes in the $BVRI$ and 2MASS photometric systems are given in the Vega system, we transformed them to the AB system for our fits. ### 3.2. Reddening and metallicity of VDB0-B195D To obtain the intrinsic SED of VDB0-B195D, its photometry must be dereddened. To date, only Perina et al. (2009) obtained reddening values for VDB0-B195D. They compared the observed CMD with theoretical isochrones and determined $E(B-V)=0.20\pm 0.03$ mag. Caldwell et al. (2009) were unable to derive the cluster’s reddening value because of the presence of a foreground field star, so they adopted $E(B-V)=0.28\pm 0.17$ mag (external rms error), equivalent to the mean reddening of the young clusters in M31. In this paper, we therefore adopt the reddening value from Perina et al. (2009). In addition, cluster SEDs are affected by age and metallicity effects. Therefore, we can only accurately constrain a cluster’s age if the metallicity is known. Perina et al. (2009) found that the CMD of VDB0-B195D, based on their HST/WFPC2 observations, is best reproduced by the solar-metallicity models of Girardi et al. (2002). We therefore adopt solar metallicity for VDB0-B195D. ### 3.3. The ‘lowest-luminosity-limit’ test The lowest-luminosity limit (LLL; Cerviño & Luridiana, 2004) implies that it is meaningless to compare a cluster with population synthesis models to obtain its age and mass if its integrated luminosity is lower than the luminosity of the most luminous star included in the model for the relevant age. The LLL method states that clusters fainter than this limit cannot be analyzed using standard procedures such as $\chi^{2}$ minimization of the observed values with respect to the mean SSP models (see also Barker et al., 2008). Below the LLL, cluster ages and masses cannot be obtained self-consistently. To take into account the effects on the integrated luminosities of statistically sampling the stellar IMF (e.g., Cerviño et al., 2000, 2002; Cerviño & Luridiana, 2004), we used the theoretical Padova isochrones at http://stev.oapd.inaf.it/cmd (CMD2.2). This interactive Web interface provides isochrones for a number of photometric systems, including optical broad-band, 2MASS, and the BATC data used here. We obtained the solar-metallicity ($Z=0.019$) isochrones of Marigo et al. (2008), as recommended by CMD2.2, based on the (Salpeter, 1955) IMF so as to match the IMF selection for our age and mass determinations of VDB0-B195D in §3.4 (see §3.1 for details). Figure 3 shows the LLL values as a function of age for the different filters used in this paper. These luminosities are obtained by identifying the most luminous star on each isochrone for the relevant passband. The gray area shows the cluster’s absolute luminosity, assuming a distance modulus of $(m-M)_{0}=24.47$ mag (785 kpc) for M31 (McConnachie et al., 2005). The upper luminosity limit has been corrected for extinction, based on a reddening value of $E(B-V)=0.20$ mag. The interstellar extinction curve, $A_{\lambda}$, is taken from Cardelli et al. (1989), $R_{V}=A_{V}/E(B-V)=3.1$. We see that VDB0-B195D does not lie below the LLL in any of the passbands used here. This means that, in general, VDB0-B195D can host the most luminous star that would be present theoretically for the given age of the cluster. Figure 3.— Lowest-luminosity limit for the filters used in this paper. The curves indicate the luminosities of the most luminous star on each isochrone for the relevant passband. The light-gray area shows the absolute magnitudes of VDB0-B195D based on a reddening value of $E(B-V)=0.20$ mag (Perina et al., 2009). We used a distance modulus of $(m-M)_{0}=24.47$ mag (785 kpc) for M31 (McConnachie et al., 2005) to calculate the absolute magnitudes. ### 3.4. Fit results In the previous section, the LLL test proves that the luminosity of VDB0-195D is higher than the luminosity of its brightest star expected for a given cluster age, i.e., that using SSP models is not completely meaningless. In addition, the bright absolute magnitude of VDB0-195D allows us to consider a possibility that the cluster is massive enough and IMF sampling effects should not strongly impact the fitting results. So we will determine the cluster’s age and mass estimates based on direct comparisons with SSP mean values in this section. However, we should keep in mind that this approach is a compromise. In fact, the fitting results (Fig. 4 and Table 5) show probable problem even for relative massive clusters. We use a $\chi^{2}$ minimization test to determine which galev SSP models are most compatible with the observed SEDs, $\chi^{2}=\sum_{i=1}^{22}{\frac{[m_{\nu_{i}}^{\rm intr}-m_{\nu_{i}}^{\rm mod}(t)]^{2}}{\sigma_{i}^{2}}},$ (2) where $m_{\nu_{i}}^{\rm mod}(t)$ is the integrated magnitude in the $i^{\rm th}$ filter of a theoretical SSP at age $t$ (for solar metallicity), $m_{\nu_{i}}^{\rm intr}$ is the intrinsic, integrated magnitude, and $\sigma_{i}$ is the magnitude uncertainty, defined as $\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm mod},i}^{2}+(R_{\lambda_{i}}*\sigma_{\rm red})^{2}+\sigma_{{\rm md},i}^{2}.$ (3) Here, $\sigma_{{\rm obs},i}$ is the observational uncertainty from column (6) of Table 1 and column (2) of Table 3, $\sigma_{{\rm mod},i}$ is the uncertainty associated with the model itself, $\sigma_{\rm red}$ is the uncertainty in the reddening value, and $R_{\lambda_{i}}=A_{\lambda_{i}}/E(B-V)$, where $A_{\lambda_{i}}$ is taken from Cardelli et al. (1989), $R_{V}=A_{V}/E(B-V)=3.1$, and $\sigma_{{\rm md},i}$ is the uncertainty in the distance modulus, for the $i^{\rm th}$ filter. Charlot et al. (1996) estimated the uncertainty associated with the term $\sigma_{{\rm mod},i}$ by comparing the colors obtained from different stellar evolutionary tracks and spectral libraries. Following Ma et al. (2007, 2009), we adopt $\sigma_{{\rm mod},i}=0.05$ mag. Perina et al. (2009) pointed out that VDB0-B195D is a particularly extended object and that the photometric measurements of van den Bergh (1969), Battistini et al. (1987), King & Lupton (1991), and Sharov et al. (1995) did not include all of its flux. Therefore, we adopt the photometry of Perina et al. (2009) to fit the observed SED with theoretical SSPs for our age determination. The fit yielding the minimum $\chi^{2}$ value ($\chi^{2}({\rm min})$) was adopted as the best fit and we adopted the corresponding age value, $60.0\pm 8.0$ Myr. In addition, our best-fitting age estimate of $60.0\pm 10.0$ Myr results from using the (redder) $k$–$p$ and $IJHK_{\rm s}$ photometry; using only the blue part of the cluster’s SED ($B,a$–$e$, where any effects caused by stochasticity may be smaller) yields an age of $72.0\pm 34.0$ Myr. The uncertainty was estimated using confidence limits. If $\chi^{2}/{\nu}<\chi^{2}({\rm min})/{\nu}+1$, the resulting age is within the 68.3% probability range; here, ${\nu}=21$ is the number of free parameters, i.e., the number of observational data points minus the number of parameters used in the theoretical model. Therefore, the accepted age range is derived from those fits that have $\chi^{2}({\rm min})/{\nu}<\chi^{2}/{\nu}<\chi^{2}({\rm min})/{\nu}+1$. The best reduced-$\chi^{2}$—defined as $\chi_{\nu}^{2}({\rm min})=\chi^{2}({\rm min})/{\nu}$—and age are listed in Table 4. The best fit to the SED of VDB0-B195D is shown in Fig. 4, where we display the intrinsic cluster SED (symbols with error bars), as well as the integrated SED (open circles) and spectrum of the best-fitting model. From Fig. 4, we note that the observational data in the $b$, $d$, $o$, and $p$ BATC filters and in the $K_{\rm s}$ band do not match the best-fitting model very well (the difference is approximately 0.3 mag). Photometric uncertainties in these filters may cause some differences, although this might not be the main reason for the discrepancy. As we know, observational star clusters’ SEDs are affected by age, metallicity and reddening. If the reddening value and metallicity adopted in this paper are not problematic, discrepancy between our observations and the best-fitting model may reflect the difficulty in achieving an appropriate (but formal) fit of an SED of a single, real cluster by SSP models. However, as we will see below, the reddening value adopted in this paper may be bigger than the actual reddening of VDB0-B195D. In addition, the differences between the photometric data and the model in Fig. 4 show a somewhat systematic behavior with wavelength: in bluer passbands the cluster seems to be more luminous than predicted by the model, while in redder passbands it is fainter than the corresponding model predictions. A blue excess and red deficiency in the observed SED with respect to the model predictions may indicate a shortage of red giants (RGs), which can occur when the cluster is either younger or less massive (or both) than the corresponding best-fitting model suggests. In other words, IMF discreteness may play a role: due to a relatively longer main-sequence (MS) phase and shorter RG phase, a random young cluster is typically bluer than predicted by SSP models. At the same time, we find that the reddening value adopted affects the fitting result greatly. In fact, the best fit to the SED of VDB0-B195D improves a great deal when adopting a smaller reddening value such as $E(B-V)=0.1$: $\chi_{\nu}^{2}({\rm min})=0.73$; the resulting age ($64.0\pm 8.0$ Myr) is nearly the same as one ($60.0\pm 8.0$ Myr) obtained with $E(B-V)=0.2$. We next determined the mass of VDB0-B195D. The galev models include absolute magnitudes (in the Vega system) in 77 filters for SSPs of $10^{6}~{}M_{\odot}$, including 66 filters of the HST, Johnson $UBVRI$ (see for details Landolt, 1983), Cousins $RI$ (see for details Landolt, 1983), and $JHK$ (Bessell & Brett, 1988) systems. The difference between the intrinsic absolute magnitudes and those given by the model provides a direct measurement of the cluster mass, in units of $10^{6}~{}M_{\odot}$. However, we should keep in mind that this is only correct for cluster masses above $10^{6}~{}M_{\odot}$. We estimated the mass of VDB0-B195D using magnitudes in all of the $BVRI$ and $JHK_{\rm s}$ bands. Therefore, we transformed the 2MASS $JHK_{\rm s}$ magnitudes to the photometric system of Bessell & Brett (1988) using the equations given by Carpenter (2001). The resulting mass determinations for VDB0-B195D are listed in Table 5 with their $1\sigma$ uncertainties including contributions from uncertainties in extinction and distance modulus. From Table 5, we see that the mass of VDB0-B195D obtained based on the magnitudes in different filters is very different. (The highest mass obtained, based on the $B$-band magnitude, is $~{}0.5\times 10^{5}~{}M_{\odot}$ more massive than that obtained using the $K_{s}$ magnitude.) In addition, the mass estimates differ systematically with filters. Provided that VDB0-B195D is massive enough to be fitted by SSP models, a systematic trend of masses based on different passbands may indicate a problem with reddening value adopted for the cluster. If the actual reddening is smaller than the adopted value, the actual luminosity would be overestimated. This effect is small in redder filters but strong in bluer filters. As discussed in age estimation, a smaller reddening value can improve the fitting result greatly. In fact, a smaller reddening value can reduce the mass discrepancies based on the magnitudes in different filters. When we adopted $E(B-V)=0.1$, the mass of VDB0-B195D obtained based on the magnitudes in different filters is the same within 1$\sigma$. We list these estimates in Table 7. From Table 5, we know that the mass of VDB0-B195D obtained in paper is between $(1.1-1.6)\times 10^{5}~{}M_{\odot}$ when the reddening value is adopted to be $(B-V)=0.2$. Figure 4.— Best-fitting, integrated theoretical galev SEDs compared to the intrinsic SED of VDB0-B195D. The photometric measurements are shown as symbols with error bars (vertical for uncertainties and horizontal for the approximate wavelength coverage of each filter). Open circles represent the calculated magnitudes of the model SED for each filter. We used a distance modulus of $(m-M)_{0}=24.47$ mag (785 kpc) for M31 (McConnachie et al., 2005) to calculate the absolute magnitudes. ## 4\. Summary and discussion VDB0-B195D was previously shown to be a massive cluster based on HST/WFPC2 observations. Its color is extremely blue and it is very bright, particularly in blue bands. In addition, VDB0-B195D is an extended object, and most previous photometric measurements did not include its full flux distribution (see Perina et al., 2009, for details). In this paper, we obtained the cluster’s SED in the 15 BATC intermediate-band filters. We subsequently determined its age and mass by comparing our multicolor photometry with theoretical stellar population synthesis models. Our multicolor photometric data consist of 15 intermediate-band filters obtained in this paper, and broad-band $BVRI$ and 2MASS $JHK_{\rm s}$ from Perina et al. (2009), covering a wavelength range from 3000 to 20,000 Å. Our results show that VDB0-B195D is a genuine YMC in M31. To understand the real nature of the BLCCs, Perina et al. (2009, 2010) performed an HST imaging survey of 20 BLCCs in M31’s disk. As a test case, Perina et al. (2009) presented details of the data-reduction pipeline that will be applied to all survey data and describe its application to VDB0-B195D. They estimated the object’s age, by comparison of the observed CMD with theoretical isochrones from Girardi et al. (2002), at $\simeq 25$ Myr. In addition, they constrained realistic upper and lower limits to the cluster’s age, independent of the adopted metallicity, within the relatively narrow range from 12 to 63 Myr. Using Maraston’s SSP models of solar metallicity (Maraston, 1998, 2005), Salpeter (1955) and Kroupa (2001) IMFs, and photometric values in the $V$ and 2MASS $J$, $H$, and $K_{\rm s}$ bands, Perina et al. (2009) concluded that the mass of VDB0-B195D is $>2.4\times 10^{4}~{}M_{\odot}$, with their best estimates in the range $\simeq(4-9)\times 10^{4}~{}M_{\odot}$. Caldwell et al. (2009) presented an updated catalog of 1300 objects in M31, including spectroscopic and imaging surveys, based on images from either the LGGS or the DSS and spectra taken with the Hectospec fiber positioner and spectrograph on the 6.5 m MMT. They derived ages and reddening values for 140 young clusters by comparing their observed spectra with model spectra from the Starburst99 SSP suite (Leitherer et al., 1999). The results show that these clusters are less than 2 Gyr old, while most have ages between $10^{8}$ and $10^{9}$ yr (the age of VDB0-B195D they derive is $\log{\rm age/yr}=7.6$). In addition, Caldwell et al. (2009) also estimated the masses of these young clusters using $V$-band photometry and model mass-to-light ratios (Leitherer et al., 1999) corresponding to the derived spectroscopic ages. This resulted in masses ranging from $2.5\times 10^{2}$ to $1.5\times 10^{5}~{}M_{\odot}$. The mass of VDB0-B195D obtained by Caldwell et al. (2009) is $\log M_{\rm cl}/M_{\odot}=5.1$ (no uncertainty quoted). We compare the various age and mass estimates of VDB0-B195D in Table 6\. Our newly obtained age is older than the estimates of both Perina et al. (2009) and Caldwell et al. (2009), while the mass obtained in this paper is higher than the estimate of Perina et al. (2009) and consistent with the determination of Caldwell et al. (2009). However, our results are in agreement with those of both Perina et al. (2009) and Caldwell et al. (2009) within $3\sigma$. The age and mass obtained in this paper confirms that VDB0-B195D is genuinely a YMC in M31. As we know, SSP models describe a very special case of a continuous distribution of stellar mass (or light) along isochrones. This is well approximated by clusters with masses larger than $10^{6}~{}M_{\odot}$. Also, for cluster masses of about $10^{5}~{}M_{\odot}$, SSP models can probably still be applied since a systematic difference between SSP models and observations should, on average, be smaller than 0.05 mag for clusters older than 10 Myr (see Fig. 3 in Piskunov et al. 2009). However, from the results of this paper, we may conclude that, probably, a formal fitting of SSP models to observed SEDs cannot be used without caution even for relatively massive (or apparently massive) clusters, and it is highly doubtful that this approach can be applied in a routine work providing accurate cluster parameters. The relative accuracy of 10% for age and 20% found for the mass of VDB0-B195D seems to be rather formal and not very confident. In addition, observational star clusters’ SEDs are affected by reddening, an effect that is also difficult to separate from the combined effects of age and metallicity (Calzetti 1997; Vazdekis et al. 1997; Origlia et al. 1999). Only the metallicity and reddening are derived accurately (and, ideally, independently), these degeneracies are largely (if not entirely) reduced, and ages can then also be estimated accurately based on a comparison of multicolor photometry spanning a significant wavelength range (de Grijs et al. 2003b; Anders et al. 2004) with theoretical stellar population synthesis models. It is true that the discrepancy between our observations and the best-fitting model is great, and the mass of VDB0-B195D obtained based on the magnitudes in different filters is very different. However, when we adopt a smaller reddening value, the results improve greatly. So, we conclude that the actual reddening value of VDB0-B195D may be smaller than $E(B-V)=0.2$. We are indebted to the anonymous referee for very carefully reading our manuscript, and for many thoughtful comments and insightful suggestions that improved this paper significantly. 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Filter | Central wavelength | Bandwidth | Number of images | Exposure time | Magnitude ---|---|---|---|---|--- | (Å) | (Å) | | (hours) | $a$ | 3360 | 222 | 6 | 2.0 | $15.51\pm 0.14$ $b$ | 3890 | 187 | 6 | 2.0 | $14.73\pm 0.11$ $c$ | 4210 | 185 | 3 | 0.8 | $14.70\pm 0.07$ $d$ | 4550 | 222 | 3 | 1.0 | $14.49\pm 0.10$ $e$ | 4920 | 225 | 3 | 1.0 | $14.67\pm 0.05$ $f$ | 5270 | 211 | 3 | 1.0 | $14.65\pm 0.05$ $g$ | 5795 | 176 | 3 | 1.0 | $14.59\pm 0.04$ $h$ | 6075 | 190 | 3 | 1.0 | $14.56\pm 0.02$ $i$ | 6660 | 312 | 3 | 1.0 | $14.50\pm 0.02$ $j$ | 7050 | 121 | 5 | 1.7 | $14.51\pm 0.06$ $k$ | 7490 | 125 | 3 | 1.0 | $14.47\pm 0.05$ $m$ | 8020 | 179 | 3 | 1.0 | $14.43\pm 0.07$ $n$ | 8480 | 152 | 6 | 2.0 | $14.49\pm 0.05$ $o$ | 9190 | 194 | 6 | 2.0 | $14.49\pm 0.05$ $p$ | 9745 | 188 | 6 | 2.0 | $14.50\pm 0.05$ Table 2Comparison of broad-band photometry of VDB0-B195D. Filter | ${\rm Mag}^{a}$ | ${\rm Mag}^{b}$ | ${\rm Mag}^{c}$ | ${\rm Mag}^{d}$ | ${\rm Mag}^{e}$ | ${\rm Mag}^{f}$ ---|---|---|---|---|---|--- $U$ | 14.66 | 15.11 | $15.12\pm 0.012$ | $14.97\pm 0.01$ | 15.110 | 15.140 $B$ | 15.14 | 15.39 | $15.49\pm 0.010$ | $15.31\pm 0.01$ | 15.410 | 15.510 $V$ | 14.94 | 15.19 | $15.32\pm 0.013$ | $15.06\pm 0.01$ | 15.190 | 15.280 $R$ | | 15.27 | | | 14.920 | avan den Bergh (1969); bBattistini et al. (1987); cKing & Lupton (1991), uncertainties are the median uncertainties in the mean for all sample cluster measurements; dSharov et al. (1995); ePhotometry from Galleti et al. (2004), based on Battistini et al. (1987); fPhotometry from Galleti et al. (2004), based on Sharov et al. (1995). Table 3Recently determined photometry for VDB0-B195D. Filter | ${\rm Mag}^{a}$ | ${\rm Mag}^{b}$ | ${\rm Mag}^{c}$ ---|---|---|--- $U$ | | | $14.37\pm 0.22$ $B$ | $14.94\pm 0.09$ | | $14.62\pm 0.13$ $V$ | $14.67\pm 0.05$ | | $14.67\pm 0.05$ $R$ | $14.45\pm 0.11$ | | $14.60\pm 0.06$ $I$ | $14.01\pm 0.11$ | | $14.19\pm 0.10$ $J$ | $13.26\pm 0.07$ | $13.78\pm 0.03$ | $H$ | $12.76\pm 0.12$ | $13.15\pm 0.04$ | $K_{s}$ | $12.77\pm 0.15$ | $12.96\pm 0.03$ | aPerina et al. (2009); bGalleti et al. (2004); cThis paper. Table 4Age estimate of VDB0-B195D based on the the galev models. Age | log (Age) | $\chi_{\nu}^{2}({\rm min})$ ---|---|--- (Myr) | [yr] | (per degree of freedom) $60.0\pm 8.0$ | $7.78\pm 0.05$ | 2.2 Table 5Mass estimates (and uncertainties) of VDB0-B195D based on the galev models. $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$ ---|---|---|---|---|---|--- | | | Mass $(10^{5}~{}M_{\odot})$ | | | $1.6\pm 0.18$ | $1.6\pm 0.13$ | $1.4\pm 0.17$ | $1.4\pm 0.16$ | $1.4\pm 0.13$ | $1.3\pm 0.16$ | $1.1\pm 0.17$ Table 6Comparison of age and mass estimates of VDB0-B195D. Age1 | Age2 | log (Age)3 | log (Age)2 | Mass1 | Mass2 | log (Mass)3 | log (Mass)2 ---|---|---|---|---|---|---|--- (Myr) | (Myr) | [yr] | [yr] | ($10^{4}~{}M_{\odot}$) | ($10^{5}~{}M_{\odot}$) | [$M_{\odot}$] | [$M_{\odot}$] 25 | $60.0\pm 8.0$ | 7.6 | $7.78\pm 0.05$ | $4-9$ | $1.1-1.6$ | $5.1$ | $5.0-5.2$ 1Perina et al. (2009); 2This paper; 3Caldwell et al. (2009). Table 7Mass estimates (and uncertainties) of VDB0-B195D based on the galev models with $E(B-V)=0.1$. $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$ ---|---|---|---|---|---|--- | | | Mass $(10^{5}~{}M_{\odot})$ | | | $1.1\pm 0.12$ | $1.2\pm 0.10$ | $1.1\pm 0.14$ | $1.2\pm 0.14$ | $1.3\pm 0.12$ | $1.2\pm 0.16$ | $1.1\pm 0.17$
arxiv-papers
2011-01-08T03:10:28
2024-09-04T02:49:16.226184
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jun Ma (1,2), Song Wang (1,3), Zhenyu Wu (1), Zhou Fan (1), Yanbin\n Yang (1), Tianmeng Zhang (1) and Jianghua Wu (1) et al. ((1) National\n Astronomical Observatories, Chinese Academy of Sciences, (2) Key Laboratory\n of Optical Astronomy, National Astronomical Observatories, Chinese Academy of\n Sciences)", "submitter": "Jun Ma", "url": "https://arxiv.org/abs/1101.1567" }
1101.1569
Vol.0 (200x) No.0, 000–000 11institutetext: 1National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, P. R. China 2Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China 11email: majun@nao.cas.cn # Detailed study of B037 based on HST images Jun Ma 1122 (Received 2001 month day; accepted 2001 month day) ###### Abstract B037 is of interest because it is both the most luminous and the most highly reddened cluster known in M31. Images of deep observations and of highly spatial resolutions with the Advanced Camera for Surveys on the Hubble Space Telescope (HST) firstly show that this cluster is crossed by a dust lane. Photometric data in the F606W and F814W filters obtained in this paper provide that, colors of ($\rm{F606W-F814W}$) in the dust lane are redder $\sim 0.4$ mags than ones in the other regions of B037. The HST images show that, this dust lane seems to be contained in B037, not from the M31 disk or the Milky Way. As we know, the formation of dust requires gas with a rather high metallicity. However, B037 has a low metallicity to be $\rm[Fe/H]=-1.07\pm 0.20$. So, it seems improbable that the observed dust lane is physically associated with B037. It is clear that the origin of this dust lane is worthy of future study. In addition, based on these images, we present the precise variation of ellipticity and position angle, and of surface brightness profile, and determine the structural parameters of B037 by fitting a single- mass isotropic King model. In the F606W filter, we derive the best-fitting scale radius, $r_{0}=0.56\pm 0.02\arcsec~{}(=2.16\pm 0.08~{}\rm{pc})$, a tidal radius, $r_{t}=8.6\pm 0.4\arcsec~{}(=33.1\pm 1.5~{}\rm{pc})$, and a concentration index $c=\log(r_{t}/r_{0})=1.19\pm 0.02$. In the F814W filter, we derive $r_{0}=0.56\pm 0.01\arcsec~{}(=2.16\pm 0.04~{}\rm{pc})$, $r_{t}=8.9\pm 0.3\arcsec~{}(=34.3\pm 1.2~{}\rm{pc})$, and $c=\log(r_{t}/r_{0})=1.20\pm 0.01$. The extinction-corrected central surface brightness is $\mu_{0}=13.53\pm 0.03~{}{\rm mag~{}arcsec^{-2}}$ in the F606W filter, and $12.85\pm 0.03~{}{\rm mag~{}arcsec^{-2}}$ in the F814W filter, respectively. We also calculate the half-light radius, at $r_{h}=1.05\pm 0.03\arcsec(=4.04\pm 0.12~{}\rm{pc})$ in the F606W filter and $r_{h}=1.07\pm 0.01\arcsec(=4.12\pm 0.04~{}\rm{pc})$ in the F814W filter, respectively. In addition, we derived the whole magnitudes of B037 in $V$ and $I$ bands by transforming the magnitudes from the ACS system to the standard system, which are in very agreement with the previous ground-based broad-band photometry. ###### keywords: galaxies: evolution – galaxies: individual (M31) – globular cluster: individual (B037) ## 1 Introduction Globular clusters (GCs) are effective laboratories for studying stellar evolution and stellar dynamics, and they are ancient building blocks of galaxies which can help us to understand the formation and evolution of their parent galaxies. In addition, GCs exhibit surprisingly uniform properties, suggesting a common formation mechanism. The closest other populous GC system beyond the halo of our Galaxy is that of M31. The study of M31 has been and continues to be a keystone of extragalactic astronomy (Barmby et al., 2000), and the study of GCs in M31 can be traced back to Hubble (1932). M31 GC B327 (B for ‘Baade’) or Bo37 (Bo for ‘Bologna’, see Battistini 1987), which, in the nomenclature introduced by Huchra et al. (1991) is referred to as B037, a designation from the Revised Bologna Catalogue (RBC) of M31 GCs and candidates (Galleti et al. 2004, 2006, 2007), which is the main catalog used in studies of M31 GCs. The extremely red color of B037 was firstly noted by Kron & Mayall (1960), who suggested that this cluster must be highly reddened. Two years later, Vetes̆nik (1962a) determined magnitudes of 257 M31 GC candidates including B037 in the $UBV$ photometric system, and then Vetes̆nik (1962b) studied the intrinsic colors of M31 GCs, and found that B037 was the most highly reddened with $E(B-V)=1.28$ in his sample of M31 GC candidates based on the photometric catalog of Vetes̆nik (1962a). With low-resolution spectroscopy, Crampton et al. (1985) also found that B037 is the most highly reddened GC candidate in M31 to have $E(B-V)=1.48$. Based on a large database of multicolor photometry, Barmby et al. (2000) determined the reddening value for each individual M31 GC including B037 using the correlations between optical and infrared colors and metallicity by defining various “reddening-free” parameters, and the reddening value of B037 is $E(B-V)=1.38\pm 0.02$ (which is kindly given us by P. Barmby). Again, Barmby et al. (2002b) derived the reddening value for this cluster to be $E(B-V)=1.30\pm 0.04$, using the spectroscopic metallicity to predict the intrinsic colors. Ma et al. (2006a) also determine the reddening of B037 by comparing independently obtained multicolor photometry with theoretical stellar population synthesis models to be $E(B-V)=1.360\pm 0.013$, which is in good agreement with the other results. Following the methods of Barmby et al. (2000), Fan et al. (2008) (re-)determined reddening values for 443 clusters and cluster candidates including B037, and the redding value of B037 obtained by Fan et al. (2008) is $E(B-V)=1.21\pm 0.03$, which is a little smaller than the previous determinations. The brightest GCs in M31 are more luminous than the most brightest Galactic cluster, $\omega$ Centauri. Among these are B037 (van den Bergh, 1968) and G1 (see details from Barmby et al., 2002b). These two clusters are both considered as the possible remnant core of a former dwarf galaxy which lost most of its envelope through tidal interactions with M31 (Meylan & Heggie, 1997; Meylan et al., 2001; Mackey & van den Bergh, 2005; Ma et al., 2006b, 2007). In this paper, we will present the photometric data of B037 using its deep images obtained with the Advanced Camera for Survey (ACS) on the HST. The deep images of highly spatial resolutions showed that this cluster is crossed by a dust lane. Our results provide that colors of F814W$-$F606W in the dust lane are redder $\sim 0.4$ mags than ones of the other regions. In addition, we studied structures of B037 in detail based on these images. ## 2 Observations and Data Reduction ### 2.1 HST images of B037 We searched the HST archive and found B037 to have been observed with the ACS- Wide Field Channel (WFC) in the F606W and the F814W filters, which were observed on 2004 August 2 and on 2004 July 4, respectively. The exposure time is 2370.0 seconds for both bands. The HST ACS-WFC resolution is $0.05\arcsec$ per pixel. The images in F606W and F814W both show that B037 is crossed by a dust lane. Fig. 1 clearly shows the dust lane, which crosses B037. If the dust lane is true, its color should be different from ones of the other regions. If not otherwise stated, the magnitudes are always on the VEGAMAG scale as defined by Sirianni et al. (2005). The relevant zero-point for this system is 26.398 and 25.501 for WFC F606W and WFC F814W, respectively. A distance to M31 of 780 kpc ($1{\arcsec}$ subtends 3.85 pc) is adopted in this paper. Figure 1: The images of GC B037 observed in the F606W and F814W filters of ACS/HST. The images clearly show that the cluster is crossed by a dust lane. The image size is $17.5\arcsec\times 17.5\arcsec$ for each panel. ### 2.2 Color difference between the dust lane and the other regions In order to study whether the color difference between the dust lane and the other regions in B037 exists, we select nine points, three of which (No. 7, 8 and 9) are located in the dust lane, the other six are randomly located in the other regions (see Figure 2). For each sample point, the PHOT routine in DAOPHOT (Stetson, 1987) is used to obtain magnitude. We adopt an aperture of a diameter of 4 pixels. The photometric data for these nine sample points are given in Table 1, in conjunction with the $1\sigma$ magnitude uncertainties from daophot. Column 4 gives the color of ($\rm{F606W-F814W}$). From Table 1, we can see that colors of ($\rm{F606W-F814W}$) in the dust lane are redder $\sim 0.4$ mags than ones of the other regions. Figure 2: The sample positions of photometry (black circles) are showed in the image of GC B037 observed in the F606W filter of ACS/HST. An aperture of radii of 2 pixels is adopted for photometry. The image size is $17.5\arcsec\times 17.5\arcsec$. Table 1: Photometric data for B037 Source | F606W | F814W | F606W$-$F814W ---|---|---|--- No. | (mag) | (mag) | (mag) 1 | $23.29\pm 0.26$ | $21.35\pm 0.16$ | $1.94$ 2 | $23.03\pm 0.23$ | $21.14\pm 0.15$ | $1.89$ 3 | $22.95\pm 0.22$ | $21.00\pm 0.14$ | $1.95$ 4 | $23.22\pm 0.25$ | $21.25\pm 0.15$ | $1.97$ 5 | $23.17\pm 0.25$ | $21.36\pm 0.16$ | $1.81$ 6 | $21.15\pm 0.10$ | $19.10\pm 0.06$ | $2.05$ 7 | $22.93\pm 0.22$ | $20.53\pm 0.11$ | $2.40$ 8 | $23.41\pm 0.28$ | $20.94\pm 0.13$ | $2.47$ 9 | $24.66\pm 0.49$ | $22.31\pm 0.25$ | $2.35$ ### 2.3 Surface brightness profiles We used the iraf task ellipse to obtain F606W and F814W surface brightness profiles for B037. B037 center position was fixed at a value derived by object locator of ellipse task, however an initial center position was determined by centroiding. Elliptical isophotes were fitted to the data, with no sigma clipping. We ran two passes of ellipse task, the first pass was run in the usual way, with ellipticity and position angle allowed to vary with the isophote semimajor axis. In the second pass, surface brightness profiles on fixed, zero-ellipticity isophotes were measured, since we choose to fit circular models for the intrinsic cluster structure and the point spread function (PSF) as Barmby et al. (2007) did (see §2.4 for details). The background value was derived as the mean of a region of $100\times 100$ pixels in “empty” areas far away from the cluster. #### 2.3.1 Ellipticity and position angle Tables 2 and 3 give the ellipticity, $\epsilon=1-b/a$, and the position angle (P.A.) as a function of the semi-major axis length, $a$, from the center of annulus in the F606W and F814W filter bands, respectively. These observables have also been plotted in Figures 3 and 4, respectively; the errors were generated by the iraf task ellipse, in which the ellipticity errors are obtained from the internal errors in the harmonic fit, after removal of the first and second fitted harmonics. From Table 3, and Figs. 3 and 4, we can see that, the values of ellipticity and position angle cannot be obtained within $0.1448\arcsec$ in the F814W filter because of very high ellipticity ($>1.0$). Ma et al. (2006b) analyzed the same F606W image of B037 used here, fitting a King (1962) model to a surface brightness profile made from a PSF-deconvolved image. They also plotted the distributions of ellipticity and the position angle as a function of the semi-major axis length. Comparison of Fig. 2 of Ma et al. (2006b) and Figs. 3 and 4 shows that, the general trend of the cluster’s ellipticity as a function of semimajor axis radius is similar between Ma et al. (2006b) and the present paper. The comparison also shows that uncertainties in the exact value of the PA are only of secondary importance for the general trend in ellipticity observed, given that the PA determination between Ma et al. (2006b) and the present paper differs somewhat greatly. There are a number of possible reasons for the offsets in PA observed between these two studies. The main reason is that, Ma et al. (2006b) used the PSF-deconvolved image. Other reasons include those related to the positions of the centering of isophotes and the different geometrical parameters set when fitting. In addition, Fig. 3 shows that the ellipticity varies significantly with position along the semimajor axis radius, especially smaller than $0.5\arcsec$. In the F814W filter band, the ellipticity is larger than 1.0 along the semimajor axis radius smaller than $0.1448\arcsec$. Table 2: B037: Ellipticity, $\epsilon$, and position angle (P.A.) as a function of the semimajor axis, $a$, in the F606W filter of HST ACS-WFC $a$ | $\epsilon$ | P.A. | $a$ | $\epsilon$ | P.A. ---|---|---|---|---|--- (arcsec) | | (deg) | (arcsec) | | (deg) 0.0260 | $0.638\pm 0.228$ | $92.9\pm 15.5$ | 0.3757 | $0.177\pm 0.031$ | $69.8\pm 5.6$ 0.0287 | $0.638\pm 0.229$ | $93.2\pm 15.6$ | 0.4132 | $0.151\pm 0.029$ | $64.7\pm 5.9$ 0.0315 | $0.639\pm 0.230$ | $93.4\pm 15.7$ | 0.4545 | $0.090\pm 0.027$ | $60.3\pm 9.1$ 0.0347 | $0.640\pm 0.232$ | $93.7\pm 15.8$ | 0.5000 | $0.005\pm 0.025$ | $172.6\pm 30.0$ 0.0381 | $0.642\pm 0.233$ | $94.0\pm 15.9$ | 0.5500 | $0.060\pm 0.020$ | $155.5\pm 9.8$ 0.0420 | $0.643\pm 0.235$ | $94.4\pm 16.0$ | 0.6050 | $0.117\pm 0.015$ | $156.3\pm 3.9$ 0.0461 | $0.645\pm 0.236$ | $94.7\pm 16.0$ | 0.6655 | $0.174\pm 0.012$ | $157.2\pm 2.2$ 0.0508 | $0.647\pm 0.182$ | $95.2\pm 12.3$ | 0.7321 | $0.233\pm 0.011$ | $157.2\pm 1.5$ 0.0558 | $0.599\pm 0.159$ | $96.8\pm 11.2$ | 0.8053 | $0.278\pm 0.011$ | $159.1\pm 1.3$ 0.0614 | $0.546\pm 0.142$ | $98.5\pm 10.6$ | 0.8858 | $0.322\pm 0.011$ | $160.8\pm 1.1$ 0.0676 | $0.503\pm 0.127$ | $100.2\pm 10.0$ | 0.9744 | $0.358\pm 0.012$ | $162.1\pm 1.2$ 0.0743 | $0.458\pm 0.099$ | $102.2\pm 8.3$ | 1.0718 | $0.380\pm 0.021$ | $164.5\pm 2.1$ 0.0818 | $0.400\pm 0.059$ | $104.3\pm 5.5$ | 1.1790 | $0.367\pm 0.022$ | $168.0\pm 2.2$ 0.0899 | $0.410\pm 0.050$ | $101.0\pm 4.6$ | 1.2969 | $0.343\pm 0.025$ | $169.2\pm 2.7$ 0.0989 | $0.428\pm 0.044$ | $98.7\pm 3.9$ | 1.4266 | $0.319\pm 0.025$ | $166.8\pm 2.8$ 0.1088 | $0.437\pm 0.046$ | $97.6\pm 3.9$ | 1.5692 | $0.252\pm 0.022$ | $165.1\pm 3.0$ 0.1197 | $0.428\pm 0.028$ | $96.6\pm 2.5$ | 1.7261 | $0.239\pm 0.021$ | $165.7\pm 2.9$ 0.1317 | $0.410\pm 0.027$ | $96.5\pm 2.5$ | 1.8987 | $0.211\pm 0.026$ | $163.6\pm 4.0$ 0.1448 | $0.400\pm 0.031$ | $96.3\pm 3.0$ | 2.0886 | $0.201\pm 0.029$ | $152.8\pm 4.6$ 0.1593 | $0.364\pm 0.023$ | $95.1\pm 2.3$ | 2.2975 | $0.188\pm 0.037$ | $150.1\pm 6.3$ 0.1752 | $0.352\pm 0.027$ | $94.8\pm 2.8$ | 2.5272 | $0.182\pm 0.033$ | $149.9\pm 5.8$ 0.1928 | $0.337\pm 0.027$ | $93.2\pm 2.9$ | 2.7800 | $0.180\pm 0.034$ | $145.6\pm 6.0$ 0.2120 | $0.311\pm 0.027$ | $92.5\pm 3.0$ | 3.0580 | $0.191\pm 0.031$ | $137.5\pm 5.2$ 0.2333 | $0.287\pm 0.026$ | $90.6\pm 3.1$ | 3.3638 | $0.143\pm 0.034$ | $125.4\pm 7.2$ 0.2566 | $0.258\pm 0.026$ | $88.6\pm 3.3$ | 3.7001 | $0.180\pm 0.041$ | $121.1\pm 7.1$ 0.2822 | $0.233\pm 0.027$ | $85.4\pm 3.8$ | 4.0701 | $0.257\pm 0.033$ | $121.6\pm 4.1$ 0.3105 | $0.207\pm 0.029$ | $81.2\pm 4.5$ | 4.4772 | $0.233\pm 0.048$ | $121.6\pm 6.5$ 0.3415 | $0.189\pm 0.030$ | $75.3\pm 5.0$ | 4.9249 | $0.237\pm 0.063$ | $116.1\pm 8.5$ Table 3: B037: Ellipticity, $\epsilon$, and position angle (P.A.) as a function of the semimajor axis, $a$, in the F814W filter of HST ACS-WFC $a$ | $\epsilon$ | P.A. | $a$ | $\epsilon$ | P.A. ---|---|---|---|---|--- (arcsec) | | (deg) | (arcsec) | | (deg) 0.0260 | | | 0.4132 | $0.031\pm 0.030$ | $77.8\pm 28.3$ 0.0287 | | | 0.4545 | $0.031\pm 0.029$ | $20.2\pm 27.5$ 0.0315 | | | 0.5000 | $0.044\pm 0.027$ | $161.2\pm 17.6$ 0.0347 | | | 0.5500 | $0.084\pm 0.023$ | $150.3\pm 8.3$ 0.0381 | | | 0.6050 | $0.127\pm 0.021$ | $150.3\pm 5.0$ 0.0420 | | | 0.6655 | $0.175\pm 0.019$ | $153.6\pm 3.4$ 0.0461 | | | 0.7321 | $0.220\pm 0.016$ | $157.4\pm 2.3$ 0.0508 | | | 0.8053 | $0.247\pm 0.013$ | $161.8\pm 1.7$ 0.0558 | | | 0.8858 | $0.251\pm 0.014$ | $167.8\pm 1.8$ 0.0614 | | | 0.9744 | $0.263\pm 0.017$ | $170.0\pm 2.1$ 0.0676 | | | 1.0718 | $0.293\pm 0.034$ | $170.8\pm 4.0$ 0.0743 | | | 1.1790 | $0.297\pm 0.035$ | $172.5\pm 4.1$ 0.0818 | | | 1.2969 | $0.230\pm 0.028$ | $171.5\pm 4.0$ 0.0899 | | | 1.4266 | $0.216\pm 0.025$ | $166.7\pm 3.7$ 0.0989 | | | 1.5692 | $0.198\pm 0.031$ | $165.5\pm 5.1$ 0.1088 | | | 1.7261 | $0.198\pm 0.025$ | $169.5\pm 4.0$ 0.1197 | | | 1.8987 | $0.188\pm 0.029$ | $167.5\pm 5.0$ 0.1317 | | | 2.0886 | $0.139\pm 0.031$ | $166.9\pm 6.9$ 0.1448 | | | 2.2975 | $0.117\pm 0.031$ | $117.8\pm 8.0$ 0.1593 | $0.908\pm 0.117$ | $89.9\pm 6.9$ | 2.5272 | $0.100\pm 0.034$ | $148.1\pm 10.2$ 0.1752 | $0.878\pm 0.026$ | $90.9\pm 1.5$ | 2.7800 | $0.118\pm 0.051$ | $141.0\pm 13.3$ 0.1928 | $0.827\pm 0.151$ | $90.8\pm 9.6$ | 3.0580 | $0.094\pm 0.043$ | $115.8\pm 13.7$ 0.2120 | $0.749\pm 0.034$ | $90.3\pm 2.3$ | 3.3638 | $0.094\pm 0.035$ | $132.7\pm 11.1$ 0.2333 | $0.731\pm 0.037$ | $89.4\pm 2.6$ | 3.7001 | $0.103\pm 0.056$ | $121.3\pm 16.4$ 0.2566 | $0.695\pm 0.041$ | $87.1\pm 2.9$ | 4.0701 | $0.127\pm 0.061$ | $120.1\pm 14.6$ 0.2822 | $0.624\pm 0.031$ | $84.4\pm 2.3$ | 4.4772 | $0.162\pm 0.031$ | $115.7\pm 5.9$ 0.3105 | $0.546\pm 0.035$ | $80.4\pm 2.6$ | 4.9249 | $0.091\pm 0.054$ | $131.4\pm 17.7$ 0.3415 | $0.401\pm 0.035$ | $72.4\pm 3.2$ | 5.4174 | $0.150\pm 0.065$ | $135.5\pm 13.3$ 0.3757 | $0.258\pm 0.029$ | $63.4\pm 3.8$ | 5.9591 | $0.188\pm 0.044$ | $161.8\pm 7.4$ Figure 3: Ellipticity as a function of the semimajor axis in the F606W and F814W filters of ACS/HST. Figure 4: P.A. as a function of the semimajor axis in the F606W and F814W filters of ACS/HST. ### 2.4 Point spread function At a distance of 780 kpc, the ACS/WFC has a scale of $\rm{0.05~{}arcsec=0.19~{}pc~{}pixel^{-1}}$, and thus M31 clusters are clearly resolved with it. Their observed core structures, however, are still affected by the PSF. We chose not to deconvolve the data, instead fitting structural models after convolving them with a simple analytic description of the PSF as Barmby et al. (2007) did. To estimate the PSF for the WFC, Barmby et al. (2007) used the iraf task ellipse with circular symmetry enforced to produce intensity profiles out to radii of about $2^{\prime\prime}$ (40 pixels) for a number of isolated stars on a number of images, and combined them to produce a single, average PSF. This was done separately for the F606W and F814W filters. They originally tried to fit these with simple Moffat profiles (with backgrounds added), but found that a better description was given by a function of the form below. For the combination of the WFC and F606W filter, $I_{\rm PSF}=I_{0}\left[1+\left(R/0\farcs 0686\right)^{3}\right]^{-1.23}\ ,$ (1) which has a full width at half-maximum of ${\rm FWHM}=0\farcs 125$, or about 2.5 px; for the combination of the WFC and F814W filter, $I_{\rm PSF}=I_{0}\left[1+\left(R/0\farcs 0783\right)^{3}\right]^{-1.19}\ ,$ (2) which has a full width at half-maximum of ${\rm FWHM}=0\farcs 145$, or about 2.9 px. In addition, since this PSF formula is radially symmetric and the models of King (1966) we fit are intrinsically spherical, the convolved models to be fitted to the data are also circularly symmetric. ### 2.5 Extinction When we fit models to the brightness profiles of B037, we will correct the inferred magnitude parameters for extinction. The reddening law from Cardelli et al. (1989) is employed in this paper. The effective wavelengths of the ACS F606W and F814W filters are $\lambda_{\rm eff}=5918$ and 8060 Å (Sirianni et al., 2005), so that from Cardelli et al. (1989), $A_{\rm{F606W}}\simeq 2.8\times E(B-V)$ and $A_{\rm{F814W}}\simeq 1.8\times E(B-V)$ (see Barmby et al., 2007; McLaughlin et al., 2008, for details). The reddening value of $E(B-V)=1.360\pm 0.013$ from Ma et al. (2006a) is adopted in this paper. ### 2.6 Magnitudes of B037 in F606W and F814W filters We derived the total flux of B037 in F606W and F814W filter bands using the iraf task phot in dapphot as below: measuring aperture magnitudes in concentric apertures with an interval of $0.1\arcsec$, drawing magnitude growth curves, and paying attention to where the flux does not increase. At last, we obtained the magnitudes of B037 in F606W and F814W to be $16.21\pm 0.010$ and $14.16\pm 0.006$, respectively. In the photometry, we derived the background value as the mean of a region far away from the cluster (see §2.3 for details). We use VEGAMAG photometric system. In order to allow a meaningful comparison with the previous ground-based broad-band photometry of Barmby et al. (2000), we transformed the magnitudes from the ACS system to the standard broad-band photometric system by following the transformation equations and coefficients of Table 22 of Sirianni et al. (2005). The results are $m_{V}{(\rm ACS)}=16.83$ (this paper) versus $m_{V}=16.82$ (Barmby et al., 2000), and $m_{I}{(\rm ACS)}=14.15$ (this paper) versus $m_{I}=14.16$ (Barmby et al., 2000). Our results are in good agreement with Barmby et al. (2000). ## 3 Models and Fits ### 3.1 Structural models After elliptical galaxies, GCs are the best understood and most thoroughly modelled class of stellar systems. For example, a large majority of the $\sim 150$ Galactic GCs have been fitted by the simple models of single-mass, isotropic, lowered isothermal spheres developed by Michie (1963) and King (1966) (hereafter “King models”), yielding comprehensive catalogs of cluster structural parameters and physical properties (see McLaughlin & van der Marel, 2005, and references therein). For extragalactic GCs, HST imaging data have been used to fit King models to a large number of GCs in M31 (e.g., Barmby et al., 2002a, 2007, and references therein), in M33 (Larsen et al., 2002), and in NGC 5128 (e.g., Harris et al., 2002; McLaughlin et al., 2008, and references therein). In this paper, we fit the usual King models to the density profile of B037 observed with ACS/WFC. ### 3.2 Observed data Tables 4 and 5 list the surface brightness, $\mu$, of B037, and its integrated magnitude, $m$, as a function of radius in the F606W and F814W filters, respectively. The errors in the surface brightness were also generated by the iraf task ellipse, in which they are obtained directly from the root mean square scatter of the intensity data along the zero-ellipticity isophotes. In addition, the surface photometries at radii where the ellipticity and position angle cannot be measured, are obtained based on the last ellipticity and position angle as the iraf task ellipse is designed. Table 4: B037: Surface brightness, $\mu$, and integrated magnitude, $m$, as a function of the radius in the F606W filter of HST ACS-WFC $R$ | $\mu$ | $m$ | $R$ | $\mu$ | $m$ ---|---|---|---|---|--- (arcsec) | (mag) | (mag) | (arcsec) | (mag) | (mag) 0.0260 | $17.327\pm 0.007$ | 23.827 | 0.3757 | $17.792\pm 0.040$ | 18.456 0.0287 | $17.328\pm 0.008$ | 23.827 | 0.4132 | $17.863\pm 0.046$ | 18.264 0.0315 | $17.328\pm 0.008$ | 23.827 | 0.4545 | $17.944\pm 0.049$ | 18.123 0.0347 | $17.329\pm 0.009$ | 23.827 | 0.5000 | $18.040\pm 0.049$ | 17.962 0.0381 | $17.330\pm 0.010$ | 23.827 | 0.5500 | $18.148\pm 0.048$ | 17.827 0.0420 | $17.331\pm 0.011$ | 23.827 | 0.6050 | $18.267\pm 0.048$ | 17.672 0.0461 | $17.331\pm 0.012$ | 23.827 | 0.6655 | $18.389\pm 0.053$ | 17.549 0.0508 | $17.332\pm 0.014$ | 22.086 | 0.7321 | $18.495\pm 0.061$ | 17.414 0.0558 | $17.334\pm 0.015$ | 22.086 | 0.8053 | $18.598\pm 0.070$ | 17.295 0.0614 | $17.338\pm 0.016$ | 22.086 | 0.8858 | $18.716\pm 0.077$ | 17.165 0.0676 | $17.341\pm 0.018$ | 22.086 | 0.9744 | $18.854\pm 0.079$ | 17.039 0.0743 | $17.346\pm 0.020$ | 21.452 | 1.0718 | $19.006\pm 0.077$ | 16.927 0.0818 | $17.351\pm 0.022$ | 21.452 | 1.1790 | $19.193\pm 0.107$ | 16.822 0.0899 | $17.356\pm 0.024$ | 21.452 | 1.2969 | $19.440\pm 0.105$ | 16.731 0.0989 | $17.363\pm 0.027$ | 21.452 | 1.4266 | $19.721\pm 0.116$ | 16.642 0.1088 | $17.372\pm 0.027$ | 21.062 | 1.5692 | $20.001\pm 0.113$ | 16.570 0.1197 | $17.382\pm 0.029$ | 20.552 | 1.7261 | $20.293\pm 0.116$ | 16.505 0.1317 | $17.395\pm 0.028$ | 20.552 | 1.8987 | $20.597\pm 0.122$ | 16.451 0.1448 | $17.411\pm 0.028$ | 20.370 | 2.0886 | $20.872\pm 0.128$ | 16.401 0.1593 | $17.429\pm 0.029$ | 19.964 | 2.2975 | $21.224\pm 0.105$ | 16.358 0.1752 | $17.450\pm 0.028$ | 19.964 | 2.5272 | $21.457\pm 0.112$ | 16.320 0.1928 | $17.475\pm 0.026$ | 19.764 | 2.7800 | $21.719\pm 0.138$ | 16.283 0.2120 | $17.504\pm 0.026$ | 19.528 | 3.0580 | $22.082\pm 0.154$ | 16.251 0.2333 | $17.538\pm 0.025$ | 19.337 | 3.3638 | $22.603\pm 0.164$ | 16.225 0.2566 | $17.578\pm 0.026$ | 19.091 | 3.7001 | $23.042\pm 0.225$ | 16.206 0.2822 | $17.624\pm 0.026$ | 19.009 | 4.0701 | $23.694\pm 0.467$ | 16.191 0.3105 | $17.675\pm 0.030$ | 18.802 | 4.4772 | $24.509\pm 0.571$ | 16.182 0.3415 | $17.732\pm 0.034$ | 18.634 | 4.9249 | $25.173\pm 1.342$ | 16.172 Table 5: B037: Surface brightness, $\mu$, and integrated magnitude, $m$, as a function of the radius in the F814W filter of HST ACS-WFC $R$ | $\mu$ | $m$ | $R$ | $\mu$ | $m$ ---|---|---|---|---|--- (arcsec) | (mag) | (mag) | (arcsec) | (mag) | (mag) 0.0260 | $15.301\pm 0.010$ | 21.800 | 0.4132 | $15.772\pm 0.032$ | 16.190 0.0287 | $15.302\pm 0.011$ | 21.800 | 0.4545 | $15.863\pm 0.036$ | 16.048 0.0315 | $15.303\pm 0.012$ | 21.800 | 0.5000 | $15.967\pm 0.038$ | 15.889 0.0347 | $15.303\pm 0.013$ | 21.800 | 0.5500 | $16.078\pm 0.037$ | 15.754 0.0381 | $15.304\pm 0.015$ | 21.800 | 0.6050 | $16.190\pm 0.033$ | 15.598 0.0420 | $15.305\pm 0.016$ | 21.800 | 0.6655 | $16.300\pm 0.036$ | 15.474 0.0461 | $15.306\pm 0.018$ | 21.800 | 0.7321 | $16.407\pm 0.046$ | 15.338 0.0508 | $15.308\pm 0.019$ | 20.061 | 0.8053 | $16.543\pm 0.053$ | 15.218 0.0558 | $15.310\pm 0.021$ | 20.061 | 0.8858 | $16.706\pm 0.054$ | 15.094 0.0614 | $15.313\pm 0.024$ | 20.061 | 0.9744 | $16.847\pm 0.054$ | 14.974 0.0676 | $15.317\pm 0.026$ | 20.061 | 1.0718 | $16.995\pm 0.061$ | 14.868 0.0743 | $15.321\pm 0.030$ | 19.430 | 1.1790 | $17.185\pm 0.105$ | 14.767 0.0818 | $15.326\pm 0.033$ | 19.430 | 1.2969 | $17.459\pm 0.086$ | 14.681 0.0899 | $15.331\pm 0.037$ | 19.430 | 1.4266 | $17.717\pm 0.107$ | 14.596 0.0989 | $15.337\pm 0.041$ | 19.430 | 1.5692 | $18.006\pm 0.101$ | 14.527 0.1088 | $15.347\pm 0.043$ | 19.039 | 1.7261 | $18.279\pm 0.102$ | 14.464 0.1197 | $15.360\pm 0.043$ | 18.531 | 1.8987 | $18.567\pm 0.086$ | 14.411 0.1317 | $15.367\pm 0.045$ | 18.531 | 2.0886 | $18.833\pm 0.092$ | 14.359 0.1448 | $15.375\pm 0.048$ | 18.350 | 2.2975 | $19.167\pm 0.095$ | 14.316 0.1593 | $15.396\pm 0.046$ | 17.940 | 2.5272 | $19.460\pm 0.094$ | 14.277 0.1752 | $15.411\pm 0.046$ | 17.940 | 2.7800 | $19.649\pm 0.156$ | 14.240 0.1928 | $15.428\pm 0.045$ | 17.738 | 3.0580 | $20.075\pm 0.137$ | 14.207 0.2120 | $15.447\pm 0.046$ | 17.496 | 3.3638 | $20.538\pm 0.103$ | 14.181 0.2333 | $15.466\pm 0.048$ | 17.301 | 3.7001 | $21.002\pm 0.138$ | 14.160 0.2566 | $15.501\pm 0.046$ | 17.046 | 4.0701 | $21.399\pm 0.203$ | 14.142 0.2822 | $15.531\pm 0.043$ | 16.960 | 4.4772 | $21.964\pm 0.228$ | 14.129 0.3105 | $15.580\pm 0.037$ | 16.745 | 4.9249 | $22.519\pm 0.240$ | 14.117 0.3415 | $15.632\pm 0.031$ | 16.573 | 5.4174 | $23.311\pm 0.588$ | 14.106 0.3757 | $15.695\pm 0.029$ | 16.388 | 5.9591 | $23.508\pm 0.782$ | 14.096 ### 3.3 Fits Our fitting procedure involves computing in full large numbers of King structural models, spanning a wide range of fixed values of the appropriate shape parameter $W_{0}$ (see McLaughlin & van der Marel, 2005, in detail). And then the models are convolved with the ACS/WFC PSF for the F606W and F814W filters of equations of (1) and (2): $\widetilde{I}_{\rm mod}^{*}(R|r_{0})=\int\\!\\!\\!\int_{-\infty}^{\infty}\widetilde{I}_{\rm mod}(R^{\prime}/r_{0})\times\widetilde{I}_{\rm PSF}\left[(x-x^{\prime}),(y-y^{\prime})\right]\ dx^{\prime}\,dy^{\prime}\ ,$ (3) where $\widetilde{I}_{\rm mod}\equiv I_{\rm mod}/I_{0}$; and $\widetilde{I}_{\rm PSF}$ is the PSF profile normalized to unit total luminosity (see McLaughlin et al., 2008, in detail). We changed the luminosity density to surface brightness $\widetilde{\mu}_{\rm mod}^{*}=-2.5\,\log\,[\widetilde{I}_{\rm mod}^{*}]$ before fitting them to the observed surface-brightness profile of B037, $\mu=\mu_{0}-2.5\,\log\,[I(R/r_{0})/I_{0}]$, finding the radial scale $r_{0}$ and central surface brightness $\mu_{0}$ which minimize $\chi^{2}$ for every given value of $W_{0}$. The $(W_{0},r_{0},\mu_{0})$ combination that yields the global minimum $\chi_{\rm min}^{2}$ over the grid used defines the best- fit model of that type: $\chi^{2}=\sum_{i}{\frac{\left[\mu_{\rm obs}(R_{i})-\widetilde{\mu}_{\rm mod}^{*}(R_{i}|r_{0})\right]^{2}}{\sigma_{i}^{2}}},$ (4) in which $\sigma_{i}$ is the error in the surface brightness. Estimates of the one-sigma uncertainties on these basic fit parameters follow from their extreme values over the subgrid of fits with $\chi^{2}/\nu\leq\chi_{\rm min}^{2}/\nu+1$, here $\nu$ is the number of free parameters. Figure 5 shows our best King fits to B037. In Fig. 5, open squares are ellipse data points included in the least-squares model fitting, and the asterisks are points not used to constrain the fit. These observed data points shown by asterisks are included in the radius of $R<2~{}\rm{pixels}=0\farcs 1$, and the isophotal intensity is dependent on its neighbors. As Barmby et al. (2007) pointed out that, the ellipse output contains brightnesses for 15 radii inside 2 pixel, but they are all measured from the same 13 central pixels and are not statistically independent. So, to avoid excessive weighting of the central regions of B037 in the fits, we only used intensities at radii $R_{\rm min}$, $R_{\rm min}+(0.5,1.0,2.0~{}{\rm pixels})$, or $R>2.5~{}{\rm pixels}$ as Barmby et al. (2007) used. Table 6 summarizes the results obtained in this paper. Figure 5: Surface brightness profile of B037 measured in the F606W and F814 filters. Dashed curve (blue) trace the PSF intensity profiles and solid (red) curves are the PSF-convolved best-fit models. Open squares are ellips data points included in the $\chi^{2}$ model fitting, and the asterisks are points not used to constrain the fits (see the text in detail). Table 6: Structural parameters of B037 Parameters | F606W | F814W ---|---|--- $r_{0}$ | $0.56\pm 0.02\arcsec~{}(=2.16\pm 0.08~{}\rm{pc})$ | $0.56\pm 0.01\arcsec~{}(=2.16\pm 0.04~{}\rm{pc})$ $r_{t}$ | $8.6\pm 0.4\arcsec~{}(=33.1\pm 1.5~{}\rm{pc})$ | $8.9\pm 0.3\arcsec~{}(=34.3\pm 1.2~{}\rm{pc})$ $c=\log(r_{t}/r_{0})$ | $1.19\pm 0.02$ | $1.20\pm 0.01$ $r_{h}$ | $1.05\pm 0.03\arcsec(=4.04\pm 0.12~{}\rm{pc})$ | $1.07\pm 0.01\arcsec(=4.12\pm 0.04~{}\rm{pc})$ $\mu_{0}$ (${\rm mag~{}arcsec^{-2}}$) | $13.53\pm 0.03$ | $12.85\pm 0.03$ ### 3.4 Comparison to previous results Ma et al. (2006b) analyzed the same F606W image of B037 used here, fitting a King (1962) model to a surface brightness profile derived from a PSF- deconvolved image. They derived the scale radius $r_{0}=0\farcs 72$ (it is called the core radius in Ma et al. (2006b)), half-light radius $r_{h}=1\farcs 11$, concentration index $c=0.91$, and central surface brightness $\mu(0)=17.21~{}{\rm mag~{}arcsec^{-2}}$ (using the value for extinction adopted in this paper, this becomes $\mu_{0}=13.40~{}{\rm mag~{}arcsec^{-2}}$). Comparing the results of Ma et al. (2006b) with Table 6 of this paper, we find that our model fits produce a somewhat higher concentration and smaller scale radius. These differences come from: (i) using different models (King (1962) vs King (1966)), (ii) the observed data are obtained with different ways. In (ii), Ma et al. (2006b) derived the surface brightness profile from a PSF-deconvolved image; in addition, Ma et al. (2006b) derived the surface brightness profile with ellipticity and position angle allowed to vary with the isophote semimajor axis, however, in this paper, we derived the surface brightness profile on fixed, zero-ellipticity isophotes, since we choose to fit circular models for the intrinsic cluster structure and the PSF as Barmby et al. (2007) did (see §2.4 for details). In fact, from Fig. 5 of this paper and Fig. 3 of Ma et al. (2006b), we can see that, the observed data are somewhat different between Ma et al. (2006b) and this paper. Barmby et al. (2007) analyzed the same F606W and F814W images of B037 used here with nearly the same observed data and method. The results of comparison are listed in Table 7 (Table 5 of Barmby et al. (2007) in the electronic edition did not list the results of B037 in the F814W filter.), from which we can see that the results obtained in this paper are in good agreement with ones of Barmby et al. (2007) (about the central surface brightnesses, we have corrected them using the value for extinction adopted in this paper). Table 7: Results of comparison | F606W | F814W ---|---|--- Parameters | Barmby et al. (2007) | This paper | Barmby et al. (2007) | This paper $r_{0}$ | 056 | 056 | 059 | 056 $c=\log(r_{t}/r_{0})$ | 1.23 | 1.19 | 1.18 | 1.20 $r_{h}$ | 109 | 105 | | 107 $\mu_{0}$ (${\rm mag~{}arcsec^{-2}}$) | 13.45 | 13.53 | 12.75 | 12.85 ## 4 Discussion and summary As discussed in §3.1, it is impossible that the dust lane comes from the Milky Way. Another possibility is that the dust lane is contained in B037 itself. As we know, the formation of dust requires gas with a rather high metallicity. Perrett et al. (2002) presented metallicities for more than 200 GCs in M31 including B037, using the Wide Field Fibre Optic Spectrograph at the 4.2 m William Herschel Telescope in La Palma, Canary Islands, which provides a total spectral coverage of $\sim$ 3700-5600 Å with two gratings. One grating (H2400B 2400 line) yielded a dispersion of 0.8 Å${\rm~{}pixel^{-1}}$ and a spectral resolution of 2.5 Å over the range 3700-4500 Å covering the CN feature at 3883 Å, the H and K lines of calcium, $\rm H\delta$, the CH G band and the 4000 Å continuum break, and the other grating (R1200R 1200 line) presented a dispersion of 1.5 Å${\rm~{}pixel^{-1}}$ and a spectral resolution of 5.1 Å over the range 4400-5600 Å to add absorption features such as $\rm H\beta$, the Mg $b$ triplet, and two iron lines near 5300 Å. Then, Perrett et al. (2002) calculated 12 absorption-line indices based on the prescription of Brodie & Huchra (1990). By the comparison of the line indices with the published M31 GC [Fe/H] values from the previous literature (Bònoli et al., 1987; Brodie & Huchra, 1990; Barmby et al., 2000), the results of linear fits were obtained. Final cluster metallicities were determined from an unweighted mean of the [Fe/H] values calculated from the CH (G), Mg $b$, and Fe53 line strengths. For B037, Perrett et al. (2002) obtained its metallicity to be $\rm[Fe/H]=-1.07\pm 0.20$. It is clear that B037 has a low metallicity. So, it is intricate that where is the dust lane from? In this paper, using the images of deep observations and of highly spatial resolutions with the ACS/HST, we firstly present that the GC B037 in M31 is crossed by a dust lane. Photometric data in the F606W and F814W bands provide that, colors of ($\rm{F606W-F814W}$) in the dust lane are redder $\sim 0.4$ mags than ones in the other regions of B037. From the HST images, this dust lane seems to be contained in B037, not from the Milky Way. However, the formation of dust requires gas with a rather high metallicity. So, it seems impossible that the observed dust lane is physically associated with B037 itself, which has a low metallicity to be $\rm[Fe/H]=-1.07\pm 0.20$ from Perrett et al. (2002). So, that the observed dust lane in the view of B037 is from B037 itself or from the Milky Way needs to be confirmed in the future. In addition, based on these images, we present the precise variation of ellipticity and position angle, and of surface brightness profile, and determine the structural parameters of B037 by fitting a single-mass isotropic King model. ###### Acknowledgements. I am indebted to Daming Chen, Zhou Fan, Tianmeng Zhang and Song Wang for their helps in finishing this paper. I am also grateful to the referee for the important comments. This work was supported by the Chinese National Natural Science Foundation grands No. 10873016, and 10633020, and by National Basic Research Program of China (973 Program), No. 2007CB815403. ## References * Barmby et al. 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arxiv-papers
2011-01-08T03:21:55
2024-09-04T02:49:16.235370
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jun Ma (1,2) ((1) National Astronomical Observatories, Chinese Academy\n of Sciences, (2) Key Laboratory of Optical Astronomy, National Astronomical\n Observatories, Chinese Academy of Sciences)", "submitter": "Jun Ma", "url": "https://arxiv.org/abs/1101.1569" }
1101.1580
# Infall and outflow motions in the high-mass star forming complex G9.62+0.19 Tie Liu11affiliation: Department of Astronomy, Peking University, 100871, Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn , Yuefang Wu11affiliation: Department of Astronomy, Peking University, 100871, Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn , Sheng-Yuan Liu22affiliation: Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, Taiwan , Sheng-Li Qin33affiliation: I. Physikalisches Institut, Universität zu Köln, Zülpicher Str. 77, 50937 Köln, Germany 44affiliation: National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012 , Yu-Nung Su22affiliation: Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, Taiwan , Huei-Ru Chen55affiliation: Institute of Astronomy and Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 22affiliation: Institute of Astronomy and Astrophysics, Academia Sinica, Taipei, Taiwan and Zhiyuan Ren11affiliation: Department of Astronomy, Peking University, 100871, Beijing China; liutiepku@gmail.com, ywu@pku.edu.cn ###### Abstract We present the results of a high resolution study with the Submillimeter Array towards the massive star forming complex G9.62+0.19. Three sub-mm cores are detected in this region. The masses are 13, 30 and 165 M☉ for the northern, middle and southern dust cores, respectively. Infall motions are found with HCN (4-3) and CS (7-6) lines at the middle core (G9.62+0.19 E). The infall rate is $4.3\times 10^{-3}~{}M_{\odot}\cdot$yr-1. In the southern core, a bipolar-outflow with a total mass about 26 M☉ and a mass-loss rate of $3.6\times 10^{-5}~{}M_{\odot}\cdot$yr-1 is revealed in SO ($8_{7}-7_{7}$) line wing emission. CS (7-6) and HCN (4-3) lines trace higher velocity gas than SO ($8_{7}-7_{7}$). G9.62+0.19 F is confirmed to be the driving source of the outflow. We also analyze the abundances of CS, SO and HCN along the redshifted outflow lobes. The mass-velocity diagrams of the outflow lobes can be well fitted by a single power law. The evolutionary sequence of the cm/mm cores in this region are also analyzed. The results support that UC Hii regions have a higher blue excess than their precursors. Massive core:pre-main sequence-ISM: molecular-ISM: kinematics and dynamics- ISM: jets and outflows-stars: formation ††slugcomment: Accepted to ApJ ## 1 Introduction High-mass stars play a major role in the evolution of the Galaxy. They are the principal sources of heavy elements and UV radiation (Zinnecker & Yorke, 2007). However, the formation and evolution of high-mass stars are still unclear. A possible evolution sequence of high-mass stars from infrared dark clouds to classic Hii regions has been suggested (Van der Tak & Menten, 2005). But one of the major topics whether high-mass stars form through accretion- disk-outflow, like low-mass ones (Shu, Adams & Lizano, 1987), or form via collision-coalescence (Wolfire, & Cassinelli, 1987; Bonnell et al., 1998) is still far from solved. Yet more and more observations at various resolutions seem to support the accretion-disk-outflow models rather than collision-coalescence models. Disks are detected in several high-mass star forming regions (Patel et al., 2005; Jiang et al., 2005; Sridharan, Williams, & Fuller, 2005). Outflows are found with a high detection rate as in low-mass cores in single-dish surveys (Wu et al., 2004; Zhang et al., 2005; Qin et al., 2008a). High resolution studies have also confirmed that molecular outflows are common in high-mass star forming regions (Su, Zhang, & Lim, 2004; Qiu et al., 2007; Qin et al., 2008b, c; Qiu et al., 2009). Searching for inflow motions also has made large progress in recent years (Wu & Evans, 2003; Fuller, Williams, & Sridharan, 2005; Wyrowski et al., 2006; Klaassen, & Wilson, 2007; Wu et al., 2007, 2009; Furuya, Cesaroni, & Shinnaga, 2011). Both infall and outflow motions in the massive core JCMT 18354-0649S are detected (Wu et al., 2005), and further confirmed by higher resolution observations (Liu et al., 2011). Although accretion-disk-outflow systems are found in high-mass star forming regions, there may be differences between low- and high-mass formation. The infall motion can be detected via ”blue profile”, a double-peaked profile with the blueshifted peak being stronger for optically thick lines and a single peak at the absorption part of optically thick lines for optically thin lines, which is caused by self absorption of the cooler outer infalling gas towards the warmer central region (Zhou et al., 1993). In contrast, the ”red profile” where the redshifted peak of a double-peaked profile being stronger for optically thick lines is suggested as indicators for outflow motions. Mardones et al. (1997) defined the ”blue excess” in a survey, E, as E = (NB- NR)/NT (Mardones et al. 1997), where NT is number of sources, NB and NR mark the number of sources with blue and red profiles, respectively. The blue excess seems to be no significant differences among the low-mass cores in different evolutionary phases. However, using the IRAM 30 m telescope, Wu et al. (2007) found that UC Hii regions show a higher blue excess than their precursors, indicating fundamental differences between low- and high-mass-star forming conditions. The searches need to be expanded. Located at a distance of 5.7 kpc (Hofner et al., 1994), G9.62+0.19 is a well studied high-mass star forming region containing a cluster of Hii regions, which are probably at different evolutionary stages. Multiwavelength VLA observations have identified nine radio continuum sources (denoted from A-I) (Garay et al., 1993; Testi et al., 2000), and components C-I are very compact ($<5\arcsec$ in diameter) (Garay et al., 1993; Testi et al., 2000). As revealed in NH3 (4,4), (5,5) and CH3CN (J=6-5), component F is a hot molecular core (HMC) and hence likely the youngest source in the region (Cesaroni et al., 1994; Hofner et al., 1994, 1996b). G9.62+0.19 E is a young massive star surrounded by a very small UC Hii region and a dusty envelope (Hofner et al., 1996b), while G9.62+0.19 D a small cometary UC Hii region excited by a B0.5 ZAMS star (Hofner et al., 1996b; Testi et al., 2000). Both G9.62+0.19 E and G9.62+0.19 D seem to be at a more evolved stage than G9.62+0.19 F. Thus G9.62 complex is an ideal sample to examine massive star forming activities including outflow and infall motions. Maser emissions of NH3, H2O, OH, and CH3OH, as well as the strong thermal NH3 emissions were detected along a narrow region with projected length $20\arcsec$ and width$\leq 2\arcsec$ (Hofner et al., 1994). A possible explanation for this alignment is compression of the molecular gas by shock front originating from an even more evolved Hii region to the west of the star-forming front (Hofner et al., 1994). High-velocity molecular outflows also have been detected in this region, and G9.62+0.19 F is believed to be the driving source (Gibb, Wyrowski, & Mundy , 2004; Hofner, Wiesemeyer, & Henning , 2001; Su et al., 2005). However, most of previous work was carried out at low frequencies, probing low excitation conditions. To exam the hot dust/gas environment and dynamical processes in this region, higher resolution studies at high frequencies are needed. In this paper we report the results of the Submillimeter Array (SMA111Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.) observations toward G9.62+0.19 region at 860 $\micron$. ## 2 Observations The observations of G9.62+0.19 with the SMA were carried out on July 9th, 2005 with seven antennas in its compact configuration at 343 GHz for the lower sideband (LSB) and 353 GHz for the upper sideband (USB). The Tsys ranges from 210 to 990 K with a typical value of 380 K at both sidebands during the observations. The observations had two fields for the G9.62+0.19 complex to cover the entire region with emissions. One phase reference center was RA(J2000) = 18h06m14.21s and DEC(J2000) = -$20\arcdeg 31\arcmin 46.2\arcsec$, and the other was RA(J2000) = 18h06m15.00s and DEC(J2000) = -$20\arcdeg 31\arcmin 34.20\arcsec$. Uranus and Neptune were observed for antenna-based bandpass calibration. QSOs 1743-038 and 1911-201 were employed for antenna- based gain correction. Neptune was used for flux-density calibration. The frequency spacing across the spectra band was 0.8125 MHz, corresponding to a velocity resolution of $\sim$0.7 km s-1. MIRIAD was employed for calibration and imaging (Sault et al., 1995). The imaging was done to each field separately and the mosaic continuum map was made using a linear mosaicing algorithm (task ”linmos” in MIRIAD). The 860 $\micron$ continuum data was acquired by averaging over all the line-free channels in both sidebands. The spectral cubes were constructed using the continuum-subtracted spectral channels smoothed into a velocity resolution of 1 km s-1. Additional self-calibration with models of the clean components from previous imaging process was performed on the continuum data in order to remove residual errors due to phase and amplitude problems, and the gain solutions obtained from the continuum data were applied to the line data. The synthesized beam size of the continuum emission with robust weighting of 0.5 is $2.76\arcsec\times 1.88\arcsec$ (P.A.=$21.4\arcdeg$). ## 3 Results ### 3.1 Continuum emission The 860 $\micron$ continuum image combining the visibility data from both sidebands is shown in Figure 1 . Three sub-mm cores are detected. The known cm and mm continuum components (Testi et al., 2000) of B, C, D, E, F, G, H, and I are marked by plus signs. Water masers (Hofner, & Churchwell, 1996a) are marked by open squares and methanol masers (Norris et al., 1993) by triangles. The near-IR sources (Persi et al., 2003; Testi et al., 1998; Linz et al., 2005) are marked by filled circles. IRAC sources are taken from the database of Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE) 222http://irsa.ipac.caltech.edu/data/SPITZER/GLIMPSE/ and labeled with asterisks. The northern core is located at south-east of G9.62+0.19 C, and the middle core is associated with G9.62+0.19 E. The 860 $\micron$ continuum emission at the southern core is concentrated on the hot molecular core G9.62+0.19 F and extends to G9.62+0.19 D in the south and to G9.62+0.19 G in the north. Gaussian fits were made to the continuum. The northern core seems to be a point-like source. The middle core is very compact with a deconvolved size of $\sim 1.4\arcsec$. The southern core is found to be elongated from north to south with an average size of $2.4\arcsec$, containing at least three sources, D, F, and G. F is at its peak position. The peak positions, sizes, peak intensities and total fluxes of these three sub-mm cores are listed in Column 2-5 in Table 1. The physical properties of these cores will be further discussed in section 4.2. ### 3.2 Line emission Tens of molecular transitions including hot molecular lines CH3OH, HCOOCH3, and CH3OCH3 are detected toward both the middle and southern sub-mm cores, indicating these two cores are hot and dense (Qin et al., 2010). Figure 2 presents the full LSB and USB spectra in the UV domain over the shortest baseline. The strongest lines are identified and labeled on the plots. Only HCN (4-3) and CS (7-6) line emissions are detected towards the northern core with our sensitivity. Thus we mainly focus on the middle and southern cores in this paper. The systemic velocities of 2.1 km s-1 for the middle core and 5.2 km s-1 for the southern core are obtained by averaging the Vlsr of multiple singly peaked lines. Six transitions of the thioformaldehyde (H2CS) and molecular transitions SO ($8_{7}$-77), CS (7-6), HC15N (4-3) and HCN (4-3) are analyzed here, while the others will be discussed in another paper. We have made gaussian fits to the beam-averaged spectra, and present the observed parameters of these lines in Table 2. #### 3.2.1 Line emission at the middle sub-mm core The integrated intensity maps of four transitions of H2CS towards the middle core are shown in the upper panels of Figure 3. From (a) to (d), the upper level energy of H2CS transitions varies from $\sim 90$ K to $\sim 400$ K. The H2CS emission is spatially coincident with continuum emission of the middle core very well. The Position-Velocity (P-V) diagram and first moment map of H2CS (102,8-92,7) emission are presented in Figure 4. The P-V diagram is constructed across the peak of the continuum along the N-S direction. From P-V diagram two emission peaks are clearly revealed. The velocities of the two emission peaks are at 1 and 3 km s-1 with $1.5\arcsec$ spatial separation, indicating a velocity gradient in N-S direction. The first moment map also shows velocity changes in N-S direction. The small velocity gradient detected in H2CS (102,8-92,7) emission may indicate a disk with a low inclination along the line of sight, which requires further confirmation with higher angular resolution observations and other molecular line tracers. The spectra and integrated intensity maps of HC15N (4-3) and SO ($8_{7}$-77) are presented in Figure 5. The two spectra seem to be symmetric, and their cores are associated with that of the continuum emission very well. Figure 6 presents the spectra and P-V diagrams of HCN (4-3) and CS (7-6) emissions of the middle core. HCN (4-3) and CS (7-6) show asymmetric profile. The blue and red emission peaks of HCN (4-3) are around 0 km s-1 and 6 km s-1, respectively. The blueshifted emission of CS (7-6) peaks around 1 km s-1, while the redshifted around 4 km s-1. We can see the blueshifted emission of both HCN (4-3) and CS (3-2) is always stronger than the redshifted emission and the absorption is also redshifted, which are blue profiles (see Sect. 1). Besides the ”blue profile”, some weak absorption dips are found around 10 km s-1 in both the spectra and P-V diagrams of HCN (4-3) and CS (7-6), and further observations are needed to determine the properties of these absorption dips. In this paper we only pay attention to the ”blue-profile” found in CS (7-6) and HCN (4-3) emission. The integrated intensity maps of HCN (4-3) and CS (7-6) towards the middle core are presented in Figure 7. The HCN (4-3) and CS (7-6) are associated with the dust emission. #### 3.2.2 Line emission at the southern core The integrated intensity maps of four transitions of H2CS at the southern core are shown in the lower panels of Figure 3. The upper level energy of H2CS transitions varies from $\sim 90$ K to $\sim 400$ K from panel (e) to panel (h). As the upper level energy increases, the emission peak of the different transitions of H2CS moves from S-E to N-W, indicating a temperature gradient in the southern core. Averaged spectra of SO ($8_{7}$-77), HC15N (4-3), HCN (4-3) and CS (7-6) at the southern core are presented in Figure 8. The spectra of SO ($8_{7}$-77) and HC15N are averaged over a region of 4$\arcsec$, while HCN (4-3) and CS (7-6) are averaged over a region of 6$\arcsec$. SO ($8_{7}$-77) emission has a total velocity extent of larger than 20 km s-1. From gaussian fit to the spectrum, the peak velocity of SO emission is $5.1\pm 0.1$ km s-1, coincident very well with the systemic velocity 5.2 km s-1. HC15N (4-3) has a velocity extent of about 15 km s-1. The velocity extents of CS (7-6) and HCN (4-3) are as high as 40 km s-1 and 60 km s-1, respectively. Emission wings are clearly detected from the spectra of the four lines. A ”red-profile” is significantly exhibited in the spectra of CS (7-6) and HCN (4-3), of which the redshifted emission is always stronger than the blueshifted emission with an absorption dip at the blueshifted side of the systemic velocity (5.2 km s-1). This profile is caused by absorption of the colder blueshifted gas in front of the hot core, indicating outflow motions. The ”red-profile” is consistent with that detected using single dish observations (see Figure 6 of Hofner, Wiesemeyer, & Henning (2001)). The integrated intensity maps of HC15N (4-3) and SO ($8_{7}$-77) at the southern core are presented in Figure 9. To avoid the influence of outflow motions, both the maps are integrated from 2 km s-1 to 8 km s-1. Both the emission of HC15N (4-3) and SO ($8_{7}$-77) coincides with the cm/mm component F, and extends from D to G. As shown in the left panels of Figure 10, the high velocity gas of HC15N (4-3) and SO ($8_{7}$-77) can be identified by the vertically dashed lines in the P-V diagrams. For SO ($8_{7}$-77), we integrate from -4 km s-1 $\leq$ V $\leq$ 0 km s-1 for the blue wing and 10 km s-1 $\leq$ V $\leq$ 14 km s-1 for the red wing, and present the contour map in (c) of Figure 10. For HC15N (4-3), only the red wing emission is presented in (d) of Figure 10. The high velocity emission of both HC15N (4-3) and SO ($8_{7}$-77) is associated with core F, indicating that core F is the driven source of the outflow. The blue and red wings of SO ($8_{7}$-77) overlap to a large extent in the contour maps, and hence the molecular outflow revealed by SO ($8_{7}$-77) is observed close to its flow axis. Figure 11 presents the channel maps of CS (7-6) emission. The redshifted high- velocity gas seems to be elongated from north-east to south-west, while the blueshifted high-velocity gas from north to south. The high velocity gas revealed by CS (7-6) is also very obvious in the P-V diagram in Figure 13(d). As shown in the P-V diagram, the blueshifted high-velocity gas extends about 8$\arcsec$ from north to south. The high-velocity emission integrated over the wings (-12 km s-1 $\leq$ V $\leq$ -5 km s-1 for the blue wing and 15 km s-1 $\leq$ V $\leq$ 22 km s-1 for the red wing) is presented in Figure 13(e). Figure 12 is the channel maps of HCN (4-3) emission. The maximum of absorptions appears at around 0 km s-1. The redshifted high-velocity gas seems to be elongated from west to east, while the blueshifted high-velocity gas from north to south. At very high velocity channels (V$\leq$ -16 km s-1), the blueshifted emission is totally located at south-east. By comparing the channel maps and P-V diagrams (see Figure 13) of HCN (4-3) and CS (7-6) at velocity intervals -12 km s-1 $\leq$ V $\leq$ -5 km s-1 and 15 km s-1 $\leq$ V $\leq$ 22 km s-1, we find similar structures in CS (7-6) and HCN (4-3) emissions. The high-velocity emission of HCN (4-3) integrated from -12 km s-1 to -5 km s-1 for the blue wing and from 15 km s-1 to 22 km s-1 for the red wing is presented in the panel (b) of Figure 13\. As of CS (7-6), the blueshifted gas revealed by HCN (4-3) is elongated from north to south with the emission center located between G9.62+0.19 F and G9.62+0.19 D, while the redshifted gas from north-east to south-west. Two clumps are found in the blueshifted high-velocity emission of HCN (4-3), which locate at north-west and south-east of F, respectively. In order to reveal the very high velocity emission traced by HCN (4-3) but not CS (7-6), we integrate over the wings at much higher velocities (-20 km s-1 $\leq$ V $\leq$ -13 km s-1 for the blue wing and 23 km s-1 $\leq$ V $\leq$ 39 km s-1 for the red wing), and present the integrated emission map in Figure 13(c). The redshifted emission is elongated from north-east to south-west with the emission center located between G9.62+0.19 F and G9.62+0.19 G, while the blueshifted emission center located between G9.62+0.19 F and G9.62+0.19 D. It is clearly seen that the high velocity gas traced by SO ($8_{7}$-77), CS (7-6) and HCN (4-3) have different spatial distributions, which should be caused by the complicated interactions between the outflow and the ambient gas. It may also indicate a change of the outflow axis. The change of outflow axis is also found in IRAS 20126+4104 (Su et al., 2007) and JCMT 18354-0649S (Liu et al., 2011). From the integrated emission maps of SO ($8_{7}$-77), HCN (4-3) and CS (7-6) high velocity gas, it is clearly seen that G9.62+0.19 F is located at the middle of the redshifted and blueshifted lobes, suggesting G9.62+0.19 F is the outflow driving source. ## 4 Discussion ### 4.1 Rotational temperature of H2CS transitions Six transitions of H2CS have been detected in the middle and southern cores, enabling us to estimate the rotational temperature. Under the assumptions that the gas is optically thin under local thermodynamic equilibrium and the gas emission fills the beam, the rotation temperature and beam-averaged column density can be estimated using the Rotational Temperature Diagram (RTD) by (Cummins, Linke, &Thaddeus, 1986; Turner et al., 1991; Liu, et al., 2002) $\textrm{ln}(\frac{N_{u}}{g_{u}})=\textrm{ln}(\frac{N_{T}}{Q_{rot}})-\frac{E_{u}}{T_{rot}}=\textrm{ln}[2.04\times 10^{20}\frac{\int~{}I(Jy~{}beam^{-1})dv(km~{}s^{-1})}{\theta_{a}\theta_{b}(arcsec^{2})g_{I}g_{K}\nu^{3}(GHz^{3})S\mu^{2}(debye^{2})}]$ (1) where Nu is the observed column density of the upper energy level, gu is the degeneracy factor in the upper energy level, NT is the total beam-averaged column density, Qrot is the rotational partition function, Eu is the upper level energy in K, Trot is the rotation temperature, $\int$ I dv is the integrated intensity of the specific transition, $\theta_{a}$ and $\theta_{b}$ are the FWHM beam size, gK is the K-ladder degeneracy, gI is the degeneracy due to nuclear spin, $\nu$ is the rest frequency, and S is line strength and $\mu$ the permanent dipole moment. For H2CS, the interchangeable nuclei are spin $\frac{1}{2}$, leading to ortho- and para-forms with gI equaling $\frac{3}{4}$ and $\frac{1}{4}$, respectively (Blake et al., 1987; Turner et al., 1991). The partition function Qrot of H2CS is (Blake et al., 1987) $Q_{rot}=2[\frac{\pi(kT_{rot})^{3}}{h^{3}ABC}]^{\frac{1}{2}}$ (2) where k and h are the Boltzmann and Planck constants, respectively, and A, B, and C are the rotation constants. Thus the rotation temperature Trot and total column density NT can be estimated by least-squares fitting to the multiple transitions. We applied the RTD method towards D, E, F, G (see Figure 14), and the fitting results are listed in the second and third columns of Table 3\. The rotational temperature of the middle core (E) is 83$\pm$21 K. In the southern core, the rotational temperature estimated decreases from G (91 K) to F (83 K) and D (43 K), suggesting the temperature gradient in the southern core. The total column density of H2CS ranges from 1.3$\times$1015 (G) to 3.8$\times$1015 cm-2 (D). However, the filling factor and the optical depth correction were not taken account of in the RTD method. To investigate their effect we applied the Population Diagram (PD) analysis (Goldsmith, & Langer, 1999; Wang et al., 2010). In the PD analysis, we have $\textrm{ln}(\frac{\hat{N_{u}}}{g_{u}})=\textrm{ln}(\frac{N_{T}}{Q_{rot}})-\frac{E_{u}}{T_{rot}}+\textrm{ln}(f)-\textrm{ln}(\frac{\tau}{1-e^{-\tau}})$ (3) where $\hat{N_{u}}$ is the inferred column density of the upper energy level from the PD analysis, f is the source filling factor and $\tau$ is the optical depth. The optical depth $\tau$ can be expressed by (Remijan et al., 2004) $\tau=\frac{8\pi^{3}S\mu^{2}\nu}{3k\Delta\textrm{v}T_{rot}}\frac{N_{T}}{Q_{rot}}e^{-\frac{E_{u}}{T_{rot}}}$ (4) where $\Delta$v is the FWHM line width. Under LTE, the upper-level populations, $\hat{N_{u}}$, can be predicted according to the right-hand side of Equation (3) for a given set of total column density, NT, rotational temperature, Trot, and source filling factor, f. The expected $\hat{N_{u}}$ were evaluated for the parameter space of Trot = 10-500 K, NT = 1014-1017 cm-2, and f between 0.01 and 1.0. To compare the observed $N_{u}$ and the inferred $\hat{N_{u}}$, we calculate the $\chi^{2}$ as: $\chi^{2}=\sum(\frac{N_{u}-\hat{N_{u}}}{\delta~{}N_{u}})^{2}$ (5) where $\delta~{}N_{u}$ is the 1 $\sigma$ error of observed upper-state column density. Although the $\chi^{2}$ is a good representation of the goodness of fit, the parameter set with the lowest $\chi^{2}$ may not actually represent physical parameters very well due to the uncertainties of the observed data. In order to find a representative parameter set, we compute a weighted mean and standard deviation for all the parameters, with the weights being the inverse of the $\chi^{2}$. All the parameter sets where the inferred upper- level population $\hat{N_{u}}$ corresponds with the observed upper-level population $N_{u}$ within 3 $\sigma$ are used to compute the weighted means and standard deviations. The derived rotational temperature, total column density and filling factor of each component are list in the [3-5] columns of Table 3. The inferred optical depths of each line transition are listed in the last six columns of Table 3. The rotational temperatures of D, E, F, G are estimated to be 42$\pm$34, 92$\pm$74, 51$\pm$23 and 105$\pm$37 K, respectively. A temperature gradient in the southern core is also revealed as in the RTD method. The four components D, E, F, G has similar total column densities as high as 4$\times 10^{16}$ cm-2, about an order of magnitude higher than those obtained from RTD method, which are mainly due to the small source filling factor ($<$0.5). The optical depths of H2CS (100,10-90,9) at the four components are all much larger than one, while the other transitions are always optically thin except H2CS (102,9-92,8) line at G. ### 4.2 Core properties In the optically thin case, the total dust and gas masses of the three sub-mm cores can be obtained with the formula $M=S_{\nu}D^{2}/\kappa_{\nu}RB_{\nu}(T_{d})$ (Hildebrand , 1983), where $S_{\nu}$ is the flux at 860 $\micron$, D is the distance, R=0.01 is the mass ratio of dust to gas, and $\kappa_{\nu}$ is dust opacity per unit dust mass. $B_{\nu}(T_{d})$ is the Planck function at a dust temperature of Td. We assume that Td equals the rotational temperature of H2CS. For the northern core, since only CS (7-6) (upper energy Eu = 65.8 K) and HCN (4-3) (Eu = 42.5 K) exhibit strong emission lines, we assume Td to be 50 K. Together with the measurements at centimeter and millimeter wavelengths, Su et al. (2005) extrapolated the ionized gas emission at mm/submm wavelengths, and found that the 0.85 mm continuum associated with components D, E, and F are dominated by thermal dust emission. They have derived opacity index $\beta$ of components E and F to be 1.2, and 0.8, respectively. For the northern sub-mm core, ${\beta}=1.5$ is assumed. Using the above dust opacity indexes, we adopt $\kappa_{\nu}$=2.0, 1.8, and 1.5 cm2g-1 for the northern, middle and southern cores, respectively (Ossenkopf & Henning, 1994). At the distance of 5.7 kpc , we get the total dust and gas masses for these three cores, and list all the parameters in Table 1. The deduced masses for the northern, middle and southern cores are 13, 30, 165 M☉, respectively. The column density of H2 are 1.2$\times 10^{24}$ and 2.1$\times 10^{24}$ cm-2 for the middle and southern sub-mm cores, respectively. ### 4.3 Infall properties in the middle core In the middle core, both CS(7-6) and HCN(4-3) emission exhibits ”blue profile” feature, indicating infall motions of the gas envelope toward the central star (Keto, Ho,& Haschick, 1988; Zhou et al., 1993; Zhang, Ho, & Ohashi, 1998; Wu & Evans, 2003; Wu et al., 2005, 2007; Fuller, Williams, & Sridharan, 2005; Wyrowski, 2007; Sun, & Gao, 2008). The velocity difference (0.9 km s-1) between the absorption dip in CS (7-6) spectrum (3 km s-1) and the systemic velocity (2.1 km s-1) is taken as the infall velocity $V_{in}$. Since both HCN (4-3) and CS (7-6) emissions are not resolved towards the middle core, we simply take the dust core size as the radius of the infall region, which may underestimate the infall rate derived below. The kinematic mass infall rate can be calculated using dM/dt=$4{\pi}R_{in}^{2}nmV_{in}$. n=1.5$\times 10^{7}$cm-3 is the number density of this dust core. Taking Helium into account, the mean molecular mass m is 1.36 times of H2 molecule mass. The infall rate calculated is $4.3\times 10^{-3}~{}M_{\odot}\cdot$yr-1. For comparison, the $V_{in}$ from pure free-infall assumption is also derived with the formula $V_{in}^{2}=2GM/R_{in}$. The pure free-infall velocity is $V_{in}=3.6~{}$km s-1 and thus the ”gravitational” mass infall rate is $1.7\times 10^{-2}~{}M_{\sun}\cdot$yr-1, which is larger than the kinematic infall rate. ### 4.4 Outflow properties in the southern core #### 4.4.1 Shock chemistry in the outflow region of the southern core Observations have suggested that there are important differences in molecular abundances in different outflow regions (Bachiller et al., 1997; Choi et al., 2004; Jörgensen, Schöier, & van Dishoeck, 2004; Codella et al., 2005). Significant abundance enhancements are found in the shocked region for sulfur- bearing molecules (Bachiller et al., 1997; Jörgensen, Schöier, & van Dishoeck, 2004), and the abundance of HCN in outflow regions is related to atomic carbon abundance (Choi, 2002). However, previous studies of the chemical impact of outflows are confined to the well collimated outflows around Class 0 sources, while such studies especially high resolution studies on massive outflows are rare (Bachiller et al., 1997; Jörgensen, Schöier, & van Dishoeck, 2004; Arce et al., 2007). A red and bright IRAC source is found to be associated with the southern core. The magnitudes of the IRAC source at 3.6 $\micron$, 4.5 $\micron$ and 5.8 $\micron$ are $10.102\pm 0.093$, $8.361\pm 0.108$ and $7.778\pm 0.302$ mag, respectively. The [3.6-4.5] color is as large as 1.74, indicating shocked emission in the southern core (Takami et al., 2010). Maser emissions of NH3, H2O, OH, and CH3OH, as well as the strong thermal NH3 emissions also uncover the existence of the shocked gas (Hofner et al., 1994). Outflows can be revealed from shocked H2 emission probed by the strong and extended emission at the 4.5 $\micron$ band (Qiu et al., 2008; Takami et al., 2010). Thus the massive outflow in the southern core of G9.62 complex provides an ideal sample to study shock chemistry. The fractional abundance of a certain molecule is defined as $\chi=N_{T}/N_{H_{2}}$, where $N_{T}$ is the total column density of a specific molecule and $N_{H_{2}}$ is the H2 column density. Assuming that the gas is optically thin and the emission fills the beam, the beam-averaged total column density of a specific molecule can be obtained from: $N_{T}=2.04\times 10^{20}\frac{\int~{}I(Jy~{}beam^{-1})dv(km~{}s^{-1})Q_{rot}e^{E_{u}/T_{rot}}}{\theta_{a}\theta_{b}(arcsec^{2})g_{I}g_{K}\nu^{3}(GHz^{3})S\mu^{2}(debye^{2})}$ (6) Assuming that Trot of HC15N equals to that of H2CS and the gas is optically thin, $N_{T}$ of HC15N is calculated to be $3.0\times 10^{13}$ cm-2 at the core region. At the galactocentric distance of 3 kpc for G9.62+0.19 (Scoville et al.1987,Hofner et al.1994), the abundance ratio [14N]/[15N]$~{}\approx~{}350$ (Wilson and Rood. 1994). Thus the total column density of HCN at the core region should be $1.1\times 10^{16}$ cm-2. Therefore, the fractional abundance of HCN relative to H2 at the core region is $5.2\times 10^{-9}$. HCN appears to be greatly enhanced in the outflow regions of the L1157 (Bachiller et al., 1997), while has similar abundances in the outflow region and the ambient cloud of NGC 1333 CIRAS 2A (Jörgensen, Schöier, & van Dishoeck, 2004). Owing to the lack of a direct estimation of the H2 column density towards the outflow region, the fractional abundance of HCN in the outflow region is also assigned to $5.2\times 10^{-9}$ in calculating the outflow parameters. Since the HC15N emission traces outflowing gas at much lower velocity than HCN, perhaps HCN could be more enhanced in the high velocity component. With the possibility of higher opacity and the lack of direct H2 column density measurement, the derived fractional abundance perhaps is a lower limit anyway. Su et al. (2007) estimate an HCN abundance of $\sim 1-2\times 10^{-8}$ in the massive outflow lobes of IRAS 20126+4104, which is comparable to our estimation here. Since the blueshifted outflow gas traced by CS (7-6) and HCN (4-3) suffers self-absorption, the abundance ratios among SO ($8_{7}-7_{7}$), CS (7-6), and HCN (4-3) were inferred from the beam-averaged spectra taken from the redshifted outflow lobe. The abundance ratio as a function of flow velocity (the outflow velocity relative to the systemic velocity) of [CS/SO] is obtained assuming five different excitation temperatures in the left panel of Figure 15\. It can be seen that the abundance ratio of [CS/SO] increases with the excited temperature. At each excitation temperature, the abundance ratio of [CS/SO] has lower values at flow velocities less than 6 km s-1, and higher values when Vflow larger than 8 km s-1, whereas the abundance ratio seems to be constant at flow velocities between 6 km s-1 and 8 km s-1. There are two reasons for the lower abundance ratio when V${}_{flow}~{}<$ 6 km s-1: first, the flux missing of CS (7-6) due to the interferometer is more serious than SO ($8_{7}-7_{7}$); second, CS (7-6) may be more optically thick at lower flow velocities than SO ($8_{7}-7_{7}$). As shown in the P-V diagrams, the emission region of CS (7-6) is much larger than SO ($8_{7}-7_{7}$) at high velocities. The higher abundance ratio when V${}_{flow}~{}>$ 8 km s-1 is due to the smaller filling factor of SO ($8_{7}-7_{7}$) emission. We propose the mean observed value between 6 km s-1 and 8 km s-1 can represent the actual abundance ratio of [CS/SO]. Assuming a typical excitation temperature of Tex=30 K (Wu et al., 2004), the abundance ratio of [CS/SO] at the redshifted lobe is inferred as 0.7. Nilsson et al. (2000) find that the [SO/CS] abundance ratios are strongly enhanced in the Orion A and NGC 2071 outflows where the [SO/CS] ratios are estimated to be about 24 and 2.2, respectively. However, the [SO/CS] abundance ratio in the outflow of G9.62+0.19 is found to be 1.4, much lower than that found in Orion A outflow. As shown in the right panel of Figure 15, the abundance ratio of [CS/HCN] decreases linearly with the flow velocity. To avoid the missing flux difficulty, the abundance ratio is calculated at high flow velocities larger than 7 km s-1. The decreasing of the abundance ratio with velocity is because that the emission region traced by CS (7-6) is always smaller than HCN (4-3), leading to smaller filling factor for CS (7-6), which can be verified easily by comparing the channel maps between CS (7-6) in Figure 11 and HCN (4-3) in Figure 12 at high velocities. We fitted the observed data with a linear function, and adopted the value at flow velocity of 10 km s-1 as the actual abundance ratio of [CS/HCN] in the outflow region, which is [CS/HCN]=1.2. Since HCN fractional abundance is $5.2\times 10^{-9}$, the fractional abundances of CS and SO are deduced to be $6.2\times 10^{-9}$ and $8.9\times 10^{-9}$, respectively. #### 4.4.2 Properties of the bipolar-outflow traced by SO ($8_{7}-7_{7}$) emission The SO ($8_{7}-7_{7}$) emission in the southern core shows line wings, suggesting outflow motions. From the integrated intensity map in Figure 10(c), we find the outflow lobes revealed by SO ($8_{7}-7_{7}$) emission peak at different position with different position angle compared with previously reported H2S ($2_{2,0}-2_{1,1}$) (Gibb, Wyrowski, & Mundy , 2004) and HCO+ (1-0) data (Hofner, Wiesemeyer, & Henning , 2001). But in the same sense, the blue- and red-lobes revealed by SO overlap to a large extent as well as HCO+ (1-0) and H2S ($2_{2,0}-2_{1,1}$) data, consistent with the argument of the outflow being viewed pole-on (Hofner, Wiesemeyer, & Henning , 2001). The total mass of each outflow lobe is given by: $M_{flow}=1.04~{}\times~{}10^{-4}D^{2}\frac{Q_{rot}e^{E_{u}/T_{rot}}}{\chi\nu^{3}S\mu^{2}}\int\frac{\tau}{1-e^{-\tau}}S_{\nu}dv$ (7) where Mflow, D, Sν, $\chi$, and $\tau$ are the outflow gas mass in M☉, source distance in kpc, line flux density in Jy, relative abundance to H2, and optical depth. The other parameters have the same units as in equation (1). The fractional abundance of SO is taken as $8.9\times 10^{-9}$ (see Sec.4.4.1). Assuming an excitation temperature of 30 K and the outflowing gas is optically thin, the inferred outflow masses are 13 M☉ for each of red and blueshifted lobes. Thus, the momentum can be calculated by $P=\sum$M(v)dv, and the energy by $E=\sum{\frac{1}{2}}$M(v)v2dv, where $v$ is the flow velocity. The derived parameters are listed in Table 4. The momentum and energy of the red lobe are 82 M${}_{\sun}\cdot$km s-1 and $5.4\times 10^{45}$ erg. For the blue lobe, the momentum and energy are calculated to be 86 M${}_{\sun}\cdot$km s-1 and $5.8\times 10^{45}$ erg. The dynamical timescale tdyn is estimated as R/Vchar, where R ($\sim$ 0.06 pc) is adopted as the mean size of the outflow lobes assuming a collimation factor of unity, and Vchar ($\sim$ 5.5 km s-1) is assumed as the mass weighted mean velocity. Thus, the dynamic timescale is estimated to be $1\times 10^{4}$ year, which may be underestimated due to the uncertainty of the outflow scale. The mechanical luminosity L, and the mass- loss rate $\dot{M}$ are calculated as L=E/t, $\dot{M}=P/(tV_{w})$, where the wind velocity Vw is assumed to be 500 km s-1 (Lamers et al., 1995). The mechanical luminosity L and the total mass-loss rate are estimated to be 9.3 L☉ and $3.6\times 10^{-5}$ M☉$\cdot$yr-1, respectively. #### 4.4.3 Very high-velocity gas detected in CS (7-6) emission The CS (7-6) emission at the southern core shows ”red-profile” with wide wings. We take $6.2\times 10^{-9}$ as the fractional abundance of CS relative to H2 along the outflow lobes. Assuming Tex = 30 K, we derive the parameters for the CS outflow (Table 4) with the same method used for SO ($8_{7}-7_{7}$). The outflow masses at very high velocities (v${}_{flow}~{}>~{}$10 km s-1) are 3.7 M☉ and 5.5 M☉ for the blueshifted and redshifted lobes, respectively. The momentum and energy of the blueshifted lobe at very high velocities are calculated to be 47 M${}_{\sun}\cdot$km s-1 and $6.0\times 10^{45}$ erg. For the redshifted lobe, the momentum and energy at extremely high velocities are calculated to be 68 M${}_{\sun}\cdot$km s-1 and $8.7\times 10^{45}$ erg, which are similar to the blueshifted lobe. #### 4.4.4 Very high-velocity gas detected in HCN (4-3) emission As discussed before, HCN (4-3) has a velocity extent of at least 60 km s-1, which traces extremely high-velocity (EHV) gas. Adopting an excited temperature of 30 K, and an HCN-to-H2 abundance ratio of $5.2\times 10^{-9}$, the parameters of the outflow are calculated and listed in Table 4. The outflow mass at very high velocities (v${}_{flow}~{}>~{}$10 km s-1) are 5.2 M☉ and 17.6 M☉ for the blueshifted and redshifted lobes, respectively. The momentum and energy of the blueshifted lobe at very high velocities are 85 M${}_{\sun}\cdot$km s-1 and $1.4\times 10^{46}$ erg. For the redshifted lobe, the momentum and energy at very high velocities are 294 M${}_{\sun}\cdot$km s-1 and $5.5\times 10^{46}$ erg, which are larger than the blueshifted lobe. #### 4.4.5 Mass-Velocity diagrams A broken power law, $dM(v)/dv\propto v^{-\gamma}$ usually exhibits in molecular outflows near young stellar objects (Chandler et al., 1996; Lada, & Fich, 1996; Ridge, & Moore, 2001; Su, Zhang, & Lim, 2004; Qiu et al., 2007, 2009). The slope, $\gamma$, typically ranging from 1 to 3 at low outflow velocities, and often steepens at velocities larger than 10 km s-1 — with $\gamma$ as large as 10 in some cases (Arce et al., 2007). Assuming optically thin, the mass-velocity diagrams of the outflow at the southern core of G9.62+0.19 complex are shown in Figure 16. SO ($8_{7}-7_{7}$), CS (7-6), HCN (4-3) results were all used in the mass spectra. We calculate the outflow mass traced by CS (7-6) and HCN (4-3) from Vflow of 10 km s-1 to avoid the absorption of the spectra. Instead of broken power law appearance, the mass- velocity diagram of blueshifted lobe can be well fitted by a single power law with a power indexes of $2.28\pm 0.23$. The mass-velocity diagram of redshifted lobe can be well fitted by a single power law with a power indexes of $1.70\pm 0.17$ even though the mass drops more rapidly after 25 km s-1. As marked by the dashed ellipse in the right panel, the outflow mass revealed by CS (7-6) is much lower than that revealed by HCN (4-3) at very high velocities. Despite the CS data, the mass-velocity diagram of redshifted lobe at velocities smaller than 25 km s-1 can be fitted by a single power law with a much smaller power indexes of $1.08\pm 0.09$. However, no significant slope changes are found in both the red- and blue-shifted lobes of the outflow at the southern core, which are very different from those previous works. ### 4.5 Different evolutionary stages of the three dust cores The northern core has the smallest diameter and mass among the three cores. It seems likely to be a point source after deconvolution. It is located south of the nominal radio UC Hii region G9.62+0.19 C. In this region, eight near-IR sources are detected in a diffuse near-IR nebulosity at the west of the radio emission peak (Persi et al., 2003). The reddest one c7 (18h06m14.34s,-20$\arcdeg$31$\arcmin$25.0$\arcsec$) is located within $1\arcsec$ of the radio peak, while the faintest one c8 (18h06m14.42s,-20$\arcdeg$31$\arcmin$27.4$\arcsec$) seems to be associated with the sub-mm core detected in SMA observation. Source c8 is too faint to be detected even at H band and also shows no emission at 12.5 $\micron$. In contrast to the bright, rich molecular spectrum forest in the middle and southern sub-mm cores, the northern sub-mm core lacks strong molecular emissions. There is also no other early star forming signature such as masers associated with it. Since it is with near-IR emission and at the edge of the UC Hii region G9.62+0.19 C, the northern core may be just a remnant core in the envelope of UC Hii region G9.62+0.19 C, which needs further observations. The middle core is associated with the hyper-compact Hii region G9.62+0.19 E (Garay et al., 1993; Kurtz, & Franco, 2002). OH, H2O, and NH3 (5,5) masers have been detected near the radio emission peak (Forster & Caswell, 1989; Hofner et al., 1994; Hofner, & Churchwell, 1996a). Periodic class II methanol masers are also found in G9.62+0.19 E (van der Walt, Goedhart, & Gaylard, 2009; Goedhart, Gaylard, & van der Walt, 2005; Norris et al., 1993). Methanol masers are believed to be a good tracer of young massive star forming regions at stages earlier than relatively evolved UC Hii regions (Longmore et al., 2007). No infrared source coincides with G9.62+0.19 E (Persi et al., 2003). Hot molecular CH3CN lines are detected in this region, and a kinematic temperature of Tk = 108 K was obtained from CH3CN emission with LVG model (Hofner et al., 1996b), which is coincident with the rotational temperature (Trot = 92 K) obtained from H2CS emission. A spectra forest including hot molecular lines, such as CH3OH, is detected towards G9.62+0.20 E, suggesting this core is in a hot phase. Infall motions are traced by CS (7-6) and HCN (4-3) lines, indicating active star forming in this region. All above suggest that G9.62+0.20 E is forming a massive young star. The 860 $\micron$ dust emission of the southern core peaks at G9.62+0.19 F, and extends from north to south. A hump structure is found to the southeast of the emission peak, indicating another possible sub-mm core. The previously recognized mm/cm cores (G9.62+0.19 D, G) are at the edges of the southern core. G9.62+0.19 G is a weak radio source (Testi et al., 2000), while G9.62+0.19 D is consistent with an isothermal UC Hii region excited by a B0.5 star (Hofner et al., 1996b). Weaker radio emission was found at core F. H2O and OH masers are found across the whole sub-mm core from north to south (Forster & Caswell, 1989; Hofner, & Churchwell, 1996a). A near-IR source with large NIR excess is found to be associated with G9.62+0.19 F (Testi et al., 1998; Persi et al., 2003). With higher resolution observations Linz et al. (2005) found four near-IR objects (F1-F4) in this core. F4 is with little emission at K band but becomes redder at longer wavelengths, which seems to correspond to the bright IRAC source with large excess at 4.5 $\micron$. This object is the dominating and closest associated source of core F. Core F is also confirmed to be the driving source of an active outflow. All of above imply that G9.62+0.19 F is a very young massive star forming region. ### 4.6 Blue excess in high-mass star forming regions Wu et al. (2007) found that UC Hii regions show a higher blue excess than UC Hii precursors with the IRAM 30 m telescope. Wyrowski et al. (2006) also detected large blue excess in UC Hii regions. ”Blue profile” was detected with CS (7-6) and HCN (4-3) lines in UC Hii region G9.62+0.19 E, while ”red profile” in hot molecular core G9.62+0.19 F, which coincides with their argument. The detection of infall signature in G9.62+0.19 E also coincides the interpretation that material is still accreted during the UC Hii phase (Wu et al., 2007; Keto, 2002). Around younger cores, the outflow is more active and cold than UC Hii regions, which leads to more ”red profile”. While in UC Hii regions, the outflows become weak. The surrounding gas of UC Hii regions is thermalized and the temperature gradient towards the central star is more likely to cause ”blue profile”, which results in the higher blue excess than UC Hii precursors. ## 5 Summary We have observed the G9.62+0.19 complex with the Submillimeter Array (SMA) both in the 860 $\micron$ continuum and molecular lines emission. The main results of this study are as follows: 1\. Dust continuum at 860 $\micron$ reveals three sub-mm cores in G9.62+0.19 star forming complex. With H2CS as the rotational temperature prober, the temperatures of E and F are estimated to be 92$\pm$74 and 51$\pm$23 K, respectively. The mass calculated are 13, 30, and 165 M☉ for the northern, middle and southern core. 2\. In the middle core, HCN (4-3) and CS (7-6) spectra exhibit infall signature. The infall rate calculated is $4.3\times 10^{-3}$ M${}_{\sun}\cdot$yr-1. The detection of infall signature in G9.62+0.19 E coincides the interpretation that material is still accreted after the onset of the UC Hii phase (Wu et al., 2007). 3\. In the southern core, high-velocity gas is detected in SO ($8_{8}-7_{7}$), CS (7-6) and HCN (4-3) lines. A bipolar-outflow with a total mass about 26 M☉ and a mass-loss rate of $3.6\times 10^{-5}$ M${}_{\sun}\cdot$yr-1 is revealed in SO ($8_{8}-7_{7}$) line wing emission. G9.62+0.19 F is confirmed to be the driving source of the outflows in the southern sub-mm core. The abundance ratios of [CS/SO] and [CS/HCN] in the outflow region are found to be 0.7 and 1.2, respectively. The abundance ratio [CS/HCN] decreases with the flow velocity, indicating smaller outflow regions revealed by CS (7-6) than that revealed by HCN (4-3). The mass-velocity diagrams of the blueshifted and redshifted outflow lobes can be well fitted by a single power law. The power indexes for the blueshifted and redshifted lobes are $2.28\pm 0.23$ and $1.70\pm 0.17$. No significant slope changes are found in the mass-velocity diagrams. 4\. The evolutionary sequence of the cm/mm cores in this region are also analyzed. The northern core may be just a remnant core in the envelope of UC Hii region G9.62+0.19 C, which needs further observations. The middle core (G9.62+0.19 E) is in a hyper-compact Hii region. Core G9.62+0.19 F is confirmed to be a hot molecular core. 5\. 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(1993) Zhou, S., Evans, N. J. II., Koempe, C., & Walmsley, C. M., 1993, ApJ, 404, 232 * Zinnecker & Yorke (2007) Zinnecker, H., & Yorke, H. W., 2007, ARA&A, 45, 481 Figure 1: The 860 $\micron$ continuum emission image. The contour levels are from 0.03 Jy beam-1 (3$\sigma$) in steps of 0.06 Jy beam-1 (6$\sigma$). The known cm and mm continuum components (Testi et al., 2000) of B, C, D, E, F, G, H, and I are marked by plus signs. Water masers (Hofner, & Churchwell, 1996a) are marked by open squares and methanol masers (Norris et al., 1993) by triangles. The near-IR sources (Persi et al., 2003; Testi et al., 1998; Linz et al., 2005) are marked by filled circles. IRAC sources are marked with asterisks. Figure 2: The full LSB and USB spectra in the UV domain over the shortest baseline. The strongest lines are identified and labeled on the plots. Figure 3: Integrated intensity maps of four transitions of H2CS at the middle (upper panels) and southern cores (lower panels). The known cm and mm continuum components are marked by plus signs as in the continuum map. The contour levels in all the panels are from 3$\sigma$ in steps of 3$\sigma$. The rms levels are 0.3, 0.3, 0.3 and 0.2 Jy beam${}^{-1}\cdot$km s-1 for H2CS (100,10-90,9) in panels (a) and (e), H2CS (102,8-92,7) in panels (b) and (f), H2CS (103,7-93,6) in panels (c) and (g), and H2CS (105-95) in panels (d) and (h), respectively. Figure 4: The P-V diagram (left) and First moment map (right) of H2CS (102,8-92,7) emission at the middle core. (a) The contours of the P-V diagram are from 0.6 to 1.4 in steps of 0.2 Jy beam-1 (1$\sigma$). (b) contour plot of H2CS (102,8-92,7) integrated intensity image overlayed on the first moment map. The contours are from 0.9 (3$\sigma$) in steps of 0.9 Jy beam${}^{-1}\cdot$km s-1. The First moment map is constructed from the data after imposing a cutoff of 3$\sigma$. Figure 5: Spectra and integrated intensity maps of HC15N (4-3) (upper panels) and SO ($8_{7}-7_{7}$) (lower panels) at the middle core. The systemic velocity is marked with the thick vertical dashed lines at the spectra panels. The known cm and mm continuum components of C, and E are marked by plus signs at the integrated maps as the continuum map. (a) the beam-averaged spectrum of HC15N (4-3) at E, (b) the integrated intensity map of HC15N (4-3). The contour levels are -1.2 (6$\sigma$), 1.2, 2.4, 4.2, 6.6, 9.6 Jy beam${}^{-1}\cdot$km s-1, (c) the beam-averaged spectrum of SO ($8_{7}-7_{7}$) at E. (d) the integrated intensity map of SO ($8_{7}-7_{7}$). The contour levels are -1.2 (6$\sigma$), 1.2, 2.4, 4.2, 6.6, 9.6, 13.2, 17.4, 22.2 Jy beam${}^{-1}\cdot$km s-1. Figure 6: Beam-averaged spectra and Position-Velocity (P-V) diagrams of HCN (4-3) (upper panels) and CS (7-6) (lower panels) at the middle core. The P-V diagrams are cut along a position angle of 0$\arcdeg$. (a) the beam-averaged spectrum of HCN (4-3) at E, (b) the P-V diagram of HCN (4-3). The contour levels are -1.5 (5$\sigma$), -0.9, 0.9, 1.5, 2.1, 2.7, 3,3, 3.9, 4.5 Jy beam-1, (c) the beam-averaged spectrum of CS (7-6) at E. (d) the P-V diagram of CS (7-6). The contour levels are -1.5 (5$\sigma$), -0.9, 0.9, 1.5, 2.1, 2.7, 3,3, 3.9, 4.5 Jy beam-1. Figure 7: Integrated intensity maps of HCN (4-3) (left) and CS (7-6) (right) at the middle core. The contour levels in both maps are from 1.5 (5$\sigma$) in steps of 3 Jy beam${}^{-1}\cdot$km s-1. HCN (4-3) is integrated from -3 to 7 km s-1, while CS (7-6) from -1 to 6 km s-1 Figure 8: Averaged spectra of SO ($8_{7}-7_{7}$) (upper-left), HC15N (4-3) (lower-left), HCN (4-3) (upper-right) and CS (7-6) (lower-right) at the southern core. The spectra of SO ($8_{7}-7_{7}$) and HC15N (4-3) are averaged over a region of 4$\arcsec$, while HCN (4-3) and CS (7-6) are averaged over a region of 6$\arcsec$. HCN $\nu 2=1$ (4-3) emission is marked by the arrow in the upper-right panel, which can be clearly distinguished from the red wing of HCN (4-3). Figure 9: The integrated intensity maps of HC15N (4-3) (left panel) and SO ($8_{7}-7_{7}$) (right panel) at the southern core. To avoid the influence of outflow motions, both the maps are integrated from 2 km s-1 to 8 km s-1. The contour levels are (a) -1.2 (6$\sigma$), 1.2, 2.4, 4.2, 6.6, 9.6, 13.2, 17.4 Jy beam${}^{-1}\cdot$km s-1 for HC15N (4-3), (b) -1.2 (6$\sigma$), 1.2, 2.4, 4.2, 6.6, 9.6, 13.2, 17.4, 22.2, 27.6, 33.6 Jy beam${}^{-1}\cdot$km s-1 for SO ($8_{7}-7_{7}$) Figure 10: P-V diagrams and integrated intensity maps of SO ($8_{7}-7_{7}$) (upper panels), and HC15N (4-3) (lower panels) at the southern core. The P-V diagrams are cut along N-S direction. The vertical solid line in P-V diagrams labels the systemic velocity. The dashed and solid contours in the right panels show the red- and blue-shifted emission, respectively. The integral velocity intervals are marked by thick dashed lines in the P-V diagrams. For both SO ($8_{7}-7_{7}$) and HC15N (4-3), the blue- shifted emission is integrated from -4 km s-1 to 0 km s-1, while the red- shifted emission from 10 km s-1 to 14 km s-1 in the integrated intensity maps. (a) P-V diagram of SO ($8_{7}-7_{7}$). The contours are from 0.6 (3$\sigma$) in steps of 0.6 Jy beam-1. (b) P-V diagram of HC15N (4-3). The contours are from 0.6 (3$\sigma$) in steps of 0.4 Jy beam-1. (c) Integrated intensity maps of SO ($8_{7}-7_{7}$) at line wings. The contours are from 1 (5$\sigma$) in steps of 1 Jy beam${}^{-1}\cdot$km s-1 for both red- and blue-shifted emission. (d) Integrated intensity maps of HC15N (4-3) at red wing. The contours are 0.6 (3$\sigma$), 1.2, 2, 3 Jy beam${}^{-1}\cdot$km s-1. Figure 11: CS (7-6) channel maps at the southern core, which is smoothed to a velocity resolution of 3 km s-1. The contours are -0.6 (3$\sigma$), 0.6, 1.2, 2.4, 4.8, 7.2, 9.6 Jy beam-1. Figure 12: HCN (4-3) channel maps at the southern core, which is smoothed to a velocity resolution of 4 km s-1. The contours are -0.6 (3$\sigma$), 0.6, 1.2, 2.4, 3.6, 4.8, 7.2 Jy beam-1. Figure 13: P-V diagrams and integrated intensity maps of HCN (4-3) (upper panels), and CS (7-6) (lower panels) at the southern core. The P-V diagrams are cut along N-S direction. The vertical solid line in P-V diagrams labels the systemic velocity. The dashed and solid contours in the right panels show the red- and blue-shifted emission, respectively. The blue- and red-shifted emission in the integrated maps are integrated from -12 km s-1 to -5 km s-1 and 15 km s-1 to 22 km s-1, respectively in (b) and (e) panels. (a) P-V diagram of HCN (4-3). The contours are from 0.9 (3$\sigma$) in steps of 1.2 Jy beam-1. (b) Integrated intensity maps of HCN (4-3) at line wings. The contours are 1.5 (5$\sigma$), 4.5, 7.5, 10.5 Jy beam${}^{-1}\cdot$km s-1. (c) The integrated intensity maps of HCN (4-3) at extremely high velocities. The blue- and red-shifted emission in the integrated maps are integrated from -20 km s-1 to -13 km s-1 and 23 km s-1 to 39 km s-1, respectively. The contours are from 1.5 (5$\sigma$) in steps of 3 Jy beam${}^{-1}\cdot$km s-1 for both blue- and red-shifted emission. (d) P-V diagram of CS (7-6). The contours are from 0.9 (3$\sigma$) in steps of 1.2 Jy beam-1. (e) Integrated intensity maps of CS (7-6) at line wings. The contours are 1.5 (5$\sigma$), 4.5, 7.5, 10.5 Jy beam${}^{-1}\cdot$km s-1 Figure 14: Population diagrams of H2CS towards four cm/mm cores. The names of the cores are labeled on the upper-right corner of each panel. Open circles in blue represent the observed data. The vertical bars present 3$\sigma$ errors of ln(Nu/gu) due to the uncertainties of integrated intensities. The solid line shows the linear least-squares fitting using the Rotational Temperature Diagram method. Crosses in red mark the weighted mean results from Population Diagram analysis. The inferred parameters from the Population Diagram analysis are presented on the upper-right corners of each panel. Figure 15: Abundance ratios of [CS/SO] (left) and [CS/HCN] (right) versus flow velocity along the redshifted lobe. We range the excitation temperature from 10 K to 50 K to derive the abundance ratios of [CS/SO]. The excitation temperature in calculation of abundance ratios of [CS/HCN] is assumed to be 30 K. The solid line in the right panel is the linear least-squares fitting, and the fitting results are presented in the upper-right corner. Figure 16: Mass-Velocity relationships for the outflow lobes. Left : blueshifted lobe; right: redshifted lobe. The solid lines in both panels show the power law fit towards all the data. The dashed line in the right panel shows the power law fit towards the HCN and SO data up to Vflow = 25 km s-1. The fitting results are presented in the lower-left corners. Table 1: Parameters of 860 $\micron$ continuum emission | R.A. | Decl. | Deconvolution sizes | Ipeak | Sν | TdaaThe dust temperature is assumed to be the same as the rotational temperature of H2CS transitions | $\beta$aaThe dust temperature is assumed to be the same as the rotational temperature of H2CS transitions | Mass | N${}_{H_{2}}$ ---|---|---|---|---|---|---|---|---|--- Name | (J2000) | (J2000) | ($\arcsec~{}\times~{}\arcsec$) | (Jy beam-1) | (Jy) | (K) | | (M☉) | ($10^{24}$ cm-2) Northern core | 18:06:14.447 | -20:31:28.253 | Point source | 0.20$\pm$0.02 | 0.26 | 50 | 1.5 | 13 | Middle core | 18:06:14.668 | -20:31:31.830 | $1.48\arcsec\times 1.29\arcsec$ (P.A.=$-37.8\arcdeg$) | 0.76$\pm$0.04 | 1.07 | 92 | 1.2 | 30 | 1.2 Southern core | 18:06:14.889 | -20:31:40.149 | $4.71\arcsec\times 1.26\arcsec$ (P.A.=$-20.2\arcdeg$) | 0.95$\pm$0.12 | 2.52 | 51 | 0.8 | 165 | 2.1 Table 2: Observed parameters of the lines Molecule | Transition | Frequency | Eu | rms | VlsrbbThe opacity index $\beta$ is obtained from Su et al. (2005) | IntensitybbThe Vlsr, Intensity and FWHM of each transition are derived from single gaussian fit towards the beam-averaged spectra. | FWHMbbThe Vlsr, Intensity and FWHM of each transition are derived from single gaussian fit towards the beam-averaged spectra. ---|---|---|---|---|---|---|--- | | (GHz) | (K) | (Jy beam-1) | (km s-1) | (Jy beam-1) | (km s-1) | | | | | D | E | F | G | D | E | F | G | D | E | F | G H2CS | 100,10-90,9 | 342.946 | 90.6 | 0.3 | 5.5$\pm$0.2 | 2.7$\pm$0.2 | 5.9$\pm$0.2 | 6.0$\pm$0.5 | 1.9$\pm$0.3 | 1.4$\pm$0.2 | 1.7$\pm$0.1 | 0.9$\pm$0.2 | 2.7$\pm$0.4 | 3.1$\pm$0.5 | 4.3$\pm$0.6 | 3.8$\pm$1.1 | 102,9-92,8ccfootnotemark: | 343.322 | 143.3 | 0.3 | 4.0$\pm$0.5 | 2.2$\pm$1.1 | | 4.9$\pm$0.3 | 0.9$\pm$0.2 | 1.4$\pm$0.3 | | 1.1$\pm$0.3 | 4.0$\pm$0.6 | 4.4$\pm$1.6 | | 1.9$\pm$0.6 | 102,8-92,7 | 343.813 | 143.3 | 0.3 | 6.0$\pm$0.3 | 2.5$\pm$0.4 | 4.9$\pm$0.5 | | 1.3$\pm$0.2 | 0.9$\pm$0.1 | 0.8$\pm$0.2 | | 3.0$\pm$0.7 | 5.5$\pm$0.9 | 3.9$\pm$1.3 | | 103,8-93,7ddBlended with H2CS (103,7-93,6) | 343.410 | 209.1 | 0.3 | 6.2$\pm$0.8 | 2.4$\pm$0.7 | 5.0$\pm$0.3 | | 1.1$\pm$0.7 | 1.1$\pm$0.1 | 2.1$\pm$0.3 | | 3.3$\pm$0.7 | 4.1$\pm$0.5 | 2.7$\pm$0.6 | | 103,7-93,6eeBlended with H2CS (103,8-93,7) | 343.414 | 209.1 | 0.3 | 5.8$\pm$2.8 | 2.6$\pm$0.2 | 5.5$\pm$0.3 | 5.8$\pm$0.2 | 0.7$\pm$0.2 | 2.0$\pm$0.2 | 2.4$\pm$0.3 | 1.5$\pm$0.3 | 5.9$\pm$0.8 | 4.0$\pm$0.5 | 2.9$\pm$0.6 | 2.3$\pm$0.5 | 105,6/5-95,5/4ffThe two transitions of H2CS (105,6-95,6) and (105,6-95,6) have the same frequency, line strength and permanent dipole moment. Therefore they has same contributions to the observed line profile. | 343.203 | 419.2 | 0.2 | | 1.8$\pm$0.4 | 5.9$\pm$0.3 | 5.9$\pm$0.2 | | 0.6$\pm$0.2 | 0.7$\pm$0.2 | 0.9$\pm$0.2 | | 3.2$\pm$1.0 | 2.4$\pm$0.8 | 1.5$\pm$0.4 SO | 88-77 | 344.311 | 87.5 | 0.2 | 4.9$\pm$0.1 | 2.2$\pm$0.1 | 5.0$\pm$0.1 | 4.4$\pm$0.1 | 2.9$\pm$0.1 | 4.0$\pm$0.1 | 5.5$\pm$0.1 | 3.5$\pm$0.1 | 3.9$\pm$0.2 | 5.3$\pm$0.2 | 9.0$\pm$0.2 | 8.2$\pm$0.3 HC15N $\nu$=0 | 4-3 | 344.200 | 41.3 | 0.2 | 6.1$\pm$0.4 | 2.6$\pm$0.1 | 5.2$\pm$0.1 | 4.8$\pm$0.3 | 0.6$\pm$0.1 | 2.1$\pm$0.1 | 2.8$\pm$0.1 | 1.1$\pm$0.1 | 3.6$\pm$1.0 | 4.8$\pm$0.3 | 8.0$\pm$0.3 | 7.5$\pm$0.8 CS | 7-6 | 342.883 | 65.8 | 0.3 | | | | | | | | | | | | HCN $\nu$=0 | 4-3 | 354.505 | 42.5 | 0.3 | | | | | | | | | | | | aafootnotetext: Not all the detected lines are listed in this table. The others will be presented in another paper. bbfootnotetext: Blended with H${}_{2}^{13}$CO (51,4-41,4) at 343.325713 GHz. Table 3: The physical parameters of H2CS transitions obtained with Rotational Temperature Diagram (RTD) method and Population Diagram (PD) analysis Core | RTD | | PD ---|---|---|--- | Trot (K) | Ntot (1015 cm-2) | | Trot (K) | Ntot (1016 cm-2) | f | $\tau$ | | | | | | | (100,10-90,9) | (102,9-92,8) | (102,8-92,7) | (103,8-93,7) | (103,7-93,6) | (105,6/5-95,6/4) D | 43$\pm$9 | 3.8$\pm$2.9 | | 42$\pm$34 | 4.2$\pm$2.9 | 0.46$\pm$0.24 | 6.4$\pm$4.4 | 0.7$\pm$0.5 | 0.9$\pm$0.6 | 0.4$\pm$0.5 | 0.2$\pm$0.3 | E | 83$\pm$21 | 2.5$\pm$1.6 | | 92$\pm$74 | 3.6$\pm$3.0 | 0.26$\pm$0.23 | 4.1$\pm$4.3 | 0.7$\pm$0.7 | 0.6$\pm$0.6 | 0.7$\pm$0.8 | 0.7$\pm$0.8 | 0.1$\pm$0.1 F | 83$\pm$7 | 2.6$\pm$0.6 | | 51$\pm$23 | 4.0$\pm$2.9 | 0.34$\pm$0.23 | 3.9$\pm$3.1 | | 1.1$\pm$0.9 | 1.1$\pm$1.2 | 1.1$\pm$1.2 | 0.0$\pm$0.1 G | 91$\pm$17 | 1.3$\pm$0.7 | | 105$\pm$37 | 3.7$\pm$3.1 | 0.12$\pm$0.18 | 2.5$\pm$2.7 | 2.4$\pm$2.3 | | | 2.4$\pm$2.1 | 0.3$\pm$0.3 aafootnotetext: The rotational temperature and total column density of H2CS transitions derived from RTD analysis are presented in the second and third columns, while those derived from PD analysis are shown in the forth and fifth columns. The sixth column gives the filling factor of each source inferred from PD analysis. The last six columns exhibit the optical depth of each transition using PD analysis. Table 4: Outflow parameters of the southern core Molecule | Velocity interval | M | P | E ---|---|---|---|--- | (km s-1) | (M☉) | (M${}_{\sun}~{}\cdot~{}km~{}s^{-1}$) | ($10^{45}$erg) Component | Blue | Red | Blue | Red | Blue | red | Blue | Red SO | [-4,0] | [10,14] | 13 | 13 | 86 | 82 | 5.8 | 5.4 CS | [-12,-5] | [15,22] | 3.7 | 5.5 | 47 | 68 | 6.0 | 8.7 HCN | [-20,-5] | [15,39] | 5.4 | 17.6 | 85 | 294 | 14.1 | 54.6
arxiv-papers
2011-01-08T09:31:31
2024-09-04T02:49:16.244323
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Tie Liu, Yuefang Wu, Sheng-Yuan Liu, Sheng-Li Qin, Yu-Nung Su, Huei-Ru\n Chen and Zhiyuan Ren", "submitter": "Tie Liu", "url": "https://arxiv.org/abs/1101.1580" }
1101.1682
# Detecting gross alignment errors in the Spoken British National Corpus ###### Abstract The paper presents methods for evaluating the accuracy of alignments between transcriptions and audio recordings. The methods have been applied to the Spoken British National Corpus, which is an extensive and varied corpus of natural unscripted speech. Early results show good agreement with human ratings of alignment accuracy. The methods also provide an indication of the location of likely alignment problems; this should allow efficient manual examination of large corpora. Automatic checking of such alignments is crucial when analysing any very large corpus, since even the best current speech alignment systems will occasionally make serious errors. The methods described here use a hybrid approach based on statistics of the speech signal itself, statistics of the labels being evaluated, and statistics linking the two. Index Terms: ASPA, HMM, phonetic, transcription, label, segment, alignment, accuracy, quality assessment, error detection ## 1 Introduction In linguistic and phonetic research, increasing emphasis is being placed on the analysis of very large corpora, to give more general, stronger conclusions. However, the analysis of such databases often requires aligned textual and audio data, and performing such alignments manually is extremely time-consuming and hence expensive. Automatic alignment can be performed by transcribing the words spoken in the corpus and then aligning them to the speech via standard HMM techniques (e.g. [1]). These Automatic Speech-to- Phoneme Alignment (ASPA) systems can produce accurate estimates of word and phoneme locations, but often fail in cases such as: 1. 1. The speech has been recorded in an environment with non-stationary background noise, competing un-transcribed speech, distortion, and/or reverberation. 2. 2. The phonemic transcription of the speech is not accurate. The phonemic transcription is usually obtained by a dictionary look-up process. Citation forms are often used, and are often unrealistic, especially when dealing with spontaneous speech. In natural, unscripted speech, people will sometimes talk simultaneously so there may be no sequence of words or phonemes that can correctly represent the audio. Even if the words can be identified and transcribed individually, they cannot be organised into a simple sequence, and so require more complex techniques (e.g. [2]), which are impractical for large corpora. 3. 3. The word-level transcription is inaccurate and/or inconsistent in its handling of nonspeech sounds, backchannels, mumbles, and speech-like noises (e.g. dog barks). 4. 4. The speech is only available in long continuous recordings: the accumulation of HMM probabilities over extended periods of time can introduces numerical errors which distort the alignment process [3]. As a result, automatic alignment of large quantities of spontaneous unconstrained speech is invariably error-prone. When labelling ``speech in the wild'' such as the Spoken British National Corpus (BNC) [4], the above four conditions frequently occur, and lead to failures of the alignment system. Identifying where such an alignment succeeds and where it fails allows bad regions to be avoided or realigned. Previous work on this topic is sparse. Traditionally, aligner accuracy is assessed by comparison with manually estimated labels in one way or another [5]. However, the task here is fundamentally different in that we operate under conditions where any aligner may fail, regardless of whether it is generally accurate or not. [6] have addressed our problem in a different context (clean speech that is intended for use in TTS systems) with some success. They flagged 24% of the segments as suspicious and detected 43% of the total errors, which would have led to a modest reduction in the effort required to verify a corpus. [7] looked at the related problem of finding errors in the lexicon used in the alignment process. Some text-to-speech system builders may also have used similar ideas (e.g. [8]) but details are lacking. Some preliminary work was done by [9], using ideas related to our improbable and unexpected features. The work described here is part of a project [10] to label speech from the Spoken BNC, originally recorded on analogue cassette tapes between 1991 and 1994. The data consists of recently-digitised recordings with an associated word-level transcription of the audio data. The recordings are mostly of unscripted, spontaneous speech, and include a diverse range of recording conditions, accents, and microphone positions relative to the speakers. Each track of the original cassette recordings has been digitised to a single file, so the data to be aligned is generally just over either 45 or 60 minutes long. Background noise varies widely, both in terms of amplitude and character (competing speech, mechanical noise, music and speech from television, radio or other sources, microphone-handling noise, etc.). Our earlier work [11, 12] investigated alignment errors by comparing the alignments produced by a large number of ASPA systems. However, in that work we were attempting to assess the general ability of aligners to identify particular phone-transitions, and the general quality of alignments produced by a specific system, respectively. ## 2 Methods We used human evaluations of overall alignment quality for each of a set of recordings, to construct algorithms that should be able to identify suspicious regions, then to compare these to the human evaluations. ### 2.1 Alignment Procedures To evaluate our methods, we used the Penn Phonetics Lab Forced Aligner, P2FA[13] to analyse 46 recordings taken from the BNC, each consisting of a full audio session recorded on one side of an audio tape without a break. P2FA is an automatic phonetic aligner based on HTK, and developed at the Phonetics Laboratory of the University of Pennsylvania. It employs monophone Gaussian Mixture Model based HMMs which were trained using 39 perceptual linear prediction (PLP) coefficients. We used the 39 phone set from the Carnegie-Mellon University Pronouncing Dictionary, CMUdict [14], with the addition of OH for British English // distinct from //, including lexical stress marking for the vowels. The current CMU Pronouncing Dictionary was extended to include all the out-of-vocabulary words and to include a range of common British English word pronunciations. This extension was performed using semi-automatic methods by experienced phoneticians. ### 2.2 Human Evaluations We observed that some of the alignments were much more successful than others, and we quantified that with a rating procedure. One of the authors examined 5-second long regions, approximately once every 60 seconds throughout the files, and checked whether that region was correctly aligned at a word level. The overall score of the file was subjective on a scale between 0 (very poor) and 10 (very good), but was intended to reflect the number and magnitude of alignment problems. ### 2.3 Algorithms We developed five algorithms to indicate potential problems with an alignment, listed below. Each one takes the aligner output (segment label times, and log- probabilities) and optionally the audio file, and identifies a list of suspicious locations. In these descriptions, $L_{i}$ is the aligner's HMM log- probability value for phoneme instance $i$, $p_{i}$ is the phoneme (i.e. /a/, /t/, //, …), and $\delta_{i}$ is the duration. The algorithms were developed without reference to the human evaluations, except for the setting of each algorithm's threshold. Unexpected Log(P): This algorithm builds a prediction of the aligner's log- probability score per unit length from the corpus as a whole (except the data file under analysis). It then computes the difference between $L_{i}/\delta_{i}$ and the prediction111We drop phones with $\delta_{i}=0$ or with $\delta_{i}\geq 1s$ on the grounds that the scaling of $L_{i}$ with $\delta_{i}$ may not be accurate on these phones.. It operates on the assumption that most of the audio in the corpus is correctly aligned so that its predictions correspond to good alignment. Thus, when the aligner is doing worse than usual, and $L_{i}$ is low, the difference will be substantially negative. We have observed that when the aligner fails, it typically fails for a relatively large region: a word or more. To make use of this knowledge, we smooth the difference over a 1 second long region. When this smoothed difference of $log(P)$ is more negative than a threshold, the algorithm has identified a suspicious region. The predictor for $L_{i}$ starts with the median value for that phoneme, $\lambda(p_{i})=\mathrm{median}(L_{j}/\delta_{j}\;\mathrm{if}\;p_{j}=p_{i})$. It then adds in a 5-term linear prediction. The independent variable in the first term is $log(\delta_{i}/D(p_{i}))$, where $D$ is the median duration of a phoneme class, and $D(p_{i})=\mathrm{median}(\delta_{j}\;\mathrm{if}\;p_{j}=p_{i})$. The remaining four terms capture some information on the phoneme sequence. The second captures the typical difference in duration between the phone class under consideration and the previous phone class: $log(D(p_{i})/D(p_{i-1}))$. The third captures the typical difference in $log(P)$ between the phoneme class under consideration and the preceding phoneme class: $\lambda(p_{i})-\lambda(p_{i-1})$. The fourth and fifth are the same, except they refer to the succeeding phoneme. The predictor (along with the medians) is trained on phonemes with $0.04s\leq\delta_{i}\leq 0.18s$. The output of this algorithm becomes the ``unexpected'' feature. Word Log(P): we consider words with four or more phonemes to avoid variations due to the vagaries of individual phonemes. The individual $L_{i}$ values for each phoneme are summed over the respective word, then normalised by dividing by the word duration. This log probability per unit time provides a stable indication of the goodness of fit of the observed data to the HMM. The final stage of of the Word Log(P) method compares the normalised log probability with a fixed threshold, yielding the ``improbable'' feature. This method is complemementary to the Unexpected Log(P) method, above, in that it combines data from a whole word, whereas the Unexpected Log(P) method utilises features at the individual phoneme level, before smoothing them, i.e. the Unexpected Log(P) algorithm normalises $L_{i}$ by duration over a larger unit. In the Unexpected Log(P) algorithm, the threshold was set by experiment, to identify about 100 suspicious regions per hour on files that had substantial alignment problems. The Word Log(P) threshold was set to identify a similar number of events on poorly aligned files, and typically fewer on well-aligned ones. Extremes of Amplitude: Many alignment systems will produce erroneous alignments when several consecutive speech labels bunch-up into a short region, with the remaining speech labelled as an extended silence. This misidentification of speech and silence can be detected by examining the amplitude of signals in each labelled phoneme. A contiguous region of quiet, of a length comparable to a short word within a segment labelled as speech, indicates that an error may have occurred. Similarly, an error is likely if there is a word-length region of high amplitude within a segment labelled as silence. High amplitude ``silence'' regions should be marked for human inspection even if they do not contain speech, because they represent high- amplitude background noise, which is itself a potential cause of problems in real-world data. The thresholds for ``quiet'' and ``high amplitude'' were set as the $3^{\mathrm{rd}}$ and the $97^{\mathrm{th}}$ percentile of the amplitudes observed over the whole of the recording. The nominal length of a ``short word'' was set to $\nicefrac{{1}}{{4}}$ second, assuming four phonemes with an average duration of $\nicefrac{{1}}{{16}}$ second each. These parameters were estimated by experiment, and chosen to give a relatively small number of false positives. This algorithm produces two factors (``loud'', and ``quiet'') as it reports the extremes separately. Word Duration: this algorithm simply examines the durations of the segments, and if there are any which are unexpectedly long or short, indicates an error. It is difficult (simply from their duration) to detect periods of silence which have become extended or merged due to an alignment error. But the durations of segments labelled as speech can be of great help. This algorithm takes a word-based approach to detecting unusual segment durations. Individual phoneme durations are not reliable indicators because of variabilities in pronunciation due either to dialect, style of speaking, or the effects of transient background noise. Thus we take all words with four or more phonemes, calculate the duration of the region labelled as the word, normalise it by dividing by the number of phonemes, and compare it with two thresholds representing the largest and the smallest average phoneme duration. Any result outside the range $\nicefrac{{1}}{{32}}\;\mathrm{s}<\mbox{mean duration}<\nicefrac{{1}}{{8}}\;\mathrm{s}$ is flagged. The lower threshold is just above the minimum possible duration of a phoneme label for our HMMs (which use 3 left-to-right states per phone). This algorithm yields two features (``short'' and ``long''), as it reports the two extremes separately. Duration Mismatch: This algorithm builds a duration model for phonemes and then measures how far each phoneme222We do not compute a result for phonemes with $\delta_{i}=0$ or with $\delta_{i}\geq 1s$ on the grounds that they are outside the range of validity of the duration model, and are almost all silences, anyway. However, these phones may be used as neighbours in the computation of other phones. deviates from the model. Regions are identified as suspicious if the smoothed absolute value of the deviation is large enough to exceed a threshold. The duration model predicts the log of the phoneme duration as $d_{i}$. It starts with a value typical of that phone: $\Delta_{i}=\mathrm{median}(\log(\delta_{j})\;\mathrm{if}\;p_{i}=p_{j})$. It then adds on a 25-term linear predictor: the constant term captures a constant offset from $\Delta_{i}$. Then, twelve terms capture the length of nearby phonemes relative to their median durations (six neighbours on each side), via factors that are $\mathrm{q}(D_{i}\delta_{i+k},D_{i+k}\delta_{i}),$ where k specifies which neighbour333In practice, these first 12 terms amount to a normalisation of the duration for changes in the local speech rate: the model adjusts the phone duration by 36% of the average change in nearby durations. The $\mathrm{q}(a,b)=\begin{Bmatrix}2(a/b)^{0.5}-2,\;\mathrm{if}\;a\leq b\\\ 2-\mathrm{q}(b,a),\;\mathrm{else}\end{Bmatrix}$ function is a sigmoid whose domain is $[0,\infty]$, and it is well-behaved at the endpoints, an important property since some phoneme durations are zero. The final 12 terms similarly capture the differences between the typical durations of neighbouring phones. The are represented by features $\mathrm{q}(D_{i},D_{i+k})$. This duration model is trained to match $\mathrm{log}(\delta_{i})$ as in the Unexpected Log(P) algorithm. Finally, each phone is scored by $S_{i}=|\delta_{i}-d_{i}|/m_{i}$, where $m_{i}=\mathrm{median}(|\log(\delta_{j})-\Delta_{i}|\;\mathrm{if}\;p_{j}=p_{i})$ is the mean absolute deviation of the log duration. The scores are then smoothed and thresholded as in the Unexpected Log(P) algorithm. This produces the ``badlength'' feature. ## 3 Results Figures 1 and 2 show two examples of regions correctly identified as misaligned. Many such identifications are correct. We combined the results for each audio file to give an overall score based on the total number or duration of suspected of regions. The outputs of the above seven features were then correlated with the evaluations using a linear regression, via the ``glm'' method of the R software package [15]. The lengths of the audio files varied, as did the amount of speech, so we devised three ways to define a score, and since we had no clear criterion to pick one over the other, we computed separate linear regressions with each. The first, $s^{nd}$, is the number of identified regions divided by the duration of the audio file; the second, $s^{nw}$, is the number of regions divided by the number of words in the audio file (as determined from the BNC transcriptions); and the third, $s^{dd}$, is the total duration of the regions divided by the duration of the audio file. Figure 1: Quiet segment error: the ``quiet'' detector has identified a region (shaded), labelled as the word ``it's''. The vowel is omitted from the labelled region, and the end of the region extended into silence. The whole region is very quiet. The top tier is the spectrogram, then phoneme and word labels, respectively. Figure 2: Long word error. Displayed as per Figure 1, it shows where the ``long'' detector has identified a region (shaded in the Figure) that was aligned as a single word, ``wasn't'', but actually included several words. The distributions of the evaluation variable and the various $s$ variables were strongly non-Gaussian, with a maxima at one edge. Transforming the independent variables by raising them to the power 0.3 made the distribution subjectively more normal, as did squaring the evaluations. However, these transforms were not more than partially successful, so we elected to regress both with and without the transforms. This led to four regressions for each choice of $s$, or 12 regressions in all. Of those 12 regressions, short was statistically significant on 10 ($P<0.01$), badlength was significant on 6, loud on 4.444Note that with 12 regressions, we expect 3 false significances at the $P<0.05$ level and one at the $P<0.01$ level. Therefore, of the 9 significances reported at the $P<0.05$ level (4 for badlength, 3 for loud, 2 for long), half are probably spurious. For simplicity, we will ignore them all. At least one of those factors was significant at the $P<0.01$ level in each regression. Pearson's $R^{2}$ averaged 0.66 with a standard deviation of 0.13 over the regressions, indicating that a combination of the algorithms was reasonably effective at matching the human judgements of overall alignment accuracy. Of the best three fits ($R^{2}=0.81$, $0.81$, $0.87$), short was significant on each at $P<0.01$, along with badlength and loud once each. Of these, one used $s^{dd}$ and did not transform the data at all; the other two used $s^{nw}$ and did not transform the independent variables. In the second analysis, the correlations between the various algorithms were calculated for each recording. Figure 3 shows the results for one recording. Each pair of algorithms scored a point when they identified a pair of regions within 5 s of each other. We compared the number of such pairs to the number of accidental pairs that would occur if the algorithm's outputs were uncorrelated with each other. Figure 3: Identified regions: each row shows where an algorithm flagged a potential alignment error. The audio file was rated 8 for overall alignment success. NB: the improbable detector never fired on this file. There is a pattern of correlation between some of the detectors in Figure 3. Several pairs of algorithms were strongly correlated: notably badlength and unexpected, badlength and long, and long and unexpected. These coincided 3.6 to 5.1 times more often than chance, with statististical significances well beyond $P<0.001$. Several of the pairs of algorithms were anticorrelated, notably loud vs. short, loud vs. badlength, and loud vs. long. This is due to the designs of the algorithms: specifically, loud triggers only on silences, while the others trigger only on speech sounds. As a result, they never pick the same phoneme, and only occasionally pick phonemes within 5 s of each other. The remaining pairs were either nearly independent (loud & unexpected, badlength & short, long & short, short & unexpected) or did not have enough occurrences to draw any reliable conclusion. Duration-based measurements seem to perform best (i.e. short and badlength). One of the most useful indicators of a gross alignment error was also the simplest: the short algorithm detected a sequence of phonemes whose durations were at the minimum allowed by their state topology (here, 3 states or 30 milliseconds). The relative lack of success of the improbable and unexpected algorithms was unexpected: good and bad alignments have similar distributions of Log(P) scores. ## 4 Conclusions We have shown that automated techniques can usefully identify regions of bad alignment. The regions identified by some of our algorithms correlate well with human evaluations of the overall quality of the alignment. In general it appears that the most reliable method for judging the quality is simply to consider the statistics of the segment durations, either over a fixed time window, or a linguistic unit (e.g. a word). This research should allow semi- automatic evaluation of the alignment of large speech corpora, which will be important for their future use in speech research. ## 5 Acknowledgements We thank JISC (in the UK) and NSF (in the USA) for their support of Mining a Year of Speech, under the Digging into Data programme. This work is also partly supported by the UK ESRC (awards RES-062-23-2566, RES-062-23-1172, and RES-062-23-1323). We thank John Coleman and Ranjan Sen for the dictionaries, and John Coleman for his comments. ## References * [1] Sjölander, K., ``An HMM-based system for automatic segmentation and alignment of speech'', Umeå University, Department of Philosophy and Linguistics, _PHONUM 9_ , pp. 93-96, 2003. * [2] M. Cooke, J. R. Hershey, S. J. Rennie, ``Monaural speech separation and recognition challenge'', _Computer Speech and Language_ 24(1), January, 2010 * [3] Toth, A. R., ``Forced Alignment for Speech Synthesis Databases Using Duration and Prosodic Phrase Breaks'', in _Proc. 5th ISCA Speech Synthesis Workshop_ , Pittsburgh, June, 2004. * [4] ``The British National Corpus'', http://www.natcorp.ox.ac.uk/ * [5] De Villiers, E., ``Automatic alignment and error detection for phonetic transcriptions in the African speech technology project databases'', MScEng (Electrical and Electronic Engineering) Thesis, University of Stellenbosch, 2006. * [6] Barnard, E and Davel, M., ``Automatic error detection in alignments for speech synthesis'', _17th Annual Symposium of the Pattern Recognition Association of South Africa (PRASA)_ , Parys, South Africa, 29 Nov - 1 Dec, pp. 53-56, 2006. * [7] Davel, M. and Barnard, E., ``Bootstrapping Pronunciation Dictionaries: Practical Issues'', in _Proc. Interspeech 2005_ , Lisboa, Portugal, pp. 1561–1564, 2005. * [8] Huang, X., Acero, A., Adcock, J., Hon, H.-w., Goldsmith, J., Liu, J., and Plumpe, M., ``Whistler: A Trainable Text-To-Speech System'', _Proc. ICSLP 96_ , Philadelphia, PA, October 1996, pp. 2387-2390. * [9] Das, R., Izak, J., Yuan, J., Liberman, M., ``Forced Alignment Under Adverse Conditions'', University of Pennsylvania, CIS Dept. Senior Design Project Report, 2010. * [10] Coleman, J., Liberman, M., Kochanksi, G., Burnard, L., and Yuan. J., ``Mining a Year of Speech,'' _New Tools and Methods for Very-Large-Scale Phonetics Research Workshop_ , University of Pennsylvania, January 28-31, 2011. * [11] Baghai-Ravary, L., Kochanski, G. and Coleman J.,``Precision of Phoneme Boundaries Derived using Hidden Markov Models'', _Proc. Interspeech 2009_. ISSN 1990-9772, Brighton, UK, pp. 2879-2882, 2009. * [12] Baghai-Ravary, L., Kochanski, G., and Coleman, J., ``Objective Optimisation of Automatic Speech-to-Phoneme Alignment Systems'', in _Human Language Technologies as a Challenge for Computer Science and Linguistics_ , Vetulani, Z., (ed.), 2009. * [13] Yuan, J. and Liberman, M., ``Speaker identification on the SCOTUS corpus'', in _Proc. Acoustics '08_ , pp. 5687-5690, 2008. * [14] Carnegie Mellon University Pronouncing Dictionary, available from http://www.speech.cs.cmu.edu/cgi-bin/cmudict * [15] ``The R Reference Index'', http://www.r-project.org/
arxiv-papers
2011-01-09T23:02:52
2024-09-04T02:49:16.258494
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ladan Baghai-Ravary, Sergio Grau, Greg Kochanski", "submitter": "Greg P. Kochanski", "url": "https://arxiv.org/abs/1101.1682" }
1101.1703
# Identifying and Characterizing Nodes Important to Community Structure Using the Spectrum of the Graph Yang Wang, Zengru Di, Ying Fan111yfan@bnu.edu.cn Department of Systems Science, School of Management and Center for Complexity Research, Beijing Normal University, Beijing 100875, China ###### Abstract Background: Many complex systems can be represented as networks, and how a network breaks up into subnetworks or communities is of wide interest. However, the development of a method to detect nodes important to communities that is both fast and accurate is a very challenging and open problem. Methodology/Principal Findings: In this manuscript, we introduce a new approach to characterize the node importance to communities. First, a centrality metric is proposed to measure the importance of network nodes to community structure using the spectrum of the adjacency matrix. We define the node importance to communities as the relative change in the eigenvalues of the network adjacency matrix upon their removal. Second, we also propose an index to distinguish two kinds of important nodes in communities, i.e., “community core” and “bridge”. Conclusions/Significance: Our indices are only relied on the spectrum of the graph matrix. They are applied in many artificial networks as well as many real-world networks. This new methodology gives us a basic approach to solve this challenging problem and provides a realistic result. ###### pacs: 89.75.Hc, 89.75.-k, 89.75.Fb ## I Introduction Networks, despite their simplicity, represent the interaction structure among components in a wide range of real complex systems, from social relationships among individuals, to interactions of proteins in biological systems, to the interdependence of function calls in large software projects. The network concept has been developed as an important tool for analyzing the relationship of structure and function for many complex systems in the last decadesRev.Mod.Phys.74 ; SIAM Rev.45 ; Science 286 ; Nature 393 ; Phys. Rep. 424 . Many real-world systems show the existence of structural modules that play significant and defined functional roles, such as friend groups in social networks, thematic clusters on the world wide web, functional groups in biochemical or neural networksPNAS99 . Exploring network communities is important for the reasons listed belowPlos1 : 1) communities reveal the network at a coarse level, 2) communities provide a new aspect for understanding dynamic processes occurring in the network and 3) communities uncover relationships among the nodes that, although they can typically be attributed to the function of the system, are not apparent when inspecting the graph as a whole. As a result, it is not surprising that recent years have witnessed an explosion of research on community structure in graphs, and a huge number of methods or techniques have been designedPNAS99 ; Phys. Rev. E 74 ; Physics Reports486 ; Eur.Phys.J.B.38 ; Phys. Rev. E 72 ; Proc. Natl. Acad. 103 ; Phys. Rev. E 72 ; arXiv:1002.2007v1 ; arXiv:0902.3331v1 ; Rhys. Rev. E 77 ; arXiv:0907.3708 (seePhysics Reports486 as a review). It is believed that community structure is important to the function of a systemProc. Natl. Acad. Sci.100 ; Physica A 384 ; Europhys. Lett.72 . In many situations, it might be desirable to control the function of modular networks by adjusting the structure of communities. For example, in biological systems, one might like to identify the nodes that are key to communities and protect them or disrupt them, such as in the case of lung cancerPhysica A 384 . In epidemic spreading, one would like to find the important nodes to understand the dynamic processes, which could yield an efficient method to immunize modular networksEurophys. Lett.72 . Such strategies would greatly benefit from a quantitative characterization of the node importance to community structure. Some important work related to this topic has been proposed. In 2006, Newman proposed a community-based metric called “Community Centrality” to measure node importance to communitiesPhys. Rev. E 74 . His basic idea relies on the modularity function $Q$. Those vertices that contribute more to $Q$ are more important for the communities than those vertices that contribute less. Kovacs et al. also proposed an influence function to measure the node importance to communitiesPLOSONE . In fact, the important nodes can have distinct functions with respect to community structure. Some previous studies have also revealed such classifications. Guimera et al. have proposed a classification of the nodes based on their roles within communities, using their within-module degree and their participation coefficientNature433 . They divided the hubs into three categories: provincial hubs, connector hubs and kinless hubs. Other approaches have also been suggested to discuss the connection between nodes and modularity in biological networks, by dividing hub nodes into two categories called “party hubs” and “date hubs”Nature430 ; PLoS Bioo ; PLoS Bio . When removed from the network, party and date hubs have strikingly distinct effects on the overall topology of the network. Recently, Kovacs et al. proposed an interesting approach. They introduced an integrative method family to detect the key nodes, overlapping communities and “date” and “party” hubsPLOSONE . In a very recent work, the authors mentioned that modular networks naturally allow the formation of clusters, and hubs connecting the modules would enhance the integration of the whole network, such as in the case of neuron networksPRE82 . As a result, it is intuitive that nodes that are important to communities can be divided into “community cores” and “bridges”. However, there is one problem. Before using the participation coefficient and the influence function to distinguish these two kinds of vertices, the exact communities of the network must first be given. In contrast, it is interesting to characterize node importance to communities before the division of the network. It is understood that the adjacency matrix contains all the information of the network. Developing methods based only on the adjacency matrix of the network to detect important nodes to communities and then distinguish them as either “community core” or “bridge” is an interesting and important problem in network research. In this manuscript, based only on the adjacency matrix of the network, we try to access the fundamental questions: how to evaluate the node importance to communities and how to distinguish different kinds of important nodes? It is implied that in many cases the spectrum of the adjacency matrix gives an indication of the community structure in the networkPRE80 . If the network has $c$ strong communities, the $c$ largest eigenvalues of the adjacency matrix are significantly larger than the magnitudes of all the other eigenvalues. These large eigenvalues are key quantities to the community structure. For this reason, we suggest a basic approach to solve the above open problem using the spectrum of the graph. We define the importance of nodes to communities as the relative change in the $c$ largest eigenvalues of the network adjacency matrix upon their removal. Furthermore, using the eigenvectors of the graph Laplacian, we divide the important nodes into community cores and bridges. We apply our method to many networks, including artificial networks and real-world networks. This new methodology gives us a basic approach to solve this challenging problem and provides a realistic result. The organization of this paper is as follows. In section II, the centrality metric identifying the important nodes to communities is proposed using the spectrum of the adjacency matrix. An index to distinguish the two kinds of important nodes using the corresponding eigenvector of the graph Laplacian is introduced in section III. In section IV, our method is applied to artificial networks and some real-world networks, and we obtain some interesting results. In section V, we extend our method into weighted networks. Finally, concluding remarks are presented in section V. ## II Centrality Metric Based on the Spectrum of the Adjacency Matrix We consider a binary network $G=(V,E)$ with $N$ nodes. The adjacency matrix $A$ is the matrix with elements $A_{ij}=1$ if there is an edge joining vertices $i$ and $j$, otherwise $0$. We denote each eigenvalue of $A$ by $\lambda$ and the corresponding eigenvector by v, such that $A\textit{{v}}=\lambda\textit{{v}}$. The eigenvector is orthogonal and normalized. The eigenvalues are ordered by decreasing magnitude: $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$. It is easy to show that $A$ is symmetric and the eigenvalues of $A$ are real. Consider the case of networks that have $c$ communities. It is implied that when these communities are disconnected, each one has its own largest eigenvalues. With proper labeling of the nodes, the matrix $A$ will have a block matrix structure with $c\times c$ blocks. Blocks on the diagonal correspond to the adjacency matrices of the individual communities, while the off-diagonal blocks correspond to the edges between communities; in other words, we can consider them as a perturbation. Therefore, $A$ can be written as $A=A_{0}+\delta A,$ (1) where $A_{0}$ is a matrix whose diagonal block elements are the diagonal block elements of $A$ and whose off-diagonal block elements are zeros, while $\delta A$ is a matrix with zeros on its diagonal blocks and with the off-diagonal blocks of $A$ as its off-diagonal block elements. Chauhan et al.PRE80 have proved that if the perturbation strength is small, the largest eigenvalues of disconnected communities are perturbed more weakly than the perturbation applied. The spectrum of the adjacency matrix of a network gives a clear indication of the number of communities in the network. If the network has $c$ strong communities, the $c$ largest eigenvalues are well separated from others. These eigenvalues are key quantities to the community structure. For this reason, we define the importance of node $k$ to communities as the relative change in the $c$ largest eigenvalues of the network adjacency matrix upon its removal: $I_{k}=-\sum\limits_{i=1}^{c}{\frac{{\Delta\lambda_{i}}}{{\lambda_{i}}}},$ (2) where $c$ is the number of communities. To avoid the computational cost, we use perturbation theory to provide approximations of $I_{k}$ in terms of the corresponding eigenvector v. Let us denote the matrix before the removal of the node by $A$ and the matrix after the removal by $A+\Delta A$; the eigenvalue of this matrix is $\lambda+\Delta\lambda$, and the corresponding eigenvector is $\textit{{v}}+\Delta\textit{{v}}$. For large matrices, it is reasonable to assume that the removal of a node has a small effect on the whole matrix and the spectral properties of the network, so that $\Delta A$ and $\Delta\lambda$ are small. We obtain $(A+\Delta A)(\textit{{v}}+\Delta\textit{{v}})=(\lambda+\Delta\lambda)(\textit{{v}}+\Delta\textit{{v}}).$ (3) The effect on the adjacency matrix $A$ of removing node $k$ is given by $(\Delta A)_{ij}=-A_{ij}(\delta_{ik}+\delta_{jk})$. We cannot assume that the $\Delta\textit{{v}}$ is small because $\Delta v_{k}=-v_{k}$, so we set $\Delta\textit{{v}}=\delta\textit{{v}}-v_{k}\widehat{e}_{k}$ where $\delta\textit{{v}}$ is small and $\widehat{\textit{{e}}}$ is the unit vector for the $k$ component. Left multiplying (3) by $\textit{{v}}^{T}$ and neglecting second order terms $\textit{{v}}^{T}\Delta A\delta\textit{{v}}$ and $\textit{{v}}^{T}\Delta\lambda\delta\textit{{v}}$, we obtain $\Delta\lambda=\frac{{\textit{{v}}^{T}\Delta A\textit{{v}}-\textit{{v}}^{T}v_{k}\Delta A\widehat{e}_{k}}}{{\textit{{v}}^{T}\textit{{v}}-v_{k}^{2}}}.$ (4) For a large network ($N\gg 1$), we know that $\textit{{v}}^{T}\textit{{v}}\gg v_{k}^{2}$; therefore, we can write $\Delta\lambda\approx\frac{{\textit{{v}}^{T}\Delta A\textit{{v}}-\textit{{v}}^{T}\textit{{v}}_{k}\Delta A\widehat{\textit{{e}}}_{k}}}{{\textit{{v}}^{T}\textit{{v}}}}$ (5) Because $(\Delta A)_{ij}=-A_{ij}(\delta_{ik}+\delta_{jk})$, we obtain $\textit{{v}}^{T}\Delta A\textit{{v}}=-2\lambda v_{k}^{2},\textit{{v}}^{T}v_{k}\Delta A\widehat{e}_{k}=-\lambda v_{k}^{2}.$ (6) Finally, the importance of node $k$ to the community structure is obtained by $I_{k}=-\sum\limits_{i=1}^{c}{\frac{{\Delta\lambda_{i}}}{{\lambda_{i}}}}\approx\sum\limits_{i=1}^{c}{\frac{{v_{ik}^{2}}}{{\textit{{v}}_{i}^{T}\textit{{v}}_{i}}}},$ (7) where $c$ is the number of communities, $v_{ik}$ is the kth element of $\textit{{v}}_{i}$ and $I_{k}$ lies in the interval $[0,1]$. If $I_{k}$ is large, node k is important to the community structure; otherwise, $k$ is on the periphery of the community. Using this metric $I$, we can quantify the node importance to the community structure. If the node is important to the community structure, when we remove it from the network, the relative changes of the $c$ largest eigenvalues are large; otherwise, the changes are small. Before applying $I$, the value of $c$ needs to be determined. The determination of the number of communities is an important but challenging question in community analysis. Here we use the method proposed by Ref.PRE80 . This method is based on the properties of the spectrum of the graph and is independent of the partition algorithms, so our metric is quite convenient to use. ## III distinguish two kinds of important nodes As mentioned above, there are two kinds of nodes that are important to communities. One is the “community core”, and the other is the “bridge” between communities. Each will affect communities deeply upon its removal. When we remove the “community core”, the community structure in the network will become fuzzy, while the community structure will become clear when we remove the “bridge”. See Fig. 1 for an example. Vertices 1 and 8 are the “community cores”, and they organize their respective communities. Meanwhile, node 15 is the “bridge” between the two communities. The “community core” is the leader in the community, and it can organize the function of each community. In contrast, the “bridge” connects the modules and can enhance the integration of the whole network. It is believed that a combination of both segregation and integration, as in neural systems, is crucialPRE82 . It is clear that effectively disconnected and fully non-synchronous regions cannot allow collective or integrative action of the elements. Similarly, a fully synchronized regime does not allow separated or segregated performance of the elements. Therefore, both situations are biologically unrealistic, as can be seen from the existence of related conditions, such as epileptic seizures (collective phenomena) and Parkinson’s disease (segregated phenomena)Neuro . For this reason, both the “community core” and the “bridge” are important to communities, but they play different roles. The metric we proposed in SectionII can determine the nodes that are important to communities, but now a method to distinguish these two kinds of important nodes is needed. In agreement with earlier findingsPLOSONE ; Nature430 ; PLoS Bioo ; PLoS Bio , we assumed that bridge nodes should have more inter-modular positions than community cores. The existence of bridge nodes often leads to some inter- modular edges. Given a graph, the simplest and most direct way to construct a partition of the graph is to solve the mincut problem (minimize the number of edges between communities $R$)CMJ23 . In practice, however, this method often does not lead to satisfactory partitions. The problem is that, in many cases, the solution of mincut simply separates one individual vertex from the rest of the graph. Of course, this is not what we want to achieve in clustering, as clusters should be reasonably large groups of points. Due to this shortcoming in the mincut problem, one common objective function to encode the desired information is RatioCutRatiocut : $RatioCut(C_{1},\cdots C_{c})\buildrel\textstyle.\over{=}\sum\limits_{i=1}^{c}{\frac{{R(C_{i},\bar{C}_{i})}}{{|C_{i}|}}},$ (8) where $|C_{i}|$ is the size of community $C_{i}$. If the sizes of the communities are almost the same, the RatioCut problem reduces to the mincut problem. ### III.1 The Condition of $c=2$ If the network is divided into only two communities ($c=2$), we define an index vector s with $N$ elements: $s_{i}=\left\\{\begin{array}[]{l}\sqrt{{{|\bar{C}|}\mathord{\left/{\vphantom{{|\bar{C}|}{|C|}}}\right.\kern-1.2pt}{|C|}}}\quad\quad{\rm{if\quad vertex}}\quad i\in C,\\\ -\sqrt{{{|C|}\mathord{\left/{\vphantom{{|C|}{|\bar{C}|}}}\right.\kern-1.2pt}{|\bar{C}|}}}\quad{\rm{if\quad vertex}}\quad i\in\bar{C}.\\\ \end{array}\right.$ (9) Then the RatioCut function is obtained as followsTutorial : $RatioCut(C,\bar{C})=\frac{1}{{|V|}}\textit{{s}}^{T}\textit{{L}}\textit{{s}},$ (10) where $|V|$ is the number of vertices in the network and L is the graph Laplacian. L is defined as $L_{ij}=-A_{ij}$ for $i\neq j$ and $L_{ii}=k_{i}$, where $k_{i}$ is the degree of node $i$. We also have two constraints on s: $\sum\limits_{i=1}^{n}{s_{i}}=0$ and $\sum\limits_{i=1}^{n}{s_{i}^{2}}=n$. Here the partition problem is equal to the problem $\min{\rm{}}\textit{{s}}^{T}\textit{{L}}\textit{{s}};\ {\rm{subject\ to}}\ \sum\limits_{i=1}^{n}{s_{i}}=0,\sum\limits_{i=1}^{n}{s_{i}^{2}}=n.$ (11) If the components of the vector s are allowed to take arbitrary values, it can be seen immediately that the solution of this problem is given by the vector s that is the eigenvector corresponding to the second-smallest eigenvalue of L, denoted by $\textit{{u}}_{2}$. So we can approximate a minimizer of RatioCut by the second eigenvector of L. Unfortunately, the components of s are only allowed to take two particular values. Thus, the simplest solution is achieved by assigning vertices to one of the groups according to the sign of the eigenvector $\textit{{u}}_{2}$. In other words, we assign vertices as follows: if $\textit{{u}}_{2}^{i}>0$, we assign vertex $i$ to community $C$; otherwise, we assign it to $\bar{C}$. Assignation priority begins with the most positive and the most negative; the node with the most positive magnitude is first to be assigned to $C$, then the second and so on, while the node with the most negative magnitude is similarly the first to be assigned to $\bar{C}$. If a node’s corresponding element is close to zero, it may have nearly equal membership in both communities, and we can assign it to both communities. In conclusion, if the network is divided into only two communities, we can use this method to characterize which are the “community cores” and which are the “bridge” between communities. If node $i$ is a “community core”, $|\textit{{u}}_{2}^{i}|$ is relatively large; otherwise, $|\textit{{u}}_{2}^{i}|$ is near zero. ### III.2 The Condition of $c>2$ Consider the division of a network into $c$ nonoverlapping communities, where $c$ is the number of communities. We define an $n\times c$-index matrix S with one column for each community, $\textit{{S}}=(\textit{{s}}_{1}|\textit{{s}}_{2}|\cdots|\textit{{s}}_{c})$, by $s_{i,j}=\left\\{\begin{array}[]{l}{{\rm{1}}\mathord{\left/{\vphantom{{\rm{1}}{\sqrt{|C_{j}|}}}}\right.\kern-1.2pt}{\sqrt{|C_{j}|}}}\quad{\rm{if\quad vertex}}\quad i\in C_{j},\\\ {\rm{0\quad otherwise}}.\\\ \end{array}\right.$ (12) Following the previous section, we obtain $RatioCut=Tr(\textit{{S}}^{T}\textit{{L}}\textit{{S}}),$ (13) where $Tr$ is the trace of a matrix and $\textit{{S}}^{T}$ is the transpose matrix of S. L is a semi-positive and symmetric matrix. We can write $\textit{{L}}=\textit{{U}}\textit{{D}}\textit{{U}}^{T}$, where U is the eigenvector of L, $\textit{{U}}=(\textit{{u}}_{1}|\textit{{u}}_{2}|\cdots|\textit{{u}}_{n})$ and D is the diagonal matrix of eigenvalues $D_{ii}=\beta_{i}$. We therefore obtain $RatioCut=\sum\limits_{j=1}^{n}{\sum\limits_{k=1}^{c}{\beta_{j}(u_{j}^{T}s_{k})^{2}}}.$ (14) It can also be written as $RatioCut=\sum\limits_{k=1}^{c}{\sum\limits_{j=1}^{n}{\beta_{j}[\sum\limits_{i=1}^{n}{U_{ij}S_{ik}}]^{2}}}.$ (15) Now we define the vertex vector of $i$ as $r_{i}$, and let $[r_{i}]_{j}=U_{ij}.$ (16) If the network has almost equal-sized communities, then equation (15) can be written as $RatioCut\approx\frac{{\sum\limits_{k=1}^{c}{\sum\limits_{j=1}^{n}{\beta_{j}[\sum\limits_{i\in G_{k}}{[r_{i}]_{j}}]^{2}}}}}{{|C|}},$ (17) where $G_{k}$ is the set of vertices belonging to community $k$ and $|C|$ is the community size. Minimizing the RatioCut can be equated with the task of choosing the nonnegative quantities so as to place as much of the weight as possible in the terms corresponding to the low eigenvalues and as little as possible in the terms corresponding to the high eigenvalues. This equates to the following maximization problem: $Max\ \sum\limits_{k=1}^{c}{\sum\limits_{j=1}^{p}{\beta_{j}[\sum\limits_{i\in G_{k}}{[r_{i}]_{j}}]^{2}}},$ (18) where $p$ is a parameter. We could choose $p=c$ if the community structure was clear. To this end, we propose an easy way to distinguish two kinds of important nodes using the theory of the graph Laplacian. If the community structure is quite clear, we focus on the vertex vector magnitude $|r_{i}|$ in the first $p$ terms, denoted by the $w$-score: $w_{i}=\sqrt{\sum\limits_{j=1}^{p}{[r_{i}]_{j}^{2}}}.$ (19) If the $w$-score of a given vertex is close to zero, we believe that this vertex has nearly equal membership in more than one community, and it is likely to be the “bridge” of these communities. This discrimination process equates to the “fuzzy” division of the network into communities. In many cases, this type of fuzzy division could result in a more accurate picture of real-world networks. ## IV Results Now we test the validity of our indices introduced in section II and section III in various artificial networks and real-world networks. ### IV.1 Artificial Networks First, we consider a sketch composed of 15 nodes (see Fig. 1) forming two communities. It is intuitive that vertices 1, 8 and 15 are important to the community structure in this sketch. Vertices 1 and 8 are the so-called “community cores”, and they form both the communities. Vertex 15 is the “bridge” between communities, and it connects these two communities. As we discussed before, removing vertex 1 or 8 will make the community structure fuzzy, and removing vertex 15 will make it clear. Figure 1: Sketch of a network composed of 15 nodes. The diameter of one vertex is proportional to the centrality metric $I$. Moreover, the color of one vertex is related to the index $w$-score. Red vertices behave like “overlapping” nodes or “bridges” between communities, and yellow vertices often lie inside their own communities. Here we use the index $H$ proposed by Hu et al.arXiv:1002.2007v1 to measure the significance of communities: $H=\frac{n}{{\bar{k}\sum\limits_{j=c+1}^{n}{\frac{1}{{|\overline{\beta}-\beta_{j}|}}}}},$ (20) where $\beta$ is the eigenvalue of the graph Laplacian, $\overline{\beta}$ is the average value of $\beta_{2}$ through $\beta_{c}$, $\bar{k}$ is the average degree of the network and $n$ is the number of vertices in the network. In networks with strong communities (many links are within communities with very sparse connections outside), $H$ is always large. Here we focus on the change of $H$ due to the removal of vertices, denoted by $\Delta H$. We also use the centrality metric proposed by NewmanPhys. Rev. E 74 , which we denote here by $M$. The results are shown in Tab. 1. Through $\Delta H$, it is implied that vertices 1 and 8 are more important than other vertices because the magnitude of $\Delta H$ is relatively larger than others. Moreover, their removal makes the communities fuzzy, while vertex 15 acts like a ”bridge” between the communities, and its removal makes the communities clear. We can see that our centrality metric performs quite well; it can identify not only the “community cores”, but also the “bridge” between communities. $M$ can also identify the “community cores”, but it has some problems. One issue is that its values tend to span a rather small dynamic range from largest to smallest. Moreover, in some cases (such as this sketch), $M$ cannot recognize important vertices among communities. In calculating the index $H$, we need to go through every vertex in the network, incurring significant computational cost. In contrast, our method provides a more efficient way, requiring less computational cost, and yields the correct answer. Table 1: Centrality metrics of the example sketched in Fig. 1. Vertex Label | $I$ | $M$ | $\Delta H$ | $w$-score ---|---|---|---|--- 1 | 0.32 | 0.758 | -0.145 | 0.2405 8 | 0.32 | 0.758 | -0.145 | 0.2405 15 | 0.173 | 0.69 | 0.116 | 0.00 2,7,9,14 | 0.09 | 0.704 | 0.04 | 0.198 3,6,10,13 | 0.1 | 0.7535 | -0.021 | 0.285 4,5,11,12 | 0.105 | 0.7327 | -0.054 | 0.3175 Here we use the classical GN benchmark presented by Girvens and Newman to test the measurementsProc. Natl. Acad. 103 . Each network has $N=128$ nodes that are divided into four communities (c = 4) with 32 nodes each. Edges between two nodes are introduced with different probabilities, which depend on whether the two nodes belong to the same community or not. Each node has $<k_{in}>$ links on average with its fellows in the same community and $<k_{out}>$ links with the other communities, and we impose $<k_{in}>+<k_{out}>=16$. The communities become fuzzier and thus more difficult to identify as $k_{out}$ increases. Because the GN benchmark is a homogenous network, there should not be any nodes that are important to the community structure. To check whether our conjecture is correct or not, we let $<k_{in}>=12$ so that the community structure is quite clear and average the result for the GN benchmark over 100 configurations of networks. From the result, about 120 nodes’ importances lie in the interval $[0.03,0.04]$, while others lie in the interval [0.02,0.03]. The mean value of $I$ is 0.0312, and the standard deviation is 0.0014. It can be concluded that, in the GN benchmark, there are no nodes that are important to the community structure. We may also test the method on the more challenging LFR benchmark presented by Lancichinetti et al.Phys. Rev. E 78 . In the LFR benchmark, the degree distribution obeys a power-law distribution $p(k)\propto k^{-\alpha}$, and the sizes of the communities are also taken from a power-law distribution with an exponent $\gamma$. Moreover, each node shares a fraction $1-\mu$ of its links with other nodes of its own community and a fraction $\mu$ with others in the rest of the network. The community structure can be adjusted by the mixing parameter $\mu$. Without loss of generality, we let $\alpha=2.5,\gamma=1.0,\mu=0.25$ and the size of the network $N=1000$. Our numerical results in the LFR benchmark are shown in Fig. 2. In this case, there is no “bridge” between communities because $\mu=0.25$. We may also calculate the $w$-score, of which the mean value is 0.1736 and the standard deviation is 0.0292. Moreover, the centrality metric is positively correlated with node degree ($r^{2}=0.7329$), but some vertices have quite high centrality while having relatively low degree, and thus the correlation index is not very high. Figure 2: (a) The Zipf plot of the nodes’ centrality to communities. (b) The centrality metric we propose is correlated with node degree. The parameters in the LFR benchmark are as follows: $\alpha=2.5,\gamma=1.0,\mu=0.25$ and the size of the network $N=1000$. ### IV.2 Real-world Networks We apply our method to some real-world networks, such as the Zachary club networkJAR33 , the word association networkSOUTH , the scientific collaboration networkwebsite , and the C. elegans neural networkTRS . First, we consider a famous example of a social network, the Zachary’s karate club network. This network represents the pattern of friendships among members of a karate club at a North American university. It contains 34 vertices, and the links between vertices are the friendships between people. The nodes labeled as 1 and 34 correspond to the club instructor and the administrator, respectively. They had a conflict which resulted in the breakup of the club. Most other nodes have a relationship with node 1, node 34, or both. In this network, $c=2$. The numerical results are shown in Fig. 3 and Fig. 4. In Fig. 3(a), we can see that nodes 1 and 34 are the most important nodes in the communities. Our method to distinguish important nodes are shown in Fig. 3(b). From the result, we can see that nodes 1 and 34 are the so-called “community cores”, and they have many connections in their own communities. Furthermore, we compare our method with Newman’s. This result is also shown in Fig. 3(a), and the two metrics are normalized by $x_{nor}=\frac{{x-<x>}}{{\sigma_{x}}},$ (21) where $<x>$ is the average value of each index and $\sigma_{x}$ is the standard deviation of each index. It is implied that these two methods have some differences. In our method, nodes 1 and 34 are absolutely more important than other nodes, while in Newman’s method, nodes 2 and 33 are also quite important, even more than node 1. In this network, the modularity function $Q$ reaches its maximum value when the network is divided into 4 communities; this fact may be the cause of the differences between the results of these two methods. The visualization of the karate network with our two measurements is sketched in Fig. 4. The diameter of each vertex is proportional to the centrality metric $I$. A large diameter indicates an important vertex. Additionally, the color of each vertex is related to the index $w$-score. Red vertices behave like “overlapping” nodes or “bridges” between communities, and yellow vertices often lie inside their own communities. Figure 3: It is shown that our method works quite well in the Zachary’s karate club network. Nodes 1 and 34 are the instructor and the administrator, respectively. In Fig. 3(a), we can see that these two nodes are more important to the community structure than other nodes. We also compare our method with Newman’s and find that the two methods exhibit some differences. In Fig. 3(b), we shown that nodes 1 and 34 are the so-called “community cores”. Figure 4: The Zachary’s karate club network, which is composed of 34 vertices. Vertex diameters indicate the community centrality $I$. The color of each vertex is proportional to the index $w$-score. Second, we analyze the word association network starting from the word “Bright”. This network was built on the University of South Florida Free Association NormsSOUTH . An edge between words A and B indicates that some people associate the word B to the word A. The graph displays four communities, corresponding to the categories Intelligence, Astronomy, Light, Colors. The word Bright is related to all of them by construction. We applied our method to this network, and the results are shown in Fig. 5. From the results, we can observe that our method considers Bright, Sun, Smart, Moon as important nodes to the community structure. It may be inferred from the result that Moon and Smart are the “community cores”, while Bright and Sun are the “bridges” between communities. Indeed, our metric yields the correct answer. For example, Smart is the core of the community Intelligence, while Moon is the core of the community Astronomy. Meanwhile, the $w$-score of node Bright is 0.08, which is close to zero. We would therefore conclude that it is a “bridge” between communities, and Bright is in fact the “bridge” among these four communities, as the network was originally derived from it. Moreover, we may investigate the effect of node removal on the modularity function $Q$. “Community cores” and “bridges” have different effects on community structure. When a “community core” is removed, the communities become clear. For example, the removal of the node “bright” makes the modularity function $Q$ increase by 0.03, which is the largest increase caused by the removal of any single node, while the removal of node “Moon” causes $Q$ to decrease by 0.015. These results are averaged over 20 trials. We can see from our results that important nodes (i.e., nodes with large $I$) affect the communities considerably. For example, the removal of the node “Smart” decreases $Q$ by 0.0152, while the removal of the node “Gifted”, which seems to be a peripheral node, decreases $Q$ by only 0.0048. Figure 5: Index $I$ and $\omega$-score for the nodes of the word association network. The node importance versus vertex rank is shown in (a). In (b), we distinguish “community cores” and “bridges” using the index $w$-score. Figure 6: The centrality metric $I$ and $w$-score for the scientist collaboration network (a,b). The centrality metric $I$ and $w$-score are also calculated in the C. elegans neural network (c,d). We may also apply our method to social networks, such as the scientist collaboration networkwebsite , and neural networks, such as the C. elegans neural networkTRS . We analyzed the largest connected component of each network. The scientist collaboration network represents scientists whose research centers on the properties of networks of one kind or another. There are 379 vertices, representing scientists who are divided into 12 communities. Edges are placed between scientists who have published at least one paper together. The neural network of C. elegans contains 302 neurons and 2,359 links. This network is divided into 3 communities, with each node representing a neuron and each link representing a synaptic connection between neurons. Here we consider the C. elegans neural network to be undirected. The results are shown in Fig. 6. In the scientist collaboration network, our centrality metric $I$ identifies “group leaders”, such as M. Newman, S. Boccaletti, and A. Barabasi. Their $w$-scores are not very large because they often have some collaboration between scientists outside their own communities. We can also find so-called “community cores” based on our method, such as R. Sole, and “bridge” vertices among some communities, such as B. Kahng. As we know, the C. elegans neural networks are composed of sensory neurons, interneurons and motor neurons. The neurons with high centrality metrics often have the most important functions, and all of them are interneurons, such as $AVA$, $AVB$, $AVD$, and $AVE$. These classes, which synapse onto motor neurons in the ventral cord, are among the most prominent neurons in the whole nervous system. They generally have larger-diameter processes than other neurons and have many synaptic connectionsTRS ; JN . As a result, they have larger $I$ than other vertices, while the typical $w$-score in these classes is quite small (smaller than 0.05). In the C. elegans neural network, connection between communities is more necessary and frequent due to some special functions. ## V Applications in Weighted networks Our method can be generalized to weighted networks because the adjacency matrix in an undirected weighted network is real and symmetric. Thus, in weighted networks, the importance of a node and its role in communities are also characterized by its $I$ and $w$-score. Let us first consider an artificial weighted network. We use similarity weight in this weighted network. A higher weight means a closer relationship between vertices. At first, 10 nodes form a complete network and are divided into two communities with 5 nodes each. We assign vertices 4 and 9 as the core of each community, each of which has links with weight 2 connecting to vertices within its community and weight 0.2 connecting to outside vertices. All other intra- connections have weight 1, and all other interconnections have weight 0.2. Then we introduce vertex 11 as the bridge between the two communities. It connects to all 10 nodes with weight 1. The index $I$ and $w$-score for each node are given in Tab. 2. The results indicate that vertices 4, 9 and 11 are more important than the other vertices, while vertex 11 is a “bridge” between these two communities. Our method works quite well in this small artificial weighted network. Table 2: Centrality metrics $I$ and $w$-score in a complete weighted network. Vertex Label | I | | $w$-score ---|---|---|--- 4 | 0.295 | | 0.316 9 | 0.295 | | 0.316 11 | 0.16 | | 0.00 others | 0.156 | | 0.316 Figure 7: Sketch of the SFI scientific collaboration network as a weighted, undirected network. It has 118 scientists. Vertex diameters indicate the community centrality $I$. The color of each vertex is proportional to the index $w$-score. As an example of a real-world weighted network, we investigate the collaboration network among scientists working at the Santa Fe Institute (the SFI network). Here we consider it as a weighted, undirected network. Collaboration events between the scientists can be repeated again and again, and a higher frequency of collaboration usually indicates a closer relationship. Furthermore, weights can be assigned to the scientists’ collaboration quite naturally: an article with $n$ authors corresponds to a collaboration act of weight $\frac{1}{{n-1}}$ between every pair of its authorsPRE73 . The results for the SFI collaboration network are sketched in Fig. 7. Vertex diameters indicate the community centrality $I$. The color of each vertex is proportional to the index w-score. Red vertices behave like “overlapping” nodes or “bridges” between communities, and yellow vertices often lie inside their own communities. We do not know the specific names; however, we observe that the positions of the large vertices are just like the “group leaders”. Vertices 2, 12 and 24 are so-called “community cores” in communities because their $w$-scores are quite large. In fact, they are the group leaders in the fields of Mathematical Ecology, Statistical Physics and Structure of RNA, respectively. However, vertices 1, 9 and 11 are the “bridges” between communities, and they have relative small $w$-scores. Interestingly, the result in the weighted network is different from the one in the corresponding unweighted network. It can be concluded that the edge weight may affect the result. For example, vertex 9 and vertex 11 collaborate quite often; this makes both of them quite important in a weighted network, while in an unweighted network, neither of them is very important to the community structure. ## VI Conclusion And Discussion In this paper, we characterize the node importance to community structure using the spectrum of the graph. The eigenspectrum of the adjacency matrix gives a clear indication of the number of “dominant” communities in a networkPRE80 . We give a centrality metric based on the spectrum of the adjacency matrix of the graph, and it can identify the nodes important to the community structure in many cases. In addition, we propose an index to distinguish the two kinds of important nodes that we term “community cores” and “bridges” using the spectrum of the graph Laplacian. We demonstrate a variety of applications of our method to both artificial and real-world networks representing social and neural networks. Our method works well in many cases without knowing the exact community structure, although the number of communities should be known. However, a limitation of this method arises when one or more of the communities is much smaller than the largest community, or when a community has very sparse intra-community connections compared to other communities. This may happen when $N_{small}^{2}<N_{large}$PRE80 . Even in the absence of perturbation, the maximum eigenvalue of a smaller community can lie inside the cloud of non- Perron-Frobenius eigenvalues of the largest community. But, with the understanding that the intent of our method is to find the important nodes in the community structure, the nodes in very small communities may be ignored. Even so, if the community structure is so fuzzy that we cannot identify the number of communities, our method is not accurate. Our method can also be used in weighted networks. From our result in the SFI network, it can be inferred that edge weight may affect the result. Furthermore, it may generalize to directed networks because the Perron- Frobenius eigenvalues are often real and positiveSIAMR . We have yet to treat the case of directed networks. The identification of such key nodes is important and could potentially be used to identify the organizer of the community in social networks, to develop an immunization strategy in an epidemic process, to identify key nodes in biological networks and so on. We hope our results may be helpful to future research. ## ACKNOWLEDGEMENTS The authors thank Di Huan, An Zeng, and Hongzhi You for their helpful suggestions. This work is supported by the NSFC under grants No. 70771011 and No. 60974084, NCET-09-0228, and fundamental research funds for the Central Universities of Beijing Normal University. ## References * (1) Albert R, Barabasi A-L (2002), Statistical mechanics of complex networks. Rev. Mod. 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B 314: 1-340. * (37) Tsalik EL and Hobert OL, Neurobiol J (2003). Functional Mapping of Neurons That Control Locomotory Behavior in Caenorhabditis elegans. 56: 178-197. * (38) Ramasco JJ and Morris SA (2006), Social inertia in collaboration networks. Phys. Rev. E 73: 016122. * (39) MacCluer CR (2000), The many proofs and applications of Perron’s Theorem. SIAM Rev. 42: 487-498.
arxiv-papers
2011-01-10T03:31:15
2024-09-04T02:49:16.265875
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yang Wang, Zengru Di, Ying Fan", "submitter": "Wang Yang", "url": "https://arxiv.org/abs/1101.1703" }
1101.1927
# A Finite State Model for Time Travel Hwee Kuan Lee Bioinformatics Institute, 30 Biopolis Street, #07-01, Matrix, Singapore 138671 ###### Abstract A time machine that sends information back to the past may, in principle, be built using closed time-like curves. However, the realization of a time machine must be congruent with apparent paradoxes that arise from traveling back in time. Using a simple model to analyze the consequences of time travel, we show that several paradoxes, including the grandfather paradox and Deutsch’s unproven theorem paradox, are precluded by basic axioms of probability. However, our model does not prohibit traveling back in time to affect past events in a self-consistent manner. ## I Background The possibility of building a time machine has been proposed by many authors friedman ; gott ; godel ; bonnor ; morris ; politzer ; boulware ; hartie ; politzer2 ; deutsch ; novikov ; lloyd ; pegg ; svetlichny . Two common approaches are through closed time-like curves (CTC) friedman ; gott ; godel ; bonnor ; morris ; politzer ; boulware ; hartie ; politzer2 and quantum phenomena deutsch ; lloyd ; pegg ; svetlichny . Although the general theory of relativity allows for CTCs, it is not clear if the laws of physics permit their existence hawking ; carroll ; deser ; carroll2 . Hence the possibility of traveling back to the distant past remains an open question. Paradoxical thought experiments have been devised to suggest that traveling back in time may lead to violations of causality, and hence is not possible. The most famous paradox is the grandfather paradox, in which an agent travels back in time to kill his grandfather before his father was conceived. In this case, the agent will not exist at the current time and hence cannot travel back in time to kill his grandfather. An alternative version of the grandfather paradox is autoinfanticide, where an agent travels back in time to kill himself as an infant. This paradox plays a central role in the argument against traveling back in time. Another paradox is the Deutsch’s unproven theorem paradox lloyd , in which an agent travels back in time to reveal the proof of a mathematical theorem. The proof is then recorded in a document that the agent reads in future time. Another version of Deutsch’s unproven paradox is what we call the chicken-and-egg paradox. A hen travels back in time to lay an egg. The egg hatches into the hen herself. Without the egg, the hen would not exist but without the hen traveling back in time, the egg would not be laid. In this paper, a simple model is used in an attempt to solve time travel paradoxes and help set the logical foundations of traveling back in time. Our approach is quite different from approaches that focus on how a time machine can be built (in principle) lloyd . We suppose that a time machine can be built and then analyze what could be possible (or impossible) in time travel. We use a simple directed cyclic graph to explain causal relationships in different scenarios of time travel. Our conclusion is that, assuming traveling back in time is feasible, an agent who travels back in time is unable to kill himself although he may be able to alter the past in other ways; in a self- consistent manner. The self-consistency principle was proposed by Wheeler and Feynman feynman , Novikov et al novikov and Lloyd et al lloyd . It states that traveling back in time may be possible, but it cannot happen in a way that violates causality. Causality in this case includes events that happen in the future affecting the past. This principle precludes time travel paradoxes but does not forbid traveling back in time. Due to space limitations, the reader is referred to feynman ; novikov ; lloyd for detailed discussion of the self- consistency principle. ## II Model Our model can be considered as a simple case of graphical models. Graphical models have been extensively studied and are applicable in many fields such as in econometric models, social sciences, artificial intelligence and even in medical studies. Publications on graphical models are so numerous that we can only provide a non-exhaustive list richardson1997 ; richardson1996 ; schmidt ; spirtes ; lacerda ; pearlbk ; rebane1987 ; lauritzen ; lauritzen1 ; salmon ; morgan ; spirtesbk ; cooper1991 . Although directed acyclic graphs have been at the center stage of graphical models, directed cyclic graphical models have also received significant attention schmidt ; richardson1997 ; richardson1996 ; spirtes ; lacerda . Two important components in graphical models are intervention and the do calculus. The theory of graphical models has few constraints built in on what is physically possible. This leaves the theory very general. $\sigma_{1}\rightarrow\sigma_{2}\rightarrow\sigma_{3}\rightarrow\sigma_{4}\cdots\rightarrow\sigma_{i}\cdots\rightarrow\sigma_{k}\cdots\rightarrow\sigma_{n}$ Figure 1: A simple graphical model for a Markov Chain We use a simple directed cyclic graph to study traveling back in time. First, we build constraints into our model as follows. Consider physical states evolving on a timeline as shown in Fig. 1. The graph is a one dimensional chain, and branching is excluded. Traveling back in time introduces a loop as in Fig. 2. We do not include intervention and do calculus because this enables us to simplify our analysis, while capturing the important physics for a closed system. $\sigma_{1}\rightarrow\sigma_{2}\rightarrow\sigma_{3}\rightarrow\sigma_{4}\cdots\rightarrow\sigma_{i}\cdots\rightarrow\sigma_{k}\cdots\rightarrow\sigma_{n}$ Figure 2: A simple cyclic graph to model traveling back in time from $t=k$ to $t=i$. At each time $t$, the state of the system $\sigma_{t}$ is a random variable. Time is also discretized and the arrows connect events at neighboring times $\sigma_{t}\rightarrow\sigma_{t+1}$. The probability of transition from $\sigma_{t}$ to $\sigma_{t+1}$ is given by $T_{t+1}(\sigma_{t+1}|\sigma_{t})$. In this case, the conditional probabilities can be interpreted as a transition matrix, and the graph as a Markov Chain. The following assumptions are used based on physical considerations: 1. 1. The statistical time flows in the same direction as the physical time. 2. 2. Local normalization constraint is enforced, i.e. $\sum_{\sigma_{t+1}}T_{t+1}(\sigma_{t+1}|\sigma_{t})=1$. Given that the system is in a state $\sigma_{t}$ at time $t$, the system has to take on a state at $t+1$. In general, we can condition on more than one variable, e.g. $T_{t+1}(\sigma_{t+1}|\sigma_{i},\sigma_{j},\cdots)$, then the local normalization condition is $\sum_{\sigma_{t+1}}T_{t+1}(\sigma_{t+1}|\sigma_{i},\sigma_{j},\cdots)=1$. 3. 3. Basic probability axioms are satisfied. Let $A_{i}$ be a set of states and $P(A_{i})$ be its probability measure, then, $0\leq P(A_{i})\leq 1$ (1) $P(\Omega)=1,\mbox{\hskip 14.22636pt}$ (2) $P(A_{i}\cup A_{j})=P(A_{i})+P(A_{j}),$ (3) $\Omega$ is the set of all possible states and $A_{i}$ and $A_{j}$ are mutually exclusive. Clearly, for discrete events if $\sigma_{i}\in\Omega$ and $\sigma_{j}\in\Omega$, $\sigma_{i}\neq\sigma_{j}$, then $P(\sigma_{i}\cup\sigma_{j})=P(\sigma_{i})+P(\sigma_{j})$. Here, we use a shorthand notation $\sigma_{i}\equiv\\{\sigma_{i}\\}$. A sequence of states $\pi_{n}$ is shown in Fig. 1. If the set of all possible states is given by $\Omega$, then the set of all possible sequences is given by $\mathbf{\Pi}=\Omega^{n}$. The probability of obtaining $\pi_{n}$ is, $P_{mc}(\pi_{n})=p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})T_{3}(\sigma_{3}|\sigma_{2})\cdots T_{n}(\sigma_{n}|\sigma_{n-1})$ (4) $p(\sigma_{1})$ is the probability of sampling the initial state $\sigma_{1}$. The conditional probabilities encode the physics of how the system evolve from state to state. It can be shown that for $P_{mc}(\pi_{n})$, basic axioms of probabilities hold. In the case of traveling back in time, the causal relationship has an arrow that loops back into the past (Fig. 2). To model traveling back in time, we condition on two states instead of one, $\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})$ where $\sigma_{k}$ is an event in the future with respect to time $i$. In this case, $P(\pi_{n})=p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})\cdots T_{n}(\sigma_{n}|\sigma_{n-1})$ (5) All the conditional probabilities $T_{j}(\sigma_{j}|\sigma_{j-1})$ are the same as in Eq. (4) except for $\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})$. Making such a generalization is non-trivial because we need to check that the basic axioms of probabilities continue to hold. At this point, we would like to emphasize some key points that are important in this paper, 1. 1. Time travel consists of sending a signal back to the past. The signal causes an effect only at one time point $t=i$ as in Fig. 2. The signal could contain a set of instructions to carry out some tasks or be an agent that travels back in time. 2. 2. The conditional probabilities $T_{j}$, $j=1,2,\cdots$, $j\neq i$ in Eq. (4) are determined by the physics of how the system evolves forward in time. 3. 3. The term $\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})$ is special as it is the only term in Eq. (21) that encodes the effects of traveling back in time. 4. 4. Our framework is probabilistic, in which many sequences of states can happen with non-zero probability, in contrast to a deterministic view where only one sequence is possible. Given any sequence $\pi_{n}$, its probability of occurrence can be calculated using Eq. (21). 5. 5. A paradox be represented by many different sequences of states. Our objective is to show that either all these sequences happen with zero probability, or they result in violation of the basic axioms of probability. Consider $\hat{T}_{i}$ to be a function of three discrete variables, $\sigma_{i-1},\sigma_{i}$ and $\sigma_{k}$. This function has to satisfy, $0\leq\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})\leq 1$ (6) $\sum_{\\{\pi_{n}\\}}P(\pi_{n})=1$ (7) $\sum_{\sigma_{i}}\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})=1$ (8) The first two conditions are analogous to Eq. (1) and (2). The last condition is the local normalization condition. Eq. (7) can be reduced to, $\sum_{\sigma_{i},\sigma_{k}}\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},\sigma_{k})V(\sigma_{k}|\sigma_{i})=1$ (9) $V(\sigma_{k}|\sigma_{i})$ is the conditional probability of $\sigma_{k}$ given $\sigma_{i}$ summed over all possible intermediate states $\sigma_{i+1}\cdots\sigma_{k-1}$. Detailed derivation of Eq. (9) is given in Appendix A. This is an important equation. We will use this equation together with Eq. (6) and (8) to show that the grandfather paradox, Deutsch’s unproven theorem paradox and chicken-and-egg paradox have to be precluded in time travel. ### II.1 Two-state system For a two-state system, $\sigma$ takes the values $\\{0,1\\}$. Using Eq. (9) and (8) and summing over four combinations $\sigma_{i+1},\sigma_{k}\in\\{0,1\\}$, we obtain, $[\hat{T}_{i}(0|\tilde{\sigma}_{i-1},1)-\hat{T}_{i}(0|\tilde{\sigma}_{i-1},0)][V(1|0)-V(1|1)]=0$ (10) We must have $V(1|0)=V(1|1)$ or $\hat{T}_{i}(0|\tilde{\sigma}_{i},1)=\hat{T}_{i}(0|\tilde{\sigma}_{i},0)$. For the case when $V(1|0)\neq V(1|1)$, the transition matrix $\hat{T}_{i}$ does not depend on $\sigma_{k}$. In this case, the backward loop in Fig. 2 has no effect. We can’t change the probability distribution of the past. For the case $V(1|0)=V(1|1)$, we could have $\hat{T}_{i}(0|\tilde{\sigma}_{i-1},1)\neq\hat{T}_{i}(0|\tilde{\sigma}_{i-1},0)$ and the transition probabilities at $t=i$ could be affected by a signal from future time ($t=k$). ### II.2 Grandfather paradox in a two-state system The grandfather paradox can be used to illustrate the physical implications of Eq. (10). The basic assumptions we will use are (i) resurrection is impossible, and (ii) basic axioms of probabilities must be satisfied. Consider an agent sending a signal back in time to kill himself. Let us denote the dead state as $\sigma=0$ and alive state as $\sigma=1$. No resurrection implies that $V$ is of the form, $V=\left(\begin{array}[]{cc}1&\beta^{*}\\\ 0&\beta\end{array}\right),$ $\beta^{*}=1-\beta$. Let $\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},1)=S(\sigma_{i}|\tilde{\sigma}_{i-1})$ be the transition probabilities for the scenario in which the agent sends a signal from the future to kill himself. Let $\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},0)=N(\sigma_{i}|\tilde{\sigma}_{i-1})$ be the transition probabilities for the sequences of events the agent is dead at $t=k$ and hence no signal is sent from the future to kill himself. Hence $S$ (the “killing” matrix) and $N$ are of the form, $S=\left(\begin{array}[]{cc}1&1\\\ 0&0\end{array}\right)\mbox{\hskip 17.07182pt}N=\left(\begin{array}[]{cc}1&b^{*}\\\ 0&b\end{array}\right)$ (11) $b^{*}=1-b$ is the probability of dying at $t=i$. Substituting values of $N$, $S$ and $V$ into Eq. (10), we obtain $[1-b^{*}]\beta=0$. Either $b^{*}=1$ or $\beta=0$. When $b^{*}=1$ then $N=S$, the agent dies at $t=i$ with probability 1. If $\beta=0$, the agent dies sometime between $t=i$ and $t=k$ with probability 1. In either case, the scenario in which the agent is alive at $t=k$ and thus able to send the signal occurs with zero probability. Note that we are analyzing probabilities rather than specific events. The conclusion comes about because resurrection is impossible ($V(1|0)=0$). Suppose resurrection is possible, $V(1|0)=\alpha^{*}<1$, the paradox goes away when $\alpha^{*}=\beta$. Intuitively, if we allow resurrection, the agent could send a signal back in time from $t=k$ to kill himself at $t=i<k$. Between the time $t=i$ and $t=k$, the agent is resurrected and hence could again send the signal at $t=k$. There is no contradiction in this case. Another way to resolve the paradox is to relax the assumption that the agent always succeeds to kill himself. In this case, the matrix $S$ is $\left(\begin{array}[]{cc}1&\lambda^{*}\\\ 0&\lambda\end{array}\right)$, $\lambda>0$. Eq. (10) gives, $\beta(\lambda-b)=0$. If $\beta=0$, then the agent dies sometime between $t=i$ and $t=k$. If $\lambda=b$ then $S=N$, the signal from the future could not change the transition probability at $t=i$. The agent cannot change his own fate by sending a signal to the past. ### II.3 Deutsch’s unproven theorem paradox An agent sends a signal containing the proof of a mathematical theorem back in time. The signal is encoded in a document that the agent reads in future time. Denote the existence of the proof as $\sigma=0$ and absence of the proof as $\sigma=1$. A general form of $V$ is, $V=\left(\begin{array}[]{cc}\alpha&\beta^{*}\\\ \alpha^{*}&\beta\end{array}\right),$ $\alpha^{*}=1-\alpha$, $\beta^{*}=1-\beta$. The basic assumptions we use are (i) the transition from $\sigma=1$ to $\sigma=0$ (transition of absence of proof to existence of proof) happens solely through the signal traveling back in time, and (ii) the transition from $\sigma=0$ to $\sigma=1$ happens with zero probability (once the proof is obtained, it never gets lost). Hence $\beta=1$ and $\alpha^{*}=0$. The transition probabilities are $\hat{T}_{i}(\sigma_{i}=0|\tilde{\sigma}_{i-1}=1,\sigma_{k}=1)=0$ representing no signal sent if proof does not exist at $t=k$ ($\sigma_{k}=1$). $\hat{T}_{i}(\sigma_{i}=0|\tilde{\sigma}_{i-1}=1,\sigma_{k}=0)=1$ represents a signal being sent when the proof exists at $t=k$. These basic assumptions contradict with Eq. (10), $[\hat{T}_{i}(0|1,1)-\hat{T}_{i}(0|1,0)](\alpha^{*}-\beta)=(0-1)(0-1)\neq 0$. Hence the assumptions are false and Deutsch’s unproven theorem paradox is precluded. The paradox can be resolved if we relax the assumptions. Suppose we allow the possibility that the proof can get lost ($\alpha^{*}\geq 0$) and that the proof can be derived by some brilliant mathematician $\beta\leq 1$. Then Eq. (10) can be satisfied if $\alpha^{*}=\beta$. There is no paradox here because the proof can be sent back in time and subsequently be lost. It can be re- derived again and be sent back to the past. $N(1|\tilde{\sigma}_{i})-S(1|\tilde{\sigma}_{i})$$N(2|\tilde{\sigma}_{i})-S(2|\tilde{\sigma}_{i})$$-a/b$ Figure 3: Shaded region shows the possible values of $N(1|\tilde{\sigma}_{i},0)-S(1|\tilde{\sigma}_{i})$, (x-axis) and $N(2|\tilde{\sigma}_{i},0)-S(2|\tilde{\sigma}_{i})$, (y-axis). ### II.4 Three-state system For a three-state system, $\sigma$ takes the values $\\{0,1,2\\}$. For simplicity, let $\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},0)=N(\sigma_{i}|\tilde{\sigma}_{i-1})$ and $\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},1)=\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},2)=S(\sigma_{i}|\tilde{\sigma}_{i-1})$. Using Eq. (8) and (9), $\displaystyle[N(1|\tilde{\sigma}_{i-1})-S(1|\tilde{\sigma}_{i-1})][V(0|1)-V(0|0)]+\mbox{}$ (12) $\displaystyle[N(2|\tilde{\sigma}_{i-1})-S(2|\tilde{\sigma}_{i-1})][V(0|2)-V(0|0)]=$ $\displaystyle 0$ this is an equation of the form $xa+yb=0$ given $a=[V(0|1)-V(0|0)]$ and $b=[V(0|2)-V(0|0)]$, $x=[N(1|\tilde{\sigma}_{i-1})-S(1|\tilde{\sigma}_{i-1})]$ and $y=[N(2|\tilde{\sigma}_{i-1})-S(2|\tilde{\sigma}_{i-1})]$ can be solved. There are in general infinitely many solutions. From Eq. (6), the range of $[N(1|\tilde{\sigma}_{i-1})-S(1|\tilde{\sigma}_{i-1})]$ and $[N(2|\tilde{\sigma}_{i-1})-S(2|\tilde{\sigma}_{i-1})]$ is bounded by the shaded region in Fig. 3. Given $a$ and $b$, the set of solutions for $x$ and $y$ contains all the points on the line shown in Fig. 3. The slope of the line is given by $-a/b$. $N\neq S$ implies that transition to the state $\sigma_{i}$ depends on future state $\sigma_{k}$, that is, signals from the future can affect the probability distribution of the past. ### II.5 The grandfather paradox in a three-state system Consider the three states represent healthy ($\sigma=2$), sick ($\sigma=1$) and dead ($\sigma=0$). First, we lay down our assumptions, 1. 1. Assume resurrection is impossible so that transition from $\sigma=0$ to $\sigma\neq 0$ happens with zero probability. Then the matrix $V$ is of the form, $V=\left(\begin{array}[]{ccc}1&\alpha_{0}&\beta_{0}\\\ 0&\alpha_{1}&\beta_{1}\\\ 0&\alpha_{2}&\beta_{2}\end{array}\right)$ (13) with $\alpha_{0}+\alpha_{1}+\alpha_{2}=1$ and $\beta_{0}+\beta_{1}+\beta_{2}=1$. 2. 2. The agent is able to send a signal back in time to kill himself only if he is not dead at $t=k$. $\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},1)$ and $\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},2)$ are the conditional probabilities that the agent is alive and sends a signal back in time to kill himself. Let, $\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},1)=\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},2)=S(\sigma_{i}|\tilde{\sigma}_{i-1})$. $\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},0)$ is the conditional probability that the agent is dead at $t=k$ and can not send a signal back in time to kill himself. Let $\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},0)=N(\sigma_{i}|\tilde{\sigma}_{i-1})$. Hence $S$ (the “killing” matrix) $N$ are, $S=\left(\begin{array}[]{ccc}1&1&1\\\ 0&0&0\\\ 0&0&0\end{array}\right)\mbox{\hskip 8.5359pt}N=\left(\begin{array}[]{ccc}1&a_{0}&b_{0}\\\ 0&a_{1}&b_{1}\\\ 0&a_{2}&b_{2}\end{array}\right)$ (14) We have from Eq. (12), $\displaystyle a_{1}(1-\alpha_{0})+a_{2}(1-\beta_{0})$ $\displaystyle=$ $\displaystyle 0$ (15) $\displaystyle b_{1}(1-\alpha_{0})+b_{2}(1-\beta_{0})$ $\displaystyle=$ $\displaystyle 0$ There are four cases in which Eq. (15) is satisfied. 1. 1. $\alpha_{0}=1$ and $\beta_{0}=1$. Then $V=S$ which means the agent is dead at $t=k$ with probability 1 (recall that $S$ is the killing matrix). 2. 2. $\alpha_{0}=1$ and $\beta_{0}<1$. To satisfy Eq. (15), $a_{2}=b_{2}=0$. In this case the agent is dead at $t=k$ with probability 1 (see Appendix B for the proof). 3. 3. $\alpha_{0}<1$ and $\beta_{0}=1$. To satisfy Eq. (15), $a_{1}=b_{1}=0$. In this case the agent is dead at $t=k$ with probability 1 (see Appendix B for the proof). 4. 4. $\alpha_{0}<1$ and $\beta_{0}<1$. Then $a_{1}=a_{2}=b_{1}=b_{2}=0$ and $N=S$ which means the agent is dead at $t=i$ with probability 1. In all cases, the agent is dead with probability 1 at $t=k$ and hence never has a chance to send a signal back in time to kill himself. Suppose $S$ is not the killing matrix (Eq. (14)) or resurrection is possible, then this argument does not hold, and the agent is able to alter his fate by changing the probability of being healthy, sick or dead. ### II.6 The chicken-and-egg paradox Consider the chicken-and-egg paradox in which at time $t=k$, a hen travels back in time to $t=i$ to lay an egg. The egg hatches into the hen herself. At this time point, there are two copies of the hen, the older self and the younger self (the chick). As both copies travel to time $t=k$, the chick grow older and travels back in time to lay the egg. This paradox seems “self- consistent” in the sense that there is no contradiction in existence of the hen and chick from one time point to another. However the problem is the hen seems to pop out from nowhere. There are three possible states, hen and chick ($\sigma=0$), hen only ($\sigma=1$) and no hen and no chick ($\sigma=2$). We exclude the state of chick only, otherwise we would need four states. There are no hen and no chick initially, hence $\tilde{\sigma}_{i-1}=2$. Let $\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},1)=\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},2)=N(\sigma_{i}|\tilde{\sigma}_{i-1})$. This is the case when no chick travels back in time and hence there remains no hen and no chick at $t=i$. Let $\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},0)=S(\sigma_{i}|\tilde{\sigma}_{i-1})$, the chick travels back in time from $t=k$ to $t=i$. The matrices $S$ and $N$ are, $S=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&1\\\ 0&0&0\end{array}\right)\mbox{\hskip 11.38092pt}N=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right)$ (16) The matrix $V$ is of the form, $V=\left(\begin{array}[]{ccc}\alpha_{0}&\beta_{0}&0\\\ \alpha_{1}&\beta_{1}&0\\\ \alpha_{2}&\beta_{2}&1\end{array}\right)$ (17) The first two columns are general expressions with $\sum_{j=0}^{2}\alpha_{j}=1$ and $\sum_{j=0}^{2}\beta_{j}=1$. The last column is $(0,0,1)^{T}$ because when there is no hen and no chick at time $t=i$, then there will be no hen and no chick at $t=k$. Now consider the probability, $P(\tilde{\sigma}_{i-1},\sigma_{i},\sigma_{k})=p(\tilde{\sigma}_{i-1})\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},\sigma_{k})V(\sigma_{k}|\sigma_{i})$ (18) $p(\tilde{\sigma}_{i-1})$ is the probability of sampling the state $\tilde{\sigma}_{i-1}$. Since $\tilde{\sigma}_{i-1}=2$, $p(\tilde{\sigma}_{i-1})=\delta_{\tilde{\sigma}_{i-1},2}$. We remind the reader that the probability distribution $V$ is the sum of probabilities over all possible intermediate sequences. The chicken-and-egg paradox requires both hen and chick to be present at $t=k$ ($\sigma_{k}=0$) and the chick to appear at $t=i$ ($\sigma_{i}=1$), all intermediate states can take arbitrary values. Reading off entries from matrices $S$ and $V$, $P(\tilde{\sigma}_{i-1}=2,\sigma_{i}=1,\sigma_{k}=0)=\beta_{0}$ (19) Using Eq. (12) we can calculate what $\beta_{0}$ should be, $\displaystyle[1-0](\beta_{0}-\alpha_{0})-[0-1]\alpha_{0}$ $\displaystyle=$ $\displaystyle 0$ (20) $\displaystyle\Rightarrow\beta_{0}$ $\displaystyle=$ $\displaystyle 0$ The sum of probabilities of all possible sequences of states that represent the chicken-and-egg paradox equals zero. Therefore the chicken-and-egg event happens with zero probability. ## III Discussion We have shown, using a graphical model with a loop back into the past, that the grandfather paradox, Deutsch’s unproven theorem paradox and the chicken- and-egg paradox are precluded in time travel. We have also demonstrated that changing the probability distributions of the past is possible when no contradicting events are present. For the paradoxes we discussed in this paper, we gave scenarios in which the paradoxes are resolved. Our analysis is based on isolated two- and three-state systems. For future work, it would be useful to generalize our formalism to arbitrary systems. Lastly, in cases when the causal relationship between events at different times are very complex, the existence of time travel paradoxes in these cases may be very subtle. We hope that our mathematical framework can be used to uncover new time travel paradoxes, especially those that are embedded in complex interactions of events and are not obvious. The author would like to thank Mui Leng Seow and Ivana Mihalek for proofreading this article. ## IV Appendix A: Derivation of Eq. (9) Probability of a sequence $\pi_{n}$ is given by, $P(\pi_{n})=p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})\cdots T_{n}(\sigma_{n}|\sigma_{n-1})$ (21) Summing over all sequences, $\displaystyle\sum_{\\{\pi_{n}\\}}P($ $\displaystyle\pi_{n})=$ (22) $\displaystyle\sum_{\sigma_{1},\sigma_{2},\cdots,\sigma_{n}}$ $\displaystyle p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})\cdots T_{n}(\sigma_{n}|\sigma_{n-1})$ Since $\sum_{\sigma_{j}}T_{j}(\sigma_{j}|\sigma_{j-1})=1\forall j$, summation can be evaluated recursively between $\sigma_{k+1}$ and $\sigma_{n}$. That is, $\sum_{\sigma_{k+1},\cdots\sigma_{n}}T_{k+1}(\sigma_{k+1}|\sigma_{k})\cdots T_{n}(\sigma_{n}|\sigma_{n-1})=1$ (23) Next define, $U(\sigma_{i-1})=\sum_{\sigma_{1},\cdots\sigma_{i-2}}p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots T_{i-1}(\sigma_{i-1}|\sigma_{i-2})$ (24) $V(\sigma_{k}|\sigma_{i})=\sum_{\sigma_{i+1},\cdots\sigma_{k-1}}T_{i+1}(\sigma_{i+1}|\sigma_{i})\cdots T_{k}(\sigma_{k}|\sigma_{k-1})$ (25) Then Eq. (22) becomes, $\sum_{\\{\pi_{n}\\}}P(\pi_{n})=\sum_{\sigma_{i-1},\sigma_{i},\sigma_{k}}U(\sigma_{i-1})\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},\sigma_{k})V(\sigma_{k}|\sigma_{i})$ (26) The objective is to find the conditions in which $\sum_{\\{\pi_{n}\\}}P(\pi_{n})=1$. $U(\sigma_{i-1})$ is the probability of sampling the state $\sigma_{i-1}$, it depends on the conditional probabilities $T_{j}$, $j\leq i-1$ and the initial condition $p(\sigma_{1})$. We therefore have the freedom to choose $U$ for example, by choosing different initial conditions. Holding $T$ and $V$ fixed, we require $\sum P(\pi_{n})=1$ for different choices of $U$, in which we arrive at, $\sum_{\sigma_{i},\sigma_{k}}\hat{T}_{i}(\sigma_{i}|\tilde{\sigma}_{i-1},\sigma_{k})V(\sigma_{k}|\sigma_{i})=1$ (27) ## V Appendix B: The grandfather paradox in a three-state system We present the proof that for the grandfather paradox in a three-state system, the probability that the agent is dead at $t=k$ is one. We consider cases II and III in which Eq. (15) is satisfied, ### V.1 Case II: $\alpha_{0}=1$ and $\beta_{0}<1$ In this case, $a_{2}=b_{2}=0$ and, $V=\left(\begin{array}[]{ccc}1&1&\beta_{0}\\\ 0&0&\beta_{1}\\\ 0&0&\beta_{2}\end{array}\right)$ (28) $N=\left(\begin{array}[]{ccc}1&a_{0}&b_{0}\\\ 0&a_{1}&b_{1}\\\ 0&0&0\end{array}\right)$ (29) We calculate the probability that the agent is dead, $\displaystyle P(\sigma_{k}=0)$ $\displaystyle=$ $\displaystyle\sum_{\sigma_{1},\cdots,\sigma_{k-1}}p(\sigma_{1})T_{2}(\sigma_{2}|\sigma_{1})\cdots$ (30) $\displaystyle=$ $\displaystyle\sum_{\sigma_{i-1},\sigma_{i}}U(\sigma_{i-1})\hat{T}_{i}(\sigma_{i}|\sigma_{i-1},0)V(0|\sigma_{i})$ $\displaystyle=$ $\displaystyle\sum_{\sigma_{i-1},\sigma_{i}}U(\sigma_{i-1})N(\sigma_{i}|\sigma_{i-1})V(0|\sigma_{i})$ Reading off the entries of matrices $V$ and $N$ in Eq. (28) and (29), we get $\sum_{\sigma_{i}}N(\sigma_{i}|\sigma_{i-1})V(0|\sigma_{i})=1$ for all $\sigma_{i-1}$. Hence $P(\sigma_{k}=0)=1$. ### V.2 Case III: $\alpha_{0}<1$ and $\beta_{0}=1$ In this case, $a_{1}=b_{1}=0$ and, $V=\left(\begin{array}[]{ccc}1&\alpha_{0}&1\\\ 0&\alpha_{1}&0\\\ 0&\alpha_{2}&0\end{array}\right)$ (31) $N=\left(\begin{array}[]{ccc}1&a_{0}&b_{0}\\\ 0&0&0\\\ 0&a_{2}&b_{2}\\\ \end{array}\right)$ (32) We calculate the probability that the agent is dead, using Eq. (30) and reading off the entries of matrices $V$ and $N$ in Eq. (31) and (32), we get $\sum_{\sigma_{i}}N(\sigma_{i}|\sigma_{i-1})V(0|\sigma_{i})=1$ for all $\sigma_{i-1}$. Hence $P(\sigma_{k}=0)=1$. ## References * (1) J. Friedman, M. S. Morris, I. D. Novikov, F. Echeverria, G. Klinkhammer, K. S. Thorne, U. Yurtsever, Phys. Rev. D 42, 1915 (1990) * (2) J. R. Gott, Phys. Rev. Lett. 66, 1126 (1991) * (3) K. Godel, Rev. Mod. Phys. 21, 447-450 (1949) * (4) W. B. Bonnor, J. Phys. A 13, 2121 (1980) * (5) M. S. Morris, K. S. Thorne, U. Yurtsever, Phys. Rev. Lett. 61, 1446-1449 (1988) * (6) H. D. Politzer, Phys. Rev. D 49, 3981 (1994) * (7) D. G. Boulware, Phys. Rev. D 46, 4421-4441 (1992) * (8) J. B. Hartle, Phys. Rev. D 49, 6543-6555 (1994) * (9) H. D. Politzer, Phys. Rev. D 46, 4470-4476 (1992) * (10) D. Deutsch, Phys. Rev. D 44, 3197-3217 (1991) * (11) S. Lloyd, L. Maccone et al, arXiv:1005.2219, arXiv:1007.2615 * (12) D. T. Pegg, arXiv:quant-ph/0506141v1. * (13) G. Svetlichny, arXiv:0902.4898v1. * (14) I. D. Novikov, Phys. Rev. D 45, 1989 (1992) * (15) S. M. Carroll, E. Farhi, A. H. Guth, Phys. Rev. Lett. 68, 263 (1992) * (16) S. W. Hawking, Phys. Rev. D 46, 603 (1992) * (17) S. Deser, R. Jackiw, G. ’t Hooft, Phys. Rev. Lett. 68, 267-269 (1992) * (18) S. M. Carroll, E. Farhi, A. H. Guth, K. D. Olum, Phys. Rev. D 50, 6190-6206 (1994) * (19) J. A. Wheeler, R. P. Feynman, Rev. Mod. Phys. 21, 425 (1949) * (20) J. Pearl, Causality (Cambridge University Press 2000) * (21) P. Sprites, C. Glymour, R. Scheines, Causation, Prediction and Search (MIT Press, Cambridge 2000) * (22) W. C. Salmon, Scientific Explanation and the Causal Structure of the World (Princeton University Press, Princeton, 1984). * (23) S. L. Lauritzen, Graphical Models (Oxford University Press, New York, 1996). * (24) S. L. Morgan and C. Winship, Counterfactuals and Causal Inference: Methods and Principles for Social Research (Cambridge University Press, Cambridge, England, 2007). * (25) G. Rebane, J. Pearl, Proc. Uncertainty in Artificial Intelligence Conf. 222-228 (1987) * (26) S. Lauritzen, In Complex Stochastic Systems (eds. Barndorff-Nielsen, Cox and Kluppelberg) (63-107 CRC Press 2000) * (27) G. F. Cooper, E. Herskovits, Proc. Uncertainty in Artificial Intelligence Conf. 86-94 (1991) * (28) T. Richardson, Int. J. Approx. Reasoning 17, 107-162 (1997) * (29) T. Richardson, Proc. Uncertainty in Artificial Intelligence Conf. 462-469 (1996) * (30) M. Schmidt, K. Murphy Proc. Uncertainty in Artificial Intelligence Conf. (2009) * (31) P. Spirtes Proc. Uncertainty in Artificial Intelligence Conf. 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arxiv-papers
2011-01-10T19:13:01
2024-09-04T02:49:16.281471
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hwee Kuan Lee", "submitter": "Hwee Kuan Lee", "url": "https://arxiv.org/abs/1101.1927" }
1101.1942
arxiv-papers
2011-01-10T20:27:45
2024-09-04T02:49:16.288116
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Mehdi Rafie-Rad", "submitter": "Mehdi Rafie-Rad", "url": "https://arxiv.org/abs/1101.1942" }
1101.2094
On Integrability of Type 0A Matrix model in the presence of D brane Chandrima Paul 111mail:chandrima@phy.iitb.ac.in _Department of Physics, Indian Institute of Technology Bombay_, _Mumbai 400 076, India _ We consider type 0A matrix model in the presence of spacelike D brane which is localized in matter direction at any arbitrary point. In string theory, the boundary state which in matrix model corresponds to the Laplace transform of the macroscopic loop operator, is known to obey the operator constraints corresponding to open string boundary condition. When we analyze MQM as well as the respective collective field theory and compare it with dual string theory it appears that consistency of the theory requires a condition equivalent to a constraint on the matter part that needed to be imposed in the matrix model. We identified this condition and observed that this has only effect into constraining the macroscopic loop operator so that it projects the Hilbert space generated by the operator to its physical sector at the point of insertion while keeping the bulk matrix model remains unaffected, thereby describing a situation parallel to string theory. We analyzed the theory with uncompactified time and have shown explicitly that the matrix model predictions are in good agreement with the relevant string theory. Next we considered the theory with compactified time, analyzed MQM on a circle in the presence of D brane. We evaluated the partition function along with the constrained macroscopic loop operator in the grand canonical ensemble and showed the free energy corresponds to that of a deformed Fermi surface. We have compared the matrix model features with that of the relevant string theory. We have also shown that the path integral in the presence of D brane can be expressed as the Fredholm determinant. We have studied the fermionic scattering in a semiclassical regime. Finally we considered the compactified theory in the presence of the D brane with tachyonic background. From the collective field theory analysis we have predicted the right structure of the theory in the presence of D brane. We evaluated the free energy in the grand canonical ensemble. We have shown the integrable structure of the respective partition function and it corresponds to the tau function of Toda hierarchy. We have also analyzed the dispersionless limit. ###### Contents 1. 1 Introduction 2. 2 Type 0A MQM in the presence of the D-brane 1. 2.1 Type 0A MQM 2. 2.2 Type 0A MQM in the presence of the D brane : The constraint from dual string theory 3. 2.3 String theoretical interpretation 3. 3 Type 0A MQM on a circle in the presence of D brane 1. 3.1 Evaluation of the free energy of Type 0A matrix model on a circle 2. 3.2 Free energy of Type 0A matrix model on a circle with D brane 3. 3.3 Evaluation of the thermal partition function 4. 3.4 String theoretical interpretation 4. 4 Fermionic scattering and semiclassical analysis 5. 5 Perturbation by momentum modes 1. 5.1 Collective field theory analysis 2. 5.2 Lax Formalism 3. 5.3 String theory on a circle with the D brane in the presence of tachyonic background 4. 5.4 Lax formalism for Type 0A MQM in the presence of D brane 5. 5.5 Representation in terms of a bosonic field 6. 5.6 The dispersionless (quasiclassical) limit 6. 6 Conclusion 7. A Appendix 1. A.1 Orthogonality Condition 2. A.2 Biorthogonality Relation ## 1 Introduction The two dimensional string theory (see e.g. [1], [2], [3] for reviews) is a very instructive model when we would like to understand the nature of string theory as a complete theory of quantum gravity. This theory has a powerful dual description of $c=1$ matrix model defined by the simple quantum mechanics of a Hermitian matrix $\Phi$ with the inverse harmonic oscillator potential $U(\Phi)=-\Phi^{2}$ after the double scaling limit.Matrix model is successfully used to describe 2D string theory in the simplest linear dilaton background as well as to incorporate perturbations. In last decade the $c=1$ matrix quantum mechanics has received lots of attention because of its new interpretation as the decoupled world volume theory of unstable D0-branes [4, 6, 5]. The matrix model dual to type 0 string theories were also proposed in [8, 9]. In particular, the matrix model dual of the two dimensional type 0 string gives a non-perturbatively well-defined formulation. For example, the type 0B model is defined by the hermitian matrix model with two Fermi surfaces. The type 0B matrix quantum mechanics (MQM) describes open string tachyons living on the unstable D0-branes, whereas the type 0A MQM describes tachyonic open strings stretched between stable D0- and anti-D0-branes. Upon compactification on Euclidean time, these two matrix models are conjectured to be T-dual to each other. The exact agreement in free energy was found in [9]. Matrix model dual of type 0 string in the flux background was explored in [29, 30]. However, unlike $c=1$ matrix model which can be derived from discretizing the Polyakov action on the string world sheet, such a derivation is not known for type 0 matrix models. An attempt was made in [13] to obtain the exact form of the macroscopic loop operator in Type 0 string theory. If we consider the bosonic string partition function $\int D{\phi}DX\,\,exp\left[-\int{d^{2}}z[\,{\frac{1}{4\pi}}({\partial{X}\partial{X}}+{\partial{\phi}\partial{\phi}})+QR{\phi}+{\mu}e^{2b\phi}]-\,\int_{\partial\Sigma}{d\xi}\,\,[{\frac{Qk\phi}{2\pi}}+{\mu_{B}}e^{b\phi}\,]\,\,\right],$ (1.1) the macroscopic loop operator inserts an operator ${W(t,l)\sim\delta\left(\int_{\partial\Sigma}e^{\phi}-l\right)\cdot\delta(X^{0}-t)},$ (1.2) within the path integral [10, 11]. $\langle W(l)\rangle=Z(l)=\int DXD\phi D[\,{\rm ghost}\,]\,\,\delta\left(\int_{\partial\Sigma}e^{\phi}-l\right)\cdot\delta(X^{0}-t)\,f(x,\phi)\,Z(\phi(\sigma),X(\sigma),[{\rm ghost}]),$ (1.3) where f is some wave function for matter ghost and Liouville. The physical meaning of this operator in two dimensional string theory is the presence of a ‘Euclidean D-brane’ localized in the time direction. To be more precise after we take the Laplace transformation $\int d\phi e^{-\mu_{B}e^{\phi}}$, we get a D-brane with the Neumann boundary condition in the Liouville direction and the Dirichlet one in the time direction $\int{\frac{dl}{l}}e^{-\mu_{B}l}\ W_{bos}(t,l)\simeq|B_{(FZZT)}(\mu_{B})\rangle_{\phi}\otimes|D\rangle_{X^{0}}.$ (1.4) when we impose the condition that boundary Liouville term is zero. ${\partial_{n}}{\phi}+{\mu_{B}}{e^{b\phi}}=0$ (1.5) Where ${\partial_{n}}{\phi}$ denotes the Liouville momentum normal to boundary while along the boundary we have ${\partial_{t}}{X^{o}}=0$. Now consider 2D superstring action obtained from extending the bosonic fields to their superspace and expanding the 2D superspace action in terms of the component field, $S={\frac{1}{2\pi}}\int{d^{2}}z[{\delta_{\mu\nu}}({\partial}{X^{\mu}}{\overline{\partial}}{X^{\nu}}+{\psi^{\mu}}{{\overline{\partial}}{\psi^{\nu}}}+{\overline{\psi^{\mu}}}{\partial}{\overline{\psi^{\nu}}})+{\frac{Q}{4}}R{X^{1}}]+2i\mu{b^{2}}\int{d^{2}}z({\psi^{1}}{\overline{\psi^{1}}}+2\pi\mu{e^{\phi}}):e^{\phi}:$ (1.6) We can also consider the macroscopic loop operator which is the superspace analogue of $W_{bos}(t,l)$, inserts the boundary condition on the fermionic coordinate ${\overline{\psi}}(\overline{z})={\eta}{\psi}(z)$ where ${\eta}=\pm 1$ describes the RR and NS NS sector. Laplace transform of this operator inside the string path integral describes the boundary states NS NS and RR sector. However depending on helicity, in each sector we have two types of boundary states ${\epsilon=\pm}$ so that we have four types of macroscopic loop operator given by ${W_{NS}^{+}},{W_{NS}^{-}},{W_{R}^{+}},{W_{R}^{-}}$. The parameter ${\mu_{B}}$ corresponds to the boundary cosmological constant in the boundary state. Indeed we can show this relation [33, 34] by computing one point function on the brane or equally annulus amplitude as shown in [12]. For $c=1$ matrix model the expression of these operators were obtained and its equivalence to string theory is verified in [10, 11, 12]. Author of [13] obtained the expressions of macroscopic loop operator in Type 0B matrix model and also for ${NS}$ sector of Type 0A matrix model which was verified by calculating the one point function. Now once we understand the duality between noncritical string theory in the linear dilaton background and Matrix model, its natural to ask whether we can understand the string theory with nontrivial background which has an obvious realization in matrix model by adding perturbations which survive in the double scaling limit. There are two ways to change the background of string theory: either to consider strings propagating in a non-trivial target space or to introduce the perturbations . In the first case one arrives at a complicated sigma-model. Not many examples are known when such a model turns out to be solvable. Besides, it is extremely difficult to construct a matrix model realization of a general sigma-model since not much known about matrix operators explicitly perturbing the metric of the target space. Thus, we lose the possibility to use the powerful matrix model machinery to tackle our problems. On the other hand, following the second way, we find that the integrability of the theory in the trivial background is preserved by the perturbations. Also when we study the theory in a nontrivial background in most of the cases the target space metric of such backgrounds is curved and often it incorporates the black hole singularities. In the superstring theories, the supersymmetry allows for some interesting nontrivial solutions which are stable and exact. But the string theory on such backgrounds is usually an extremely complicated sigma-model, very difficult even to formulate it explicitly, not to mention studying quantitatively its dynamics. The two-dimensional bosonic string theory as well as Type 0 theory are the rare cases of sigma-model where such a dynamics is integrable, at least for some particular backgrounds, including the dilatonic black hole background. A physically transparent way to study the perturbative (one loop) string theory around such a background is provided by the CFT approach. However once we try to understand higher loops or multipoint correlators, we have to address ourselves to the matrix model approach to the 2D string theory . The 2D string theory has been constructed as the collective field theory [25],[28], in which the only excitation, the massless tachyon, was related to the eigenvalue density of the matrix field. Now consequence of the deformation in eigenvalue density corresponding to deformation in string background at classical limit was studied in [26]. $C=1$ string theory perturbed by tachyonic mode studied in [32]. Vortex perturbation and its equivalence to sine -Liouville theory was studied in [36],[37]. Its shown that partition function is integrable and have Toda structure. Toda structure and Lax formalism in the context of matrix model described in [31]. Many more works in this direction was done in [15, 16, 17, 18, 19, 21, 22, 23]. Now its an interesting question to ask that can we study this nontrivial background in the context of dual matrix model in the presence of D brane which are just the Laplace transform of the macroscopic loop operator as we discussed. Open close duality predicts that partition function must have integrable structure. So we consider Type 0A MQM with simplest macroscopic loop operator, which is the operator in NS sector(as prescribed in [13]), which is localized in time direction and with it we show that partition function indeed have an integrable structure.We obtain the string equation. The plan of this work is in section 2 we are going to consider basic Type 0A MQM in the presence of D brane, with uncompactified time, We are going to introduce a no leakage condition to matrix model which is equivalent to some constraint to boundary state of string theory. We are going to explain its origin as well as its string theoretical interpretation. In section 3 we consider MQM compactified on a circle in the presence of D brane. From path integral approach we are going to show that the partition function can be expressed as Fredholm determinant. We have explicitly evaluated the thermal partition function and have shown that without application of this constraint the partition function diverge. In section 4 we have discussed scattering in semiclassical regime. In section 5 we have considered string theory in the presence of momentum modes and have shown that the partition function in this background have an integrable structure if we apply this constraint. ## 2 Type 0A MQM in the presence of the D-brane ### 2.1 Type 0A MQM Let us start with the MQM of type 0A theory in two dimensions, which is the decoupled world volume theory of (stable) D0 –brane and anti D0–branes. A spacelike D0 –$\overline{D0}$ pair, i.e with Neumann boundary condition in Liouville direction and Dirichlet in matter direction, gives a macroscopic loop observable of the matrix model after Laplace transformation [9]. We are going to consider Type 0A MQM in the presence of this operator and study the relevant physics. Here is a brief review of the Type 0A MQM. In the background with no net D0-brane charges, the matrix model has $U(N)\times U(N)$ gauge symmetry. This is the case we are going to consider. We have the $U(N)\times U(N)$ gauge field $A_{0}$ and bifundamental tachyon $\Phi$, ${A_{o}}=\left(\begin{array}[]{cc}A&0\\\ 0&\tilde{A}\\\ \end{array}\right),$ (2.1) ${\Phi}=\left(\begin{array}[]{cc}0&M\\\ {M^{\dagger}}&0\\\ \end{array}\right).$ (2.2) The action is $\int dtTr\left[{({D_{o}}M)}^{\dagger}{{D_{o}}M}+{\frac{1}{2\alpha^{\prime}}}{M^{\dagger}}M\right],$ (2.3) where ${D_{o}}M={\partial_{o}}M+iAM-iM{\overline{A}}.$ (2.4) As M is a complex matrix so we denote M by Z, ${M^{\dagger}}=\overline{Z}$ $\displaystyle{D_{o}}Z={\partial_{o}}Z+iAZ-iZ{\tilde{A}}$ $\displaystyle\quad;\quad$ $\displaystyle{({D_{o}}M)}^{\dagger}={{\overline{D}}_{o}}{\overline{Z}}={\partial_{o}}\overline{Z}+i{\tilde{A}}\overline{Z}-i\overline{Z}A.$ (2.5) Now define ${{Z}_{\pm}}={\frac{1}{\sqrt{2\alpha^{\prime}}}}Z\pm{D_{o}}Z\quad;\quad{{\overline{Z}}_{\pm}}={\frac{1}{\sqrt{2\alpha^{\prime}}}}\overline{Z}\pm{\overline{D}_{o}}\overline{Z}.$ (2.6) The Type 0A matrix model action in terms of the light cone variable $S=\int dtTr\left[{\overline{Z}_{+}}{D_{A}}{Z_{-}}+Z_{+}\overline{({D_{A}}Z)}+{\frac{1}{2}}(\overline{Z}_{-}Z_{+}+\overline{Z}_{+}Z_{-})\right].$ (2.7) The gauge field A acts as a lagrange multiplier which projects the theory onto singlet wave functions. Its shown in [15, 9] that type 0A MQM when projected to singlet sector can be represented by non-relativistic free fermions moving in a two dimensional upside-down harmonic oscillator potential $\hat{H}={\frac{1}{2}}(\hat{p}_{x}^{2}+\hat{p}_{y}^{2})-{\frac{1}{4\alpha^{\prime}}}(\hat{x}^{2}+\hat{y}^{2}).$ (2.8) The theory has different independent sectors labeled by net D0-brane charge q, which is the same as the angular momentum $\hat{J}=\hat{x}\hat{p}_{y}-\hat{y}\hat{p}_{x}$ [9]. Here we will consider the case where there is no net D0 –brane charge, namely the $J=0$ sector. Now with $z,{\overline{z}}=x\pm iy;$ and light cone variable are as defined in (2.6 ) we have the hamiltonian $\displaystyle{H_{o}}$ $\displaystyle=$ $\displaystyle-{\frac{1}{2}}({\hat{z}_{+}}{\hat{\overline{z}}_{-}}+{\hat{\overline{z}}_{+}}{\hat{z}_{-}}-{\frac{2i}{\sqrt{2{\alpha^{\prime}}}}})$ (2.9) $\displaystyle=$ $\displaystyle{\mp}{\frac{i}{\sqrt{2{\alpha^{\prime}}}}}[{z_{\pm}}{\frac{\partial}{\partial{z_{\pm}}}}+{\overline{z}_{\pm}}{\frac{\partial}{\partial{\overline{z}_{\pm}}}}+1].$ The commutation relation satisfied by these operators $\displaystyle[{\hat{z}_{+}},{\hat{\overline{z}}_{-}}]=[{\hat{\overline{z}}_{+}},{\hat{z}_{-}}]=2{\frac{i}{\sqrt{2{\alpha^{\prime}}}}},$ $\displaystyle[{\hat{z}_{+}},{\hat{z}_{-}}]=[{\hat{\overline{z}}_{+}},{\hat{\overline{z}}_{-}}]=0,$ (2.10) so that ${\hat{\overline{z}}_{+}}=-{\frac{\partial}{\partial{z_{-}}}}{\quad\quad;\quad\quad}{\hat{z}_{-}}={\frac{\partial}{\partial{\overline{z}_{+}}}}.$ (2.11) We have Schrodinger equation $\displaystyle i{\frac{\partial}{\partial t}}{\Psi}({\overline{z}_{+}},{z_{+}},t)$ $\displaystyle=$ $\displaystyle{\mp}{\frac{i}{\sqrt{2{\alpha^{\prime}}}}}[({z_{+}}{\frac{\partial}{\partial{z_{+}}}}+{\overline{z}_{+}}{\frac{\partial}{\partial{\overline{z}_{+}}}}+1]{\Psi}({z_{+}},{\overline{z}_{+}},t).$ (2.12) Note, here we have absorbed the Vandermonde determinant in the wave function so that the wave function ${\psi}$ in (2.12) describes a fermion. ### 2.2 Type 0A MQM in the presence of the D brane : The constraint from dual string theory Consider the type 0A matrix model in the presence of D brane which arises when we insert an operator ${e^{\int dtW(t)\delta(t-{t_{o}})}}$ in the matrix model path integral where $W(t)$ is the Laplace transform of the macroscopic loop operator([13], [18]). In the dual two dimensional type 0A theory this means that there is one Euclidean $D0\textendash\overline{D0}$ brane is localized at time $t_{0}$. The branes extends along the Liouville direction after the Laplace transformation. The macroscopic operators can be divided into NSNS and RR sector part such that they correspond to the NSNS and RR sector part of the D-brane boundary state. Moreover,since we know that there are two types of (FZZT-like) boundary states $|B(\epsilon)\rangle$ according to the spin structures there should be two macroscopic operators $W^{(\epsilon)}$ with $\epsilon=\pm$ in each sector. First consider the simplest expression of macroscopic loop operator which is the one in NS NS sector as prescribed in [13] and expressed as ${e^{-l{M^{\dagger}}M({t_{o}})}}$. Now, consider the Laplace transform of the operator $\displaystyle\int{\frac{dl}{l}}e^{-{\mu_{B}^{2}}l}{e^{-l{M^{\dagger}}M}}$ $\displaystyle=$ $\displaystyle- Tr\log(1+{\frac{{M^{\dagger}}M}{\mu_{B}^{2}}})$ (2.13) $\displaystyle=$ $\displaystyle-Tr\log(1+{\frac{\overline{Z}Z}{\mu_{B}^{2}}})$ $\displaystyle=$ $\displaystyle-\sum\log(1+{\frac{\overline{z}z}{\mu_{B}^{2}}})$ $\displaystyle=$ $\displaystyle-\sum\log(1+{\frac{({{z}_{+}}+{{z}_{-}})({{{\overline{z}}}_{+}}+{{{\overline{z}}}_{-}})}{\mu_{B}^{2}}})$ $\displaystyle=$ $\displaystyle-\sum\log(1+{\frac{{{{\overline{z}}}_{+}}{{z}_{+}}+{{{\overline{z}}}_{-}}{{z}_{-}}+{{{\overline{z}}}_{+}}{{z}_{-}}+{{{\overline{z}}}_{-}}{{z}_{+}}}{\mu_{B}^{2}}})$ $\displaystyle=$ $\displaystyle W({\overline{z}_{+}},{z_{+}},{\overline{z}_{-}},{z_{-}}).$ (2.15) (Here $\sum$ implies sum over the eigenvalues ). Now the macroscopic loop operator for ${{NS}^{-}}$ sector can be expressed as $\displaystyle W$ $\displaystyle=$ $\displaystyle-\sum\log(1+{\frac{{{{\overline{z}}}_{+}}{{z}_{+}}+{{{\overline{z}}}_{-}}{{z}_{-}}+{{{\overline{z}}}_{+}}{{z}_{-}}+{{{\overline{z}}}_{-}}{{z}_{+}}}{\mu_{B}^{2}}})$ (2.16) $\displaystyle=$ $\displaystyle-\sum\\{{\displaystyle\sum_{n=1}^{\infty}}{\frac{{(-1)}^{n}}{n}}{{[{\frac{{{{\overline{z}}}_{+}}{{z}_{+}}+{{{\overline{z}}}_{-}}{{z}_{-}}+{{{\overline{z}}}_{+}}{{z}_{-}}+{{{\overline{z}}}_{-}}{{z}_{+}}}{\mu_{B}^{2}}}]}^{n}}\\}.$ The path integral over the Euclidean time in the presence of D brane is expressed as $\int\prod{dZ_{+}}d{Z_{-}}d{\overline{Z}_{+}}d{\overline{Z}_{-}}dAd{\tilde{A}}{e^{-\int dt[\beta L(Z_{+},\overline{Z}_{+},Z_{-},\overline{Z}_{-},A,{\tilde{A}})-W\delta(t-{t_{o}})]}},$ (2.17) apparently implies a shift222Note that as $W({\overline{z}_{+}},{z_{+}},{\overline{z}_{-}},{z_{-}})$ in any sector expressed in light cone variable, so does not involve any derivative. Hence we can just add it to hamiltonian or lagrangian as a potential localized in $t_{o}$ 333Note, when we are adding the term $W\delta(t-{t_{o}})$ in the expression of the hamiltonian from the operator ${e^{W({t_{o}})}}$ it is supposed to add in the hamiltonian the terms like $[\beta\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}dtH,\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}d{t^{\prime}}W\delta({t^{\prime}}-{t_{o}})]$ \+ ….higher commutators $=e^{-W(t_{o})}[\,\beta\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}dtH\,]e^{W(t_{o})}+...O({\beta\epsilon})^{2}..\,$. However as around the ${\delta-{\rm function}}$, $\epsilon$ can be made arbitrarily small i.e $\epsilon<<{\frac{1}{\beta}}$, so upto quasiclassical limit one can just put these terms to zero while in the classical limit these terms are trivially zero. in the free single fermion hamiltonian $\beta H\rightarrow\beta H+W\delta(t-{t_{o}})$ (in Euclidean time). Note here we are going to consider complete quantum theory along with the macroscopic loop operator W. However before proceeding note that the operator $e^{\int dtW\delta(t-{t_{o}})}$ which is localized at $t={t_{o}}$, in general breaks the time translation symmetry of MQM action. So as far as matrix model action in the presence of brane is concerned (as considered in [13]) this is describing leakage of energy exactly at $t_{o}$. Now to be more precise consider MQM path integral in the presence of an operator $e^{W(t_{o})}$, for which under any infinitesimal variation in time $t\rightarrow t+\epsilon(t)$, Ward identity implies 444Here we have used the fact that equation of motion in the presence of an insertion $e^{W(t_{o})}$ is satisfied $\displaystyle\delta(t-{t_{o}})\delta\langle e^{W(t_{o})}\rangle$ $\displaystyle=$ $\displaystyle-\partial_{t}\langle{H_{o}}(t)\,e^{W(t_{o})}\rangle$ (2.18) $\displaystyle\Rightarrow$ $\displaystyle\langle\,\delta W({t_{o}})\,e^{W(t_{o})}\,\rangle+\lim_{\epsilon\rightarrow 0}\int_{t_{o}-\epsilon}^{t_{o}+\epsilon}\partial_{t}\langle{H_{o}}(t)\,e^{W(t_{o})}\,\rangle,$ where $\delta W$ is the variation of W due to infinitesimal time translation at fixed time ${t_{o}}$. The above identity arises when we consider the first order(in $\epsilon$) variation. However $e^{W(t_{o})}$ is a coherent source of the operator $W(t_{o})$ and on expansion generates infinitely many source W in the path integral. So in principle we sum up the contribution from every order of $\epsilon$ this gives rise an operator in the path integral $\langle{e^{\left[\epsilon\partial_{t}{H_{o}}(t)+\epsilon\delta W(t)\delta(t-{t_{o}})+\\{W(t_{o})+\epsilon[\partial_{t}{H_{o}}(t),W(t_{o})]+...{\rm higher}\,{\rm commutators}\\}\right]}}\rangle$ (with proper time ordering of operators). However integrating the argument of exponential over an infinitesimally small interval ${t_{o}}-\epsilon$ to ${t_{o}}+\epsilon$,time translation invariance implies $\int_{t_{o}-\epsilon}^{t_{o}+\epsilon}dt\,\partial_{t}\langle\,{H_{o}}(t)\,\rangle+\delta\langle\,W({t_{o}})\,\rangle=0.$ (2.19) (one can verify the commutator terms in this integration will give zero because of time ordering) which is an operator constraint. Note the term $W(t_{o})$ which is completely localized at a point $t_{o}$ essentially creates the effect of boundary in the matrix model action which is defined on infinite real line in the time direction. So any variation of the expectation value of the hamiltonian (from (2.18)) exactly at $t_{o}$ due to the interaction $[{H_{o}},W\delta(t-{t_{o}})]$ with the source $W(t_{o})$ is the signal of leakage of MQM hamiltonian ${H_{o}}$ exactly at $t_{o}$. So we conclude that the time translation invariance of the path integral implies555Note the effect of leakage is observed within an interval $t_{o}-\epsilon$ to $t_{o}+\epsilon$. Once we move slightly away from $t_{o}$ system will evolve according to the conserved hamiltonian $H_{o}$ that in an infinitesimal small interval around $t_{o}$ we have $\langle\,{H_{o}}(t)\,\rangle{|_{t_{o}-\epsilon}^{t_{o}+\epsilon}}+\delta\langle W(t_{o})\,\rangle=0$, where $\delta W({t_{o}})=[\int_{t_{o}-\epsilon}^{t_{o}+\epsilon}[{H_{o}},W(t)\delta(t-{t_{o}})]$ is the variation of $W(t_{o})$ due to time translation $t_{o}\rightarrow t_{o}+\epsilon(t_{o})$, which indicates the leakage of energy of the fermionic system. However the inclusion of an operator $W(t_{o})$ in MQM action has an interpretation in dual string theory is to create a boundary to string world sheet by insertion of a macroscopic loop localized at $X^{o}\equiv i{t_{o}}$ or presence of a boundary state in closed string channel. So the above phenomenon in string theory implies that closed string hamiltonian is undergoing a leakage while being scattered from the boundary state localized at $X={X^{o}}=i{t_{o}}$ ! This means that energy from the bulk is flowing out across the boundary or in other words the bulk hamiltonian is not conserved in the presence of boundary! This effect can be visualized in matrix model from the consideration of collective field theory. Note the path integral with the operator $e^{W(t_{o})}$ does have an interpretation that free fermionic state are getting scattered from an operator $e^{W(t_{o})}$. The free fermionic states after being scattered becomes permanently changed due to the action of an operator which is the function of the leakage factor $\int dt[\delta W(t)]\delta(t-{t_{o}})$ at $t\geq{t_{o}}$. Clearly the scattered state will differ from the incoming state with a term which is function of $\delta W(t_{o})$. As the term $\delta W(t_{o})$ is not present in the effective hamiltonian 666which is given by $W\delta(t-{t_{o}})+e^{-W(t_{o})}[\,\beta\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}dtH\,]e^{W(t_{o})}\sim\beta{H_{o}}+W\delta(t-{t_{o}})$ as we mentioned or cannot arise in by the time evolution of $\langle{H_{o}}\rangle$ w.r.t the complete hamiltonian 777as $W(t_{o})\rightarrow W(t_{o})+\delta W(t_{o})$ is an instantaneous process so it describes a leakage. These deformed states although evolve according to the free fermionic hamiltonian ${H_{o}}$ but they can be considered as the superposition of the states which are stationary w.r.t a hamiltonian deformed from ${H_{o}}$ where the deformation is caused by the leakage as we discussed. Now in collective field theory the fluctuation of collective field from its static value gives a field which corresponds to 2D spacetime tachyon. The action for this fluctuation gives the propagator of a 2D massless scalar. Now in the presence of the operator $e^{W(t_{o})}$, the wave function of this scalar field above $t\geq{t_{o}}$ although evolve according to the hamiltonian of a 2D massless scalar but will be deformed from the one at $t\leq{t_{o}}$ by the action of an operator which is function of the leakage factor $\delta W(t_{o})$. As this operator is not present in the path integral so the deformation of the wave function above $t\geq{t_{o}}$ must show up as the modification of the propagator from that of a 2D massless scalar ! This can be easily seen from the canonical quantization of 2D massless field and considering the deformation of the wave function. So we see that although the time translation invariance is maintained by extending Ward’s principle to every order but its not giving the right string theory picture! This is because the closed strings which are getting scattered from D brane the scattered states remain the same on shell states w.r.t the hamiltonian same as that for incoming one! So it appears that we must need to impose a constraint in matrix model side in order to extract right string theory from it. Lets briefly go through the string theory scenario and try to understand string theoretical origin of such constraint. Note as far as the open string world sheet is concerned Dirichlet boundary condition which fixes the matter coordinate ${X}={X^{o}}\equiv{it_{o}}$ at the boundary implies nonconservation of the momentum associated with the matter direction. However the open string action ${S_{\rm open}}$ and the open string path integral $Z_{\rm open}$ remains invariant under an infinitesimal variation of X which is ensured from the boundary condition ${\delta{X}}({X^{o}})=0\quad\Rightarrow\quad{\delta_{X}}{S_{\rm open}}=0\quad,\quad{\delta_{X}}[Z_{\rm open}]=0,$ (2.20) where ${\delta_{X}}$ implies infinitesimal variation in X at every point of the world sheet. Also the conservation of the string hamiltonian is associated with the boundary condition $T(z)-\overline{T}(\overline{z})=0,$ (2.21) which ensures there is no leakage of energy at the boundary 888 To explain a bit more, in the presence of the D brane we know the energy of the incoming state is (associated with closed string )is not same as that of the outgoing state in the direction with Dirichlet boundary condition as D brane act as a source. However (2.20) implies we can consider the incoming and the outgoing state as the separate conserved system evolve according to same conserved hamiltonian (but different state) with none suffering any leakage at the boundary [40, 39]. Moreover we have the constraint from the current algebra and superpartners of all the above conditions which ensures the conservation of the symmetry generators. Now the string path integral with a Laplace transformed macroscopic loop (which creates the boundary localized at $X={X^{o}}$ with the imposed wave function giving right string one point function ) must obey the conditions (2.20,2.21) where the condition (2.20) follows from the function $\delta(X_{\rm boundary}-{X^{o}})$ present in the macroscopic loop functional. Hence these conditions must show up in the dual matrix model with a D brane. To state more precisely the 2D path integral on a manifold with the macroscopic loop (for the bosonic case which is given by (1.3) ) corresponds to a physical state where the respective wave function satisfy WdW equation [10]. WdW equation implies invariance of the wave function under the action of the generator of $\tau$(worldsheet time) translation. However the conservation of these symmetry generators follows from these conditions and hence the respective state must have information about it. So to gain the insight about what these constraints correspond in the matrix model let us look at the closed string channel and express the constraints in terms of the boundary state. Now in the minisuperspace approximation only the zero mode part of the constraints will be relevant and can be expressed as $(L_{o}-\overline{L}_{o})|B\rangle=0\quad;\quad\delta X_{\rm boundary}|B\rangle=0.$ (2.22) Note that both the above constraints are followed by their superpartners however as far as zero mode is concerned we have already applied such constraints when we classified macroscopic boundary state according to their spin structure [9]. The second condition essentially describes the zero mode condition $(\hat{X}-{X^{o}})|B\rangle=0\quad\Rightarrow\quad|B\rangle\equiv\delta(\hat{X}-{X^{o}})|0\rangle.$ (2.23) In closed siring channel path integral with a Laplace transformed macroscopic loop corresponds to boundary states [39]. This can be seen by expressing the path integral functional $\Psi(X,\phi)$ as a sum over operators(in Minkowskian signature) ${O_{i}}$ by state operator mapping where we know that these operators corresponds to Ishibashi states and the wave function $\langle{\Psi}|O_{i}\rangle$ gives the one point function. So $|\Psi\rangle$ must be annihilated by the constraints from (2.23) [42]. So the same constraints must be imposed on the state associated with matrix model path integral in the presence of D brane. This is because macroscopic loop operator in matrix model can equivalently be expressed in terms of operators along with the respective wave function where each component is in one to one correspondence with the one in string theory side. So we conclude that the kind of leakage we discussed at the beginning of the section is caused due to absence of any condition equivalent to (2.22) and must be cured once we impose an equivalent condition to matrix model. Lets try to find out the constraint in matrix model. From the first condition in (2.22) along with the one from the zero mode part of the current algebra in Dirichlet boundary state for matter just emphasizes the fact that 2D boundary state or the macroscopic loop operator will correspond to the superposition of primary states/operators expressed in terms of the momentum modes(i.e no winding modes)[39] with a reflection symmetry $P\rightarrow-P$ in matter as well as Liouville part [33, 39] which is already known in the matrix model [2, 28]. The reason we obtain Liouville one point function in exact form from matrix model, is that in minisuperspace approximation this condition is trivially satisfied and consequently not going to impose any constraint in matrix model side. However the second condition in (2.22), (2.23) or (2.20) is not yet properly understood in the matrix model. More precisely in the presence of macroscopic operator the state from the path integral is represented by an wave function expressed as a functional of bulk d.o.f $\Psi=\Psi(\\{X\\},\\{\phi\\}...)$ and under any infinitesimal transformation $X\rightarrow X+\delta X$ we must have $\delta\Psi=\int_{\rm boundary}\delta{\hat{X}}J(\hat{X})\,\Psi=0$ (2.24) which ensures the bulk conformal invariance and implication of (2.23). Note $\Psi$ is the wave function $\Psi=\Psi(X_{\rm boundary},\phi_{\rm boundary})$ which is an eigenfunction of complete string hamiltonian, representing BRST invariance. Similarly its discussed in [4, 6] that matrix model path integral in the presence of an operator $e^{W(t_{o})}$ ( which arises by including a probe eigenvalue) is an wave function $\psi=\psi({\overline{z}_{\pm}}{z_{\pm}}(t_{o})\,)$which satisfied the Schrodinger equation. However considering the fact that this operator also creates an effect of boundary and $\psi$ is a functional of MQM variables $\psi=\psi(\\{{\overline{z}_{\pm}}{z_{\pm}}(t)\\})$ we must have an condition equivalent to (2.24) in matrix model which ensures conservation of MQM hamiltonian in presence of such operator. Naturally no such constraint arises from Liouville d.o.f for the reason as we discussed. Here first we will find such constraint in the matrix model from somewhat intuitive way, solve it in the context of the matrix model path integral and show its consequence. Finally with the help of it we will come to more rigorous analogy between the string theory and the matrix model scenario in the next subsection. First note that in the matrix model the matter coordinate X is getting mapped to time(Minkowskian) coordinate, $X\rightarrow it$. So we can guess that string theory boundary condition must be reflected in MQM as an overall invariance of the path integral under time translation i.e $\delta X\equiv\delta t$ with no leakage. Note when there is no leakage, under infinitesimal time translation $t\rightarrow t+\epsilon(t)$ the variation of path integral is given by $\langle{\delta}e^{\int dtW({t})\delta(t-{t_{o}})}\rangle$, where $\delta$ defines the variation of the operator due to infinitesimal time translation at fixed time $t={t_{o}}$. So string theory boundary condition (2.20) must be reflected in the following constraint in matrix model $\delta\langle e^{\int dtW({t})\delta(t-{t_{o}})}\rangle=\langle\delta e^{\int dtW({t})\delta(t-{t_{o}})}\rangle=0.$ (2.25) Indeed in the string theory path integral if we expand the macroscopic loop in terms of operators which corresponds to Ishibashi state one can verify that this is the consequence of Ward identity under an infinitesimal transformation $X\rightarrow X+\delta X$ which arises on application of the second condition in (2.20) and we have already mentioned it in an alternative way in (2.24). In next subsection we will show that the consequence of this condition are in exact agreement with that of string theory. To understand the impact of this condition in MQM first we need to write down the Schrodinger equation and study the Hilbert space. We have the time dependent Schrodinger equation for a single fermion 999When we consider the insertion of $\beta$ factor, for the macroscopic loop operator we have the expression ${\frac{1}{\beta}}{\hat{W}}$, however when we rewrite the Schrodinger equation in terms of the eigenvalues x,y which corresponds to the real and imaginary part of eigenvalue z we have the Schrodinger equation $[{\frac{1}{\beta^{2}}}{\frac{\partial^{2}}{\partial{x^{2}}}}+{x^{2}}+{\frac{1}{\beta^{2}}}{\frac{\partial^{2}}{\partial{y^{2}}}}+{y^{2}}+{\frac{1}{\beta}}{\hat{W}}\delta(t-{t_{o}})]\psi=E\psi$ In the double scaling limit we take $x,y\rightarrow{\sqrt{\beta}}x,{\sqrt{\beta}}y$ [1], which gives the Schrodinger equation (2.12 ) in Minkowskian time $\displaystyle[i{\frac{\partial}{{\partial}t}}$ $\displaystyle-$ $\displaystyle i{\delta}(t-{t_{0}})W({\overline{z}_{+}},{z_{+}},{\overline{z}_{-}},{z_{-}})]{\Psi}({\overline{z}_{\pm}}{z_{\pm}},t)$ (2.26) $\displaystyle=$ $\displaystyle{\mp}{\frac{i}{\sqrt{2{\alpha^{\prime}}}}}[({z_{+}}{\frac{\partial}{\partial{z_{\pm}}}}+{\overline{z}_{+}}{\frac{\partial}{\partial{\overline{z}_{\pm}}}}+1{]}{\Psi}({z_{\pm}},{\overline{z}_{\pm}},t).$ For t away from ${t_{o}}$ we have time independent Schrodinger equation $\displaystyle i{\frac{\partial}{\partial t}}{\psi}({\overline{z}_{\pm}}{z_{\pm}},t)$ $\displaystyle=$ $\displaystyle{\mp}{\frac{i}{\sqrt{2{\alpha^{\prime}}}}}[({z_{\pm}}{\frac{\partial}{\partial{z_{\pm}}}}+{\overline{z}_{\pm}}{\frac{\partial}{\partial{\overline{z}_{\pm}}}}+1]{\Psi}({\overline{z}_{\pm}}{z_{\pm}},t)=E{\psi}({\overline{z}_{\pm}}{z_{\pm}},t).$ (2.27) The free fermion solution with energy E is ${\psi^{E}_{o\pm}}({z_{\pm}},t)={e^{-iEt}}{e^{\mp i{\frac{\phi_{o}(E)}{2}}}}{{({\overline{z}_{\pm}}{z_{\pm}})}^{{\pm}iE-{\frac{1}{2}}}},$ (2.28) where we have chosen ${\alpha^{\prime}}=2$ and ${\phi_{o}}(E)$ is determined from biorthogonal property (discussed in the Appendix) and given by $e^{i{\phi_{0}(E)}}={\Gamma(iE+{\frac{1}{2}})\over\Gamma(-iE+{\frac{1}{2}})}.$ (2.29) Now consider the commutation relation $\displaystyle[{H_{o}},{\hat{z}_{+}}]$ $\displaystyle=$ $\displaystyle-i{\quad\quad;\quad\quad}[{H_{o}},{\hat{\overline{z}}_{+}}]=-i,$ $\displaystyle[{H_{o}},{\hat{z}_{-}}]$ $\displaystyle=$ $\displaystyle\quad i{\quad\quad;\quad\quad}[{H_{o}},{\hat{\overline{z}}_{-}}]=i.$ (2.30) This implies ${\hat{\overline{z}}_{+}}{\hat{z}_{+}}$ and ${\hat{\overline{z}}_{-}}{\hat{z}_{-}}$ when acts on a state $|E\rangle$ expressed as ${\hat{\overline{z}}_{+}}{\hat{z}_{+}}|E\rangle=|E-i\rangle$ and ${\hat{\overline{z}}_{-}}{\hat{z}_{-}}|E\rangle=|E+i\rangle$. These states can be represented as $|E\pm ni\rangle$. These describe a different Hilbert space [20] which can be understood as their inner product with the state $|E\rangle$ either diverge or zero. These states actually can be identified with the discrete tachyonic states over the matrix model ground state [28, 20]. Now to the meaning of the constraint. First we will find the expression for the constraint and then solve it in the context of the matrix model path integral for the macroscopic loop operator we considered or in the more complicated case to reach to the right expression free energy. Consider the v.e.v of the operator in single fermionic state which is given by $\langle e^{W({t_{o}})}\rangle=\langle e^{{\rm log}(1+{\frac{{\overline{z}_{+}}{z_{+}}+{\overline{z}_{-}}{z_{-}}-2{H_{o}}}{\mu_{B}^{2}}})}\rangle=\langle(1+{\frac{{\overline{z}_{+}}{z_{+}}+{\overline{z}_{-}}{z_{-}}-2{H_{o}}}{\mu_{B}^{2}}})\rangle.$ (2.31) (Here we have shown the expectation value w.r.t the single fermionic state. This we could do because the theory is projected singlet sector and N fermionic state is just the direct product of each). So (2.25) along with (2.30) gives the constraint $\langle{\overline{z}_{+}}{z_{+}}(t_{o})-{\overline{z}_{-}}{z_{-}}(t_{o})\rangle=0.$ (2.32) As the constraint is exclusively on the variable associated with W so it must has effect in constraing the Hilbert space created by W at $t_{o}$. Note for the macroscopic loop operator in any other sector, in general the variation of $\langle e^{W({t_{o}})}\rangle$ can be expressed as $\langle e^{W({t_{o}})}\delta W({t_{o}})\rangle=0\quad\quad\Rightarrow\quad\quad\langle e^{W({t_{o}})}[{\overline{z}_{+}}{z_{+}}(t_{o})-{\overline{z}_{-}}{z_{-}}(t_{o})]\rangle=0.$ (2.33) As the path integral with the uncompactified time essentially describes the transition amplitude from the initial state $|{\overline{z}_{\pm}}{z_{\pm}}({t_{i}})\rangle$ to the final state $|{\overline{z}_{\pm}}{z_{\pm}}({t_{f}})\rangle$, so the condition (2.25) remain true for any arbitrary variation in time implies that we have $\langle{\psi_{f}}|\delta e^{{W}({t_{o}})}|\psi_{i}\rangle=0,$ (2.34) between any two physical states $|\psi_{i}\rangle$, $|{\psi_{f}}\rangle$ which according to the Hilbert space described by (2.30) will be of either form $|{\overline{z}_{+}}{z_{+}},t\rangle$ or $|{\overline{z}_{-}}{z_{-}},t\rangle$. Now recall the expression of the inserted macroscopic loop operator $W(t_{o})$ in (2.15) which is expressed in the form (2.31). While the ${\hat{H}_{o}}$ in the expression of W keeps the underlying state invariant the ${\hat{\overline{z}}_{+}}{\hat{z}_{+}}$ and ${\hat{\overline{z}}_{-}}{\hat{z}_{-}}$ act on the vacuum to create the states $|E=-i\rangle$ , $|E=i\rangle$ or in other words the macroscopic loop operator (2.15) due to its $\overline{z}z$ field acts on the vacuum to create a left moving as well as right moving state. So the constraint essentially relates the two exactly at $t_{o}$ by putting the following constraint inside the path integral, ${\overline{z}_{+}}(t_{o})-{\overline{z}_{-}}(t_{o})=0\quad;\quad{z_{+}}(t_{o})-{z_{-}}(t_{o})=0,$ (2.35) The physical meaning of the above condition is that the quantum fluctuations of the variables ${z_{+}}({\overline{z}_{+}})$ and ${z_{-}}({\overline{z}_{-}})$ around its classical value appears to be identical at $t_{o}$. This in turn implies that the wave functions of the states which are being created by the action of ${\hat{\overline{z}}_{+}}{\hat{z}_{+}}$ and ${\hat{\overline{z}}_{-}}{\hat{z}_{-}}$ over the ground state in ${\overline{z}_{+}},{z_{+}}$ and ${\overline{z}_{-}}.{z_{-}}$ representations respectively , appeared to be indistinguishable exactly at $t_{o}$. Note that in the above constraint (2.35 ), the l.h.s will not remain invariant once we move away from $t_{o}$. In otherwords if we express them in terms of the respective operators(which acts on vacuum), we see that l.h.s does not commute with the free hamiltonian ${H_{o}}$ (2.9). So once we move away from the point $t={t_{o}}$ we have our physical Hilbert space described by the free hamiltonian ${H_{o}}$ and the relation (2.30) so that again the left and right moving states created by $\hat{W}$ will appear to be distinguishable. In next few steps we will see that the condition (2.35)when acts inside the path integral it just has a meaning to project the operator W and the Hilbert space generated by it to its physical sector while keeping the bulk physics unaffected! So with the constraint we can express the path integral somewhat schematically as 101010above identity can arise by inserting $\delta({\overline{z}_{+}}(t_{o})-{\overline{z}_{-}}(t_{o}))\delta({z_{+}}(t_{o})-{z_{-}}(t_{o}))$ in the path integral, operating ${\overline{z}_{-}}{z_{-}}={\frac{\partial}{\partial{z_{+}}}}{\frac{\partial}{\partial{\overline{z}_{+}}}}$ on the inner product $\langle{\overline{z}_{+}}{z_{+}}(t_{o}+dt_{o})|{\overline{z}_{+}}{z_{+}}(t_{o})\rangle$. Its important to note that due to $\delta(t-{t_{o}})$, inside the path integral we can make the time interval around ${t_{o}}$ arbitrarily small i.e $dt<<{\frac{1}{\beta}}$ so that around $t_{o}$, $e^{-\beta\int_{{t_{o}}-\epsilon}^{{t_{o}}+\epsilon}dtL}$ will just be identity and hence the effect the constraint is only to modify W without affecting the MQM lagrangian L and hence no boundary condition will be imposed on the variables in original lagrangian at $t={t_{o}}$. Finally integrating over ${z_{-}}(t_{o}),{{\overline{z}_{-}}}(t_{o})$ where the $\delta-{\rm function}$ gives ${\overline{z}_{-}},{z_{-}}={\overline{z}_{+}},{z_{+}}$, leads to the above expression of W. As W becomes independent of ${\overline{z}_{-}},{z_{-}}$ so we can just express the integral as a continuous integral. Also note in the expression of $W({\hat{\overline{z}}_{+}}{\hat{z}_{+}},{H_{o}})$,the action of $H_{o}$ on any intermediate state in the path integral gives the constant and it can be alternatively given by ${\frac{\partial}{\partial t}}$. So it will not be affected by the constraint. 111111note that here path integral is expressed in a schematic way omitting the angular factors. One can verify that even the inclusion of angular factor will not change the picture $\displaystyle\int d{\overline{z}_{+}}d{z_{+}}d{\overline{z}_{-}}d{z_{-}}{e^{-\int_{-\infty}^{\infty}dt[\beta L-W({\overline{z}_{+}},{z_{+}},{\overline{z}_{-}},{z_{-}};{t_{o}})]}}$ $\displaystyle=$ $\displaystyle\int\displaystyle\prod_{t\leq{t_{o}}}d{\overline{z}_{+}}d{z_{+}}d{\overline{z}_{-}}d{z_{-}}{e^{-\int_{t_{o}}^{\infty}dtL}}$ $\displaystyle\int d{\overline{z}_{+}}(t_{o})d{z_{+}}(t_{o})d{\overline{z}_{-}}(t_{o})d{z_{-}}(t_{o})$ $\displaystyle\delta({\overline{z}_{+}}({t_{o}})-{\overline{z}_{-}}({t_{o}}))\delta({z_{+}}({t_{o}})-{z_{-}}({t_{o}}))e^{W({\overline{z}_{+}}{z_{+}},{\overline{z}_{-}}{z_{-}},H_{o};\,{t_{o}})}$ $\displaystyle\int\displaystyle\prod_{t\geq{t_{o}}}d{\overline{z}_{+}}d{z_{+}}d{\overline{z}_{-}}d{z_{-}}{e^{-\int_{-\infty}^{t_{o}}dtL}}$ $\displaystyle=$ $\displaystyle\int d{\overline{z}_{+}}d{z_{+}}d{\overline{z}_{-}}d{z_{-}}{e^{-\int_{-\infty}^{\infty}dt[\beta L-W({\overline{z}_{\pm}}{z_{\pm}}(t_{o}),H_{o})]}}.$ (2.36) Its important to note that the effect of the constraint is only to project W at its physical sector without imposing any boundary condition to original lagrangian which happens due to $\delta(t-t_{o})$ factor as we explained. Projection implies we can get same path integral expression or the transition amplitude by expressing W either in ${\overline{z}_{+}}{z_{+}}$ or in ${\overline{z}_{-}}{z_{-}}$ mode which happens due to the fact that wave function associated with the either mode appears to be same at $t_{o}$. Its also important to note that when we consider the original expression of W (2.15), the quantum fluctuations of the varibles ${z_{+}}({z_{-}})$ and ${\overline{z}_{+}}({\overline{z}_{-}})$ are constrained by (2.35) and in that sense when we consider the complete Hilbert space as described by (2.30). ${\hat{z}_{+}}({\hat{z}_{-}})$ and ${\hat{\overline{z}}_{+}}({\hat{\overline{z}}_{-}})$ are not ordinary operators. However the implication of (2.36) is that we can replace the theory with the expression of W (as in (2.15) ) by the projected one $W\rightarrow W_{\rm proj}=W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o})$ and in the theory with $W_{\rm proj}$ these operators act as ordinary operators and one can evaluate the partition function in a formal way in MQM by using $W_{\rm proj}$ instead of W to give the right transition amplitude as in the original theory with the constraint(2.35). Now we will show that how the constraint is solved in the context of the path integral leads to the right expression of the partition function. Note in (2.34) $|\psi_{i},t_{i}\rangle$ can be expressed in the free fermionic Hilbert space $|E\rangle$ while the constraint (2.25) implies the macroscopic loop operator which has expression $W({\overline{z}_{+}}{z_{+}}(t_{o}),{H_{o}})$ or $W({\overline{z}_{-}}{z_{-}}(t_{o}),{H_{o}})$ leads to the most general expression of the final state $|\psi_{f}\rangle$ is $\sum{c_{o;E}}|E\rangle+\sum{c_{n;E}}|E+ni\rangle$ or $\sum{c_{o;E}}|E\rangle+\sum{c_{-n;E}}|E-ni\rangle$. As the tachyonic states have imaginary energy so the out state with indefinite number of tachyons will contribute to the path integral. So (2.34) now can be expressed as a trivial identity $\displaystyle\sum_{n=0}^{\infty}{c_{-n;E}}\langle E-ni;f|{\rm exp}[W({\overline{z}_{+}}{z_{+}},{H_{o}},(t_{o}))]|E\rangle=\displaystyle\sum_{n=0}^{\infty}{c_{n;E}}\langle E+ni;f|{\rm exp}[W({\overline{z}_{-}}{z_{-}},{H_{o}},(t_{o}))]|E\rangle.$ (2.37) As $\psi({\overline{z}_{+}}{z_{+}},E)$ and $\psi({\overline{z}_{-}}{z_{-}},E)$ are the same wave function in different representation so we see in the above l.h.s and r.h.s are the same expression, expressed in different representation. Now before coming to the string theoretical interpretation of all the matrix model events first we will derive the expression of the wave function(physical) for $t\geq{t_{o}}$. Integrating (2.26) over the infinitesimally small interval around $t={t_{o}}$ and following the constraint from (2.35,2.36) we have $\displaystyle i{\Psi}({z_{\pm}},{\overline{z}_{\pm}},t){|_{{t_{0}}+{\epsilon}}}-i{\Psi}({z_{\pm}},{\overline{z}_{\pm}},t){|_{{t_{0}}-{\epsilon}}}$ $\displaystyle=$ $\displaystyle iW_{\rm proj}\Psi({z_{\pm}},{\overline{z}_{\pm}},{t_{0}})$ (2.38) . $\displaystyle=$ $\displaystyle-\sum i\log(1+{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}+{\hat{\overline{z}}_{+}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}{\hat{z}_{+}}}{\mu_{B}^{2}}}){\Psi}({z_{\pm}},{\overline{z}_{\pm}},{t_{o}})$ So exactly at $t={t_{o}}$ the wave function is given by $\lim_{\epsilon\to 0}{\Psi_{>}}({z_{\pm}},{\overline{z}_{\pm}},t_{o})=(1-W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})){\Psi_{o}}({z_{\pm}},{\overline{z}_{\pm}},t_{o})$ (2.40) So for the macroscopic loop operator in the NS NS sector we have the physical wave function for $t\geq{t_{o}}$ ${\Psi_{>}}({\overline{z}_{\pm}}{z_{\pm}},t_{o})=[1-{\rm log}(1-{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}(t_{o})-2{H_{o}}}{\mu_{B}^{2}}})]{\Psi_{o}}({\overline{z}_{\pm}}{z_{\pm}},t)$ (2.41) Now we need to find the wave function at $t>{t_{o}}$. In order to do so first note that for $t\geq{t_{o}}$ wave function evolves according to the free hamiltonian ${H_{o}}$. So at the first sight it appears that the wave function at $t\geq{t_{o}}$ is given by the one obtained from the time evolution from ${\Psi_{>}}({\overline{z}_{\pm}}{z_{\pm}},t_{o})$. It is given by ${\Psi_{>}}({\overline{z}_{\pm}}{z_{\pm}},t)=[1-{\rm log}(1-{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}(t)-2{H_{o}}}{\mu_{B}^{2}}})]{\Psi_{o}}({\overline{z}_{\pm}}{z_{\pm}},t)$ (2.42) Now although exactly at $t={t_{o}}$, $\hat{W}$ is expressed in the form $W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})$ but once we move above $t_{o}$, W can in principle be expressed in terms of both ${\hat{\overline{z}}_{+}}{\hat{z}_{+}}$ and ${\hat{\overline{z}}_{-}}{\hat{z}_{-}}$ as both of them are related by the constraint (2.35) at ${t_{o}}$. More precisely these states are given by $|{\psi_{f}}\rangle=\sum{c_{mn}}(E){({\hat{\overline{z}}_{+}}{\hat{z}_{+}})}^{m}{({\hat{\overline{z}}_{-}}{\hat{z}_{-}})}^{n}|E\rangle,$ (2.43) So following the previous discussion $|{\psi_{f}}\rangle$ is expected to be given by $(1-W({\hat{\overline{z}}_{+}}{\hat{z}_{+}}(t),{\hat{\overline{z}}_{+}}{\hat{z}_{+}}(t){H_{o}})),{\Psi_{o}}({z_{\pm}},{\overline{z}_{\pm}},t).$ (2.44) We can express these states as the one time evoluted from the wave function from $t={t_{o}}$. Although it appears that exactly at $t={t_{o}}$, $|{\psi_{f}}\rangle$ and $|{\Psi_{>}}\rangle$ are of similar structure but they are not same as time evolution property of ${\hat{\overline{z}}_{+}}$ and ${\hat{z}_{+}}$ are different and once we move away from ${t_{o}}$ we have our original Hilbert space. Finally note that at $t={t_{o}}$ fermionic states are being converted from ${\psi_{o}}$ to $\psi_{>}$ so the change of fermion number N must be measured from the variation of the transition amplitude at $t={t_{o}}$ i.e in terms of the macroscopic loop operator described in (2.15) we have ${\frac{dN}{dt}}={\frac{d}{dt}}\langle{\psi_{>}}({t_{f}})|{\psi_{o}({t_{i}})}\rangle=\langle{\frac{dW}{dt}}\rangle{|_{t={t_{o}}}},$ (2.45) Where ${t_{i}}\rightarrow-\infty$ and ${t_{f}}\rightarrow\infty$. The average $\langle{\frac{dW}{dt}}\rangle{|_{t={t_{o}}}=0}$ described above is w.r.t the path integral. So in order to have the fermion number unchanged we must need to impose the condition $\langle{\frac{dW}{dt}}\rangle{|_{t={t_{o}}}}=0.$ (2.46) Note that this is just a condition parallel to (2.25) and have an effect to project W as described above. In the next part of this section we are going to see that all these matrix model events are exactly in one to one correspondence to the string theory. ### 2.3 String theoretical interpretation In this section we will see that the constraint (2.25,2.35) we imposed leads to right string theoretical result. First The boundary state of 2D spacelike brane is expressed as $|B_{\rm(SuperFZZT)}(\mu_{B})\rangle_{\phi}\otimes|D\rangle_{X^{0}}$ where $|D\rangle_{X^{0}}$ for NS NS sector [39] is expressed as ${\cal N}\int_{-\infty}^{\infty}dPe^{iP{X^{o}}}|P\rangle$ ( ${\cal N}$ is the normalization factor) which in terms of the vertex operator can be written as $|B{\rangle_{X}}={\cal N}\int_{-\infty}^{\infty}d{P_{m}}[e^{i{P_{m}}(X-{X^{o}})}]|0\rangle+{\rm descendants}.$ (2.47) The string endpoints are localized at $X=X^{o}$ and the matter part of the boundary state form the representation of $\delta(\hat{X}-{X^{o}})|0\rangle$, satisfying (2.22). Boundary state of Super Liouville NS NS sector is given by $|B{\rangle_{L}}={\cal N}_{l}{\int_{0}^{\infty}}dPU(P_{l})|P_{l}\rangle\quad;\quad U(P_{l})={\frac{\pi{\rm cos}2\pi s{P_{l}}}{{\rm sin}h(\pi{P_{l}})}}\quad;\quad|P_{l}\rangle=(1+{\frac{L_{-1}\tilde{L}_{-1}}{P_{l}^{2}+M^{2}}}+.....)|v_{P_{l}}\rangle$ (2.48) where $|v_{P_{l}}\rangle$ primary macroscopic state associated with a vertex $e^{({\frac{Q}{2}}+i{P_{l}})\phi}$, M is the mass of intermediate propagating mode. The boundary state is the direct product of the matter and Liouville part along with the ghost factors. Setting ${P_{m}}={P_{l}}=P$ we have the primary part of the boundary state without ghost excitation mode, which is the superposition of the tachyonic field can be expressed in terms of the operators from the state operator mapping as $\int dPU(P)[e^{{iP(X-{X^{o}})}}+e^{{-iP(X-{X^{o}})}}]{e^{({\frac{Q}{2}}+iP)\phi}}.$ (2.49) Now in the 2nd quantized matrix model we can express the macroscopic loop operator as $W(l,t)=\int e^{l{\overline{z}{z}}}{\psi^{\dagger}}{\psi}=\int d\tau e^{l{\rm cosh}^{2}\tau}{\partial_{\tau}}\eta(l,t),$ (2.50) where $\tau$ is the time of flight coordinate obtained from reparameterization ${\overline{z}{z}}={r^{2}}\quad;\quad{r^{2}}\sim{\rm cos}^{2}{h\tau}$ , ${\psi^{\dagger}}{\psi}\sim{\partial\tau}\eta(l,t)$ and $\eta$ is the massless bosonic field $({\partial_{t}^{2}}-{\partial_{\tau}^{2}})\eta(t,\tau)=0$ which corresponds to the tachyon in the string theory at asymptotic $\tau$[25, 2, 28]. $\eta$ corresponds to the fluctuation of the collective field $\phi={\phi_{o}}+{\partial_{\tau}}\eta$. $\eta$ satisfies Dirichlet boundary condition in $\tau$ direction [11], [28]. However note that when associated with the Laplace transformed macroscopic loop operator operator W which correspond to D brane boundary state, $\eta$ is no longer an ordinary state but it corresponds to an Ishibashi state. Here we show that the constraint we imposed (2.25,2.35) leads to the right matter one point function from $\eta$ what is expected from string theory. Consider $\eta(\tau,t)=\int_{-\infty}^{\infty}{\frac{dp}{p}}{\tilde{\eta}}(p)(a(p)e^{-ipt}+b(p)e^{ipt}){\rm sin}(p\tau).$ (2.51) In order to find the exact t dependence consider the fact that the implication of (2.46) in the context of collective field theory is ${\delta_{t}}{\phi}{|_{t=t_{o}}}=0,$ (2.52) where as before $\delta_{t}$ implies variation of the collective field $\phi$ due to infinitesimal variation in t, at fixed t. So (2.52) implies in (2.51) we have $a=e^{ipt_{o}}\quad;\quad b=e^{-ipt_{o}}$ (2.53) So (2.51) is expressed as $\eta(\tau,t)=\int_{-\infty}^{\infty}{\frac{dp}{p}}{\tilde{\eta}}(p)(e^{-ip(t-{t_{o}})}+e^{ip(t-{t_{o}})}){\rm sin}p\tau$. Now integrating over $\tau$ we have $W(p,t)={\frac{e^{-l\mu}}{2}}p{K_{ip}}(\mu l){\tilde{\eta}}(p)(e^{-ip(t-{t_{o}})}+e^{ip(t-{t_{o}})})$ where ${K_{ip}}(\mu l)$ is the Bessel’s function which is the macroscopic wave function satisfying WdW equation [13]. This on Laplace transform $\int{\frac{dl}{l}}e^{-\mu_{B}^{2}l}$ can be expressed as $W(p,t)=U(p)\tilde{\eta}(p,t)\quad;\quad U(p)={\frac{\pi{\rm cos}2\pi sp}{{\rm sin}h(\pi p)}}$ (2.54) where we have $\mu_{B}^{2}=2\sinh^{2}(\pi s)|\mu|\ \ \ (\epsilon\cdot{\rm sign}(\mu)<0),\ \ \ \ \ \mu_{B}^{2}=2\cosh^{2}(\pi s)|\mu|\ \ \ (\epsilon\cdot{\rm sign}(\mu)>0).$ (2.55) Note (2.25) and (2.52)are just the same constraint expressed in the 1rst and 2nd quantized formalism. Given the form of $\eta$, on Laplace transform of (2.50) and on inverse Fourier transform of (2.54) we can express W as $W_{p}(\tau,t)=U(p){\tilde{\eta}}(p)[e^{ip(t-{t_{o}})}+e^{-ip(t-{t_{o}})}]=U(p)[{\partial_{\tau}}{\eta_{p}}(\tau+(t-{t_{o}}))+{\partial_{\tau}}{\eta_{p}}(\tau-(t-{t_{o}}))]$ (2.56) where the suffix p implies pth component. We know $\eta$ corresponds to the space time tachyonic field in the asymptotic $\tau$ region which in the string theory side describes asymptotics of Liouville field $\phi$ which describes vanishing Liouville wall. So we see that each component with momentum p in the expansion of W describes the tachyonic operator ${\tilde{\eta}}(p)$ dressed with Liouville and matter wave function (2.47,2.48) where W is symmetric under $P\rightarrow-P$. So these are just in one to one correspondence with the state obtained in the expansion of the boundary state in (2.49). So (2.56) is just the same as (2.49) when we identify the physical d.o.f. More interestingly if we try to extrapolate $\eta$ in the region $P_{m}\neq P_{L}$ with the same $\tau$ dependence as in (2.51), from (2.52,2.51,2.56) we have $\displaystyle\int{\frac{dl}{l}}{e^{-\mu_{B}l}}W(l)$ $\displaystyle=$ $\displaystyle[\int_{0}^{\infty}dP_{l}U(P_{l}){\eta_{l}}(P_{l})]\times[\int_{-\infty}^{\infty}d{P_{m}}e^{i{P_{m}}(t-{t_{o}})}]$ (2.57) $\displaystyle=$ $\displaystyle[\int_{0}^{\infty}dP_{l}U(P_{l}){\eta_{l}}(P_{l})]\times\delta(t-{t_{o}}),$ where $\eta_{l}$ implies Liouville part of $\eta$. This has a very similar structure with that of 2D boundary state $|B\rangle=|B_{\rm Liouville}\rangle\otimes|B_{\rm matter}\rangle=\int_{0}^{\infty}dP_{l}U(P_{l})|P\rangle\otimes\delta(\hat{X}-{X^{o}})|0\rangle.$ (2.58) However this is a mere extrapolation without any physical justification. Finally we see the effect of the constraint (2.25) is only to give the correct form of the matter wave function while keeping the Liouville wave function unchanged as we discussed in section 2.2. Finally lets explain the meaning of the projection $W\rightarrow W_{\rm proj}$ as described in (2.36),(2.37). Note both Liouville as well as the matter part of the boundary state is symmetric under interchange of $P\rightarrow-P$ (2.48). So projecting the theory to either right moving or left moving part i.e in the boundary state keeping either the left moving or the right moving part while expressing the superposition of left and right moving state as only left or only right moving state (i.e flipping the sign of p where necessary)in the incoming or the outgoing sector yield the same transition amplitude as the original one. Overall time translation invariance of the projected theory follows from the consideration of either left or the right moving part as the image of the other. Lets come to a precise analogy between MQM, 2nd quantized matrix model or collective field theory and string theory, which arises as a consequence of the constraint. We know the macroscopic loop operator can be expanded in terms of microscopic operators. To see that in dual matrix model consider $W_{\rm proj}({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o})$,acting on ground state ${\rm log}(1+{\frac{{\hat{\overline{z}}_{+}}{\hat{z}_{+}}+{\hat{\overline{z}}_{-}}{\hat{z}_{-}}-2{H_{o}}}{\mu_{B}^{2}}})|\mu\rangle=\sum\displaystyle{a_{mn}}(\mu)[{\frac{{\hat{\overline{z}}_{+}}{\hat{z}_{+}}}{\mu_{B}^{2}}}{]^{n}}[{\frac{{\hat{\overline{z}}_{-}}{\hat{z}_{-}}}{\mu_{B}^{2}}}{]^{m}}|\mu\rangle$ (2.59) Essentially the above which is described in Minkowskian time corresponds to a coherent state. Also the above is an expansion in terms of microscopic operator. This is evident from the correspondence between the operator ${({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}})}^{n}$ and the vertex operator for discrete tachyonic states.[28]. From the previous discussion it follows that when we consider the space time wave function for each component in W, we found an effect of boundary exactly at $X={X^{o}}({\rm or,}\,t={t_{o}}$). More precisely at this point the state associated with each component of boundary state in (2.49) or the states appear on expansion of W (2.56) in collective field theory, expressed with an wave function which has only Liouville part. So the left and right moving state appears to be identical at $X={X^{o}}({\rm or}\,t={t_{o}}$). Now in order to understand this effect in MQM note that the macroscopic loop operator $W({\hat{\overline{z}}_{+}}{\hat{z}_{+}},{\hat{\overline{z}}_{-}}{\hat{z}_{-}},{H_{o}})$ acting on ground state, essentially create a left moving as well as right moving state (2.59). The constraint (2.35) inside the path integral have an interpretation in MQM that at $t={t_{o}}$ we have $[{{{\hat{\overline{z}}_{+}}{\hat{z}_{+}}}}{]^{n}}||\mu\rangle\equiv[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{n}}|\mu\rangle.$ (2.60) i.e the left and right moving states which are created by the action of $\hat{W}$ on MQM ground state appear to be exactly identical at $t_{o}$ (but different when we move away from $t_{o}$). This leads to the projection $W\rightarrow{W_{\rm proj}}=W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}(t_{o}),{H_{o}})$ in the path integral (2.36). More explicitly the created states obey ${\psi_{+}}({\overline{z}_{+}}{z_{+}}(t_{o});E=ni)={\psi_{-}}({\overline{z}_{-}}{z_{-}}(t_{o});E=-ni)$. This is supported from (2.35) and give the justification for (2.37). Again lets emphasize on the fact that these constraints no way affects the free fermionic states as well as the discrete tachyonic states of the original Type 0A matrix model. This distinction just corresponds to that of the closed string states associated with the boundary states and the free closed string states in the bulk. Now once move away from $t_{o}$ we are back to our original scenario described by (2.30) with the distinguishable states described by (2.41,2.44). An important point to note here is that when a bulk operator approaches to the boundary we encounter a singularity [40]. Similarly in MQM we have the ordering ambiguity of the operators arises when we move from the region $t={t_{o}}$ to $t\neq{t_{o}}$ on time evolution of (2.40) to (2.44). If we consider the entire space of N fermions $N\rightarrow\infty$ this leads to singularity. Now in order to find the exact coefficient of superposition of the states from (2.41) and (2.43) above $t_{o}$ we must remember the fact that these states just show up as indistinguishable at $t_{o}$ while away from $t_{o}$ they are distinct. So the coherent state above $t_{o}$ must be given by (2.44). From the discussion of [11, 28, 2] these operators corresponds to special tachyonic states and higher Virasoro primaries. Now in this context we need to identify the open string operator or the boundary operator. Note that exactly at $t={t_{o}}$ for the states/operators from the string theory(2.47), collective field theory (2.56) or from the MQM (2.60) (when we set the origin of time at ${t_{o}}$), they do not show any explicit time dependence exactly at $t_{o}$. These operators can either be viewed as the one obtained from the one at $t\neq{t_{o}}$ by time evolution as we discussed throughout or the one without any explicit time dependent part. The time independent one must correspond to the states/operators which are extended in Liouville direction but localized in matter direction. Hence they should be identified with the open string operators. Finally let us briefly tell about matrix model string theory correspondence of the complete scenario we just obtained. Theory describes free fermionic state as well as coherent state where the coherent state is strongly localized at $t={t_{o}}$. The closed string state turned into a coherent state at $t_{o}$ as described by (LABEL:scrh2), which follows the view of [44, 5]. The transition amplitude can be read from collective field theory. As the collective field correspond to a spacetime tachyon so from (2.56) we obtain the closed string emission/absorption amplitude, which is giving the one point function in mini superspace approximation $\langle\hat{W}\,\hat{V}(P)\rangle=U(P)e^{-iP{t_{o}}.}$ (2.61) where V(P) is the operator representing tachytonic vertex and the above relation can be viewed on evaluation of the above in 2nd quantized field theory [2]. Now the state in $(\ref{ps})$ while show up as time dependent state w.r.t the free hamiltonian ${H_{o}}$, they are stable w.r,t an effective hamiltonian $H_{\rm eff}={e^{-W}}{H_{o}}{e^{W}}$. The effect of the macroscopic loop operator is to cause transition of a fermion from Fermi level to above. At the double scaling limit $\beta\rightarrow\infty$ the transition amplitude for a single fermion is given from (2.28,2.40) by $\langle{\psi_{>}}({E^{\prime}},t_{f})|{\psi}(E,t_{i})\rangle\sim\langle{\psi_{o}}(E^{\prime})|W|{\psi_{o}}(E)\rangle,$ (2.62) with ${t_{i}}(t_{f})={-\infty}(\infty)$. The presence of D brane will change the Fermi level $\mu\rightarrow\mu^{\prime}$. So from the 2nd quantized theory we can see that the transition amplitude $|\mu\rangle\rightarrow|\mu^{\prime}\rangle$ is in accordance with [4],[38]. $\langle{\mu^{\prime}}|W(l,t)|\mu\rangle=e^{-i\delta(\mu-{\mu^{\prime}})}e^{i(\mu-{\mu^{\prime}})t}K_{i(\mu-{\mu^{\prime}})}(\sqrt{\mu l}).$ (2.63) Note the transition amplitude is time independent which is evident from matter one point function (2.56) and it signifies the fact that we have stable D brane. ## 3 Type 0A MQM on a circle in the presence of D brane ### 3.1 Evaluation of the free energy of Type 0A matrix model on a circle We consider Type 0A matrix model compactified on a circle of radius R. As considered in the previous section, there is no net background D0 brane charge and hence it is described with $U(N)\times U(N)$ gauge symmetry. The partition function in terms of the light cone variables is given by $\displaystyle\int d{Z_{+}}d{Z_{-}}d{\overline{Z}_{+}}d{\overline{Z}_{-}}dAd{\tilde{A}}e^{{-\beta\int_{0}^{2{\pi}R}}dtTr\left[{\overline{Z}_{+}{D_{A}}}{Z_{-}}+{Z_{+}\overline{({D_{A}}Z_{-})}}+{\frac{1}{2}}(\overline{Z}_{-}Z_{+}+\overline{Z}_{+}Z_{-})\right]},$ (3.1) where ${D_{A}}Z={\partial_{t}}Z+i[{A}Z-Z{\tilde{A}}]$ and A is as given in (2.1). Now we fix the gauge ${\partial_{t}}A=0$, which sets $A$ and $\tilde{A}$ to their zero modes $A^{(0)}\equiv X/2\pi\alpha^{\prime}$ and $\tilde{A}^{(0)}\equiv\tilde{X}/2\pi\alpha^{\prime}$, where in the T-dual theory X and ${\tilde{X}}$ corresponds to collective coordinate of D0 and anti D0 brane [9]. As before the gauge fixing introduces the FP determinant [15] $\int db\,dc\exp({\rm Tr}b\partial_{t}D_{t}c)={\prod_{i<j}}{\left({\frac{\sin[(x_{i}-x_{j})R/2]}{(x_{i}-x_{j})R/2}}\right)}^{2}{\left({\frac{\sin[(\tilde{x}_{i}-\tilde{x}_{j})R/2]}{(\tilde{x}_{i}-\tilde{x}_{j})R/2}}\right)}^{2},$ (3.2) where $x_{i}$ and $\tilde{x}_{i}$ are the eigenvalues of $X$ and $\tilde{X}$ respectively. Now the denominator gets canceled with the respective Vandermonde determinant so that the usual measure factor $\Delta(x)^{2}\Delta(\tilde{x})^{2}$ is converted to the measure for unitary matrices ${\prod_{i<j}\sin^{2}({(x_{i}-x_{j})R\over 2})\sin^{2}({(\tilde{x}_{i}-\tilde{x}_{j})R\over 2})}.$ (3.3) Note these are the measure for unitary matrices $U=e^{\frac{iXR}{2}},~{}\tilde{U}=e^{\frac{i\tilde{X}R}{2}}$.which are holonomy factors (we chose ${\alpha^{\prime}=2}$). Therefore the natural variables to be integrated over are the “holonomies” $U=e^{2i\pi A^{(0)}R},~{}\tilde{U}=e^{2\pi i\tilde{A}^{(0)}R}$. Once we gauge fix A and $\tilde{A}$ The partition function depends on the gauge field only through the global holonomy factor, given by the unitary matrix $\Omega=\hat{T}e^{i\int_{0}^{2\pi R}A(t)dt}\quad;\quad\tilde{\Omega}=\hat{T}e^{i\int_{0}^{2\pi R}\tilde{A}(t)dt}.$ (3.4) In the $A=const$ gauge, in the path integral, the constant modes of A can be absorbed by redefining the fields ${Z_{-}},{\overline{Z}_{-}}$ as $Z_{-}(t)\rightarrow{e^{-iAt}Z_{-}(t)e^{i\tilde{A}t}}\quad;\quad\overline{Z}_{-}(t)\rightarrow{e^{-i\tilde{A}t}\overline{Z}_{-}(t)e^{iAt}},$ (3.5) which replaces the periodic boundary condition ${Z_{\pm}}(2{\pi}R)={Z_{\pm}}(0)\quad;\quad{\overline{Z}_{\pm}}(2{\pi}R)={\overline{Z}_{\pm}}(0)$ by a $SU(N)$-twisted one [14],[17] $\displaystyle Z_{+}(2{\pi}R)={Z_{\pm}}(0)$ $\displaystyle\quad;\quad\overline{Z}_{+}(2{\pi}R)=\overline{Z}_{+}(0)$ $\displaystyle Z_{-}(2{\pi}R)={\Omega}Z_{-}(0){\tilde{\Omega}^{-1}}$ $\displaystyle\quad;\quad\overline{Z}_{-}(2{\pi}R)={\tilde{\Omega}}\overline{Z}_{-}(0){\Omega^{-1}},$ (3.6) So in the constant A gauge integration with respect to the fields ${Z_{\pm}}(x),{\overline{Z}_{\pm}}(x)$ is Gaussian with the determinant of the quadratic form equal to one. Therefore it is reduced to the integral with respect to the initial values ${Z_{\pm}},{\overline{Z}_{\pm}}={Z_{\pm}},{\overline{Z}_{\pm}}(0)$ of the action evaluated along the classical trajectories, which satisfy the twisted periodic boundary condition (3.6). Therefore the canonical partition function of the matrix model can be reformulated as an ordinary matrix integral with respect to the hermitian matrices $Z_{+}$ , $Z_{-}$ ; $\overline{Z}_{+}$ , $\overline{Z}_{-}$ and the unitary matrices $\Omega,\tilde{\Omega}$, : ${\cal{Z}}_{N}=\int dZ_{+}dZ_{-}d\overline{Z}_{+}d\overline{Z}_{-}d\Omega d{\tilde{\Omega}}e^{i\beta Tr(\overline{Z}_{+}Z_{-}+Z_{+}\overline{Z}_{-}-q\overline{Z}_{-}\Omega Z_{+}\tilde{\Omega}^{-1}-qZ_{-}\tilde{\Omega}\overline{Z}_{+}\tilde{\Omega})},$ (3.7) where we denote $q=e^{2i\pi R}.$ (3.8) Now note the above expression can be written as $\displaystyle\cal{Z}_{N}$ $\displaystyle=$ $\displaystyle\int dZ_{+}dZ_{-}d\overline{Z}_{+}d\overline{Z}_{-}d\Omega d{\tilde{\Omega}}e^{i\beta{\rm Tr}(\overline{Z}_{+}Z_{-}+Z_{+}\overline{Z}_{-}-q\overline{Z}_{+}\Omega Z_{-}\tilde{\Omega}^{-1}-qZ_{+}\tilde{\Omega}\overline{Z}_{-}{{\Omega}^{-1}})}$ (3.9) $\displaystyle=$ $\displaystyle\int dZ_{+}dZ_{-}d\overline{Z}_{+}d\overline{Z}_{-}d\Omega d{\tilde{\Omega}}e^{i\beta{\rm Tr}(\overline{Z}_{+}Z_{-}+Z_{+}\overline{Z}_{-}-q\overline{Z}_{+}(\Omega Z_{-}{\Omega}^{-1})\Omega\tilde{\Omega}^{-1}-qZ_{+}\tilde{\Omega}{\Omega}^{-1}({\Omega}\overline{Z}_{-}{{\Omega}^{-1}})}$ $\displaystyle=$ $\displaystyle\int dZ_{+}dZ_{-}d\overline{Z}_{+}d\overline{Z}_{-}d\Omega d{\tilde{\Omega}}e^{i\beta{\rm Tr}(\overline{Z}_{+}Z_{-}+Z_{+}\overline{Z}_{-}-q\Omega\tilde{\Omega}^{-1}\overline{Z}_{+}(\Omega Z_{-}{\Omega}^{-1})-qZ_{+}\tilde{\Omega}{\Omega}^{-1}({\Omega}\overline{Z}_{-}{{\Omega}^{-1}})}$ $\displaystyle=$ $\displaystyle\int dZ_{+}^{\prime}dZ_{-}d\overline{Z}_{+}^{\prime}d\overline{Z}_{-}d\Omega d{\tilde{\Omega}}e^{i\beta{\rm Tr}(\tilde{\Omega}{\Omega}^{-1}{\overline{Z}_{+}^{\prime}}Z_{-}+{\overline{Z}_{-}}{Z_{+}^{\prime}}\Omega\tilde{\Omega}^{-1}-q\overline{Z}_{+}^{\prime}\Omega Z_{-}{\Omega}^{-1}-qZ_{+}^{\prime}{\Omega}\overline{Z}_{-}{{\Omega}^{-1}}})$ $\displaystyle=$ $\displaystyle\int dZ_{+}^{\prime}dZ_{-}d\overline{Z}_{+}^{\prime}d\overline{Z}_{-}d\Omega d{\Omega^{\prime}}e^{i\beta{\rm Tr}(\Omega^{\prime}{\overline{Z}_{+}^{\prime}}Z_{-}+{\overline{Z}_{-}}{Z_{+}^{\prime}}{{\Omega^{\prime}}^{-1}}-q\overline{Z}_{+}^{\prime}\Omega Z_{-}{\Omega}^{-1}-qZ_{+}^{\prime}{\Omega}\overline{Z}_{-}{{\Omega}^{-1}}}),$ where we define $Z_{+}\tilde{\Omega}{\Omega}^{-1}=Z_{+}^{\prime}\quad;\quad\Omega\tilde{\Omega}^{-1}\overline{Z}_{+}={\overline{Z}_{+}^{\prime}}$ ; ${{\Omega^{\prime}}}=\tilde{\Omega}{\Omega}^{-1}$.The last expression implies replacing $\tilde{X}$ by $\tilde{X}-X$, both running over the infinite real line. The redefinition of the variables will keep the measure invariant. So by generalizing Harishchandra-Itzykson-Zuber integral we can write the above partition function as 121212$(\int dUe^{iTrUX{U^{-1}}Y}=Const.{\frac{\det{e^{i{x_{k}}{y_{l}}}}}{\Delta(x)\Delta(y)}}$where ${x_{k}}$ and ${y_{l}}$ are eigenvalues of X and Y and ${\Delta(x)}$ ${\Delta(y)}$ are Vandermonde determinant given by $\Delta(x)={\displaystyle\prod_{i\leq j}}({x_{k}}-{x_{l}})$ , 131313To express the part involving ${\Omega^{\prime}}$ we used the fact that in the integral $(\int dUe^{i{\rm Tr}UX{U^{-1}}DY}$ where D is a complex diagonal matrix with ${D^{-1}}={D^{\dagger}}$ and $Y=VyV^{-1}$ where y is the eigenvalue of Y and V is the unitary matrix diagonalizing Y. Now we can write $DY={V^{\prime}}d\,y{{V^{\prime}}^{-1}}$ for some other diagonalizing matrix ${V^{\prime}}$ which exploits the fact that ${\rm Diag}({{(DY)}^{\dagger}}DY)={\rm Diag}({{Y}^{\dagger}}Y)={\rm Diag}({V}{y^{*}}y{V^{\dagger}})$ which implies the above expression (d is eigenvalue of D).So following the formal derivation of the integral we can write $(\int dUe^{iTrUX{U^{-1}}DY}=Const.{\frac{\det{e^{i{x_{k}}{d_{kl}}{y_{l}}}}}{\Delta(x)\Delta(y)}},$ which is nonzero only when $k=l$. for any diagonal matrix D.Note,we are not summing over k and l.Denominators gets canceled with the Vandermonde determinants appearing from $\overline{Z}_{+},Z_{+},\overline{Z}_{-},Z_{-}$.[14]. $\displaystyle{\cal{Z}}_{N}(t)$ $\displaystyle=$ $\displaystyle\int\limits_{-\infty}^{\infty}\prod_{k=1}^{N}[d{z_{+}}_{k}][d{z_{-}}_{k}][d{\overline{z}_{+}}_{k}][d{\overline{z}_{-}}_{k}][d{\Omega^{\prime}}_{kk}]$ (3.10) $\displaystyle[\det_{jk}\left(e^{i{{\Omega^{\prime}}_{jk}}{\overline{z}_{+}}_{j}{z_{-}}_{k}}\right){\rm det}_{jk}\left(e^{-iq{\overline{z}_{+}}_{j}{z_{-}}_{k}}\right){\rm det}_{jk}\left(e^{i{{\Omega^{\prime}}^{-1}}_{jk}{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)$ $\displaystyle{\rm det}_{jk}\left(e^{-iq{z_{+}}_{j}{{\overline{z}_{-}}_{k}}}\right)],$ where ${\Omega^{\prime}_{jk}}$ which has only diagonal elements nonzero. Now we show that the grand canonical partition function can be written as a Fredholm determinant $Z(\mu,t)={\rm Det}(1+e^{-2{\pi}R\beta\mu}K_{+}K_{-}),$ (3.11) where $\displaystyle[{K_{+}}f]({{\overline{z}_{-}}{z_{-}}})$ $\displaystyle=$ $\displaystyle\int[d{\overline{z}_{+}}][d{z_{+}}]dt{e^{i(t{\overline{z}_{+}}{z_{-}}+{t^{-1}}{\overline{z}_{-}}{z_{+}})}}f({{\overline{z}_{+}}{z_{+}}}),$ $\displaystyle[{K_{-}}f]({{\overline{z}_{+}}{z_{+}}})$ $\displaystyle=$ $\displaystyle\int[d{\overline{z}_{-}}][d{z_{-}}]{e^{-iq({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})}}f({{\overline{z}_{-}}{z_{-}}}).$ (3.12) ${K_{+}}{K_{-}}f({\overline{z}_{+}}{z_{+}})=\int[d{\overline{z}_{-}}][d{z_{-}}]dte^{i(t{\overline{z}_{+}}{z_{-}}+{t^{-1}}{\overline{z}_{-}}{z_{+}})}\int[d{\overline{z}_{+}}^{\prime}][d{z_{+}}^{\prime}]e^{-iq({\overline{z}_{+}}^{\prime}{z_{-}}+{\overline{z}_{-}}{z_{+}}^{\prime})}f({{\overline{z}_{+}}^{\prime}}{{z_{+}}^{\prime}}).$ (3.13) Note that t and ${t^{-1}}$ denote the diagonal elements of ${\Omega^{\prime}}$ and ${{\Omega^{\prime}}^{-1}}$ corresponding to ${\overline{z}_{\pm}},{z_{\pm}}$. Now note when we evaluate the determinant in a diagonalizable basis which is naturally given by $f({\overline{z}_{\pm}}{z_{\pm}})$, ${K_{+}}{K_{-}}f({\overline{z}_{\pm}}{z_{\pm}})$ will be independent of t i.e $\int dt$ will come out as an overall factor. So following the analysis of [14] the grand canonical partition function $\displaystyle\sum_{N}{e^{-2{\pi}R\beta N\mu}{\cal{Z}}_{N}}$ can be expressed as the Fredholm determinant (3.11) which is same as that of $c=1$ matrix model. Now following [24] we can express the partition function as $Trexp[-2{\pi}R\beta H]$. The gauge field A project the theory to singlet sector so that in the absence of perturbation, the grand canonical partition function is given by the Fredholm determinant ${\cal{Z}}(\mu)={\rm Det}(1+e^{-2{\pi}R\beta(\mu+H_{0})}),$ (3.14) which must be same as (3.11). This can be interpreted as the grand canonical finite-temperature partition function for a system of non-interacting fermions in the inverse Gaussian potential. The Fredholm determinant can be computed once we know a complete set of eigenfunctions for the one-particle Hamiltonian ${H_{o}}$. Now in order to evaluate the free energy we need to find the density of states, it is conventional to introduce a cutoff $\Lambda$. There is no momentum flow through the wall $\overline{z}z={x^{2}}+{y^{2}}=|{\Lambda}|^{2}$ is implied by the condition $({\hat{x}}{\hat{p}_{y}}+{\hat{x}}{\hat{p}_{y}}){\psi_{\pm}}(x,y){|_{({x^{2}}+{y^{2}}=|{\Lambda}|^{2})}}=({\hat{\overline{z}}_{+}}{\hat{z}_{+}}-{\hat{\overline{z}}_{-}}{\hat{z}_{-}}){\psi_{\pm}}({\overline{z}},z){|_{({\overline{z}}z=|{\Lambda}|^{2})}}=0$, which has a solution $\psi^{E}_{+}({\Lambda})=\psi^{E}_{-}({\Lambda}).$ (3.15) This condition is satisfied for a discrete set of energies $E_{n}(n\in Z)$ defined by $\phi_{0}(E_{n})-E_{n}\log\Lambda+2\pi n=0.$ (3.16) From (3.16) we can find the density of the energy levels in the confined system $\rho(E)={\frac{{\rm log}\Lambda}{2\pi\beta}}-{\frac{1}{2\pi\beta}}{\frac{d\phi_{0}(E)}{dE}},$ (3.17) as derived in [14] Now we can calculate free energy ${\cal{F}}(\mu,R)={\rm log}{\cal{Z}}(\mu,R)$ as ${\cal F}(\mu,R)=\int_{-\infty}^{\infty}dE\,\rho(E)\log\left[1+e^{-2{\pi}R\beta(\mu+E)}\right],$ (3.18) with the density (3.17). Integrating by parts in and dropping out the $\Lambda$-dependent piece, we get ${\cal{F}}(\mu,R)=-{1\over{2\pi\beta}}\int d\phi_{0}(E)\log\left(1+e^{-2{\pi}R\beta(\mu+E)}\right)=-R\int_{-\infty}^{\infty}dE{\phi_{0}(E)\over 1+e^{2{\pi}R\beta(\mu+E)}}..$ (3.19) We close the contour of integration in the upper half plane and take the integral as a sum of residues. This gives for the free energy ${\cal{F}}=-i\sum_{r=n+{\frac{1}{2}}>0}\phi_{o}\left(ir/R-\mu\right).$ (3.20) As the Fredholm determinant is similar to that of $c=1$ MQM so following the analysis of [14], [15] we can see From (3.20) it follows that [19] $2{\rm sin}{\frac{\partial_{\mu}}{2{\beta R}}}\cdot{\cal F}(\mu)=\phi_{o}(-\mu).$ (3.21) Also its shown that the free energy can be expressed as ${\cal F}_{\rm pert}(\mu)_{\\{t_{k}=0\\}}=-\frac{R}{2}\mu^{2}\log\frac{\mu}{\Lambda}-\frac{R+{1\over R}}{24}\log\frac{\mu}{\Lambda}+R\sum\limits_{h=2}^{\infty}\mu^{2-2h}c_{h}(R),$ (3.22) ### 3.2 Free energy of Type 0A matrix model on a circle with D brane In this section we will consider the type 0A matrix model path integral in the presence of a D brane and show the grand canonical partition function can be expressed as the Fredholm determinant. We consider the brane in the NS NS sector and show that how to generalize the analysis for the brane in any other sector. Consider the path integral in the presence of the macroscopic loop operator localized at ${t_{o}}$, which is the generalization of (3.1). The classical action will remain periodic even in the presence of D brane so we can express the $\displaystyle\int d{Z_{+}}d{Z_{-}}d{\overline{Z}_{+}}d{\overline{Z}_{-}}dAd{\tilde{A}}e^{{-\beta\int_{0}^{2{\pi}R}}dt{\rm Tr}\left[{\overline{Z}_{+}{D_{A}}}{Z_{-}}+Z_{+}{D_{A}}{\overline{Z}_{-}}+{\frac{1}{2}}(\overline{Z}_{-}Z_{+}+\overline{Z}_{+}Z_{-})\right]+{\rm Tr}{W(t_{o})}},$ (3.23) The macroscopic loop operator depends on diagonal elements only, so the partition function (3.7) can be expressed as $\displaystyle{{\cal{Z}}_{N}}(t)=\int\limits_{-\infty}^{\infty}\prod_{k=1}^{N}$ $\displaystyle[d{z_{+}}_{k}][d{z_{-}}_{k}][d{\overline{z}_{+}}_{k}][d{\overline{z}_{-}}_{k}][dt_{k}]{\rm det}_{jk}\left(e^{i{t^{-1}_{jk}}{z_{-}}_{j}{\overline{z}_{+}}_{k}}\right){\rm det}_{jk}{\left(e^{-iq{z_{-}}_{j}{\overline{z}_{+}}_{k}}\right)}{\rm det}_{jk}\left(e^{i{t_{jk}}{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)$ $\displaystyle{\rm det}_{jk}{\left(e^{-iq{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)}$ $\displaystyle exp[\displaystyle\sum_{i}log(1+{\frac{{\overline{z}_{+}}_{i}{z_{+}}_{i}+{\overline{z}_{-}}_{i}{z_{-}}_{i}+{\overline{z}_{+}}_{i}{z_{-}}_{i}+{\overline{z}_{-}}_{i}{z_{+}}_{i}}{\mu_{B}^{2}}})],$ (3.24) (when we have off-diagonal t is zero , also we are not summing over j,k) where $\displaystyle\sum_{i}$ is coming from Trace and again above can be expressed as $\displaystyle{\cal{Z}}_{N}(t)$ $\displaystyle=$ $\displaystyle\int\limits_{-\infty}^{\infty}\prod_{k=1}^{N}[d{z_{+}}_{k}][d{z_{-}}_{k}][d{\overline{z}_{+}}_{k}][d{\overline{z}_{-}}_{k}]{\rm det}_{jk}\left(e^{i{t^{-1}_{jk}}{z_{-}}_{j}{\overline{z}_{+}}_{k}}\right){\rm det}_{jk}\left(e^{-iq{z_{-}}_{j}{\overline{z}_{+}}_{k}}\right){\rm det}_{jk}\left(e^{i{t_{jk}}{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)$ (3.25) $\displaystyle{\rm det}_{jk}\left(e^{-iq{\overline{z}_{-}}_{j}{z_{+}}_{k}}\right)\prod_{r=1}^{N}(1+{\frac{{\overline{z}_{+}}_{r}{z_{+}}_{r}+{\overline{z}_{-}}_{r}{z_{-}}_{r}+{\overline{z}_{+}}_{r}{z_{-}}_{r}+{\overline{z}_{-}}_{r}{z_{+}}_{r}}{\mu_{B}^{2}}}).$ Now in order to write the above expression we have used the fact that the classical action is periodic even in the presence of the macroscopic loop operator $Z(2\pi R)=Z(0)$. However at the quantum level there is a discontinuity of state $|\psi(2{\pi}R-\epsilon)\rangle\neq|\psi(0)+\epsilon\rangle$. This causes the absence of the vortex d.o.f. Now if $f({{\overline{z}_{\pm}}{z_{\pm}}})$ is the function which form the representation of $K_{+}$ and $K_{-}$ (3.12, 3.13), the action of W on f is given by ${\hat{W}}f({{\overline{z}_{\pm}}{z_{\pm}}})=(1+{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}+{\hat{\overline{z}}_{+}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}{\hat{z}_{+}}}{\mu_{B}^{2}}})f({{\overline{z}_{\pm}}{z_{\pm}}}).$ (3.26) Now, the operator ${\hat{W}}$ does not introduce any interaction between the fermions, so no off diagonal terms from W. Now from (3.11) and (3.24) we can write the grand canonical partition $\sum_{N}{e^{-2{\pi}R\beta N}}{\cal{Z}}_{N}$ as ${\cal{Z}}(\mu)={\rm det}(1+e^{-2\pi R\beta\mu}WK).$ (3.27) Now in order to evaluate (3.27) following (3.13) a representation of K is formed by the basis $f({{\overline{z}_{+}}_{i}}{{z_{+}}_{i}})$ with i runs from 1 to N. Also from (3.13) it follows that $f({\overline{z}_{+}}{z_{+}})\sim{{({\overline{z}_{+}}{z_{+}})}^{n}}$. So when we evaluate the expectation value of WK in this basis, in the expression of W we see that $\langle{\overline{z}_{\pm}}{z_{\pm}}\rangle=0$ as on a closed contour the angular integral will vanish. in the inner product, the other term ${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}$ expresses nothing but the hamiltonian of which f is an eigenfunction. So if ${\psi_{n}}$ are the set of functions which diagonalizes K we can write (3.27) as $\displaystyle\sum_{n}{\rm log}\langle{\psi_{n}}|(1+e^{-2\pi R\beta\mu}\hat{W}\hat{K})|{\psi_{n}}\rangle,$ (3.28) where $\displaystyle\hat{W}{K_{+}}{K_{-}}f({\overline{z}_{+}}{z_{+}})$ $\displaystyle=$ $\displaystyle\int[d{\overline{z}_{-}}][d{z_{-}}]e^{i({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})}[d{\overline{z}_{+}}^{\prime}][d{z_{+}}^{\prime}]e^{-iq({\overline{z}_{+}}^{\prime}{z_{-}}+{\overline{z}_{-}}{z_{+}}^{\prime})}$ (3.29) $\displaystyle(1+{\frac{{\overline{z}_{+}}^{\prime}{z_{-}}+{\overline{z}_{-}}{z_{+}}^{\prime}}{\mu_{B}^{2}}})f({{\overline{z}_{+}}^{\prime}}{{z_{+}}^{\prime}}).$ As the expression depends on $({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})$ which is the expression for free hamiltonian ${H_{o}}$ so comparison with (3.14), Fredholm determinant is expected to be given by ${\cal{Z}}(\mu)={\rm det}(1+e^{-2{\pi}R\beta(\mu+H_{0})-\log(1-{\frac{2{H_{o}}}{\mu_{B}^{2}}})}).$ (3.30) This is,we are going to analyze in the next part of this section. ### 3.3 Evaluation of the thermal partition function In this section we are going to study type 0A MQM in the presence of D brane with time t compactified on a circle, evaluate and analyze the free energy. In the absence of the brane when we compactify string theory on a circle of radius R, in the dual MQM the Schrodinger equation have periodic solution i.e ${\psi}(t)={\psi}(t+2{\pi}R)$, which implies $E={\frac{n}{R}}$. Now consider the theory with D brane which can be accomplished by including a macroscopic loop operator localized at $t={t_{o}}=0\equiv 2{\pi}R$ (say) to the action. From previous discussion it follows that in the presence of the operator Schrodinger equation will have well defined solution only in the region $0\leq t\leq 2{\pi}R$ when discontinuity occur at the respective point and we have $\psi({2{\pi}R-\epsilon})\neq\psi({2{\pi}R+\epsilon})$ in the limit $\epsilon\rightarrow 0$. This is consistent with the fact that the presence of a spacelike brane breaks the winding symmetry and apparently the theory correspond to that of an open string. At the end we will see how the closed string scenario arise in this picture. Now in a compact time we must have the condition ${\psi}(t)={\psi}(2{\pi}R+t)$. So effectively we can view the theory as MQM on a line of length ${2{\pi}R}$ with two ${\delta}\textendash{\rm potential}$ along with the operator $\hat{W}$ (where one is the image of the other, superimposed) placed at its two ends. When we cross the boundary on either side, situation repeats 141414note it never implies periodicity, its just similar to the situation of an open string in 2D with Dirichlet boundary condition in compact direction and identification of the matter direction with t. It winds along the circle m times although ends are not identified. The open string which wraps m times a circle of length $2\pi{R^{\prime}}$ with $2\pi m{R^{\prime}}=2{\pi}R$, we can define same theory on either of the slices $2\pi(n-1)R\leq t\leq 2\pi nR$, crossing the boundary of the slice implies going back from that end of the string to the other and hence the situation repeats, i.e we can define the theory on any of the slices $2\pi(n-1)R\leq t\leq 2\pi nR$. So effectively we have the time dependent Schrodinger equation with double delta potential well as: $\displaystyle[\,\,i{\frac{\partial}{{\partial}t}}$ $\displaystyle-$ $\displaystyle\\{{\delta}(t)+{\delta}(t-2{\pi}R)\\}W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}(t),{H_{o}})]{\Psi}({\overline{z}_{\pm}}{z_{\pm}},t)$ (3.31) $\displaystyle=$ $\displaystyle{\mp}i\left[{z_{\pm}}{\frac{\partial}{\partial{z_{\pm}}}}+{{\overline{z}_{\pm}}}{\frac{\partial}{\partial{\overline{z}_{\pm}}}}+1\right]{\Psi}({\overline{z}_{\pm}}{z_{\pm}},t),$ We have the discontinuity $\displaystyle{\psi}({\epsilon})-{\psi}({-\epsilon})$ $\displaystyle=$ $\displaystyle{\hat{W}}{\psi_{o}}(t=0)$ $\displaystyle{\psi}({2{\pi}R}+{\epsilon})-{\psi}({2{\pi}R-\epsilon})$ $\displaystyle=$ $\displaystyle{\hat{W}}{\psi_{o}}(t={2{\pi}R}).$ (3.32) In order to evaluate the partition function we must need to know what is the right Hilbert space describe the wave function $\psi$ on the circle. This is because we know that the Hilbert space $\\{|E\rangle\\}$ and $\\{|E\pm ni\rangle\\}$ cannot be mapped to each other. So, to answer this note that when we define the Schrodinger equation in the double delta potential well in an uncompactified direction we have one type of the solution inside the well, while the solution at the left and the rightside of the well differs from the same, decided by the discontinuity (3.32). Now compactification on a circle of length $2{\pi}R$ imply the outside region of the well is just squeezed to a point $t=0\equiv 2{\pi}R$ and the wave function at the left and right side of the double delta-well are given by the wave function at the right and left $\epsilon\textendash{\rm neighborhood}$ of that point. Now in uncompactified time we had free fermion wave function (2.28) in the region $-\infty$ to ${t_{o}}$ while from ${t_{o}}$ to $\infty$ the wave function is described by (2.44). So in the compactified time we have the ambiguity that which one should describe the fermionic wave function in the region $0\leq t\leq 2{\pi}R$. To resolve recall the macroscopic loop operator $\hat{W}$ ( 2.15) and the constraint (2.25) are symmetric under ${\overline{z}_{+}},{z_{+}}\rightarrow{\overline{z}_{-}},{z_{-}}$ . So in the Schrodinger equation(2.12) we have the symmetry, $t\rightarrow{2{\pi}R-t};\quad\hat{W}\rightarrow-\hat{W},$ (3.33) which takes an wave function ${\overline{z}_{+}},{z_{+}}\rightarrow{\overline{z}_{-}},{z_{-}}$ representation151515 this is from the definition of ${\overline{z}_{\pm}},{z_{\pm}}$ (2.6) and the reversal of the sign of W is explained from (3.32),the transformation changes the sign in the r.h.s of (3.32) because the time interval $\epsilon\rightarrow-\epsilon$ under the transformation and hence the relation will remain unchanged.Note this transformation is also associated with the reversal of the sign of the gauge field A. but we chose axial gauge $a=\tilde{a}$,where $a,\tilde{a}$ are the zero modes of gauge fields $A,\tilde{A}$, so this is not affected along with reversal of the sign of the energy $E\rightarrow-E$ in the free fermionic wave function as introduced in (2.28). As both ${\overline{z}_{+}},{z_{+}}$ and ${\overline{z}_{-}},{z_{-}}$ describes same wave function in different representation so it must be a symmetry inside the double delta well (note this is never a symmetry outside the well where the fermion can see only one potential barrier). The condition (2.25) remains unaffected by this symmetry and we can project the wave function $t={t_{o}}=0\equiv 2{\pi}R$ to the physical sector. Now under the transformation (3.33) the free fermion wave function (2.28) is just changed by a phase ${e^{2iE\pi R}}$ whereas according to (2.28,2.30,2.41) the wave function describes a state $|E\pm ni\rangle$ goes from $e^{-iEt\pm nt}\rightarrow e^{iE(2{\pi}R-t)\mp(2\pi nR-nt)}$. Although the transformation (3.33) take the wave function from ${z_{+}}$ to ${z_{-}}$ representation but the both have the same time dependent part. So when we consider the wave function (2.41) we see it does not respect the symmetry. Note unlike the closed string momentum modes which respects the symmetry in Euclidean time because of their periodicity on the circle, $W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})$ generates discrete shift in energy $E\pm ni$ at any R, which leads to the violation of symmetry. So we conclude the wave function inside the well which is our compact time $0\leq t\leq 2{\pi}R$ must corresponds to that of a free fermion (2.28). We will come to the string theoretical interpretation in the next subsection. The wave function at $t=0\equiv 2{\pi}R$ corresponds to (2.41) so the partition function corresponds to the transition amplitude $\displaystyle\lim_{\epsilon\to 0}\,\langle{\psi_{o}}{(\epsilon)}|\psi_{>}(2{\pi}R-\epsilon)\rangle$ $\displaystyle=$ $\displaystyle\langle{\psi_{o}}[{e^{-{\beta\int_{0}^{2{\pi}R}}dt{\hat{H}_{o}}}}{e^{{\hat{W}}({t_{o}})}}]{\psi_{o}}\rangle$ (3.34) $\displaystyle=$ $\displaystyle{{\rm Tr}_{\psi_{o}^{E}}}[{e^{-{2{\pi}R\beta}{\hat{H}_{o}}}}[{e^{{\hat{W}}({t_{o}})}}]\ ]$ $\displaystyle=$ $\displaystyle{{\rm Tr}_{\psi_{o}^{E}}}\left[{e^{-{2{\pi}R\beta}[{\hat{H}_{o}}-{\frac{1}{2{\pi}R\beta}}\hat{W}({t_{o}})]}}\right],$ where ${{\rm Tr}_{\psi_{o}^{E}}}$ implies the summation over all free fermion eigenfunctions and $\psi_{>}$ corresponds to (2.41). As $\beta\rightarrow\infty$ at double scaling limit, so inside the partition function we can replace it by $\psi_{>}\rightarrow e^{-W{(t_{o})}}{\psi_{o}}\,$ 161616In order to reach from the 1rst to 2nd step in (3.34)we utilize the fact that we can scale the time $t\rightarrow\beta t$ so that the term with the macroscopic loop operator $\int dtW(t)\delta(t-{t_{o}})$ will get a factor ${\frac{1}{\beta}}$. Hence in the double scaling limit where $\beta\rightarrow\infty$ and with Euclidean time, we can lift up the term to the exponential and the exponent gives an exact expression what we have obtained from the path integral (3.23) . Also following the discussion of section 2 we can directly add $W(t_{o})$ in the expression of hamiltonian in Euclidean time to get the expression (3.34) . Note when we consider Schrodinger equation, the contribution from the term ${\frac{1}{2{\pi}R\beta}}\hat{W}({t_{o}})$ in the expression of hamiltonian (3.34) at double scaling limit will not be negligible due to a transformation of variable which leads to the physics at the vicinity of the top of the potential, as we discussed section $2.1$ and can be found [1]. From (2.15) we know that in a single fermionic state the presence of D brane implies implies the insertion of the following operator ${e^{{\hat{W}}({t_{o}})}}=1+{\frac{{\hat{\overline{z}}_{+}}{\hat{z}_{+}}+{\hat{\overline{z}}_{-}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}{\hat{z}_{+}}+{\hat{\overline{z}}_{+}}{\hat{z}_{-}}}{\mu_{B}^{2}}}.$ (3.35) Here first we will evaluate the partition function for single fermionic d.o.f in order to understand the behaviour of the system in the presence of a brane. Next we will derive the grand canonical partition function. Now following the discussion in section 2 we can represent the wave function (2.40) at $t=2{\pi}R-\epsilon$ in (3.34) with the operator $\hat{W}$ expressed either in terms of ${\hat{\overline{z}}_{+}},{\hat{z}_{+}}$ or ${\hat{\overline{z}}_{-}},{\hat{z}_{-}}$ representation. We have shown in the Appendix that $\langle{\overline{z}_{+}}{z_{+}}|{\hat{\overline{z}}_{+}}{\hat{z}_{+}}|{\overline{z}_{+}}{z_{+}}\rangle$ and $\langle{\overline{z}_{-}}{z_{-}}|{\hat{\overline{z}}_{-}}{\hat{z}_{-}}|{\overline{z}_{-}}{z_{-}}\rangle$ diverge. So we must express $\hat{W}$ as $W({{\hat{\overline{z}}_{-}}{\hat{z}_{-}},H_{o}})$ for the basis $|{\overline{z}_{+}}{z_{+}},E\rangle$ basis and vice versa. So applying (A.7) we can write (3.34) as: ${{\rm Tr}_{\psi_{o}}}\left[{e^{-{2{\pi}R\beta}[{\hat{H}_{o}}-{\frac{1}{2{\pi}R\beta}}{\rm log}(1-{\frac{2\hat{H}_{o}}{\mu_{B}^{2}}})]}}\right]={{\rm Tr}_{\psi_{o}}}\left[e^{-2{\pi}R\beta H_{o}^{\prime}}\right],$ (3.36) Where $\hat{H}_{o}^{\prime}={\hat{H}_{o}}-{\frac{1}{2{\pi}R\beta}}{\rm log}(1-{\frac{2\hat{H}_{o}}{\mu_{B}^{2}}})\quad;\quad{E_{o}^{\prime}}={E}-{\frac{1}{2{\pi}R}}{\rm log}(1-{\frac{2{E}}{\mu_{B}^{2}}}),$ (3.37) with $E_{o}^{\prime}$ is the eigenvalue of $H_{o}^{\prime}$(note that we omitted $\beta$ factor from the expression of ${E_{o}^{\prime}}$ following the discussion of section 2, which can be done at double scaling limit by redefinition of the variable). Before any further analysis let us make the comment that here we have considered the macroscopic loop operator in the NS NS sector. For any other sector we can use analysis of section 2, expressing W in terms of ${\hat{\overline{z}}_{-}}.{\hat{z}_{-}}({\hat{\overline{z}}_{+}},{\hat{z}_{+}})$ in ${\overline{z}_{+}},{z_{+}}({\overline{z}_{-}},{z_{-}})$ representation and using (A.7) to express $W({t_{o}})$ complete in terms of ${H_{o}}$ within the trace. Although we will have a very different expression of ${H_{o}^{\prime}}$ but the analysis will remain same. Also if we did not apply this condition (2.25), note we will have the term $\langle{\hat{\overline{z}}_{-}}{\hat{z}_{-}}\rangle_{-}(\,\,\langle{\hat{\overline{z}}_{+}}{\hat{z}_{+}}\rangle_{+}\,\,)$ in the partition function from the expression of $\hat{W}$. This gives rise to an infinite contribution to the partition function $\lim_{r\to\infty}r{e^{i\phi(E-i)}}$ (where $\phi$ is the phase of the wave function) and so the partition function diverge. This is the signature of the presence of an unphysical degree of freedom leads to instability of the system due to leakage. Now according to the discussion of section 2, at the double scaling limit the partition function (3.36) can be expressed as the sum over ${E_{o}^{\prime}}$the eigenvalue of ${H_{o}^{\prime}}$ as: ${{\rm Tr}_{\psi_{o}}}\left[e^{-2{\pi}R\beta H_{o}^{\prime}}\right]=\sum_{E}\left[{e^{-{2{\pi}R\beta}[{E}-{\frac{1}{2{\pi}R}}{\rm log}(1-{\frac{2{E}}{\mu_{B}^{2}}})]}}\right]$ (3.38) Now note that ${E_{o}^{\prime}}$,the eigenvalue of ${H_{o}^{\prime}}$, has branch cut at $E={\frac{\mu_{B}^{2}}{2}}$ so we need to subtract a small cut- off ${\rm log}\epsilon$ in order to have an well defined expression of the energy and after subtraction $E\rightarrow{E^{\prime}}$ is an one to one mapping, Also note at the singular point, $E={\frac{1}{2}}{\mu_{B}^{2}}$, $e^{-2{\pi}R\beta{E^{\prime}}}$ is trivially zero and so it will not contribute any pole to integrand. Now the string theory compactified at time interval $2{\pi}R$ is described by the grand canonical partition function of fermion at finite temperature ${\frac{1}{2{\pi}R}}$ and chemical potential ${\mu}$. So the free energy ${{\cal F}}={\rm log}{\cal{Z}}$ in the presence of Dbrane is given by ${{\cal F}}({\mu})={\int_{\infty}^{\infty}}dE{\rho}(E)log(1+{e^{-{\beta}({\mu}+{E^{\prime}}(E))}}),$ (3.39) where$\rho(E)$ is given in (3.17) $e^{i\phi_{0}(E)}=R(E)={\frac{\Gamma(iE+1/2)}{\Gamma(-iE+1/2)}}.$ (3.40) Now we can calculate free energy ${\cal{F}}(\mu,R)={\rm log}{\cal{Z}}(\mu,R)$ as. ${\cal{F}}(\mu,R)=\int_{-\infty}^{\infty}dE\rho(E)\log[1+e^{-\beta(\mu+{E^{\prime}}(E))}].$ (3.41) with the density (3.17),and from (3.39,3.37) the free energy is given by $\displaystyle{\cal F}(\mu,R)$ $\displaystyle=$ $\displaystyle-{\frac{1}{2\pi}}\int d{\phi_{0}}(E){\rm log}\left(1+{e^{-2{\pi}R\beta({\mu}+{E^{\prime}}(E))}}\right)$ (3.42) $\displaystyle=$ $\displaystyle-R{\int_{-\infty}^{\infty}}d{E^{\prime}}{\frac{\phi_{0}(E({E^{\prime}}))}{1+e^{-2{\pi}R\beta(\mu+{E^{\prime}}(E)))}}}$ $\displaystyle=$ $\displaystyle-i{\sum_{r=n+{\litfont{1\over 2}}>0}\phi_{o}(E({E^{\prime}}={\frac{ir}{\beta R}}-\mu))}$ $\displaystyle=$ $\displaystyle-i{\sum_{r=n+{\litfont{1\over 2}}>0}}{\phi_{o}^{\prime}}({\frac{ir}{R}}-\mu),$ where we have ${\phi_{0}^{\prime}}({E^{\prime}})={\phi_{0}}(E)$ (3.43) So from the expression of the density of the energy eigenstates (3.17) (ignore the $\Lambda$ factor ) this implies the number of energy eigenstates between E to E+dE is same as that of between ${E^{\prime}}$ to ${E^{\prime}}+d{E^{\prime}}$. So the partition function on a circle in the presence of D brane corresponds to a deformation in static Fermi sea where the deformation is expressed as $E=-\mu\,\Rightarrow\,E^{\prime}=-\mu$ with all the energy eigenstates are in one to one mapping. Note that the partition function is getting contribution only from the deformation of Fermi surface instead of excitation modes. This is because we are in compact dimension and its only the trace over energy eigenstates contribute, which we will explain more in the next subsection. Finally the expression of the free energy ${\cal F}(\mu,R)$ in (3.42) along with (3.43) suggests the following $\displaystyle{{\rm Tr}_{\psi_{o}}}\left[{e^{-{2{\pi}R\beta}[{\hat{H}_{o}}-{\frac{1}{2{\pi}R\beta}}{\hat{W}}({t_{o}})]}}\right]$ $\displaystyle=$ $\displaystyle{{\rm Tr}_{\psi_{o}}}[{e^{-{2{\pi}R\beta}\\{{H_{o}}-{\frac{1}{2{\pi}R\beta}}{\log}(1-{\frac{2{H_{o}}}{\mu_{B}^{2}}})\\}}}]$ (3.44) $\displaystyle=$ $\displaystyle{{\rm Tr}_{\psi_{o}}}[{e^{-{2{\pi}R\beta}\\{{H_{o}}-{\frac{1}{2{\pi}R\beta}}f({H_{o}})\\}}}]$ $\displaystyle=$ $\displaystyle{{\rm Tr}_{\psi_{o}}}[{e^{-{2{\pi}R\beta}({H_{o}^{\prime}})}}]$ $\displaystyle=$ $\displaystyle{{\rm Tr}_{\psi_{o}^{\prime}}}[{e^{-{2{\pi}R\beta}({H_{o}})}}],$ where $f({H_{o}})={\log}(1-{\frac{2{H_{o}}}{\mu_{B}^{2}}}){\quad;\quad}{H_{o}^{\prime}}={H_{o}}-{\frac{1}{2{\pi}R\beta}}f({H_{o}}).$ (3.45) and in the last step we made a transformation from the basis ${\psi^{\pm}_{o}}(E)={e^{\mp{i\phi_{o}}(E)}}{e^{-iEt}}{{({\overline{z}_{\pm}}{z_{\pm}})}^{\pm iE-{\frac{1}{2}}}}\rightarrow{\psi^{\pm}_{o}}({E^{\prime}})={e^{\mp{i\phi_{o}^{\prime}}({E^{\prime}})}}{e^{-i{E^{\prime}}t}}{{({\overline{z}_{\pm}}{z_{\pm}})}^{\pm i{E^{\prime}}-{\frac{1}{2}}}}$ with ${E^{\prime}}=E-{\frac{1}{2{\pi}R}}{\log}(1-{\frac{2E}{\mu_{B}^{2}}})$ and also ${\phi_{o}^{\prime}}({E^{\prime}})={\phi_{o}}(E)$. Note as we discussed in section 2, although the contribution from the macroscopic loop operator has a factor ${\frac{1}{\beta}}$ however in the double scaled hamiltonian it cannot be ignored and the shifted energy will be given by $E^{\prime}$. Above expression implies that the Type 0A MQM on a circle in the presence of a D brane can be viewed as a free theory with the free hamiltonian ${H_{o}}$ with the wave function replaced by the above one. This point will be relevant in section 5. Note the relation (3.21,3.22) can be expressed as $2{\rm sin}{\frac{\partial_{\mu}}{2\beta R}}\cdot{\cal F}(\mu)=\phi_{o}^{\prime}(-\mu).$ (3.46) Also note that from nonlinear relation between the effective hamiltonian ${H_{o}^{\prime}}$ and the free hamiltonian ${H_{o}}$ its evident that in the genus expansion of free energy in the relation (3.22) will have both odd and even powers of $\mu$( this is because the new free energy corresponds to replacing $\mu\rightarrow E(E^{\prime}){|_{E^{\prime}=\mu}}$ which follows from (3.39)). This is the signature of the presence of surface with boundary which is the implication from MQM/string theory duality. ### 3.4 String theoretical interpretation Let us very briefly say about the string theory side of the above story. The matrix model partition function (3.44) corresponds to the disk amplitude $\langle B|q^{(L_{o}+\overline{L}_{o})}|I\rangle$ (3.47) Where $|B\rangle$ stands for boundary state and the above expression resembles (3.34) on the matrix model side. The invariance of theory under the symmetry (3.33) is related to the symmetry of the boundary state (2.47,2.48) under interchange of the left and right moving tachyonic components in W $U(k)e^{ik(X+\phi)-\sqrt{2}\phi}\leftrightarrow U(-k)e^{ik(X-\phi)-\sqrt{2}\phi}$ (3.48) as well as other Virasoro primaries obtained from discrete shift of matter and Liouville momentum of tachyonic modes. One can see that in absence of D brane this is a symmetry in MQM. The symmetry is implemented from the interchange $X\rightarrow 2{\pi}R-X$ followed by a sign reversal of momentum $k\rightarrow-k$ which is in accordance (3.33) with $k={\frac{n}{R}}$ and $X\rightarrow it$, U(k) is the wave function. This keeps the matter part ( temporal part in matrix model) of the operator/state invariant. The reason for this symmetry in matrix model is that the tachyonic states obtained from the 2nd quantized free fermionic theory (ground state) or the collective field theory are actually in one to one to one correspondence with the above operators(3.48), as we explained in section 2.3. Consequently we found that the symmetry (3.33) projects the Hilbert space between $0<t<2{\pi}R$ to free fermionic ground state. This is because the discrete tachyonic modes arises from Minkowskian theory with shift $X\rightarrow iX\,;\,k\rightarrow ik\,\,\Rightarrow e^{ikX}(e^{ikt})\rightarrow e^{ikX}(e^{ikt})$. However the states created by the action of the operator ${({\hat{\overline{z}}_{-}}{\hat{\overline{z}}_{+}})}^{n}{({\hat{\overline{z}}_{-}}{\hat{\overline{z}}_{+}})}^{m}$ on ground state, although describes a state with imaginary energy but remains in Minkowskian time (unperiodic) and so projected out by the symmetry within $0<t<2{\pi}R$. However these states arise on expansion of $W({\hat{\overline{z}}_{-}}{\hat{z}_{-}},{\hat{\overline{z}}_{+}}{\hat{z}_{+}},{H_{o}})$ which actually causes the excitation of a fermion over free fermionic ground state,creates a coherent state. So we have only free fermionic ground state in the region $0<t<2{\pi}R$ which in string theory indicates that we have only free closed string modes in the region $0<X<2{\pi}R$. Coherent states are strongly localized at the point of insertion of $\hat{W}$. In matrix model partition function, the absence of the excitation modes has an explanation in the fact that on a circle in the presence of a brane, partition function corresponds to transition $\psi\rightarrow{\hat{W}}\psi$ exactly at $t={t_{o}}=0\equiv 2{\pi}R$, where $|\psi\rangle$ is the free fermionic ground state. So the operators from W which generates excitation naturally will not contribute in the trace. This, alongwith the condition (2.25)(which ensures the conservation of fermion number) implies that the partition function will corresponds to that of a deformed Fermi surface. The poles of the partition function (3.42) corresponds to free closed string tachyons in the bulk. In section 5 we will see that same features will be reflected even when we turn on the tachyonic background. ## 4 Fermionic scattering and semiclassical analysis In this section we will study scattering of fermions in the presence of D brane and tachyonic background at quasiclassical limit. The scattering amplitude is given by $S=\langle\beta,{t\rightarrow\infty}|\alpha,{t\rightarrow-\infty}\rangle,$ (4.1) where ${\alpha}$ and ${\beta}$ denotes the incoming and outgoing state. As the single incoming and outgoing state is given by $|{\overline{z}_{+}}{z_{+}}\rangle$ and $|{\overline{z}_{-}}{z_{-}}\rangle$ respectively [28], so $S=\langle{\overline{z}_{-}}{z_{-}},{\rm out}|{\overline{z}_{+}}{z_{+}},{\rm in}\rangle.$ (4.2) Now note that ${\overline{z}_{+}}{z_{+}}$ and ${\overline{z}_{-}}{z_{-}}$ representations are related by a unitary operator $\hat{S}$, which in our case is nothing but the Fourier transformation on the complex plane. Recall, the energy eigenstates in absence of the D brane are given by (2.28). The wave functions in $(z_{+},\bar{z}_{+})$ and $(z_{-},\bar{z}_{-})$ representations are related by $\displaystyle\psi_{-}(z_{-},\bar{z}_{-})$ $\displaystyle=$ $\displaystyle\hat{S}{\psi_{+}}(z_{-},\bar{z}_{-})$ (4.3) $\displaystyle=$ $\displaystyle\int dz_{+}d\bar{z}_{+}K(\bar{z}_{-},z_{+})K(z_{-},\bar{z}_{+})\psi_{+}(z_{+},\bar{z}_{+}),$ where $K(z_{-},z_{+})={1\over\sqrt{2\pi}}e^{iz_{-}z_{+}}$. Acting on energy eigenstates, we have $\hat{S}\psi_{+}^{E}={\cal{R}}(E)\psi_{-}^{E},~{}~{}~{}~{}{\cal{R}}(E)={\frac{\Gamma(iE+{\frac{1}{2}})}{\Gamma(-iE+{\frac{1}{2}})}}.$ (4.4) The factor ${\cal R}(E)$ is a pure phase $\overline{{\cal R}(E)}{\cal R}(E)={\cal R}(-E){\cal R}(E)=1,$ (4.5) which proves the unitarity of the operator $\hat{S}$. Now in absence of the D brane wave function (2.28) evolve according to free hamiltonian ${H_{o}}$ so from orthonormality of the wave functions (4.2) is given by ${\cal R}(E){\delta}({E_{+}}-{E_{-}})$. The operator $\hat{S}$ relates the incoming and the outgoing waves and therefore can be interpreted as the fermionic scattering matrix. The factor ${\cal R}(E)$ is identical to the the reflection coefficient. This condition can also be expressed as the orthonormality of in and out eigenfunctions $\langle\Psi^{{}_{E_{{-}}}}_{-}|K|\Psi^{{}_{E_{{+}}}}_{+}\rangle=\delta(E_{+}-E_{-}),$ (4.6) with respect to the scalar product. We usually absorb the factor${\cal R}(E)$ in phase by defining $e^{i{\phi(E)}}={\cal R}(E)$ (4.7) to make the wave function biorthogonal, where ${\phi(E)}$ is the phase of the incoming and the outgoing wave function(2.28). Now consider the presence of D brane. For a single fermionic state, (4.2) is given by $\langle{\overline{z}_{+}}{z_{+}},t=\infty|{e^{\hat{W}(t)}}|{\overline{z}_{-}}{z_{-}},t=\infty\rangle=\langle{\overline{z}_{+}}{z_{+}},{\rm out}|(1-{\frac{{\overline{z}_{+}}{z_{+}}+{\overline{z}_{-}}{z_{-}}-2{H_{o}}}{\mu_{B}^{2}}})|{\overline{z}_{-}}{z_{-}},{\rm in}\rangle.$ (4.8) Now according to (2.32) W will be expressed either in ${\overline{z}_{+}}{z_{+}}$ or ${\overline{z}_{-}}{z_{-}}$ mode $\displaystyle\langle{z_{+}},E_{+},{\rm out}|e^{{\hat{W}}({t_{o}})}|{{z_{-}}},{E_{-}},{\rm in}\rangle$ $\displaystyle=$ $\displaystyle\langle{{z_{+}}},E_{+}|(1+{\frac{2{\overline{z}_{-}}{z_{-}}-2{H_{o}}}{\mu_{B}^{2}}}{)_{t={t_{o}}}}|{z_{-}},{E_{-}}{\rm in}\rangle$ (4.9) $\displaystyle=$ $\displaystyle\langle{z_{+}},E_{+}|(1+{\frac{-2{H_{o}}}{\mu_{B}^{2}}})|{z_{-}},{E_{-}}\rangle$ $\displaystyle=$ $\displaystyle\langle{z_{+}},E_{+}|(1-{\frac{2E}{\mu_{B}^{2}}})|{z_{-}},{E_{-}}\rangle$ $\displaystyle=$ $\displaystyle{\cal R}(E_{+}){e^{log(1-{\frac{2E}{\mu_{B}^{2}}})}}\delta(E_{+}-{E_{-}}),$ where in the 2nd step we have $\langle{{\overline{z}_{+}}{z_{+}}},E_{+};{\rm out}|{\hat{\overline{z}}_{-}}{\hat{z}_{-}}|{{\overline{z}_{-}}{z_{-}}},{E_{-}},{\rm in}\rangle=0$ from (A.6) . Now the presence of the D brane will modify the phase of the outgoing state over the incoming, which is given by the factor ${e^{log(1-{\frac{2E}{\mu_{B}^{2}}})}}$. So for the change of phase $\delta\phi(E)$ we can write $e^{-i\frac{\delta\phi(E)}{2}}={e^{log(1-{\frac{2E}{\mu_{B}^{2}}})}}$ (4.10) Now in (4.10) using the relation (4.7) will leave us with the amplitude $e^{-i\frac{\delta\phi(E)}{2}}$. In [43] its explained that the complex phase in the wave function is the signature of tunneling and we can presume the above factor accounts for the same. Also note that instead of the above macroscopic loop operator in NS sector, if we took the macroscopic loop operator in some other sector given by ${W^{\prime}}({\frac{{\overline{z}_{+}}{z_{+}},{\overline{z}_{-}}{z_{-}},{H_{o}}}{\mu_{B}^{2}}})$ according to the discussion in Appendix we will have the phase shift of the outgoing state ${\frac{\phi(E)}{2}}+i{W^{\prime}(1-{\frac{2E}{\mu_{B}^{2}}})}$. Lets consider scattering of a tachyonic state (which are being created from the action of ${({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}})}^{n}$ on fermionic ground state) from D brane. This is better understood from the collective field theory where scattering to a single tachyonic state with energy E is given by $\sim U(E)$ where U(E) is given by (2.48). This supports the fact that the D brane act as a coherent source of closed strings. Now we consider the classical limit $\beta\rightarrow\infty$. At this limit the ground state of MQM is obtained by filling all energy levels up to some fixed Fermi energy which we choose to be $E_{F}=-\mu$. Quasiclassically every energy level corresponds to a certain trajectory in the phase space of ${\overline{z}_{+}}{z_{+}},{\overline{z}_{-}}{z_{-}}$ variables and they are separated by a factor ${\frac{1}{\beta}}$. The Fermi sea can be viewed as a stack of all classical trajectories with $E\leq E_{F}$ and the ground state is completely characterized by the curve representing the trajectory of the fermion with highest energy $E_{F}$. For the Hamiltonian ${H_{o}}$ all trajectories are hyperboles ${\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}}=-E$ and the profile of the Fermi sea is given by ${\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}}=-\mu.$ (4.11) First consider the theory without D brane. Then the low lying collective excitations are represented by deformations of the Fermi surface, ${\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}}=M({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}}).$ (4.12) In order to study the scattering with such deformed background we will follow the analysis of [16]. The perturbed wave functions are related to the old ones by a phase factor ${\psi^{E}_{\pm}}({\overline{z}_{\pm}}{z_{\pm}})=e^{\mp i{\varphi_{\pm}}({\overline{z}_{\pm}}{z_{\pm}};E)}{\psi^{E}_{\pm}}({\overline{z}_{\pm}}{z_{\pm}}),$ (4.13) whose asymptotics at large ${\overline{z}_{\pm}}{z_{\pm}}$ characterizes the incoming/outgoing tachyon state. We split the phase into three terms $\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}};E)=V_{\pm}({\overline{z}_{\pm}}{z_{\pm}})+{\frac{1}{2}}\phi(E)+v_{\pm}({\overline{z}_{\pm}}{z_{\pm}};E),$ (4.14) where the potentials $V_{\pm}$ are fixed smooth functions vanishing at ${\overline{z}_{\pm}}{z_{\pm}}=0$, while the term $v_{\pm}$ vanishing at infinity and the constant $\phi$ are to be determined. Now in order to understand the time-dependent profile of Fermi sea first consider the situation in the absence of the brane as described in [16]. $\langle\Psi^{{}_{E_{-}}}_{-}|\Psi^{{}_{E_{+}}}_{+}\rangle={\cal N}{e^{-i\phi}}\int_{0}^{\infty}{\frac{d{\overline{z}_{+}}d{\overline{z}_{-}}d{z_{-}}d{z_{+}}}{\sqrt{{\overline{z}_{+}}{z_{+}}}\sqrt{{\overline{z}_{-}}{z_{-}}}}}{e^{i({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}}e^{-i{\varphi_{+}}({z_{+}})-i{\varphi_{-}}({z_{-}})}{({\overline{z}_{+}}{z_{+}})}^{iE_{-}}{({\overline{z}_{-}}{z_{-}})}^{iE_{+}},$ (4.15) where ${\cal N}$ is the normalization. At $\beta\rightarrow\infty$ Fermi profile can be obtained from saddle point approximation which is given by ${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}=-E_{\pm}+({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}}).$ (4.16) So following [16] it appears the perturbed state will be an eigenstate of the deformed hamiltonian $H={H_{o}}+{H_{p}}$ where $H_{p}$ is given by ${H_{p}}=({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}};H)$ (4.17) Now in the presence of the D brane scattering matrix element will be given by $\langle{\overline{z}_{+}}{z_{+}},E_{+},{\rm out}|e^{\hat{W}({t_{o}})}|,{E_{-}},{\overline{z}_{-}}{z_{-}},{\rm in}\rangle$ . S–matrix element is expressed as $\displaystyle S_{perturb}$ $\displaystyle=$ $\displaystyle{e^{-i\phi}{\cal N}\int\limits_{0}^{\infty}{\frac{d{\overline{z}_{+}}d{\overline{z}_{-}}d{z_{-}}d{z_{+}}}{\sqrt{{\overline{z}_{+}}{z_{+}}}\sqrt{{\overline{z}_{-}}{z_{-}}}}}{e^{i({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}}e^{-i{\varphi}{({\overline{z}_{-}}{z_{-}})}}{({\overline{z}_{-}}{z_{-}})}^{iE_{-}}}$ $\displaystyle[e^{-i{\int_{t_{o}}^{\infty}}{\hat{H}_{o}}}[e^{-i{W_{\rm proj}}({t_{o}})}]e^{i{\int_{-\infty}^{t_{o}}}{\hat{H_{o}}}}]e^{-i{\varphi_{+}}({\overline{z}_{+}}{z_{+}})}{({\overline{z}_{+}}{z_{+}})}^{iE_{+}}$ $\displaystyle\sim$ $\displaystyle e^{-i\phi}{\cal N}\int\limits_{0}^{\infty}{\frac{d{\overline{z}_{+}}d{\overline{z}_{-}}d{z_{-}}d{z_{+}}}{{\sqrt{{\overline{z}_{+}}{z_{+}}}}\sqrt{{\overline{z}_{-}}{z_{-}}}}}{e^{i({\overline{z}_{+}}{z_{-}}(t)+{z_{+}}{\overline{z}_{-}}(t))}}$ $\displaystyle e^{-i{\varphi}({\overline{z}_{-}}{z_{-}}(t))}{({\overline{z}_{-}}{z_{-}}(t))}^{iE_{-}}[e^{iW(t)({\overline{z}_{\pm}}{z_{\pm}},{H_{o}})}]e^{-i{\varphi_{+}}({\overline{z}_{+}}{z_{+}}(t))}{({\overline{z}_{+}}{z_{+}}(t))}^{iE_{+}}$ So in the presence of the D brane, in the classical regime from (LABEL:pertu) we can write the Fermi profile in the presence of D brane as ${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}=-E_{\pm}+({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}})+({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})W({\overline{z}_{\pm}}{z_{\pm}},E)$ (4.19) The perturbed hamiltonian for the deformed state is given by ${H_{p}^{\prime}}={H_{p}}+({z_{\pm}}\partial_{\pm}+{\overline{z}_{\pm}}{\overline{\partial}_{\pm}})W({\overline{z}_{\pm}}{z_{\pm}};H)$ (4.20) So essentially the Dbrane act as a source at $t_{o}$. So as we see in the presence of D brane Fermi profile develops instability. ## 5 Perturbation by momentum modes ### 5.1 Collective field theory analysis In this subsection we will consider type 0A matrix model with the time t compactified on a circle of radius R, perturbed by momentum modes ${V_{\frac{n}{R}}}$ in the presence of D brane. We know that the presence of D brane change the tachyonic background [18] so that we need to go through collective field theory analysis to understand how the MQM wave function in a perturbed background with a D brane is related to the one without a brane. Here first we briefly review the scenario without D brane and then study what happens when we consider the theory in the presence of D brane. The tachyon modes of the closed string theory are the asymptotic states of collective field theory [25]. The discrete tachyonic operator ${\cal T}_{n}\sim\int_{\rm worldsheet}e^{\pm inx/R}e^{(|n|/R-2)\phi}$ corresponds to the following operator in matrix model[28, 26], ${V_{{\pm}n/R}}=e^{-{\frac{n}{R}}t}{{({\overline{z}_{\pm}}{z_{\pm}})}^{n/R}}.$ (5.1) These operators creates a discrete tachyonic state of momenta ${\frac{n}{R}}$ over the matrix model ground state and are periodic in Euclidean time. $[{H_{o}},{V_{{\pm}n/R}}]=\mp i{\frac{n}{R}}{V_{{\pm}n/R}}\quad;\quad k\geq 1.$ (5.2) So ${V_{{\pm}n/R}}$ shift the energy $E\rightarrow E\mp i{\frac{n}{R}}$ cause a time-dependent perturbation to Fermi sea. The perturbed state in general can be expressed as $\Psi_{\pm}^{E}=e^{\mp i\varphi(z_{\pm}\bar{z}_{\pm};E)}\psi_{o\pm}^{E}\equiv{\cal W}_{\pm}\psi_{o\pm}^{E},$ (5.3) where the phases $\varphi_{\pm}$ have Laurent expansion ${\varphi_{\pm}}({z_{\pm}}\overline{z}_{\pm};E)={\frac{1}{2}}\phi(E)+R\sum_{k\geq 1}t_{\pm k}{(z_{\pm}\overline{z}_{\pm})}^{k/R}-R\sum_{k\geq 1}{\frac{1}{k}}v_{\pm k}({z_{\pm}}\overline{z}_{\pm})^{-k/R}..$ (5.4) $t_{\pm k}$ parametrize the asymptotic perturbation by momentum modes of NS-NS scalars, corresponding to the operator introduced (5.1), Note the above wave function asymptotically behave as $\Psi^{E}_{\pm}({\overline{z}_{\pm}}{z_{\pm}})\sim{({\overline{z}_{\pm}}{z_{\pm}})}^{\pm iE-{\litfont{1\over 2}}}\ e^{\mp{\litfont{1\over 2}}i\phi(E)}\ e^{iU_{\pm}({\overline{z}_{\pm}}{z_{\pm}})}\quad;\quad U_{\pm}({\overline{z}_{\pm}}{z_{\pm}})=\sum_{k\geq 1}{|{\overline{z}_{\pm}}{z_{\pm}}|}^{\frac{k}{R}}.$ (5.5) From the above its evident that tachyonic perturbation can be achieved by deforming the integration measures $d[{\overline{z}_{\pm}}{z_{\pm}}]$ to [14] $[d{\overline{z}_{\pm}}{z_{\pm}}]\rightarrow[d{\overline{z}_{\pm}}{z_{\pm}}]{\rm exp}\left(\pm i{U_{\pm}}({\overline{z}_{\pm}}{z_{\pm}})\right).$ (5.6) Extending the discussions of section 3, these wave functions (5.3) diagonalizes the deformed kernel (5.6). While the perturbed wave function evolves in time with ${H_{o}}$, but it can be seen as the stationary state w.r.t an effective hamiltonian $H={H_{o}}+{H_{p}}(H)$, where the expressions for perturbed hamiltonian ${H_{p}}$ from semiclassical analysis is obtained in [16] as we reviewed in section 4. The partition function is given by ${\rm Tr}e^{-2{\pi}R\beta H}$, following section 3 which can also be expressed as Fredholm determinant. We have the free energy${\cal F}=-i\sum_{r\geq 1/2}\phi\left(ir/R-\mu\right)$ where $\phi(E)$ is the phase described by (5.4). It satisfies the equation $\phi(-\mu)=2{\rm sin}\left({\frac{\partial_{\mu}}{2\beta R}}\right){\cal F}(\mu,R)..$ (5.7) Having given a brief review of type 0A MQM perturbed by the momentum modes lets consider the theory with D brane. Note that type 0A MQM without any net D0 brane background charge, in the double scaling limit can be viewed as a pair of noninteracting fermions moving in inverted harmonic oscillator potential in x and y direction respectively (2.8). So from the analysis of [25] its apparent that the respective collective field theory will be the generalization of the one for $c=1$ case to two (target space ) dimension(x,y) and the collective field expressed as $\phi(x,y,t)=\phi(z,\overline{z},t)$, give the eigenvalue density in two dimension with appropriate normalization. Collective field hamiltonian will be the sum of the hamiltonian for $\phi(x,t)$ and $\phi(y,t)$. The fluctuation of the collective field over the static value $\phi={\phi_{o}}+{\partial_{\tau}}\eta({\tau},t)$,where $\pi\phi_{o}={p_{o}}=\sqrt{\mu_{F}-x^{2}-y^{2}}$ with $p_{o}^{2}=p_{ox}^{2}+p_{oy}^{2}\,;\,p_{ox}={\frac{dx}{d\tau}}\,;\,p_{oy}={\frac{dy}{d\tau}}$ and ${\partial_{\tau}}\eta({\tau},t)={\psi^{\dagger}}\psi$ where $\psi$ corresponds to the 2nd quantized fermionic field. Now the presence of D brane implies, inclusion of a macroscopic loop operator W (Laplace transformed, localized in time) to the collective field theory action which essentially creates a coherent state over MQM ground state. In linearized approximation we have $W(z,\overline{z},t_{o})$ $\sim\int dt\int d\tau{e^{-l\overline{z}z}}{\partial_{\tau}}{\eta}(\tau,t)\delta(t-t_{o})$. So the presence of the D brane implies a field independent source term in the collective field equation of motion (which can be viewed as the back-reaction due to the D-brane). Consequently we obtain the solution for collective field at $t\geq{t_{o}}$ as $\phi={\phi_{\rm free}}+{\phi_{\pm{\rm perturbed}}}$ where ${\phi_{\rm perturbed}}=\int d\tau j(t_{o},\tau)G({t_{o}},\tau)$ with $j(t_{o})$ is the current associated with the macroscopic loop operator and G is the Green’s function. As ${\phi_{\rm perturbed}}$ is independent of $\phi$ so we see that the effect of macroscopic loop operator in the action is to change the momentum associated with $\eta$ by the external current or in other words the classical solution for $\eta$ will get a field independent(but profile ($x(t_{o}),y(t_{o})$) dependent) shift. The stationary field $\phi_{o}$ will also be shifted due to the change of the potential. Essentially from string theory side we can just identify the current j (transformed from $\tau$ to $\phi$ space,and Fourier transformed to momentum space) with the overlap amplitude $\langle V(p)|B\rangle$ and the interaction term introduced in the collective field action $\int dtd\tau\,j\,{\partial_{\tau}}\eta\delta(t-{t_{o}})$ as $\int\phi_{\rm cl}(p)\langle V|B\rangle$ where $\phi_{\rm cl}$ can be viewed as closed string field. On this identification we see that quadratic action for $\eta$ [25, 28] along with the source term resembles the closed string field action in the presence of D brane $S=S_{\rm closed}(\phi_{\rm cl})+\phi_{\rm cl}(X,\phi)\langle V(X,\phi)|B\rangle$ where $\phi_{\rm cl}$ is the closed string field. Change in momentum of $\eta$ due to interaction with the localized source has an explanation in the fact that closed string momentum is not conserved in the direction of spacelike boundary condition of D brane. Now the meaning of the constraint (2.25) is that we have to ensure the fact that while the collective field is in interaction with the localized source $L_{\rm int}=\int d\tau d{t}\,j\,(\partial_{\tau}\eta)(t.\tau)\,\delta(t-{t_{o}})$, the hamiltonian for $\eta$ will always remain conserved which implies time translation invariance of complete action with no leakage from the bulk by making $\delta_{t}L_{\rm int}|_{\rm t={t_{o}}}\,=0$ . So the quantized action for $\eta$ always gives the propagator have a pole corresponding to 2D massless scalar [25] i.e resembles the tachyon. So from string theoretical point of view we see that the constraint act as a no leakage condition ensures bulk conformal invariance as we mentioned in section 2. . Now lets find out the exact form of MQM wave function in the presence of D brane in the background perturbed by momentum modes from Collective field theory(which in the absence of Dbrane is given by (5.3) ). Lets proceed in the following way. The collective field equation of motion implies the classical solution for left and right moving field $\alpha_{x\pm}\,,\,\alpha_{y\pm}\,$ (given by $\phi(x)+{\alpha^{\prime}}\partial_{x}\Pi(x)\,\,,\,\,\phi(y)+{\alpha^{\prime}}\partial_{y}\Pi(y)$ where $\Pi$ is the collective field momentum), correspond to fermionic momentum density at the edge of the Fermi sea [26, 28]. From the expression of the hamiltonian (2.8), Fermi surface is described by ${\frac{1}{2}}({p}_{x}^{2}+{p}_{y}^{2})-{\frac{1}{4\alpha^{\prime}}}(\hat{x}^{2}+\hat{y}^{2})=-\mu.$ (5.8) The collective field equation corresponds to two separate equation for two fermions described by x,y as no interaction exists among them. So the momentum density ${p_{\pm x}},{p_{\pm y}}$ at the edge of the Fermi surface evolves with time t as [26] $\partial_{t}{p_{\pm x}}+{p_{\pm x}}{\partial_{x}}{p_{\pm x}}-x=0\quad;\quad\partial_{t}{p_{\pm y}}+{p_{\pm y}}{\partial_{y}}{p_{\pm y}}-y=0.$ (5.9) If we consider the fluctuation of collective field around its classical solution $p_{o}\,(p_{ox},p_{oy})$. $\alpha_{\pm}\rightarrow{p_{o}}+{\epsilon_{\pm}}$ defining ${\epsilon_{\pm}}={\frac{1}{p_{o}}}{\xi_{\pm}}$, ${\xi_{\pm}}$ is shown to correspond the right and left moving tachyonic fluctuations [28]. Now at classical limit the macroscopic loop operator contributes a source term to the above $\partial_{t}{p_{\pm x}}+{p_{\pm x}}{\partial_{x}}{p_{\pm x}}-x=\left[\,{\frac{\partial W(x,y)}{\partial x}}{|_{y}}\,\right]{\delta(t-{t_{o}})}\quad;\quad\partial_{t}{p_{\pm y}}+{p_{\pm y}}{\partial_{y}}{p_{\pm y}}-y=\left[\,{\frac{\partial W(x,y)}{\partial y}}{|_{x}}\,\right]{\delta(t-{t_{o}})}.$ (5.10) This essentially gives a discontinuity $p_{\pm{x_{i}}}{|_{{t_{o}}+\epsilon}}-p_{\pm{x_{i}}}{|_{{t_{o}}-\epsilon}}={\frac{\partial W(x,y)}{\partial{x_{i}}}}{|_{t_{o}}}\quad,$ (5.11) where in (5.11) $x_{i}$ stands for x,y for $i=1,2$. So the effect of D brane is to change the Fermi profile above $t\geq{t_{o}}$ to ${p_{o}}\rightarrow{p_{o}^{\prime}}={p_{o}}+{\frac{\partial W(x,y)}{\partial{x_{i}}}}({t_{o}})$. As above $t_{o}$, $p_{\pm}$ evolves according to free hamiltonian ${H_{o}}$ so we can express the fluctuation of the collective field for $t\geq{t_{o}}$ as ${\epsilon^{\prime}_{\pm}}={\frac{1}{p_{o}^{\prime}}}{\xi^{\prime}_{\pm}}$ to see ${\xi^{\prime}_{\pm}}$ corresponds to tachyonic fluctuation mode. However the redefinition above ${t_{o}}$ implies a nonlinear shift of collective field momenta $p_{o}^{\prime}=p_{o}^{\prime}(p_{o},x,y)$ and the fluctuation $\xi^{\prime}=\xi^{\prime}(\xi,p_{o},x,y)$. ${\epsilon^{\prime}_{\pm x}}$ ,${\epsilon^{\prime}_{\pm y}}$ combined to give complex tachyonic field. So from the viewpoint of collective field scenario lets find out the picture in MQM. The shift of fluctuation mode above $\xi\rightarrow\xi^{\prime}$ implies a shift in the perturbing phase (5.4) . ${\psi^{\prime}_{p\pm>}}^{E}=e^{\mp i\varphi_{w}(z_{\pm}\overline{z}_{\pm};E)}\psi_{o\pm}^{E}={{\cal{W}}^{\prime}}\psi_{o\pm}^{E},$ (5.12) where $\displaystyle{\varphi_{w\pm}}({z_{\pm}}\overline{z}_{\pm};E)$ $\displaystyle=$ $\displaystyle{\frac{1}{2}}\phi(E)+R\sum_{k\geq 1}t_{\pm k}(t_{m\pm},v_{n\pm},{\frac{r}{R}},E){f_{tk}}(E,{\overline{z}_{\pm}}{z_{\pm}})\,{\left[z_{\pm}\overline{z}_{\pm}\right]}^{k/R}$ (5.13) $\displaystyle-$ $\displaystyle R\sum_{k\geq 1}{\frac{1}{k}}v_{\pm k}(t_{m\pm},v_{n\pm},{\frac{r}{R}},E){f_{vk}}(E,{\overline{z}_{\pm}}{z_{\pm}})\,\left[{z_{\pm}}\overline{z}_{\pm}\right]^{-k/R},$ ${f_{vk}}$ and ${f_{tk}}$is the extra factor arises due to the nonlinear shift in $\xi$ from ${\overline{z}_{\pm}}{z_{\pm}}$ factor which arises due to the action of W. These factors give nonperiodic shift to momentum by integer numbers i.e ${\frac{k}{R}}\rightarrow{\frac{k}{R}}+n$ with $t_{\pm k}=t_{\pm k}(t_{m\pm},v_{n\pm},E)\,,\,v_{\pm k}=v_{\pm k}(t_{m\pm},v_{n\pm},E)$. Note that once we try to interpret the consequence of (5.11) in MQM we need to replace $W\rightarrow W_{\rm proj}$ (2.41). The fact that in absence of $\hat{W}$ we get back our original wave function so (5.3) implies that the shifted dressing operator ${\cal W}^{\prime}$ is of the form ${\cal W}^{\prime}=F(\hat{W}){\cal W}$ with $F\rightarrow 1$ in absence of $\hat{W}$. Exploiting the fact that (LABEL:scrh2) is just the quantum version of (5.11) and the indication of the semiclassical analysis (4.19) implies ${\cal W}^{\prime}$ can be expressed as : $\Psi^{\prime E}_{p\pm>}=e^{\mp i\varphi_{w}({\overline{z}_{\pm}}{z_{\pm}};E)}\psi_{o}^{E}=(1\mp\hat{W}_{\rm proj})e^{\mp i\varphi({\overline{z}_{\pm}}{z_{\pm}};E)}\psi_{o\pm}^{E}=(1\mp\hat{W}_{\rm proj}){\cal{W}}\psi_{o\pm}^{E},$ (5.14) This is because at double scaling limit $\beta\rightarrow\infty$ the wave function is effectively given by $e^{-\hat{W}}{\cal{W}}\psi_{o\pm}^{E}$(as we have seen in (3.34) ). This is the solution from (2.40) if the initial free fermionic wave function (2.28) is replaced by the dressed one (5.3). Finally as before we will express $W_{\rm proj}$ in terms of ${\hat{\overline{z}}_{-}}{\hat{z}_{-}}\,({\hat{\overline{z}}_{+}}{\hat{z}_{+}})$ in ${\overline{z}_{+}}{z_{+}}\,({\overline{z}_{-}}{z_{-}})$ basis as this will express (5.13) in the following form $\displaystyle{\varphi_{wp\pm}}({z_{\pm}}\overline{z}_{\pm};E)$ $\displaystyle=$ $\displaystyle{\frac{1}{2}}\phi(E)+R\sum_{k\geq 1}t_{\pm k}(t_{m\pm},v_{n\pm},{\frac{r}{R}},E){f_{tk}}(E,{a_{r}}{({\overline{z}_{\pm}}{z_{\pm}})}^{-r})\,{\left[z_{\pm}\overline{z}_{\pm}\right]}^{k/R}$ (5.15) $\displaystyle-$ $\displaystyle R\sum_{k\geq 1}{\frac{1}{k}}v_{\pm k}(t_{m\pm},v_{n\pm},{\frac{r}{R}},E){f_{vk}}(E,{b_{r}}{({\overline{z}_{\pm}}{z_{\pm}})}^{-r})\,\left[{z_{\pm}}\overline{z}_{\pm}\right]^{-k/R},`$ where r is a positive integer. In next subsection we will see that this expression of ${\varphi^{\prime}}$ lead to convergence of the partition function. The tachyonic perturbation can be introduced in the path integral by deforming the kernel as in (5.6) and consequently the string partition function can be expressed as the Fredholm determinant as in (3.27). We can evaluate the Fredholm determinant with a set of diagonalizing wave function which is given by (5.12). In the next part of this section we will evaluate the gran canonical partition function in the hamiltonian formalism. We will show that the tachyonic deformation in the presence of the D brane is generated by a system of commuting flows $H_{n}$ associated with the coupling constants $t_{\pm n}$. The associated integrable structure of the partition function is that of a constrained Toda Lattice hierarchy. Now in order to see the Toda structure of the partition function we need to review Lax formalism. ### 5.2 Lax Formalism Here we will briefly review the Lax formalism in the context of Type 0A matrix model. Consider the operator $(\hat{z}_{\pm}\hat{\bar{z}}_{\pm})$ which can be represented as shift operators $\hat{\omega}^{\pm 1}$, where $\hat{\omega}$ acts on energy eigenstates as $\hat{\omega}^{\pm 1}\psi_{\pm}^{E}=\psi_{\pm}^{E\mp i}$. We have $\hat{\omega}=e^{-i\partial_{E}}$ shifts the variable E by $i$. The operators $\hat{\omega}$ and $\hat{E}$ satisfy the Heisenberg-Weyl commutation relation $[\hat{\omega},-\hat{E}]=i\hat{\omega},\qquad[\hat{\omega}^{-1},-\hat{E}]=-i\hat{\omega}^{-1}.$ (5.16) Now let us consider the representation of these commutation relations in the perturbed theory. The dressing operators ${\cal W}_{\pm}$ (5.5)are now exponents of series in $\hat{\omega}$ with $\hat{E}$-dependent coefficient $\hat{\cal W}_{\pm}=e^{iR\sum_{n\geq 1}t_{\pm n}\hat{\omega}^{n/R}}\ e^{\mp i\phi(E)}e^{iR\sum_{n\geq 1}v_{\pm n}({E})\ \hat{\omega}^{-n/R}}.$ (5.17) The operators $\displaystyle L_{+}$ $\displaystyle=$ $\displaystyle{\cal W}_{+}\hat{\omega}{\cal W}^{-1}_{+},\quad L_{-}={\cal W}_{-}\hat{\omega}^{-1},{\cal W}^{-1}_{-},$ $\displaystyle M_{+}$ $\displaystyle=$ $\displaystyle-{{\cal W}_{+}}\hat{E}{{\cal W}^{-1}_{+}}\quad{M_{-}}=-{{\cal W}_{-}}{\hat{E}},{\cal W}^{-1}_{-}.$ (5.18) known as Lax and Orlov-Schulman operators satisfy the same commutation relations as the operators $\hat{\omega}$ and $\hat{E}$ $[L_{+},M_{+}]=iL_{+}\quad,\quad[L_{-},M_{-}]=-iL_{-}.$ (5.19) The Lax operators $L_{\pm}$ represent the canonical coordinates ${\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}$ in the basis of perturbed wave functions $\langle E|e^{\pm i{\frac{\phi_{0}}{2}}}\hat{\cal W}_{\pm}L_{\pm}|{\overline{z}_{\pm}}{z_{\pm}}\rangle=\langle E|e^{\pm i{\frac{\phi_{0}}{2}}}\hat{{\cal W}}_{\pm}{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}|{\overline{z}_{\pm}}{z_{\pm}}\rangle,$ (5.20) while the Orlov-Shulman operators $M_{\pm}$ represent hamiltonian $H_{0}=-{\frac{1}{2}}({\hat{\overline{z}}_{+}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}\hat{\hat{z}_{+}})$. Therefore the L and M operators are related also by $M_{+}=M_{-},~{}~{}~{}[L_{+},L_{-}]=2iM_{\pm},~{}~{}~{}\\{L_{+},L_{-}\\}=2M_{\pm}^{2}-{1\over 2}.$ (5.21) The last identity is not satisfied automatically in the Toda system and represent an additional constraint analogous to the string equations. The operators $M_{\pm}$ can be expanded as infinite series of the $L$-operators. Indeed, as they act to the dressed wave functions as $\displaystyle\langle E|$ $\displaystyle e^{\pm i\phi_{0}}\hat{\cal W}_{\pm}\;{M_{\pm}}|{\overline{z}_{\pm}}{z_{\pm}}\rangle=\pm i({z_{\pm}}\partial_{{z_{\pm}}}+{\overline{z}_{\pm}}\partial_{{\overline{z}_{\pm}}}+1)\Psi^{E}_{\pm}({\overline{z}_{\pm}}{z_{\pm}})$ (5.22) $\displaystyle=$ $\displaystyle\left(\sum_{k\geq 1}kt_{\pm k}{({\overline{z}_{\pm}}{z_{\pm}})}^{k/R}+\mu+\sum_{k\geq 1}v_{\pm k}{{\overline{z}_{\pm}}{z_{\pm}}}^{-k/R}\right)\Psi^{E}_{\pm}(({\overline{z}_{\pm}}{z_{\pm}})).$ we can write $M_{\pm}=\sum_{k\geq 1}kt_{\pm k}L_{\pm}^{k/R}+\mu+\sum_{k\geq 1}v_{\pm k}L_{\pm}^{-k/R}.$ (5.23) In order to exploit the Lax equations and the string equations we need the explicit form of the two operators. It follows from that $L_{\pm}$ can be represented as series of the form $\displaystyle{L_{+}}$ $\displaystyle=$ $\displaystyle e^{-i\phi/2}\left(\omega+\sum_{k\geq 1}a_{k}\omega^{1-n/R}\right)e^{i\phi/2},$ $\displaystyle{L_{-}}$ $\displaystyle=$ $\displaystyle e^{i\phi/2}\left(\omega^{-1}+\sum_{k\geq 1}a_{-k}\omega^{-1+n/R}\right)e^{-i\phi/2}.$ (5.24) Recall that the dressing operators ${\cal W}_{\pm}$ in terms of $\hat{E}$ and $\hat{\omega}$ are of the for ${\cal{W}}_{\pm}=e^{\mp i\phi/2}\left(1+\sum_{k\geq 1}w_{\pm k}\hat{\omega}^{\mp k/R}\right)e^{\mp iR\sum_{k\geq 1}t_{\pm k}\hat{\omega}^{\pm k/R}}$ (5.25) Studying the evolution laws of the Orlov–Shulman operators, one can find that [20] ${\partial v_{k}\over\partial t_{l}}={\partial v_{l}\over\partial t_{k}}.$ (5.26) It means that there exists a generating function $\tau_{s}[t]$ of all coefficients $v_{\pm k}$ $v_{k}(s)={({\frac{1}{\beta}})^{2}}\,{\partial\log\tau_{s}[t]\over\partial t_{k}}.$ (5.27) It is called $\tau$-function of Toda hierarchy. It also allows to reproduce the zero mode $\phi$ and, consequently, the first coefficient in the expansion of the Lax operators $e^{\beta\phi(s)}={\tau_{s}\over\tau_{s+{\frac{1}{\beta}}}},\qquad r^{2}(s-{\frac{1}{\beta}})={\tau_{s+{\frac{1}{\beta}}}\tau_{s-{\frac{1}{\beta}}}\over\tau_{s}^{2}}.$ (5.28) We are going to show that the partition function coincides with $\tau$–function (5.27). Finally note as the partition function is described in terms of the Fermi level ${\mu}$. So in the description of Lax formalism we will replace E by ${\mu}$. Now let us discuss about the integrable flow. Let us identify the integrable flows associated with the coupling constants $t_{n}$. $\partial_{t_{n}}L_{\pm}=[H_{n},L_{\pm}],$ (5.29) where from (5.18), the operators $H_{n}$ are related to the dressing operators as $H_{n}=({\partial_{t_{n}}}{\cal W}_{+}){\cal W}_{+}^{-1}=({\partial_{t_{n}}}{\cal W}_{-}){{\cal W}_{-}^{-1}}.$ (5.30) it is clear that $H_{n}=W_{+}\hat{\omega}^{n/R}W_{+}^{-1}+$ negative powers of $\hat{\omega}^{1/R}$, which implies expression of ${H_{n}}$ can be given by [14] $H_{\pm n}=(L_{\pm}^{n/R})_{{}^{>}_{<}}+{\frac{1}{2}}(L_{\pm}^{n/R})_{0},\qquad n>0,$ (5.31) $\partial_{t_{m}}H_{n}-\partial_{t_{n}}H_{m}-[H_{m},H_{n}]=0.$ (5.32) Equations (5.32,5.30,5.31) imply that the perturbed theory possesses the Toda lattice integrable structure. The Toda structure implies an infinite hierarchy of PDE’s for the coefficients $v_{n}$ of the dressing operators, the first of which is the Toda equation for the phase $\phi(\mu)\equiv\phi(E=-\mu)$ $i{\frac{\partial}{\partial t_{1}}}{\frac{\partial}{\partial t_{-1}}}\phi(\mu)=e^{i\phi(\mu)-i\phi(\mu-i/R)}-e^{i\phi(\mu+i/R)-i\phi(\mu)}.$ (5.33) ### 5.3 String theory on a circle with the D brane in the presence of tachyonic background In this section we are going to evaluate the free energy of type 0A MQM in the presence of D brane and with tachyonic background, in the grand canonical ensemble and try to understand the relevant string theory. Recall in section 3 we have seen that in the absence of the momentum modes, within the time circle $0\leq t\leq 2\pi R$ the solution of the Schrodinger equation corresponds to the free fermionic wave function. This in the string theory side giving a picture that we have free closed string states along the circle and the coherent states are strongly localized at $t={t_{o}}\equiv i{X^{o}}$ . So with the same view in the presence of tachyonic background, within the circle $0\leq t\leq 2{\pi}R$ the wave function must be given by (5.3). The perturbed wave function, while time dependent w.r.t the free hamiltonian ${H_{o}}$ it is stationary w.r.t an effective hamiltonian H, similar as discussed in section 5.1. So lets consider the perturbed MQM with the effective hamiltonian $H={H_{o}}+{H_{p}}(H)$ in the presence of D brane. First consider the partition function (3.7) where now we replace the integration kernels with the deformed measures (5.6). So as a generalization of (3.7), in the perturbed background, the Matrix model partition function in the presence of D brane (3.23, 3.24,3.25 ) will be with the deformed kernel as $\displaystyle{\cal{Z}}_{N}(t)$ $\displaystyle=$ $\displaystyle\int\limits_{-\infty}^{\infty}\prod_{k=1}^{N}[d{z_{+}}_{k}][d{z_{-}}_{k}][d{\overline{z}_{+}}_{k}][d{\overline{z}_{-}}_{k}][d{t_{k}}]{e^{i{t_{n\pm}}{({\overline{z}_{\pm}}{z_{\pm}})}^{{\frac{n}{R}}}}}{\rm det}_{jk}\left(e^{i{t_{jk}}{\overline{z}_{+}}_{j}{z_{-}}_{k}}\right)$ (5.34) $\displaystyle{\rm det}_{jk}\left(e^{-iq{z_{+}}_{j}{\overline{z}_{-}}_{k}}\right){\rm det}_{jk}\left(e^{i{t^{-1}_{jk}}{\overline{z}_{+}}_{j}{z_{-}}_{k}}\right)\ {\rm det}_{jk}\left(e^{-iq{z_{+}}_{j}{\overline{z}_{-}}_{k}}\right)$ $\displaystyle{\rm exp}[\displaystyle\sum_{i}{\rm log}(1+{\frac{{\overline{z}_{+}}_{i}{z_{+}}_{i}+{\overline{z}_{-}}_{i}{z_{-}}_{i}+{\overline{z}_{+}}_{i}{z_{-}}_{i}+{\overline{z}_{-}}_{i}{z_{+}}_{i}}{\mu_{B}^{2}}})].$ The partition function will be given by Fredholm determinant ${\rm Det}(1+e^{-\beta\mu}WK))$ (3.27) where in order to evaluate the determinant we need to choose the basis which diagonalizes K (i.e (3.13) with deformed measure as given in (5.34) ) and we evaluate the expectation value of ${\hat{W}}$ in the same. In order to evaluate free energy in the presence of D brane we will proceed in the following way. First consider the scenario without D brane. Recall the expression of free energy which is expressed in terms of the phase of wave function [14, 15](which is of the same form of (3.42), expressed in the absence of brane). In a perturbed background the phase $\phi(E)$ will be replaced by that of the perturbed wave function (5.3) in the expression of free energy[14]. So for the effective hamiltonian H ( $H={H_{o}}+{H_{p}}(H)$ where $H_{p}$in the semiclassical limit obtained in (4.17) ) of which (5.3) is an eigenfunction, the analysis of section 3.3 implies that free energy of the perturbed system in grand canonical ensemble is given by ${\cal F}={\rm log}{\cal{Z}}$ with ${\cal{Z}}=\displaystyle\sum_{N=0}^{\infty}e^{-2{\pi}R\beta\mu N}\\{{\rm Tr}e^{-2{\pi}R\beta H}{\\}_{N}}={\rm Det}\,(1+e^{-2{\pi}R\beta(\mu+H)})$. This is supported from the view of [16] where in the semiclassical regime the explicit expression of ${\cal{Z}}$ is obtained in this form. The Fredholm determinant (3.13) in a perturbed background is given by ${\cal{Z}}$ [14]. In the presence of D brane we have the Fredholm determinant (3.27) which in perturbed background is expressed in (5.34). So as in section 3.3 free energy must be obtained from the thermal partition function in the presence of D brane i.e by insertion of the operator $e^{W(t_{o})}$ in the partition function and evaluating the expectation value. So here in hamiltonian formalism we will evaluate the grand canonical partition function ${\rm Det}(1+e^{-\beta(H+\mu)})$ in the basis (5.3) with the insertion of the operator and it must be same as the Fredholm determinant (5.34). Here we are going to show that if we consider the prtojected theory as described in section 2, the above mentioned grand canonical partition function have the integrable structure of tau function of Toda hierarchy. Now the partition function with the momentum modes in the presence of the D brane is given by the transition amplitude from the initial state ${\cal{W}}{\psi_{o}}$ to the final state${\cal{W}}^{\prime}\psi_{o>}$, where $\psi_{o>}$ is given in (2.41) and they represent the fermionic wave function before and after being scattered from the D brane and the corresponding dressing operator is ${\cal{W}}^{\prime}$ .Now note that in a compact dimension just before being scattered, the wave function at $t=2{\pi}R-\epsilon$ must be given by the one at $t=\epsilon$ with a time evolution $2{\pi}R$. This leads to the identity ${\cal{W}}^{\prime}\psi_{o>}(t_{o})={\cal{W}}^{\prime}(1-W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})){\psi_{o}}(t_{o})=\left(1-W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}})\right){\cal{W}}{\psi_{o}}(t_{o}),$ (5.35) where ${t_{o}}=0\equiv 2{\pi}R$. The above relation can also be viewed from (2.40) in the presence of tachyonic background, if we replace the initial fermionic wave function (2.28) by the dressed one (5.3). Hence the partition function on the circle corresponds to the transition amplitude $\displaystyle{\cal{Z}}$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\to 0}\,\langle{{\cal{W}}\psi_{o}}{(\epsilon)}|{\cal{W}}^{\prime}\psi_{o>}(2{\pi}R-\epsilon)\rangle$ (5.36) $\displaystyle=$ $\displaystyle\lim_{\epsilon\to 0}\,\langle{{\cal{W}}\psi_{o}}{(\epsilon)}|(1-W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},{H_{o}}))|{\cal{W}}\psi_{o}(2{\pi}R-\epsilon)\rangle$ $\displaystyle=$ $\displaystyle{\rm Tr}_{{\cal{W}}\psi_{o}}\\{e^{{-\beta}[{\int_{\epsilon}^{2{\pi}R-{\epsilon}}}dtH+{\int_{-\epsilon}^{\epsilon}}dtH]+{\int_{-\epsilon}^{\epsilon}}dtW{\delta}(t)\\}}\\}$ $\displaystyle=$ $\displaystyle{\rm Tr}_{{\cal{W}}\psi_{o}}\\{e^{{-\beta}[{\int_{\epsilon}^{2{\pi}R-{\epsilon}}}dtH]}e^{W(t=0)}\\}$ $\displaystyle=$ $\displaystyle{\rm Tr}_{{\cal{W}}\psi_{o}}\\{e^{{-2{\pi}R\beta}H}e^{W(t=0)}\\}.$ Where the partition function is evaluated in Euclidean time and ${\rm Tr}_{{\cal{W}}\psi_{o}}$ denotes the trace taken w.r.t (5.3)171717In order to reach from the 2nd to 3rd step in (5.36)we utilize the fact that we can scale the time $t\rightarrow\beta t$ so that the term with the macroscopic loop operator $\int dtW(t)\delta(t-{t_{o}})$ will get a factor ${\frac{1}{\beta}}$ so that in the double scaling limit where $\beta\rightarrow\infty$ and with Euclidean time, we can lift up the term to the exponential and the exponent gives an exact expression what we have obtained from the path integral (2.36). Grand canonical Partition function will be given by the following expression where we will have the contribution from singlet states only ${\displaystyle\prod_{E}}\\{1+e^{{-2\beta}{\pi R}(\mu+E)}\langle{\psi_{p}^{E}}|e^{\hat{W}(t=0)}|{\psi_{p}^{E}}\rangle\\}\\\ ={\displaystyle\prod_{E}}\\{1+e^{{-2\beta}{\pi R}(\mu+E)}\langle{\psi_{p}^{E}}|e^{\hat{W}}|{\psi_{p}^{E}}\rangle\\},$ Note, we could write the above expression for grand canonical partition function only because $\hat{W}$ can be expressed as the direct product of the operators for the single fermionic states. Now following (2.15, 2.41) the partition function (5.36) can be expressed as $\displaystyle{\displaystyle\prod_{E}}$ $\displaystyle[1+e^{{-2\beta}{\pi R}(\mu+E)}\langle{\psi_{p}^{E}}|e^{{\rm log}(1+{\frac{2{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}-2{H_{o}}}{\mu_{B}^{2}}})}|{\psi_{p}^{E}}\rangle]$ (5.37) $\displaystyle=$ $\displaystyle{\displaystyle\prod_{E}}[1+e^{{-2\beta}{\pi R}(\mu+E)}\langle{\psi_{p}^{E}}|(1+{\frac{{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}-2{H_{o}}}{\mu_{B}^{2}}})|{\psi_{p}^{E}}\rangle].$ Now we have shown in the appendix that $\langle{\overline{z}_{+}}{z_{+}}|{\hat{\overline{z}}_{+}}{\hat{z}_{+}}|{\overline{z}_{+}}{z_{+}}\rangle$ and $\langle{\overline{z}_{-}}{z_{-}}|{\hat{\overline{z}}_{-}}{\hat{z}_{-}}|{\overline{z}_{-}}{z_{-}}\rangle$ diverge. So we must express $\hat{W}$ as $W({{\hat{\overline{z}}_{-}}{\hat{z}_{-}},H_{o}})$ for the basis $|{\overline{z}_{+}}{z_{+}},E\rangle$ basis and vice versa. Now note according to the commutation relation (2.30) and from the form of the wave function (5.3) ${\hat{\overline{z}}_{-}}{\hat{z}_{-}}={e^{-i{\frac{\varphi{(\hat{E+i})}}{2}}}}{{\frac{\partial}{\partial{\overline{z}_{+}}}}{\frac{\partial}{\partial{z_{+}}}}}{e^{i{\frac{\varphi{(\hat{E})}}{2}}}}$ . So according to the analysis Appendix, $\langle{\psi_{p}^{E}}|{\frac{2{\hat{\overline{z}}_{-}}{\hat{z}_{-}}}{\mu_{B}^{2}}}|{\psi_{p}^{E}}\rangle=0$ except when R is an integer181818This is because we can write the integral as $\langle{\psi_{o}}|{({\overline{z}_{+}}{z_{+}})}^{{\frac{n}{R}}-1}|{\psi_{o}}\rangle$ which following the analysis of Appendix-A contributes only at pole.. However for R an integer it contributes a constant term independent of $\phi,E$, in the partition function and can be ignored by subtracting out an overall constant from the hamiltonian which amounts to multiplying the partition function by an overall factor. For the macroscopic loop operator in any other sector, we can proceed in the same way . So we can write the partition function as $\displaystyle{\prod_{E}}[1$ $\displaystyle+$ $\displaystyle\langle{\psi_{p}^{E}}|(1+{\frac{{\hat{\overline{z}}_{+}}{\hat{z}_{-}}+{\hat{\overline{z}}_{-}}{\hat{z}_{+}}}{\mu_{B}^{2}}})e^{{-2\beta}{\pi R}(\mu+E)})|{\psi_{p}^{E}}\rangle]$ (5.38) $\displaystyle=$ $\displaystyle{\prod_{E}}[1+\langle{\psi_{p}^{E}}|e^{{\rm log}(1-2{\frac{H_{0}}{\mu_{B}^{2}}})}e^{{-2\beta}{\pi R}(\mu+E)})|{\psi_{p}^{E}}\rangle]$ $\displaystyle=$ $\displaystyle{\prod_{E}}[1+\langle{\psi_{p}^{E}}|e^{{-2\beta}{\pi R}(\mu+H-{\frac{1}{2{\pi}{\beta}R}}{\rm log}(1-2{\frac{H_{o}}{\mu_{B}^{2}}})}|{\psi_{p}^{E}}\rangle]$ $\displaystyle=$ $\displaystyle{\prod_{E}}[1+\langle{\psi_{p}^{E}}|e^{{-2\beta}{\pi R}(\mu+{H_{o}^{\prime}}+{H_{p}})}|{\psi_{p}^{E}}\rangle]$ $\displaystyle=$ $\displaystyle{\rm Tr}_{\psi_{p}^{E}}[e^{{-2\beta}{\pi R}(\mu+{H_{o}^{\prime}}+{H_{p}})}]$ where $H_{o}^{\prime}$ is as discussed in (3.45), given by ${H_{o}^{\prime}}={H_{o}}-{\frac{1}{2{\pi}R\beta}}{\rm log}(1-{\frac{2H_{o}}{\mu_{B}^{2}}})$; ${H_{p}}={H_{p}}({\overline{z}_{\pm}}{z_{\pm}},H)$ is the effective perturbation in the presence of momentum modes 191919 In order to reach from 2nd to 3rd step we used the same tricks of section 3 which implies that around a delta function in time, we can make the time interval infinitesimally small so that we can ignore the commutator terms ($[{H_{o}^{\prime}},{H_{o}}]$ +….higher commutators) what can arise on exponential as a consequence of Baker Hausdorff formula . First note that $|{\psi_{p}^{E}}\rangle$, the eigenstate of $H={H_{o}}+{H_{p}}$ does not diagonalize the complete effective hamiltonian $H_{\rm eff}=H-{\frac{1}{2{\pi}{\beta}R}}{\rm log}(1-2{\frac{H_{o}}{\mu_{B}^{2}}})$. This is the indication from collective field theory as we discussed in section 5.1 that the effect of putting a macroscopic loop operator in MQM action is to deform the static Fermi sea as well as the tachyonic background which shows up as a nonlinear shift of the perturbing phase $\varphi\rightarrow\varphi_{wp}$ (5.15,5.14). Clearly the eigenfunction which diagonalizes the complete effective hamiltonian ${H_{o}^{\prime}}+{H_{p}}$ will be given by a shift ${\cal W}\rightarrow{\cal W}^{\prime}=e^{i{\frac{\phi^{\prime}}{2}}}$ where ${i\phi^{\prime}}$ will be of similar form of (5.15) and so we will denote it by $\phi_{wp}$. Now we can express ${\cal W}^{\prime}$ as the product ${\cal W}^{\prime}=U({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o}){\cal W}$ where U is the factor responsible for the presence of macroscopic loop operator so that in absence of the operator we should have ${\cal W}^{\prime}={\cal W}$. Now in order to diagonalize lets recall the tricks we used in (3.44). If ${\cal W}^{\prime}{\psi_{o}}$ diagonalize the deformed hamiltonian in (5.38) then we can replace the partition function with deformed hamiltonian $H={H_{o}}^{\prime}+{H_{p}^{\prime}}(H,{\overline{z}_{\pm}}{z_{\pm}})$ evaluated in the basis ${\cal W}^{\prime}{\psi_{o}}(E)$ wlth the one in the shifted basis ${\cal W}^{\prime}{\psi_{o}}(E)\rightarrow{\cal W}{\psi_{o}}(E^{\prime})$ evaluated w.r.t the hamiltonian $H={H_{o}}+{H_{p}}(H,{\overline{z}_{\pm}}{z_{\pm}})$. By this shift we can identify $U({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o})$ with an operator which has an effect to shift $E^{\prime}\rightarrow E$ in the eigenfunction $\psi^{p}({E^{\prime}})$. So the operator U will be of the form $U(\sum_{n}{a_{n}}{({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}})}^{in}..){U_{1}}$, where ${U_{1}}$ is an operator making unitary transformation to ${\cal W}$. Clearly ${\cal W}$ and ${\cal W}^{\prime}$ are not related by unitary transformation and so ${\cal W}\psi_{o}$ and ${\cal W}^{\prime}\psi_{o}$ defines the basis of completely different Hilbert space. So in the presence of the Brane we need to evaluate the partition function in the basis ${\cal W}^{\prime}\psi_{o}$ which we can view as deformed tachyonic background caused by the presence of the brane. One can verify the expectation value of macroscopic loop operator $W({\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}},H_{o})$ in this basis effectively gets contribution from its hamiltonian part i.e $W(H_{o})$ as in the case for undeformed basis (5.3) and following similar steps we will get the partition function (5.38) with a shifted basis ${\cal W}{\psi_{o}}\rightarrow{\cal W}^{\prime}{\psi_{o}}$. The perturbing phase $\phi^{\prime}$ of the shifted basis in principle can have a complex part for which we need to choose appropriate normalization. However in order to evaluate the partition function we will follow (3.44). This partition function is exactly given by the one with a shift $\psi^{\prime E}_{p}={\cal W}^{\prime}{\psi_{o}}(E)\,\,\Rightarrow\,\,\psi_{p}^{E^{\prime}}={\cal W}{\psi_{o}}(E^{\prime})$ and the perturbing phase $\phi_{wp}(E)\rightarrow\phi(E(E^{\prime}))=\phi^{\prime}(E^{\prime})$ evaluated w.r.t the effective hamiltonian $H={H_{o}}+{H_{p}}(H,{\overline{z}_{\pm}}{z_{\pm}})$ but without insertion of the macroscopic loop operator W, where $\phi_{wp}(E)$ is as introduced in (5.15) and $\psi^{\prime E}_{p}$ is the basis which diagonalizes the deformed hamiltonian ${H_{o}^{\prime}}+{H_{p}}$ , as we discussed. So following (3.44) we can evaluate the partition function (5.38)in the shifted basis $\displaystyle{\psi_{p}^{\prime E}}$ $\displaystyle\rightarrow$ $\displaystyle{\psi_{p}^{E^{\prime}}}={\cal{W}}_{s}{\psi_{o}^{E^{\prime}}}=e^{{\frac{1}{2}}\phi(E({E^{\prime}}))+R\sum_{k\geq 1}t_{\pm k}{(z_{\pm}\bar{z}_{\pm})}^{k/R}+\sum_{k\geq 1}{1\over k}v_{\pm k}(E({E^{\prime}})){(z_{\pm}\bar{z}_{\pm})}^{-k/R}}$ $\displaystyle{\psi_{o}^{E^{\prime}}}=$ $\displaystyle e^{{\frac{1}{2}}(\phi^{\prime})({E^{\prime}})+R\sum_{k\geq 1}t_{\pm k}(z_{\pm}\bar{z}_{\pm})^{k/R}-R\sum_{k\geq 1}{1\over k}{v^{\prime}}_{\pm k}({E^{\prime}})(z_{\pm}\bar{z}_{\pm})^{-k/R}}{\psi_{o}^{E^{\prime}}},$ (5.39) where the shifted dressing operator is given by ${\cal{W}}_{s}=e^{{\frac{1}{2}}\phi(E({E^{\prime}}))+R\sum_{k\geq 1}t_{\pm k}(z_{\pm}\bar{z}_{\pm})^{k/R}+\sum_{k\geq 1}{1\over k}v_{\pm k}(E({E^{\prime}})){(z_{\pm}\bar{z}_{\pm})}^{-k/R}}$ and $\phi(E)$ is the pase for perturbed wave function (5.4). So following (3.42) we have the free energy given by $\displaystyle{\cal{F}}(\mu,R)$ $\displaystyle=$ $\displaystyle\phi(E({E^{\prime}}={\frac{ir}{\beta R}}-\mu))$ (5.40) $\displaystyle=$ $\displaystyle-i{\sum_{r=n+{\frac{1}{2}}\geq 0}}{\phi^{\prime}}({\frac{ir}{\beta R}}-\mu),$ where we have ${\phi^{\prime}}({E^{\prime}})={\phi}(E)$ (5.41) So we see the partition function in the presence of D brane in a background perturbed by momentum modes with compactified time is obtained from the one without D brane by the shift $E\rightarrow{E^{\prime}}\quad;\quad{\cal{W}}\rightarrow{\cal{W}}_{s}$ which defines a deformed Fermi surface.. ### 5.4 Lax formalism for Type 0A MQM in the presence of D brane Our lesson from the previous discussion is that Toda structure for Type 0A MQM perturbed by tachyonic modes ,in the presence of D brane can be obtained when we replace $\displaystyle{\cal{W}}$ $\displaystyle\rightarrow$ $\displaystyle{\cal{W}}_{s}=e^{iR\sum_{n\geq 1}t_{\pm n}\omega^{n/R}}\ e^{\mp i{\phi^{\prime}}(E^{\prime})}\ e^{iR\sum_{n\geq 1}{v_{\pm n}^{\prime}}(E^{\prime})\ \omega^{-n/R}}.$ $\displaystyle\psi_{o}^{\prime E}$ $\displaystyle\rightarrow$ $\displaystyle\psi_{o}^{E^{\prime}}={\psi_{o}}{(E-{\frac{1}{2\pi R}}\log(1-{\frac{2E}{\mu_{B}^{2}}}))},$ (5.42) $\displaystyle{L^{\prime}}_{+}$ $\displaystyle=$ $\displaystyle{{\cal{W}}_{s}}_{+}\omega{{\cal{W}}_{s}}^{-1}_{+},\quad L_{-}={{\cal{W}}_{s-}}\omega^{-1}{{\cal{W}}_{s-}}^{-1}_{-},$ $\displaystyle{M^{\prime}}_{+}$ $\displaystyle=$ $\displaystyle{{\cal{W}}_{s+}}{\hat{E}}{{\cal{W}}_{s+}}^{-1},\quad M_{-}={{\cal{W}}_{s-}}{\hat{E}}{{\cal{W}}_{s-}}.$ (5.43) Note the operator algebra (5.19) remains same. $\langle E|e^{\pm i{\phi^{\prime}}}{\hat{\cal{W}}_{s\pm}}{L^{\prime}}_{\pm}|{\overline{z}_{\pm}}{z_{\pm}}\rangle=\langle E|e^{\pm i{\phi^{\prime}}_{0}}\hat{\cal{W}}_{s\pm}{\hat{\overline{z}}_{\pm}}{\hat{z}_{\pm}}|{\overline{z}_{\pm}}{z_{\pm}}\rangle,$ (5.44) where ${\phi^{\prime}}=\phi({E^{\prime}})$. Expression of ${M^{\prime}}$ is as described in (5.21) $\displaystyle\langle E|$ $\displaystyle e^{\pm i{\phi^{\prime}}}\hat{\cal W}_{\pm}\;{{M^{\prime}}_{\pm}}|{\overline{z}_{\pm}}{z_{\pm}}\rangle=\pm i({z_{\pm}}\partial_{{z_{\pm}}}+{\overline{z}_{\pm}}\partial_{{\overline{z}_{\pm}}}+1)\Psi^{E^{\prime}}_{\pm}({\overline{z}_{\pm}}{z_{\pm}})$ (5.45) $\displaystyle=$ $\displaystyle\left(\sum_{k\geq 1}k\ t_{\pm k}\ {({\overline{z}_{\pm}}{z_{\pm}})}^{k/R}+{E^{\prime}}+\sum_{k\geq 1}{v^{\prime}}_{\pm k}\ {({\overline{z}_{\pm}}{z_{\pm}})}^{-k/R}\right)\Psi^{E^{\prime}}_{\pm}(({\overline{z}_{\pm}}{z_{\pm}})).$ As the partition function described in terms of Fermi level ${\mu}$ ${M^{\prime}_{\pm}}=\sum_{k\geq 1}kt_{\pm k}{L^{\prime}}_{\pm}^{k/R}+\hat{\mu}+\sum_{k\geq 1}{v^{\prime}}_{\pm k}{L^{\prime}_{\pm}}^{-k/R}.$ (5.46) The structure of the integrable flow remain same. The Toda flow equation will be given by ${\phi^{\prime}}(\mu)\equiv\phi({E^{\prime}}=-\mu)$ $i{\partial\over\partial t_{1}}{\partial\over\partial t_{-1}}{\phi^{\prime}}(\mu)=e^{i{\phi^{\prime}}(\mu)-i{\phi^{\prime}}(\mu-i/R)}-e^{i{\phi^{\prime}}(\mu+i/R)-i\phi(\mu)}.$ (5.47) Now in order to see that partition function is a tau function of Toda lattice hierarchy first note that ${\cal{Z}}(\mu,t)=\prod\limits_{n\geq 0}\exp\left[{i\beta}\phi\left(i{\frac{1}{\beta}}{n+{\litfont{1\over 2}}\over R}-\mu\right)\right].$ (5.48) with Fermi level ${E^{\prime}}=-\mu$. Now on the other hand, the zero mode of the perturbing phase is actually equal to the zero mode of the dressing operators (5.42). Hence it is expressed through the $\tau$-function as in (5.28). Since the shift in the discrete parameter n is equivalent to an imaginary shift of the chemical potential $\mu$, so (5.48) implies $e^{{i\beta}\phi(-\mu)}={\frac{{{\cal{Z}}}(\mu+{\frac{i}{2R\beta}})}{{{\cal{Z}}}(\mu-{\frac{i}{2R\beta}})}}.$ (5.49) However from (5.28) we have $e^{{i\beta}\phi(-\mu)}={\frac{{\tau_{o}}(\mu+{\frac{i}{2R\beta}})}{{\tau_{o}}(\mu-{\frac{i}{2R\beta}})}}$ (5.50) So one concludes that ${\cal{Z}}(\mu,t)=\tau_{0}(\mu,t).$ (5.51) ### 5.5 Representation in terms of a bosonic field Here we will study the classical limit following the analysis of [14] The momentum modes can be described as the oscillator modes of a bosonic field $\varphi({\overline{z}_{+}}{z_{+}},{\overline{z}_{-}}{z_{-}})=\varphi_{+}({\overline{z}_{+}}{z_{+}})+\varphi_{-}({\overline{z}_{-}}{z_{-}})$. The bosonization formula is $\Psi^{{E^{\prime}}=-\mu-i}_{\pm}({\overline{z}_{\pm}}{z_{\pm}})={\cal{Z}}^{-1}e^{\pm i\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}})}\cdot{\cal{Z}}.$ (5.52) (Note here in the presence of FZZT brane ${\mu}$ corresponds to the deformed Fermi surface) where ${\cal{Z}}$ is the partition function and $\varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}})=+R\sum_{k\geq 1}t_{k}{({\overline{z}_{\pm}}{z_{\pm}})}^{k/R}+{1\over R}\partial_{\mu}+\mu\log{\overline{z}_{\pm}}{z_{\pm}}-R\sum_{k\geq 1}{1\over k}{({\overline{z}_{\pm}}{z_{\pm}})}^{-k/R}{\partial\over\partial t_{k}}.$ (5.53) Then from (5.45) the operators $M_{\pm}$ are represented by the currents ${\overline{z}_{\pm}}{z_{\pm}}\partial_{\pm}\varphi$ ${M_{\pm}^{\dagger}}\Psi_{\pm}^{E}({\overline{z}_{\pm}}{z_{\pm}})|_{E=-\mu-i}={\cal{Z}}^{-1}{\overline{z}_{\pm}}{z_{\pm}}\partial_{\pm}\varphi\cdot{\cal{Z}}.$ (5.54) ### 5.6 The dispersionless (quasiclassical) limit We consider the quasiclassical limit $\beta\rightarrow\infty$. In this limit the integrable structure described above reduces to the dispersionless Toda hierarchy where the operators $\hat{\mu}$ and $\hat{\omega}$ can be considered as a pair of classical canonical variables with Poisson bracket $\\{\omega,\mu\\}=\omega$ (5.55) Similarly, all operators become $c$-functions of these variables. The Lax operators can be identified with the classical phase space coordinates ${\overline{z}_{\pm}}{z_{\pm}}$, which satisfy $\\{{\overline{z}_{+}},{z_{-}}\\}=\\{{z_{+}},{\overline{z}_{-}}\\}=1$ (5.56) The shape of the Fermi sea is determined by the classical trajectory corresponding to the Fermi level ${E^{\prime}}=-\mu$. So we have ${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}-{\frac{1}{2{\pi}R\beta}}\log(1-{\frac{2({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}{\mu_{B}^{2}}})-\log{\epsilon}=-\mu.$ (5.57) Where $\log{\epsilon}$ is the cut-off cancelling the singular contribution from the point $(1-{\frac{2({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}{\mu_{B}^{2}}})=0$. In the perturbed theory the classical trajectories are of the form ${\overline{z}_{\pm}}{z_{\pm}}={L^{\prime}}_{\pm}(\omega,\mu).$ (5.58) where the functions $L_{\pm}$ are of the form ${L^{\prime}}_{\pm}(\omega,\mu)=e^{{\frac{1}{2}}\partial_{\mu}{\phi^{\prime}}}\ \omega^{\pm 1}\left(1+\sum_{k\geq 1}{a^{\prime}}_{\pm k}(\mu)\ \omega^{\mp k/R}\right).$ (5.59) The flows $H_{n}$ become Hamiltonians for the evolution along the ‘times’ $t_{n}$. The unitary operators ${\cal W}_{\pm}$ becomes a pair of canonical transformations between the variables $\omega,\mu$ and $L_{\pm},M_{\pm}$. Their generating functions are given by the expectation values $S_{\pm}={\cal{Z}}^{-1}\ \varphi_{\pm}({\overline{z}_{\pm}}{z_{\pm}})\ \cdot{\cal{Z}}$ of the chiral components of the bosonic field $\phi$ $S_{\pm}=\pm R\sum_{k\geq 1}t_{\pm k}\ {({\overline{z}_{\pm}}{z_{\pm}})}^{k/R}+\mu\log({\overline{z}_{\pm}}{z_{\pm}})-{\phi^{\prime}}\pm R\sum_{k\geq 1}{1\over k}{v_{k}^{\prime}}\ {({\overline{z}_{\pm}}{z_{\pm}})}^{-k/R},$ (5.60) where $v_{k}=\partial{\cal F}/\partial t_{k}$. The differential of the function $S_{\pm}$ is ${dS}_{\pm}=M_{\pm}d{\rm log}({\overline{z}_{\pm}}{z_{\pm}})+{\rm log}\omega\ d\mu+R\sum_{n\neq 0}{H_{n}}{dt_{n}}.$ (5.61) If we consider the coordinate $\omega$ as a function of either ${\overline{z}_{+}}{z_{+}}$ or ${\overline{z}_{-}}{z_{-}}$, then $\omega=e^{{\partial_{\mu}}{S_{+}}({\overline{z}_{+}}{z_{+}})}=e^{{\partial_{\mu}}{S_{-}}({\overline{z}_{-}}{z_{-}})}.,$ (5.62) The classical string equation ${\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}}-{\frac{1}{2\pi R}}\log(1-{\frac{2({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}{\mu_{B}^{2}}})={M_{+}}={M_{-}},$ (5.63) can be written as $\displaystyle{\overline{z}_{+}}{z_{-}}$ $\displaystyle+$ $\displaystyle{\overline{z}_{-}}{z_{+}}-{\frac{1}{2\pi R}}{\rm log}(1-{\frac{2({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}{\mu_{B}^{2}}})$ (5.64) $\displaystyle=$ $\displaystyle\sum_{k\geq 1}kt_{k}{({\overline{z}_{+}}{z_{+}})}^{k/R}+\mu+\sum_{k\geq 1}v_{k}{({\overline{z}_{+}}{z_{+}})}^{-k/R}.$ ## 6 Conclusion Here we have studied Type 0A matrix model in the presence of spacelike D brane which are localized in matter direction. In matrix model this is expressed by insertion of an operator $e^{W(t_{o})}$ into the path integral. When we studied the respective MQM we found by application of Ward identity that the time translation invariance of the path integral in the presence of such operator gives the signal of leakage of MQM hamiltonian. However in dual string theory this phenomenon has a meaning that closed string hamiltonian is undergoing a leakage when the string is getting scattered from Dbrane! in order to obtain right string theory picture we impose a constraint (2.25) on matrix model path integral in the presence of D brane. We explained that this condition has an effect to constrain the Hilbert space generated by macroscopic loop operator while keeping type 0A MQM unaffected. We have shown that when we impose the constraint we get the matter one point function from collective field theory. We have further shown that exactly at the point of insertion of the brane ( which in string theory correspond to the point where open string ends are localized) the wave function for the right and left moving component of boundary state with any momentum appears to be identical which can be seen in matrix model as a consequence of this constraint. We also found right transition amplitude from a free fermionic state to coherent state. Next we consider type 0A MQM with the time t compactified on a circle. We have shown matrix model path integral in the presence of Dbrane can be expressed as Fredholm determinant. We evaluated the thermal partition function in grand canonical ensemble. As the theory is defined on a circle so the partition function correspond to that of a deformed Fermi surface. We have further shown that in absence of any such constraint, the partition function diverges. Theory on the circle also posses a symmetry (3.33), which is parallel to string theory as we discussed in section 2.4. This symmetry clearly indicates that coherent states are strongly localized at the point of insertion of macroscopic loop operator. Finally we considered type 0A MQM in the background of momentum modes. First in section 4 we made a semiclassical analysis, studied fermionic scattering in the presence of D brane. We found the effective hamiltonian in the perturbed background from semiclassical analysis. Its known that the presence of D brane change the tachyonic background. So from collective field theory analysis we found the right expression of MQM wave function i.e exact modification of the dressing operator in the presence of Dbrane. We derived the grand canonical partition function in the perturbed background in the presence of D brane. We have shown the partition function corresponds to tau function of Toda hierarchy. We have also analyzed the theory in dispersionless limit. Its interesting to study T duality between type 0A and type 0B MQM in the presence of D brane. One can also study the theory in the presence of flux background and see the consequence of the constraint. Acknowledgments The author wishes to express her deep sense of gratitude to Raghava Varma for constant support, motivation and encouragement to pursue the research at IIT Bombay. Its a pleasure to thank Satchidananda Naik for useful discussions which lead to the finding of this problem and valuable suggestions during the progress of the work. The author is greatly indebted to P. Ramadevi for her generous support and encouragement, necessary advices, and comments on the draft. The author is grateful to Sunil Mukhi and Koushik Ray for valuable discussions and important comments. The author would also like to thank S. Uma Sankar and Soumitra Sengupta for their inputs. Special thanks to Ankhi Roy and Rajeeb R. Mallick for their sincere cooperation during the work in IIT Bombay. The author is also thankful to Partha Pratim Pal and Colina Dutta for proof- reading the manuscript. Finally, the author wants to thank all the research scholars in the High energy Physics group at IIT Bombay, in particular, Reetanjali Moharana, Sushant k. Raut, Sasmita Mishra,Amal Sarkar, Ravi Manohar, Neha Shah, Himani Bhatt, Kabita Chandwani, Suprabh Prakash for discussions and help. This work is supported by funds from IRCC, IITB and the research development fund of Prof. Raghava Varma. Appendix ## Appendix A Appendix Here we will show that type 0A MQM wave functions in the presence of D brane satisfy orthogonality and biorthogonality conditions. ### A.1 Orthogonality Condition The wave function is expected to show orthonormal property. 1\. For $t<{t_{o}}$ we have the wave function ${\psi_{o}}$ given in (2.28). One can check the orthonormality property by considering the contour integral ,and it is given by [15] $\langle{\psi_{E^{\prime}}}|{\psi}_{E}\rangle={\delta}(E-{E^{\prime}}).$ (A.1) For ${t>{t_{o}}}$, first consider the wave function ${\psi_{+}}({z_{+}},{\overline{z}_{+}},t;E)$ given in (2.42). we have $\displaystyle\langle{\psi}({E^{\prime}},t){\psi}(E,t)\rangle$ $\displaystyle=$ $\displaystyle\int d{{z_{+}}}d{{\overline{z}_{+}}}[\\{1-{{\rm log}(1+{\frac{W({\overline{z}_{-}}{z_{-}},2{H_{o}})}{\mu_{B}^{2}}})\\}{e^{i{E^{\prime}}(t-{t_{o}})}}{e^{i{{\frac{\phi_{o}}{2}}}(E^{\prime})}}{({\overline{z}_{-}}{z_{-}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}]$ (A.2) $\displaystyle[\\{1-{\rm log}(1+{\frac{W({\overline{z}_{-}}{z_{-}},2{H_{o}})}{\mu_{B}^{2}}})\\}{e^{-iE(t-{t_{o}})}}{e^{-i{\frac{\phi_{o}}{2}}(E)}}{({\overline{z}_{+}}{z_{+}})}^{iE-{\frac{1}{2}}}].$ Now in order to see the orthonormal property first recall the commutation relation (2.30) Note that ${\hat{z}_{+}}$ and ${\hat{z}_{-}}$ shifts E by -i and +i respectively.So we can write ${\hat{\overline{z}_{+}}}{\hat{z}_{+}}={e^{\frac{-i\phi_{o}}{2}}}{e^{-i{\partial_{E}}}}{e^{\frac{i\phi_{o}}{2}}}$ (A.3) Similarly ${\hat{\overline{z}}_{-}}{\hat{z}_{-}}={e^{\frac{-i\phi_{o}}{2}}}{e^{i{\partial_{{E}}}}}{e^{\frac{i\phi_{o}}{2}}}.$ (A.4) Now to evaluate (A.2) first consider the expression $\displaystyle\langle{\psi_{+}}|[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{m}}|{\psi_{+}}\rangle$ $\displaystyle=$ $\displaystyle\int d{z_{+}}d{\overline{z}_{+}}\\{{e^{i{E^{\prime}}t}}{e^{{\frac{i}{2}}{\phi_{o}}(E^{\prime})}}{{({z_{+}}{\overline{z}_{+}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}\\}$ (A.5) $\displaystyle[{\frac{\partial}{\partial{\overline{z}_{+}}}}{\frac{\partial}{\partial{z_{+}}}}{]^{m}}\\{{e^{-iEt+mt}}{e^{-{\frac{i}{2}}{\phi_{o}}(E-mi)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}}\\}.$ For $E\neq{E^{\prime}}$ this can just be written as $\displaystyle\langle{\psi_{+}}|[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{m}}|{\psi_{+}}\rangle$ $\displaystyle=$ $\displaystyle{e^{-{\frac{i}{2}}{\phi_{o}}}}{e^{im{\partial_{E}}}}{e^{{{\frac{i}{2}}\phi_{o}}}}\int d{z_{+}}d{\overline{z}_{+}}\\{{e^{i{E^{\prime}}t}}{e^{{\frac{i}{2}}{\phi_{o}}(E^{\prime})}}{{({z_{+}}{\overline{z}_{+}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}\\}$ (A.6) $\displaystyle\\{{e^{-iEt}}{e^{-{\frac{i}{2}}{\phi_{o}}(E)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}}\\}$ $\displaystyle=$ $\displaystyle{e^{-{\frac{i}{2}}{\phi_{o}}}}{e^{im{\partial_{E}}}}{e^{{\frac{i}{2}}{\phi_{o}}}}\langle{\psi_{+}}({E^{\prime}})|{\psi_{+}}(E){\rangle_{E\neq{E^{\prime}}}}$ $\displaystyle=$ $\displaystyle 0.$ For $E={E^{\prime}}$ (A.2) takes the form $\displaystyle\langle{\psi_{+}}|[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{m}}|{\psi_{+}}\rangle$ $\displaystyle=$ $\displaystyle\\{{e^{i{\phi_{o}}(E)}}{e^{-i{\phi_{o}}(E-mi)}}\\}\int d{z_{+}}d{\overline{z}_{+}}[{\overline{z}_{+}}{z_{+}}{]^{-m-1}}{e^{-mt}}.$ (A.7) From contour integral which is 0 for $m\geq 1$. Also we conclude that $\langle{\psi_{+}}|[{\hat{\overline{z}}_{+}}{\hat{z}_{+}}{]^{m}}|{\psi_{+}}\rangle$ and $\langle{\psi_{-}}|[{\hat{\overline{z}}_{-}}{\hat{z}_{-}}{]^{m}}|{\psi_{-}}\rangle$ diverge. Here before going to show the orthogonality lets consider the situation when m is not an integer. This kind of integration we had in the section 5 in the expression $\langle{\psi_{o}}|{({\overline{z}_{+}}{z_{+}})}^{{\frac{n}{R}}-1}|{\psi_{o}}\rangle$. We can consider this as the product over two branh cut integrals ${z_{+}}$ and ${\overline{z}_{+}}$ and the each branch cut integral can be expressed as the sum of two standard contour integral with the cut on right and left side of the real axis $0\geq x\geq\infty$ and $-\infty\geq x\geq 0$ respectively and with a pole at $x=0$. One can see that the integral turns out to be zero for any noninteger ${\frac{m}{R}}$. We have nonzero contribution only when R is an integer and the contributing term is m=R.This term corresponds to a pure pole and give a constant contribution to the integral, Now back to the question of orthogonality. So using (A.4 ,A.7), we can be written (A.2) as $\displaystyle\langle{\psi}({E^{\prime}},t)|{\psi}(E,t)\rangle$ $\displaystyle=$ $\displaystyle\int d{z_{+}}d{\overline{z}_{+}}[\\{1-{\rm log}(1+{\frac{f(0,2{H_{o}})}{\mu_{B}^{2}}})\\}{e^{i{E^{\prime}}(t-{t_{o}})}}{e^{{\frac{i}{2}}{\phi_{o}}(E^{\prime})}}{{({z_{+}}{\overline{z}_{+}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}]$ (A.8) $\displaystyle[\\{1-{\rm log}(1+{\frac{f(0,2{H_{o}})}{\mu_{B}^{2}}})\\}{e^{-iE(t-{t_{o}})}}{e^{-{\frac{i}{2}}{\phi_{o}}(E)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}}]$ $\displaystyle=$ $\displaystyle\int d{z_{+}}d{\overline{z}_{+}}[\\{1-{\rm log}(1+{\frac{f(0,2E)}{\mu_{B}^{2}}})\\}{e^{i{E^{\prime}}(t-{t_{o}})}}{e^{{\frac{i}{2}}{\phi_{o}}(E^{\prime})}}{{({z_{+}}{\overline{z}_{+}})}^{-i{E^{\prime}}-{\frac{1}{2}}}}]$ $\displaystyle[\\{1-{\rm log}(1+{\frac{f(0,2E)}{\mu_{B}^{2}}})\\}{e^{-iE(t-{t_{o}})}}{e^{-{\frac{i}{2}}{\phi_{o}}(E)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}}].$ Now write ${\phi_{o}}={\phi_{oRe}}+i{\phi_{oIm}}$. Shifting ${\phi_{oIm}}\rightarrow{\phi_{oIm}}-{\frac{i}{2}}{\rm log}[1-{\rm log}(1+{\frac{f(0,2E)}{\mu_{B}^{2}}})]$ we get the orthogonal property (A.1) with ${\psi_{+}}(E,t)={e^{iE(t-{t_{o}})}}{e^{-{\frac{i}{2}}{\phi_{0+}}(E)}}{{({z_{+}}{\overline{z}_{+}})}^{iE-{\frac{1}{2}}}},$ (A.9) where ${\phi_{o+}}(E)={\phi_{o}}(E)-i{\rm log}[1-log(1+{\frac{f(0,2E}{\mu_{B}^{2}}})].$ (A.10) Similarly for the wave function in ${z_{-}}$ representation we find ${\phi_{o-}}(E)={\phi_{o}}(E)+ilog[1-log(1+{\frac{f(0,2E}{\mu_{B}^{2}}})]$ (A.11) So we see the consequence of the insertion of macroscopic loop operator is that the phase of the wave function develops an imaginary part which is associated with tunneling. ### A.2 Biorthogonality Relation The wave function is expected to satisfy the following biorthogonality condition which has a consequence in the evaluation of scattering amplitude [14, 15, 16]. $\int d{{z_{+}}}d{{z_{-}}}d{{\overline{z}_{+}}}d{{\overline{z}_{-}}}{\overline{\psi^{E}_{+}}}({\overline{z}_{+}}{z_{+}},t){e^{i({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})}}{\psi^{E^{\prime}}_{-}}({\overline{z}_{-}}{z_{-}},t)={\delta}(E-{E^{\prime}})$ (A.12) Now for $t\leq{t_{o}}$ we have ${\psi}={\psi_{o}}$ and for which biorthogonality relation is already derived in [14],[15] giving $e^{i\phi_{0}(E)}={\frac{\Gamma(iE+1/2)}{\Gamma(iE+1/2)}}$ [15]. For $t\geq{t_{o}}$ , .Biorthogonality relation takes the form $\displaystyle\int d{{z_{+}}}d{{z_{-}}}d{{\overline{z}_{+}}}d{{\overline{z}_{-}}}$ $\displaystyle{\overline{\psi}^{E}_{+}}({\overline{z}_{+}}{z_{+}},t)\\{1-{\hat{W}}({{\hat{z}_{+}},{\hat{\overline{z}}_{+}},{H_{o}},t})\\}{e^{i({\overline{z}_{+}}{z_{-}}+{z_{+}}{\overline{z}_{-}})}})(1-{\hat{W}}({{\hat{z}_{+}},{\hat{\overline{z}}_{+}},{H_{o}},t})){\psi^{E^{\prime}}_{-}}$ (A.13) $\displaystyle=$ $\displaystyle{\delta}(E-{E^{\prime}}),$ In order to show this note that $\displaystyle\int d{{z_{+}}}d{{z_{-}}}d{{\overline{z}_{+}}}d{{\overline{z}_{-}}}$ $\displaystyle{\overline{\psi^{E}_{+}}}({{z_{+}}},{{\overline{z}_{+}}},t)\\{1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t})\\}{e^{i({z_{+}}{z_{-}}+{\overline{z}_{+}}{\overline{z}_{-}})}})(1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t})){\psi^{E^{\prime}}_{-}}$ $\displaystyle=$ $\displaystyle\int d{{z_{+}}}d{{\overline{z}_{+}}}{\overline{\psi^{E}_{+}}}({{z_{+}}},{{\overline{z}_{+}}},t)\\{1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t})\\}(1-{\hat{W}}({{\frac{\partial}{\partial{z_{+}}}},{\frac{\partial}{\partial{\overline{z}_{+}}}},{H_{o}},t}))$ $\displaystyle\int d{{z_{-}}}d{{\overline{z}_{-}}}{e^{i({\overline{z}_{+}}{z_{-}}+{\overline{z}_{-}}{z_{+}})}}){\psi^{E^{\prime}}_{-}}({{\overline{z}_{-}}{z_{-}}},t)$ $\displaystyle=$ $\displaystyle\int d{{z_{+}}}d{{\overline{z}_{+}}}{\overline{\psi^{E}_{+}}}({{z_{+}}},{{\overline{z}_{+}}},t)\\{1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t})\\}(1-{\hat{W}}({{\hat{z}_{-}},{\hat{\overline{z}}_{-}},{H_{o}},t}))$ $\displaystyle\int d{{z_{-}}}d{{\overline{z}_{-}}}{e^{i({z_{+}}{z_{-}}+{\overline{z}_{+}}{\overline{z}_{-}})}}){\psi^{E^{\prime}}_{-}}({{z_{-}}},{{\overline{z}_{-}}},t)$ $\displaystyle=$ $\displaystyle R(E)\int d{{z_{+}}}d{{\overline{z}_{+}}}{\overline{\psi^{E}_{+}}}({{z_{-}}},{{\overline{z}_{-}}},t)\\{1-{\hat{W}}({{z_{-}},{\overline{z}_{-}}{H_{o}},t})\\}(1-{\hat{W}}({{z_{-}},{\overline{z}_{-}},{H_{o}},t}))$ $\displaystyle{\psi^{E^{\prime}}_{-}}({{z_{+}}},{{\overline{z}_{+}}},t)$ (A.14) $\displaystyle=$ $\displaystyle{R}(E){e^{i{\phi_{o+}}}}{\delta}(E-{E^{\prime}}).$ Where in order to come from 2nd to 3rd step we used the fact that in ${z_{+}}$ representation we have ${\hat{z}_{-}},{\hat{\overline{z}}_{-}}={\frac{\partial}{\partial{z_{+}}}},{\frac{\partial}{\partial{\overline{z}_{+}}}}$ The integral in the 4th step, we have evaluated in (A.8) leads to the last step. In order to get the biorthogonality relation(A.13) we must need to set $e^{i{\phi_{o+}}(E)}=R(E)$. Compared to the $t\leq{t_{o}}$ case note the shift of ${\phi_{o}}(E)$ due to the insertion of macroscopic loop operator. $R(E){\psi^{E^{\prime}}_{+}}({z_{+}},{\overline{z}_{+}},t)={\int_{-\infty}^{\infty}}d{{z_{-}}}d{{\overline{z}_{-}}}{e^{i({z_{+}}{z_{-}}+{\overline{z}_{+}}{\overline{z}_{-}})}}){\psi^{E^{\prime}}_{-}}({z_{-}},{\overline{z}_{-}},t)$ (A.15) R(E) getting absorbed to decide ${\phi_{o}}$ and shifting ${\phi_{o}}$ according to (A.11) we get the above result. ## References * [1] Igor R. Klebanov, String Theory in Two Dimensions, hep-th/9108019 * [2] P. Ginsparg, Gregory Moore, Lectures on 2D gravity and 2D string theory, (TASI 1992), hep-th/9304011 * [3] Joseph Polchinski, What is String Theory , hep-th/9411028 * [4] John McGreevy, Herman Verlinde, Strings from Tachyons, hep-th/0304224 * [5] Igor R. 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arxiv-papers
2011-01-11T11:09:26
2024-09-04T02:49:16.301618
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Chandrima Paul", "submitter": "Chandrima Paul", "url": "https://arxiv.org/abs/1101.2094" }
1101.2130
020010 2010 V. Lakshminarayanan S. Roy, Dayalbagh Educational Institute, Agra, India. 020010 Live cell imaging using metallic nanoparticles as tags is an emerging technique to visualize long and highly dynamic processes due to the lack of photobleaching and high photon rate. However, the lack of excited states as compared to fluorescent dyes prevents the use of resonance energy transfer and recently developed super resolution methods to measure distances between objects closer than the diffraction limit. In this work, we experimentally demonstrate a technique to determine subdiffraction distances based on the near field coupling of metallic nanoparticles. Due to the symmetry breaking in the scattering cross section, not only distances but also relative orientations can be measured. Single gold nanoparticles were prepared on glass, statistically yielding a small fraction of dimers. The sample was sequentially illuminated with two wavelengths to separate background from nanoparticle scattering based on their spectral properties. A novel total internal reflection illumination scheme in which the polarization can be rotated was used to further minimize background contributions. In this way, radii, distance and orientation were measured for each individual dimer, and their statistical distributions were found to be in agreement with the expected ones. We envision that this technique will allow fast and long term tracking of relative distance and orientation in biological processes. # Experimental determination of distance and orientation of metallic nanodimers by polarization dependent plasmon coupling H. E. Grecco [inst1, inst2] O. E. Martínez[inst1] E-mail: hgrecco@df.uba.arE- mail: oem@df.uba.ar ($22$ April 2010; 2 December 2010) ††volume: 2 99 inst1 Laboratorio de Electrónica Cuántica, Universidad de Buenos Aires. Buenos Aires, Argentina. inst2 Current address: Department of Systemic Cell Biology Max Planck Institute of Molecular Physiology. Dortmund, Germany ## 1 Introduction Microscopy is an example of the ongoing symbiotic relationship between physics and biology: as early microscopes allowed fundamental discoveries like microorganisms or DNA; the need to see smaller, faster and deeper has pushed the development of a plethora of optical concepts and microscopy techniques. Today, fluorescence microscopy is an essential tool in biology as it can visualize the spatio-temporal dynamics of intracellular processes. However, many important mechanisms, like protein interaction, clustering or conformational changes, occur at length scales smaller than the resolution limit of conventional microscopy and therefore cannot be assessed by standard imaging. Unraveling the dynamics of such inter- and intramolecular mechanisms that provides function richness to molecules and molecular complexes is essential to understand key biological processes such as cellular signal propagation. Subdiffraction distances have been determined by exploiting quantum and near field properties of the interaction between light and matter in the nanometer scale. For example, Fluorescence/Förster Resonance Energy Transfer (FRET) [5, 6, 7, 14] has proven to be a valuable technique as it provides an optical signal directly related to the proximity of the molecules. The desire to extend this technique to other biological systems with different time and length scales has been hindered by the inherent limitations of fluorescent dyes (i.e. lack of photostability, low brightness and short range of interaction). Super resolution techniques such as STED [23] or PALM [2] have recently gained momentum to directly observe fluorescent molecules spaced closer than the diffraction limit. Although much work has been done to increase the total acquisition time and frame rate, these methods are still limited by the lack of photostability and the need to image a single resolvable structure per diffraction limited spot at a time. In the past, it has been shown that scattering microscopy using metallic nanoparticles can complement its fluorescence sibling as it uses an everlasting tag with no rate-limited amount of photons [10]. Metallic nanoparticles are stable, biocompatible and easy to synthesize and conjugate to biological targets and thereby ideal as contrast agents. A landmark example of the biological application of such techniques was the direct observation of receptors hopping across previously unknown membrane domains. This provided valuable insight into the spatial regulation of signaling complexes and closed a 30-year controversy about the diffusion coefficients of membrane proteins [21]. While previous experiments using fluorescent tags yielded a diffusion coefficient in biological membranes much slower than the one observed in synthetic membranes, the fast acquisition speed (40 103 frames per second) enabled by scattering microscopy showed that this is the result of a fast diffusion and a slow hopping rate between domains [9]. In addition, the presence of a plasmon, i.e. a collective oscillation of the free electrons within the nanoparticle, converts metallic nanoparticles into very effective scatterers when illuminated at their resonance optical frequency. The resulting strong electromagnetic enhancement in the vicinity of the particle provides a near field effect that can be used to sense information about their surroundings such as the effective index of refraction or the presence of other scatterers [8, 16]. For example, it has been experimentally shown that the shift in the plasmon resonance can be used to determine the length of DNA molecules attached to a metallic nanoparticle [13]. Moreover, the coupling between two nanoparticles in close proximity produces an alteration of the plasmon spectra. This alteration has been used as a nanometric ruler to determine the distance between them [20, 19]. In a previous work, [4] we theoretically showed that the coupling between two nanoparticles is highly sensitive to the polarization of the external field. The scattering cross section ($C_{sca}$) is maximum when the incident polarization is parallel to the dimer orientation due to the reinforcement of the external field by the induced dipoles [Fig. 1(a)]. As the coupling decreases monotonically with the distance between nanoparticles, so does the average $C_{sca}$ over all polarizations ($v_{m})$ [Fig. 1(b)] and the anisotropy [Fig. 1(c)] defined as: $\displaystyle\eta=\frac{C_{sca}^{\parallel}-C_{sca}^{\perp}}{C_{sca}^{\parallel}+C_{sca}^{\perp}}.$ (1) We have proposed, in our previous work, that by measuring the scattering cross section as a function of the incident polarization angle, the axis of the dimer and the distance between nanoparticles could be determined. In this work, we provide experimental evidence supporting this concept by measuring gold nanodimers on a glass surface and we introduce a novel total internal reflection experimental setup that provides polarized illumination with a high NA objective. Figure 1: Conceptual idea of the technique. (a) Two metallic nanoparticles of radii $a$ located at a distance $d$ are illuminated with a linearly polarized electromagnetic field. (b) Theoretical results as a function of the interparticle distance for the average $C_{sca}$ over all polarizations ($v_{m}$) and (c) the anisotropy. Parameters of the calculation: $a$ = 20 nm, wavelength of the light: 532 nm. ## 2 Materials and methods ### 2.1 Sample preparation Coverslips were cleaned by sonication at 50∘ for 20 minutes in Milli-Q water, and then sequentially immersed for 5 seconds in HFL 5%, sodium bicarbonate and acetone (analytic level). After the cleaning process, coverslips were dried and stored in a chamber overpressurized with nitrogen until further use. Before sample preparation, a Parafilm chamber was assembled on top of the coverslip. To create a hydrophilic surface, bovine serum albumina (BSA) in phosphate buffer solution (PBS) was incubated for 15 min and then rinsed with PBS. Fluorescein-streptavidin in PBS (50 mg/ml) was then incubated for 30 min and rinsed with PBS, to obtain an adsorbed layer that was verified using confocal fluorescence microscopy. Finally, a solution of biotinylated gold nanoparticles, nominal radius ($20\pm 5$) nm (GB-01-40. EY Laboratories, USA), was incubated for 15 min and then rinsed by washing 5 times with PBS. The concentration and incubation time where empirically chosen to provide a concentration about 1 nanoparticle/10 $\mu$m2. As the particles are randomly distributed, it is expected to find many monomers, some dimers, and very few trimers and higher n-mers. A negative control sample was prepared in the same way but omitting the incubation of gold nanoparticles. ### 2.2 Dual color scheme Spurious reflections and scattering centers other than gold will produce unwanted bright spots in the images. Even thresholding the image taken at the resonance peak (532 nm) will result in many false positive regions. The presence of a plasmon resonance in the scattering spectrum of gold nanoparticles was used as a signature to distinguish them. The ratio between the scattering cross section at 532 nm and 473 nm was found to be larger than 1.4 for gold monomers [Fig. 2(a), solid thin line] using Mie theory [3] and even larger for dimers (dashed line) as calculated using GMMie, a multiparticle extension of the Mie theory [11, 12]. In contrast, non metallic scattering centers lack of a plasmon resonance and therefore yield a smaller ratio between 532 nm and 473 nm $C_{sca}$ (solid thick line). Therefore, by imaging at these two wavelengths and thresholding the ratio image above 1.4, the pixels containing gold nanoparticles were further segmented. Figure 2: Experimental Setup. (a) Comparison of spectra averaging over all polarizations. While the spectrum of dielectric particles (thick solid line) decreases monotonically, the spectra of metallic monomers and dimers (thin lines) show a plasmon resonance. (b) The polarization of the refracted wave is dependent on the direction of the incident polarization with respect to the displacement in the back focal plane (BFP) of the objective which defines the plane of incidence. (c) The sample is illuminated in Total Internal Reflection and imaged using a cooled CCD. Laser light is tightly focused off-center in the back focal plane of the objective and the angle $\beta$ is moved using a pair of galvanometer scanners moving in orthogonal directions. ### 2.3 Polarization control in Total Internal Reflection We used Total Internal Reflection (TIR) microscopy [1] to restrict the illumination to the surface of the coverslip using an evanescent wave. In objective-based TIR, the beam is focused off-axis in the back focal plane (BFP) of the objective to achieve critical illumination. The components of evanescent field are defined by the angle of incidence ($\theta$) and the incident polarization with respect to the plane of incidence. Indeed, rotating the excitation polarization before entering the microscope does not produce a constant intensity in the sample plane as the transmission efficiency for the parallel polarization [Fig. 2(b), left] will be much smaller than for the perpendicular one [Fig. 2(b), center]. A circularly polarized beam before the objective results in an “elliptically”111The electromagnetic field in the sample cannot be said to be strictly elliptically polarized as an evanescent field (not a propagating beam) is generated after the interface. Nevertheless, an elliptical rotation of the electric field is achieved. polarized field which has the minor axis in the plane of incidence [Fig. 2(b), right]. The ratio between the major and minor axis of this evanescent “elliptical” beam depends on $\theta$ and if the plane of incidence is changed, the ellipse will rotate with it. We therefore modified a wide-field inverted microscope (IX71. Olympus, Japan) using a TIRF objective (Olympus TIRFM 63X/1.45 PlanApo Oil) to allow rotating the plane of incidence [Fig. 2(c)] by changing the position in which the beam is focused in the BFP. Two lasers were used: one near the gold particle plasmon resonance (532 nm, Compass C315M. Coherent Inc., USA) and another shifted towards shorter wavelengths (473 nm, VA-I-N-473. Viasho Technology, China). The power of the lasers after the objective was 13 $\mu$W. Circularly polarized light was achieved at the BFP by inserting a quarter and a half wave plate in the beam path adjusted to precompensate for the polarization dependent transmission of the beam splitter, filters and mirrors. The beam was expanded and filtered to achieve a diffraction limited spot in the BFP. In order to displace the beam in the BFP and therefore change the plane of incidence, a pair of computer controlled galvanometer scanners (SC2000 controller, Minisax amplifier and M2 galvanometer. GSI Group, USA) were used. The polarization distortion due to the change in the angle of incidence onto the mirrors of the scanners (while moving) was verified to be negligible. Images of the sample were acquired using a cooled monochrome CCD camera (Alta U32. Apogee Instruments, USA. $2148\times 1472$ pixels each $6.8\times 6.8$ $\mu$m2) through a dichroic filter for the fluorescence sample (XF2009 550DCLP. Chroma Technology, USA) or a 30/70 beam splitter for the gold nanoparticles (21009. Chroma Technology, USA). ### 2.4 Intrinsic anisotropy determination To assure a constant ratio between the two polarizations of the beam and a uniform intensity, the beam needed to be moved on the BFP in a circle centered in the optical axis. Failure to do this would have reduced the dynamic range of the system by introducing an intrinsic anisotropy. To minimize this value, the path of the beam was iteratively modified while measuring the anisotropy (see below) of a diluted solution of Rhodamine 101. The emission of such a sample is independent of the excitation polarization and thus the measured anisotropy can be assigned only to the system. After optimization, the obtained anisotropy for the 532 and 473 channels was 0.06 and 0.05 in a region of $50\times 50$ $\mu$m2 ($440\times 440$ pixel2). It is worth noting than these values are five times smaller than the expected anisotropy for a 20 nm homodimer. ### 2.5 Image acquisition and processing The acquisition process consisted in sequentially imaging at 473 nm and 532 nm while changing the angle $\beta$ in 20 discrete steps over $2\pi$ to sample different polarizations. An image with both lasers off was also acquired to account for ambient light and dark counts of the camera. Each image was background corrected and normalized by the excitation power and detection efficiency at the corresponding wavelength. Mean images ($m_{473}$ and $m_{532}$) were obtained by averaging over all polarizations and, from these, the ratio image $m=m_{532}/m_{473}$ was calculated. The scattering image at the resonance peak ($m_{532}$) was segmented by Otsu’s thresholding and masked with the ratio image thresholded above 1.4 to detect pixels containing gold. A connected region analysis was performed to keep only those regions with area between 3 and 15 pixels. The upper bound was chosen to be slightly bigger than the airy diffraction limited spot for the system, but still much smaller than the mean distance between gold nanoparticles. Regions containing single gold nanoparticles should have a constant intensity over the stack of frames acquired for different polarization orientation, while dimers should provide an oscillating signal with period $\pi$. Therefore, a Fourier analysis [Eq. (2)] was performed. For each pixel, the following coefficients were calculated: $\displaystyle\tilde{c}_{2}$ $\displaystyle=\frac{2}{\sum_{\beta}I(\beta)}\sum_{\beta}I(\beta)cos(2\beta)$ (2a) $\displaystyle\tilde{s}_{2}$ $\displaystyle=\frac{2}{\sum_{\beta}I(\beta)}\sum_{\beta}I(\beta)sin(2\beta)$ (2b) $\displaystyle\eta$ $\displaystyle=\sqrt{\tilde{c}_{n}^{2}+\tilde{s}_{n}^{2}}$ (2c) $\displaystyle tan(\delta)$ $\displaystyle=\frac{\tilde{s}_{2}}{\tilde{c}_{2}}$ (2d) This was done for each wavelength obtaining values for the anisotropy ($\eta_{473}$ and $\eta_{532}$) and orientation ($\delta_{473}$ and $\delta_{532}$) in each pixel. The acquisition and analysis process were repeated for 30 and 10 fields of view of the sample and negative control sample respectively. Retrieval uncertainty was estimated by performing the same numerical analysis on simulated data calculated by adding two terms to the theoretical response for different dimers. The first, an oscillating term with $2\pi$ periodicity, emulated a small misalignment that produced a non- constant illumination while rotating the beam in the BFP. The amplitude of this term was obtained from the Rhodamine calibration. The second term simulated coherent background and its values for each pixel were drawn from a Gaussian distribution obtained from the control images. Figure 3: Experimental results (a) Representative images of a field of view for single wavelength (left) and a ratio (right) imaging. Some points (circles) are bright in both images (gold) while others squares faint in the ratio image (not gold). Gold containing regions were segmented by finding bright pixels in both images. Images size: 50x50 ${\mu m}^{2}$. 440x440 pixels. (b) Anisotropy vs. mean value. A strong correlation is observed between anisotropy and mean value for both wavelengths as expected. The 473 nm data shows a plateau due to the intrinsic anisotropy of the system. ## 3 Results and discussion The need for a two color approach is evident when comparing single wavelength with ratio images: while regions with and without nanoparticles [Fig. 3(a), circles and squares respectively] were bright due to the high background in the single channel image, only gold nanoparticles were above the threshold in the ratio image. Importantly, in the negative control stack, no pixel was found above the a priori defined threshold. For the 35 regions identified as gold monomers/dimer, a strong correlation between anisotropy and mean value was found as expected [Fig. 3(b)]. The variance over each region for all values was below the corresponding retrieval uncertainty. A plateau was observed for the 473 nm channel due to the intrinsic anisotropy of the system. In this set of candidates for dimers, eight points presented an unexpected high individual anisotropy and hence were rejected. Although the exact origin of this eight scattering centers could not be established, it is worthwhile noting that it is extremely relevant in a tracking experiment to avoid false positives that would severely distort the retrieved information, and this ability to reject scattering centers based on their response is an additional advantage of the technique. Figure 4: Comparison between experimental data and homodimer model. The mean value ratio is plotted against the anisotropy ratio for the regions segmented from the images (blue dots). Theoretical calculations for different homodimers are also plotted. The vertical lines show the results keeping constant the surface to surface distance ($d_{ss}$) while changing the radii ($a$). The opposite is shown in the horizontal lines. Remarkably, experimental data distributed close to the curve for 20 nm homodimers (solid line) as expected as this is in fact the mean radii of the particles used. Notice that this is not a fit (no free parameters), but the predictions from the homodimer model superimposed to the experimental data. The recovered scattering parameters were compared with the expected results for a homodimer configuration (Fig. 4) obtaining a good correspondence with the nominal size of the nanoparticles used (20 nm). Indeed, the mesh shown in Fig. 4 was calculated using only the photophysical and geometrical properties of the dimer (no fitted parameters). The ability of the technique to blindly recover the correct size of the particles was a cross-check for its reliability. The actual configuration of each dimer was obtained by fitting the theoretical model to the experimental values. The in-plane orientation was directly obtained as a weighted average of these values. To fit the radii of each particle and the distance between them, the values of $\eta_{473}$, $\eta_{532}$ and $m$ were used. The values were first fitted using the analytical solution of a homodimer configuration in the dipole-dipole approach, in which the induced dipole moment $\vec{p}$ of each particle in an incident field $\vec{E}_{inc}$ can be expressed as: $\displaystyle\vec{p}_{\parallel}=\frac{1}{1-\frac{\alpha}{2d^{3}\pi}}\epsilon_{m}\alpha\vec{E}_{inc}$ (3a) $\displaystyle\vec{p}_{\perp}=\frac{1}{2+\frac{\alpha}{2d^{3}\pi}}\epsilon_{m}\alpha\vec{E}_{inc}$ (3b) $\epsilon_{m}$ being the dielectric constant of the medium and $\alpha$ the polarizability of the sphere which is proportional to the cube of its radius [4, 3]. This homodimer configuration was used as an initial value in the time consuming iterative process of finding a heterodimer configuration compatible with the experimental data using GMMie calculation. The traveling wave approximation of the evanescent field was used as the particles are small compared to the decay length of the field [18, 22] and the collection efficiency is much smaller for the dipole induced in the optical axis orientation than for the one perpendicular to it. In this way, the two radii, orientation and distance for each dimer were obtained (Fig. 5). The distribution of radii was found to be centered in 20 nm, compatible with the nominal size of the particles. For the interparticle distances, the distribution showed an increase as expected but then a decrease for distances at which the anisotropy is close to the intrinsic anisotropy of the system. This mismatch at larger distances is due to the conservative criterion to separate dimers from monomers that fails to identify correctly nanoparticles that couple weakly. The dimer orientation was uniformly distributed between $-\pi$ and $\pi$, as expected. To further test this, we compared the experimental and simulated distributions using a Kolmogorov- Smirnov [15] statistical test. The level of significance set at the usual value of 5%. The expected distributions (Fig. 5, right column) were obtained from the nominal radius of the nanoparticles and Monte Carlo simulation of the adsorption process. The experimental and simulated distributions for radii and orientation were found in close agreement. For the interparticle distance, the distributions were found compatible when compared up to 70 nm. Figure 5: Fit to a heterodimer configuration using GMMie. For each dimer (left column, x axis), the radii (top), the distance (middle) and the angle (bottom) were obtained. Experimental and simulated histograms are shown for each magnitude (middle and right columns). ## 4 Conclusions We have experimentally shown that distance between two nanoparticles, as well as their individual radii, can be obtained by measuring the intensity of spot as a function of the incident polarization. Additionally, the in-plane orientation of the dimer was obtained with less than 10∘ uncertainty. The presented method strongly exploits the particular spectroscopic properties of metallic nanoparticles to sense their environment. It is worth noting that the distance in which the technique is sensitive scales with the radii of the used particles. By using nanoparticles with radius between 4 and 20 nm, the gap between FRET and standard super-resolution techniques (10 nm to 50 nm) could be bridged. This fact, together with the ability to recover orientation, makes this approach unique. A non-uniform anisotropic illumination is the major source of uncertainty as it will mask the anisotropy of the dimers, specially for those in which the distance is much larger than the radius of the particles. This should be properly controlled by measuring an isotropic sample as it was done in this work. Additionally, it is important to mention that various factors such as a non-monodisperse or non-spherical population of particles will have an incidence in the recovery of dimer distance, size and orientation from model based fittings. However, having a multiparametric readout (i.e. $m_{532}$, $m_{473}$, $\eta_{473}$ and $\eta_{532}$) with a non-trivial dependence of the physical parameters (i.e. distance, size, shape) provides a way to control for this and exclude points that do no match the expected relations between photophysical properties. Such conservative criterion would be recommended for tracking experiments where false negatives have minor impact as they only reduce the amount of information gathered per frame. If the yield of dimers can be raised and several dozens of dimers can be imaged in the same field of view, we expect that the presented technique will be useful to add information about the relative movement of the two particles to already existing tracking assays. Numerical simulations showed that if coherent background can be diminished, an order of magnitude (i.e. by the use of broader band excitation source), distance and orientation could be tracked at 100 Hz. If just the rotation and the movement of the center of mass is desired, the retrieval can be performed much faster as only one wavelength (532 nm) would be necessary after an initial identification of the dimers is made by the two color method. As other scattering based techniques, the lack of photobleaching constitutes a major advantage of this approach. Moreover, the absence of saturation in the light scattering of metallic nanoparticles, as compared to the absorption of fluorescent molecules, provides an acquisition rate only limited by the detector speed. The combination of these two aspects means that scattering based microscopy does not need to make compromises between experiment length and temporal resolution. We have also demonstrated that the use of two color imaging can provide an efficient way to detect scattering centers that have plasmon resonances. Recent work by Olk et al. [17] has shown that upon illumination with a wideband light source, a modulation of the spectra due to far-field effects can be observed as a function of the incident polarization. The combination of the two techniques could lead to a more robust detection of both, orientation and distance. Finally, the novel illumination setup introduced in this work provides a robust way to change the polarization in TIR, allowing the implementation of anisotropy based techniques in fluorescence and scattering microscopy. Additionally, the same scheme permits a fast switching between TIR and standard wide-field as well as sweeping of multiple evanescent field penetration depths. ###### Acknowledgements. HEG was funded by the Universidad de Buenos Aires. ## References * [1] D Axelrod, T P Burghardt, N L Thompson, Total internal reflection fluorescence, Annu. Rev. Biophys. Bio. 13, 247 (1984). * [2] E Betzig, G H Patterson, R Sougrat, O W Lindwasser, S Olenych, J S Bonifacino, M W Davidson, J Lippincott-Schwartz, H F Hess, Imaging intracellular fluorescent proteins at nanometer resolution, Science 313, 1642 (2006). * [3] C F Bohren, D R Huffman, Absorption and scattering of light by small particles, Whiley (1983). * [4] H E Grecco, O E Martínez, Distance and orientation measurement in the nanometric scale based on polarization anisotropy of metallic dimers, Opt. Express 14, 8716 (2006). * [5] F G Haj, P J Verveer, A Squire, B G Neel, P I H Bastiaens, Imaging sites of receptor dephosphorylation by PTP1B on the surface of the endoplasmic reticulum, Science 295, 1708 (2002). * [6] E A Jares-Erijman, T M Jovin, FRET imaging, Nat. Biotechnol. 21, 1387 (2003). * [7] Z Kam, T Volberg, B Geiger, Mapping of adherens junction components using microscopic resonance energy transfer imaging, J. Cell Sci. 108, 1051 (1995). * [8] K L Kelly, E Coronado, L L Zhao, G C Schatz, The optical properties of metal nanoparticles: The influence of size, shape, and dielectric environment, J. Phys. Chem. B 107, 668 (2003). * [9] A Kusumi, C Nakada, K Ritchie, K Murase, K Suzuki, H Murakoshi, R S Kasai, J Kondo, T Fujiwara, Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: High-speed single-molecule tracking of membrane molecules, Annu. Rev. Bioph. Biom. 34, 351 (2005). * [10] D Lasne, G Blab, S Berciaud, M Heine, L Groc, D Choquet, L Cognet, B Lounis, Single nanoparticle photothermal tracking (SNaPT) of 5-nm gold beads in live cells, Biophys. J. 91, 4598 (2006). * [11] Y lin Xu, Electromagnetic scattering by an aggregate of spheres, Appl. Optics 34, 4573 (1995). * [12] Y lin Xu, Electromagnetic scattering by an aggregate of spheres: Far field, Appl. Optics 36, 9496 (1997). * [13] G L Liu, et al., A nanoplasmonic molecular ruler for measuring nuclease activity and dna footprinting, Nat. Nanotechnol. 1, 47 (2006). * [14] N P Mahajan, K Linder, G Berry, G W Gordon, R Heim, B Herman, Bcl-2 and bax interactions in mitochondria probed with green fluorescent protein and fluorescence resonance energy transfer, Nat. Biotechnol. 16, 547 (1998). * [15] F J Massey, The Kolmogorov–Smirnov test for goodness of fit, J. Am. Stat. Assoc. 46, 68 (1951). * [16] A D McFarland, R P V Duyne, Single silver nanoparticles as real-time optical sensors with zeptomole sensitivity, Nano Lett. 3, 1057 (2003). * [17] P Olk, J Renger, M T Wenzel, L M Eng, Distance dependent spectral tuning of two coupled metal nanoparticles, Nano Lett. 8, 1174 (2008). * [18] M Quinten, A Pack, R Wannemacher, Scattering and extinction of evanescent waves by small particles, Appl. Phys. B: Lasers O. 68, 87 (1999). * [19] B M Reinhard, M Siu, H Agarwal, A P Alivisatos, J Liphardt, Calibration of dynamic molecular rulers based on plasmon coupling between gold nanoparticles, Nano Lett. 5, 2246 (2005). * [20] C Sönnichsen, B M Reinhard, J Liphardt, A P Alivisatos, A molecular ruler based on plasmon coupling of single gold and silver nanoparticles, Nature Biotech. 23, 741 (2005). * [21] K Suzuki, K Ritchie, E Kajikawa, T Fujiwara, A Kusumi, Rapid hop diffusion of a g-protein-coupled receptor in the plasma membrane as revealed by single-molecule techniques, Biophys. J. 88, 3659 (2005). * [22] R Wannemacher, A Pack, M Quinten, Resonant absorption and scattering in evanescent fields, Appl. Phys. B: Lasers O. 68, 225 (1999). * [23] V Westphal, S O Rizzoli, M A Lauterbach, D Kamin, R Jahn, S W Hell, Video-rate far-field optical nanoscopy dissects synaptic vesicle movement, Science 320, 246 (2008).
arxiv-papers
2011-01-11T14:07:56
2024-09-04T02:49:16.323120
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "H. E. Grecco, O. E. Mart\\'inez", "submitter": "Luis Ariel Pugnaloni", "url": "https://arxiv.org/abs/1101.2130" }
1101.2181
# CLASSIFICATION OF 5-DIMENSIONAL MD-ALGEBRAS HAVING NON-COMMUTATIVE DERIVED IDEALS Le Anh ${\bf Vu}^{*}$, Ha Van ${\bf Hieu}^{**}$ and Tran Thi Hieu ${\bf Nghia}^{***}$ ∗Department of Mathematics and Economic Statistics, University of Economics and Law Vietnam National University - Ho Chi Minh City, Viet nam E-mail: vula@uel.edu.vn ∗∗ E-mail: havanhieu88@gmail.com ∗∗∗ E-mail: hieunghiatoan1a@gmail.com ###### Abstract The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e. five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbits) are orbits of zero or maximal dimension. The main result of the paper is the classification up to an isomorphism of all MD5-algebras having non-commutative derived ideals. AMS Mathematics Subject Classification: Primary 22E45, Secondary 46E25, 20C20. Key words: Lie group, Lie algebra, MD5-group, MD5-algebra, K-orbits. ### INTRODUCTION In 1962, studying theory of representations, A. A. Kirillov introduced the Orbit Method (see [2]). This method quickly became the most important method in the theory of representations of Lie groups. Using the Kirillov’s Orbit Method, we can obtain all the unitary irreducible representations of solvable and simply connected Lie Groups. The importance of Kirillov’s Orbit Method is the co-adjoint representation (K-representation). Therefore, it is meaningful to study the K-representation in the theory of representations of Lie groups. After studying the Kirillov’s Orbit Method, Do Ngoc Diep in 1980 suggested to consider the class of Lie groups and Lie Algebras MD such that the $C^{*}-algebras$ of them can be described by using KK-functors (see [1]). Let G be an n-dimensional real Lie group. G is called an MDn-group if and only if its orbits in the K-representation (i.e. K-orbits) are orbits of dimension zero or maximal dimension. The corresponding Lie algebra of G is called an MDn-algebra. Thus, classification and studying of K-representation of the class of MDn-groups and MDn-algebras is the problem of interest. Because all Lie algebras of n dimension (with $n\leq 3$) were listed easily, we have to consider MDn-groups and MDn-algebras with $n\geq 4$. In 1990, all MD4-algebras were classified up to an isomorphism by Vu - the first author (see [5]). Recently, Vu and some his colleagues have continued studying MD5-groups and MD5-algebras having commutative derived ideals (see [6], [7], [8]). In 2008, a classification of all MD5-algebras having commutative derived ideals was given by Vu and Kar Ping Shum (see [9]). In this paper, we shall give the classification up to an isomorphism of all MD5-algebras $\mathcal{G}$ whose derived ideals ${\mathcal{G}}^{1}:=[\mathcal{G},\mathcal{G}]$ are non-commutative. This classification is the main result of the paper. The paper is organized as follows: The first section deals with some preliminary notions, section 2 is devoted to the discussion of some results on MDn-algebras, in particular, the main result of the paper is given in this section. ## 1 PRELIMINARIES We first recall in this section some preliminary results and notations which will be used later. For details we refer the reader to the book [2] of A. A. Kirillov and the book [1] of Do Ngoc Diep. ### 1.1 The K-representation and K-orbits Let G be a Lie group, $\cal G$ =Lie(G) be the corresponding Lie algebra of G and ${\cal G}^{*}$ be the dual space of $\cal G$. For every $g\in G$, we denote the internal automorphism associated with g by $A_{(g)}$, and whence, $A_{(g)}:G\to G$ can be defined as follows $A_{(g)}:=g.x.g^{-1},\forall x\in G$. This automorphism induces the following map ${A_{\left(g\right)}}_{*}:{\cal G}\to{\cal G}$ which is defined as follows ${A_{\left(g\right)}}_{*}\left(X\right):=\frac{d}{{dt}}\left[{g.\exp\left({tX}\right).g^{-1}}\right]\left|{{}_{t=0}}\right.$, $\forall X\in{\cal G}.$ This map is called the tangent map of $A_{(g)}$. We now formulate the definitions of K-representation and K-orbit. ###### Definition 1.1.1. The action $K:G\longrightarrow Aut(\mathcal{G}^{*})$ $g\longmapsto K_{(g)}$ such that $\left\langle{K_{\left(g\right)}(F),X}\right\rangle:=\left\langle{F,{A_{\left(g^{-1}\right)}}_{*}\left(X\right)}\right\rangle,\forall F\in{\cal G}^{*},\,\forall X\in{\cal G}$ is called the co-adjoint representation or K-representation of G in $\mathcal{{G}^{*}}$. ###### Definition 1.1.2. Each orbit of the co-adjoint representation of G is called a K-orbit of G. We denote the K-orbit containing F by $\Omega_{F}$. Thus, for every $F\in{\cal G}^{*}$, we have $\Omega_{F}:=\left\\{{K\left(g\right)(F)|g\in G}\right\\}$. The dimension of every K-orbit of an arbitrary Lie group G is always even. In order to define the dimension of the K-orbits $\Omega_{F}$ for each F from the dual space ${\cal G}^{*}$ of the Lie algebra $\cal G$ = Lie(G), it is useful to consider the following skew-symmetric bilinear form $B_{F}$ on $\cal G$: $B_{F}\left({X,Y}\right)=\left\langle{F,\left[{X,Y}\right]}\right\rangle,\forall X,Y\in{\cal G}$. We denote the stabilizer of F under the co-adjoint representation of G in ${\cal G}^{*}$ by $G_{F}$ and ${\cal G}_{F}:=$ Lie($G_{F}$). We shall need in the sequel of the following result. ###### Proposition 1.1.3 (see [2, Section 15.1]). $KerB_{F}={\cal{G}}_{F}$ and $dim{\Omega}_{F}=dim{\mathcal{G}}-dim{\mathcal{G}}_{F}=rankB_{F}.$ $\square$ ### 1.2 MD-groups and MD-algebras ###### Definition 1.2.1 (see [1, Chapter 2]). An MD-group is a real solvable Lie group of finite dimension such that its K-orbits are orbits of dimension zero or maximal dimension (i.e. dimension k, where k is some even constant and no more than the dimension of the considered group). When the dimension of considered group is n (n is a some positive integer), the group is called an MDn-group. The Lie algebra of an MD-group (MDn-group, respectively) is called an MD-algebra (MDn-algebra, respectively). The following proposition gives a necessary condition for a Lie algebra belonging to the class of MD-algebras. ###### Proposition 1.2.2 (see [3, Theorem 4]). Let $\mathcal{G}$ be an MD-algebra. Then its second derived ideal ${\mathcal{G}}^{2}:=[[\mathcal{G},\mathcal{G}],[\mathcal{G},\mathcal{G}]]$ is commutative. $\square$ We point out here that the converse of the above result is in general not true. In other words, the above necessary condition is not a sufficient condition. So, we now only consider the real solvable Lie algebras having commutative second derived ideals. Thus, they could be MD-algebras. ###### Proposition 1.2.3 (see [1, Chapter 2, Proposition 2.1]). Let $\cal{G}$ be an MD-algebra with F (in ${\cal{G}}^{*}$) is not vanishing perfectly in ${\cal{G}}^{1}:=[\mathcal{G},\mathcal{G}]$, i.e. there exists $U\in{\cal{G}}^{1}$ such that $\langle F,U\rangle\neq 0.$ Then the K-orbit ${\Omega}_{F}$ is one of the K-orbits having maximal dimension. $\square$ ## 2 THE CLASS OF MD5-ALGEBRAS HAVING NON-COMMUTATIVE DERIVED IDEALS ### 2.1 Some Results on the Class of MD-algebras In this subsection, we shall present some results on general MDn-algebras ($n\geq 4$). Firstly, we consider a real solvable Lie algebra $\cal{G}$ of dimension $n$ such that $dim{\cal G}^{1}=n-k$ ($k$ is some integer constant, $1\leq k<n$), ${\cal G}^{2}$ is non - trivial commutative and $dim{\cal G}^{2}=dim{\cal G}^{1}-1=n-k-1$. Without loss of generality, we may assume that ${\cal G}\,\,\,=gen\left(X_{1},X_{2},...,X_{n}\right),\,(n\geq 4)$, ${{\cal G}^{1}}=gen\left(X_{k+1},X_{k+2},...,X_{n}\right),\,(n>k\geq 1)$, ${{\cal G}^{2}}=gen\left(X_{k+2},...,X_{n}\right)$, with the Lie brackets are given by $\left[X_{i},X_{j}\right]=\sum\limits_{l=k+1}^{n}{C_{ij}^{l}{X_{l}}},\,1\leq i<j\leq n,$ where $C_{ij}^{l}\left({1\leq i<j\leq n},k+1\leq l\leq n\right)$ are constructional constants of $\cal{G}$. ###### Theorem 2.1.1. There is no MD-algebra $\cal G$ such that its second derived ideal ${\cal G}^{2}$ is not trivial and less than its first derived ideal ${\cal G}^{1}$ by one dimension: $dim{\cal G}^{2}=dim{\cal G}^{1}-1.$ In order to prove this theorem, we need some lemmas. ###### Lemma 2.1.2. The operator $ad_{X_{k+1}}$ restricted on ${\cal G}^{2}$ is an automorphism. ###### Proof. Since ${\cal G}^{2}$ is commutative, $\left[{{X_{i}},{X_{j}}}\right]=0,\,\,\forall i,j\geq k+2$. Hence, $\begin{array}[]{l}\quad\,gen\left({X_{k+2}},{X_{k+3}},\cdots,{X_{n}}\right)={{\cal G}^{2}}=\left[{{{\cal G}^{1}},{{\cal G}^{1}}}\right]\\\ =gen\left(\left[X_{k+1},X_{k+2}\right],\left[X_{k+1},X_{k+3}\right],\cdots,\left[X_{k+1},X_{n}\right]\right)\\\ =gen\left(ad_{X_{k+1}}\left(X_{k+2}\right),\cdots,ad_{X_{k+1}}\left(X_{n}\right)\right).\end{array}$ It follows that $ad_{X_{k+1}}$ restricted on ${\cal G}^{2}$ is automorphic. ∎ ###### Lemma 2.1.3. Without any restriction of generality, we can always suppose right from the start that $\left[X_{i},X_{k+1}\right]=0$ for all indices $i$ such that $1\leq i\leq k$. ###### Proof. Firstly, we remark that $\left[X_{1},X_{k+1}\right]\in{\cal G}^{1}$, so there exists $X\in{\cal G}^{2}$ such that $\left[X_{1},X_{k+1}\right]=C_{1,k+1}^{k+1}X_{k+1}+X$. Since ${ad_{X_{k+1}}}$ restricted on ${\cal G}^{2}$ is automorphic, there exists $Y\in{\cal G}^{2}$ such that $ad_{X_{k+1}}\left(Y\right)=X$. By changing $X^{\prime}_{1}=X_{1}+Y$, we get $\left[X^{\prime}_{1},X_{k+1}\right]=C_{1,k+1}^{k+1}{X_{k+1}}$. Using the Jacobi identity for $X^{\prime}_{1},X_{k+1}$, and an arbitrary element $Z\in{\cal G}^{2}$ we obtain $ad_{X_{1}}ad_{X_{k+1}}-ad_{X_{k+1}}ad_{X_{1}}=\alpha ad_{X_{k+1}}$, where $\alpha$ is some real constant. Since $ad_{X_{k+1}}$ is automorphic on ${\cal G}^{2}$, $\alpha$ must be zero. Therefore, $C_{1,k+1}^{k+1}=0$, i.e. $\left[X^{\prime}_{1},X_{k+1}\right]=0$. So, we can suppose that $\left[X_{1},X_{k+1}\right]=0$. By the same way, we can suppose $\left[X_{2},X_{k+1}\right]=\cdots=\left[X_{k},X_{k+1}\right]=0.$ ∎ ###### Lemma 2.1.4. $\left[X_{i},X_{j}\right]=C_{ij}^{k+1}X_{k+1}$ for all pairs of indices i, j such that $1\leq i<j\leq k$. ###### Proof. Consider an arbitrary pair of indices $i,j$ such that $1\leq i<j\leq k$. Note that $\left[X_{i},X_{j}\right]=\sum\limits_{l=k+1}^{n}{C_{ij}^{l}{X_{l}}}=C_{ij}^{k+1}X_{k+1}+\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}}$. By using the Jacobi identity, we have $\left[X_{i},\left[X_{j},X_{k+1}\right]\right]+\left[X_{j},\left[X_{k+1},X_{i}\right]\right]+\left[X_{k+1},\left[X_{i},X_{j}\right]\right]=0$ $\Longrightarrow\left[X_{k+1},\left[X_{i},X_{j}\right]\right]=\left[X_{k+1},\,C_{ij}^{k+1}X_{k+1}+\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}}\right]=0$ $\Longrightarrow ad_{X_{k+1}}{\left(\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}}\right)}=0$ $\Longrightarrow\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}}=0$ (because $ad_{X_{k+1}}$ is automorphic on ${\cal{G}}^{2}$) $\Longrightarrow\left[{{X_{i}},{X_{j}}}\right]=C_{ij}^{k+1}{X_{k+1}};1\leq i<j\leq k$. ∎ We now prove Theorem 2.1.1. Namely, we will prove that if $\cal G$ is a real solvable Lie algebra such that ${\cal G}^{2}$ is non - trivial commutative and $dim{\cal G}^{2}=dim{\cal G}^{1}-1=n-k-1,\,1\leq k<n$, then $\cal G$ is not an MD-algebra. ###### Proof of Theorem 2.1.1. According to above lemmas, we can choose a suitable basis $\left(X_{1},X_{2},\cdots,X_{n}\right)$ of $\cal G$ which satisfies the following conditions: $\left[X_{i},X_{j}\right]=C_{ij}^{k+1}X_{k+1},\,\,1\leq i<j\leq k$; $\left[X_{i},X_{k+1}\right]=0,\,\,1\leq i\leq k$; $\left[X_{i},X_{j}\right]=\sum\limits_{l=k+2}^{n}{C_{ij}^{l}{X_{l}}},\,\,1\leq i\leq k+1,\,k+2\leq j\leq n$. Moreover, the constructional constants $C_{ij}^{k+1}$ can not concomitantly vanish and the matrix $A={\left(C_{j,k+1}^{l}\right)}_{k+2\leq j,l\leq n}$ is invertible because $ad_{X_{k+1}}$ restricted on ${\cal{G}}^{2}$ is automorphic. Since A is invertible, there exist $\alpha_{k+2},\cdots,\alpha_{n}\in{\mathbb{R}}$, which are not concomitantly vanished, such that $A\left[\begin{array}[]{l}\alpha_{k+2}\\\ \,\,\,\,\vdots\\\ \alpha_{n}\end{array}\right]=\left[\begin{array}[]{l}1\\\ 0\\\ \vdots\\\ 0\end{array}\right]\in{\mathbb{R}}^{n-k-1}.$ Let $\left(X_{1}^{*},X_{2}^{*},\cdots,X_{n}^{*}\right)$ is the dual basis in ${\cal G}^{*}$ of $\left(X_{1},X_{2},\cdots,X_{n}\right)$. We choose ${F_{1}}=X_{k+1}^{*}$ and ${F_{2}}=X_{k+1}^{*}+\alpha_{k+2}X_{k+2}^{*}+\cdots+{\alpha_{n}}X_{n}^{*}$ in ${\cal G}^{*}$ . It can easily be seen that $F_{1},F_{2}$ are not perfectly vanishing in ${\mathcal{G}}^{1}$. In the view of Proposition 1.2.3, if $\cal G$ is an MD-algebra then $\Omega_{F_{1}},\Omega_{F_{2}}$ are orbits of maximal dimension, in particular we have $rankB_{F_{2}}=dim\Omega_{F_{1}}=dim\Omega_{F_{2}}=rankB_{F_{2}}.$ But it is easy to verify that $rankB_{F_{2}}\geq rankB_{F_{1}}+2$. This contradiction proves that $\cal G$ is not an MD-algebra and the proof of Theorem 2.1.1 is therefore complete. $\square$ Now we consider an arbitrary real solvable Lie algebra $\cal{G}$ of dimension $n$ ($n\geq 5$) such that $dim{\cal G}^{1}=n-1$. It is obvious that we can choose one basis $\left(X_{1},X_{2},\cdots,X_{n}\right)$ of $\cal G$ such that ${\cal G}^{1}=gen\left(X_{2},X_{3},\cdots,X_{n}\right)$, $\,{\cal G}^{2}\subset gen\left(X_{3},\cdots,X_{n}\right)$ and ${\cal G}^{2}$ is commutative. Let $C_{ij}^{l}\left({1\leq i<j\leq n},2\leq l\leq n\right)$ are constructional constants of $\cal{G}$. Then the Lie brackets are given by the following formulas $\left[{{X_{i}},{X_{j}}}\right]=\sum\limits_{l=2}^{n}{C_{ij}^{l}{X_{l}}}\left({1\leq i<j\leq n}\right).$ ###### Theorem 2.1.5. Let ${\cal G}$ be a real solvable Lie algebra of dimension $n$ such that its first derived ideal ${\cal G}^{1}$ is $(n-1)$-dimensional ($n\geq 5$) and its second derived ideal ${\cal G}^{2}$ is commutative. Then ${\cal G}$ is MDn- algebra if and only if $\,{\cal G}^{1}$ is commutative. In order to prove this theorem, once again, we also need some lemmas. ###### Lemma 2.1.6. If $\,\cal G$ is an MD-algebra of dimension $n\,$ ($n\geq 5$) such that $\,\dim{\cal G}^{1}=n-1$ then $\dim{\Omega_{F}}\in\\{0,2\\}\,$ for every $F\in{\cal G}^{*}$. ###### Proof. Let $ad_{X_{1}}=\left(a_{ij}\right)_{n-1}\in End\left({\cal G}^{1}\right)$. With $F_{0}=X_{2}^{*}\in{\cal G}^{*}$, the matrix of the bilinear form $B_{F_{0}}$ in the chosen basis as follows $B_{F_{0}}=\left[\begin{array}[]{l}0\,\,\,\,\,\,\,-{a_{12}}\,\,\,\,\,\,-{a_{13}}\,\,\,....\,\,-{a_{1n}}\\\ {a_{12}}\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,0\\\ {a_{13}}\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,0\\\ ....\,\,\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,....\,\,\,\,\,\,....\\\ {a_{1n}}\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,....\,\,\,\,\,\,\,\,0\end{array}\right].$ It is plain that $rank\,B_{F_{0}}=2$. Since ${\cal G}$ is an MD-algebra, we get $\dim{\Omega_{F}}\in\\{0,2\\}\,$ for every $F\in{\cal G}^{*}$. ∎ ###### Lemma 2.1.7. Suppose that ${\cal G}=gen\left(X_{1},X_{2},\cdots,X_{n}\right)$ is a real solvable Lie algebra of dimension n such that ${\cal G}^{1}=gen\left(X_{2},X_{3},\cdots,X_{n}\right)$ and $\,{\cal G}^{2}=gen\left(X_{k+1},\cdots,X_{n}\right)$, $k>1$. Let $A=\left({\begin{array}[]{*{20}{c}}{C_{12}^{2}}&\ldots&{C_{1k}^{2}}\\\ \vdots&\ddots&\vdots\\\ {C_{12}^{k}}&\cdots&{C_{1k}^{k}}\end{array}}\right)$ be the matrix established by the constructional constants ${C_{1j}^{l}}$ $\left(2\leq j,l\leq k\right)$ of $\cal G$. Then A is invertible. ###### Proof. Since ${\cal G}^{1}=\left[\cal G,\cal G\right]$, there exist real numbers $\alpha_{ij},\,1\leq i<j\leq n$, such that ${X_{2}}=\sum\limits_{1\leq i<j\leq n}{\alpha_{ij}}\left[{{X_{i}},{X_{j}}}\right]$ $=\sum\limits_{j=2}^{k}{{\alpha_{1j}}\left[{{X_{1}},{X_{j}}}\right]}+\sum\limits_{j=k+1}^{n}{{\alpha_{1j}}\left[{{X_{1}},{X_{j}}}\right]}+\sum\limits_{2\leq i<j\leq n}{{\alpha_{{ij}}}\left[{{X_{i}},{X_{j}}}\right]}$ $=\sum\limits_{j=2}^{k}{{\alpha_{1j}}\left[{{X_{1}},{X_{j}}}\right]+LC_{1}\left({{{\cal G}^{2}}}\right)}$ $=\sum\limits_{j=2}^{k}{{\alpha_{1j}}{\left(\sum\limits_{l=2}^{n}{C_{1j}^{l}{X_{l}}}\right)}+LC_{1}\left({\cal G}^{2}\right)}$ $=\sum\limits_{j=2}^{k}{{\alpha_{1j}}{\left(\sum\limits_{l=2}^{k}{C_{1j}^{l}{X_{l}}}+\sum\limits_{l=k+1}^{n}{C_{1j}^{l}{X_{l}}}\right)}+LC_{1}\left({\cal G}^{2}\right)}$ $=\sum\limits_{l=2}^{k}{\sum\limits_{j=2}^{k}{C_{1j}^{l}{\alpha_{1j}}X_{l}}+LC_{2}\left({\cal G}^{2}\right)},$ where $LC_{1}\left({\cal G}^{2}\right),\,LC_{2}\left({\cal G}^{2}\right)$ are linear combinations of some definite vectors from the chosen basis of ${\cal G}^{2}.$ This implies that there exists columnar vector ${Y_{2}}\in{{\mathbb{R}}^{k-1}}$ such that $AY_{2}=\left[\begin{array}[]{l}1\\\ 0\\\ \vdots\\\ 0\end{array}\right]\in{\mathbb{R}}^{k-1}.$ Similarly, there exist columnar vectors $Y_{3},\cdots,{Y_{k}}\in{{\mathbb{R}}^{k-1}}$ such that $AY_{3}=\left[\begin{array}[]{l}0\\\ 1\\\ 0\\\ \vdots\\\ 0\end{array}\right],\,\cdots,\,AY_{k}=\left[\begin{array}[]{l}0\\\ 0\\\ \vdots\\\ 0\\\ 1\end{array}\right]\in{\mathbb{R}}^{k-1}.$ Thus, there is a real matrix $P$ such that $A.P=I$, where $I$ is the identity $(k-1)$-matrix. So $A$ is invertible and Lemma 2.1.7 is proved completely. ∎ ### Proof of Theorem 2.1.5 Firstly, we shall prove that, if ${\cal G}=gen\left(X_{1},X_{2},\cdots,X_{n}\right)\,\left(n\geq 5\right)$ such that ${\cal G}^{1}=gen\left(X_{2},X_{3},\cdots,X_{n}\right)$ is non-commutative, then ${\cal G}$ is not an MD-algebra. Let ${{\cal G}^{2}}=gen\left(X_{k+1},\cdots,X_{n}\right),\,2\leq k<n$. We need consider some cases which contradict each other as follows. * 1. $k=2$. Then, $\dim{\cal G}^{2}=\dim{\cal G}^{1}-1$. According to Theorem 2.1.1, G is not an MD-algebra. * 2. $k=3$. That means that ${\cal G}^{2}=gen\left(X_{4},\cdots,X_{n}\right)$. Assume that $\cal G$ is an MD-algebra. Remember that $\left[X_{1},X_{2}\right]=\sum\limits_{l=2}^{n}{C_{12}^{l}{X_{l}}},\,\left[X_{1},X_{3}\right]=\sum\limits_{l=2}^{n}{C_{13}^{l}{X_{l}}},\,\left[X_{i},X_{j}\right]=\sum\limits_{l=4}^{n}{C_{ij}^{l}{X_{l}}}\scriptsize{\footnotesize\textbullet},$ for all $j>4$ when $i=1$, $j\geq 3$ when $i=2$ and $j>3$ when $i=3$. According to Lemma 2.1.7, the matrix $P=\left[{\begin{array}[]{*{20}{c}}{C_{12}^{2}}&{C_{12}^{3}}\\\ {C_{13}^{2}}&{C_{13}^{3}}\end{array}}\right]$ is invertible. Let $F={\alpha_{1}}X_{1}^{*}+{\alpha_{2}}X_{2}^{*}+\cdots+{\alpha_{n}}X_{n}^{*}$ be an arbitrary element of ${\cal G}^{*}$, where $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\in\mathbb{R}$. The matrix of the bilinear form ${B_{F}}$ is ${B_{F}}=\left[{\begin{array}[]{*{20}{c}}0&{-F\left({\left[{{X_{1}},{X_{2}}}\right]}\right)}&\cdots&{-F\left({\left[{{X_{1}},{X_{n}}}\right]}\right)}\\\ {F\left({\left[{{X_{1}},{X_{2}}}\right]}\right)}&0&\cdots&{-F\left({\left[{{X_{2}},{X_{n}}}\right]}\right)}\\\ \cdots&\cdots&\cdots&\cdots\\\ {F\left({\left[{{X_{1}},{X_{n}}}\right]}\right)}&{F\left({\left[{{X_{2}},{X_{n}}}\right]}\right)}&\cdots&0\end{array}}\right].$ Now we consider the $4$-submatrices of ${B_{F}}$ established by the elements which are on the rows and the columns of the same numbers 1, 2, 3, i ($i>3$). Because $\cal G$ is an MD-algebra, so according to Lemma 2.1.6, we get $rank(B_{F})\in\\{0,2\\}$, this implies that the determinants of these $4$-submatrices are zero for any $F\in{{\cal G}^{*}}$. By direct computations, using the following obvious result of Linear Algebra: the determinant of any skew-symmetric real $4$-matrix $\left(a_{ij}\right)_{4}$ is equal to zero if and only if ${a_{12}}.{a_{34}}-{a_{13}}.{a_{24}}+{a_{14}}.{a_{23}}=0$, we get $C_{2i}^{l}=C_{3i}^{l}=0,l\geq 4$. This implies $\left[X_{2},X_{i}\right]=\left[X_{3},X_{i}\right]=0,i\geq 4$. Note that ${\cal G}^{2}$ is commutative. So we have ${\cal G}^{2}=\left[{\cal G}^{1},{\cal G}^{1}\right]=gen\left(X_{4},\cdots,X_{n}\right)$ $=gen\left(\left[X_{i},X_{j}\right];i,j\geq 2\right)$ $=gen\left(\left[X_{2},X_{3}\right]\right)$. Thus, $n-3=\dim{\cal G}^{2}\leq 1$, i.e. $n\leq 4$. This contradicts the hypothesis $n\geq 5$. That means $\cal G$ is not an MD-algebra. * 3. $k\geq 4$. By an argument analogous to that used above, we also prove that $\cal G$ is not an MD-algebra. Conversely, assume that ${\cal G}$ is a real solvable Lie algebra of dimension $n$ such that its first derived ideal is $(n-1)$-dimensional and commutative, i.e. ${{\cal G}^{1}}\equiv{\mathbb{R}}.{X_{2}}\oplus{\mathbb{R}}.{X_{3}}\oplus...\oplus{\mathbb{R}}.{X_{n}}\equiv{{\mathbb{R}}^{n-1}}$. We need show that $\cal G$ is an MD-algebra. Let $F={\alpha_{1}}X_{1}^{*}+{\alpha_{2}}X_{2}^{*}+\cdots+{\alpha_{n}}X_{n}^{*}\equiv\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\right)\in{\mathbb{R}}^{n}$ be an arbitrary element from ${\cal G}^{*}\equiv{\mathbb{R}}^{n}$, where $\alpha_{1},\alpha_{2},\cdots,\alpha_{n}\in{\mathbb{R}}$. By simple computation, we can see that the matrix of the bilinear form ${B_{F}}$ is ${B_{F}}=\left[{\begin{array}[]{*{20}{c}}0&{-F\left({\left[{{X_{1}},{X_{2}}}\right]}\right)}&\cdots&{-F\left({\left[{{X_{1}},{X_{n}}}\right]}\right)}\\\ {F\left({\left[{{X_{1}},{X_{2}}}\right]}\right)}&0&\cdots&0\\\ \cdots&\cdots&\cdots&\cdots\\\ {F\left({\left[{{X_{1}},{X_{n}}}\right]}\right)}&0&\cdots&0\end{array}}\right].$ It is clear that $rankB_{F}\in\left\\{{0,2}\right\\}$. Hence, ${\cal G}$ is an MDn-algebra and Theorem 2.1.5 is proved completely. $\square$ ### 2.2 Classification of MD5-algebras having non-commutative derived ideals The following theorem is the main result of the paper. It gives the classification up to an isomorphism of MD5-algebras having non-commutative derived ideals. ###### Theorem 2.2.1. Let ${\cal G}$ be an MD5-algebra such that the first derived ideal ${\cal G}^{1}=\left[{\cal G},{\cal G}\right]$ is non-commutative. Then the following assertions hold. * (i) If ${\cal G}$ is decomposable, then ${\cal G}\cong{\cal H}\oplus{\cal K}$, where ${\cal H}$ and ${\cal K}$ are MD-algebras of dimensions which are no more than 4. * (ii) If ${\cal G}$ is indecomposable, then we can choose a suitable basis $(X_{1},X_{2},X_{3},$ $X_{4},X_{5})$ of ${\cal G}$ such that ${\cal G}^{1}=gen\left(X_{3},X_{4},X_{5}\right),\,\left[X_{3},X_{4}\right]=X_{5}$; operators $ad_{X_{1}},ad_{X_{2}}$ act on ${\cal G}^{1}$ as the following endomorphisms $ad_{X_{1}}=\left({\begin{array}[]{*{20}{c}}1&0&0\\\ 0&1&0\\\ 0&0&2\end{array}}\right),\,\,ad_{X_{2}}=\left({\begin{array}[]{*{20}{c}}0&-1&0\\\ 1&0&0\\\ 0&0&0\end{array}}\right)$ and the other Lie brackets are trivial. We need to prove some lemmas before we prove Theorem 2.2.1. ###### Lemma 2.2.2. Let ${\cal G}$ be a real solvable Lie algebra. For any $Z\in{\cal G}$ we consider $ad_{Z}$ as an operator acted on ${\cal G}^{1}$. Then we have $Trace\left(ad_{Z}\right)=0$ for all $Z\in{\cal G}^{1}$. ###### Proof. Using the Jacobi identity for $X,Y\in{\cal G}$ and an arbitrary element $Z\in{{\cal G}^{1}}$, we have $\left[{X,\left[{Y,Z}\right]}\right]+\left[{Y,\left[{Z,X}\right]}\right]+\left[{Z,\left[{X,Y}\right]}\right]=0$. So, $a{d_{X}}\circ a{d_{Y}}-a{d_{Y}}\circ a{d_{X}}=a{d_{\left[{X,Y}\right]}}$. This implies $Trace\left({a{d_{\left[{X,Y}\right]}}}\right)=0$. Note that ${{\cal G}^{1}}=\left[{{\cal G},{\cal G}}\right]$ and $a{d_{Z}}$ is a linear map. So we get $Trace\left({a{d_{Z}}}\right)=0$ for all $Z\in{{\cal G}^{1}}$. ∎ ###### Lemma 2.2.3. If ${\cal G}$ is a real solvable Lie algebra with $\dim{\cal G}^{1}=2$ then ${{\cal G}^{1}}$ is commutative. ###### Proof. We choose a basis $(X,Y)$ of ${{\cal G}^{1}}$. Assume that $\left[X,Y\right]=aX+bY$. So we have $ad_{X}=\left({\begin{array}[]{*{20}{c}}0&a\\\ 0&b\end{array}}\right),ad_{Y}=\left({\begin{array}[]{*{20}{c}}{-a}&0\\\ {-b}&0\end{array}}\right)\in End({\cal G}^{1})$. According to Lemma 2.2.2, we get $a=b=0$. Hence, ${{\cal G}^{1}}$ is commutative. ∎ Now we are ready to prove Theorem 2.2.1 - The main result of the paper. ###### Proof of Theorem 2.2.1. It is clear that assertion (i) of Theorem 2.2.1 holds obviously. We only need to prove assertion (ii). Let ${\cal G}$ be an indecomposable MD5-algebra with the first derived ideal ${\cal G}^{1}=\left[{\cal G},{\cal G}\right]$ is non - commutative and the second derived ideal ${\cal G}^{2}=\left[{\cal G}^{1},{\cal G}^{1}\right]$ is commutative. According to Theorems 2.1.1, 2.1.5 and Lemma 2.2.3, the dimensions of ${\cal G}^{1}$ and ${\cal G}^{2}=\left[{\cal G}^{1},{\cal G}^{1}\right]$ must be 3 and 1, respectively. We choose a basis $\left(X_{1},X_{2},X_{3},X_{4},X_{5}\right)$ such that ${\cal G}^{1}=gen\left(X_{3},X_{4},X_{5}\right)$ and ${\cal G}^{2}=gen\left(X_{5}\right)$ with the Lie brackets are given by $\begin{array}[]{l}\left[X_{1},X_{2}\right]={a_{3}}{X_{3}}+{a_{4}}{X_{4}}+{a_{5}}{X_{5}},\\\ \left[X_{1},X_{3}\right]={b_{3}}{X_{3}}+{b_{4}}{X_{4}}+{b_{5}}{X_{5}},\\\ \left[X_{1},X_{4}\right]={c_{3}}{X_{3}}+{c_{4}}{X_{4}}+{c_{5}}{X_{5}},\\\ \left[X_{1},X_{5}\right]={d_{3}}{X_{3}}+{d_{4}}{X_{4}}+{d_{5}}{X_{5}},\\\ \left[X_{2},X_{3}\right]={e_{3}}{X_{3}}+{e_{4}}{X_{4}}+{e_{5}}{X_{5}},\\\ \left[X_{2},X_{4}\right]={f_{3}}{X_{3}}+{f_{4}}{X_{4}}+{f_{5}}{X_{5}},\\\ \left[X_{2},X_{5}\right]={k_{3}}{X_{3}}+{k_{4}}{X_{4}}+{k_{5}}{X_{5}},\\\ \left[X_{3},X_{4}\right]={g_{5}}{X_{5}},\left[X_{3},X_{5}\right]={h_{5}}{X_{5}},\left[X_{4},X_{5}\right]={l_{5}}{X_{5}},\end{array}$ where $a_{i},b_{i},c_{i},d_{i},e_{i},f_{i},k_{i}\,(i=3,4,5)$ and $g_{5},h_{5},l_{5}$ are the definite real numbers. Now we give some useful remarks as follows. * a. According to Lemma 2.2.2, $Trace\left(ad_{X_{3}}\right)=Trace\left(ad_{X_{4}}\right)=0$. That means ${h_{5}}={l_{5}}=0$. * b. $g_{5}\neq 0$ because ${\cal G}^{2}=gen(X_{5})$. By changing ${X_{3}}$ with ${X^{\prime}_{3}}={\frac{1}{g_{5}}}X_{3}$, we get $\left[X^{\prime}_{3},X_{4}\right]=X_{5}$. So, we can suppose right from the start that $\left[X_{3},X_{4}\right]=X_{5}$, i.e. $g_{5}=1$. * c. ${d_{3}}={d_{4}}={k_{3}}={k_{4}}=0$ because ${\cal G}^{2}=\mathbb{R}.X_{5}$ is an ideal of ${\cal G}$. So, we get $\left[X_{1},X_{5}\right]={d_{5}}{X_{5}},\,\left[X_{2},X_{5}\right]={k_{5}}{X_{5}}.$ If $k_{5}\neq 0$, by changing $X^{\prime}_{2}=X_{1}-{\frac{d_{5}}{k_{5}}}X_{2}$ we get $\left[X^{\prime}_{2},X_{5}\right]=0$. So, we can always assume that $k_{5}=0$. * d. By changing ${X_{1}}$ with ${X^{\prime}_{1}}={X_{1}}-{c_{5}}{X_{3}}+{b_{5}}{X_{4}}$ and ${X_{2}}$ with ${X^{\prime}_{2}}={X_{2}}-{f_{5}}{X_{3}}+{e_{5}}{X_{4}}$, we get $\left[X^{\prime}_{1},X_{3}\right]={b_{3}}{X_{3}}+{b_{4}}{X_{4}}$, $\left[X^{\prime}_{1},X_{4}\right]={c_{3}}{X_{3}}+{c_{4}}{X_{4}}$, $\left[X^{\prime}_{2},X_{3}\right]={e_{3}}{X_{3}}+{e_{4}}{X_{4}}$, $\left[X^{\prime}_{2},X_{4}\right]={f_{3}}{X_{3}}+{f_{4}}{X_{4}}$. Thus, we can suppose right from the start that ${b_{5}}={c_{5}}={e_{5}}={f_{5}}=0$. Using the Jacobi identity for triads $X_{1},X_{2},X_{i}\,(i=3,4,5)$, we obtain $(I)\left\\{\begin{array}[]{l}{a_{3}}={a_{4}}=0,\\\ {e_{4}}{c_{3}}={b_{4}}{f_{3}},\\\ {e_{3}}{b_{4}}+{e_{4}}{c_{4}}={b_{3}}{e_{4}}+{b_{4}}{f_{4}},\\\ {f_{3}}{b_{3}}+{f_{4}}{c_{3}}={c_{3}}{e_{3}}+{c_{4}}{f_{3}},\\\ {b_{3}}+{c_{4}}={d_{5}},\\\ {e_{3}}+{f_{4}}=0.\end{array}\right.$ So we can reduce the Lie brackets as follows $\begin{array}[]{l}\left[{{X_{1}};{X_{2}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{a_{5}}{X_{5}},\\\ \left[{{X_{1}};{X_{3}}}\right]={b_{3}}{X_{3}}+{b_{4}}{X_{4}},\\\ \left[{{X_{1}};{X_{4}}}\right]={c_{3}}{X_{3}}+{c_{4}}{X_{4}},\\\ \left[{{X_{1}};{X_{5}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left({{b_{3}}+{c_{4}}}\right){X_{5}},\\\ \left[{{X_{2}};{X_{3}}}\right]={e_{3}}{X_{3}}+\,{e_{4}}{X_{4}},\\\ \left[{{X_{2}};{X_{4}}}\right]={f_{3}}{X_{3}}-{e_{3}}{X_{4}},\\\ \left[{{X_{3}};{X_{4}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{X_{5}}.\end{array}$ Thus, Relations $(I)$ can be rewritten as follows $(II)\left\\{\begin{array}[]{l}{e_{4}}{c_{3}}={b_{4}}{f_{3}},\\\ 2{e_{3}}{b_{4}}={e_{4}}\left({{b_{3}}-{c_{4}}}\right),\\\ 2{c_{3}}{e_{3}}={f_{3}}\left({{b_{3}}-{c_{4}}}\right).\end{array}\right.$ Now we need consider the following cases which contradict each other. * Case 1: ${e_{3}}={e_{4}}=0.$ $(II)\Leftrightarrow\left\\{\begin{array}[]{l}{b_{4}}{f_{3}}=0,\\\ {f_{3}}\left({{b_{3}}-{c_{4}}}\right)=0.\end{array}\right.$ * 1.1. Assume that ${f_{3}}=0$. Then, Relations $(II)$ is automatically satisfied. * By choosing $F_{1}=X_{3}^{*}\in{\cal G}^{*}$, we get $rank{B_{{F_{1}}}}=2$. * Now we choose $F_{2}=X_{5}^{*}\in{\cal G}^{*}$. By simple computation, we obtain ${B_{{F_{2}}}}=\left[{\begin{array}[]{*{20}{c}}0&{-{a_{5}}}&0&0\\\ {{a_{5}}}&0&0&0\\\ 0&0&0&{-1}\\\ {\begin{array}[]{*{20}{c}}0\\\ {{b_{3}}+{c_{4}}}\end{array}}&{\begin{array}[]{*{20}{c}}0\\\ 0\end{array}}&{\begin{array}[]{*{20}{c}}1\\\ 0\end{array}}&{\begin{array}[]{*{20}{c}}0\\\ 0\end{array}}\end{array}\,\,\,\,\,\,\,\begin{array}[]{*{20}{c}}{-\left({{b_{3}}+{c_{4}}}\right)}\\\ 0\\\ 0\\\ {\begin{array}[]{*{20}{c}}0\\\ 0\end{array}}\end{array}}\right]$ Since ${\cal G}$ is an MD-algebra, this implies that $rank{B_{{F_{2}}}}=rank{B_{{F_{1}}}}=2$. That fact implies that ${a_{5}}={b_{3}}+{c_{4}}=0$, so ${\cal G}$ is decomposable, which is a contradiction. Thus, this case cannot happen. * 1.2. Now we assume that ${f_{3}}\neq 0$. Then $b_{4}=0,b_{3}=c_{4}$. By changing $X_{1}$ and $X_{2}$ with ${X^{\prime}_{1}}={X_{1}}-{c_{3}}{X^{\prime}_{2}}$ and $X^{\prime}_{2}={\frac{1}{f_{3}}X_{2}}$ we can suppose right from the start that $f_{3}=1,c_{3}=0$. Because of the dimension of ${\cal G}^{1}$ is 3, $b_{3}\neq 0$. By changing $X_{1}$ with $X^{\prime}_{1}={\frac{1}{b_{3}}}{X_{1}}$, we can always assume that $b_{3}=1$. By changing $X_{2}$ with $X^{\prime}_{2}=X_{2}-\frac{a_{5}}{2}{X_{5}}$, we can assume that $a_{5}=0$. Now, we choose $F_{3}=X_{3}^{*}$ and $F_{4}=X_{4}^{*}$ from ${\cal G}^{*}$, we get $rank{B_{{F_{3}}}}=2,\,rank{B_{{F_{4}}}}=4$. This cannot happen because ${\cal G}$ is an MD-algebra. * Case 2: ${e_{4}}=0,{e_{3}}\neq 0$ $\left({II}\right)\Leftrightarrow\left\\{\begin{array}[]{l}{b_{4}}{f_{3}}=0,\\\ {b_{4}}{e_{3}}=0,\\\ 2{c_{3}}{e_{3}}={f_{3}}\left({{b_{3}}-{c_{4}}}\right).\end{array}\right.\Leftrightarrow\left\\{\begin{array}[]{l}{b_{4}}=0,\\\ 2{c_{3}}{e_{3}}={f_{3}}\left({{b_{3}}-{c_{4}}}\right).\end{array}\right.$ By changing $X_{2}$ with $X^{\prime}_{2}={\frac{1}{e_{3}}}{X_{2}}$, we can assume that $e_{3}=1$. Now Relations $(II)$ can be rewritten as follows $(III)\left\\{\begin{array}[]{l}{b_{4}}=0,\\\ 2{c_{3}}={f_{3}}\left({{b_{3}}-{c_{4}}}\right).\end{array}\right.$ By changing $X_{4}$ with $X^{\prime}_{4}=X_{4}-\frac{f_{3}}{2}{X_{3}}$ and $X_{1}$ with $X^{\prime}_{1}=X_{1}-{b_{3}}{X_{2}}$, we can suppose that $b_{3}=f_{3}=0$. From Relations (III) we get $b_{4}=c_{3}=0$. Let $F=\alpha\,X_{1}^{*}+\beta X_{2}^{*}+\gamma X_{3}^{*}+\delta X_{4}^{*}+\sigma X_{5}^{*}\equiv\left({\alpha,\beta,\gamma,\delta,\sigma}\right)$ be an arbitrary element from ${{\cal G}^{*}}\equiv{\mathbb{R}^{5}}$; $\alpha,\beta,\gamma,\delta,\sigma\in\mathbb{R}$. By simple computation, we obtain the matrix of the bilinear form ${B_{F}}$ as follows ${B_{F}}=\left[{\begin{array}[]{*{20}{c}}0&{-{a_{5}}\sigma}&0&{-{c_{4}}\delta}\\\ {{a_{5}}\sigma}&0&{-\gamma}&\delta\\\ 0&\gamma&0&{-\sigma}\\\ {\begin{array}[]{*{20}{c}}{{c_{4}}\delta}\\\ {{c_{4}}\sigma}\end{array}}&{\begin{array}[]{*{20}{c}}\delta\\\ 0\end{array}}&{\begin{array}[]{*{20}{c}}\sigma\\\ 0\end{array}}&{\begin{array}[]{*{20}{c}}0\\\ 0\end{array}}\end{array}\,\,\,\,\,\,\,\begin{array}[]{*{20}{c}}{-{c_{4}}\sigma}\\\ 0\\\ 0\\\ {\begin{array}[]{*{20}{c}}0\\\ 0\end{array}}\end{array}}\right]$ Since ${\cal G}$ is an MD-algebra, $rank{B_{F}}$ only get two values zero or two. Hence, it is easy to prove that $c_{4}=a_{5}=0$. But this implies that ${\cal G}$ is decomposable. This is a contradiction. Thus, this case cannot happen. * Case 3: ${e_{4}}\neq 0$. By changing $X_{4}$ with $X^{\prime}_{4}={e_{4}}{X_{4}}+{e_{3}}{X_{3}}$, we can assume that $e_{3}=0,e_{4}=1$. Now, Relations $(II$) can be rewritten as follows $\left\\{\begin{array}[]{l}{c_{3}}={b_{4}}{f_{3}},\\\ {b_{3}}={c_{4}}.\end{array}\right.$ By changing $X_{1}$ with $X^{\prime}_{1}=X_{1}-{b_{4}}{X_{2}}$, we can assume that $b_{4}=0$. Putting this into the relation above, we get $c_{3}=0$. Then, the Lie brackets in $\cal G$ can be reduced as follows $\begin{array}[]{l}\left[X_{1},X_{2}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\lambda{X_{5}},\\\ \left[X_{1},X_{3}\right]=\mu{X_{3}},\\\ \left[X_{1},X_{4}\right]=\,\,\,\,\,\,\,\,\,\mu{X_{4}},\\\ \left[X_{1},X_{5}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\mu{X_{5}},\\\ \left[X_{2},X_{3}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,{X_{4}},\\\ \left[X_{2},X_{4}\right]=\,\theta{X_{3}},\\\ \left[X_{3},X_{4}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{X_{5}}.\end{array}$ Let $F=\alpha\,X_{1}^{*}+\beta X_{2}^{*}+\gamma X_{3}^{*}+\delta X_{4}^{*}+\sigma X_{5}^{*}\in{\cal G}^{*}$ be an arbitrary element from ${\cal G}^{*}\equiv{\mathbb{R}^{5}}$. Then by simple computation, we see that ${B_{F}}=\left[\begin{array}[]{l}0\,\,\,\,\,-\lambda\sigma\,\,\,\,-\mu\gamma\,\,\,\,-\mu\delta\,\,\,\,-2\mu\sigma\\\ \lambda\sigma\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,-\delta\,\,\,\,\,\,-\theta\gamma\,\,\,\,\,\,\,\,\,\,\,\,0\\\ \mu\gamma\,\,\,\,\,\,\,\,\delta\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,-\sigma\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\\ \mu\delta\,\,\,\,\,\,\,\,\theta\gamma\,\,\,\,\,\,\,\,\,\,\,\,\sigma\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\\ 2\mu\sigma\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array}\right].$ * 3.1. Assume $\mu\neq 0$. * When $\sigma\neq 0$, we have $rank{B_{F}}=4$. Because ${\cal G}$ is an MD5-algebra, we get $rank{B_{F}}\in\\{0,4\\}$ for all $\alpha,\beta,\delta,\gamma,\sigma\in\mathbb{R}$. But this only can happen if $\theta<0$. By changing $X_{1},X_{2},X_{4},X_{5}$ with $X^{\prime}_{1}=\mu{X_{1}},X^{\prime}_{2}=\sqrt{-\theta}{X_{2}},X^{\prime}_{4}=\sqrt{-\theta}{X_{4}},X^{\prime}_{5}=\sqrt{-\theta}{X_{5}}$, we can assume that $\theta=-1$. By changing $X_{2}$ with $X^{\prime}_{2}=X_{2}-\frac{\lambda}{2}{X_{5}}$, we can assume that $\lambda=0$. Hence, the Lie brackets of $\cal G$ can be reduced as follows $\begin{array}[]{l}\left[{{X_{1}};{X_{3}}}\right]={X_{3}},\\\ \left[{{X_{1}};{X_{4}}}\right]=\,\,\,\,\,\,\,\,\,{X_{4}},\\\ \left[{{X_{1}};{X_{5}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2{X_{5}},\\\ \left[{{X_{2}};{X_{3}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,{X_{4}},\\\ \left[{{X_{2}};{X_{4}}}\right]=\,-{X_{3}},\\\ \left[{{X_{3}};{X_{4}}}\right]=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{X_{5}}.\end{array}$ * 3.2 Assume $\mu=0$. By the same way, we consider the matrix of the bilinear form ${B_{F}}$ and obtain $\lambda=0$. But this shows that ${\cal G}$ is decomposable. This construction show that this case can not happen. The theorem 2.2.1 is proved completely. So there is only one MD5-algebra having non-commutative derived ideal.$\hfill\square$ ### CONCLUDING REMARK Let us recall that each real Lie algebra $\cal G$ defines only one connected and simply connected Lie group G such that Lie(G) = $\cal G$. Therefore we obtain only one connected and simply connected MD5-group corresponding to the MD5-algebra given in Theorem 2.2.1. In the next paper, we shall describe the geometry of K-orbits of considered MD5-group, describe topological properties of MD5-foliation formed by the generic K-orbits of this MD5-group and give the characterization of the Connes’s $C^{*}$-algebra associated to this MD5-foliation. ## References * [1] Do Ngoc Diep, Method of Noncommutative Geometry for Group $C^{*}$-algebras, Chapman and Hall / CRC Press Reseach Notes in Mathematics Series, # 416, 1999. * [2] A.A. Kirillov, Elements of the Theory of Representations, Springer-Verlag, Berlin – Heidenberg – New York, 1976. * [3] Vuong Manh Son et Ho Huu Viet, Sur La Structure Des $C^{*}-alg{\grave{e}}bres$ D’une Classe De Groupes De Lie, J. Operator Theory, 11 (1984), 77-90. * [4] Kristopher Tapp, Matrix Groups for Undergraduates, AMS, 2005. * [5] Le Anh Vu, On the foliations formed by the generic K-orbits of the MD4-groups, Acta Math. Vietnam, $N^{0}$ 2 (1990), 39 - 45. * [6] Le Anh Vu, On a Subclass of 5-dimensional Lie Algebras Which have 3-dimensional Commutative Derived Ideals, East-West J. of Mathematics, Vol.7, $N^{0}$ 1 (2005), 13 - 22. * [7] Le Anh Vu, Classification of 5-dimensional MD-algebras Which have 4-dimensional Commutative Derived Ideals, Scientific Journal of University of Pedagogy of Ho Chi Minh City, $N^{0}$ 12 (46) (2007), 3 - 15 (In Vietnamese). * [8] Le Anh Vu and Duong Quang Hoa, The Geometricaly Picture of K-orbits of Connected and Simply Connected MD5-groups such that their MD5-algebras have 4-dimensional Commutative Derived Ideals, Scientific Journal of University of Pedagogy of Ho Chi Minh City, $N^{0}$ 12 (46) (2007), 16 - 28 (In Vietnamese). * [9] Le Anh Vu and Kar Ping Shum, On a Subclass of 5-dimentional Solvable Lie Algebras Which Have Commutative Derived Ideal, Advances in Algebra and Combinatorics, World Scientific Publishing Co. (2008), pp 353 - 371.
arxiv-papers
2011-01-11T19:11:39
2024-09-04T02:49:16.331659
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Le Anh Vu, Ha Van Hieu and Tran Thi Hieu Nghia", "submitter": "Vu Le Anh", "url": "https://arxiv.org/abs/1101.2181" }
1101.2314
††thanks: Corresponding author # Decoherence in Attosecond Photoionization Stefan Pabst Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany Department of Physics,University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany Loren Greenman Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA Phay J. Ho Argonne National Laboratory, Argonne, Illinois 60439, USA David A. Mazziotti Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA Robin Santra robin.santra@cfel.de Center for Free-Electron Laser Science, DESY, Notkestrasse 85, 22607 Hamburg, Germany Department of Physics,University of Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany ###### Abstract The creation of superpositions of hole states via single-photon ionization using attosecond extreme-ultraviolet pulses is studied with the time-dependent configuration interaction singles (TDCIS) method. Specifically, the degree of coherence between hole states in atomic xenon is investigated. We find that interchannel coupling not only affects the hole populations, it also enhances the entanglement between the photoelectron and the remaining ion, thereby reducing the coherence within the ion. As a consequence, even if the spectral bandwidth of the ionizing pulse exceeds the energy splittings among the hole states involved, perfectly coherent hole wave packets cannot be formed. For sufficiently large spectral bandwidth, the coherence can only be increased by increasing the mean photon energy. ###### pacs: 32.80.Aa, 42.65.Re, 03.65.Yz The typical time scale of electronic motion in atoms, molecules, and condensed matter systems ranges from a few attoseconds ($1~{}\text{as}=10^{-18}$ s) to tens of femtoseconds ($1~{}\text{fs}=10^{-15}$ s) Krausz and Ivanov (2009); Zewail (2000); Cavalieri et al. (2007). In the last decade the remarkable progress in high harmonic generation Doumy et al. (2009); Dudovich et al. (2006); López-Martens et al. (2005); Gibson et al. (2004); Schafer et al. (1993) made it possible to generate attosecond pulses as short as 80 as Goulielmakis et al. (2008). Attosecond pulses have opened the door to real- time observations of the most fundamental processes on the atomic scale Krausz and Ivanov (2009); Pfeifer et al. (2008). For instance, the generation of attosecond pulses was utilized to determine spatial structures of molecular orbitals Haessler et al. (2010); an interferometric technique using attosecond pulses was used to characterize attosecond electron wave packets Mauritsson et al. (2010); and attosecond pulse trains Singh et al. (2010) and isolated attosecond pulses Sansone et al. (2010), in combination with an intense few- cycle infrared pulse, enabled the control of electron localization in molecules. Attosecond technology demonstrated the ability to follow, on a subfemtosecond time scale, processes such as photoionization Skantzakis et al. (2010), Auger decay Drescher et al. (2002), and valence electron motion driven by relativistic spin-orbit coupling Goulielmakis et al. (2010). Furthermore, the availability of attosecond pulses fuelled a broad interest in exploring charge transfer dynamics following photoexcitation or photoionization Sansone et al. (2010). In this Letter, we analyze the creation of hole states via single-photon ionization using a single extreme-ultraviolet attosecond pulse. We investigate the impact of the freed photoelectron on the remaining ion and demonstrate that the interaction between the photoelectron and the ion cannot be neglected for currently available state-of-the-art attosecond pulses. In particular, the interchannel coupling of the initially coherently excited hole states greatly enhances the entanglement between the photoelectron and the ionic states. Interchannel coupling is mediated by the photoelectron and mixes different ionization channels, i.e., hole configurations, with each other. Consequently, the degree of coherence among the ionic states is strongly reduced, making it impossible to describe the subsequent charge transfer in the ion with a pure quantum mechanical state. Experiments on photosynthetic systems Lee et al. (2007); Sarovar et al. (2010); Collini et al. (2010); Harel et al. (2010) have revealed a correlation between highly efficient energy transport and coherent dynamics in molecules (nuclear and electronic dynamics in this case). Similarly, high degrees of coherence in nonstationary hole states may be necessary for efficient charge transport within molecules. In the last decade, much work has been done in the realm of hole migration Breidbach and Cederbaum (2003); Nest et al. (2008); Kuleff et al. (2010). It was shown that electronic motion can be triggered solely by electron correlation Breidbach and Cederbaum (2003). Charge transfers mediated by electronic correlations typically take place in a few femtoseconds and are thus faster than electronic dynamics initiated by nuclear motion Kröner et al. (2007); Muskatel et al. (2009). Recent experiments Weinkauf et al. (1997); Schlag et al. (2007) have demonstrated that electronically excited ionic states can modify site-selective reactivity within tens of femtoseconds, making hole migration processes a promising tool to control chemical reactions. Up to now, theoretical calculations Breidbach and Cederbaum (2003); Kuleff et al. (2010) investigating hole migration phenomena have neglected the interaction between the parent ion and the photoelectron and assumed a perfectly coherent hole wave packet. As long as the photoelectron departs sufficiently rapidly from the parent ion, this assumption is appropriate Cederbaum et al. (1986). However, for attosecond pulses with large spectral bandwidths, the enhanced production of slow photoelectrons will affect (mainly via interchannel coupling) both the final hole populations and the coherence among these hole states. Furthermore, recent results in high harmonic spectroscopy suggest that interchannel coupling may be the missing link to understand hole dynamics occurring in high harmonic generation processes before the ejected electron recombines with the parent ion Mairesse et al. (2010). We investigate the creation of hole states via attosecond photoionization using the implementation of the time-dependent configuration-interaction singles (TDCIS) approach described in Ref. Greenman et al. (2010) (see also Schlegel et al. (2007); Klinkusch et al. (2009)). TDCIS allows us to study ionization dynamics beyond the single-channel approximation and to understand systematically the relevance of interchannel coupling in the hole creation process. The TDCIS wave function for the entire system is $\displaystyle\left|\Psi(t)\right>=\alpha_{0}(t)\,\left|\Phi_{0}\right>+\sum_{a,i}\alpha^{a}_{i}(t)\,\left|\Phi^{a}_{i}\right>,$ (1) where $\left|\Phi_{0}\right>$ is the Hartree-Fock ground state and $\left|\Phi^{a}_{i}\right>=\hat{c}^{\dagger}_{a}\hat{c}_{i}\left|\Phi_{0}\right>$ is a one-particle–one-hole excitation ($\hat{c}^{\dagger}_{a}$ and $\hat{c}_{i}$ are creation and annihilation operators for an electron in orbitals $a$ and $i$, respectively). The corresponding coefficients $\alpha_{0}(t)$ and $\alpha^{a}_{i}(t)$, respectively, are functions of time and describe the dynamics of the system. Throughout, indices $i,j,$ are used for occupied orbitals in $\left|\Phi_{0}\right>$; indices $a,b,$ stand for unoccupied orbitals. We focus our discussion on the case where single-photon ionization is the dominant effect and higher order processes can be neglected. Our model system is atomic xenon. The corresponding Hamiltonian (neglecting spin-orbit coupling) is $\displaystyle\hat{H}(t)$ $\displaystyle=$ $\displaystyle\hat{H}_{0}+\hat{H}_{1}+E(t)\,\hat{z},$ (2a) where $E(t)$ is the electric field, $\hat{z}$ the dipole operator, and $\hat{H}_{0}$ is the mean- field Fock operator, which is diagonal with respect to the basis used in Eq. (1). The residual Coulomb interaction, $\displaystyle\hat{H}_{1}$ $\displaystyle=$ $\displaystyle\hat{V}_{c}-\hat{V}_{\text{MF}},$ (2b) is defined such that $\hat{H}_{0}+\hat{H}_{1}$ gives the exact nonrelativistic Hamiltonian for the electronic system in the absence of external fields ($\hat{V}_{c}$ is the electron–electron interaction). We study the impact of different approximations for $\hat{H}_{1}$ on the hole state as follows. The Coulomb-free model, the simplest approximation, removes the residual Coulomb interaction ($\hat{H}_{1}=0$) between the excited electron and the parent ion. In this approximation, the excited electron always sees a neutral atom via the $\hat{V}_{\text{MF}}$ potential Szabo and Ostlund (1996). A more realistic approximation is the intrachannel model including direct and exchange contributions of the Coulomb interaction only within a given channel. In this second model, the excited electron can only interact with the occupied orbital from which it originates. Interactions between different occupied orbitals are neglected, i.e. we set $\left<\Phi^{a}_{i}\right|\hat{H}_{1}\left|\Phi^{b}_{j}\right>=0$ for $i\neq j$. The third and final model describes the Coulomb interaction exactly within the TDCIS framework. We refer to this as the full model. Note that the exact nonrelativistic Hamiltonian $\hat{H}_{0}+\hat{H}_{1}$ is diagonal with respect to the ionic one-hole states $\left|\Phi_{i}\right>=\hat{c}_{i}\left|\Phi_{0}\right>$. In the full model, the photoelectron can couple the hole states, as $\hat{H}_{1}$ in the particle-hole space is not diagonal with respect to the hole index (i.e., $\left<\Phi^{a}_{i}\right|\hat{H}_{1}\left|\Phi^{b}_{j}\right>$ generally differs from zero). This type of photoelectron-mediated interaction is called interchannel coupling Starace (1980). As a consequence, in the full model the hole index is not a good quantum number, whereas in the Coulomb-free and intrachannel models, excited eigenstates of $\hat{H}_{0}+\hat{H}_{1}$ are characterized by a well-defined hole index. To describe the hole states of the remaining ion, we employ the ion density matrix Greenman et al. (2010) $\displaystyle\hat{\rho}^{\text{IDM}}_{i,j}(t)$ $\displaystyle=$ $\displaystyle\text{Tr}_{a}[\left|\Psi(t)\right>\left<\Psi(t)\right|]_{i,j}=\sum_{a}\left<\Phi^{a}_{i}|\Psi(t)\right>\left<\Psi(t)|\Phi^{a}_{j}\right>,$ (3) where $\text{Tr}_{a}$ stands for the trace over the photoelectron. The properties of the ion density matrix can be measured using attosecond transient absorption spectroscopy Goulielmakis et al. (2010). A description of the cationic eigenstates in terms of one-hole configurations is a physically meaningful approximation for noble-gas atoms such as xenon Buth et al. (2003). Figure 1: (color online) The $4d_{0}$ [panel (a)] and $5s$ [panel (b)] hole populations of xenon as a function of time are shown for three different residual Coulomb interaction approximations: (1) the full model (red solid line), (2) the intrachannel model (green dotted line), and (3) the Coulomb- free model (blue dash-dotted line). The attosecond pulse has a peak field strength of 25 GV/m, a pulse duration of 20 as, a (mean) photon energy of 136 eV, and is centered at $t=0$ as. In Fig. 1 the hole populations $\rho^{\text{IDM}}_{5s,5s}(t)$ and $\rho^{\text{IDM}}_{4d_{0},4d_{0}}(t)$ of the xenon $5s$ and $4d_{0}$ orbitals, respectively, are shown for all three interaction models ($4d_{0}$ stands for the $4d$ orbital with $m=0$). The ionizing, gaussian-shaped attosecond pulse is linearly polarized and has a peak field strength of 25 GV/m, a pulse duration of $\tau=20$ as, and a (mean) photon energy of $\omega_{0}=136$ eV. The hole dynamics of the Coulomb-free and intrachannel models are alike. In both cases, the population is constant after the pulse, since the hole index is a good quantum number within these models. The extension to the exact Coulomb interaction changes the situation. Interchannel coupling causes the hole populations to remain nonstationary as long as the photoelectron remains close to the ion. As the distance between the photoelectron and the ion increases, the interchannel coupling weakens and the populations $\rho^{\text{IDM}}_{i,i}(t)$ become stationary (see Fig. 1). We confine our discussion to the first hundreds of attoseconds after the pulse, allowing us to neglect decay processes, which start to take place after a few femtoseconds. As we will see in the following, interchannel coupling not only affects the hole populations but also the coherence between the created hole states. The degree of coherence between $\left|\Phi_{i}\right>$ and $\left|\Phi_{j}\right>$ is given by $\displaystyle g_{i,j}(t)=\frac{|\rho^{\text{IDM}}_{i,j}(t)|}{\sqrt{\rho^{\text{IDM}}_{i,i}(t)\rho^{\text{IDM}}_{j,j}(t)}}.$ (4) Totally incoherent statistical mixtures result in $g_{i,j}(t)=0$. The fact that the density matrix is positive semidefinite implies the Cauchy-Schwarz relations $|\rho^{\text{IDM}}_{i,j}(t)|^{2}\leq\rho^{\text{IDM}}_{i,i}(t)\rho^{\text{IDM}}_{j,j}(t)$, which bound the maximum achievable (perfect) coherence ($g_{i,j}(t)=1$). To investigate the effect of interchannel coupling on the coherence between the orbitals $4d_{0}$ and $5s$ in xenon, we restrict the definition of the $4d_{0}$ hole population to the events where the photoelectron has angular momentum $l=1$. The other possible angular momentum for the $4d_{0}$ photoelectron, $l=3$, does not contribute to the coherence, since the photoelectron from $5s$ can only have $l=1$. For a similar reason, it is impossible to create a coherent superposition of $5p$ and $5s$ (or $4d$) hole states via one-photon absorption in the electric dipole approximation. Figure 2: (color online) The time evolution of the coherence between the $4d_{0}$ and $5s$ hole states in xenon is shown for the full Coulomb interaction model. The photon energy is 136 eV and the pulse duration varies from 5–60 as. Figure 2 illustrates the time evolution of the coherence between $4d_{0}$ and $5s$ in xenon for different pulse durations and fixed photon energy ($\omega_{0}=136$ eV). Here, we use the full interaction model. Directly after the ionizing pulse is over, the initial degree of coherence (at $t\approx 0$ as) rises with decreasing pulse duration, i.e., increasing spectral bandwidth, and converges to a value close to unity. (The difference of the ionization potentials, $\varepsilon_{5s}-\varepsilon_{4d_{0}}$, is $\approx 50$ eV.) At $t\approx 0$ as, the photoelectron is still in immediate contact with the parent ion. Therefore, the coherence properties of the system of interest—the parent ion—are affected by its interaction with the bath represented by the photoelectron. The system–bath interaction leads to a reduction in the coherence of the system Breuer and Petruccione (2002), which can be seen by the rapid drops in all curves in Fig. 2 within tens of attoseconds after the pulse. With time, as the photoelectron departs from the ion, the Coulomb (“system-bath”) interaction becomes less important and the coherence converges to a stationary value. The maximum for this stationary value is obtained with a 25 as pulse ($g_{4d_{0},5s}\approx 0.6$). For pulses shorter than 25 as, oscillations in $g_{4d_{0},5s}$ occur that persist for hundreds of attoseconds, and the final degree of coherence reached falls below 0.6. The spin-orbit dynamics associated with the fine-structure within the $4d$ shell is slow in comparison to the time scale of the decoherence between $4d_{0}$ and $5s$, and is, therefore, not considered here. Figure 3: (color online) The time evolution of the coherence between the $4d_{0}$ and $5s$ hole states, calculated with the full Coulomb interaction model, is shown for different photon energies. The pulse duration is in all cases 20 as. We see in Fig. 3 that when holding the pulse duration fixed ($\tau=20$ as), the degree of coherence rises with increasing $\omega_{0}$. The magnitude of the oscillations decreases as the final coherence (at $t\approx 1$ fs) increases. This trend indicates less system-bath interactions occur with higher photoelectron energies keeping the degree of coherence among the hole states high. In Fig. 4 we compare the impact of the different Coulomb approximations on the final coherence. The drops in coherence that occur for the full model for short pulses [Fig. 4(a)] and low photon energies [Fig. 4(b)] cannot be seen in the Coulomb-free and intrachannel models—which both neglect interchannel coupling. Hence, the decay of coherence is solely driven by the interchannel coupling due to the slow photoelectron. As a comparison to the Coulomb-free model shows, intrachannel coupling affects the coherence in an insignificant way. Figure 4: (color online) The dependence of the coherence between the $4d_{0}$ and $5s$ hole states as function of the pulse duration (a) and as function of the photon energy (b) are shown for all three interaction approximations. In the limit of long pulse durations (small spectral bandwidths), the coherence vanishes for all models, since photoelectrons from the $4d_{0}$ and $5s$ become energetically distinguishable and cannot contribute to a coherent statistical mixture of hole states. The slight drop in the coherence for the Coulomb-free and intrachannel models with increasing $\omega_{0}$ [Fig. 4(b)] is related to the reduced factorizability of the numerator of Eq. (4). In contrast, the trend in the full model for increasing $\omega_{0}$ is dominated by the gain in coherence due to higher photoelectron energy resulting in less system-bath interaction. In conclusion, we demonstrated that the coherence of the ionic states produced via attosecond photoionization is not solely determined by the bandwidth of the ionizing pulse, but greatly depends on the kinetic energy of the photoelectron, which can be controlled by the (mean) photon energy. Interchannel coupling leads to an enhanced entanglement between the photoelectron and the parent ion resulting in a reduced coherence in the ionic states. This reduction can be mitigated with higher photon energies, thereby sacrificing high photon cross sections and the possibility of controlling independently the relative populations of the various hole states in the statistical mixture. Our results have far-reaching consequences beyond the atomic case. Molecules will be even more strongly affected by interchannel coupling due to the reduced symmetry and smaller energy splittings between the cation many- electron eigenstates. Interchannel coupling is also likely to be significant for inner-valence hole configurations in molecules, which show strong mixing to configurations outside the TDCIS model space. The present study suggests that interchannel coupling accompanying the hole creation process will affect attosecond experiments investigating charge transfer processes in photoionized systems. The control of decoherence requires widely tunable attosecond sources, thus offering a new opportunity for x-ray free-electron lasers Zholents and Fawley (2004). ###### Acknowledgements. P.J.H. was supported by the Office of Basic Energy Sciences, U.S. Department of Energy under Contract No. DE-AC02-06CH11357. L.G. thanks Martha Ann and Joseph A. Chenicek and their family. D.A.M. gratefully acknowledges the NSF, the Henry-Camille Dreyfus Foundation, the David-Lucile Packard Foundation, and the Microsoft Corporation for their support. ## References * Krausz and Ivanov (2009) F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009). * Zewail (2000) A. H. Zewail, J. Phys. Chem. A 104, 5660 (2000). * Cavalieri et al. (2007) A. L. Cavalieri et al., Nature 449, 1029 (2007). * Doumy et al. (2009) G. Doumy et al., Phys. Rev. Lett. 102, 093002 (2009). * Dudovich et al. (2006) N. Dudovich et al., Nature Phys. 2, 781 (2006). * López-Martens et al. (2005) R. López-Martens et al., Phys. Rev. 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arxiv-papers
2011-01-12T10:31:27
2024-09-04T02:49:16.341562
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Stefan Pabst, Loren Greenman, Phay J. Ho, David A. Mazziotti, Robin\n Santra", "submitter": "Stefan Pabst", "url": "https://arxiv.org/abs/1101.2314" }
1101.2323
# Semileptonic charmed $B$ meson decays in Universal Extra Dimension Model M. Ali Paracha paracha@phys.qau.edu.pk Physics Department, Quaid-i-Azam University, Islamabad, Pakistan Ishtiaq Ahmed ishtiaq@ncp.edu.pk National Centre for Physics, Quaid-i-Azam University, Islambad, Pakistan M. Jamil Aslam jamil@phys.qau.edu.pk Physics Department, Quaid-i-Azam University, Islamabad, Pakistan. ###### Abstract Form factors parameterizing the semileptonic decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ ($l=\mu,\tau$) are calculated using the frame work of Ward Identities. These form factors are then used to calculate the physical observables like branching ratio and helicity fractions of final state $D_{s}^{\ast}$ meson in these decay modes. The analysis is then extended to the the universal extra dimension (UED) model where the dependence of above mentioned physical variables to the compactification radius R, the only unknown parameter in UED model, is studied. It is shown that the helicity fractions of $D_{s}^{\ast}$ are quite sensitive to the UED model especially when have muons as the final state lepton. Therefore, these can serve as a useful tool to establish new physics predicted by the UED model. ## I Introduction Living in the LHC era, it is hoped to either verify the Standard Model (SM) or to explore the properties of more accurate underlying theory that describes the theory of weak scale. Flavor Changing Neutral Current (FCNC) decays of $B$-meson are an important tool to investigate the structure of weak interactions and also provide us a frame work to look for the physics beyond the Standard Model (SM). This lies in the fact that FCNC decays are not allowed at tree level in the SM and occur only at the loop level 1 ; 2 ; 2a and makes them quite sensitive to possible small corrections that may be result of any modification to the SM, or from the new interactions. This gives us solid reason to study these decays both theoretically and experimentally. Since the CLEO observations of the rare radiative $b\rightarrow s\gamma$ transition 3 , there have been intensive studies on rare semileptonic, radiative and leptonic decays of $B_{u,d,s}$ mesons induced by FCNC transitions of $b\rightarrow s,d$ 4 . The study will be even more complete if one consider the similar decays of the charmed $B$ mesons $(B_{c})$. The charmed $B_{c}$ meson is a bound state of two heavy quarks, bottom $b$ and charm $c$, and was first observed in 1998 at Tevatron in Fermilab 8 . Because of two heavy quarks, the $B_{c}$ mesons are rich in phenomenology compared to the other $B$ mesons. At the Large Hadron Collider (LHC) the expected number of events for the production of $B_{c}$ meson are about $10^{8}-10^{10}$ per year 9 ; 10 which is a reasonable number to work on the phenomenology of the $B_{c}$ meson. In literature, some of the possible radiative and semileptonic exclusive decays of $B_{c}$ mesons like $B_{c}\rightarrow\left(\rho,K^{\ast},D_{s}^{\ast},B_{u}^{\ast}\right)\gamma,B_{c}\rightarrow l\nu\gamma$ $,B_{c}\rightarrow B_{u}^{\ast}l^{+}l^{-},B_{c}\rightarrow D_{1}^{0}l\nu,B_{c}\rightarrow D_{s0}^{\ast}l^{+}l^{-}$ and $B_{c}\rightarrow D_{s,d}^{\ast}l^{+}l^{-}$ have been studied using the frame work of relativistic constituent quark model 11 , QCD Sum Rules and the Light Cone Sum Rules 12 . The focus of the present work is the study of exclusive $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ decay. While working on the exclusive $B$-meson decays the main job is to calculate the form factors which are the non perturbative quantities and are the scalar functions of the square of momentum transfer. In literature the form factors for $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ decay were calculated using different approaches, such as light front constituent quark models and a relativistic quark model 11 ; 13 . In this work we calculate the form factors for the above mentioned decay in a model independent way through Ward identities, which was earlier applied to $B\rightarrow\rho,\gamma$ 14 ; MJAR and $B\rightarrow K_{1}$ decays 15 . This approach enables us to make a clear separation between the pole and non pole type contributions, the former is known in terms of a universal function $\xi_{\perp}(q^{2})\equiv g_{+}(q^{2})$ which is introduced in the Large Energy Effective Theory (LEET) of heavy to light transition form factors 16 . The residue of the pole is then determined in a self consistent way in terms of $g_{+}(0)$ which will give information about the couplings of $B_{s}^{\ast}(1^{-})$ and $B_{sA}^{\ast}(1^{+})$ with $B_{c}D_{s}^{\ast}$ channel. The above mentioned coupling arises at lower pole masses because the higher pole masses of $B_{c}$ meson do not contribute for the decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}.$ The form factors are then determine in terms of a known parameter $g_{+}(0)$ and the pole masses of the particles involved, which will then be used to calculate different physical observables like the branching ratio and the helicity fractions of $D_{s}^{\ast}$ for these decays. At the quark level the semileptonic decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ is governed by the FCNC transition $b\rightarrow sl^{+}l^{-},$ therefore it is an important candidate to look for physics in and beyond the SM. Many investigations for the physics beyond the SM are now being performed in various areas of particle physics which are expected to get the direct or indirect evidence at high energy colliders such as LHC. During the last couple of years there have been an increased interest in models with extra dimensions, since they solve the hierarchy problem and they can provide the unified framework of gravity and other interactions together with a connection to the string theory 17 . Among them the special role plays the one with universal extra dimensions (UED) as in this model all SM fields are allowed to propagate in available all dimensions. The economy of UED model is that there is only one additional parameter to that of SM which is the radius $R$ of the compactified extra dimension. Now above the compactification scale $1/R$ a given UED model becomes a higher dimensional field theory whose equivalent description in four dimensions consists of SM fields and the towers of KK modes having no partner in the SM. A simplest model of this type was proposed by Appelquist,Cheng and Dobrescu (ACD) 18 . In this model all the masses of the KK particles and their interactions with SM particles and also among themselves are described in terms of the inverse of compactification radius $R$ and the parameters of the SM 19 . The most important property of ACD model is the conservation of parity which implies the absence of tree level contributions of KK states to the low energy processes taking place at scale $\mu<<1/R.$ This brings interest towards the FCNC transitions, $b\rightarrow s$ as mentioned earlier that these transitions occur at loop level in SM and hence the one loop contribution due to KK modes to them could in principly be important. These processes are used to constrain the mass and couplings of the KK states, i.e, the compactification radius $1/R$ 19 ; 20 . Buras et al. have computed the effective Hamiltonian of several FCNC processes in ACD model, particularly in $b$ sector, namely $B_{s,d}$ mixing and $b\rightarrow s$ transition such as $b\rightarrow s\gamma$ and $b\rightarrow sl^{+}l^{-}$ decay 19 . The implications of physics with UED are examined with data from Tevatron experiments and the bounds on the inverse of compactification radius are found to be $1/R\geq 250-300$ GeV 21 . There exists some studies in the literature on different $B$ to light meson decays in ACD model, where the dependence of different physical observables like branching ratio, forward-backward asymmetry, lepton polarization asymmetry and the helicity fractions of final state mesons on $1/R$ is examined 21 ; 22 ; 23 . In this work we will study the branching ratio and helicity fractions of $D_{s}^{\ast}$ meson in $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ decay both in the SM and ACD model using the framework of $B\rightarrow(K^{\ast},K_{1})l^{+}l^{-}$ decays described in refs. 22 ; 23 . The paper is organized as follows. In Sec. II we present the effective Hamiltonian for the decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}.$ Section III contains the definitions as well as the detailed calculation of the form factors using Ward Identities. In Sec. IV we present the basic formulas for physical observables like decay rate and helicity fractions of $D_{s}^{\ast}$ meson where as the numerical analysis of these observables will be given in Section V. Section VI gives the summary of the results. ## II Effective Hamiltonian and Matrix Elements At quark level, the semileptonic decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ is governed by the transition $b\rightarrow sl^{+}l^{-}$ for which the effective Hamiltonian can be written as $\displaystyle H_{eff}$ $\displaystyle=$ $\displaystyle-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{\ast}\bigg{[}\sum\limits_{i=1}^{10}C_{i}(\mu)O_{i}\bigg{]},$ (1) where $O_{i}(\mu)$ $(i=1,...,10)$ are the four quark operators and $C_{i}(\mu)$ are the corresponding Wilson coefficients at the energy scale $\mu$ 24 which was usually take to be the $b$-quark mass $\left(m_{b}\right)$. The theoretical uncertainties related to the renormalization scale can be reduced when the next to leading logarithm corrections are included. The explicit form of the operators responsible for the decay $B_{c}^{-}\rightarrow D_{s}^{\ast-}l^{+}l^{-}$ is $\displaystyle O_{7}$ $\displaystyle=$ $\displaystyle\frac{e^{2}}{16\pi^{2}}m_{b}(\bar{s}\sigma_{\mu\nu}Rb)F^{\mu\nu}$ (2) $\displaystyle O_{9}$ $\displaystyle=$ $\displaystyle\frac{e^{2}}{16\pi^{2}}\left(\bar{s}\gamma_{\mu}Lb\right)\bar{l}\gamma^{\mu}l$ (3) $\displaystyle O_{10}$ $\displaystyle=$ $\displaystyle\frac{e^{2}}{16\pi^{2}}\left(\bar{s}\gamma_{\mu}Lb\right)\bar{l}\gamma^{\mu}\gamma^{5}l$ (4) with $L,R=\left(1\mp\gamma^{5}\right)/2$. Using the effective Hamiltonian given in Eq.(1) the free quark amplitude for $b\rightarrow sl^{+}l^{-}$ can be written as $\displaystyle\mathcal{M(}b\rightarrow sl^{+}l^{-})$ $\displaystyle=$ $\displaystyle-\frac{G_{F}\alpha}{\sqrt{2}\pi}V_{tb}V_{ts}^{\ast}\bigg{[}{C_{9}^{eff}\left(\mu\right)(\bar{s}\gamma_{\mu}Lb)(\bar{l}\gamma^{\mu}l)+C_{10}(}\bar{s}\gamma_{\mu}Lb)(\bar{l}\gamma^{\mu}\gamma^{5}l)$ (5) $\displaystyle-2C_{7}^{eff}\left(\mu\right)\frac{m_{b}}{q^{2}}(\bar{s}i\sigma_{\mu\nu}q^{\nu}Rb)\bar{l}\gamma^{\mu}l\bigg{]}$ where $q^{2}$ is the square of momentum transfer. Note that the operator $O_{10}$ given in Eq.(4) can not be induced by the insertion of four quark operators because of the absence of $Z$-boson in the effective theory. Therefore, the Wilson coefficient $C_{10}$ does not renormalize under QCD corrections and is independent on the energy scale $\mu.$ Additionally the above quark level decay amplitude can get contributions from the matrix element of four quark operators, $\sum_{i=1}^{6}\left\langle l^{+}l^{-}s\left|O_{i}\right|b\right\rangle,$ which are usually absorbed into the effective Wilson coefficient $C_{9}^{eff}(\mu)$ and can be written as 25 ; 26 ; 27 ; 28 ; 29 ; 30 ; 31 $C_{9}^{eff}(\mu)=C_{9}(\mu)+Y_{SD}(z,s^{\prime})+Y_{LD}(z,s^{\prime}).$ where $z=m_{c}/m_{b}$ and $s^{\prime}=q^{2}/m_{b}^{2}$. $Y_{SD}(z,s^{\prime})$ describes the short distance contributions from four-quark operators far away from the $c\bar{c}$ resonance regions, and this can be calculated reliably in the perturbative theory. However the long distance contribution $Y_{LD}(z,s^{\prime})$ cannot be calculated by using the first principles of QCD, so they are usually parameterized in the form of a phenomenological Breit-Wigner formula making use of the vacuum saturation approximation and quark hadron duality. Therefore, one can not calculate them reliably so we we will neglect these long distance effects for the case of $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$. The expression for the short distance contribution $Y_{SD}(z,s^{\prime})$ is given as $\displaystyle Y_{SD}(z,s^{\prime})$ $\displaystyle=$ $\displaystyle h(z,s^{\prime})(3C_{1}(\mu)+C_{2}(\mu)+3C_{3}(\mu)+C_{4}(\mu)+3C_{5}(\mu)+C_{6}(\mu))$ (6) $\displaystyle-\frac{1}{2}h(1,s^{\prime})(4C_{3}(\mu)+4C_{4}(\mu)+3C_{5}(\mu)+C_{6}(\mu))$ $\displaystyle-\frac{1}{2}h(0,s^{\prime})(C_{3}(\mu)+3C_{4}(\mu))+{\frac{2}{9}}(3C_{3}(\mu)+C_{4}(\mu)+3C_{5}(\mu)+C_{6}(\mu)),$ with $\displaystyle h(z,s^{\prime})$ $\displaystyle=$ $\displaystyle-{\frac{8}{9}}\mathrm{ln}z+{\frac{8}{27}}+{\frac{4}{9}}x-{\frac{2}{9}}(2+x)|1-x|^{1/2}\left\\{\begin{array}[]{l}\ln\left|\frac{\sqrt{1-x}+1}{\sqrt{1-x}-1}\right|-i\pi\quad\mathrm{for}{{\ }x\equiv 4z^{2}/s^{\prime}<1}\\\ 2\arctan\frac{1}{\sqrt{x-1}}\qquad\mathrm{for}{{\ }x\equiv 4z^{2}/s^{\prime}>1}\end{array}\right.,$ (9) $\displaystyle h(0,s^{\prime})$ $\displaystyle=$ $\displaystyle{\frac{8}{27}}-{\frac{8}{9}}\mathrm{ln}{\frac{m_{b}}{\mu}}-{\frac{4}{9}}\mathrm{ln}s^{\prime}+{\frac{4}{9}}i\pi\,\,.$ (10) Also the non factorizable effects from the charm loop brings further corrections to the radiative transition $b\rightarrow s\gamma,$ and these can be absorbed into the effective Wilson coefficients $C_{7}^{eff}$ which then takes the form 32 ; 33 ; 34 ; 35 ; 36 $C_{7}^{eff}(\mu)=C_{7}(\mu)+C_{b\rightarrow s\gamma}(\mu)$ with $\displaystyle C_{b\rightarrow s\gamma}(\mu)$ $\displaystyle=$ $\displaystyle i\alpha_{s}\left[\frac{2}{9}\eta^{14/23}(G_{1}(x_{t})-0.1687)-0.03C_{2}(\mu)\right]$ (11) $\displaystyle G_{1}(x_{t})$ $\displaystyle=$ $\displaystyle\frac{x_{t}\left(x_{t}^{2}-5x_{t}-2\right)}{8\left(x_{t}-1\right)^{3}}+\frac{3x_{t}^{2}\ln^{2}x_{t}}{4\left(x_{t}-1\right)^{4}}$ (12) where $\eta=\alpha_{s}(m_{W})/\alpha_{s}(\mu),$ $x_{t}=m_{t}^{2}/m_{W}^{2}$ and $C_{b\rightarrow s\gamma}$ is the absorptive part for the $b\rightarrow sc\bar{c}\rightarrow s\gamma$ rescattering. The new physics effects manifest themselves in rare $B$ decays in two different ways, either through new contribution to the Wilson coefficients or through the new operators in the effective Hamiltonian, which are absent in the SM. Being minimal extension of SM the ACD model is the most economical one because it has only additional parameter $R$ i.e. the radius of the compactification leaving the operators basis same as that of the SM. Therefore, the whole contribution from all the KK states is in the Wilson coefficients which are now the functions of the compactification radius $R$. At large value of $1/R$ the new states being more and more massive and will be decoupled from the low-energy theory,therefore one can recover the SM phenomenology. The modified Wilson coefficients in ACD model contain the contribution from new particles which are not present in the SM and comes as an intermediate state in penguin and box diagrams. Thus, these coefficients can be expressed in terms of the functions $F\left(x_{t},1/R\right)$, $x_{t}=\frac{m_{t}^{2}}{M_{W}^{2}}$, which generalize the corresponding SM function $F_{0}\left(x_{t}\right)$ according to: $F\left(x_{t},1/R\right)=F_{0}\left(x_{t}\right)+\sum_{n=1}^{\infty}F_{n}\left(x_{t},x_{n}\right)$ (13) with $x_{n}=\frac{m_{n}^{2}}{M_{W}^{2}}$ and $m_{n}=\frac{n}{R}$ 44 . The relevant diagrams are $Z^{0}$ penguins, $\gamma$ penguins, gluon penguins, $\gamma$ magnetic penguins, Chormomagnetic penguins and the corresponding functions are $C\left(x_{t},1/R\right)$, $D\left(x_{t},1/R\right)$, $E\left(x_{t},1/R\right)$, $D^{\prime}\left(x_{t},1/R\right)$ and $E^{\prime}\left(x_{t},1/R\right)$ respectively. These functions are calculated at next to leading order by Buras et al. 19 and can be summarized as: $\bullet C_{7}$ In place of $C_{7},$ one defines an effective coefficient $C_{7}^{(0)eff}$ which is renormalization scheme independent 45 : $C_{7}^{(0)eff}(\mu_{b})=\eta^{\frac{16}{23}}C_{7}^{(0)}(\mu_{W})+\frac{8}{3}(\eta^{\frac{14}{23}}-\eta^{\frac{16}{23}})C_{8}^{(0)}(\mu_{W})+C_{2}^{(0)}(\mu_{W})\sum_{i=1}^{8}h_{i}\eta^{\alpha_{i}}$ (14) where $\eta=\frac{\alpha_{s}(\mu_{W})}{\alpha_{s}(\mu_{b})}$, and $C_{2}^{(0)}(\mu_{W})=1,\mbox{ }C_{7}^{(0)}(\mu_{W})=-\frac{1}{2}D^{\prime}(x_{t},\frac{1}{R}),\mbox{ }C_{8}^{(0)}(\mu_{W})=-\frac{1}{2}E^{\prime}(x_{t},\frac{1}{R});$ (15) the superscript $(0)$ stays for leading logarithm approximation. Furthermore: $\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle\frac{14}{23}\mbox{ \quad}\alpha_{2}=\frac{16}{23}\mbox{ \quad}\alpha_{3}=\frac{6}{23}\mbox{ \quad}\alpha_{4}=-\frac{12}{23}$ $\displaystyle\alpha_{5}$ $\displaystyle=$ $\displaystyle 0.4086\mbox{ \quad}\alpha_{6}=-0.4230\mbox{ \quad}\alpha_{7}=-0.8994\mbox{ \quad}\alpha_{8}=-0.1456$ $\displaystyle h_{1}$ $\displaystyle=$ $\displaystyle 2.996\mbox{ \quad}h_{2}=-1.0880\mbox{ \quad}h_{3}=-\frac{3}{7}\mbox{ \quad}h_{4}=-\frac{1}{14}$ $\displaystyle h_{5}$ $\displaystyle=$ $\displaystyle-0.649\mbox{ \quad}h_{6}=-0.0380\mbox{ \quad}h_{7}=-0.0185\mbox{ \quad}h_{8}=-0.0057.$ (16) The functions $D^{\prime}$ and $E^{\prime}$ are $D_{0}^{\prime}(x_{t})=-\frac{(8x_{t}^{3}+5x_{t}^{2}-7x_{t})}{12(1-x_{t})^{3}}+\frac{x_{t}^{2}(2-3x_{t})}{2(1-x_{t})^{4}}\ln x_{t},$ (17) $E_{0}^{\prime}(x_{t})=-\frac{x_{t}(x_{t}^{2}-5x_{t}-2)}{4(1-x_{t})^{3}}+\frac{3x_{t}^{2}}{2(1-x_{t})^{4}}\ln x_{t},$ (18) $\displaystyle D_{n}^{\prime}(x_{t},x_{n})$ $\displaystyle=$ $\displaystyle\frac{x_{t}(-37+44x_{t}+17x_{t}^{2}+6x_{n}^{2}(10-9x_{t}+3x_{t}^{2})-3x_{n}(21-54x_{t}+17x_{t}^{2}))}{36(x_{t}-1)^{3}}$ $\displaystyle+\frac{x_{n}(2-7x_{n}+3x_{n}^{2})}{6}\ln\frac{x_{n}}{1+x_{n}}$ $\displaystyle-\frac{(-2+x_{n}+3x_{t})(x_{t}+3x_{t}^{2}+x_{n}^{2}(3+x_{t})-x_{n})(1+(-10+x_{t})x_{t}))}{6(x_{t}-1)^{4}}\ln\frac{x_{n}+x_{t}}{1+x_{n}},$ $\displaystyle E_{n}^{\prime}(x_{t},x_{n})$ $\displaystyle=$ $\displaystyle\frac{x_{t}(-17-8x_{t}+x_{t}^{2}+3x_{n}(21-6x_{t}+x_{t}^{2})-6x_{n}^{2}(10-9x_{t}+3x_{t}^{2}))}{12(x_{t}-1)^{3}}$ (20) $\displaystyle+-\frac{1}{2}x_{n}(1+x_{n})(-1+3x_{n})\ln\frac{x_{n}}{1+x_{n}}$ $\displaystyle+\frac{(1+x_{n})(x_{t}+3x_{t}^{2}+x_{n}^{2}(3+x_{t})-x_{n}(1+(-10+x_{t})x_{t}))}{2(x_{t}-1)^{4}}\ln\frac{x_{n}+x_{t}}{1+x_{n}}.$ Following reference 19 , one gets the expressions for the sum over $n:$ $\displaystyle\sum_{n=1}^{\infty}D_{n}^{\prime}(x_{t},x_{n})$ $\displaystyle=$ $\displaystyle-\frac{x_{t}(-37+x_{t}(44+17x_{t}))}{72(x_{t}-1)^{3}}$ (21) $\displaystyle+\frac{\pi M_{w}R}{2}[\int_{0}^{1}dy\frac{2y^{\frac{1}{2}}+7y^{\frac{3}{2}}+3y^{\frac{5}{2}}}{6}]\coth(\pi M_{w}R\sqrt{y})$ $\displaystyle+\frac{(-2+x_{t})x_{t}(1+3x_{t})}{6(x_{t}-1)^{4}}J(R,-\frac{1}{2})$ $\displaystyle-\frac{1}{6(x_{t}-1)^{4}}[x_{t}(1+3x_{t})-(-2+3x_{t})(1+(-10+x_{t})x_{t})]J(R,\frac{1}{2})$ $\displaystyle+\frac{1}{6(x_{t}-1)^{4}}[(-2+3x_{t})(3+x_{t})-(1+(-10+x_{t})x_{t})]J(R,\frac{3}{2})$ $\displaystyle-\frac{(3+x_{t})}{6(x_{t}-1)^{4}}J(R,\frac{5}{2})],$ $\displaystyle\sum_{n=1}^{\infty}E_{n}^{\prime}(x_{t},x_{n})$ $\displaystyle=$ $\displaystyle-\frac{x_{t}(-17+(-8+x_{t})x_{t})}{24(x_{t}-1)^{3}}$ (22) $\displaystyle+\frac{\pi M_{w}R}{2}[\int_{0}^{1}dy(y^{\frac{1}{2}}+2y^{\frac{3}{2}}-3y^{\frac{5}{2}})\coth(\pi M_{w}R\sqrt{y})]$ $\displaystyle-\frac{x_{t}(1+3x_{t})}{(x_{t}-1)^{4}}J(R,-\frac{1}{2})$ $\displaystyle+\frac{1}{(x_{t}-1)^{4}}[x_{t}(1+3x_{t})-(1+(-10+x_{t})x_{t})]J(R,\frac{1}{2})$ $\displaystyle-\frac{1}{(x_{t}-1)^{4}}[(3+x_{t})-(1+(-10+x_{t})x_{t})]J(R,\frac{3}{2})$ $\displaystyle+\frac{(3+x_{t})}{(x_{t}-1)^{4}}J(R,\frac{5}{2})],$ where $J(R,\alpha)=\int_{0}^{1}dyy^{\alpha}[\coth(\pi M_{w}R\sqrt{y})-x_{t}^{1+\alpha}\coth(\pi m_{t}R\sqrt{y})].$ (23) $\bullet C_{9}$ In the ACD model and in the NDR scheme one has $C_{9}(\mu)=P_{0}^{NDR}+\frac{Y(x_{t},\frac{1}{R})}{\sin^{2}\theta_{W}}-4Z(x_{t},\frac{1}{R})+P_{E}E(x_{t},\frac{1}{R})$ (24) where $P_{0}^{NDR}=2.60\pm 0.25$ 12 and the last term is numerically negligible. Besides $\displaystyle Y(x_{t},\frac{1}{R})$ $\displaystyle=$ $\displaystyle Y_{0}(x_{t})+\sum_{n=1}^{\infty}C_{n}(x_{t},x_{n})$ $\displaystyle Z(x_{t},\frac{1}{R})$ $\displaystyle=$ $\displaystyle Z_{0}(x_{t})+\sum_{n=1}^{\infty}C_{n}(x_{t},x_{n})$ (25) with $\displaystyle Y_{0}(x_{t})$ $\displaystyle=$ $\displaystyle\frac{x_{t}}{8}[\frac{x_{t}-4}{x_{t}-1}+\frac{3x_{t}}{(x_{t}-1)^{2}}\ln x_{t}]$ $\displaystyle Z_{0}(x_{t})$ $\displaystyle=$ $\displaystyle\frac{18x_{t}^{4}-163x_{t}^{3}+259x_{t}^{2}-108x_{t}}{144(x_{t}-1)^{3}}$ (26) $\displaystyle+[\frac{32x_{t}^{4}-38x_{t}^{3}+15x_{t}^{2}-18x_{t}}{72(x_{t}-1)^{4}}-\frac{1}{9}]\ln x_{t}$ $C_{n}(x_{t},x_{n})=\frac{x_{t}}{8(x_{t}-1)^{2}}[x_{t}^{2}-8x_{t}+7+(3+3x_{t}+7x_{n}-x_{t}x_{n})\ln\frac{x_{t}+x_{n}}{1+x_{n}}]$ (27) and $\sum_{n=1}^{\infty}C_{n}(x_{t},x_{n})=\frac{x_{t}(7-x_{t})}{16(x_{t}-1)}-\frac{\pi M_{w}Rx_{t}}{16(x_{t}-1)^{2}}[3(1+x_{t})J(R,-\frac{1}{2})+(x_{t}-7)J(R,\frac{1}{2})]$ (28) $\bullet C_{10}$ $C_{10}$ is $\mu$ independent and is given by $C_{10}=-\frac{Y(x_{t},\frac{1}{R})}{\sin^{2}\theta_{w}}.$ (29) The normalization scale is fixed to $\mu=\mu_{b}\simeq 5$ GeV. ## III Matrix Elements and Form Factors The exclusive $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$decay involves the hadronic matrix elements which can be obtained by sandwiching the quark level operators give in Eq. (5) between initial state $B_{c}$ meson and final state $D_{s}^{\ast}$ meson. These can be parameterized in terms of form factors which are scalar functions of the square of the four momentum transfer($q^{2}=(p-k)^{2}).$ The non vanishing matrix elements for the process $B_{c}\rightarrow D_{s}^{\ast}$ can be parameterized in terms of the seven form factors as follows $\displaystyle\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}b\right|B_{c}(p)\right\rangle$ $\displaystyle=$ $\displaystyle\frac{2\epsilon_{\mu\nu\alpha\beta}}{M_{B_{c}}+M_{D_{s}^{\ast-}}}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}V(q^{2})$ (30) $\displaystyle\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}\gamma_{5}b\right|B_{c}(p)\right\rangle$ $\displaystyle=$ $\displaystyle i\left(M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}\right)\varepsilon^{\ast\mu}A_{1}(q^{2})$ $\displaystyle-i\frac{(\varepsilon^{\ast}\cdot q)}{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}\left(p+k\right)^{\mu}A_{2}(q^{2})$ $\displaystyle-i\frac{2M_{D_{s}^{\ast-}}}{q^{2}}q^{\mu}(\varepsilon^{\ast}\cdot q)\left[A_{3}(q^{2})-A_{0}(q^{2})\right]$ where $p$ is the momentum of $B_{c}$, $\varepsilon$ and $k$ are the polarization vector and momentum of the final state $D_{s}^{\ast}$ meson. Here, the form factor $A_{3}(q^{2})$ can be expressed in terms of the form factors $A_{1}(q^{2})$ and $A_{2}(q^{2})$ as $A_{3}(q^{2})=\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{2M_{D_{s}^{\ast-}}}A_{1}(q^{2})-\frac{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}{2M_{D_{s}^{\ast-}}}A_{2}(q^{2})$ (32) with $A_{3}(0)=A_{0}(0)$ In addition to the above form factors there are some penguin form factors, which we can write as $\displaystyle\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}b\right|B_{c}(p)\right\rangle$ $\displaystyle=$ $\displaystyle-\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}2F_{1}(q^{2})$ (33) $\displaystyle\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}\gamma^{5}b\right|B_{c}(p)\right\rangle$ $\displaystyle=$ $\displaystyle i\left[\left(M_{Bc^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}\right)\varepsilon_{\mu}-(\varepsilon^{\ast}\cdot q)(p+k)_{\mu}\right]F_{2}(q^{2})$ $\displaystyle+(\varepsilon^{\ast}\cdot q)i\left[q_{\mu}-\frac{q^{2}}{M_{Bc^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}}(p+k)_{\mu}\right]F_{3}(q^{2})$ with $F_{1}(0)=F_{2}(0)$ Now the different form factors appearing in Eqs. (30-LABEL:13b) can be related to each other with the help of Ward identities as follows 14 $\displaystyle\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}b\right|B_{c}(p)\right\rangle$ $\displaystyle=$ $\displaystyle(m_{b}+m_{s})\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}b\right|B_{c}(p)\right\rangle$ (35) $\displaystyle\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}\gamma^{5}b\right|B_{c}(p)\right\rangle$ $\displaystyle=$ $\displaystyle-(m_{b}-m_{s})\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{\mu}\gamma_{5}b\right|B_{c}(p)\right\rangle$ (36) $\displaystyle+(p+k)_{\mu}\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}\gamma_{5}b\right|B_{c}(p)\right\rangle$ By putting Eq.(30-LABEL:13b) in Eq.(35) and (36) and comparing the coefficients of $\varepsilon_{\mu}^{\ast}$ and $q_{\mu}$ on both sides, one can get the following relations between the form factors: $\displaystyle F_{1}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{(m_{b}+m_{s})}{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}V(q^{2})$ (37) $\displaystyle F_{2}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{m_{b}-m_{s}}{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}A_{1}(q^{2})$ (38) $\displaystyle F_{3}(q^{2})$ $\displaystyle=$ $\displaystyle-(m_{b}-m_{s})\frac{2M_{D_{s}^{\ast-}}}{q^{2}}\left[A_{3}(q^{2})-A_{0}(q^{2})\right]$ (39) The results given in Eqs. (37, 38, 39) are derived by using Ward identities and therefore are the model independent. The universal normalization of the above form factors at $q^{2}=0$ are obtained by defining 14 $\displaystyle\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\alpha\beta}b\right|B_{c}(p)\right\rangle$ $\displaystyle=$ $\displaystyle-i\epsilon_{\alpha\beta\rho\sigma}\varepsilon^{\ast\rho}\left[(p+k)^{\sigma}g_{+}+q^{\sigma}g_{-}\right]-(\varepsilon^{\ast}\cdot q)\epsilon_{\alpha\beta\rho\sigma}(p+k)^{\rho}q^{\sigma}h$ (40) $\displaystyle-i\left[(p+k)_{\alpha}\varepsilon_{\beta\rho\sigma\tau}\varepsilon^{\ast\rho}(p+k)^{\sigma}q^{\tau}-\alpha\leftrightarrow\beta\right]h_{1}$ Making use of the Dirac identity $\sigma^{\mu\nu}\gamma^{5}=-\frac{i}{2}\epsilon^{\mu\nu\alpha\beta}\sigma_{\alpha\beta}$ (41) in Eq.(40), we get $\displaystyle\left\langle D_{s}^{\ast}(k,\varepsilon)\left|\bar{s}i\sigma_{\mu\nu}q^{\nu}\gamma^{5}b\right|B_{c}(p)\right\rangle$ $\displaystyle=$ $\displaystyle\varepsilon_{\mu}^{\ast}\left[(M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2})g_{+}+q^{2}g_{-}\right]$ (42) $\displaystyle-q\cdot\varepsilon^{\ast}\left[q^{2}(p+k)_{\mu}g_{+}-q_{\mu}g_{-}\right]$ $\displaystyle+q\cdot\varepsilon^{\ast}\left[q^{2}(p+k)_{\mu}-(M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2})q_{\mu}\right]h$ On comparing coefficents of $q_{\mu},\varepsilon_{\mu}^{\ast}$ and $\epsilon_{\mu\nu\alpha\beta}$ from Eqs.(33), (LABEL:13b), (40) and (42), we have $\displaystyle F_{1}(q^{2})$ $\displaystyle=$ $\displaystyle\left[g_{+}(q^{2})-q^{2}h_{1}(q^{2})\right]$ (43) $\displaystyle F_{2}(q^{2})$ $\displaystyle=$ $\displaystyle g_{+}(q^{2})+\frac{q^{2}}{M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}}g_{-}(q^{2})$ (44) $\displaystyle F_{3}(q^{2})$ $\displaystyle=$ $\displaystyle- g_{-}(q^{2})-(M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2})h(q^{2})$ (45) One can see from Eq. (43) and Eq. (44) that at $q^{2}=0,$ $F_{1}(0)=F_{2}(0).$ The form factors $V(q^{2}),A_{1}(q^{2})$ and $A_{2}(q^{2})$ can be written in terms of $g_{+},g_{-}$ and $h$ as $\displaystyle V(q^{2})$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}+m_{s}}\left[g_{+}(q^{2})-q^{2}h_{1}(q^{2})\right]$ (46) $\displaystyle A_{1}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}\left[g_{+}(q^{2})+\frac{q^{2}}{M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}}g_{-}(q^{2})\right]$ (47) $\displaystyle A_{2}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}\left[g_{+}(q^{2})-q^{2}h(q^{2})\right]-\frac{2M_{D_{s}^{\ast-}}}{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}A_{0}(q^{2})$ (48) By looking at Eq. (46) and Eq. (47) it is clear that the normalization of the form factors $V$ and $A_{1}$ at $q^{2}=0$ is determined by a single constant $g_{+}(0),$ where as from Eq. (48) the form factor $A_{2}$ at $q^{2}=0$ is determined by two constants i.e. $g_{+}(0)$ and $A_{0}(0).$ ### III.1 Pole Contribution In $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ decay, there will be a pole contribution to $h_{1},g_{-},h$ and $A_{0}$ from $B_{s}^{\ast}(1^{-}),B_{sA}^{\ast}(1^{+})$ and $B_{s}(0^{-})$ mesons which can be parameterized as $\displaystyle h_{1}|_{pole}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\frac{g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}}{M_{B_{s}^{\ast}}^{2}}\frac{f_{T}^{B^{\ast}}}{1-q^{2}/M_{B^{\ast}}^{2}}=\frac{R_{V}}{M_{B_{s}^{\ast}}^{2}}\frac{1}{1-q^{2}/M_{B_{s}^{\ast}}^{2}}$ (49) $\displaystyle g_{-}|_{pole}$ $\displaystyle=$ $\displaystyle-\frac{g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}{M_{B_{sA}^{\ast}}^{2}}\frac{f_{T}^{B_{sA}^{\ast}}}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}=\frac{R_{A}^{S}}{M_{B_{sA}^{\ast}}^{2}}\frac{1}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}$ (50) $\displaystyle h|_{pole}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{f_{B_{SA}^{\ast}B_{c}D_{s}^{\ast}}}{M_{B_{sA}^{\ast}}^{2}}\frac{f_{T}^{B_{SA}^{\ast}}}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}=\frac{R_{A}^{D}}{M_{B_{sA}^{\ast}}^{2}}\frac{1}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}$ (51) $\displaystyle A_{0}(q^{2})|_{pole}$ $\displaystyle=$ $\displaystyle\frac{g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}}{M_{B_{s}^{\ast}}^{2}}f_{B_{s}}\frac{q^{2}/M_{B}^{2}}{1-q^{2}/M_{B}^{2}}=R_{0}\frac{q^{2}/M_{B_{s}}^{2}}{1-q^{2}/M_{B_{s}}^{2}}$ (52) where the quantities $R_{V},R_{A}^{S},R_{A}^{D}$ and $R_{0}$ are related to the coupling constants $g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}},g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}$ and $g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}$, respectively. Here we would like to mention that the above mentioned couplings aries as the lower pole mass, because the higher pole masses of $B_{c}$ meson do not contribute for the $B_{c}\to D_{s}^{\ast}l^{+}l^{-}$ decay. The form factors $A_{1}(q^{2}),A_{2}(q^{2})$ and $V(q^{2})$ can be written in terms of these quantities as $\displaystyle V(q^{2})$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast}}}{m_{b}+m_{s}}\left[g_{+}(q^{2})-\frac{R_{V}}{M_{B_{s}^{\ast}}^{2}}\frac{q^{2}}{1-q^{2}/M_{B_{s}^{\ast}}^{2}}\right]$ (53) $\displaystyle A_{1}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}\left[g_{+}(q^{2})+\frac{q^{2}}{M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}}\tilde{g}_{-}(q^{2})+\frac{R_{A}^{S}}{M_{B_{sA}^{\ast}}^{2}}\frac{q^{2}}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}\right]$ (54) $\displaystyle A_{2}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}\left[g_{+}(q^{2})-\frac{R_{A}^{D}}{M_{B_{As}^{\ast}}^{2}}\frac{q^{2}}{1-q^{2}/M_{B_{sA}^{\ast}}^{2}}\right]-\frac{2M_{D_{s}^{\ast-}}}{M_{B_{c}}-M_{D_{s}^{\ast-}}}A_{0}(q^{2})$ (55) Now, the behavior of $g_{+}(q^{2}),\tilde{g}_{-}(q^{2})$ and $A_{0}(q^{2})$ is known from LEET and their form is 14 $\displaystyle g_{+}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{\xi_{\bot}(0)}{(1-q^{2}/M_{B}^{2})^{2}}=-\tilde{g}_{-}(q^{2})$ (56) $\displaystyle A_{0}(q^{2})$ $\displaystyle=$ $\displaystyle\left(1-\frac{M_{D_{s}^{\ast-}}^{2}}{M_{B_{c}}E_{D_{s}^{\ast-}}}\right)\xi_{\|}(0)+\frac{M_{D_{s}^{\ast-}}}{M_{B_{c}}}\xi_{\perp}(0)$ (57) $\displaystyle E_{D_{s}^{\ast}}$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}}}{2}\left(1-\frac{q^{2}}{M_{B_{c}}^{2}}+\frac{M_{D_{s}^{\ast}}^{2}}{M_{B_{c}}^{2}}\right)$ (58) $\displaystyle g_{+}(0)$ $\displaystyle=$ $\displaystyle\xi_{\bot}(0)$ (59) The pole terms given in Eqs.(53-55) dominate near $q^{2}=M_{B_{s}^{\ast}}^{2}$ and $q^{2}=M_{B_{sA}^{\ast}}^{2}$. Just to make a remark that relations obtained from the Ward identities can not be expected to hold for the whole $q^{2}.$ Therefore, near $q^{2}=0$ and near the pole following parametrization is suggested 14 $F(q^{2})=\frac{F(0)}{\left(1-q^{2}/M^{2}\right)(1-q^{2}/M^{\prime 2})}$ (60) where $M^{2}$ is $M_{B_{s}^{\ast}}^{2}$ or $M_{B_{sA}^{\ast}}^{2}$, and $M^{\prime}$ is the radial excitation of $M.$ The parametrization given in Eq. (60) not only takes into account the corrections to single pole dominance suggested by the dispersion relation approach 37 ; 38 ; 39 but also give the correction of off-mass shell-ness of the couplings of $B_{s}^{\ast}$ and $B_{sA}^{\ast}$ with the $B_{c}D_{s}^{\ast}$ channel. Since $g_{+}(0)$ and $\tilde{g}_{-}(q^{2})$ have no pole at $\ q^{2}=M_{B_{s}^{\ast}}^{2},$ hence we get $V(q^{2})(1-\frac{q^{2}}{M_{B^{\ast}}^{2}})|_{q^{2}=M_{B^{\ast}}^{2}}=-R_{V}\left(\frac{M_{B_{c}}+M_{D_{s}^{\ast}}}{m_{b}-m_{s}}\right)$ This becomes $R_{V}\equiv-\frac{1}{2}g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}f_{B^{\ast}_{s}}=-\frac{g_{+}(0)}{1-M_{B^{\ast}}^{2}/M_{B^{\ast}}^{\prime 2}}$ (61) and similarly $R_{A}^{D}\equiv\frac{1}{2}f_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}f^{B^{\ast}_{sA}}_{T}=-\frac{g_{+}(0)}{1-M_{B_{sA}^{\ast}}^{2}/M_{B_{sA}^{\ast}}^{\prime 2}}$ (62) We cannot use the parametrization given in Eq.(60) for the form factor $A_{1}(q^{2}),$ since near $q^{2}=0,$ the behavior of $A_{1}(q^{2})$ is $g_{+}(q^{2})\left[1-q^{2}/\left(M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast-}}^{2}\right)\right],$ therefore we can write $A_{1}(q^{2})$ as follows $A_{1}(q^{2})=\frac{g_{+}(0)}{\left(1-q^{2}/M_{B_{sA}^{\ast}}^{2}\right)\left(1-q^{2}/M_{B_{sA}^{\ast}}^{\prime 2}\right)}\left(1-\frac{q^{2}}{M_{B_{c}^{-}}^{2}-M_{D_{s}^{\ast}}^{2}}\right)$ (63) The only unkonown parameter in the above form factors calculation is $g_{+}(0)$ and its value can be extracted by using the central value of branching ratio for the decay $B_{c}^{-}\rightarrow D_{s}^{\ast-}\gamma$ 41 . From the formula of decay rate $\Gamma\left(B_{c}\rightarrow D_{s}^{\ast}\gamma\right)=\frac{G_{F}^{2}\alpha}{32\pi^{4}}\left|V_{tb}V_{ts}^{\ast}\right|^{2}m_{b}^{2}M_{B_{c}}^{3}\times\left(1-\frac{M_{D_{s}^{\ast}}^{2}}{M_{B_{c}}^{2}}\right)^{3}\left|C_{7}^{eff}\right|^{2}\left|g_{+}(0)\right|^{2}$ (64) and by putting the values of everything one can find the value of unknown parameter $g_{+}(0)=0.32\pm 0.1$. In the forthcoming analysis we use the value of $g_{+}(0)=0.42$ which was calculated in ref. 41 . Using $f_{B_{c}}=0.35$ GeV we have prediction from Eq.(61) that $g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}=10.38GeV^{-1}.$ (65) Similarly the ratio of $S$ and $D$ wave couplings are predicted to be $\frac{g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}{f_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}=-0.42GeV^{2}$ (66) The different values of the $F(0)$ are $\displaystyle V(0)$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}+m_{s}}g_{+}(0)$ (67) $\displaystyle A_{1}(0)$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}g_{+}(0)$ (68) $\displaystyle A_{2}(0)$ $\displaystyle=$ $\displaystyle\frac{M_{B_{c}^{-}}+M_{D_{s}^{\ast-}}}{m_{b}-m_{s}}g_{+}(0)-\frac{2M_{D_{s}^{\ast-}}}{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}A_{0}(0)$ (69) The calculation of the numerical values of $V(0)$ and $A_{1}(0)$ is quite trivial but for the value of $A_{2}(0),$ the value of $A_{0}(0)$ has to be known. Although LEET does not give any relationship between $\xi_{||}(0)$ and $\xi_{\perp}(0)$, but in LCSR $\xi_{||}(0)$ and $\xi_{\perp}(0)$ are related due to numerical coincidence 42 $\xi_{||}(0)\simeq\xi_{\perp}(0)=g_{+}(0)$ (70) From Eq. (57) we have $A_{0}(0)=1.12g_{+}(0)$ The value of the form factors at $q^{2}=0$ is given in Table-1 Table 1: Values of the form factors at $q^{2}=0$. $V(0)$ | $A_{1}(0)$ | $\tilde{A}_{2}(0)$ | $A_{0}(0)$ | | ---|---|---|---|---|--- $0.51\pm 0.17$ | $0.28\pm 0.08$ | $0.22\pm 0.07$ | $0.35\pm 0.11$ | | and can be extrapolated for the other values of $q^{2}$ as follows: $\displaystyle V(q^{2})$ $\displaystyle=$ $\displaystyle\frac{V(0)}{(1-q^{2}/M_{B_{s}^{\ast}}^{2})(1-q^{2}/M_{B_{s}^{\ast}}^{\prime 2})}$ (71) $\displaystyle A_{1}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{A_{1}(0)}{(1-q^{2}/M_{B_{sA}^{\ast}}^{2})(1-q^{2}/M_{B_{sA}^{\ast}}^{\prime 2})}$ (72) $\displaystyle A_{2}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{\tilde{A}_{2}(0)}{(1-q^{2}/M_{B_{sA}^{\ast}}^{2})(1-q^{2}/M_{B_{sA}^{\ast}}^{\prime 2})}$ $\displaystyle-\frac{2M_{D_{s}^{\ast-}}}{M_{B_{c}^{-}}-M_{D_{s}^{\ast-}}}\frac{A_{0}(0)}{(1-q^{2}/M_{B_{s}}^{2})(1-q^{2}/M_{B_{s}}^{\prime 2})}$ The behavior of form factors $V(q^{2}),$ $A_{1}(q^{2})$ and $A_{2}(q^{2})$ are shown in Fig. 1. (a)(b)(c) | | ---|---|--- Figure 1: Form factors are plotted as a function of $q^{2}$. Solid line, dashed line and long-dashed line correspond to $g_{+}(0)$ equal to 0.42, 0.32 and 0.22 respectively. ## IV Physical Observables for $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ In this section we will present the calculations of the physical observables like the decay rates and the helicity fractions of $D_{s}^{\ast}$ meson. From Eq. (5) it is straightforward to write $\displaystyle\mathcal{M}_{B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}}$ $\displaystyle=$ $\displaystyle-\frac{G_{F}\alpha}{2\sqrt{2}\pi}V_{tb}V_{ts}^{\ast}\left[T_{\mu}^{1}(\bar{l}\gamma^{\mu}l)+T_{\mu}^{2}\left(\bar{l}\gamma^{\mu}\gamma^{5}l\right)\right]$ (74) where $\displaystyle T_{\mu}^{1}$ $\displaystyle=$ $\displaystyle f_{1}(q^{2})\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}+if_{2}(q^{2})\varepsilon_{\mu}^{\ast}+if_{3}(q^{2})(\varepsilon^{\ast}\cdot q)P_{\mu}$ (75) $\displaystyle T_{\mu}^{2}$ $\displaystyle=$ $\displaystyle f_{4}(q^{2})\epsilon_{\mu\nu\alpha\beta}\varepsilon^{\ast\nu}p^{\alpha}k^{\beta}+if_{5}(q^{2})\varepsilon_{\mu}^{\ast}+if_{6}(q^{2})(\varepsilon^{\ast}\cdot q)P_{\mu}$ (76) The functions $f_{1}$ to $f_{6}$ in Eq.(75) and Eq. (76) are known as auxiliary functions, which contains both long distance (Form factors) and short distance (Wilson coefficients) effects and these can be written as $\displaystyle f_{1}(q^{2})$ $\displaystyle=$ $\displaystyle 4(m_{b}+m_{s})\frac{C_{7}^{eff}}{q^{2}}F_{1}(q^{2})+C_{9}^{eff}\frac{V(q^{2})}{M_{B_{c}}+M_{D_{s}^{\ast}}}$ $\displaystyle f_{2}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{C_{7}^{eff}}{q^{2}}4(m_{b}-m_{s})F_{2}(q^{2})\left(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}\right)+C_{9}^{eff}A_{1}(q^{2})\left(M_{B_{c}}+M_{D^{\ast}}\right)$ $\displaystyle f_{3}(q^{2})$ $\displaystyle=$ $\displaystyle-\left[C_{7}^{eff}4(m_{b}-m_{s})\left(F_{2}(q^{2})+q^{2}\frac{F_{3}(q^{2})}{\left(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}\right)}\right)+C_{9}^{ff}\frac{A_{2}(q^{2})}{M_{B_{c}}+M_{D_{s}^{\ast}}}\right]$ $\displaystyle f_{4}(q^{2})$ $\displaystyle=$ $\displaystyle C_{10}\frac{V(q^{2})}{M_{B_{c}}+M_{D_{s}^{\ast}}}$ $\displaystyle f_{5}(q^{2})$ $\displaystyle=$ $\displaystyle C_{10}A_{1}(q^{2})\left(M_{B_{c}}+M_{D_{s}^{\ast}}\right)$ $\displaystyle f_{6}(q^{2})$ $\displaystyle=$ $\displaystyle- C_{10}\frac{A_{2}(q^{2})}{M_{B_{c}}+M_{D_{s}^{\ast}}}$ $\displaystyle f_{0}(q^{2})$ $\displaystyle=$ $\displaystyle C_{10}A_{0}(q^{2})$ (77) The next task is to calculate the decay rate and the helicity fractions of $D_{s}^{\ast}$ meson in terms of these auxiliary functions. ### IV.1 The Differential Decay Rate of $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ In the rest frame of $B_{c}$ meson the differential decay width of $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ can be written as $\displaystyle\frac{d\Gamma(B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-})}{dq^{2}}$ $\displaystyle=$ $\displaystyle\frac{1}{\left(2\pi\right)^{3}}\frac{1}{32M_{B_{c}}^{3}}\int_{-u(q^{2})}^{+u(q^{2})}du\left|\mathcal{M}_{B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}}\right|^{2}$ (78) where $\displaystyle q^{2}$ $\displaystyle=$ $\displaystyle(p_{l^{+}}+p_{l^{-}})^{2}$ (79) $\displaystyle u$ $\displaystyle=$ $\displaystyle\left(p-p_{l^{-}}\right)^{2}-\left(p-p_{l^{+}}\right)^{2}$ (80) Now the limits on $q^{2}$ and $u$ are $\displaystyle 4m_{l}^{2}$ $\displaystyle\leq$ $\displaystyle q^{2}\leq(M_{B_{c}}-M_{D_{s}^{\ast}})^{2}$ (81) $\displaystyle-u(q^{2})$ $\displaystyle\leq$ $\displaystyle u\leq u(q^{2})$ (82) with $u(q^{2})=\sqrt{\lambda\left(1-\frac{4m_{l}^{2}}{q^{2}}\right)}$ (83) where $\lambda\equiv\lambda(M_{B_{c}}^{2},M_{D_{s}^{\ast}}^{2},q^{2})=M_{B_{c}}^{4}+M_{D_{s}^{\ast}}^{4}+q^{4}-2M_{B_{c}}^{2}M_{D_{s}^{\ast}}^{2}-2M_{D_{s}^{\ast}}^{2}q^{2}-2q^{2}M_{B_{c}}^{2}$ The decay rate of $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ can easily obtained in terms of auxiliary function by integrating on $u$ (c.f. Eq. (78)) as $\displaystyle\frac{d\Gamma(B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-})}{dq^{2}}$ $\displaystyle=$ $\displaystyle\frac{G_{F}^{2}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\alpha^{2}}{2^{11}\pi^{5}3M_{B_{c}}^{3}M_{D_{s}^{\ast}}^{2}q^{2}}u(q^{2})\bigg{[}24\left|f_{0}(q^{2})\right|^{2}m_{l}^{2}M_{D_{s}^{\ast}}^{2}\lambda$ $\displaystyle+8M_{D_{s}^{\ast}}^{2}q^{2}\lambda[(2m_{l}^{2}+q^{2})\left|f_{1}(q^{2})\right|^{2}-(4m_{l}^{2}-q^{2})\left|f_{4}(q^{2})\right|^{2}]$ $\displaystyle+\lambda[(2m_{l}^{2}+q^{2})\left|f_{2}(q^{2})+(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}-q^{2})f_{3}(q^{2})\right|^{2}$ $\displaystyle-(4m_{l}^{2}-q^{2})\left|f_{5}(q^{2})+(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}-q^{2})f_{6}(q^{2})\right|^{2}]$ $\displaystyle+4M_{D_{s}^{\ast}}^{2}q^{2}[(2m_{l}^{2}+q^{2})\left(3\left|f_{2}(q^{2})\right|^{2}-\lambda\left|f_{3}(q^{2})\right|^{2}\right)$ $\displaystyle-(4m_{l}^{2}-q^{2})\left(3\left|f_{5}(q^{2})\right|^{2}-\lambda\left|f_{6}(q^{2})\right|^{2}\right)]\bigg{]}$ ### IV.2 HELICITY FRACTIONS OF $D_{s}^{\ast}$ IN $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ We now discuss helicity fractions of $D_{s}^{\ast}$ in $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ which are intersting variable and are as such independent of the uncertainities arising due to form factors and other input parameters. The final state meson helicity fractions were already discussed in literature for $B\rightarrow K^{\ast}\left(K_{1}\right)l^{+}l^{-}$ decays 22 ; 23 . Even for the $K^{\ast}$ vector meson, the longitudinal helicity fraction $f_{L}$ has been measured by Babar collaboration for the decay $B\rightarrow K^{\ast}l^{+}l^{-}(l=e,\mu)$ in two bins of momentum transfer and the results are 46 $\displaystyle f_{L}$ $\displaystyle=$ $\displaystyle 0.77_{-0.30}^{+0.63}\pm 0.07,\ \ \ \ \ 0.1\leq q^{2}\leq 8.41GeV^{2}$ $\displaystyle f_{L}$ $\displaystyle=$ $\displaystyle 0.51_{-0.25}^{+0.22}\pm 0.08,\ \ \ \ \ q^{2}\geq 10.24GeV^{2}$ while the average value of $f_{L}$ in full $q^{2}$ range is $f_{L}=0.63_{-0.19}^{+0.18}\pm 0.05,\ \ q^{2}\geq 0.1GeV^{2}$ (86) The explicit expression of the helicity fractions for $B_{c}^{-}\rightarrow D_{s}^{\ast-}l^{+}l^{-}$ decay can be written as $\displaystyle\frac{d\Gamma_{L}(q^{2})}{dq^{2}}$ $\displaystyle=$ $\displaystyle\frac{G_{F}^{2}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\alpha^{2}}{2^{11}\pi^{5}}\frac{u(q^{2})}{M_{B_{c}}^{3}}\times$ (89) $\displaystyle\frac{1}{3}\frac{1}{q^{2}M_{D_{s}^{\ast}}^{2}}\left[\begin{array}[]{c}24\left|f_{0}(q^{2})\right|^{2}m_{l}^{2}M_{D_{s}^{\ast}}^{2}\lambda+(2m_{l}^{2}+q^{2})\left|\left(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}-q^{2}\right)f_{2}(q^{2})+\lambda f_{3}(q^{2})\right|^{2}\\\ +\left(q^{2}-4m_{l}^{2}\right)\left|\left(M_{B_{c}}^{2}-M_{D_{s}^{\ast}}^{2}-q^{2}\right)f_{5}(q^{2})+\lambda f_{6}(q^{2})\right|^{2}\end{array}\right]$ $\displaystyle\frac{d\Gamma_{+}(q^{2})}{dq^{2}}$ $\displaystyle=$ $\displaystyle\frac{G_{F}^{2}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\alpha^{2}}{2^{11}\pi^{5}}\frac{u(q^{2})}{M_{B_{c}}^{3}}\times$ (91) $\displaystyle\frac{4}{3}\left[\left(q^{2}-4m_{l}^{2}\right)\left|f_{5}(q^{2})-\sqrt{\lambda}f_{4}(q^{2})\right|^{2}+\left(q^{2}+2m_{l}^{2}\right)\left|f_{2}(q^{2})-\sqrt{\lambda}f_{1}(q^{2})\right|^{2}\right]$ $\displaystyle\frac{d\Gamma_{-}(q^{2})}{dq^{2}}$ $\displaystyle=$ $\displaystyle\frac{G_{F}^{2}\left|V_{tb}V_{ts}^{\ast}\right|^{2}\alpha^{2}}{2^{11}\pi^{5}}\frac{u(q^{2})}{M_{B_{c}}^{3}}\times$ (92) $\displaystyle\frac{4}{3}\left[\left(q^{2}-4m_{l}^{2}\right)\left|f_{5}(q^{2})+\sqrt{\lambda}f_{4}(q^{2})\right|^{2}+\left(q^{2}+2m_{l}^{2}\right)\left|f_{2}(q^{2})+\sqrt{\lambda}f_{1}(q^{2})\right|^{2}\right]$ where the auxiliary functions and the corresponding form factors are given in Eq.(77) and Eqs.(71-LABEL:44). Finally the longitudinal and transverse helicity amplitude becomes $\displaystyle f_{L}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{d\Gamma_{L}(q^{2})/dq^{2}}{d\Gamma(q^{2})/dq^{2}}$ $\displaystyle f_{\pm}(q^{2})$ $\displaystyle=$ $\displaystyle\frac{d\Gamma_{\pm}(q^{2})/dq^{2}}{d\Gamma(q^{2})/dq^{2}}$ $\displaystyle f_{T}(q^{2})$ $\displaystyle=$ $\displaystyle f_{+}(q^{2})+f_{-}(q^{2})$ (93) so that the sum of the longitudinal and transverse helicity amplitudes is equal to one i.e. $f_{L}(q^{2})+f_{T}(q^{2})=1$ for each value of $q^{2}$22 . ## V Numerical Analysis. In this section we present the numerical analysis of the branching ratio and helicity fractions of $D_{s}^{\ast}$ meson in $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}(l=\mu,\tau)$ both in the SM and in ACD model. One of the main input parameters are the form factors which are non perturbative quantities and are the major source of uncertainties. Here we calculated the form factors using the Ward identities and their dependence on momentum transfer $q^{2}$ is given in Section III. We have used next-to-leading order approximation for the Wilson Coefficients at the renormalization scale $\mu=m_{b}.$ It has already been mentioned that besides the contribution in the $C_{9}^{eff}$, there are long distance contributions resulting from the $c\bar{c}$ resonances like $J/\psi$ and its excited states. For the present analysis we do not take into account these long distance effects. The numerical results for the decay rates and helicity fractions of $D_{s}^{\ast}$ for the decay mode $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ both for the SM and ACD model are depicted in Figs. 2-4. Figs. 2 (a, b) shows the differential decay rate of $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}(l=\mu,\tau).$ One can see that there is a significant enhancement in the decay rate due to KK-contribution for $1/R=300$ GeV, whereas the value of the decay rate is shifted towards the SM at large value of $1/R$ , both in small and large value of momentum transfer $q^{2}.$ (a)(b) | | ---|---|--- Figure 2: Branching ratio for the $B\rightarrow D_{s}^{\ast}l^{+}l^{-}$ $(l=\mu,\tau)$ decays as functions of $q^{2}$ for different values of $1/R$. Solid line correspond to SM value,dotted line is for $1/R=300$, dashed is for $1/R=500$, long dashed line is for $1/R=700$. In general the sensitivity on $1/R$ is usually masked by the uncertainties which arises due to the number of sources. Among them the major one lies in the numerical analysis of $B_{c}\to D^{*}_{s}l^{+}l^{-}$ decay originated from the $B_{c}\to D^{*}_{s}$ transition form factors calculated in the present approach as shown in Table I, which can bring about almost $40\%$ errors to the differential decay rate of above mentioned decay, which showed that it is not a very suitable tool to look for the new physics. The large uncertainties involved in the form factors are mainly from the variations of the decay constant of $B_{c}$ meson and also there are some uncertainties from the strange quark mass $m_{s}$, which are expected to be very tiny on account of the negligible role of $m_{s}$ suppressed by the much larger energy scale of $m_{b}$. Moreover, the uncertainties of the charm quark and bottom quark mass are at the $1\%$ level, which will not play significant role in the numerical analysis and can be dropped out safely. It also needs to be stressed that these hadronic uncertainties almost have no influence on the various asymmetries including the polarization asymmetries of final state meson on account of the serious cancelation among different polarization states and this make them one of the best tool to look for physics beyond the SM. Figs. 3 (a, b) shows the longitudinal and transverse helicity fractions of $D_{s}^{\ast}$ for the decay $B_{c}\rightarrow D_{s}^{\ast}\mu^{+}\mu^{-}$ where we have used the central value of the form factors which we have calculated in Section III. Choosing the different values of compactification radius $1/R$, one can see from the graphs that the effect of extra dimensions are quite significant at a particular region of $q^{2}$. These effects are constructive for the case of transverse helicity fraction and destructive for the case of longitudinal helicity fraction. (a)(b) | | ---|---|--- Figure 3: Longitudinal Lepton polarization Fig.1a and Transverse Lepton polarization Fig.2b for the $B\rightarrow D_{s}^{\ast}\mu^{+}\mu^{-}$ decays as functions of $q^{2}$ for different values of $1/R$. Solid line correspond to SM value,dotted line is for $1/R=300$, dashed is for $1/R=500$, long dashed line is for $1/R=700$. Similarly, Figs. 4 (a,b) show the helicity fraction of $D_{s}^{\ast}$ for the decay $B_{c}\rightarrow D_{s}^{\ast}\tau^{+}\tau^{-}$ where one can see that the effects of the extra dimensions are mild as compared to the case of $B_{c}\rightarrow D_{s}^{\ast}\mu^{+}\mu^{-}$ . Moreover from Figs.2-4 it is clear that each value of momentum transfer $q^{2}$ the sum of the longitudinal and transverse helicity fractions are equal to one, i.e. $f_{L}(q^{2})+f_{T}(q^{2})=1$. (a)(b) | | ---|---|--- Figure 4: Longitudinal Lepton polarization Fig.1a and Transverse Lepton polarization Fig.2b for the $B\rightarrow D_{s}^{\ast}\tau^{+}\tau^{-}$ decays as functions of $q^{2}$ for different values of $1/R$. Solid line correspond to SM value,dotted line is for $1/R=300$, dashed is for $1/R=500$, long dashed line is for $1/R=700$. ## VI Conclusion: We investigated the semileptonic decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ $(l=\mu,\tau)$ using the Ward identities. The form factors have been calculated and we found that the normalization of the form factors in terms of a single universal constant $g_{+}(0)$. The value of $g_{+}(0)=0.42$ is obtained from the decay $B_{c}\rightarrow D_{s}^{\ast}\gamma$ 41 . Considering the radial excitation at lower pole masses $M$ ( where $M=M_{B_{s}^{\ast}}$ and $M_{B_{sA}^{\ast}})$ one can predict the coupling of $B_{s}^{\ast}$ with $B_{c}D_{s}^{\ast}$ channel as indicated in Eq.(65) which is $g_{B_{s}^{\ast}B_{c}D_{s}^{\ast}}=10.38$ GeV${}^{-1}.$ Also we predicted the ratio of $S$ and $D$ wave couplings $\frac{g_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}{f_{B_{sA}^{\ast}B_{c}D_{s}^{\ast}}}=-0.42$ $GeV^{2}$ given in Eq.(66). The form factors are summarized in Eqs.(71-LABEL:44) and their values at $q^{2}=0$ are given in Tabel-I. Using these form factors we studied the observables, i.e. the branching ratio and helicity fraction of $D_{s}^{\ast}$ in the decay $B_{c}\rightarrow D_{s}^{\ast}l^{+}l^{-}$ $(l=e,\mu)$ both in SM and in ACD model, which has one additional parameter i.e. the inverse compactification radius $1/R.$ The effects of extra dimensions to the helicity fraction of $D_{s}^{\ast}$ is very mild for the case when the tauon $(\tau)$ is taken as a final state lepton as shown in fig 3, however the effects of extra dimensions are quite significant for the case when muon ($\mu$) is taken as a final state lepton as shown in fig 2. In near future when LHC is fully operational where more data is available, will put a stringent constraint on compactification radius $R$ and gives us a deep understanding of $B$ Physics. ## Acknowledgements The authors would like to thank Profs. Riazuddin and Fayyazuddin for their valuable guidance and helpful discussions. The authors M. A. P. and M. J. 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arxiv-papers
2011-01-12T11:04:36
2024-09-04T02:49:16.347870
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. Ali Paracha, Ishtiaq Ahmed, M. Jamil Aslam", "submitter": "Muhammad Jamil Aslam", "url": "https://arxiv.org/abs/1101.2323" }
1101.2326
# Nucleon form factors and structure functions from $N_{\mathrm{f}}=2$ clover fermions QCDSF/UKQCD Collaboration: S. Collins,a M. Göckeler,a Ph. Hägler,a T. Hemmert,a R. Horsley,b Y. Nakamura,a,c A. Nobile,a H. Perlt,d ,e P.E.L. Rakow,f A. Schäfer,a G. Schierholz,e A. Sternbeck,a H. Stüben,g F. Wintera and J.M. Zanottib a Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany b School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK c Center for Computational Sciences, University of Tsukuba, Ibaraki 305-8577, Japan d Institut für Theoretische Physik, Universität Leipzig, 04109 Leipzig, Germany e Deutsches Elektronen-Synchrotron DESY, 15738 Zeuthen, Germany f Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UK g Konrad-Zuse-Zentrum für Informationstechnik Berlin, 14195 Berlin, Germany Email ###### Abstract: We give an update on our ongoing efforts to compute the nucleon’s form factors and moments of structure functions using $N_{\mathrm{f}}=2$ flavours of non- perturbatively improved Clover fermions. We focus on new results obtained on gauge configurations where the pseudo-scalar meson mass is in the range of 170-270 MeV. We will compare our results with various estimates obtained from chiral effective theories since we have some overlap with the quark mass region where results from such theories are believed to be applicable. ## 1 Introduction Over years significant efforts have been made to use lattice techniques to investigate the structure of the nucleon. Of particular interest are the Parton Distribution Functions (PDFs) and form factors. The latter encode information about charge distribution and magnetization while the PDFs tell us about the distribution of momentum and spin. While some of the related observables can be determined with good accuracy by experiments (e.g. the nucleon’s axial charge $g_{\mathrm{A}}$) other quantities are difficult to access (like the tensor charge $g_{\mathrm{T}}$). A precise determination of moments of nucleon PDFs and form factors on the lattice turned out to be rather challenging. It continues to be difficult to reach sufficient control on all systematic errors such as finite size effects, lattice artefacts and the influence of the chiral extrapolation. Simulations are performed in volumes of a size where some quantities exhibit significant finite size effects. The available lattice data of the quantities of interest show no significant discretization effects. But current simulations only probe a small window of lattice spacings thus providing us with limited control on the continuum extrapolation. From chiral effective theories (ChEFT) there are indications that the quark mass dependence close to the physical pion mass is very strong. Therefore extending lattice simulations into the region where $m_{\pi}\leq m_{\mathrm{PS}}\lesssim 300~{}\mbox{MeV}$ has become a major goal for recent calculations. ## 2 Simulation details For our simulations we use Wilson glue and $N_{\mathrm{f}}=2$ degenerate flavours of Clover fermions, where the improvement coefficient $c_{\mathrm{SW}}$ has been determined non-perturbatively. Most of our configurations have been generated using the BQCD implementation of the HMC algorithm [1]. Various algorithmic improvements have been applied which accelerate this algorithm, such as the Hasenbusch preconditioning and the use of multiple time-scales, or reducing the time spent for matrix inversion. For instance, chronological guess and the Schwarz Alternating Procedure (SAP) are used to start the inversion and to precondition the fermion matrix [2]. These algorithmic improvements plus recent increase in computing resources enabled simulations in the region of small quark masses, i.e. in a region where the pseudo-scalar mass is smaller than 300 MeV. Fig. 1a shows the parameter region of our simulations. When approaching physical quark masses larger lattices are needed to stay in the region $m_{\mathrm{PS}}L\gtrsim 3$ where finite size effects are expected to be sufficiently small (see Fig. 1b). To investigate such finite size effects we have also performed simulations with $m_{\mathrm{PS}}L<3$. (a) (b) Figure 1: The left panel shows the simulation points in the $m_{\mathrm{PS}}^{2}$ vs. $m_{\mathrm{PS}}L$ plane. In the right panel dashed lines show the lattice spacing and box sizes for which simulations have been performed. In both figures the continuous lines show where $m_{\mathrm{PS}}L=3$. We compute the quark propagators using point sources which we (Jacobi) smear to improve overlap with the ground state. For the three-point correlation functions we apply standard sequential source techniques. The distance between source and sink is about 1 fm. Throughout this paper we will ignore contributions coming from disconnected terms. While these anyhow cancel in the iso-vector cases, results for the iso-scalar case maybe affected by an uncontrolled systematic error. To set the scale we use the Sommer parameter $r_{0}/a$ which we extrapolated to the chiral limit at each beta. While on the lattice this quantity can be determined with small statistical errors, there is no experimental determination. We therefore computed the dimensionless quantity $(am_{N})(r_{0}/a)$ on the lattice and use the experimentally well known mass of the nucleon $m_{N}$ to obtain $r_{0}=0.467\,\mbox{fm}$. Most of the quantities considered in this paper need to be renormalised. The renormalisation constants have been determined using the $\mathrm{RI}^{\prime}\mathrm{-MOM}$ scheme [3], except for the vector current renormalisation constant $Z_{V}$. Here we applied the condition that the nucleon’s local vector current at zero momentum must be 1. If necessary, the results are converted into $\overline{\mathrm{MS}}$ scheme using the 4- and 2-3-loop expressions of the $\beta$ function and corresponding anomalous dimension $\gamma$, respectively. ## 3 Lowest moments of PDFs Let us first consider the lowest moment of the polarized nucleon PDF $\langle 1\rangle_{\Delta q}$ (also known as axial coupling constant $g_{\mathrm{A}}$). This quantity is determined from the renormalised axial vector current $A_{\mu}^{R}=Z_{A}\,(1+b_{A}\,am_{q})A_{\mu}$, where $am_{q}=(1/\kappa-1/\kappa^{(S)}_{c})/2$. $Z_{A}$ is known non-perturbatively [3], for $b_{A}$ we use a tadpole improved one-loop perturbation theory result. (a) (b) Figure 2: The left panel shows $g_{\mathrm{A}}$ as a function of $m_{\mathrm{PS}}^{2}$. The open and filled diamonds show the lattice results before and after correction of finite size effects, respectively. The star indicates the experimental result. The line shows a fit to the data as described in the text. The right panel shows the relative finite size effects determined on the lattice (symbols) and obtained from a fit to an expression from ChEFT. We have fitted our lattice results to an expression from ChEFT based on the SSE formalism. Using this formalism both the quark mass dependence [4] and the finite volume dependence [5] have been calculated. Since our results for different lattice spacings do not exhibit clear discretization effects we combine all our results where $m_{\mathrm{PS}}\leq 450\,\mbox{MeV}$. The fit range has been chosen such that stable fits are obtained. Our data is not sufficiently precise to determine all parameters. We therefore fix a few parameters to their phenomenological value and keep only $g_{\mathrm{A}}$ in the chiral limit, the leading $\Delta\Delta$-coupling $g_{1}$ and the SSE coupling term $B_{9}^{r}(\lambda)$ as free fit parameters. The resulting fit and the lattice data are shown in Fig. 2a. In our fit we only included results for the largest lattice at a given set of bare parameters. For some data sets we have results for different volumes. We thus can compute the relative shift $\delta_{g_{\mathrm{A}}}(L)=\frac{g_{\mathrm{A}}(L)-g_{\mathrm{A}}(\infty)}{g_{\mathrm{A}}(\infty)}$ (1) both from the fit as well as from the lattice data taking the results on the largest lattice as approximation of $g_{\mathrm{A}}(\infty)$. In Fig. 2b we compare the relative shift for different values of the quark mass with our lattice results at $m_{\mathrm{PS}}\simeq 270\,\mbox{MeV}$. The shift predicted from ChEFT only slightly underestimates the relative shift computed on the lattice. Also after correcting for finite size effects we observe a significant difference to the experimental value. It is interesting to notice that a much better agreement with the experimental value is observed for the ratio $g_{\mathrm{A}}/f_{\mathrm{PS}}$ (see Fig. 3a). In this ratio the renormalization constant $Z_{\mathrm{A}}$ drops out. In Fig. 3b we show our results for the nucleon tensor charge $\langle 1\rangle_{\delta q}=g_{\mathrm{T}}$. We observe only a very mild quark mass dependence and the data reveals no systematic discretization effects. This quantity is not well known experimentally. Our values are larger than the phenomenological results presented in [6]. (a) (b) Figure 3: The left panel shows $g_{\mathrm{A}}/f_{\mathrm{PS}}$ as a function of $m_{\mathrm{PS}}^{2}$. The right panel shows our results for $g_{\mathrm{T}}^{\overline{\mathrm{MS}}}$ at a scale $\mu=2\,\mbox{GeV}$. ## 4 $n=2$ moments of PDFs The lowest moment of the unpolarized PDF $\langle x\rangle_{q}=v_{2}$ corresponds to the momentum fraction carried by the quarks in the nucleon. Lattice results from different collaborations tend to be significantly larger than the phenomenological value. Fig. 4a and 4b show our most recent results for the iso-vector and iso-scalar channel. In the latter case disconnected contributions have been ignored. Also shown are the results from a fit to results utilizing methods of covariant Baryon Chiral Perturbation Theory (BChPT) [7]. Fits have been performed with most parameters fixed to phenomenologically known values. The iso-vector (iso-scalar) channel data is fitted with only 2 free parameters: $v_{2}$ in the chiral limit and the coupling $c_{8}$ ($c_{9}$). Near the physical light quark masses, BChPT predicts $v_{2}$ to become larger when the quark mass becomes heavier. In our data for $m_{\mathrm{PS}}\lesssim 250\,\mbox{MeV}$ we do not see any indication for a bending down when approaching the physical pion mass. It thus does not seem that a lack of results at sufficiently small quark masses could explain the large discrepancy between the phenomenological value and the lattice results. There are some indications that part of the discrepancy can be explained by excited state contamination [8]. (a) (b) Figure 4: The left and right panel show results for the second moment of the iso-vector and iso-scalar unpolarized PDFs, respectively, as a function of $m_{\mathrm{PS}}^{2}$. The solid lines show the fits to an expression from ChEFT. In Fig. 5a the results for the second moment of the polarized PDF $\langle x\rangle_{\Delta q}=a_{1}$ is shown. Discretization effects again seem to be absent in data. From a comparison of the results for different volumes it seems that also finite size effects are small. Results from Heavy Baryon Chiral Perturbation Theory (HBChPT) [9] lead to the following expression: $a_{1}^{(u-d)}(m_{\mathrm{PS}})=C\left[1-\frac{4g_{\mathrm{A}}^{2}+1}{2(4\pi f_{\mathrm{PS}})^{2}}m_{\mathrm{PS}}^{2}\ln\left(\frac{m_{\mathrm{PS}}^{2}}{\mu^{2}}\right)\right]+\cdots$ (2) In Fig. 5a we plot this expression using $\mu=m_{\mathrm{N}}$ and $C$ chosen such that it matches the phenomenological value. The bending down which we observe in our data for $m_{\mathrm{PS}}\lesssim 0.5\,\mbox{MeV}$ is much less than one would expect from this HBChPT result. ## 5 Electromagnetic form factors (a) (b) Figure 5: The left panel shows the results for the second of the polarized PDFs as a function of $m_{\mathrm{PS}}^{2}$. In the right panel the results for the Dirac form factor radius $r_{1}$ are plotted. The dashed lines show results from ChEFT as described in the text. To compute the electromagnetic form factors one makes use of the standard decomposition of the nucleon electromagnetic matrix elements $\langle p^{\prime},s^{\prime}|V_{\mu}|p,s\rangle=\overline{u}\left[\gamma_{\mu}F_{1}(Q^{2})+\frac{\sigma_{\mu\nu}\,q_{\nu}}{2m_{\mathrm{N}}}F_{2}(Q^{2})\right]u$, (in Euclidian space) where we use the local vector current $V_{\mu}$. $p$ ($s$) and $p^{\prime}$ ($s^{\prime}$) denote initial and final momenta (spins), $q=p^{\prime}-p$ the momentum transfer (with $Q^{2}=-q^{2}$). To calculate form factor radii and the anomalous magnetic we have to parametrize the lattice results. Here we use the ansatz $F_{i}(Q^{2})=\frac{F_{i}(0)}{\left[1+\frac{Q^{2}}{pm_{i}^{2}}\right]^{p}}$ (3) with $p=2$ and $p=3$ for the Dirac and Pauli form factors $F_{1}$ and $F_{2}$, respectively. Our data is not sufficiently precise to favour a particular parametrization (see [10] for another parametrization). From fits to Eq. (3) we determine the form factor radii $r_{1}$ and $r_{2}$ as well as the anomalous magnetic moment $\kappa$. The quark mass dependence of these quantities has been calculated using the SSE formalism [11]. For $r_{1}$ the parameters are known and we therefore restrict ourselves to a comparison of the SSE result and the lattice data (see Fig. 5b). While for $m_{\mathrm{PS}}\gtrsim 300\,\mbox{MeV}$ the lattice results are significantly smaller than the phenomenological value, towards smaller quark masses we observe an increase of the radius. This is consistent with predictions from ChEFT. For $r_{2}$ and $\kappa$ we find a similar behaviour. Since there are no phenomenological estimates for all parameters of the SSE expressions we perform a combined fit. The results are plotted in 6a and 6b. (a) (b) Figure 6: The left panel and right panel shows the results for the Pauli radius $r_{2}$ and the anomalous magnetic moment $\kappa$. The solid lines show fits to an expression from ChEFT. ## 6 Summary and outlook We have presented an update of QCDSF results on the lowest moments of unpolarized, polarized and tensor PDFs as well as the electromagnetic form factors. Some of our results at light quark masses with $m_{\mathrm{PS}}\lesssim 300\,\mbox{MeV}$ confirm the expectations from ChEFT that light quark mass effects are significant. However, this possibly does not explain all of the observed discrepancies from phenomenological values. ## Acknowledgements The numerical calculations have been performed on the APEmille and apeNEXT systems at NIC/DESY (Zeuthen), the BlueGene/P at NIC/JSC (Jülich), the BlueGene/L at EPCC (Edinburgh), the Dell PC-cluster at DESY (Zeuthen), the QPACE systems [12] of the SFB TR-55, the SGI Altix and ICE systems at LRZ (Munich) and HLRN (Berlin/Hannover). This work was supported in part by the DFG (SFB TR-55) and by the European Union (grants 238353, ITN STRONGnet and 227431, HadronPhysics2, and 256594). ## References * [1] Y. Nakamura and H. Stüben, PoS(LATTICE 2010)040. * [2] A. Nobile, PoS(LATTICE 2010)034. * [3] M. Göckeler et al. [QCDSF Collaboration], arXiv:1003.5756 [hep-lat]. * [4] V. Bernard et al., Nucl. Phys. A635, 121 (1998); A642, 563(E) (1998). T.R. Hemmert, M. Procura, and W. Weise, Phys. Rev. D 68, 075009 (2003). * [5] A. Ali Khan et al. [QCDSF Collaboration], Phys. Rev. D 74 (2006) 094508. * [6] M. Anselmino et al., Nucl. Phys. Proc. Suppl. 191 (2009) 98-107. * [7] M. Dorati, T. A. Gail and T. R. Hemmert, Nucl. Phys. A 798 (2008) 96. * [8] M. Göckeler et al., in preparation. * [9] D. Arndt and M.J. Savage, Nucl. Phys. A697, 429 (2002); J.W. Chen and X. Ji, Phys. Lett. B 523, 107 (2001); W. Detmold, W. Melnitchouk and A. W. Thomas, Phys. Rev. D 66 (2002) 054501. * [10] M. Göckeler et al. [QCDSF Collaboration], PoS(LATTICE 2007)161. * [11] T. R. Hemmert and W. Weise, Eur. Phys. J. A 15 (2002) 487 [arXiv:hep-lat/0204005]; M. Göckeler et al. [QCDSF Collaboration], Phys. Rev. D 71 (2005) 034508 [arXiv:hep-lat/0303019]. * [12] H. Baier et al., PoS(LATTICE 2009)001.
arxiv-papers
2011-01-12T11:19:52
2024-09-04T02:49:16.356674
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. Collins, M. G\\\"ockeler, Ph. H\\\"agler, T. Hemmert, R. Horsley, Y.\n Nakamura, A. Nobile, H. Perlt, D. Pleiter, P.E.L. Rakow, A. Sch\\\"afer, G.\n Schierholz, A. Sternbeck, H. St\\\"uben, F. Winter, J.M. Zanotti", "submitter": "Dirk Pleiter", "url": "https://arxiv.org/abs/1101.2326" }
1101.2342
# A contribution to the condition number of the total least squares problem Zhongxiao Jia 1, Bingyu Li 2,111Corresponding author. 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China jiazx@tsinghua.edu.cn (Z. Jia) mathliby@gmail.com (B. Li) ###### Abstract This paper concerns cheaply computable formulas and bounds for the condition number of the TLS problem. For a TLS problem with data $A$, $b$, two formulas are derived that are simpler and more compact than the known results in the literature. One is derived by exploiting the properties of Kronecker products of matrices. The other is obtained by making use of the singular value decomposition (SVD) of $[A\,\,b]$, which allows us to compute the condition number cheaply and accurately. We present lower and upper bounds for the condition number that involve the singular values of $[A\,\,b]$ and the last entries of the right singular vectors of $[A\,\,b]$. We prove that they are always sharp and can estimate the condition number accurately by no more than four times. Furthermore, we establish a few other lower and upper bounds that involve only a few singular values of $A$ and $[A\,\,b]$. We discuss how tight the bounds are. These bounds are particularly useful for large scale TLS problems since for them any formulas and bounds for the condition number involving all the singular values of $A$ and/or $[A\ b]$ are too costly to be computed. Numerical experiments illustrate that our bounds are sharper than a known approximate condition number in the literature. Keywords: total least squares, condition number, singular value decomposition. AMS subject classification (2000): 65F35. ## 1 Introduction For given $A\in\mathbb{R}^{m\times n}(m>n)$, $b\in\mathbb{R}^{m}$, the TLS problem can be formulated as (see, e.g., [5, 12]) $\min\|[E\,\,r]\|_{F},\text{\quad subject to \quad}b-r\in\mathcal{R}(A+E),$ (1) where $\|\cdot\|_{F}$ denotes the Frobenius norm of a matrix and $\mathcal{R}(\cdot)$ denotes the range space. Suppose that $[E_{TLS}\,\,r_{TLS}]$ solves the above problem. Then $x=x_{TLS}$ that satisfies the equation $(A+E_{TLS})x=b-r_{TLS}$ is called the TLS solution of (1). The condition number measures the worst-case sensitivity of a solution of a problem to small perturbations in the input data. Combined with backward errors, it provides an approximate local linear upper bound on the computed solution. Since the 1980’s, algebraic perturbation analysis for the TLS problem has been studied extensively; see [3, 5, 9, 15, 16] and the references therein. In recent years, asymptotic perturbation analysis and TLS condition numbers have been studied; see, e.g., [1, 8, 18]. In the present paper, we continue our work in [8] that studied the condition number of the TLS problem. We will derive a number of results. Firstly, we establish two formulas that are simpler and more suitable for computational purpose than the known results in the literature. One is derived by exploiting the properties of Kronecker products of matrices. It improves the formulas given in [8, 18], is independent of Kronecker products of matrices and makes its computation convenient. The other is obtained by making use of the SVD of $[A\,\,b]$, which can be used to compute the condition number more cheaply and accurately than that in [1]. Secondly, we present lower and upper bounds for the condition number that involve the singular values of $[A\,\,b]$ and the last entries of the right singular vectors of $[A\,\,b]$. We prove that these bounds are always sharp and can estimate the condition number accurately by no more than four times. Finally, we focus on cheaply computable bounds of the TLS condition number. We establish lower and upper bounds that involve only a few singular values of $A$ and $[A\,\,b]$. We discuss how tight the bounds are. These bounds are particularly useful for large scale TLS problems since for them any formulas and bounds for the condition number involving all the singular values of $A$ and/or $[A\ b]$ are too costly to be computed. So we can compute these bounds by calculating only a few singular values of $A$ and/or $[A\ b]$ using some iterative solvers for large SVDs. In [2], an approximate TLS condition number is presented and is applied to evaluate conditioning of the TLS problem there. In this paper, we present numerical experiments to demonstrate a possibly great improvement of one of our upper bounds over the approximate condition number in [2]. The paper is organized as follows. In Section 2, we give some preliminaries necessary. In Section 3, we present computable formulas of the TLS condition number. The straightforward bounds on the TLS condition number are considered in Section 4. In Section 5, we present numerical experiments to show the tightness of bounds for the TLS condition number. We end the paper with some concluding remarks in Section 6. Throughout the paper, for given positive integers $m,n$, $\mathbb{R}^{n}$ denotes the space of $n$-dimensional real column vectors, $\mathbb{R}^{m\times n}$ denotes the space of all $m\times n$ real matrices. $\|\cdot\|$ and $\|\cdot\|_{F}$ denote 2-norm and Frobenius norm of their arguments, respectively. Given a matrix $A$, $A(1:i,1:i)$ is a Matlab notation that denotes the $i$th leading principal submatrix of $A$, and $\sigma_{i}(A)$ denotes the $i$th largest singular value of $A$. For a vector $a$, $a(i)$ denotes the $i$th component of $a$, and ${\rm{diag}}(a)$ is a diagonal matrix whose diagonals are given as components of $a$. $I_{n}$ denotes the $n\times n$ identity matrix, $O_{mn}$ denotes the $m\times n$ zero matrix, whereas $O$ denotes a zero matrix whose order is clear from the context. For matrices $A=[a_{1},\ldots,a_{n}]=[A_{ij}]\in\mathbb{R}^{m\times n}$ and $B$, $A\otimes B=[A_{ij}B]$ is the Kronecker product of $A$ and $B$, the linear operator ${\rm{vec}}:\mathbb{R}^{m\times n}\rightarrow\mathbb{R}^{mn}$ is defined by ${\rm{vec}}(A)=[a^{T}_{1},\ldots,a^{T}_{n}]^{T}$ for $A\in\mathbb{R}^{m\times n}$. ## 2 Preliminaries Throughout the paper, we let $\hat{U}^{T}A\hat{V}={\rm{diag}}(\hat{\sigma}_{1},\ldots,\hat{\sigma}_{n})$ be the thin SVD of $A\in\mathbb{R}^{m\times n}$, where $\hat{\sigma}_{1}\geq\cdots\geq\hat{\sigma}_{n}$, $\hat{U}\in\mathbb{R}^{m\times n}$, $\hat{U}^{T}\hat{U}=I_{n}$, $\hat{V}\in\mathbb{R}^{n\times n}$, $\hat{V}^{T}\hat{V}=I_{n}$. Let $U^{T}[A\,\,b]V={\rm{diag}}(\sigma_{1},\ldots,\sigma_{n+1})$ be the thin SVD of $[A\,\,b]\in\mathbb{R}^{m\times(n+1)}$, where $\sigma_{1}\geq\cdots\geq\sigma_{n+1}$, $U=[u_{1},\ldots,u_{n+1}]\in\mathbb{R}^{m\times(n+1)}$, $U^{T}U=I_{n+1}$, $V=[v_{1},\ldots,v_{n+1}]\in\mathbb{R}^{(n+1)\times(n+1)}$, $V^{T}V=I_{n+1}$. The following result presents an existence and uniqueness condition for the TLS solution [5]. ###### Theorem 1 If $\sigma_{n+1}<\hat{\sigma}_{n},$ (2) then the TLS problem (1) has a unique solution $x_{TLS}$. Moreover, $\displaystyle x_{TLS}$ $\displaystyle=$ $\displaystyle(A^{T}A-\sigma^{2}_{n+1}I)^{-1}A^{T}b$ (3) $\displaystyle=$ $\displaystyle-\left[\frac{v_{n+1}(1)}{v_{n+1}(n+1)},\ldots,\frac{v_{n+1}(n)}{v_{n+1}(n+1)}\right]^{T}.$ (4) In the paper, it is always assumed that condition (2) holds. We note that, for a given TLS problem (1), if $\sigma_{n+1}=0$, then $b\in\mathcal{R}(A)$. In this case, the system of equations $Ax=b$ is compatible, and we can take $[E\,\,r]=O$. As in [8, 18], in the sequel, we do not consider this trivial case and assume that $0<\sigma_{n+1}<\hat{\sigma}_{n}.$ (5) We will use the following properties of the TLS problem, which are in [5]: $\sigma^{2}_{n+1}=\frac{\|r\|^{2}}{1+\|x\|^{2}}$ (6) and $A^{T}r=\frac{\|r\|^{2}}{1+\|x\|^{2}}x=\sigma^{2}_{n+1}x,$ (7) where $x=x_{TLS}$, $r=Ax-b$. By (4), it holds that $v_{n+1}=\frac{1}{\sqrt{1+\|x\|^{2}}}\left[\begin{array}[]{c}x\\\ -1\\\ \end{array}\right]$ (8) up to a sign $\pm 1$. The following basic properties of the Kronecker products of matrices are needed later and can be found in [6]: $\displaystyle(A\otimes C)(B\otimes D)=(AB)\otimes(CD),$ $\displaystyle(A\otimes B)^{T}=A^{T}\otimes B^{T},$ where $A,B,C,D$ are matrices with appropriate sizes. ## 3 Computable formulas for the TLS condition number Throughout the paper, we follow the definition of condition number in [4, 13]. Let $g:\mathbb{R}^{p}\longrightarrow\mathbb{R}^{q}$ be a continuous map in normed linear spaces defined on an open set $D_{g}\subset\mathbb{R}^{p}$. For a given $a\in D_{g}$, $a\neq 0$, with $g(a)\neq 0$, if $g$ is differentiable at $a$, then the relative condition number of $g$ at $a$ is $\kappa^{r}_{g}(a)=\frac{\|g^{\prime}(a)\|\|a\|}{\|g(a)\|}$ (9) and the absolute condition number of $g$ is $\kappa_{g}(a)=\|g^{\prime}(a)\|,$ (10) where $g^{\prime}(a)$ denotes the derivative of $g$ at $a$. Given the TLS problem (1), let $\tilde{A}=A+\Delta A$, $\tilde{b}=b+\Delta b$, where $\Delta A$ and $\Delta b$ denote the perturbations in $A$ and $b$, respectively. Consider the perturbed TLS problem $\min\|[E\,\,r]\|_{F}\,\,\,{\text{subject to}}\,\,\tilde{b}-r\in\mathcal{R}(\tilde{A}+E).$ (11) In [8], we have established the following result. ###### Theorem 2 Suppose the TLS problem (1) satisfies (5). Denote by $x=x_{TLS}$ the TLS solution, and define $r=Ax-b$, $G(x)=[x^{T}\,\,-1]\otimes I_{m}$. If $\|[\Delta A\,\,\Delta b]\|_{F}$ is small enough, then the perturbed problem (11) has a unique TLS solution $\tilde{x}_{TLS}$. Moreover, $\tilde{x}_{TLS}=x_{TLS}+K~{}\left[\begin{array}[]{c}{\rm{vec}}(\Delta A)\\\ \Delta b\\\ \end{array}\right]+\mathcal{O}(\|[\Delta A\,\,\Delta b]\|^{2}_{F}),$ (12) where $K=\left(A^{T}A-\sigma^{2}_{n+1}I_{n}\right)^{-1}\left(2A^{T}\frac{r}{\|r\|}\frac{r^{T}}{\|r\|}G(x)-A^{T}G(x)-[I_{n}\otimes r^{T}\,\,O]\right).$ (13) Denote $a={\rm{vec}}(A)$. Based on Theorem 2, in a small neighborhood of $[a^{T},b^{T}]^{T}\in\mathbb{R}^{m(n+1)}$, we can define the function $\begin{array}[]{ccc}g:\mathbb{R}^{m(n+1)}&\longrightarrow&\mathbb{R}^{n}\\\ \small{\left[\begin{array}[]{c}\tilde{a}\\\ \tilde{b}\\\ \end{array}\right]}&\longmapsto&\tilde{x}=(\tilde{A}^{T}\tilde{A}-\tilde{\sigma}^{2}_{n+1}I_{n})^{-1}\tilde{A}^{T}\tilde{b},\end{array}$ where $\tilde{a}=a+{\rm{vec}}(\Delta A)={\rm{vec}}(\tilde{A})$, $\tilde{b}=b+\Delta b$, and $\tilde{x}$ is the solution to the perturbed TLS problem (11). In particular, $g([a^{T},b^{T}]^{T})=x$. Thus, we have the following theorem. ###### Theorem 3 Let $\kappa_{g}(A,b)$ and $\kappa^{r}_{g}(A,b)$ be the absolute and relative condition numbers of the TLS problem, respectively. Then $\kappa_{g}(A,b)=\|K\|,\,\,\kappa^{r}_{g}(A,b)=\frac{\|K\|\|[A\,\,b]\|_{F}}{\|x\|},$ (14) where $K$ is defined as in (13). Proof. By the definition of $g$ and (12), we see that $g$ is differentiable at $[a^{T},b^{T}]^{T}$ and $g^{\prime}\left([a^{T},b^{T}]^{T}\right)=K$. Then the assertion follows from (9) and (10). $\Box$ The dependence of Kronecker product of matrices for $K$ makes the computation of $\kappa_{g}(A,b)$ via (14) too costly. The same are the formulas given in [8, 18]. For a computational purpose, we will present a new formula of $\kappa_{g}(A,b)$ that has a simpler and clearer form. To this end, we need a lemma. ###### Lemma 1 Let $C=A^{T}A+\sigma^{2}_{n+1}I_{n}-\frac{2\sigma^{2}_{n+1}xx^{T}}{\|x\|^{2}+1}.$ Then $C$ is positive definite. Proof. Noticing that $C=A^{T}A-\sigma^{2}_{n+1}I_{n}+2\sigma^{2}_{n+1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right),$ (15) and that $A^{T}A-\sigma^{2}_{n+1}I_{n}$ and $I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}$ are both positive definite, we complete the proof of the lemma. $\Box$ ###### Theorem 4 Let $A^{T}A+\sigma^{2}_{n+1}I_{n}-\frac{2\sigma^{2}_{n+1}xx^{T}}{\|x\|^{2}+1}=LL^{T}$ be the Cholesky factorization. Then $\kappa_{g}(A,b)=\sqrt{\|x\|^{2}+1}\left\|(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}L\right\|.$ (16) Proof. Consider expression (13) of $K$. By the properties of Kronecker product of matrices, we get $G(x)G^{T}(x)=\left([x^{T}\,-1]\otimes I_{m}\right)\left(\left[\begin{array}[]{c}x\\\ -1\\\ \end{array}\right]\otimes I_{m}\right)=(\|x\|^{2}+1)I_{m},$ $[I_{n}\otimes r^{T}\,O]G^{T}(x)=[I_{n}\otimes r^{T}\,O]\left[\begin{array}[]{c}x\otimes I_{m}\\\ -I_{m}\\\ \end{array}\right]=(I_{n}\otimes r^{T})(x\otimes I_{m})=xr^{T}$ and $[I_{n}\otimes r^{T}\,O]\left[\begin{array}[]{c}I_{n}\otimes r\\\ O\\\ \end{array}\right]=(I_{n}\otimes r^{T})(I_{n}\otimes r)=\|r\|^{2}I_{n}.$ Thus, we have $\displaystyle\left(2A^{T}\frac{r}{\|r\|}\frac{r^{T}}{\|r\|}G(x)-A^{T}G(x)-[I_{n}\otimes r^{T}\,O]\right)$ $\displaystyle\cdot\left(2G^{T}(x)\frac{r}{\|r\|}\frac{r^{T}}{\|r\|}A-G^{T}(x)A-\left[\begin{array}[]{c}I_{n}\otimes r\\\ O\\\ \end{array}\right]\right)$ $\displaystyle=$ $\displaystyle(\|x\|^{2}+1)A^{T}A+\|r\|^{2}I_{n}-xr^{T}A-A^{T}rx^{T}$ $\displaystyle=$ $\displaystyle(\|x\|^{2}+1)A^{T}A+\|r\|^{2}I_{n}-2\sigma^{2}_{n+1}xx^{T}.$ The last equality used $A^{T}rx^{T}=\sigma^{2}_{n+1}xx^{T}$, which follows from (7). Denote $P=A^{T}A-\sigma^{2}_{n+1}I_{n}$. We get $\displaystyle KK^{T}$ $\displaystyle=$ $\displaystyle P^{-1}\left((\|x\|^{2}+1)A^{T}A+\|r\|^{2}I_{n}-2\sigma^{2}_{n+1}xx^{T}\right)P^{-1}$ (20) $\displaystyle=$ $\displaystyle(\|x\|^{2}+1)P^{-1}\left(A^{T}A+\sigma^{2}_{n+1}I_{n}-\frac{2\sigma^{2}_{n+1}xx^{T}}{\|x\|^{2}+1}\right)P^{-1}.$ (21) In the last equality we used (6). From Theorem 3, we have $\kappa_{g}(A,b)=(\|x\|^{2}+1)^{\frac{1}{2}}\left\|P^{-1}\left(A^{T}A+\sigma^{2}_{n+1}I_{n}-\frac{2\sigma^{2}_{n+1}xx^{T}}{\|x\|^{2}+1}\right)P^{-1}\right\|^{\frac{1}{2}}.$ Based on Lemma 1, we complete the proof. $\Box$ Compared with the formula of $\kappa_{g}(A,b)$ in Theorem 3, the formula in Theorem 4 does not involve the Kronecker product of matrices and makes its computation convenient. However, if $\hat{\sigma}_{n}$ and $\sigma_{n+1}$ are close, then $A^{T}A-\sigma^{2}_{n+1}I_{n}$ becomes ill conditioned. Therefore, it may be hard to use (16) to calculate $\kappa_{g}(A,b)$ accurately. Next we derive a new formula that can be used to compute the condition number accurately. ###### Theorem 5 Let $U^{T}[A\,\,b]V={\rm{diag}}(\sigma_{1},\ldots,\sigma_{n+1})$ be the SVD of $[A\,\,b]$ with $V=[v_{1},\ldots,v_{n+1}]$. Denote $V_{11}=V(1:n,1:n)$. Then $\kappa_{g}(A,b)=\sqrt{\|x\|^{2}+1}~{}\|V^{-T}_{11}S\|,$ (22) where $S={\rm{diag}}([s_{1},\ldots,s_{n}])$, $s_{i}=\frac{\sqrt{\sigma^{2}_{i}+\sigma^{2}_{n+1}}}{\sigma^{2}_{i}-\sigma^{2}_{n+1}}$, $i=1,\ldots,n$. Proof. Denote $P=A^{T}A-\sigma^{2}_{n+1}I_{n}$. From (21), we have $\frac{1}{\|x\|^{2}+1}~{}KK^{T}=P^{-1}+2\sigma^{2}_{n+1}P^{-1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right)P^{-1}.$ (23) Note that $\displaystyle[A\,\,b]^{T}[A\,\,b]-\sigma^{2}_{n+1}I_{n+1}$ $\displaystyle=$ $\displaystyle\sum^{n+1}_{i=1}\sigma^{2}_{i}v_{i}v^{T}_{i}-\sigma^{2}_{n+1}\sum^{n+1}_{i=1}v_{i}v^{T}_{i}$ $\displaystyle=$ $\displaystyle\sum^{n}_{i=1}(\sigma^{2}_{i}-\sigma^{2}_{n+1})v_{i}v^{T}_{i}.$ We get $\displaystyle P=A^{T}A-\sigma^{2}_{n+1}I_{n}$ $\displaystyle=$ $\displaystyle[I_{n}\,\,0]\sum^{n}_{i=1}(\sigma^{2}_{i}-\sigma^{2}_{n+1})v_{i}v^{T}_{i}\left[\begin{array}[]{c}I_{n}\\\ 0\\\ \end{array}\right]$ (26) $\displaystyle=$ $\displaystyle[I_{n}\,\,0][v_{1},\ldots,v_{n}]\left[\begin{array}[]{ccc}\sigma^{2}_{1}-\sigma^{2}_{n+1}&&\\\ &\ddots&\\\ &&\sigma^{2}_{n}-\sigma^{2}_{n+1}\\\ \end{array}\right]\left[\begin{array}[]{c}v^{T}_{1}\\\ \vdots\\\ v^{T}_{n}\\\ \end{array}\right]\left[\begin{array}[]{c}I_{n}\\\ 0\\\ \end{array}\right]$ (35) $\displaystyle=$ $\displaystyle V_{11}\left[\begin{array}[]{ccc}\sigma^{2}_{1}-\sigma^{2}_{n+1}&&\\\ &\ddots&\\\ &&\sigma^{2}_{n}-\sigma^{2}_{n+1}\\\ \end{array}\right]V^{T}_{11}:=V_{11}\Lambda V^{T}_{11}.$ (39) Similarly, by (8), since $v_{n+1}=\frac{1}{\sqrt{1+\|x\|^{2}}}\left[\begin{array}[]{c}x\\\ -1\\\ \end{array}\right],$ we have $\displaystyle I_{n+1}-\frac{1}{{1+\|x\|^{2}}}\left[\begin{array}[]{cc}xx^{T}&-x\\\ -x&1\\\ \end{array}\right]=I_{n+1}-v_{n+1}v^{T}_{n+1}=[v_{1},\ldots,v_{n}]\left[\begin{array}[]{c}v^{T}_{1}\\\ \vdots\\\ v^{T}_{n}\\\ \end{array}\right]$ (45) and $I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}=V_{11}V^{T}_{11}.$ (46) By (46), we see that $V_{11}$ is invertible. Combining (39) and (46), we have $\displaystyle P^{-1}+2\sigma^{2}_{n+1}P^{-1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right)P^{-1}$ (47) $\displaystyle=$ $\displaystyle V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}+2\sigma^{2}_{n+1}\left(V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}\right)V_{11}V^{T}_{11}\left(V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}\right)$ $\displaystyle=$ $\displaystyle V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}+2\sigma^{2}_{n+1}V^{-T}_{11}\Lambda^{-2}V^{-1}_{11}$ $\displaystyle=$ $\displaystyle V^{-T}_{11}\left(\Lambda^{-1}+2\sigma^{2}_{n+1}\Lambda^{-2}\right)V^{-1}_{11}=\left(V^{-T}_{11}S\right)\left(V^{-T}_{11}S\right)^{T}.$ (48) Then by (23) and Theorem 3 we get the desired equality. $\Box$ By Theorem 5, we can calculate $\kappa_{g}(A,b)$ by solving a linear system with the coefficient matrix $V^{T}_{11}$. Next we show what the condition number of $V^{T}_{11}$ is exactly. ###### Theorem 6 For $V_{11}$, we have $\sigma_{1}(V_{11})=1,\ldots,\sigma_{n-1}(V_{11})=1,\sigma_{n}({V_{11}})=\frac{1}{\sqrt{1+\|x\|^{2}}}$ (49) and $\kappa(V^{T}_{11})=\frac{\sigma_{1}(V_{11})}{\sigma_{n}(V_{11})}=\sqrt{1+\|x\|^{2}}.$ (50) Proof. By the definition of $V_{11}$ and the interlacing property [17, p.103] for eigenvalues of symmetric matrices, we get $\sigma_{1}(V_{11})=1,\ldots,\sigma_{n-1}(V_{11})=1.$ Noticing that $V_{11}V^{T}_{11}x=\left(I_{n}-\frac{1}{1+\|x\|^{2}}xx^{T}\right)x=x-\frac{\|x\|^{2}}{1+\|x\|^{2}}x=\frac{1}{1+\|x\|^{2}}x,$ we know that $\frac{1}{1+\|x\|^{2}}$ is an eigenvalue of $V_{11}V^{T}_{11}$, that is, $\sigma_{n}({V_{11}})=\frac{1}{\sqrt{1+\|x\|^{2}}}$. Thus, we have proved (49) and (50). $\Box$ A different SVD-based closed formula for $\kappa_{g}(A,b)$ appears in [1]. It is shown in [1] that $\kappa_{g}(A,b)=\sqrt{\|x\|^{2}+1}\left\|\hat{D}~{}[\hat{V}^{T}\,\,O_{n,1}]~{}V~{}[D\,\,\,O_{n,1}]^{T}\right\|,$ (51) where $\displaystyle\hat{D}$ $\displaystyle=$ $\displaystyle{\rm{diag}}\left(\left[\frac{1}{\hat{\sigma}^{2}_{1}-\sigma^{2}_{n+1}},\ldots,\frac{1}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}\right]\right),$ $\displaystyle D$ $\displaystyle=$ $\displaystyle{\rm{diag}}\left(\left[\sqrt{\sigma^{2}_{1}+\sigma^{2}_{n+1}},\ldots,\sqrt{\sigma^{2}_{n}+\sigma^{2}_{n+1}}\right]\right).$ Compared with (51), our (22) is simpler and more compact. Furthermore, (51) depends on the singular values and right singular vectors of both $A$ and $[A\,\,b]$. In contrast, (22) involves only singular values and right singular vectors of $[A\,\,b]$. Therefore, the computational cost of the condition number by (22) is half of that by (51). Furthermore, the following example shows that the computed results by (22) can be more accurate than those by (51). A small example. We construct a TLS problem with $\hat{\sigma}_{n}$ and $\sigma_{n+1}$ very close. We generate $A,b$ by $[A\,\,b]={\text{\bf{generate}}}Ab\alpha(m,n,\alpha)$ (see Appendix) by taking $m=15,n=10$, $\alpha=10^{-8}$. $~{}\sigma_{n+1}/\sigma_{n}~{}$ | $~{}\sigma_{n+1}/\hat{\sigma}_{n}~{}$ | $~{}{\kappa}^{r}_{g}(\ref{knoneckercond})~{}$ | $~{}{\kappa}^{r}_{g}(\ref{mynewclosed})~{}$ | $~{}\kappa^{r}_{g}(\ref{Baboulin})~{}$ ---|---|---|---|--- $0.608$ | $~{}1-1.49\times 10^{-15}$ | $~{}-~{}$ | $1.13\times 10^{9}$ | $3.12\times 10^{8}$ In the table, $~{}\sigma_{n+1}/\sigma_{n}~{}$ and $~{}\sigma_{n+1}/\hat{\sigma}_{n}~{}$ denote the quotients of $\sigma_{n+1}$ over $\sigma_{n}$ and $\hat{\sigma}_{n}$, respectively. ${\kappa}^{r}_{g}(\ref{knoneckercond})$, ${\kappa}^{r}_{g}(\ref{mynewclosed})$ and $\kappa^{r}_{g}(\ref{Baboulin})$ denote the computed $\kappa^{r}_{g}(A,b)$ by calculating $\kappa_{g}(A,b)$ via (14), (22) and (51), respectively. $\sigma_{n+1}$ and $\hat{\sigma}_{n}$ being so close makes $A^{T}A-\sigma^{2}_{n+1}I_{n}$ numerically singular and makes ${\kappa}^{r}_{g}(\ref{knoneckercond})$ unreliable completely, so the result of ${\kappa}^{r}_{g}(\ref{knoneckercond})$ is omitted. We comment that ${\kappa}^{r}_{g}(\ref{mynewclosed})$ is reliable as, by the remark in Appendix and Theorem 6, $\kappa(V^{T}_{11})=\alpha^{-1}=10^{8}$. This means that computing $\kappa_{g}(A,b)$ via (22) amounts to solving a moderately ill-conditioned linear system. Furthermore, the right-hand side $S$ of the system can be constructed with high accuracy since $\sigma_{n+1}$ and $\sigma_{i}$ are not close: $\frac{\sigma_{n+1}}{\sigma_{i}}\leq\frac{\sigma_{n+1}}{\sigma_{n}}=0.608$, $i=1,2,\ldots,n$. In contrast, $\kappa^{r}_{g}(\ref{Baboulin})$ is inaccurate since computing $\kappa_{g}(A,b)$ via (51) involves the diagonal matrix $\hat{D}$ and the closeness of $\sigma_{n+1}$ (about $0.299$) and $\hat{\sigma}_{n}$ makes its last diagonal entry both very large (about $10^{15}$) and very inaccurate in finite precision arithmetic. ## 4 Straightforward bounds on the TLS condition number ### 4.1 Sharp lower and upper bounds based on SVD of $[A\,\,b]$ In this subsection, we further improve our result in Theorem 5 from the viewpoint of computational cost. We will show that with the SVD $U^{T}[A\,\,b]V={\rm{diag}}(\sigma_{1},\ldots,\sigma_{n+1})$ we are capable of estimating $\kappa_{g}(A,b)$ accurately based on the singular values of $[A\,\,b]$ and the last row of $V$ without calculating $\left\|V^{-T}_{11}S\right\|$, where $V_{11}=V(1:n,1:n)$, $S={\rm{diag}}([s_{1},\ldots,s_{n}])$, $s_{i}=\frac{\sqrt{\sigma^{2}_{i}+\sigma^{2}_{n+1}}}{\sigma^{2}_{i}-\sigma^{2}_{n+1}}$, $i=1,\ldots,n$, as defined in Theorem 5. From now on we denote $\alpha=\frac{1}{\sqrt{1+\|x\|^{2}}}$, which is always smaller than one for $x\not=0$. Keep (49) in mind and note that $s_{1}\leq s_{2}\leq\cdots\leq s_{n}.$ We then get $s_{n}=\sigma_{n}(V^{-T}_{11})\|S\|\leq\|V^{-T}_{11}S\|\leq\|V^{-T}_{11}\|\|S\|=\alpha^{-1}s_{n}.$ Therefore, from Theorem 5 we get $\underline{\kappa}:=\alpha^{-1}s_{n}\leq\kappa_{g}(A,b)\leq\bar{\kappa}:=\alpha^{-2}s_{n}.$ (52) So, if $\alpha\approx 1$, that is, $V_{11}$ is nearly an orthogonal matrix, the lower and upper bounds in (52) must be tight. More generally, for $\alpha$ not small, say, $\frac{1}{2}<\alpha<1$, we have $\bar{\kappa}<4s_{n}$ and $\underline{\kappa}>s_{n}$. So $\underline{\kappa}<\bar{\kappa}<4\underline{\kappa}$. Therefore, in this case, our lower and upper bounds on the condition number $\kappa_{g}(A,b)$ are very tight and can estimate the condition number accurately by no more than four times. In the following, we only need to discuss the case that $\alpha\leq\frac{1}{2}$. It will appear that we can establish some lower bound $\underline{\kappa}$ and upper bound $\bar{\kappa}$ such that $\underline{\kappa}<\bar{\kappa}<4\underline{\kappa}$ still holds. As a result, together with the above, for any $0<\alpha<1$, we can estimate $\kappa_{g}(A,b)$ accurately. ###### Lemma 2 $V$ can be written as $V=\left[\begin{array}[]{cc}V_{11}&\sqrt{1-\alpha^{2}}~{}\bar{u}_{n}\\\ \sqrt{1-\alpha^{2}}~{}\bar{v}^{T}_{n}&-\alpha\\\ \end{array}\right],$ where $\bar{u}_{n}$ and $\bar{v}_{n}$ are the left and right singular vectors associated with the smallest singular value of $V_{11}$. Proof. Based on Theorem 6, we let $V_{11}=\bar{U}\left[\begin{array}[]{cc}I_{n-1}&\\\ &\alpha\\\ \end{array}\right]\bar{V}^{T}$ be the SVD of $V_{11}$, where $\bar{U}=[\bar{u}_{1},\ldots,\bar{u}_{n}]\in\mathbb{R}^{n\times n}$, $\bar{V}=[\bar{v}_{1},\ldots,\bar{v}_{n}]\in\mathbb{R}^{n\times n}$, and $\bar{U}^{T}\bar{U}=\bar{V}^{T}\bar{V}=I_{n}$. It is easily justified from (4) that $|V(n+1,n+1)|=\alpha$. Without loss of generality, we assume $V(n+1,n+1)=-\alpha$. Then, by the theorem in Section 4 of [11], we get $\displaystyle V$ $\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}\bar{U}&\\\ &1\\\ \end{array}\right]\left[\begin{array}[]{ccc}I_{n-1}&O_{n-1,1}&O_{n-1,1}\\\ O_{1,n-1}&\alpha&\sqrt{1-\alpha^{2}}\\\ O_{1,n-1}&\sqrt{1-\alpha^{2}}&-\alpha\\\ \end{array}\right]\left[\begin{array}[]{cc}\bar{V}^{T}&\\\ &1\\\ \end{array}\right]$ (60) $\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}\bar{U}\left[\begin{array}[]{cc}I_{n-1}&\\\ &\alpha\\\ \end{array}\right]\bar{V}^{T}&~{}~{}~{}~{}\sqrt{1-\alpha^{2}}\bar{u}_{n}\\\ \sqrt{1-\alpha^{2}}\bar{v}^{T}_{n}&-\alpha\\\ \end{array}\right],$ (65) the desired form of $V$. $\Box$ Following Lemma 2 and letting $[\beta_{1},\ldots,\beta_{n},-\alpha]$ be the last row of $V$, we have $\bar{v}^{T}_{n}=\frac{1}{\sqrt{1-\alpha^{2}}}[\beta_{1},\ldots,\beta_{n}].$ (66) Noticing that ($\alpha^{-1}$, $\bar{u}_{n}$, $\bar{v}_{n}$) is the largest singular triplet of $V^{-T}_{11}$, we denote by $V^{-T}_{11}=[\bar{u}_{n},\bar{u}_{1}\ldots,\bar{u}_{n-1}]\left[\begin{array}[]{cccc}\alpha^{-1}&&&\\\ &1&&\\\ &&\ddots&\\\ &&&1\\\ \end{array}\right]\left[\begin{array}[]{c}\bar{v}^{T}_{n}\\\ \bar{v}^{T}_{1}\\\ \vdots\\\ \bar{v}^{T}_{n-1}\end{array}\right],$ which is the SVD of $V^{-T}_{11}$. Then, by (66) we have $\displaystyle V^{-T}_{11}$ $\displaystyle=$ $\displaystyle\left[\alpha^{-1}\bar{u}_{n},\bar{u}_{1},\ldots,\bar{u}_{n-1}\right]\left[\begin{array}[]{ccccc}\frac{\beta_{1}}{\sqrt{1-\alpha^{2}}}&\cdots&\frac{\beta_{k}}{\sqrt{1-\alpha^{2}}}&\cdots&\frac{\beta_{n}}{\sqrt{1-\alpha^{2}}}\\\ \bar{v}_{1}(1)&\ldots&\bar{v}_{1}(k)&\cdots&\bar{v}_{1}(n)\\\ \vdots&&\vdots&&\vdots\\\ \bar{v}_{n-1}(1)&\cdots&\bar{v}_{n-1}(k)&\cdots&\bar{v}_{n-1}(n)\\\ \end{array}\right]$ (71) $\displaystyle=$ $\displaystyle\left[\frac{\alpha^{-1}\beta_{1}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+w_{1},\ldots,\frac{\alpha^{-1}\beta_{k}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+w_{k},\ldots,\frac{\alpha^{-1}\beta_{n}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+w_{n}\right],$ (72) where $\bar{v}_{i}(k)$ denotes the $k$th component of $\bar{v}_{i}$, $w_{k}=\sum^{n-1}_{i=1}\bar{v}_{i}(k)\bar{u}_{i}$, $k=1,\ldots,n$. ###### Lemma 3 For given matrices $A_{1},A_{2}\in\mathbb{R}^{n\times n}$, if $A^{T}_{1}A_{2}=O$, then $\frac{1}{2}(\|A_{1}\|+\|A_{2}\|)\leq\|A_{1}+A_{2}\|.$ (73) Proof. For an arbitrary vector $x\in\mathbb{R}^{n}$, from $(A_{1}x)^{T}(A_{2}x)=0$ it follows that $\|A_{1}x\|,\|A_{2}x\|\leq\|A_{1}x+A_{2}x\|$ and that $\displaystyle\|A_{1}\|$ $\displaystyle=$ $\displaystyle{\rm{max}}_{\|x\|=1}\|A_{1}x\|\leq{\rm{max}}_{\|x\|=1}\|A_{1}x+A_{2}x\|=\|A_{1}+A_{2}\|,$ $\displaystyle\|A_{2}\|$ $\displaystyle=$ $\displaystyle{\rm{max}}_{\|x\|=1}\|A_{2}x\|\leq{\rm{max}}_{\|x\|=1}\|A_{1}x+A_{2}x\|=\|A_{1}+A_{2}\|.$ So, we get the desired inequality. $\Box$ To prove the main results of this section, we need the following two propositions. ###### Proposition 1 Let $[\beta_{1},\ldots,\beta_{n},-\alpha]$ be the last row of $V$, $V_{11}=V(1:n,1:n)$ and $\bar{S}={\rm{diag}}([\bar{s}_{1},\ldots,\bar{s}_{n}])$, where $\bar{s}_{1},\ldots,\bar{s}_{n}$ are arbitrary positive numbers and satisfy $0<\bar{s}_{1}\leq\bar{s}_{2}\leq\cdots\leq\bar{s}_{n}$. Then $\displaystyle\underline{c}:=\frac{1}{2}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}\right)$ $\displaystyle\leq$ $\displaystyle\left\|V^{-T}_{11}\bar{S}\right\|\leq\bar{c}:=\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\bar{s}_{n}.$ Proof. Following (72), we get $\displaystyle V^{-T}_{11}\bar{S}$ $\displaystyle=$ $\displaystyle\left[\frac{\alpha^{-1}\beta_{1}\bar{s}_{1}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+\bar{s}_{1}w_{1},\ldots,\frac{\alpha^{-1}\beta_{k}\bar{s}_{k}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+\bar{s}_{k}w_{k},\ldots,\frac{\alpha^{-1}\beta_{n}\bar{s}_{n}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}+\bar{s}_{n}w_{n}\right].$ Define $A_{1}=\left[\frac{\alpha^{-1}\beta_{1}\bar{s}_{1}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n},\ldots,\frac{\alpha^{-1}\beta_{n}\bar{s}_{n}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}\right],\,\,A_{2}=\left[\bar{s}_{1}w_{1},\ldots,\bar{s}_{n}w_{n}\right].$ Then $V^{-T}_{11}\bar{S}=A_{1}+A_{2}$. Noticing that $\bar{u}^{T}_{n}w_{k}=0,\,\,k=1,\ldots,n,$ we get $A^{T}_{1}A_{2}=O$. Thus, we have $\frac{1}{2}(\|A_{1}\|+\|A_{2}\|)\leq\left\|V^{-T}_{11}\bar{S}\right\|\leq\|A_{1}\|+\|A_{2}\|,$ (74) in which the left-hand side inequality follows from Lemma 3. Furthermore, noticing that $A_{1}=\frac{\alpha^{-1}}{\sqrt{1-\alpha^{2}}}\bar{u}_{n}\left[\beta_{1}\bar{s}_{1},\ldots,\beta_{n}\bar{s}_{n}\right]$ and $\|\bar{u}_{n}\|=1$, we have $\|A_{1}\|=\frac{\alpha^{-1}}{\sqrt{1-\alpha^{2}}}\left\|\left[\beta_{1}\bar{s}_{1},\ldots,\beta_{n}\bar{s}_{n}\right]\right\|=\frac{\alpha^{-1}}{\sqrt{1-\alpha^{2}}}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}.$ (75) In the meantime, note that $\|w_{n}\|=\sqrt{\sum^{n-1}_{i=1}\bar{v}^{2}_{i}(n)}=\sqrt{1-\frac{\beta^{2}_{n}}{1-\alpha^{2}}}=\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}},$ $\|[w_{1},\ldots,w_{n}]\|=\left\|[\bar{u}_{1},\ldots,\bar{u}_{n-1}]\left[\begin{array}[]{c}\bar{v}^{T}_{1}\\\ \vdots\\\ \bar{v}^{T}_{n-1}\\\ \end{array}\right]\right\|=1,$ and $\|\bar{S}\|=\bar{s}_{n}.$ From $\|\bar{s}_{n}w_{n}\|\leq\|A_{2}\|\leq\left\|[w_{1},\ldots,w_{n}]\right\|\|\bar{S}\|$ we get $\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}\leq\|A_{2}\|\leq\bar{s}_{n}.$ (76) Combining (74), (75) and (76), we establish the desired inequality. $\Box$ ###### Proposition 2 Suppose that $\alpha\leq\frac{1}{2}$. Then for $\underline{c}$ and $\bar{c}$ in Proposition 1, we have $\underline{c}<\bar{c}<4\underline{c}.$ (77) Proof. If $\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}<\frac{\sqrt{3}}{2}$, then it is easy to verify that $\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}>\frac{1}{2}$ and $\underline{c}>\frac{1}{4}\bar{c}.$ Thus, (77) holds. If $\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}\geq\frac{\sqrt{3}}{2}$, then $\alpha^{-1}\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}>\frac{\sqrt{3}}{2}\alpha^{-1}>1,$ so $\alpha^{-1}\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}>\bar{s}_{n}$, from which and the definitions of $\bar{c}$ and $\underline{c}$ it follows that $\displaystyle\bar{c}$ $\displaystyle<$ $\displaystyle\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\alpha^{-1}\frac{|\beta_{n}|}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}$ $\displaystyle\leq$ $\displaystyle\frac{2\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}$ $\displaystyle\leq$ $\displaystyle\frac{2\alpha^{-1}\sqrt{\beta^{2}_{1}\bar{s}^{2}_{1}+\ldots+\beta^{2}_{n}\bar{s}^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{2\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}\bar{s}_{n}=4\underline{c}.$ Thus, (77) still holds. $\Box$ Now we are in a position to derive sharp bounds on $\kappa_{g}(A,b)$. ###### Theorem 7 Let $[\beta_{1},\ldots,\beta_{n},-\alpha]$ be the last row of $V$ and $S={\rm{diag}}([s_{1},\ldots,s_{n}])$, $s_{i}=\frac{\sqrt{\sigma^{2}_{i}+\sigma^{2}_{n+1}}}{\sigma^{2}_{i}-\sigma^{2}_{n+1}}$, $i=1,\ldots,n$. Then $\displaystyle\underline{\kappa}:=\frac{1}{2}\left(\frac{\alpha^{-2}\sqrt{\beta^{2}_{1}s^{2}_{1}+\ldots+\beta^{2}_{n}s^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}\alpha^{-1}s_{n}\right)$ $\displaystyle\leq$ $\displaystyle\kappa_{g}(A,b)\leq\bar{\kappa}:=\frac{\alpha^{-2}\sqrt{\beta^{2}_{1}s^{2}_{1}+\ldots+\beta^{2}_{n}s^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\alpha^{-1}s_{n}.$ Moreover, if $\alpha\leq\frac{1}{2}$, then $\underline{\kappa}<\bar{\kappa}<4\underline{\kappa}.$ Proof. Noticing that $0<s_{1}\leq s_{2}\leq\cdots\leq s_{n}$ and using Proposition 1, we have $\displaystyle\frac{1}{2}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}s^{2}_{1}+\ldots+\beta^{2}_{n}s^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}s_{n}\right)$ $\displaystyle\leq$ $\displaystyle\left\|V^{-T}_{11}S\right\|\leq\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}s^{2}_{1}+\ldots+\beta^{2}_{n}s^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+s_{n}.$ By Theorem 5, we get the first part of the theorem. Furthermore, we have the second part of the theorem by Proposition 2. $\Box$ A small example (Continued). From Theorem 7, we have $5.65\times 10^{8}\leq\kappa^{r}_{g}(A,b)\leq 1.13\times 10^{9}.$ The lower and upper bounds estimate ${\kappa}^{r}_{g}(\ref{mynewclosed})=1.13\times 10^{9}$ accurately, as described in the second part of Theorem 7. ### 4.2 Lower and upper bounds based on a few of singular values of $A$ and $[A\,\,b]$ In [10], bounds on the condition number of the Tikhonov regularization solution are established based on a few singular values of $A$, where $A$ is the coefficient matrix of the least squares problem under consideration. This is particularly useful for large scale TLS problems since for them any formulas and bounds for the condition number involving all the singular values of $A$ and/or $[A\ b]$ are too costly to be computed. Such a bound can be obtained by computing only a few singular values of $A$ and/or $[A\ b]$. In the following theorem, we establish similar results for the condition number of the TLS problem. ###### Theorem 8 We have $\underline{\kappa}_{1}\leq\kappa_{g}(A,b)\leq\bar{\kappa}_{1},$ (78) where $\displaystyle\underline{\kappa}_{1}=\frac{\sqrt{1+\|x\|^{2}}\sqrt{\hat{\sigma}^{2}_{n-1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n-1}-\sigma^{2}_{n+1}},\,\,\bar{\kappa}_{1}=\frac{\sqrt{1+\|x\|^{2}}\sqrt{\hat{\sigma}^{2}_{n}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$ (79) Proof. Denoting $M=(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}\left((\|x\|^{2}+1)A^{T}A+\|r\|^{2}I_{n}\right)(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1},$ from (20) we have $KK^{T}=M-2\sigma^{2}_{n+1}(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}xx^{T}(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}.$ (80) Here and hereafter, $\lambda_{i}(M)$ denotes the $i$th largest eigenvalue of $M$, where $M$ is an arbitrary symmetric matrix. By the Courant-Fischer theorem [14, p.182], from (80) we get $\lambda_{2}(M)\leq\lambda_{1}(KK^{T}).$ (81) Furthermore, since $2\sigma^{2}_{n+1}(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}xx^{T}(A^{T}A-\sigma^{2}_{n+1}I_{n})^{-1}$ is nonnegative definite, the following inequality holds $\lambda_{1}(KK^{T})\leq\lambda_{1}(M).$ (82) Collecting (81) and (82) and based on (14), we have $\sqrt{\lambda_{2}(M)}\leq\kappa_{g}(A,b)\leq\sqrt{\lambda_{1}(M)}.$ It is easy to verify that the set $\left\\{\frac{(\|x\|^{2}+1)\hat{\sigma}^{2}_{j}+\|r\|^{2}}{(\hat{\sigma}^{2}_{j}-\sigma^{2}_{n+1})^{2}}\right\\}^{n}_{j=1}$ consists of all the eigenvalues of $M$. We define the function $f(\sigma)=\frac{(\|x\|^{2}+1){\sigma}^{2}+\|r\|^{2}}{({\sigma}^{2}-\sigma^{2}_{n+1})^{2}},\,\,\sigma>\sigma_{n+1},$ and differentiate it to get $f^{\prime}(\sigma)=\frac{-2\sigma^{3}(\|x\|^{2}+1)-2\sigma(\|x\|^{2}+1)\sigma^{2}_{n+1}-4\sigma\|r\|^{2}}{(\sigma^{2}-\sigma^{2}_{n+1})^{3}}.$ It is seen that $f^{\prime}(\sigma)<0$ and $f(\sigma)$ is decreasing in the interval $(\sigma_{n+1},\infty)$. Thus, we get that $\lambda_{1}(M)=\frac{(\|x\|^{2}+1)\hat{\sigma}^{2}_{n}+\|r\|^{2}}{(\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1})^{2}},\,\,\lambda_{2}(M)=\frac{(\|x\|^{2}+1)\hat{\sigma}^{2}_{n-1}+\|r\|^{2}}{(\hat{\sigma}^{2}_{n-1}-\sigma^{2}_{n+1})^{2}}$ and $\frac{\sqrt{(\|x\|^{2}+1)\hat{\sigma}^{2}_{n-1}+\|r\|^{2}}}{\hat{\sigma}^{2}_{n-1}-\sigma^{2}_{n+1}}\leq\kappa_{g}(A,b)\leq\frac{\sqrt{(\|x\|^{2}+1)\hat{\sigma}^{2}_{n}+\|r\|^{2}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$ Noticing that $\frac{\|r\|^{2}}{1+\|x\|^{2}}=\sigma^{2}_{n+1}$, we complete the proof. $\Box$ Remark. In Corollary 1 of [1], the authors prove that $\kappa_{g}(A,b)\leq\frac{\sqrt{1+\|x\|^{2}}\sqrt{{\sigma}^{2}_{1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$ Since $\hat{\sigma}_{n}\leq\hat{\sigma}_{1},\ \hat{\sigma}_{1}\leq\sigma_{1}$, we get $\bar{\kappa}_{1}\leq\frac{\sqrt{1+\|x\|^{2}}\sqrt{\hat{\sigma}^{2}_{1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}\leq\frac{\sqrt{1+\|x\|^{2}}\sqrt{{\sigma}^{2}_{1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$ Therefore, our $\bar{\kappa}_{1}$ in (79) is sharper than the above upper bound. It is seen that the lower and upper bounds on $\kappa_{g}(A,b)$ in Theorem 8 are marginally different provided that $\hat{\sigma}_{n}$ and $\hat{\sigma}_{n-1}$ are close. This means that in this case both bounds are very tight. For the case that $\hat{\sigma}_{n}$ and $\hat{\sigma}_{n-1}$ are not close, we next give a new lower bound that can be better than that in Theorem 8. ###### Theorem 9 It holds that $\underline{\kappa}_{2}\leq\kappa_{g}(A,b)\leq\bar{\kappa}_{1},$ where $\bar{\kappa}_{1}$ is defined as in Theorem 8 and $\underline{\kappa}_{2}=\frac{\sqrt{1+\|x\|^{2}}}{\sqrt{{\hat{\sigma}}^{2}_{n}-\sigma^{2}_{n+1}}}.$ Moreover, when $\hat{\sigma}_{n-1}\geq\sigma_{n+1}+\sqrt{{\hat{\sigma}}^{2}_{n}-\sigma^{2}_{n+1}}$, we have $\underline{\kappa}_{1}\leq\underline{\kappa}_{2}.$ Proof. Denote $P=A^{T}A-\sigma^{2}_{n+1}I_{n}$. From (23), we have $\frac{1}{\|x\|^{2}+1}~{}KK^{T}=P^{-1}+2\sigma^{2}_{n+1}P^{-1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right)P^{-1}.$ Noticing the second term in the right-hand side of the above relation is positive definite, we have $(\|x\|^{2}+1)\lambda_{1}(P^{-1})\leq\lambda_{1}(~{}KK^{T}),$ that is, $\frac{\|x\|^{2}+1}{{\hat{\sigma}^{2}_{n}}-\sigma^{2}_{n+1}}\leq\kappa^{2}_{g}(A,b).$ Thus, the first part of the theorem is obtained. The second part of the theorem is proved by noting $\frac{\sqrt{\hat{\sigma}^{2}_{n-1}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n-1}-\sigma^{2}_{n+1}}<\frac{1}{\hat{\sigma}_{n-1}-\sigma_{n+1}}\leq\frac{1}{\sqrt{{\hat{\sigma}}^{2}_{n}-\sigma^{2}_{n+1}}}$ under the assumption that $\hat{\sigma}_{n-1}-\sigma_{n+1}\geq\sqrt{{\hat{\sigma}}^{2}_{n}-\sigma^{2}_{n+1}}$. $\Box$ Remark 1. At first glance, the assumption in the second part of the theorem seems not so direct but we can justify that it indeed implies that $\hat{\sigma}_{n}$ and $\hat{\sigma}_{n-1}$ are not close. Actually, we can verify that the second part of Theorem 9 holds under a slightly stronger but much simpler condition that $\hat{\sigma}_{n-1}\geq 2\hat{\sigma}_{n}.$ Remark 2. From $\frac{\bar{\kappa}_{1}}{\underline{\kappa}_{2}}=\frac{\sqrt{\hat{\sigma}^{2}_{n}+\sigma^{2}_{n+1}}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}=\sqrt{\frac{1+\frac{\sigma^{2}_{n+1}}{\hat{\sigma}^{2}_{n}}}{1-\frac{\sigma^{2}_{n+1}}{\hat{\sigma}^{2}_{n}}}},$ it is seen that $\frac{\bar{\kappa}_{1}}{\underline{\kappa}_{2}}>1$ provided $\sigma_{n+1}>0$. Only for $\sigma_{n+1}=0$, $\bar{\kappa}_{1}=\underline{\kappa}_{2}$ holds. At this time, $b\in\mathcal{R}(A)$ and $r=0$. We observe that the bounds on $\kappa_{g}(A,b)$ in Theorem 9 are tight when $\frac{\sigma_{n+1}}{\hat{\sigma}_{n}}$ is small, compared with one. On the other hand, once $\frac{\sigma_{n+1}}{\hat{\sigma}_{n}}$ is not small, these bounds may not be tight. For this case, we will present new bounds that may better estimate $\kappa_{g}(A,b)$. The proof of the following theorem depends strongly on Propositions 1 and 2. ###### Theorem 10 Assume that $\alpha\leq\frac{1}{2}$. Denote $\rho=\frac{\sigma_{n+1}}{\sigma_{n}}$. Then $\displaystyle\underline{\kappa}_{2}:=\frac{\sqrt{1+\|x\|^{2}}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}\leq\kappa_{g}(A,b)$ $\displaystyle<$ $\displaystyle\bar{\kappa}_{2}:=\sqrt{\frac{1+31\rho^{2}}{1-\rho^{2}}}\frac{\sqrt{1+\|x\|^{2}}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}.$ (83) Proof. Based on Theorem 9, it suffices to prove the right-hand side of (83). From (23) and (47), we get $\displaystyle\frac{1}{\|x\|^{2}+1}~{}KK^{T}$ $\displaystyle=$ $\displaystyle P^{-1}+2\sigma^{2}_{n+1}P^{-1}\left(I_{n}-\frac{xx^{T}}{1+\|x\|^{2}}\right)P^{-1},$ (84) $\displaystyle=$ $\displaystyle V^{-T}_{11}\Lambda^{-1}V^{-1}_{11}+2\sigma^{2}_{n+1}V^{-T}_{11}\Lambda^{-2}V^{-1}_{11}:=P^{-1}+E,$ where $P=A^{T}A-\sigma^{2}_{n+1}I_{n}$, $\Lambda={\rm{diag}}([\sigma^{2}_{1}-\sigma^{2}_{n+1},\ldots,\sigma^{2}_{n}-\sigma^{2}_{n+1}])$. Denote $\displaystyle D$ $\displaystyle=$ $\displaystyle{\rm{diag}}([d_{1},\ldots,d_{n}]),\,d_{i}=\frac{\sigma_{n+1}}{\sigma^{2}_{i}-\sigma^{2}_{n+1}},i=1,\ldots,n,$ $\displaystyle T$ $\displaystyle=$ $\displaystyle{\rm{diag}}([t_{1},\ldots,t_{n}]),\,t_{i}=\frac{1}{\sqrt{\sigma^{2}_{i}-\sigma^{2}_{n+1}}},i=1,\ldots,n.$ Then $P^{-1}=\left(V^{-T}_{11}T\right)\left(TV^{-1}_{11}\right)$ and $E=2\left(V^{-T}_{11}D\right)\left(DV^{-1}_{11}\right)$. Note that $0<d_{1}\leq d_{2}\leq\cdots\leq d_{n}$ and $0<t_{1}\leq t_{2}\leq\cdots\leq t_{n}$. Applying Proposition 1, we get $\displaystyle\frac{1}{2}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}d^{2}_{1}+\ldots+\beta^{2}_{n}d^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}d_{n}\right)$ (85) $\displaystyle\leq$ $\displaystyle\left\|V^{-T}_{11}D\right\|\leq\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}d^{2}_{1}+\ldots+\beta^{2}_{n}d^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+d_{n}$ and $\displaystyle\frac{1}{2}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}t_{n}\right)$ $\displaystyle\leq$ $\displaystyle\left\|V^{-T}_{11}T\right\|\leq\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+t_{n},$ respectively, where $[\beta_{1},\ldots,\beta_{n},-\alpha]$ denotes the last row of $V$ as before. Define $k_{n}=\frac{d_{n}}{t_{n}}=\frac{\sigma_{n+1}}{\sqrt{\sigma^{2}_{n}-\sigma^{2}_{n+1}}}$. Then $\frac{d_{1}}{t_{1}}=\frac{\sigma_{n+1}}{\sqrt{\sigma^{2}_{1}-\sigma^{2}_{n+1}}}\leq k_{n}\,,\ldots,\frac{d_{n-1}}{t_{n-1}}=\frac{\sigma_{n+1}}{\sqrt{\sigma^{2}_{n-1}-\sigma^{2}_{n+1}}}\leq k_{n}.$ Thus, by (85) we have $\displaystyle\frac{1}{\sqrt{2}}\|E\|^{\frac{1}{2}}=\left\|V^{-T}_{11}D\right\|\leq k_{n}\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+t_{n}\right).$ (86) Note that for the lower and upper bounds on $\left\|V^{-T}_{11}T\right\|$ above, by Proposition 2 it holds that $\displaystyle\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+t_{n}$ $\displaystyle<$ $\displaystyle 2\left(\frac{\alpha^{-1}\sqrt{\beta^{2}_{1}t^{2}_{1}+\ldots+\beta^{2}_{n}t^{2}_{n}}}{\sqrt{1-\alpha^{2}}}+\frac{\sqrt{1-\alpha^{2}-\beta^{2}_{n}}}{\sqrt{1-\alpha^{2}}}t_{n}\right)$ (87) $\displaystyle<$ $\displaystyle 4\left\|V^{-T}_{11}T\right\|.$ Based on (86) and (87), we derive that $\frac{1}{\sqrt{2}}\|E\|^{\frac{1}{2}}<4k_{n}\left\|V^{-T}_{11}T\right\|=4k_{n}\|P^{-1}\|^{\frac{1}{2}}$ and that $\|E\|<32k^{2}_{n}\|P^{-1}\|.$ (88) Combining (88) and (84), we establish that $\displaystyle\kappa_{g}(A,b)=\|K\|=\|KK^{T}\|^{\frac{1}{2}}$ $\displaystyle<$ $\displaystyle\sqrt{1+32k^{2}_{n}}\sqrt{1+\|x\|^{2}}\|P^{-1}\|^{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{1+31\rho^{2}}{1-\rho^{2}}}\frac{\sqrt{1+\|x\|^{2}}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}.$ So, the proof of the theorem is completed. $\Box$ Remark. It is clear that the bounds in Theorem 10 are tight when $\rho=\frac{\sigma_{n+1}}{\sigma_{n}}$ is small, compared with one. The result in this theorem is of particular importance in the case that $\frac{\sigma_{n+1}}{\hat{\sigma}_{n}}$ is close to one. Recall that the lower and upper bounds in Theorem 9 differ considerably when $\frac{\sigma_{n+1}}{\hat{\sigma}_{n}}$ is close to one. Theorem 10 tells us that, if only $\frac{\sigma_{n+1}}{\sigma_{n}}$ is not so close to one, $\kappa_{g}(A,b)$ should be close to the lower bound. The improvement of $\bar{\kappa}_{2}$ to $\bar{\kappa}_{1}$ can be illustrated as follows. For $\frac{\sigma_{n+1}}{\sigma_{n}}$ small, i.e., ${\sigma_{n+1}}$ and ${\sigma_{n}}$ not close, as an upper bound of $\kappa^{r}_{g}(A,b)$, $\displaystyle\bar{\kappa}^{r}_{2}:=\frac{\bar{\kappa}_{2}}{\|x\|}\|[A\,\,b]\|_{F}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{1+31\rho^{2}}{1-\rho^{2}}}\frac{\sqrt{1+\|x\|^{2}}}{\|x\|}\frac{\|[A\,\,b]\|_{F}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}$ $\displaystyle\approx$ $\displaystyle\sqrt{\frac{1+31\rho^{2}}{1-\rho^{2}}}\frac{\|[A\,\,b]\|_{F}}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}$ is a moderate multiple of $\frac{1}{\sqrt{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}}$. In contrast, $\displaystyle\bar{\kappa}^{r}_{1}:=\frac{\bar{\kappa}_{1}}{\|x\|}\|[A\,\,b]\|_{F}$ $\displaystyle=$ $\displaystyle\frac{\sqrt{1+\|x\|^{2}}}{\|x\|}\frac{\sqrt{\hat{\sigma}^{2}_{n}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}\|[A\,\,b]\|_{F}$ $\displaystyle\approx$ $\displaystyle\frac{\sqrt{\hat{\sigma}^{2}_{n}+\sigma^{2}_{n+1}}}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}\|[A\,\,b]\|_{F}$ is a moderate multiple of $\frac{1}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}$. The improvement of $\bar{\kappa}^{r}_{2}$ over $\bar{\kappa}^{r}_{1}$ becomes significant as ${\sigma_{n+1}}$ and ${\hat{\sigma}_{n}}$ are close. Similarly, $\bar{\kappa}^{r}_{2}$ also improves the approximate condition number used in [2]: $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}:=\frac{\hat{\sigma}_{1}}{\hat{\sigma}_{n}-\sigma_{n+1}}=\frac{\hat{\sigma}_{1}(\hat{\sigma}_{n}+\sigma_{n+1})}{\hat{\sigma}^{2}_{n}-\sigma^{2}_{n+1}}.$ We will further illustrate the improvement by numerical experiments to be presented in Section 5. ## 5 Numerical experiments We present numerical experiments to illustrate how tight the bounds in Theorems 9 and 10 are, and to compare the bounds with the related result in [2]. For a given TLS problem, the TLS solution is computed by (4). All experiments were run using Matlab 7.8.0 with the machine precision $\epsilon_{\rm mach}=2.22\times 10^{-16}$ under the Microsoft Windows XP operating system. Example 1. In this example, the TLS problem comes from [7]. Specifically, an $m\times(m-2\omega)$ convolution matrix $\bar{T}$ is constructed to have the first column $t_{i,1}=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{2\pi\alpha^{2}}{\rm{exp}}\left[\frac{-(\omega-i+1)^{2}}{2\alpha^{2}}\right]}&\hbox{\,\,\,$i=1,2,\ldots,2\omega+1$,}\\\ 0&\hbox{\,\,\,otherwise,}\end{array}\right.$ and the first row $t_{1,j}=\left\\{\begin{array}[]{ll}t_{1,1}&\hbox{\,\,\,if $j=1$,}\\\ 0&\hbox{\,\,\,otherwise,}\end{array}\right.$ where $\alpha=1.25$ and $\omega=8$. A Toeplitz matrix $A$ and a right-hand side vector $b$ are constructed as $A=\bar{T}+E$ and $b=\bar{g}+e$, where $\bar{g}=[1,\ldots,1]^{T}$, $E$ is a random Toeplitz matrix with the same structure as $\bar{T}$ and $e$ is a random vector. The entries in $E$ and $e$ are generated randomly from a normal distribution with mean zero and variance one, and scaled so that $\|e\|=\gamma\|\bar{g}\|,\,\,\,\|E\|=\gamma\|\bar{T}\|,\,\,\gamma=0.001.$ Table 1: $m$ | $\sigma_{n+1}/\sigma_{n}$ | $\sigma_{n+1}/\hat{\sigma}_{n}$ | $\kappa^{r}_{g}(A,b)$ | $\underline{\kappa}^{r}_{2}$ | $\bar{\kappa}^{r}_{2}$ | $\bar{\kappa}^{r}_{1}$ | $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$ ---|---|---|---|---|---|---|--- $~{}100~{}$ | $0.981$ | $~{}1-7.85\times 10^{-9}$ | $7.70\times 10^{7}$ | $7.04\times 10^{7}$ | $2.01\times 10^{9}$ | $7.94\times 10^{11}$ | $1.03\times 10^{11}$ $~{}300~{}$ | $0.995$ | $~{}1-2.05\times 10^{-8}$ | $1.40\times 10^{8}$ | $1.26\times 10^{8}$ | $6.90\times 10^{9}$ | $8.83\times 10^{11}$ | $6.54\times 10^{10}$ $~{}500~{}$ | $0.998$ | $~{}1-5.66\times 10^{-8}$ | $9.01\times 10^{7}$ | $7.89\times 10^{7}$ | $6.56\times 10^{9}$ | $3.32\times 10^{11}$ | $1.90\times 10^{10}$ In the table, $\underline{\kappa}^{r}_{2}=\frac{\underline{\kappa}_{2}}{\|x\|}\|[A\,\,b]\|_{F},\,\,\bar{\kappa}^{r}_{2}=\frac{\bar{\kappa}_{2}}{\|x\|}\|[A\,\,b]\|_{F},\,\,\bar{\kappa}^{r}_{1}=\frac{\bar{\kappa}_{1}}{\|x\|}\|[A\,\,b]\|_{F},$ see Theorems 10 and 9, respectively. We calculate the approximate condition number used in [2]: $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}=\frac{\hat{\sigma}_{1}}{\hat{\sigma}_{n}-\sigma_{n+1}}.$ As indicated by the table, all the given TLS problems are similar in that $\sigma_{n+1}$ and $\hat{\sigma}_{n}$ are close but $\sigma_{n+1}$ and $\sigma_{n}$ are not so close. As estimates of $\kappa^{r}_{g}(A,b)$, the lower bounds $\underline{\kappa}^{r}_{2}$ are very accurate, and the upper bounds $\bar{\kappa}^{r}_{2}$ improve the corresponding $\bar{\kappa}^{r}_{1}$ and $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$ significantly by one or two orders. Example 2. In this example, the TLS problems are generated by the function described in Appendix. For given $m,n$ and $\alpha$, $A$ and $b$ are generated by $[A\,\,b]={\text{generate}}Ab\alpha(m,n,\alpha).$ A different $\alpha$ gives rise to a different TLS problem with different properties. As $\alpha$ becomes smaller, $\sigma_{n+1}$ and $\hat{\sigma}_{n}$ become closer, so that the TLS problem becomes worse conditioned. For each of the TLS problems, we calculate the same quantities as those in Example 1 and list them in Table 2 in which the first set of data is for $(m,n)=(500,350)$ and the second set is for $(m,n)=(1000,750)$. Table 2: $\alpha$ | $\sigma_{n+1}/\sigma_{n}$ | $\sigma_{n+1}/\hat{\sigma}_{n}$ | $\kappa^{r}_{g}(A,b)$ | $\underline{\kappa}^{r}_{2}$ | $\bar{\kappa}^{r}_{2}$ | $\bar{\kappa}^{r}_{1}$ | $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$ ---|---|---|---|---|---|---|--- $10^{-2}$ | $0.953$ | $1-3.05\times 10^{-4}$ | $2.55\times 10^{4}$ | $8.98\times 10^{3}$ | $1.60\times 10^{5}$ | $5.14\times 10^{5}$ | $6.29\times 10^{5}$ $10^{-3}$ | $0.980$ | $1-3.16\times 10^{-6}$ | $2.01\times 10^{5}~{}$ | $8.75\times 10^{4}$ | $2.42\times 10^{6}$ | $4.92\times 10^{7}$ | $6.03\times 10^{7}$ $10^{-5}$ | $0.953$ | $1-2.77\times 10^{-10}$ | $1.97\times 10^{7}$ | $9.78\times 10^{6}$ | $1.74\times 10^{8}$ | $5.87\times 10^{11}$ | $7.20\times 10^{11}$ $10^{-7}$ | $0.966$ | $1-1.80\times 10^{-14}$ | $3.28\times 10^{9}$ | $1.12\times 10^{9}$ | $2.38\times 10^{10}$ | $8.38\times 10^{15}$ | $1.02\times 10^{16}$ $10^{-2}$ | $0.983$ | $1-2.78\times 10^{-4}$ | $6.76\times 10^{4}$ | $1.65\times 10^{4}$ | $4.97\times 10^{5}$ | $9.90\times 10^{5}$ | $1.21\times 10^{6}$ $10^{-3}$ | $0.978$ | $1-1.95\times 10^{-6}$ | $6.70\times 10^{5}$ | $1.93\times 10^{5}$ | $5.09\times 10^{6}$ | $1.38\times 10^{8}$ | $1.69\times 10^{8}$ $10^{-5}$ | $0.968$ | $1-3.01\times 10^{-10}$ | $4.33\times 10^{7}$ | $1.60\times 10^{7}$ | $3.52\times 10^{8}$ | $9.24\times 10^{11}$ | $1.13\times 10^{12}$ $10^{-7}$ | $0.993$ | $1-3.82\times 10^{-14}$ | $1.13\times 10^{10}$ | $1.44\times 10^{9}$ | $7.02\times 10^{10}$ | $7.38\times 10^{15}$ | $9.03\times 10^{15}$ We can see from the table that, for $\alpha=10^{-2}$ in which $\hat{\sigma}_{n}$ and $\sigma_{n+1}$ are not very close, $\bar{\kappa}^{r}_{1}$ and $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$ are very tight and they estimate $\kappa^{r}_{g}(A,b)$ quite accurately; for $\alpha\leq 10^{-3}$, $\hat{\sigma}_{n}$ and $\sigma_{n+1}$ become closer with decreasing $\alpha$, $\bar{\kappa}^{r}_{1}$ and $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$ estimate $\kappa^{r}_{g}(A,b)$ increasingly more poorly. In contrast, however, for all the cases, since ${\sigma}_{n}$ and $\sigma_{n+1}$ are not so close, $\underline{\kappa}^{r}_{2}$ and $\bar{\kappa}^{r}_{2}$ estimate $\kappa^{r}_{g}(A,b)$ accurately. Particularly, for $\alpha\leq 10^{-5}$, $\bar{\kappa}^{r}_{2}$ improves $\bar{\kappa}^{r}_{1}$ and $\bar{\kappa}^{r}_{\cite[cite]{[\@@bibref{}{BjorckHeggernesMatstoms:2000}{}{}]}}$ very considerably by several orders. ## 6 Concluding Remarks In the paper, we have mainly studied the condition number of the TLS problem and its lower and upper bounds that can be numerically computed cheaply. For the TLS condition number, we have derived a new closed formula. For a computational purpose, we can use it to compute the condition number more accurately. We have derived a few bounds, which are quite sharp and can be calculated cheaply. We have confirmed our results numerically and demonstrated the tightness of our bounds by numerical experiments. ACKNOWLEDGEMENTS The work was partially supported by National Basic Research Program of China 2011CB302400 and National Science Foundation of China (No. 11071140) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20070200009) ## Appendix A Codes for generating tested TLS problems The following codes produce an $m\times(n+1)$ matrix $[A\,\,b]$, which has the SVD $[A\,\,b]=U\Sigma V^{T}$ with $V(n+1,n+1)=-\alpha$, where $0<\alpha<1$. $\displaystyle[A\,\,b]={\text{{\bf{generate}}}}{\bf{Ab}}{\bf{\alpha}}(m,n,\alpha)$ $\displaystyle\%~{}m,n:{\text{two given positive integers with }}m\geq n$ $\displaystyle\%~{}\alpha:{\text{a given positive number with $0<\alpha<1$}}$ $\displaystyle\text{Generate}~{}\tilde{V};~{}~{}\text{\% a random orthogonal matrix }~{}\text{of order}~{}n$ $\displaystyle V=\text{\bf{generateV}}(n,\tilde{V},\alpha);$ $\displaystyle B=\text{rand}(m,n+1);~{}~{}\text{\% the Matlab function rand(~{})}$ $\displaystyle[U,\Sigma,\hat{V}]=\text{svd}(B,0);~{}~{}\text{\% the Matlab function svd(~{})}$ $\displaystyle[A\,\,b]=U*\Sigma*V^{T}$ The subfunction ${\bf{generateV}}()$ is shown as follows. It is used to produce an $(n+1)\times(n+1)$ orthogonal matrix $V$ with $V(n+1,n+1)=-\alpha$, where $0<\alpha<1$. The idea of construction comes from Lemma 2. $\displaystyle[V]={\text{{\bf{generate}}}}{\bf{V}}(n,\tilde{V},\alpha)$ $\displaystyle\%~{}n:{\text{a given positive integer}}$ $\displaystyle\%~{}\tilde{V}:{\text{a given orthogonal matrix of order $n$}}$ $\displaystyle\%~{}\alpha:{\text{a given positive number with $0<\alpha<1$}}$ $\displaystyle\text{partition}~{}\tilde{V}=[\tilde{v}_{1},\ldots,\tilde{v}_{n}];$ $\displaystyle\text{generate}~{}U=[u_{1},\ldots,u_{n}];~{}~{}~{}{\text{\% a random orthogonal matrix of order $n$}}$ $\displaystyle V_{11}=[u_{1},\ldots,u_{n-1}][\tilde{v}_{1},\ldots,\tilde{v}_{n-1}]^{T}+\alpha u_{n}\tilde{v}^{T}_{n};$ $\displaystyle V=\left[\begin{array}[]{cc}V_{11}&\sqrt{1-\alpha^{2}}u_{n}\\\ \sqrt{1-\alpha^{2}}\tilde{v}^{T}_{n}&-\alpha\\\ \end{array}\right]$ (91) Remark. Lemma 4.3 in [5] gives $\frac{|\hat{u}^{T}_{n}b|}{2(\hat{\sigma}_{n}-\sigma_{n+1})}\leq\|x\|\leq\frac{\|b\|}{\hat{\sigma}_{n}-\sigma_{n+1}}.$ Equivalently, it holds that $\frac{|\hat{u}^{T}_{n}b|}{2\|x\|}\leq\hat{\sigma}_{n}-\sigma_{n+1}\leq\frac{\|b\|}{\|x\|},$ (92) where it is supposed that $x\neq 0$. Note that $V(n+1,n+1)=-\alpha$ and $\alpha=\frac{1}{\sqrt{1+\|x\|^{2}}}$. From (92) we see that a small $\alpha$ implies that $\hat{\sigma}_{n}$ and $\sigma_{n+1}$ are close in some sense. ## References * [1] M. Baboulin, S. Gratton, A contribution to the conditioning of the total least squares problem, arXiv:1012.5484v1. * [2] Å. Björck, P. Heggernes, P. Matstoms, Methods for large scale total least squares problems, SIAM J. Matrix Anal. Appl., 22 (2000) 413–429. * [3] R. D. Fierro, J. R. Bunch, Perturbation theory for orthogonal projection methods with applications to least squares and total least squares, Linear Algebra Appl., 234 (1996) 71–96. * [4] I. Gohberg, I. Koltracht, Mixed, componentwise, and structured condition numbers, SIAM J. Matrix. Anal. Appl., 14 (1993) 688–704. * [5] G. H. Golub,C. F. Van Loan, An analysis of the total least squares problem, SIAM J. Numer. Anal., 17 (1980) 883–893. * [6] A. Graham, Kronecker Products and Matrix Calculus with Application, Wiley, New York, 1981. * [7] J. Kamm and J. G. Nagy, A total least squares method for Toeplitz system of equations, BIT, 38 (1998) 560–582. * [8] B. Li, Z. Jia, Some results on condition numbers of the scaled total least squares problem, Linear Algebra Appl. (2010), doi:10.1016/j.laa.2010.07.022. * [9] X. Liu, On the solvability and perturbation analysis for total least squares problem, Acta Mathematicae Applicatae Sinica, 19 (1996) 253–262 (in Chinese). * [10] A. N. Malyshev,A unified theory of conditioning for linear least squares and Tikhonov regularization solutions, SIAM J. Matrix. Anal. Appl., 24 (2003) 1186–1196. * [11] C. C. Paige, M. A. Saunders, Towards a generalized singular value decomposition, SIAM J. Numer. Anal., 18 (1981) 398–405. * [12] C. C. Paige, Z. Strako$\check{s}$, Scaled total least squares fundamentals, Numer. Math., 91 (2002) 117–146. * [13] J. R. Rice, A theory of condition, SIAM J. Numer. Anal., 3 (1966) 287–310. * [14] Roger A. Horn, Charles R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985. * [15] M. Wei, The analysis for the total least squares problem with more than one solution, SIAM J. Matrix. Anal. Appl., 13 (1992) 746–763. * [16] M. Wei, On the perturbation of the LS and TLS problems, Mathematica Numerica Sinica, 20 (1998) 267–278 (in Chinese). * [17] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, London, 1965. * [18] L. Zhou, L. Lin, Y. Wei, S. Qiao, Perturbation analysis and condition numbers of Scaled Total Least Squares problems. Numer. Algor., 51 (2009) 381–399.
arxiv-papers
2011-01-12T12:42:18
2024-09-04T02:49:16.362561
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zhongxiao Jia and Bingyu Li", "submitter": "Li", "url": "https://arxiv.org/abs/1101.2342" }
1101.2363
In this article we review the theory of anafunctors introduced by Makkai and Bartels, and show that given a subcanonical site $S$, one can form a bicategorical localisation of various 2-categories of internal categories or groupoids at weak equivalences using anafunctors as 1-arrows. This unifies a number of proofs throughout the literature, using the fewest assumptions possible on $S$. § INTRODUCTION It is a well-known classical result of category theory that a functor is an equivalence (that is, in the 2-category of categories) if and only if it is fully faithful and essentially surjective. This fact is equivalent to the axiom of choice. It is therefore not true if one is working with categories internal to a category $S$ which doesn't satisfy the (external) axiom of choice. This is may fail even in a category very much like the category of sets, such as a well-pointed boolean topos, or even the category of sets in constructive foundations. As internal categories are the objects of a 2-category $\Cat(S)$ we can talk about internal equivalences, and even fully faithful functors. In the case $S$ has a singleton pretopology $J$ (i.e. covering families consist of single maps) we can define an analogue of essentially surjective functors. Internal functors which are fully faithful and essentially surjective are called weak equivalences in the literature, going back to [16]. We shall call them $J$-equivalences for clarity. We can recover the classical result mentioned above if we localise the 2-category $\Cat(S)$ at the class $W_J$ of $J$-equivalences. We are not just interested in localising $\Cat(S)$, but various full sub-2-categories $C \into \Cat(S)$ which arise in the study of presentable stacks, for example algebraic, topological, differentiable, etc. stacks. As such it is necessary to ask for a compatibility condition between the pretopology on $S$ and the sub-2-category we are interested in. We call this condition existence of base change for covers of the pretoplogy, and demand that for any cover $p\colon U\to X_0$ (in $S$) of the object of objects of $X\in C$, there is a fully faithful functor in $C$ with object component $p$. Let $S$ be a category with singleton pretopology $J$ and let $C$ be a full sub-2-category of $\Cat(S)$ which admits base change along arrows in $J$. Then $C$ admits a calculus of fractions for the $J$-equivalences. Pronk gives us the appropriate notion of a calculus of fractions for a 2-category in [53] as a generalisation of the usual construction for categories [29]. In her construction, 1-arrows are spans and 2-arrows are equivalence classes of bicategorical spans of spans. This construction, while canonical, can be a little unwieldy so we look for a simpler construction of the localisation. We find this in the notion of anafunctor, introduced by Makkai for plain small categories [41] (Kelly described them briefly in [34] but did not develop the concept further). In his setting an anafunctor is a span of functors such that the left (or source) leg is a surjective-on-objects, fully faithful functor.[Anafunctors were so named by Makkai, on the suggestion of Pavlovic, after profunctors, in analogy with the pair of terms anaphase/prophase from biology. For more on the relationship between anafunctors and profunctors, see below.] For a general category $S$ with a subcanonical singleton pretopology $J$ [6], the analogon is a span with left leg a fully faithful functor with object component a cover. Composition of anafunctors is given by composition of spans in the usual way, and there are 2-arrows between anafunctors (a certain sort of span of spans) that give us a bicategory $\Cat_\ana(S,J)$ with objects internal categories and 1-arrows anafunctors. We can also define the full sub-bicategory $C_\ana(J) \into \Cat_\ana(S,J)$ analogous to $C$, and there is a strict inclusion 2-functor $C \into C_\ana(J)$. This gives us our second main Let $S$ be a category with subcanonical singleton pretopology $J$ and let $C$ be a full sub-2-category of $\Cat(S)$ which admits base change along arrows in $J$, Then $C \into C_\ana(J)$ is a localisation of $C$ at the class of $J$-equivalences. So far we haven't mentioned the issue of size, which usually is important when constructing localisations. If the site $(S,J)$ is locally small, then $C$ is locally small, in the sense that the hom-categories are small. This also implies that $C_\ana(J)$ and hence any $C[W_J^{-1}]$ has locally small hom-categories i.e. has only a set of 2-arrows between any pair of 1-arrows. To prove that the localisation is locally essentially small (that is, hom-categories are equivalent to small categories), we need to assume a size restriction axiom on the pretopology $J$, called WISC (Weakly Initial Sets of Covers). WISC can be seen as an extremely weak choice principle, weaker than the existence of enough projectives, and states that for every object $A$ of $S$, there is a set of $J$-covers of $A$ which is cofinal in all $J$-covers of $A$. It is automatically satisfied if the pretopology is specified as an assignment of a set of covers to each object. Let $S$ be a category with subcanonical singleton pretopology $J$ satisfying WISC, and let $C$ be a full sub-2-category of $\Cat(S)$ which admits base change along arrows in $J$. Then any localisation of $C$ at the class of $J$-equivalences is locally essentially small. Since a singleton pretopology can be conveniently defined as a certain wide subcategory, this is not a vacuous statement for large sites, such as $\Top$ or $\Grp(E)$ (group objects in a topos $E$). In fact WISC is independent of the Zermelo-Fraenkel axioms (without Choice) [13, 54]. It is thus possible to have the theorem fail for the topos $S = \Set_{\neg AC}$ with surjections as covers. Since there have been many very closely related approaches to localisation of 2-categories of internal categories and groupoids, we give a brief sketch in the following section. Sections 3 to 6 of this article then give necessary background and notation on sites, internal categories, anafunctors and bicategories of fractions respectively. Section 7 contains our main results, while section 8 shows examples from the literature that are covered by the theorems from section 7. A short appendix detailing superextensive sites is included, as this material does not appear to be well-known (they were discussed in the recent [58], Example 11.12). This article started out based on the first chapter of the author's PhD thesis, which only dealt with groupoids in the site of topological spaces and open covers. Many thanks are due to Michael Murray, Mathai Varghese and Jim Stasheff, supervisors to the author. The patrons of the $n$-Category Café and $n$Lab, especially Mike Shulman and Toby Bartels, provided helpful input and feedback. Steve Lack suggested a number of improvements, and the referee asked for a complete rewrite of this article, which has greatly improved the theorems, proofs, and hopefully also the exposition. Any delays in publication are due entirely to the author. § ANAFUNCTORS IN CONTEXT The theme of giving 2-categories of internal categories or groupoids more equivalences has been approached in several different ways over the decades. We sketch a few of them, without necessarily finding the original references, to give an idea of how widely the results of this paper apply. We give some more detailed examples of this applicability in section 8. Perhaps the oldest related construction is the distributors of Bénabou, also known as modules or profunctors [9] (see [31] for a detailed treatment of internal profunctors, as the original article is difficult to source). Bénabou pointed out [12], after a preprint of this article was released, that in the case of the category $\Set$ (and more generally in a finitely complete site with reflexive coequalisers that are stable under pullback, see [42]), the bicategory of small (resp. internal) categories with representable profunctors as 1-arrows is equivalent to the bicategory of small categories with anafunctors as 1-arrows. In fact this was discussed by Baez and Makkai [8], where the latter pointed out that representable profunctors correspond to saturated anafunctors in his setting. The author's preference for anafunctors lies in the fact they can be defined with weaker assumptions on the site $(S,J)$, and in fact in the sequel [56], do not require the 2-category to have objects which are internal categories. In a sense this is analogous to [60], where the formal bicategorical approach to profunctors between objects of a bicategory is given, albeit still requiring more colimits to exist than anafunctors do. Bénabou has pointed out in private communication that he has an unpublished distributor-like construction that does not rely on existence of reflexive coequalisers; the author has not seen any details of this and is curious to see how it compares to anafunctors. Related to this is the original work of Bunge and Paré [16], where they consider functors between indexed categories associated to internal categories, that is, the externalisation of an internal category and stack completions thereof. This was one motivation for considering weak equivalences in the first place, in that a pair of internal categories have equivalent stack completed externalisations if and only if they are connected by a span of internal functors which are weak equivalences. Another approach is constructing bicategories of fractions à la Pronk [53]. This has been followed by a number of authors, usually followed up by an explicit construction of a localisation simplifying the canonical one. Our work here sits at the more general end of this spectrum, as others have tailored their constructions to take advantage of the structure of the site they are interested in. For example, butterflies (originally called papillons) have been used for the category of groups [50, 3, 4], abelian categories [15] and semiabelian categories [1, 42]. These are similar to the meromorphisms of [52], introduced in the context of the site of smooth manifolds; though these only use a 1-categorical approach to localisation. Vitale [62], after first showing that the 2-category of groupoids in a regular category has a bicategory of fractions, then shows that for protomodular regular categories one can generalise the pullback congruences of Bénabou in [11] to discuss bicategorical localisation. This approach can be applied to internal categories, as long as one restricts to invertible 2-arrows. Similarly, [42] give a construction of what they call fractors between internal groupoids in a Mal'tsev category, and show that in an efficiently regular category (e.g. a Barr-exact category) fractors are 1-arrows in a localisation of the 2-category of internal groupoids. The proof also works for internal categories if one considers only invertible 2-arrows. Other authors, in dealing with internal groupoids, have adopted the approach pioneered by Hilsum and Skandalis [30], which has gone by various names including Hilsum-Skandalis morphisms, Morita morphisms, bimodules, bibundles, right principal bibundles and so on. All of these are very closely related to saturated anafunctors, but in fact no published definition of a saturated anafunctor in a site other than $\Set$ ([41]) has appeared, except in the guise of internal profunctors (e.g. [31], section B2.7). Note also that this approach has only been applied to internal groupoids. The review [37] covers the case of Lie groupoids, and in particular orbifolds, while [48] treats bimodules between groupoids in the category of affine schemes, but from the point of view of Hopf algebroids. The link between localisation at weak equivalences and presentable stacks is considered in (of course) [53], as well as more recently in [18], [57], in the cases of topological and algebraic stacks respectively, and for example [61] in the case of differentiable stacks. A third approach is by considering a model category structure on the 1-category of internal categories. This is considered in [32] for categories in a topos, and in [28] for categories in a finitely complete subcanonical site $(S,J)$. In the latter case the authors show when it is possible to construct a Quillen model category structure on $\Cat(S)$ where the weak equivalences are the weak equivalences from this paper. Sufficient conditions on $S$ include being a topos with nno, being locally finitely presentable or being finitely complete regular Mal'tsev – and additionally having enough $J$-projective objects. If one is willing to consider other model-category-like structures, then these assumptions can be dropped. The proof from [28] can be adapted to show that for a finitely complete site $(S,J)$, the category of groupoids with source and target maps restricted to be $J$-covers has the structure of a category of fibrant objects, with the same weak equivalences. We note that [20] gives a Quillen model structure for the category of orbifolds, which are there defined to be proper topological groupoids with discrete hom-spaces. In a similar vein, one could consider a localisation using hammock localisation <cit.> of a category of internal categories, which puts one squarely in the realm of $(\infty,1)$-categories. Alternatively, one could work with the $(\infty,1)$-category arising from a 2-category of internal categories, functors and natural isomorphisms and consider a localisation of this as given in, say [38]. However, to deal with general 2-categories of internal categories in this way, one needs to pass to $(\infty,2)$-categories to handle the non-invertible 2-arrows. The theory here is not so well-developed, however, and one could see the results of the current paper as giving toy examples with which one could work. This is one motivation for making sure the results shown in this paper apply to not just 2-categories of groupoids. Another is extending the theory of presentable stacks from stacks of groupoids to stacks of categories [55]. § SITES The idea of surjectivity is a necessary ingredient when talking about equivalences of categories—in the guise of just essential surjectivity—but it doesn't generalise in a straightforward way from the category $\Set$. The necessary properties of the class of surjective maps are encoded in the definition of a Grothendieck pretopology, in particular a singleton pretopology. This section gathers definitions and notations for later use. A Grothendieck pretopology (or simply pretopology) on a category $S $ is a collection $J$ of families \[ \{ (U_i \to A)_{i\in I} \}_{A\in \Obj(S)} \] of morphisms for each object $A \in S$ satisfying the following properties * $(A' \stackrel{\sim}{\to} A)$ is in $J$ for every isomorphism $A'\simeq A$. * Given a map $B \to A$, for every $(U_i \to A)_{i\in I}$ in $J$ the pullbacks $B \times_A A_i$ exist and $(B \times_A A_i \to B)_{i\in I}$ is in $J$. * For every $(U_i \to A)_{i\in I}$ in $J$ and for a collection $(V_k^i \to U_i)_{k\in K_i}$ from $J$ for each $i \in I$, the family of composites \[ (V_k^i \to A)_{k\in K_i,i\in I} \] are in $J$. Families in $J$ are called covering families. We call a category $S$ equipped with a pretopology $J$ a site, denoted $(S,J)$ (note that often one sees a site defined as a category equipped with a Grothendieck topology). The pretopology $J$ is called a singleton pretopology if every covering family consists of a single arrow $(U \to A)$. In this case a covering family is called a cover and we call $(S,J)$ a unary site. Very often, one sees the definition of a pretopology as being an assignment of a set covering families to each object. We do not require this, as one can define a singleton pretopology as a subcategory with certain properties, and there is not necessarily then a set of covers for each object. One example is the category of groups with surjective homomorphisms as covers. This distinction will be important later. One thing we will require is that sites come with specified pullbacks of covering families. If one does not mind applying the axiom of choice (resp. axiom of choice for classes) then any small site (resp. large site) can be so equipped. But often sites that arise in practice have more or less canonical choices for pullbacks, such as the category of ZF-sets. The prototypical example is the pretopology $\mathcal{O}$ on $\Top$, where a covering family is an open cover. The class of numerable open covers (i.e. those that admit a subordinate partition of unity [25]) also forms a pretopology on $\Top$. Much of traditional bundle theory is carried out using this site; for example the Milnor classifying space classifies bundles which are locally trivial over numerable covers. A covering family $(U_i \to A)_{i\in I} $ is called effective if $A$ is the colimit of the following diagram: the objects are the $U_i$ and the pullbacks $U_i \times_A U_j$, and the arrows are the projections \[ U_i \leftarrow U_i \times_A U_j \to U_j. \] If the covering family consists of a single arrow $(U \to A)$, this is the same as saying $U \to A$ is a regular epimorphism. A site is called subcanonical if every covering family is effective. On $\Top$, the usual pretopology $\mathcal{O}$ of opens, the pretopology of numerable covers and that of open surjections are subcanonical. In a regular category, the class of regular epimorphisms forms a subcanonical singleton In fact we can make the following definition. For a category $S$, the largest class of regular epimorphisms of which all pullbacks exist, and which is stable under pullback, is called the canonical singleton pretopology and denoted $\can$. This is a to be contrasted to the canonical topology on a category, which consists of covering sieves rather than covers. The canonical singleton pretopology is the largest subcanonical singleton pretopology on a category. Let $(S,J)$ be a site. An arrow $P \to A$ in $S$ is called a $J$-epimorphism if there is a covering family $(U_i \to A)_{i\in I}$ and a lift \[ \xymatrix{ & P \ar[d] \\ U_i \ar@{-->}[ur] \ar[r] & A \] for every $i \in I$. A $J$-epimorphism is called universal if its pullback along an arbitrary map exists. We denote the singleton pretopology of universal $J$-epimorphisms by $J_{un}$. This definition of $J$-epimorphism is equivalent to the definition in III.7.5 in [40]. The dotted maps in the above definition are called local sections, after the case of the usual open cover pretopology on $\Top$. If $J$ is a singleton pretopology, it is clear that $J \subset J_{un}$. The universal $\mathcal{O}$-epimorphisms for the pretopology $\mathcal{O}$ of open covers on $\Diff$ form $Subm$, the pretopology of surjective submersions. In a finitely complete category the universal $triv$-epimorphisms are the split epimorphisms, where $triv$ is the trivial pretopology where all covering families consist of a single isomorphism. In $\Set$ with the axiom of choice there are all the epimorphisms. Note that for a finitely complete site $(S,J)$, $J_{un}$ contains $triv_{un}$, hence all the split epimirphisms. Although we will not assume that all sites we consider are finitely complete, results similar to ours have, and so in that case we can say a little more, given stronger properties on the pretopology. A singleton pretopology $J$ is called saturated if whenever the composite $A \stackrel{h}{\to} B \stackrel{g}{\to} C$ is in $J$, then $g\in J$. The concept of a saturated pretopology was introduced by Bénabou under the name calibration [10]. It follows from the definition that a saturated singleton pretopology contains the split epimorphisms (take $h$ to be a section of the epimorphism $g$). The canonical singleton pretopology $\can$ in a regular category (e.g. a topos) is saturated. Given a pretopology $J$ on a finitely complete category, $J_{un}$ is saturated. Sometimes a pretopology $J$ contains a smaller pretopology that still has enough covers to compute the same $J$-epimorphisms. If $J$ and $K$ are two singleton pretopologies with $J \subset K$, such that $K \subset J_{un}$, then $J$ is said to be cofinal in $K$. Clearly $J$ is cofinal in $J_{un}$ for any singleton pretopology $J$. If $J$ is cofinal in $K$, then $J_{un} = K_{un}$. We have the following lemma, which is essentially proved in [31], C2.1.6. If a pretopology $J$ is subcanonical, then so any pretopology in which it is cofinal. In particular, $J$ subcanonical implies $J_{un}$ subcanonical. As mentioned earlier, one may be given a singleton pretopology such that each object has more than a set's worth of covers. If such a pretopology contains a cofinal pretopology with set-many covers for each object, then we can pass to the smaller pretopology and recover the same results (in a way that will be made precise later). In fact, we can get away with something weaker: one could ask only that the category of all covers of an object (see definition <ref> below) has a set of weakly initial objects, and such set may not form a pretopology. This is the content of the axiom WISC below. We first give some more precise definitions. A category $C$ has a weakly initial set $\mathcal{I}$ of objects if for every object $A$ of $C$ there is an arrow $O\to A$ from some object $O\in \mathcal{I}$. For example the large category $\Fields$ of fields has a weakly initial set, consisting of the prime fields $\{\mathbb{Q},\mathbb{F}_p|p\textrm{ prime}\}$. To contrast, the category of sets with surjections for arrows doesn't have a weakly initial set of objects. Every small category has a weakly initial set, namely its set of objects. We pause only to remark that the statement of the adjoint functor theorem can be expressed in terms of weakly initial sets. Let $(S,J)$ be a site. For any object $A$, the category of covers of $A$, denoted $J/A$ has as objects the covering families $(U_i \to A)_{i\in I}$ and as morphisms $(U_i \to A)_{i\in I} \to (V_j \to A)_{j\in J}$ tuples consisting of a function $r\colon I\to J$ and arrows $U_i \to V_{r(i)}$ in $S/A$. When $J$ is a singleton pretopology this is simply a full subcategory of $S/A$. We now define the axiom WISC (Weakly Initial Set of Covers), due independently to Mike Shulman and Thomas Streicher. A site $(S,J)$ is said to satisfy WISC if for every object $A$ of $S$, the category $J/A$ has a weakly initial set of objects. A site satisfying WISC is in some sense constrained by a small amount of data for each object. Any small site satisfies WISC, for example, the usual site of finite-dimensional smooth manifolds and open covers. Any pretopology $J$ containing a cofinal pretopology $K$ such that $K/A$ is small for every object $A$ satisfies WISC. Any regular category (for example a topos) with enough projectives, equipped with the canonical singleton pretopology, satisfies WISC. In the case of $\Set$ `enough projectives' is the Presentation Axiom (PAx), studied, for instance, by Aczel [2] in the context of constructive set theory. $(\Top,\mathcal{O})$ satisfies WISC, using AC in $\Set$. Choice may be more than is necessary here; it would be interesting to see if weaker choice principles in the site $(\Set,surjections)$ are enough to prove WISC for $(\Top,\mathcal{O})$ or other concrete sites. If $(S,J)$ satisfies WISC, then so does $(S,J_{un})$. It is instructive to consider an example where WISC fails in a non-artificial way. The category of sets and surjections with all arrows covers clearly doesn't satisfy WISC, but is contrived and not a `useful' sort of category. For the moment, assume the existence of a Grothendieck universe $\mathbb{U}$ with cardinality $\lambda$, and let $\mathrm{Set}_\mathbb{U}$ refer to the category of $\mathbb{U}$-small sets. Clearly we can define WISC relative to $\mathbb{U}$, call it WISC${}_\mathbb{U}$. Let $G$ be a $\mathbb{U}$-large group and $\mathbf{B}G$ the $\mathbb{U}$-large groupoid with one object associated to $G$. The boolean topos $\mathrm{Set}_\mathbb{U}^{\mathbf{B}G}$ of $\mathbb{U}$-small $G$-sets is a unary site with the class $epi$ of epimorphisms for covers. One could consider this topos as being an exotic sort of forcing construction. If $G$ has at least $\lambda$-many conjugacy classes of subgroups, then $(\mathrm{Set}_\mathbb{U}^{\mathbf{B}G},epi)$ does not satisfy Alternatively, one could work in foundations where it is legitimate to discuss a proper class-sized group, and then consider the topos of sets with an action by this group. If there is a proper class of conjugacy classes of subgroups, then this topos with its canonical singleton pretopology will fail to satisfy WISC. Simple examples of such groups are $\mathbb{Z}^\mathbb{U}$ (given a universe $\mathbb{U}$) and $\mathbb{Z}^K$ (for some proper class $K$). Recently, [13] (relative to a large cardinal axiom) and [54] (with no large cardinals) have shown that the category of sets may fail to satisfy WISC. The models constructed in [33] are also conjectured to not satisfy WISC. Perhaps of independent interest is a form of WISC with a bound: the weakly initial set for each category $J/A$ has cardinality less than some cardinal $\kappa$ (call this WISC${}_\kappa$). Then one could consider, for example, sites where each object has a weakly initial finite or countable set of covers. Note that the condition `enough projectives' is the case $\kappa = 2$. § INTERNAL CATEGORIES Internal categories were introduced in [27], starting with differentiable and topological categories (i.e. internal to $\Diff$ and $\Top$ respectively). We collect here the necessary definitions, terminology and notation. For a thorough recent account, see [7] or the encyclopedic Fix a category $S$, referred to as the ambient category. An internal category $X$ in a category $S$ is a diagram \[ X_1 \times_{X_0} X_1 \times_{X_0} X_1\rightrightarrows X_1 \times_{X_0} X_1 \xrightarrow{m} X_1 \stackrel{s,t}{\st} X_0 \xrightarrow{e} X_1 \] in $S$ such that the multiplication $m$ is associative (we demand the limits in the diagram exist), the unit map $e$ is a two-sided unit for $m$ and $s$ and $t$ are the usual source and target. An internal groupoid is an internal category with an \[ (-)^{-1}\colon X_1 \to X_1 \] satisfying the usual diagrams for an inverse. Since multiplication is associative, there is a well-defined map $X_1 \times_{X_0} X_1 \times_{X_0} X_1 \to X_1$, which will also be denoted by $m$. The pullback in the diagram in definition <ref> is \[ \xymatrix{ X_1 \times_{X_0} X_1 \ar[r] \ar[d] & X_1 \ar[d]^-{s}\\ X_1 \ar[r]_-{t} & X_0\;. \] and the double pullback is the limit of $X_1 \stackrel{t}{\rightarrow} X_0 \stackrel{s}{\leftarrow} X_1 \stackrel{t}{\rightarrow} X_0 \stackrel{s}{\leftarrow}X_0$. These, and pullbacks like these (where source is pulled back along target), will occur often. If confusion can arise, the maps in question will be explicity written, as in $X_1 \times_{s,X_0,t} X_1$. One usually sees the requirement that $S$ is finitely complete in order to define internal categories. This is not strictly necessary, and not true in the well-studied case of $S = \Diff$, the category of smooth manifolds. Often an internal category will be denoted $X_1 \st X_0$, the arrows $m,s,t,e$ (and $(-)^{-1}$) will be referred to as structure maps and $X_1$ and $X_0$ called the object of arrows and the object of objects respectively. For example, if $S = \Top$, we have the space of arrows and the space of objects, for $S = \Grp$ we have the group of arrows and so on. If $X \to Y$ is an arrow in $S$ admitting iterated kernel pairs, there is an internal groupoid $\check{C}(X)$ with $\check{C}(X)_0 = X$, $\check{C}(X)_1 = X \times_Y X$, source and target are projection on first and second factor, and the multiplication is projecting out the middle factor in $X \times_Y X \times_Y X$. This groupoid is called the Čech groupoid of the map $X \to Y$. The origin of the name is that in $\Top$, for maps of the form $\coprod_I U_i \to Y$ (arising from an open cover), the Čech groupoid $\check{C}(\coprod_I U_i)$ appears in the definition of Čech cohomology. Let $S$ be a category with binary products. For each object $A \in S$ there is an internal groupoid $\disc(A)$ which has $\disc(A)_1 = \disc(A)_0 = A$ and all structure maps equal to $id_A$. Such a category is called discrete. There is also an internal groupoid $\codisc(A)$ with \[ \codisc(A)_0 = A,\ \codisc(A)_1 = A \times A \] and where source and target are projections on the first and second factor respectively. Such a groupoid is called Given internal categories $X$ and $Y$ in $S$, an internal functor is a pair of maps \[ f_0\colon X_0 \to Y_0 \quad\textrm{and}\quad f_1\colon X_1 \to Y_1 \] called the object and arrow component respectively. Both components are required to commute with all the structure maps. If $A\to C$ and $B\to C$ are maps admitting iterated kernel pairs, and $A \to B$ is a map over $C$, there is a functor $\check{C}(A) \to \check{C}(B)$. If $(S,J)$ is a subcanonical unary site, and $U \to A$ is a cover, a functor $\check{C}(U) \to \disc(B)$ gives a unique arrow $A\to B$. This follows immediately from the fact $A$ is the colimit of the diagram underlying $\check{C}(U)$. Given internal categories $X,Y$ and internal functors $f,g\colon X \to Y$, an internal natural transformation (or simply transformation) \[ a\colon f \Rightarrow g \] is a map $a\colon X_0 \to Y_1$ such that $s \circ a = f_0,\ t\circ a = g_0$ and the following diagram commutes \begin{equation}\label{diag:naturality} \xymatrix{ X_1 \ar[r]^-{(g_1,a\circ s)} \ar[d]_{(a \circ t,f_1)} & Y_1 \times_{Y_0} Y_1 \ar[d]^{m} \\ Y_1 \times_{Y_0} Y_1 \ar[r]^-{m} & Y_1 \end{equation} expressing the naturality of $a$. Internal categories (resp. groupoids), functors and transformations in a locally small category $S$ form a locally small 2-category $\Cat(S)$ (resp. $\Gpd(S)$) [27]. There is clearly an inclusion 2-functor $\Gpd(S) \to \Cat(S)$. Also, $\disc$ and $\codisc$, described in example <ref>, are 2-functors $S \to \Gpd(S)$, whose underlying functors are left and right adjoint to the functor \[ \Obj\colon\Cat(S)_{\leq 1} \to S,\qquad (X_1\st X_0)\mapsto X_0. \] Here $\Cat(S)_{\leq 1}$ is the 1-category underlying the 2-category $\Cat(S)$. Hence for an internal category $X$ in $S$, there are functors $\disc(X_0) \to X$ and $X \to \codisc(X_0)$, the arrow component of the latter being $(s,t):X_1\to X_0^2$. We say a natural transformation is a natural isomorphism if it has an inverse with respect to vertical composition. Clearly there is no distinction between natural transformations and natural isomorphisms when the codomain of the functors is an internal groupoid. We can reformulate the naturality diagram (<ref>) in the case that $a$ is a natural isomorphism. Denote by $-a$ the inverse of $a$. Then the diagram (<ref>) commutes if and only if the diagram \begin{equation}\label{naturality} \xymatrix{ X_0 \times_{X_0} X_1 \times_{X_0} X_0 \ar[rr]^{-a\times f_1 \times a} \ar[d]_{\simeq} &&Y_1 \times_{Y_0} Y_1 \times_{Y_0} Y_1 \ar[d]^m \\ X_1 \ar[rr]_{g_1} && Y_1 \end{equation} commutes, a fact we will use several times. If $X$ is a category in $S$, $A$ is an object of $S$ and $f,g:X \to \codisc(A)$ are functors, there is a unique natural isomorphism $f\stackrel{\sim}{\Rightarrow} g$. An internal or strong equivalence of internal categories is an equivalence in the 2-category of internal categories. That is, an internal functor $f \colon X\to Y$ such that there is a functor $f'\colon Y\to X$ and natural isomorphisms $f\circ f' \Rightarrow \id_Y$, $f'\circ f \Rightarrow \id_X$. For an internal category $X$ and a map $p:M\to X_0$ in $S$ the base change of $X$ along $p$ is any category $X[M]$ with object of objects $M$ and object of arrows given by the pullback \[ \xymatrix{ M^2 \times_{X_0^2} X_1 \ar[r] \ar[d] & X_1 \ar[d]^{(s,t)} \\ M^2 \ar[r]_{p^2} & X_0^2 \] If $C\subset \Cat(S)$ denotes a full sub-2-category and if the base change along any map in a given class $K$ of maps exists in $C$ for all objects of $C$, then we say $C$ admits base change along maps in $K$, or simply admits base change for $K$. In all that follows, `category' will mean object of $C$ and similarly for `functor' and `natural transformation/isomorphism'. The strict pullback of internal categories \[ \xymatrix{ X \times_Y Z \ar[r] \ar[d] & Z \ar[d] \\ X \ar[r] & Y \] when it exists, is the internal category with objects $X_0 \times_{Y_0} Z_0$, arrows $X_1 \times_{Y_1} Z_1$, and all structure maps given componentwise by those of $X$ and $Z$. Often we will be able to prove that certain pullbacks exist because of conditions on various component maps in $S$. We do not assume that all strict pullbacks of internal categories exists in our chosen $C$. It follows immediately from definition <ref> that given maps $N\to M$ and $M\to X_0$, there is a canonical isomorphism \begin{equation}\label{induced_cat_1} X[M][N] \simeq X[N]. \end{equation} with object component the identity map, when these base changes exist. If we agree to follow the convention that $M \times_N N = M$ is the pullback along the identity arrow $\id_N$, then $X[X_0] = X$. This also simplifies other results of this paper, so will be adopted from now on. One consequence of this assumption is that the iterated fibre product \[ M\times_M M \times_M \ldots \times_M M, \] bracketed in any order, is equal to $M$. We cannot, however, equate two bracketings of a general iterated fibred product; they are only canonically isomorphic. Let $Y\to X$ be a functor in $S$ and $j_0\colon U \to X_0$ a map. If the base change along $j_0$ exists, the following square is a strict pullback \[ \xymatrix{ Y[Y_0\times_{X_0}U] \ar[r] \ar[d] & X[U] \ar[d]^j \\ Y \ar[r] & X \] assuming it exists. Since base change along $j_0$ exists, we know that we have the functor $Y[Y_0\times_{X_0}U] \to Y$, we just need to show it is a strict pullback of $j$. On the level of objects this is clear, and on the level of arrows, we have \begin{align*} (Y_0\times_{X_0}U)^2 \times_{Y_0^2}Y_1 &\simeq U^2\times_{X_0^2} Y_1\\ &\simeq (U^2\times_{X_0^2}X_1) \times_{X_1}Y_1 \\ &\simeq X[U]_1\times_{X_1}Y_1 \end{align*} so the square is a pullback. We are interested in 2-categories $C$ which admits base change for a given pretopology $J$ on $S$, which we shall cover in more detail in section <ref>. Equivalences in $\Cat$—assuming the axiom of choice—are precisely the fully faithful, essentially surjective functors. For internal categories, however, this is not the case. In addition, we need to make use of a pretopology to make the `surjective' part of `essentially surjective' meaningful. Let $(S,J)$ be a unary site. An internal functor $f:X \to Y$ in $S$ is called * fully faithful if \[ \xymatrix{ X_1 \ar[r]^{f_1} \ar[d]_{(s,t)} & Y_1 \ar[d]^{(s,t)}\\ X_0 \times X_0 \ar[r]_{f_0 \times f_0} & Y_0 \times Y_0 \] is a pullback diagram; * $J$-locally split if there is a $J$-cover $U\to Y_0$ and a diagram \[ \xymatrix{ Y[U] \ar[d]_{\bar f} \ar@/^.5pc/[dr]_{\ }="s1"^{u}& \\ X\ar[r]_{f}^(.33){\ }="t1"&Y \ar@{=>}"s1";"t1" \] commuting up to a natural isomorphism; * a $J$-equivalence if it is fully faithful and $J$-locally split. The class of $J$-equivalences will be denoted $W_J$. If mention of $J$ is suppressed, they will be called weak equivalences. There is another defintion of full faithfulness for internal categories, namely that of a functor $f\colon Z\to Y$ being representably fully faithful. This means that for all categories $Z$, the functor \[ f_\ast\colon \Cat(S)(Z,X) \to \Cat(S)(Z,Y) \] is fully faithful. It is a well-known result that these two notions coincide, so we shall use either characterisation as needed. If $f:X \to Y$ is a fully faithful functor such that $f_0$ is in $J$, then $f$ is $J$-locally split. That is, the canonical functor $X[U] \to X$ is a $J$-equivalence whenever the base change exists. Also, we do not require that $J$ is subcanonical. We record here a useful lemma. Given a fully faithful functor $f\colon X \to Y$ in $C$ and a natural isomorphism $f \Rightarrow g$, the functor $g$ is also fully faithful. In particular, an internal equivalence is fully faithful. This is a simple application of the definition of representable full faithfulness and the fact that the result is true in $\Cat$. The first definition of weak equivalence of internal categories along the lines we are considering appeared in [16] for $S$ a regular category, and $J$ the class of regular epimorphisms (i.e. $\can$), in the context of stacks and indexed categories. This was later generalised in [28] to more general finitely complete sites to discuss model structures on the category of internal categories. Both work only with saturated singleton pretopologies. Note that when $S$ is finitely complete, the object $X_1^{iso} \into X_1$ of isomorphisms of a category $X$ can be constructed as a finite limit [16], and in the case when $X$ is a groupoid we have $X_1^{iso} \simeq X_1$. [16, 28] For a finitely complete unary site $(S,J)$ with $J$ saturated, a functor $f$ is called essentially $J$-surjective if the arrow labelled $\circledast$ below is in $J$. \[ \xymatrix{ &\ar[dl] X_0 \times_{Y_0} Y_1^{iso} \ar@/^1pc/[ddr]^\circledast \ar[d]&\\ X_0 \ar[d]_{f_0} & \ar[dl]^s Y_1^{iso} \ar[dr]_t &\\ Y_0 && Y_0 \] A functor is called a Bunge-Paré $J$-equivalence if it is fully faithful and essentially $J$-surjective. Denote the class of such maps by $W_J^{BP}$. Definition <ref> is equivalent to the one in [16, 28] in the sites they consider but seems more appropriate for sites without all finite limits. Also, definition <ref> makes sense in 2-categories other than $\Cat(S)$ or sub-2-categories thereof. Let $(S,J)$ be a finitely complete unary site with $J$ saturated. Then a functor is a $J$-equivalence if and only if it is a Bunge-Paré Let $f\colon X \to Y$ be a Bunge-Paré $J$-equivalence, and consider the $J$-cover given by the map $U := X_0 \times_{Y_0} Y_1^{iso} \to Y_0$. Denote by $\iota\colon U\to Y_1^{iso}$ the projection on the second factor, by $-\iota$ the composite of $\iota$ with the inversion map $(-)^{-1}$ and by $s_0\colon U\to X_0$ the projection on the first factor. The arrow $s_0$ will be the object component of a functor $s\colon Y[U] \to X$, we need to define the arrow component $s_1$. Consider the composite \begin{align*} Y[U]_1 \simeq U\times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{(s,\iota)\times\id\times(-\iota,s)} (X_0 \times_{Y_0} Y_1^{iso}) \times_{Y_0} Y_1 \times_{Y_0} ( Y_1^{iso} \times_{Y_0} X_0) \\ \hookrightarrow X_0 \times_{Y_0} Y_3 \times_{Y_0} X_0 \xrightarrow{\id\times m\times\id} X_0 \times_{Y_0} Y_1 \times_{Y_0} X_0 \simeq X_1 \end{align*} where the last isomorphism arises from $f$ being fully faithful. It is clear that this commutes with source and target, because these are given by projection on the first and last factor at each step. To see that it respects identities and composition, one can use generalised elements and the fact that the $\iota$ component will cancel with the $-\iota = (-)^{-1}\circ \iota$ component. We define the natural isomorphism $f\circ s \Rightarrow j$ (here $j\colon Y[U] \to Y$ is the canonical functor) to have component $\iota$ as denoted above. Notice that the composite $f_1\circ s_1$ is just \[ Y[U]_1 \simeq U \times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{\iota\times\id \times -\iota} Y_1^{iso} \times_{Y_0} Y_1 \times_{Y_0} Y_1^{iso} \hookrightarrow Y_3 \xrightarrow{m} Y_1. \] Since the arrow component of $Y[U] \to Y$ is $U \times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{\pr_2} Y_1$, $\iota$ is indeed a natural isomorphism using the diagram (<ref>). Thus a Bunge-Paré $J$-equivalence is a $J$-equivalence. In the other direction, given a $J$-equivalence $f\colon X\to Y$, we have a $J$-cover $j\colon U\to Y_0$ and a map $(\overline{f},a)\colon U \to X_0 \times Y_1^{iso}$ such that $j = (t\circ pr_2)\circ(\overline{f},a)$. Since $J$ is saturated, $(t\circ pr_2)\in J$ and hence $f$ is a Buge-Paré $J$-equivalence. We can thus use definition <ref> as we like, and it will still refer to the same sorts of weak equivalences that appear in the literature. § ANAFUNCTORS We now let $J$ be a subcanonical singleton pretopology on the ambient category $S$. In this section we assume that $C\into \Cat(S)$ admits base change along arrows in the given pretopology $J$. This is a slight generalisation of what is considered in [6], where only $C = \Cat(S)$ is considered. [41, 6] An anafunctor in $(S,J)$ from a category $X$ to a category $Y$ consists of a $J$-cover $(U \to X_0)$ and an internal functor \[ f\colon X[U] \to Y. \] Since $X[U]$ is an object of $C$, an anafunctor is a span in $C$, and can be denoted \[ (U,f)\colon X \gento Y. \] For an internal functor $f\colon X \to Y$ in $S$, define the anafunctor $(X_0,f)\colon X \gento Y$ as the following span \[ X \xleftarrow{=} X[X_0] \xrightarrow{f} Y. \] We will blur the distinction between these two descriptions. If $f=id\colon X \to X$, then $(X_0,id)$ will be denoted simply by $id_X$. If $U \to A$ is a cover in $(S,J)$ and $\mathbf{B}G$ is a groupoid with one object in $S$ (i.e. a group in $S$), an anafunctor $(U,g)\colon\disc(A) \gento \mathbf{B}G$ is the same thing as a Čech cocycle. [41, 6] Let $(S,J)$ be a site and let \[ (U,f),(V,g)\colon X \gento Y \] be anafunctors in $S$. A transformation \[ \alpha\colon (U,f) \Rightarrow (V,g) \] from $(U,f)$ to $(V,g)$ is a natural transformation \[ \xymatrix{ & \ar[dl] X[U\times_{X_0}V] \ar[dr] & \\ X[U] \ar[dr]_f & \stackrel{\alpha}{\Rightarrow} & X[V] \ar[dl]^g\\ & Y & \] If $\alpha$ is a natural isomorphism, then $\alpha$ will be called an isotransformation. In that case we say $(U,f)$ is isomorphic to $(V,g)$. Clearly all transformations between anafunctors between internal groupoids are isotransformations. Given functors $f,g\colon X \to Y$ between categories in $S$, and a natural transformation $a\colon f \Rightarrow g$, there is a transformation $a\colon (X_0,f) \Rightarrow (X_0,g)$ of anafunctors, given by the component $X_0\times_{X_0}X_0 = X_0 \xrightarrow{a} Y_1$. If $(U,g),(V,h)\colon \disc(A) \gento \mathbf{B}G$ are two Čech cocycles, a transformation between them is a coboundary on the cover $U\times_A V\to A$. Let $(U,f)\colon X \gento Y$ be an anafunctor in $S$. There is an isotransformation $1_{(U,f)}\colon (U,f) \Rightarrow (U,f)$ called the identity transformation, given by the natural transformation with component \begin{equation}\label{id_transf_component} U \times_{X_0} U \simeq (U \times U) \times_{X_0^2} X_0 \xrightarrow{id_U^2 \times e} X[U]_1 \xrightarrow{f_1} Y_1 \end{equation} [41] Given anafunctors $(U,f)\colon X\to Y$ and $(V,f\circ k)\colon X \to Y$ where $k \colon V\to U$ is a cover (over $X_0$), a renaming transformation \[ (U,f)\Rightarrow(V,f\circ k) \] is an isotransformation with component \[ 1_{(U,f)}\circ (k\times \id):V\times_{X_0} U \to U\times_{X_0} U \to Y_1. \] (We also call its inverse for vertical composition a renaming transformation.) If $k$ is an isomorphism, then it will itself be referred to as a renaming isomorphism. We define (following [6]) the composition of anafunctors as follows. Let \[ (U,f)\colon X \gento Y \quad \textrm{and} \quad (V,g)\colon Y \gento Z \] be anafunctors in the site $(S,J)$. Their composite $(V,g)\circ(U,f)$ is the composite span defined in the usual way. It is again a span in $C$: \[ \xymatrix{ && \ar[dl] X[U\times_{Y_0}V] \ar[dr]^{f^V} & \\ &\ar[dl]X[U] \ar[dr]_f & & Y[V] \ar[dl] \ar[dr]^g\\ X&& Y &&Z \] The square is a pullback by lemma <ref> (which exists because $V\to Y_0$ is a cover), and the resulting span is an anafunctor because $V \to Y_0$, hence $U\times_{Y_0}V\to X_0$, are covers, and using the isomorphism (<ref>). We will sometimes denote the composite by $(U\times_{Y_0}V,g\circ f^V)$. Here we are using the fact we have specified pullbacks of covers in $S$. Without this we would not end up with a bicategory (see theorem <ref>), but what [41] calls an anabicategory. This is similar to a bicategory, but composition and other structural maps are only anafunctors, not functors. Consider the special case when $V = Y_0$, so that $(Y_0,g)$ is just an ordinary functor. Then there is a renaming transformation (the identity transformation!) $(Y_0,g)\circ(U,f) \Rightarrow (U,g\circ f)$, using the equality $U \times_{Y_0} Y_0= U$ (by remark <ref>). If we let $g=\id_Y$, then we see that $(Y_0,\id_Y)$ is a strict unit on the left for anafunctor composition. Similarly, considering $(V,g)\circ(Y_0,\id)$, we see that $(Y_0,\id_Y)$ is a two-sided strict unit for anafunctor composition. In fact, we have also proved Given two functors $f\colon X\to Y$, $g\colon Y \to Z$ in $S$, their composition as anafunctors is equal to their composition as functors: \[ (Y_0,g)\circ(X_0,f) = (X_0,g\circ f). \] As a concrete and relevant example of a renaming transformation we can consider the triple composition of anafunctors \begin{align*} (U,f)\colon & X \gento Y,\\ (V,g)\colon & Y \gento Z,\\ (W,h)\colon & Z \gento A. \end{align*} The two possibilities of composing these are \[ \left((U\times_{Y_0} V)\times_{Z_0}W,h\circ(gf^V)^W\right)\quad \text{and}\quad \left(U \times_{Y_0} (V\times_{Z_0} W),h\circ g^W\circ f^{V\times_{Z_0}W}\right). \] The unique isomorphism $(U\times_{Y_0} V)\times_{Z_0}W \simeq U\times_{Y_0} (V \times_{Z_0} W)$ commuting with the various projections is a renaming isomorphism. The isotransformation arising from this renaming transformation is called the associator. A simple but useful criterion for describing isotransformations where one of the anafunctors involved is a functor is as follows. An anafunctor $(V,g)\colon X \gento Y$ is isomorphic to a functor $(X_0,f)\colon X \gento Y$ if and only if there is a natural isomorphism \[ \xymatrix{ & \ar[dl] X[V] \ar[dr]^g \\ X \ar@/_1.5pc/[rr]_(.6){f}& \stackrel{\sim}{\Rightarrow} & Y \] Just as there is a vertical composition of natural transformations between internal functors, there is a vertical composition of transformations between internal anafunctors [6]. This is where the subcanonicity of $J$ will be used in order to construct a map locally over some cover. Consider the following diagram \[ \xymatrix{ && \ar[dl] X[U\times_{X_0} V\times_{X_0} W] \ar[dr]\\ & \ar[dl] X[U\times_{X_0} V] \ar[dr] & & \ar[dl] X[V\times_{X_0} W] \ar[dr]\\ X[U] \ar[drr]_f & \stackrel{a}{\Rightarrow} & X[V] \ar[d]^g& \stackrel{b} {\Rightarrow} & X[W] \ar[dll]^h \\ \] We can form a natural transformation between the leftmost and the rightmost composites as functors in $S$. This will have as its component the arrow \[ \widetilde{ba}\colon U\times_{X_0} V\times_{X_0} W \xrightarrow{\id\times \Delta \times \id} U\times_{X_0}V\times_{X_0}V\times_{X_0} W \xrightarrow{a\times b} Y_1\times_{Y_0} Y_1 \xrightarrow{m} Y_1 \] in $S$. Notice that the Čech groupoid of the cover \begin{equation}\label{iterated_cover} U\times_{X_0} V\times_{X_0} W \to U \times_{X_0} W \end{equation} \[ U\times_{X_0} V\times_{X_0} V\times_{X_0} W \st U\times_{X_0} V\times_{X_0} W, \] with source and target arising from the two projections $V\times_{X_0} V \to V$. Denote this pair of parallel arrows by $s,t\colon UV^2W \st UVW$ for brevity. In [6], section 2.2.3, we find the commuting diagram \begin{equation}\label{tobys_diag} \xymatrix{ UV^2W \ar[r]^t \ar[d]_s & UVW \ar[d]^{\widetilde{ba}}\\ UVW \ar[r]_{\widetilde{ba}} & Y_1 \end{equation} (this can be checked by using generalised elements) and so we have a functor \[ \check{C}(U\times_{X_0} V\times_{X_0} W) \to \disc(Y_1). \] Our pretopology $J$ is assumed to be subcanonical, so example <ref> gives us a unique arrow $ba\colon U\times_{X_0} W \to Y_1$, which is the data for the composite of $a$ and $b$. In the special case that $U\times_{X_0} V\times_{X_0} W \to U \times_{X_0} W$ is split (e.g. is an isomorphism), the composite transformation has \[ U \times_{X_0} W\to U\times_{X_0} V\times_{X_0} W \xrightarrow{\widetilde{ba}} Y_1 \] as its component arrow. In particular, this is the case if one of $a$ or $b$ is a renaming transformation. Let $(U,f):X\gento Y$ be an anafunctor and $U'' \xrightarrow{j'} U' \xrightarrow{j} U$ successive refinements of $U \to X_0$ (e.g isomorphisms). Let $(U',f_{U'})$ and $(U'',f_{U''})$ denote the composites of $f$ with $X[U'] \to X[U]$ and $X[U''] \to X[U]$ respectively. The arrow \[ U \times_{X_0} U'' \xrightarrow{\id_U\times j\circ j'} U \times_{X_0} U \to Y_1 \] is the component for the composition of the isotransformations $(U,f) \Rightarrow(U',f_{U'}),\Rightarrow(U'',f_{U''})$ described in example <ref>. Thus we can see that the composite of renaming transformations associated to isomorphisms $\phi_1,\phi_2$ is simply the renaming transformation associated to their composite $\phi_1\circ \phi_2$. This can be used to show that the associator satisfies the necessary coherence conditions. If $a\colon f\Rightarrow g,\ b\colon g\Rightarrow h$ are natural transformations between functors $f,g,h\colon X\to Y$ in $S$, their composite as transformations between anafunctors \[ (X_0,f),(X_0,g),(X_0,h)\colon X\gento Y. \] is just their composite as natural transformations. This uses the equality \[ X_0\times_{X_0} X_0\times_{X_0} X_0= X_0\times_{X_0} X_0 = X_0, \] which is due to our choice in remark <ref> of canonical Even though we don't have pseudoinverses for weak equivalences of internal categories, one might guess that the local splitting guaranteed to exist by definition is actually more than just a splitting of sorts. This is in fact the case, if we use anafunctors. Let $f\colon X \to Y$ be a $J$-equivalence in $S$. There is an anafunctor \[ (U,\bar{f})\colon Y \gento X \] and isotransformations \begin{align*} \iota\colon (X_0,f)\circ (U,\bar{f}) & \Rightarrow id_Y\\ \epsilon\colon(U,\bar{f})\circ (X_0,f) & \Rightarrow id_X \end{align*} We have the anafunctor $(U,\bar{f})$ by definition as $f$ is $J$-locally split. Since the anafunctors $\id_X,\ \id_Y$ are actually functors, we can use lemma <ref>. Using the special case of anafunctor composition when the second is a functor, this tells us that $\iota$ will be given by a natural isomorphism \[ \xymatrix{ & X \ar[dr]^{f}_(0.2){\ }="s" & \\ Y[U] \ar[rr]^{\ }="t" \ar[ur]^{\bar{f}} && Y \ar@{=>}"s";"t" \] with component $\iota\colon U \to Y_1$. Notice that the composite $f_1\circ \bar{f}_1$ is just \[ Y[U]_1 \simeq U \times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{\iota\times\id \times -\iota} Y_1 \times_{Y_0} Y_1 \times_{Y_0} Y_1 \hookrightarrow Y_3 \xrightarrow{m} Y_1. \] Since the arrow component of $Y[U] \to Y$ is $U \times_{Y_0} Y_1 \times_{Y_0} U \xrightarrow{\pr_2} Y_1$, $\iota$ is indeed a natural isomorphism using the diagram The other isotransformation $\epsilon$ is between $(X_0\times_{Y_0} U,\bar{f}\circ \pr_2)$ and $(X_0,\id_X)$, and is given by the component \[ \epsilon\colon X_0 \times_{X_0} X_0\times_{Y_0} U = X_0\times_{Y_0} U \xrightarrow{\id\times (\bar{f}_0,\iota)} X_0\times_{Y_0} (X_0\times_{Y_0} Y_1) \simeq X_0^2 \times_{Y_0^2} Y_1 \simeq X_1 \] The diagram \[ \xymatrix{ (X_0\times_{Y_0^2} U)^2 \times_{X_0^2} X_1 \ar[d]_\simeq \ar[rr]^{\pr_2} & &X_1 \ar[dd]^\simeq\\ U \times_{Y_0} X_1 \times_{Y_0}U \ar[d]_{-\iota\times f\times\iota} & \\ (X_0 \times_{Y_0} Y_1) \times_{Y_0} Y_1 \times_{Y_0} (Y_1 \times_{Y_0} X_0) \ar[rr]_(.6){\id\times m \times \id} && X_0\times_{Y_0} Y_1 \times_{Y_0} X_0 \] commutes (a fact which can be checked using generalised elements), and using (<ref>) we see that $\epsilon$ is natural. The first half of the following theorem is proposition 12 in [6], and the second half follows because all the constructions of categories involved in dealing with anafunctors outlined above are still objects of $C$. [6] For a site $(S,J)$ where $J$ is a subcanonical singleton pretopology, internal categories, anafunctors and transformations form a bicategory $\Cat_\ana(S,J)$. If we restrict attention to a full sub-2-category $C$ which admits base change for arrows in $J$, we have an analogous full sub-bicategory $C_\ana(J)$. In fact the bicategory $C_{ana}(J)$ fails to be a strict 2-category only in the sense that the associator is given by the non-identity isotransformation from lemma <ref>. All the other structure is strict. There is a strict 2-functor $C_\ana(J) \to \Cat_\ana(S,J)$ which is an inclusion on objects and fully faithful in the strictest sense, namely being the identity functor on hom-categories. The following is the main result of this section, and allows us to relate anafunctors to the localisations considered in the next section. There is a strict, identity-on-objects 2-functor \[ \alpha_J\colon C \to C_\ana(J) \] sending $J$-equivalences to equivalences, and commuting with the respective inclusions into $\Cat(S)$ and $\Cat_\ana(S,J)$. We define $\alpha_J$ to be the identity on objects, and as described in examples <ref>, <ref> on 1-arrows and 2-arrows (i.e. functors and transformations). We need first to show that this gives a functor $C(X,Y) \to C_\ana(J)(X,Y)$. This is precisely the content of example <ref>. Since the identity 1-cell on a category $X$ in $C_\ana(J)$ is the image of the identity functor on $S$ in $C$, $\alpha_J$ respects identity 1-cells. Also, lemma <ref> tells us that $ \alpha_J$ respects composition. That $\alpha_J$ sends $J$-equivalences to equivalences is the content of lemma <ref>. The 2-category $C$ is locally small (i.e. enriched in small categories) if $S$ itself is locally small (i.e. enriched in sets), but a priori the collection of anafunctors $X\gento Y$ do not constitute a set for $S$ a large category. Let $(S,J)$ be a locally small, subcanonical unary site satisfying WISC and let $C$ admit base change along arrows in $J$. Then $C_\ana(J)$ is locally essentially small. Given an object $A$ of $S$, let $I(A)$ be a weakly initial set for $J/A$. Consider the locally full sub-2-category of $C_\ana(J)$ with the same objects, and arrows those anafunctors $(U,f):X \gento Y$ such that $U \to X_0$ is in $I(X_0)$. Every anafunctor is then isomorphic, by example <ref>, to one in this sub-2-category. The collection of anafunctors $(U,f):X \gento Y$ for a fixed $U$ forms a set, by local smallness of $C$, and similarly the collection of transformations between a pair of anafunctors forms a set by local smallness of $S$. Examples of locally small sites $(S,J)$ where $C_\ana(J)$ is not known to be locally essentially small are the category of sets from the model of ZF used in [13], the model of ZF constructed in [54] and the topos from proposition <ref>. We note that local essential smallness of $C_\ana(J)$ seems to be a condition just slightly weaker than WISC. § LOCALISING BICATEGORIES AT A CLASS OF 1-CELLS Ultimately we are interesting in inverting all $J$-equivalences in $C$ and so need to discuss what it means to add the formal pseudoinverses to a class of 1-cells in a 2-category – a process known as localisation. This was done in [53] for the more general case of a class of 1-cells in a bicategory, where the resulting bicategory is constructed and its universal properties examined. The application in loc. cit. is to show the equivalence of various bicategories of stacks to localisations of 2-categories of smooth, topological and algebraic groupoids. The results of this article can be seen as one-half of a generalisation of these results to more general sites. [53] Let $B$ be a bicategory and $W \subset B_1$ a class of 1-cells. A localisation of $B$ with respect to $W$ is a bicategory $B[W^{-1}]$ and a weak 2-functor \[ U \colon B \to B[W^{-1}] \] such that $U$ sends elements of $W$ to equivalences, and is universal with this property i.e. precomposition with $U$ gives an equivalence of bicategories \[ U^* \colon Hom(B[W^{-1}],D) \to Hom_W(B,D), \] where $Hom_W$ denotes the sub-bicategory of weak 2-functors that send elements of $W$ to equivalences (call these $W$-inverting, abusing notation slightly). The universal property means that $W$-inverting weak 2-functors $F\colon B \to D$ factor, up to an equivalence, through $B[W^{-1}]$, inducing an essentially unique weak 2-functor $\widetilde{F}\colon B[W^{-1}] \to D$. [53] Let $B$ be a bicategory with a class $W$ of 1-cells. $W$ is said to admit a right calculus of fractions if it satisfies the following conditions $W$ contains all equivalences a) $W$ is closed under composition b) If $a\in W$ and there is an isomorphism $a \stackrel{\sim}{\Rightarrow} b$ then $b\in W$ For all $w\colon A' \to A,\ f\colon C \to A$ with $w\in W$ there exists a 2-commutative square \[ \xymatrix{ P \ar[dd]^v \ar[rr]^g && A'\ar[dd]^w_{\ }="s" \\ \\ C \ar[rr]^{f}="t" & & A \ar@{=>}_{\simeq} "s"; "t" \] with $v\in W$. If $\alpha\colon w \circ f \Rightarrow w \circ g$ is a 2-arrow and $w\in W$ there is a 1-cell $v \in W$ and a 2-arrow $\beta\colon f\circ v \Rightarrow g \circ v$ such that $\alpha\circ v = w \circ \beta$. Moreover: when $\alpha$ is an isomorphism, we require $\beta$ to be an isomorphism too; when $v'$ and $\beta'$ form another such pair, there exist 1-cells $u,\,u'$ such that $v\circ u$ and $v'\circ u'$ are in $W$, and an isomorphism $\epsilon\colon v\circ u \Rightarrow v' \circ u'$ such that the following diagram commutes: \begin{equation}\label{2cf4.diag} \xymatrix{ f \circ v \circ u \ar@{=>}[rr]^{\beta\circ u} \ar@{=>}[dd]_{f\circ \epsilon}^\simeq && g\circ v \circ u \ar@{=>}[dd]^{g\circ \epsilon}_\simeq \\ \\ f\circ v' \circ u' \ar@{=>}[rr]_{\beta'\circ u'} && g\circ v' \circ u' \end{equation} For a bicategory $B$ with a calculus of right fractions, [53] constructs a localisation of $B$ as a bicategory of fractions; the 1-arrows are spans and the 2-arrows are equivalence classes of bicategorical spans-of-spans diagrams. From now on we shall refer to a calculus of right fractions as simply a calculus of fractions, and the resulting localisation constructed by Pronk as a bicategory of fractions. Since $B[W^{-1}]$ is defined only up to equivalence, it is of great interest to know when a bicategory $D$, in which elements of $W$ are sent to equivalences by a 2-functor $B \to D$, is equivalent to $B[W^{-1}]$. In particular, one might be interested in finding such an equivalent bicategory with a simpler description than that which appears in [53] A weak 2-functor $F:B \to D$ which sends elements of $W$ to equivalences induces an equivalence of bicategories \[ \widetilde{F} \colon B[W^{-1}] \xrightarrow{\sim} D \] if the following conditions hold $F$ is essentially surjective, For every 1-cell $f \in D_1$ there are 1-cells $w \in W$ and $g\in B_1$ such that $Fg \stackrel{\sim}{\Rightarrow} f \circ Fw$, $F$ is locally fully faithful. Thanks are due to Matthieu Dupont for pointing out (in personal communication) that proposition <ref> actually only holds in the one direction, not in both, as claimed in loc. cit. The following is useful in showing a weak 2-functor sends weak equivalences to equivalences, because this condition only needs to be checked on a class that is in some sense cofinal in the weak equivalences. Let $V \subset W$ be two classes of 1-cells in a bicategory $B$ such that for all $w\in W$, there exists $v\in V$ and $s\in W$ and an invertible 2-cell \[ \xymatrix{ && a \ar[dd]^w \\ & & \\ b \ar[rr]_v^{\ }="t1" \ar[uurr]^s_{\ }="s1" && c\; . \ar@{=>}"s1";"t1"^{\simeq} \] Then a weak 2-functor $F\colon B \to D$ that sends elements of $V$ to equivalences also sends elements of $W$ to equivalences. In the following the coherence arrows will be present, but unlabelled. It is enough to prove that if in a bicategory $D$ with a class of maps $M$ (in our case $M=F(W)$) such that for all $w\in M$ there is an equivalence $v$ and an isomorphism $\alpha$, \[ \xymatrix{ && a \ar[dd]^w \\ & & \\ b \ar[rr]_v^{\ }="t1" \ar[uurr]^s_{\ }="s1" && c \ar@{=>}"s1";"t1"^{\simeq}_\alpha \] where $s\in M$, then all elements of $M$ are also equivalences. Let $\bar v$ be a pseudoinverse for $v$ and let $j = s \circ \bar v$. Then there is sequence of isomorphisms \[ w\circ j \Rightarrow (w\circ s)\circ \bar v \Rightarrow v \circ \bar v \Rightarrow I. \] Since $s\in M$, there is an equivalence $u$, $t\in M$ and an isomorphism $\beta$ giving the following diagram \[ \xymatrix{ d \ar[dd]_{t} \ar[rr]^{u}_{\ }="s2" && a \ar[dd]^w \\ & & \\ b \ar[rr]_v^{\ }="t1" \ar[uurr]^s="t2"_{\ }="s1" && c \; . \ar@{=>}"s1";"t1"^\alpha \ar@{=>}"s2";"t2"_\beta \] Let $\bar u$ be a pseudoinverse of $u$. We know from the first part of the proof that we have a pseudosection $k = t\circ \bar u$ of $s$, with an isomorphism $s \circ k \Rightarrow I$. We then have the following sequence of isomorphisms: \[ j\circ w = (s\circ \bar v) \circ w \Rightarrow ((s\circ \bar v) \circ w) \circ (s \circ k) \Rightarrow s \circ ((\bar v \circ v) \circ (t\circ \bar u)) \Rightarrow (s\circ t) \circ u \Rightarrow \bar u \circ u \Rightarrow I. \] Thus all elements of $M$ are equivalences. § 2-CATEGORIES OF INTERNAL CATEGORIES ADMIT BICATEGORIES OF FRACTIONS In this section we prove the result that $C\into \Cat(S)$ admits a calculus of fractions for the $J$-equivalences, where $J$ is a singleton pretopology on $S$. The following is the first main theorem of the paper, and subsumes a number of other, similar theorems throughout the literature (see section <ref> for details). Let $S$ be a category with a singleton pretopology $J$. Assume the full sub-2-category $C \into \Cat(S)$ admits base change along maps in $J$. Then $C$ admits a right calculus of fractions for the class $W_J$ of $J$-equivalences. We show the conditions of definition <ref> hold. 2CF1. An internal equivalence is clearly $J$-locally split. Lemma <ref> gives us the rest. a) That the composition of fully faithful functors is again fully faithful is trivial. Consider the composition $g\circ f$ of two $J$-locally split functors, \[ \xymatrix{ Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}&Z[V] \ar[d]\ar@/^.5pc/[dr]_(.5){\ }="s2"^{v}& \\ X\ar[r]_{f}^(.33){\ }="t1"&Y \ar[r]_{g}^(.33){\ }="t2" & Z \ar@{=>}"s1";"t1" \ar@{=>}"s2";"t2" \] By lemma <ref> the functor $u$ pulls back to a functor $Z[U\times_{Y_0}V] \to Z[V]$. The composite $Z[U\times_{Y_0}V] \to Z$ is fully faithful with object component in $J$, hence $g\circ f$ is $J$-locally split. b) Lemma <ref> tells us that fully faithful functors are closed under isomorphism, so we just need to show $J$-locally split functors are closed under isomorphism. Let $w,f\colon X\to Y$ be functors and $a\colon w \Rightarrow f$ be a natural isomorphism. First, let $w$ be $J$-locally split. It is immediate from the diagram \[ \xymatrix{ Y[U] \ar[dd] \ar@/^.7pc/[ddrr]_{\ }="s1"^{u} \\ \\ X\ar@/^1pc/[rr]^{w}="t1"_{\ }="s2" \ar@/_1pc/[rr]_{f}^{\ }="t2" \ar@{=>}"s1";"t1" \ar@{=>}"s2";"t2"^{a} \] that $f$ is also $J$-locally split. 2CF3. Let $w\colon X\to Y$ be a $J$-equivalence, and let $f\colon Z\to Y$ be a functor. From the definition of $J$-locally split, we have the diagram \[ \xymatrix{ Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}& \\ X\ar[r]_{w}^(.33){\ }="t1"&Y \ar@{=>}"s1";"t1" \] We can use lemma <ref> to pull $u$ back along $f$ to get a 2-commuting diagram \[ \xymatrix{ & Z[U\times_{Y_0} Z_0] \ar[dr]^{v} \ar[dl] \\ Y[U] \ar[d] \ar@/^.5pc/[dr]_{\ }="s1"^{u}& &Z \ar[dl]^f\\ X\ar[r]_{w}^(.33){\ }="t1"&Y \ar@{=>}"s1";"t1" \] with $v\in W_J$ as required. Since $J$-equivalences are representably fully faithful, given \[ \xymatrix{ &Y \ar[dr]^w \\ X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\ & Y \ar[ur]_w \] where $w\in W_J$, there is a unique $a'\colon f \Rightarrow g$ such that \[ \raisebox{36pt}{ \xymatrix{ &Y \ar[dr]^w \\ X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\ & Y \ar[ur]_w \equals \raisebox{36pt}{ \xymatrix{ X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g&\Downarrow a'& Y \ar[r]^w & Z \,. \] The existence of $a'$ is the first half of 2CF4, where $v=\id_X$. Note that if $a$ is an isomorphism, so if $a'$, since $w$ is representably fully faithful. Given $v'\colon W\to X \in W_J$ such that there is a transformation \[ \xymatrix{ &X \ar[dr]^f \\ W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y\\ & X \ar[ur]_g \] \begin{align}\label{antiwhisker_eqn} \raisebox{36pt}{ \xymatrix{ &X \ar[dr]^f \\ W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y \ar[r]^w & Z\\ & X \ar[ur]_g \equals & \raisebox{36pt}{ \xymatrix{ &&Y \ar[dr]^w \\ W \ar[r]^{v'} &X \ar[ur]^f \ar[dr]_g & \Downarrow a & Z\\ && Y \ar[ur]_w } \nonumber \\ \equals & \raisebox{36pt}{ \xymatrix{ W\ar[r]^{v'}&X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g &\Downarrow a'& Y \ar[r]^w & Z }\, , \end{align} then uniqueness of $a'$, together with equation (<ref>) gives us \[ \raisebox{36pt}{ \xymatrix{ &X \ar[dr]^f \\ W \ar[ur]^{v'} \ar[dr]_{v'} & \Downarrow b & Y \\ & X \ar[ur]_g \equals \raisebox{36pt}{ \xymatrix{ W\ar[r]^{v'}&X \ar@/^1.5pc/[rr]^f \ar@/_1.5pc/[rr]_g &\Downarrow a' & Y }\, . \] This is precisely the diagram (<ref>) with $v=\id_X$, $u=v'$, $u'=\id_W$ and $\epsilon$ the identity 2-arrow. Hence 2CF4 holds. The proof of theorem <ref> is written using only the language of 2-categories, so can be generalised from $C$ to other 2-categories. This approach will be taken up in [56]. The second main result of the paper is that we want to know when this bicategory of fractions is equivalent to a bicategory of anafunctors, as the latter bicategory has a much simpler construction. Let $(S,J)$ be a subcanonical unary site and let the full sub-2-category $C\into \Cat(S)$ admit base change along arrows in $J$. Then there is an equivalence of bicategories \[ C_\ana(J) \simeq C[W_J^{-1}] \] under $C$. Let us show the conditions in proposition <ref> hold. To begin with, the 2-functor $\alpha_J\colon C \to C_{ana}(J)$ sends $J$-equivalences to equivalences by proposition <ref>. EF1. $\alpha_J$ is the identity on 0-cells, and hence surjective on objects. EF2. This is equivalent to showing that for any anafunctor $(U,f)\colon X\gento Y$ there are functors $w,g$ such that $w$ is in $W_J$ and \[ (U,f) \stackrel{\sim}{\Rightarrow} \alpha_J(g)\circ\alpha_J(w)^{-1} \] where $\alpha_J(w)^{-1}$ is some pseudoinverse for $\alpha_J(w)$. Let $w$ be the functor $X[U] \to X$ and let $g=f\colon X[U] \to Y$. First, note that \[ \xymatrix{ & \ar[dl] X[U] \ar[dr]^= &\\ X && X[U] \] is a pseudoinverse for \[ \alpha_J(w) \equals \left(\raisebox{24pt}{ \xymatrix{ & \ar[dl]_{=} X[U][U] \ar[dr] &\\ X[U] && X \] Then the composition $ \alpha_J(f)\circ\alpha_J(w)^{-1}$ is \[ \xymatrix{ & \ar[dl] X[U\times_U U \times_U U]\ar[dr]\\ X && Y\; , \] which is just $(U,f)$ (recall we have the equality $U\times_U U \times_U U = U$ by remark <ref>). EF3. If $a\colon(X_0,f)\Rightarrow(X_0,g)$ is a transformation of anafunctors for functors $f,g\colon X\to Y$, it is given by a natural transformation \[ f \Rightarrow g\colon X = X[X_0 \times_{X_0} X_0] \to Y. \] Hence we get a unique natural transformation $a\colon f\Rightarrow g$ such that $a$ is the image of $a'$ under We now give a series of results following from this theorem, using basic properties of pretopologies from section <ref>. When $J$ and $K$ are two subcanonical singleton pretopologies on $S$ such that $J_{un}=K_{un}$, for example $J$ cofinal in $K$, there is an equivalence of bicategories \[ C_\ana(J) \simeq C_\ana(K). \] The class of maps in $\Top$ of the form $\coprod U_i \to X$ for an open cover $\{U_i\}$ of $X$ form a singleton pretopology. This is because $\mathcal{O}$ is a superextensive pretopology (see the appendix). Given a site with a superextensive pretopology $J$, we have the following result which is useful when $J$ is not a singleton pretopology (the singleton pretopology $\amalg J$ is defined analogously to the case of $\Top$, details are in the appendix). Let $(S,J)$ be a superextensive site where $J$ is a subcanonical pretopology. Then \[ C[W_{J_{un}}^{-1}] \simeq C_\ana(\amalg J). \] This essentially follows by lemma <ref>. Obviously this can be combined with previous results, for example if $K$ is cofinal in $\amalg J$, for $J$ a non-singleton pretopology, $K$-anafunctors localise $C$ at the class of $J_{un}$-equivalences. Finally, given WISC we have a bound on the size of the hom-categories, up to equivalence. Let $(S,J)$ be a subcanonical unary site satisfying WISC with $S$ locally small and let $C\into \Cat(S)$ admit base change along arrows in $J$. Then any localisation $C[W_J^{-1}]$ is locally essentially small. Recall that this localisation can be chosen such that the class of objects is the same as the class of objects of $C$, and so it is not necessary to consider additional set-theoretic mechanisms for dealing with large (2-)categories here. We note that the issue of size of localisations is not touched on in [53]. even though such issues are commonly addressed in localisation of 1-categories. If we have a specified bound on the hom-sets of $S$ and also know that some WISC${}_\kappa$ holds, then we can put specific bounds on the size of the hom-categories of the localisation. This is important if examining fine size requirements or implications for localisation theorems such as these, for example higher versions of locally presentable categories. § EXAMPLES The simplest example is when we take the trivial singleton pretopology $triv$, where covering families are just single isomorphisms: $triv$-equivalences are internal equivalences and, up to equivalence, localisation at $W_{triv}$ does nothing. It is worth pointing out that if we localise at $W_{triv_{un}}$, which is equivalent to considering anafunctors with source leg having a split epimorphism for its object component, then by corollary <ref> this is equivalent to localising at $W_{triv}$, so $C_{ana}(triv_{un}) \simeq C_{ana}(triv)\simeq C$. The first non-trivial case is that of a regular category with the canonical singleton pretopology $\can$. This is the setting of [16]. Recall that $W_J^{BP}$ is the class of Bunge-Paré $J$-equivalences (definition <ref>). For now, let $C$ denote either $\Cat(S)$ or $\Gpd(S)$. Let $(S,J)$ be a finitely complete unary site with $J$ saturated. Then we have \[ C[(W_J^{BP})^{-1}] \simeq C[W_J^{-1}] \] This is merely a restatement of the fact Bunge-Paré $J$-equivalences and ordinary $J$-equivalences coincide in this case. The canonical singleton pretopology $\can$ on a finitely complete category $S$ is saturated. Hence $W_{\can}^{BP} = W_{\can}$ for this site, and \[ C[(W_{\can}^{BP})^{-1}] \simeq C[W_{\can}^{-1}]\simeq C_\ana(\can) \] We can combine this corollary with corollary <ref> so that the localisation of either $\Cat(S)$ or $\Gpd(S)$ at the Bunge-Paré weak equivalences can be calculated using $J$-anafunctors for $J$ cofinal in $\can$. We note that $\can$ does not satisfy WISC in general (see proposition <ref> and the comments following), so the localisation might not be locally essentially small. The previous corollaries deal with the case when we are interested in the 2-categories consisting of all of the internal categories or groupoids in a site. However, for many applications of internal categories/groupoids it is not sufficient to take all of $\Cat(S)$ or $\Gpd(S)$. One widely used example is that of Lie groupoids, which are groupoids internal to the category of (finite-dimensional) smooth manifolds such that source and target maps are submersions (more on these below). Other examples are used in the theory of algebraic stacks, namely groupoids internal to schemes or algebraic spaces. Other types of such presentable stacks use groupoids internal to some site with specified conditions on the source and target maps. Although it is not covered explicitly in the literature, it is possible to consider presentable stacks of categories, and this will be taken up in future work [55]. We thus need to furnish examples of sub-2-categories $C$, specified by restricting the sort of maps that are allowed for source and target, that admit base change along some class of arrows. The following lemma gives a sufficiency condition for this to be so. Let $\Cat^\mathcal{M}(S)$ be defined as the full sub-2-category of $\Cat(S)$ with objects those categories such that the source and target maps belong to a singleton pretopology $\mathcal{M}$. Then $\Cat^\mathcal{M}(S)$ admits base change along arrows in $\mathcal{M}$, as does the corresponding 2-category $\Gpd^\mathcal{M}(S)$ of groupoids. Let $X$ be an object of $\Cat^\mathcal{M}(S)$ and $f\colon M\to X_0 \in \mathcal{M}$. In the following diagram, all the squares are pullbacks and all arrows are in $\mathcal{M}$. \[ \SelectTips {cm}{}% \xymatrix{ X[M]_1 \ar[d] \ar[r] \ar @/_2.4pc/ [dd]_{s'} \ar @/^1pc/[rr]^{t'} & X_1\times_{X_0} M \ar[r] \ar[d] & M \ar[d] \\ M\times_{X_0} X_1 \ar[d] \ar[r] & X_1 \ar[r] \ar[d] & X_0 \\ M \ar[r] & X_0 \] The maps marked $s',t'$ are the source and target maps for the base change along $f$, so $X[M]$ is in $\Cat^\mathcal{M}(S)$. The same argument holds for groupoids verbatim. In practice one often only wants base change along a subclass of $\mathcal{M}$, such as the class of open covers sitting inside the class of open maps in $\Top$. We can then apply theoerems <ref> and <ref> to the 2-categories $\Cat^\mathcal{M}(S)$ and $\Gpd^\mathcal{M}(S)$ with the classes of $\mathcal{M}$-equivalences, and indeed to sub-2-categories of these, as we shall in the examples below. We shall focus of a few concrete cases to show how the results of this paper subsume similar results in the literature proved for specific sites. The category of smooth manifolds is not finitely complete so the localisation results in this section so far do not apply to it. There are two ways around this. The first is to expand the category of manifolds to a category of smooth spaces which is finitely complete (or even cartesian closed). In that case all the results one has for finitely complete sites can be applied. The other is to take careful note of which finite limits are actually needed, and show that all constructions work in the original category of manifolds. There is then a hybrid approach, which is to work in the expanded category, but point out which results/constructions actually fall inside the original category of manifolds. Here we shall take the second approach. First, let us pin down some definitions. Let $\Diff$ be the category of smooth, finite-dimensional manifolds. A Lie category is a category internal to $\Diff$ where the source and target maps are submersions (and hence the required pullbacks exist). A Lie groupoid is a Lie category which is a groupoid. A proper Lie groupoid is one where the map $(s,t)\colon X_1 \to X_0 \times X_0$ is proper. An étale Lie groupoid is one where the source and target maps are local By lemma <ref> the 2-categories of Lie categories, Lie groupoids and proper Lie groupoids admit base change along any of the following classes of maps: open covers ($\amalg\mathcal{O}$), surjective local diffeomorphisms ($\acute{e}t$), surjective submersions ($Subm$). The 2-categories of étale Lie groupoids and proper étale Lie groupoids admit base change along arrows in $\acute{e}t$ and $Subm$. We should note that we have $\amalg\mathcal{O}$ cofinal in $\acute{e}t$, which is cofinal in We can thus apply the main results of this paper to the sites $(\Diff,\mathcal{O})$, $(\Diff,\amalg\mathcal{O})$, $(\Diff,\acute{e}t)$ and $(\Diff,Subm)$ and the 2-categories of Lie categories, Lie groupoids, proper Lie goupoids and so on. However, the definition of weak equivalence we have here, involving $J$-locally split functors, is not one that apppears in the Lie groupoid literature, which is actually Bunge-Paré $Subm$-equivalence. However, we have the following result: A functor $f\colon X\to Y$ between Lie categories is a $Subm$-equivalence if and only if it is a Bunge-Paré Before we prove this, we need a lemma proved by Ehresmann. [26] For any Lie category $X$, the subset of invertible arrows, $X_1^{iso} \into X_1$ is an open Hence there is a Lie groupoid $X^{iso}$ and an identity-on-objects functor $X^{iso} \to X$ which is universal for functors from Lie groupoids. In particular, a natural isomorphism between functors with codomain $X$ is given by a component map that factors through $X_1^{iso}$, and the induced source and target maps $X_1^{iso} \to X_0$ are submersions. (proposition <ref>) Full faithfulness is the same for both definitions, so we just need to show that $f$ is $Subm$-locally split if and only if it is essentially $Subm$-surjective. We first show the forward implication. The special case of a $\amalg\mathcal{O}$-equivalence between Lie groupoids is a small generalisation of the proof of proposition 5.5 in [46], which states than an internal equivalence of Lie groupoids is a Bunge-Paré $Subm$-equivalence. Since $\amalg\mathcal{O}$ is cofinal in $Subm$, a $Subm$-equivalence is a $\amalg\mathcal{O}$-equivalence, hence a Bunge-Paré $Subm$-equivalence. For the case when $X$ and $Y$ are Lie categories, we use the fact that we can define $X_0\times_{Y_0}Y_1^{iso}$ and that the local sections constructed in Moerdijk-Mrčun's proof factor through this manifold to set up the proof as in the groupoid case. For the reverse implication, the construction in the first half of the proof of proposition <ref> goes through verbatim, as all the pullbacks used involve submersions. The need to localise the category of Lie groupoids at $W_{Subm}$ was perhaps first noted in [52], where it was noted that something other than the standard construction of a category of fractions was needed. However Pradines lacked the necessary 2-categorical localisation results. Pronk considered the sub-2-category of étale Lie groupoids, also localised at $W_{Subm}$, in order to relate these groupoids to differentiable étendues [53]. Lerman discusses the 2-category of orbifolds qua stacks [37] and argues that it should be a localisation of the 2-category of proper étale Lie groupoids (again at $W_{Subm}$). These three cases use different constructions of the 2-categorical localisation: Pradines used what he called meromorphisms, which are equivalence classes of butterfly-like diagrams and are related to Hilsum-Skandalis morphisms, Pronk introduces the techniques outlined in this paper, and Lerman uses Hilsum-Skandalis morphisms, also known as right principal Interestingly, [19] considers this localisation of the 2-category of Lie groupoids then considers a further localisation, not given by the results of this paper.[In fact this is the only 2-categorical localisation result involving internal categories or groupoids known to the author to not be covered by theorem <ref> or its sequel [56].] Colman in essence shows that the full sub-2-category of topologically discrete groupoids, i.e. ordinary small groupoids, is a localisation at those internal functors which induce an equivalence on fundamental groupoids. Our next example is that of topological groupoids, which correspond to various flavours of stacks on the category $\Top$. The idea of weak equivalences of topological groupoids predates the case of Lie groupoids, and [52] credits it to Haefliger, van Est, and [30]. In particular the first two were ultimately interested in defining the fundamental group of a foliation, that is to say, of the topological groupoid associated to a foliation, considered up to weak eqivalence. However more recent examples have focussed on topological stacks, or variants thereon. In particular, in parallel with the algebraic and differentiable cases, the topological stacks for which there is a good theory correspond to those topological groupoids with conditions on their source and target maps. Aside from étale topological groupoids (which were considered by [53] in relation to étendues), the real advances here have come from work of Noohi, starting with [49], who axiomatised the concept of local fibration and asked that the source and target maps of topological groupoids are local fibrations. A singleton pretopology $LF$ in $\Top$ is called a class of local fibrations if the following conditions hold:[We have packaged the conditions in a way slightly different to [49], but the definition is in fact identical.] * $LF$ contains the open embeddings * $LF$ is stable under coproducts, in the sense that $\coprod_{i\in I} X_i \to Y$ is in $LF$ if each $X_i\to Y$ is in $LF$ * $LF$ is local on the target for the open cover pretopology. That is, if the pullback of a map $f\colon X\to Y$ along an open cover of $Y$ is in $LF$, then $f$ is in $LF$. Conditions 1. and 2. tell us that $\amalg\mathcal{O} \subset LF$, and that $LF$ is $\amalg J$ for some superextensive pretopology $J$ containing the open embeddings as singleton `covering' families (beware the misleading terminology here: covering families are not assumed to be jointly surjective). Note that $LF$ will not be subcanonical, by condition 1. As an example, given any of the following pretopologies $K$: * Serre fibrations, * Hurewicz fibrations, * open maps, * split maps, * projections out of a cartesian product, * isomorphisms; one can define a class of local fibrations by choosing those maps which are in $K$ on pulling back to an open cover of the codomain. Such maps are then called local $K$. As an example of the usefulness of this concept, the topological stacks corresponding to topological groupoids with local Hurewicz fibrations as source and target have a nicely behaved homotopy theory. The case of étale groupoids corresponds to the last named class of maps, which give us local isomorphisms, i.e. étale maps. We can then apply lemma <ref> and theorem <ref> to the 2-category $\Grp^{LF}(\Top)$ to localise at the class $W_{\amalg \mathcal{O}}$ (as $\amalg \mathcal{O} \subset LF$), or any other singleton pretopology contained in $LF$, using anafunctors whenever this pretopology is subcanonical. Note that if $C$ satisfies WISC, so will the corresponding $LF$, although this is probably not necessary to consider in the presence of full AC. A slightly different approach is taken in [18], where the author introduces a new pretopology on the category $CGH$ of compactly generated Hausdorff spaces. We give a definition equivalent to the one in loc cit. A (not necessarily open) cover $\{V_i\into X\}_{i\in I}$ is called a $\mathcal{CG}$-cover if for any map $K\to X$ from a compact space $K$, there is a finite open cover $\{U_j \into K\}$ which refines the cover $\{V_i\times_X K\to K\}_{i\in I}$. $\mathcal{CG}$-covers form a pretopology $\mathcal{CG}$ on $CGH$. Compactly generated stacks then correspond to groupoids in $CGH$ such that source and target maps are in the pretopology $\mathcal{CG}_{un}$. Again, we can localise $\Gpd^{\mathcal{CG}}(CGH)$ at $W_{\mathcal{CG}_{un}}$ using lemma <ref> and theorem <ref>, and anafunctors can be again pressed into service. We now arrive at the more involved case of algebraic stacks (cf. the continually growing [59] for the extent of the theory of algebraic stacks), which were the first presentable stacks to be defined. There are some subtleties about the site of definition for algebraic stacks, and powerful representability theorems, but we can restrict to three main cases: groupoids in the category of affine schemes $\Aff = \Ring^{op}$; groupoids in the category $\Sch$ of schemes; and groupoids in the category $\AlgSp$ of algebraic spaces. Algebraic spaces reduce to algebraic stacks on $\Sch$ represented by groupoids with trivial automorphism groups, and the category of schemes is a subcategory of $Sh(\Aff)$, so we shall just consider the case when our ambient category is $\Aff$. In any case, all the special properties of classes of maps in all three sites are ultimately defined in terms of properties of ring homomorphisms. Note that groupoids in $\Aff$ are exactly the same thing as cogroupoid objects in $\Ring$, which are more commonly known as Hopf algebroids. Despite the possibly unfamiliar language used by algebraic geometry, algebraic stacks reduce to the following semiformal definition. We fix three singleton pretopologies on our site $\Aff$: $J$, $E$ and $D$ such that $E$ and $D$ are local on the target for the pretopology $J$. An algebraic stack then is a stack on $\Aff$ for the pretopology $J$ which `corresponds' to a groupoid $X$ in $\Aff$ such that source and target maps belong to $E$ and $(s,t)\colon X_1 \to X_0^2$ belongs to $D$. We recover the algebraic stacks by localising the 2-category of such groupoids at $W_E$ (this claim of course needs substantiating, something we will not do here for reasons of space, referring rather to [53, 57] and the forthcoming [55]). In practice, $D$ can be something like closed maps (to recover Hausdorff-like conditions) or all maps, and $E$ consists of either smooth or étale maps, corresponding to Artin and Deligne-Mumford stacks respectively. $J$ is then something like the étale topology (or rather, the singleton pretopology associated to it, as the étale topology is superextensive), and we can apply lemma <ref> to see that base change exists along $J$, along with the fact that asking for $(s,t) \in D$ is automatically stable under forming the base change. In practice, a variety of combinations of $J,E$ and $D$ are used, as well as passing from $\Aff$ to $\Sch$ and $\AlgSp$, so there are various compatibilities to check in order to know one can apply theorem <ref>. A final application we shall consider is when our ambient category consists of algebraic objects. As mentioned in section 2, a number of authors have considered localising groupoids in Mal'tsev, or Barr-exact, or protomodular, or semi-abelian categories, which are hallmarks of categories of algebraic objects rather than spatial ones, as we have been considering so far. In the case of groupoids in $\Grp$ (which, as in any Mal'tsev category, coincide with the internal categories) it is a well-known result that they can be described using crossed modules. A crossed module (in $\Grp$) is a homomorphism $t\colon G\to H$ together with a homomorphism $\alpha\colon H\to \Aut(G)$ such that $t$ is $H$-equivariant (using the conjugation action of $H$ on itself), and such that the composition $\alpha\circ t\colon G\to\Aut(G)$ is the action of $G$ on itself by conjugation. A crossed module is often denoted, when no confusion will arise, by $(G\to H)$. A morphism $(G \to H) \to (K\to L)$ of crossed modules is a pair of maps $G\to K$ and $H\to L$ making the obvious square commute, and commuting with all the action maps. Similar definitions hold for groups internal to cartesian closed categories, and even just finite-product categories if one replaces $H\to \Aut(G)$ with its transpose $H\times G\to G$. Ultimately of course there is a definition for crossed modules in semiabelian categories (e.g. [1]), but we shall consider just groups. There is a natural definition of 2-arrow between maps of crossed modules, but the specifics are not important for the present purposes, so we refer to <cit.> for details. The 2-categories of groupoids internal to $\Grp$ and crossed modules are equivalent, so we shall just work with the terminology of the latter. Given the result that crossed modules correspond to pointed, connected homotopy 2-types, it is natural to ask if all maps of such arise from maps between crossed modules. The answer is, perhaps unsurprisingly, no, as one needs maps which only weakly preserve the group structure. One can either write down the definition of some generalised form of map (<cit.>), or localise the 2-category of crossed modules ([51] considers a model structure on the category of crossed modules). To localise the 2-category of crossed modules we can consider the singleton pretopology $epi$ on $\Grp$ consisting of the epimorphisms, and localise $\Gpd(\Grp)$ at $W_{epi}$. There are potentially interesting sub-2-categories of crossed modules that one might want to consider, for example, the one corresponding to nilpotent pointed connected 2-types. These are crossed modules $t\colon G \to H$ where the cokernel of $t$ is a nilpotent group and the (canonical) action of $\coker t$ on $\ker t$ is nilpotent. The correspondence between such crossed modules and the corresponding internal groupoids is a nice exercise, as well as seeing that this 2-category admits base change for the pretopology $epi$. § SUPEREXTENSIVE SITES The usual sites of topological spaces, manifolds and schemes all share a common property: one can (generally) take coproducts of covering families and end up with a cover. In this appendix we gather some results that generalise this fact, none of which are especially deep, but help provide examples of bicategories of anafunctors. Another reference for superextensive sites is [58]. [17] A finitary (resp. infinitary) extensive category is a category with finite (resp. small) coproducts such that the following condition holds: let $I$ be a a finite set (resp. any set), then, given a collection of commuting diagrams \[ \xymatrix{ X_i \ar[r] \ar[d] &Z \ar[d] \\ A_i \ar[r] & \coprod_{i\in I} A_i\;, \] one for each $i\in I$, the squares are all pullbacks if and only if the collection $\{X_i \to Z\}_{i\in I}$ forms a coproduct diagram. In such a category there is a strict initial object: given a map $A \to 0$ with $0$ initial, we have $A \simeq 0$. $\Top$ is infinitary extensive. $\Ring^{op}$, the category of affine schemes, is finitary extensive. In $\Top$ we can take an open cover $\{U_i\}_I$ of a space $X$ and replace it with the single map $\coprod_I U_i \to X$, and work just as before using this new sort of cover, using the fact $\Top$ is extensive. The sort of sites that mimic this behaviour are called superextensive. A superextensive site is an extensive category $S$ equipped with a pretopology $J$ containing the families \[ (U_i \to \coprod_I U_i)_{i\in I} \] and such that all covering families are bounded; this means that for a finitely extensive site, the families are finite, and for an infinitary site, the families are small. The pretopology in this instance will also be called superextensive. Given an extensive category $S$, the extensive pretopology has as covering families the bounded collections $(U_i \to \coprod_I U_i)_{i\in I}$. The pretopology on any superextensive site contains the extensive pretopology. The category $\Top$ with its usual pretopology of open covers is a superextensive An elementary topos with the coherent pretopology is finitary superextensive, and a Grothendieck topos with the canonical pretopology is infinitary superextensive. Given a superextensive site $(S,J)$, one can form the class $\amalg J$ of arrows of the form $\coprod_I U_i \to A$ for covering families $\{U_i \to A\}_{i\in I}$ in $J$ (more precisely, all arrows isomorphic in $S/A$ to such arrows). The class $\amalg J$ is a singleton pretopology, and is subcanonical if and only if $J$ Since isomorphisms are covers for $J$ they are covers for $\amalg J$. The pullback of a $\amalg J$-cover $\coprod_I U_i \to A$ along $B \to A$ is a $\amalg J$-cover as coproducts and pullbacks commute by definition of an extensive category. Now for the third condition we use the fact that in an extensive category a map \[ f\colon B \to \coprod_I A_i \] implies that $B\simeq \coprod_I B_i$ and $f=\coprod_i f_i$. Given $\amalg J$-covers $\coprod_I U_i \to A$ and $\coprod_J V_j \to (\coprod_I U_i)$, we see that $\coprod_J V_j \simeq \coprod_I W_i$ for some objects $W_i$. By the previous point, the pullback \[ \coprod_I U_k \times_{\coprod_I U_{i'}} W_i \] is a $\amalg J$-cover of $U_i$, and hence $(U_k \times_{\coprod_I U_{i'}} W_i \to U_k)_{i\in I}$ is a $J$-covering family for each $k\in I$. Thus \[ (U_k \times_{\coprod_I U_{i'}} W_i \to A)_{i,k\in I} \] is a $J$-covering family, and so \[ \coprod_J V_j \simeq \coprod_{k\in I} \left( \coprod_{i\in I} U_k \times_{\coprod_I U_{i'}} W_i\right) \to A \] is a $\amalg J$-cover. The map $\coprod_I U_i \to A$ is the coequaliser of $\coprod_{I\times I} U_i \times_A U_j \st \coprod_I U_i$ if and only if $A$ is the colimit of the diagram in definition <ref>. Hence $(\coprod_I U_i \to A)$ is effective if and only if $ (U_i \to A)_{i\in I}$ is effective Notice that the original superextensive pretopology $J$ is generated by the union of $\amalg J$ and the extensive pretopology. One reason we are interested in superextensive sites is the following. In a superextensive site $(S,J)$, we have $J_{un} = (\amalg J)_{un}$. This means we can replace the singleton pretopology $J_{un}$ (e.g. local-section-admitting maps of topological spaces) with the singleton pretopology $\amalg J$ (e.g. disjoint unions of open covers) when defining anafunctors. This makes for much smaller pretopologies in practice. One class of extensive categories which are of particular interest is those that also have finite/small limits. These are called lextensive. For example, $\Top$ is infinitary lextensive, as is a Grothendieck topos. In contrast, an elementary topos is in general only finitary lextensive. We end with a lemma about WISC. If $(S,J)$ is a superextensive site, $(S,J)$ satisfies WISC if and only if $(S,\amalg J)$ One reason for why superextensive sites are so useful is the following result from Let $(S,J)$ be a superextensive site, and $F$ a stack for the extensive topology on $S$. 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[51] B. Noohi, Notes on 2-groupoids, 2-groups and crossed-modules, preprint (2005) [52] J. Pradines, Morphisms between spaces of leaves viewed as fractions, Cah. Topol. Géom. Différ. Catég. 30 (1989), no. 3, pp 229–246, [arXiv:0803.4209]. [53] D. Pronk, Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996), no. 3, pp 243–303. [54] D. M. Roberts, Con(ZF+ $\neg$WISC), preprint, (2013). [55] D. M. Roberts, All presentable stacks are stacks of anafunctors, forthcoming (A). [56] D. M. Roberts, Strict 2-sites, $J$-spans and localisations, forthcoming (B). [57] D. Schäppi, A characterization of categories of coherent sheaves of certain algebraic stacks, preprint (2012), [arXiv:1206.2764]. [58] M. Shulman, Exact completions and small sheaves, Theory and Application of Categories, 27 (2012), no. 7, pp 97–173. [59] The Stacks project authors, Stacks project, <http://stacks.math.columbia.edu>. 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arxiv-papers
2011-01-12T13:56:33
2024-09-04T02:49:16.375698
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "David M. Roberts", "submitter": "David Roberts", "url": "https://arxiv.org/abs/1101.2363" }
1101.2437
# New Physics effects on decay $B_{s}\to\gamma\gamma$ in Technicolor Model Qin XiuMei snowlotusqin@yahoo.com.cn Wujun Huo whuo@ictp.it Xiaofang Yang Department of Physics Department, Southeast University, Nanjing, Jiangsu 211189, China ###### Abstract In this paper we calculate the contributions to the branching ratio of $B_{s}\to\gamma\gamma$ from the charged Pseudo-Goldstone bosons appeared in one generation Technicolor model. We find that the theoretical values of the branching ratio, $BR(B_{s}\to\gamma\gamma)$, including the contributions of PGBs, $P^{\pm}$ and $P^{\pm}_{8}$ , are much different from the $SM$ prediction. The new physics effects can be enhance 2-3 levels to $SM$ result. It is shown that the decay $B_{s}\to\gamma\gamma$ can give the test the new physics signals from the technicolor model. ## I introduction As is well known, the rare radiative decays of $B$ mesons is in particular sensitive to contributions from those new physics beyond the standard model(SM). Both inclusive and exclusive processes, such as the decays $B_{s}\to X\gamma$, $B_{s}\to\gamma\gamma$ and $B\to X_{s}\gamma$ have been received some attention in the literature[1-14]. In this paper, we will present our results in Technicolor theories. The one generation Technicolor model (OGTM)[15-16]is the simplest and most frequently studied model which contained the parameters are less than SM. Same as other models, the OGTM has its defects such as the S parameter large and positive[17]. But we can relax the constraints on the OGTM form the $S$ parameter by introducing three additional parameters $(V,W,X)$[18]. The basic idea of the OGTM is: we introduce a new set of asymptotically free gauge interactions and the Technicolor force act on Technifermions. The Technicolor interaction at $1Tev$ become strong and cause a spontaneous breaking of the global flavor symmetry $SU(8)_{L}\times SU(8)_{R}\times U(1)_{Y}$. The result is $8^{2}-1=63$ massless Goldstone bosons. Three of the these objects replace the Higgs field and induce a mass of $W^{\pm}$ and $Z^{0}$ gauge bosons. And at the new strong interaction other Goldstone bosons acquire masses. As for the $B_{s}\to\gamma\gamma$, only the charged color single and color octets have contributions. The gauge couplings of the PGBs are determined by their quantum numbers. In Table 1 we listed the relevant couplings[19] needed in our calculation, where the $V_{ud}$ is the corresponding element of $Kobayashi- Maskawa$ matrix . The Goldstone boson decay constant $F_{\pi}$[20] should be $F_{\pi}=v/2=123GeV$, which corresponds to the vacuum expectation of an elementary Higgs field . $P^{+}P^{-}\gamma_{\mu}$ | $-ie(p_{+}-p_{-})_{\mu}$ ---|--- $P^{+}_{8a}P^{-}_{8b}\gamma_{\mu}$ | $-ie(p_{+}-p_{-})_{\mu}\delta_{ab}$ $P^{+}\;u\;d$ | $i\frac{V_{ud}}{2F_{\pi}}\sqrt{\frac{2}{3}}[M_{u}(1-\gamma_{5})-M_{d}(1+\gamma_{5})]$ $P^{+}_{8a}\;u\;d$ | $i\frac{V_{ud}}{2F_{\pi}}\lambda_{a}[M_{u}(1-\gamma_{5})-M_{d}(1+\gamma_{5})]$ $P^{+}_{8a}P^{-}_{8b}g_{c\mu}$ | $-gf_{abc}(p_{a}-p_{b})_{\mu}$ Table 1: The relevant gauge couplings and Effective Yukawa couplings for the OGTM. At the LO in QCD the effective Hamiltonian is ${\cal H}_{eff}=\frac{-4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{*}\displaystyle{\sum_{i=1}^{8}}C_{i}(M_{W}^{-})O_{i}(M_{W}^{-}).$ (1) Where, as usual, $G_{F}$ denotes the Fermi coupling constant and $V_{tb}V_{ts}^{*}$ indicates the Cabibbo-Kobayashi-Maskawa matrix element.And the current-current, QCD penguin, electromagnetic and chromomagnetic dipole operators are of the form $\displaystyle O_{1}$ $\displaystyle=$ $\displaystyle(\overline{c}_{L\beta}\gamma^{\mu}b_{L\alpha})(\overline{s}_{L\alpha}\gamma_{\mu}c_{L\beta})\;$ (2) $\displaystyle O_{2}$ $\displaystyle=$ $\displaystyle(\overline{c}_{L\alpha}\gamma^{\mu}b_{L\alpha})(\overline{s}_{L\beta}\gamma_{\mu}c_{L\beta})\;$ (3) $\displaystyle O_{3}$ $\displaystyle=$ $\displaystyle(\overline{s}_{L\alpha}\gamma^{\mu}b_{L\alpha})\sum_{q=u,d,s,c,b}(\overline{q}_{L\beta}\gamma_{\mu}q_{L\beta})\;$ (4) $\displaystyle O_{4}$ $\displaystyle=$ $\displaystyle(\overline{s}_{L\alpha}\gamma^{\mu}b_{L\beta})\sum_{q=u,d,s,c,b}(\overline{q}_{L\beta}\gamma_{\mu}q_{L\alpha})\;$ (5) $\displaystyle O_{5}$ $\displaystyle=$ $\displaystyle(\overline{s}_{L\alpha}\gamma^{\mu}b_{L\alpha})\sum_{q=u,d,s,c,b}(\overline{q}_{R\beta}\gamma_{\mu}q_{R\beta})\;$ (6) $\displaystyle O_{6}$ $\displaystyle=$ $\displaystyle(\overline{s}_{L\alpha}\gamma^{\mu}b_{L\beta})\sum_{q=u,d,s,c,b}(\overline{q}_{R\beta}\gamma_{\mu}q_{R\alpha})\;$ (7) $\displaystyle O_{7}$ $\displaystyle=$ $\displaystyle(e/16\pi^{2})m_{b}\overline{s}_{L}\sigma^{\mu\nu}b_{R}F_{\mu\nu}\;$ (8) $\displaystyle O_{8}$ $\displaystyle=$ $\displaystyle(g/16\pi^{2})m_{b}\overline{s}_{L}\sigma^{\mu\nu}T^{a}b_{R}G_{\mu\nu}^{a}\;$ (9) where $\alpha$ and $\beta$ are color indices, $\alpha=1,...,8$ labels SU(3)c generators, e and $g$ refer to the electromagnetic and strong coupling constants, while $F_{\mu\nu}$ and $G^{a}_{\mu\nu}$ denote the QED and QCD field strength tensors, respectively. The Feynman diagrams that contribute to the matrix element as the following Figure 1: Examples of Feynman diagrams that contribute to the matrix element. Figure 2: The Feynman diagrams that contribute to the Wilson coefficients C7,C8. In Fig.2 the shot-dash lines represent the charged PGBs $P^{\pm}$ and $P^{\pm}_{8}$ of OGTM. We at first integrate out the top quark and the weak $W$ bosons at $\mu=M_{W}$ scale, generating an effective five-quark theory and run the effective field theory down to b-quark scale to give the leading log QCD corrections by using the renormalization group equation. The Wilson coefficients are process independent and the coefficients $C_{i}(\mu)$ of 8 operators are calculated from the Fig.2.The Wilson coefficients are read[21] $\displaystyle C_{i}(M_{W})=0,\;\;i=1,3,4,5,6,\;\;\;C_{2}(M_{W})=1,$ (10) $\displaystyle C_{7}(M_{W})=-A(\delta)+\frac{B(x)}{3\sqrt{2}G_{F}F_{\pi}^{2}}+\frac{8B(y)}{3\sqrt{2}G_{F}F_{\pi}^{2}}$ (11) $\displaystyle C_{8}(M_{W})=-C(\delta)+\frac{D(x)}{3\sqrt{2}G_{F}F_{\pi}^{2}}+\frac{8D(y)+E(y)}{3\sqrt{2}G_{F}F_{\pi}^{2}}$ (12) with $\delta=M_{W}^{2}/m_{t}^{2}$, $x=(m(P^{\pm})/m_{t})^{2}$ and $y=(m(P^{\pm}_{8})/m_{t})^{2}$.From the $Eq(11),(12)$ , we can see the situation of the color-octet charged PGBs is more complicate than that of the color-singlet charged PGBs ,because of the involvement of the color interactions. where $\displaystyle A(\delta)$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{3}+\frac{5}{24}\delta-\frac{7}{24}\delta^{2}}{(1-\delta)^{3}}+\frac{\frac{3}{4}\delta-\frac{1}{2}\delta^{2}}{(1-\delta)^{4}}\log[\delta]$ (13) $\displaystyle B(y)$ $\displaystyle=$ $\displaystyle\frac{-\frac{11}{36}+\frac{53}{72}y-\frac{25}{72}y^{2}}{(1-y)^{3}}$ (14) $\displaystyle+$ $\displaystyle\frac{-\frac{1}{4}y+\frac{2}{3}y^{2}-\frac{1}{3}y^{3}}{(1-y)^{4}}\log[y]$ $\displaystyle C(\delta)$ $\displaystyle=$ $\displaystyle\frac{\frac{1}{8}-\frac{5}{8}\delta-\frac{1}{4}\delta^{2}}{(1-\delta)^{3}}-\frac{\frac{3}{4}\delta^{2}}{(1-\delta)^{4}}\log[\delta]$ (15) $\displaystyle D(y)$ $\displaystyle=$ $\displaystyle\frac{-\frac{5}{24}+\frac{19}{24}y-\frac{5}{6}y^{2}}{(1-y)^{3}}$ (16) $\displaystyle+$ $\displaystyle\frac{\frac{1}{4}y^{2}-\frac{1}{2}y^{3}}{(1-y)^{4}}\log[y]$ $\displaystyle E(y)$ $\displaystyle=$ $\displaystyle\frac{\frac{3}{2}-\frac{15}{8}y-\frac{15}{8}y^{2}}{(1-y)^{3}}+\frac{\frac{9}{4}y-\frac{9}{2}y^{2}}{(1-y)^{4}}\log[y]$ (17) By caculate the graphs of the exchanged $W$ boson in the SM we gained the function $A$ and $C$;And by caculate the graphs of the exchanged color-singlet and color-octet charged PGBs in OGTM we gained the function $B$, $D$ and $E$. when $\delta<1$, $x,y>>1$, the OGTM contribution $B$, $D$ and $E$ have always a relative minus sign with the SM contribution $A$ and $C$. As a result, the OGTM contribution always destructively interferes with the SM contribution. The leading-order results for the Wilson coefficients of all operators entering the effective Hamiltonian in Eq.(1) can be written in an analytic form. They are $\displaystyle C_{7}^{eff}(m_{b})$ $\displaystyle=$ $\displaystyle\eta^{16/23}C_{7}(M_{W})+\frac{8}{3}(\eta^{14/23}-\eta^{16/23})\times$ (18) $\displaystyle C_{8}(M_{W})+C_{2}(M_{W})\displaystyle\sum_{i=1}^{8}h_{i}\eta^{a_{i}}.$ With $\eta=\alpha_{s}(M_{W})/\alpha_{s}(m_{b})$, $\displaystyle h_{i}$ $\displaystyle=$ $\displaystyle(\frac{626126}{272277},-\frac{56281}{51730},-\frac{3}{7},-\frac{1}{14},-0.6494,$ (19) $\displaystyle-0.0380,-0.0186,-0.0057).$ $\displaystyle a_{i}$ $\displaystyle=$ $\displaystyle(\frac{14}{23},\frac{16}{23},\frac{6}{23},-\frac{12}{23},$ (20) $\displaystyle 0.4086,-0.4230,-0.8994,0.1456).$ To calculate $B_{s}\to\gamma\gamma$ , one may follow a perturbative QCD approach which includes a proof of factorization, showing that soft gluon effects can be factorized into $B_{s}$ meson wave function; and a systematic way of resumming large logarithms due to hard gluons with energies between 1Gev and $m_{b}$. In order to calculate the matrix element of Eq(1) for the $B_{s}\to\gamma\gamma$ , we can work in the weak binding approximation and assume that both the $b$ and the $s$ quarks are at rest in the $B_{s}$ meson, and the $b$ quarks carries most of the meson energy, and its four velocity can be treated as equal to that of $B_{s}$. Hence one may write $b$ quark momentum as $p_{b}=m_{b}v$ where is the common four velocity of $b$ and $B_{s}$. We have $\displaystyle p_{b}\cdot k_{1}$ $\displaystyle=$ $\displaystyle m_{b}v\cdot k_{1}={1\over 2}m_{b}m_{B_{s}}=p_{b}\cdot k_{2},$ $\displaystyle p_{s}\cdot k_{1}$ $\displaystyle=$ $\displaystyle(p-k_{1}-k_{2})\cdot k_{1}=$ (21) $\displaystyle-{1\over 2}m_{B_{s}}(m_{B_{s}}-m_{b})=p_{s}\cdot k_{2},$ We compute the amplitude of $B_{s}\to\gamma\gamma$ using the following relations $\displaystyle\left\langle 0|\bar{s}\gamma_{\mu}\gamma_{5}b|B_{s}(P)\right\rangle$ $\displaystyle=$ $\displaystyle-if_{B_{s}}P_{\mu},$ $\displaystyle\left\langle 0|\bar{s}\gamma_{5}b|B_{s}(P)\right\rangle$ $\displaystyle=$ $\displaystyle if_{B_{s}}M_{B},$ (22) where $f_{B_{s}}$ is the $B_{s}$ meson decay constant which is about $200$ MeV . The total amplitude is now separated into a CP-even and a CP-odd part $T(B_{s}\to\gamma\gamma)=M^{+}F_{\mu\nu}F^{\mu\nu}+iM^{-}F_{\mu\nu}\tilde{F}^{\mu\nu}.$ (23) We find that $\displaystyle M^{+}$ $\displaystyle=$ $\displaystyle{-4{\sqrt{2}}\alpha G_{F}\over 9\pi}f_{B_{s}}m_{b_{s}}V_{ts}^{*}V_{tb}\times$ (24) $\displaystyle\left(\frac{m_{b}}{m_{B_{s}}}BK(m_{b}^{2})+{3C_{7}\over 8\bar{\Lambda}}\right).$ with $B=-(3C_{6}+C_{5})/4$, $\bar{\Lambda}=m_{B_{s}}-m_{b}$, and $\displaystyle M^{-}$ $\displaystyle=$ $\displaystyle{4{\sqrt{2}}\alpha G_{F}\over 9\pi}f_{B_{s}}m_{b_{s}}V_{ts}^{*}V_{tb}\times$ (25) $\displaystyle\left(\sum_{q}A_{q}J(m_{q}^{2})+\frac{m_{b}}{m_{B_{s}}}BL(m_{b}^{2})+{3C_{7}\over 8\bar{\Lambda}}\right).$ where $\displaystyle A_{u}$ $\displaystyle=$ $\displaystyle(C_{3}-C_{5})N_{c}+(C_{4}-C_{6})$ $\displaystyle A_{d}$ $\displaystyle=$ $\displaystyle{1\over 4}\left[(C_{3}-C_{5})N_{c}+(C_{4}-C_{6})\right]$ $\displaystyle A_{c}$ $\displaystyle=$ $\displaystyle(C_{1}+C_{3}-C_{5})N_{c}+(C_{2}+C_{4}-C_{6})$ $\displaystyle A_{s}$ $\displaystyle=$ $\displaystyle{1\over 4}\left[(C_{3}+C_{4}-C_{5})N_{c}+(C_{3}+C_{4}-C_{6})\right]$ (26) $\displaystyle A_{s}$ $\displaystyle=$ $\displaystyle{1\over 4}\left[(C_{3}+C_{4}-C_{5})N_{c}+(C_{3}+C_{4}-C_{6})\right].$ (27) The functions $J(m^{2})$, $K(m^{2})$ and $L(m^{2})$ are defined by $\displaystyle J(m^{2})$ $\displaystyle=$ $\displaystyle I_{11}(m^{2}),$ $\displaystyle K(m^{2})$ $\displaystyle=$ $\displaystyle 4(I_{11}(m^{2})-I_{00}(m^{2})),$ $\displaystyle L(m^{2})$ $\displaystyle=$ $\displaystyle I_{00}(m^{2}),$ (28) with $I_{pq}(m^{2})=\int_{0}^{1}{dx}\int_{0}^{1-x}{dy}\frac{x^{p}y^{q}}{m^{2}-2xyk_{1}\cdot k_{2}-i\varepsilon}$ (29) The decay width for $B_{s}\to\gamma\gamma$ is simply $\Gamma(B_{s}\to\gamma\gamma)={m_{B_{s}}^{3}\over 16\pi}({|M^{+}|}^{2}+{|M^{-}|}^{2}).$ (30) In SM, with $C_{2}=C_{2}(M_{W})=1$ , and the other Wilson coefficients are zero, we find $\Gamma(B_{s}\to\gamma\gamma)=1.3\times 10^{-10}\ {\rm eV}$ which amounts to a branching ratio $Br(B_{s}\to\gamma\gamma)=3.5\times 10^{-7}$, for the given $\Gamma^{total}_{B_{s}}=4\times 10^{-4}\ {\rm eV}$. In numerical calculations we use the corresponding input parameters $M_{W}=80.22\;GeV$, $\alpha_{s}(m_{Z})=0.117$, $m_{c}=1.5\;GeV$, $m_{b}=4.8\;GeV$ and $|V_{tb}V_{ts}^{*}|^{2}/|V_{cb}|^{2}=0.95$ , respectively. The present experimental limit[22] on the decay $B_{s}\to\gamma\gamma$ is $\displaystyle{\rm Br}(B_{s}\to\gamma\gamma)\leq 8.6\times 10^{-6},$ (31) which is far from the theoretical results. So, we can not put the constraint to the masses of PGBs. The constraints of the masses of $P^{\pm}$ and $P^{\pm}_{8}$ can be from the decay[24] $B\to s\gamma$ : $m_{P^{\pm}_{8}}>400$GeV. Figure 3: the $Br(B_{s}\to\gamma\gamma)$ about the mass of $P_{8}^{\pm}$ under different values of $m_{P^{\pm}}$. Figure 4: the $Br(B_{s}\to\gamma\gamma)$ about the mass of $P^{\pm}$ under different values of $P_{8}^{\pm}$. Fig.3(4) denotes the $Br(B_{s}\to\gamma\gamma)$ about the mass of $P_{8}^{\pm}$ ($P^{\pm}$) under different values of $m_{P^{\pm}}$ ($P_{8}^{\pm}$). From Fig.3 and 4, we find the the curves are much different from the the SM one. It can be enhanced about 1-2 levels to the SM prediction in the reasonable region of the masses of PGBs. This gives the strong new physics signals from the Technicolor Model. The branching ratio of $B_{s}\to\gamma\gamma$ decrease along with the mass of $P_{8}^{\pm}$ and $P^{\pm}$ reduce. This is from the decoupling theorem that for heavy enough nonstandard boson. When $m(P^{\pm})$ and $m(P^{\pm}_{8})$ have large values, the contributions from OGTM is small.From the $Eq(16),(17),(18)$ ,we can see the functions $B$, $D$ and $E$ go to zero, as $x$, $y\to\infty$.The branching ratio in the Fig.(3) is changed much faster than that in the Fig.(4).This is because the contribution to $B_{s}\to\gamma\gamma$ from the color octet $P_{8}^{\pm}$ is large when compared with the contribution from color singlet $P^{\pm}$. As a conclusion, the size of contribution to the rare decay of $B_{s}\to\gamma\gamma$ from the PGBs strongly depends on the values of the masses of the charged PGBs. This is quite different from the SM case. By the comparison of the theoretical prediction with the current data one can derived out the the contributions of the PGBs: $P^{\pm}$ and $P^{\pm}_{8}$ to $B_{s}\to\gamma\gamma$ and give the new physics signals of new physics. ## References * (1) * (2) A N Mitra. Phys. Lett., 2000, B473: 297-304 * (3) Junjie Cao,Zhenjun Xiao,Gongru Lu. Phys. Rev., 2001, D64: 014012 * (4) Z.J.Xiao, C.D.L, W.J.Huo.Phys. Rev., 2003, D67: 094021 * (5) L. Reina, G. Ricciardi, A. Soni. Phys. Rev., 1997, D56: 5805 * (6) G. Hiller, E.O. Iltan. Phys. Lett., 1997, B409: 425 * (7) C.-H. Chang, G.-L. Lin, Y.-P. Yao. Phys. Lett., 1997, B415: 395-401 * (8) M.R. Ahmady, E. Kou. hep-ph/9708347 * (9) G. Hiller, E.O. Iltan, Mod.Phys.Lett., 1997.A12: 2837-2846 * (10) S. Choudlhury, J. Ellis. hep-ph/9804300 * (11) T.M. Aliev, G. Turan.Phys. Rev., 1993, D48: 1176 * (12) T.M. Aliev, G. Hiller, E.O. Iltan. Nucl.Phys.,1998, B515: 321-341 * (13) T.M. Aliev, E.O. Iltan. Phys. Rev., 1998, D58: 095014 * (14) S. Bertolini, J. Matias. Phys. Rev., 1998, D57: 4197-4204 * (15) P. Singer, D.-X. Zhang.Phys. Rev., 1997, D56: 4274 * (16) S. Dimopoulos. Nucl.Phys.,1980, B168: 69 * (17) E. Farhi, L. Susskind.Phys. Rev., 1979, D20: 3404 * (18) M.E. PeskinT. Takeuchi. Phys.Rev. Lett., 1990, 65: 964 * (19) I. Maksymyk, C.P. Burgess. Phys. Rev., 1994, D50: 529 * (20) Z.J.Xiao, L.D.Wan, J.M.Yang, etal. Phys. Rev., 1994, D49: 5949 * (21) E.Eichten,I. Hinchliffe,K. Lane,etal. Phys. Rev., 1986, D34: 1547 * (22) C.D. Lu ,Z.J. Xiao. Phys. Rev., 1996, D53: 2529 * (23) J.Wicht, I. Adachi, H. Aihara, etal. Phys.Rev. Lett., 2008, 100:121801
arxiv-papers
2011-01-12T20:18:57
2024-09-04T02:49:16.393518
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Qin XiuMei, Wujun Huo, Xiaofang Yang", "submitter": "Wujun Huo Dr", "url": "https://arxiv.org/abs/1101.2437" }
1101.2471
# CATEGORY OF FUZZY HYPER BCK-ALGEBRAS J.DONGHO ****Department of Mathematics, University of Yaounde, BP 812, Cameroon josephdongho@yahoo.fr ###### Abstract. In this paper we first define the category of fuzzy hyper BCK-algebras. After that we show that the category of hyper BCK-algebras has equalizers, coequalizers, products. It is a consequence that this category is complete and hence has pullbacks. ## 1\. Introduction The study of hyperstructure was initiated in 1934 by F. Marty at 8th congress of Scandinavian Mathematiciens. Y.B. Jun et al. applied the hyperstructures to BCK-algebras, and introduces the notion of hyper BCK-algebra. Now we follow [1,2,3,4] and introduce the category of fuzzy hyperBCK-algebra and obtain some result, as mentioned in the abstarct. ## 2\. Preliminaries We now review some basic definitions that are very useful in the paper. ###### Definition 1. $[\ref{3}]$ Let $H$ be an non empty set. A hyperoperation $*$ on $H$ is a mapping of $H\times H$ family of non-empty subsets of $H$ $\mathcal{P}^{*}(H)$ ###### Definition 2. Let $*$ be an hyperoperation on $H$ and $O$ a constant element of $H$ An hyperorder on $H$ is subset $<$ of $\mathcal{P}^{*}(H)\times\mathcal{P}^{*}(H)$ define by: for all $x,y\in H,x<y$ iff $O\in x*y$ and for every $A,B\subseteq H,A<B$ iff $\forall a\in A,\exists b\in B$ such that $a<b.$ ###### Definition 3. If $*$ is hyperoperation on $H$. For all $A,B\subseteq H,A*B:=\underset{a\in A,b\in B}{\bigcup}a*b$ ###### Definition 4. $[1]$ By hyper BCK-algebra we mean a non empty set $H$ endowed with a hyper- operation $*$ and a constant $O$ satisfying the following axioms. * (HK1) $(x*z)*(y*z)<(x*y)$ * (HK2) $(x*y)*z=(x*z)*y$ * (HK3) $x*H<\\{x\\}$ ###### Definition 5. A fuzzy hyper BCK-algebra is a pair $(\mathbf{H};\mu_{H})$ where $\mathbf{H}=(H;*;O)$ is hyper BCK-algebra and $\mu_{H}:H\longrightarrow[0,1]$ is a map satisfy the following property: $\inf(\mu_{H}(x*y))\geq\min(\mu_{H}(x),\mu_{H}(y))$ for all $x,y\in H.$ ###### Example 1. $[\ref{5}]$ Let $n\in\mathbb{N}^{*}.$ Define the hyperoperation $*$ on $H=[n,+\infty)$ as follows: $x*y=\left\\{\begin{array}[]{lllc}[n,x]&\texttt{iff}\quad x<y\\\ (n,y]&\texttt{iff}\quad x>y\neq n\\\ \\{x\\}&\texttt{iff}\quad y=n\end{array}\right.$ for all $x,y\in H$. To show that $(H,*,n)$ is hyper BCK-algebra, it suffice to show axiom $HK3.$ For all $x\in H,x*H=\underset{t\in H}{\bigcup}x*t.$ For all $x\in H$ then $x*x\subseteq x*H.$ And then $n\in[n,x]*\\{x\\}$ ## 3\. The category of fuzzyhyper BCK-algebras ###### Lemma 1. Let $(\mathbf{H};\mu_{H})$ be a fuzzy hyper BCK-algebra. For all $x\in H,\mu_{H}(O)\geq\mu_{H}(x)$ Proof. For all $x\in H,x<x;$ then $O\in x*x$. $\begin{array}[]{ccll}O\in x*x&\texttt{imply}&\mu_{H}(O)\geq\inf(\mu_{H}(x*y))\geq\min(\mu_{H}(x),\mu_{H}(y))\\\ &\texttt{i.e}&\mu_{H}(O)\geq\min(\mu_{H}(x),\mu_{H}(y))=\mu_{H}(x)\\\ &\texttt{i.e}&\mu_{H}(O)\geq\mu_{H}(x).\end{array}$ ###### Definition 6. Let $(\mathbf{H};\mu_{H})$ be a fuzzy hyperBCK-algebra. $\mu_{H}$ is called a fuzzy map. ###### Lemma 2. Let $(\mathbf{H};\mu_{H})$ a fuzzy hyper BCK-algebra.The following properties are trues: * i) If for all $x,y\in H,x<y$ imply $\mu_{H}(x)\leq\mu_{H}(y)$ then for all $x\in H,\mu_{H}(x)=\mu_{H}(O)$ * ii) If $\mu_{H}(O)=0$ then $\mu_{H}(x)=0$ Proof. * i) For all $x\in H$, $x*H<\\{x\\}$ then $x*O<x$. $O<x\Rightarrow\mu_{H}(O)\leq\mu_{H}(x)$. Then $\mu_{H}(x)\leq\mu_{H}(O)$ and $\mu_{H}(O)\leq\mu_{H}(x)$ for all $x\in H$. i.e $\mu_{H}(x)=\mu_{H}(O)$ for all $x\in H$. * ii) $\mu_{H}(O)=O\Rightarrow\mu_{H}(O)\leq\mu_{H}(x),$ for all $x\in H.$ Then $\mu_{H}(x)=\mu_{H}(O)$ for all $x\in H$. ###### Definition 7. Let $(\mathbf{H};\mu_{H})$ and $(\mathbf{F},\mu_{F})$ two fuzzy hyperBCK- algebras. An homomorphism from $(\mathbf{H},\mu_{H})$ to $(\mathbf{F},\mu_{F})$ is an homomorphism $f:\mathbf{H}\longrightarrow\mathbf{F}$ of hyper BCK-algebra such that for all $x\in H$, $\mu_{F}(f(x))\geq\mu_{H}(x)$ ###### Proposition 1. Let $(\mathbf{H},\mu_{H})$ an hyperBCK-algebra. Let $\mathbf{G},\mathbf{F}\subset H$ two hyperBCK-sualgebras of $\mathbf{H}$. If there exist $\alpha\in]0,1[$ such that $\mu_{H}(G^{*})\subset[0,\alpha[$ and $\mu_{H}(F)\subseteq]\alpha,1]$. Then any homorphism of hyper BCK-algebra $f:G\longrightarrow F$ is homomorphism of fuzzy hyperBCK-algebra. Proof. Suppose that there is $\alpha\in]0,1]$ such that $\mu_{H}(G^{*})\subset[0,\alpha[$ and $\mu_{H}(F)\subseteq]\alpha,1[.$ Let $f:G\longrightarrow F$ an homomorphism of hyper BCK-algebra. For all $x\in G^{*},f(x)\in F.$ And $\mu_{F}(f(x))>\alpha>\mu_{H}(x).$ Then $\mu_{F}(f(x))>\mu_{H}(x)$ for all $x\in G^{*}$ $f(O)=O$ then $\mu_{F}(f(O))=\mu_{F}(O)=\mu_{H}(O)$ i.e $\mu_{F}(f(O))=\mu_{H}(O).$ therefore, for all $x\in x\in G,\mu_{F}(x)\geq\mu_{H}(x)$ ###### Example 2. $[\ref{1}]$ Define the hyper operation $"*"$ on $H=[1;+\infty]$ as follow. $x*y=\left\\{\begin{array}[]{ccccll}&[1,x]&&\texttt{if}&\quad x\leq y\\\ &(1,y]&&\texttt{if}&\quad x>y\neq 1\\\ &\\{x\\}&&\texttt{if}&\quad y=1\end{array}\right.$ For all $x,y\in H,(\mathbf{H},*,1)$ is hyperBCK-algebra. Define the fuzzy structure $\mu_{H}$ on $H$ by: $\begin{array}[]{lllcc}\mu_{H}:&H&\longrightarrow&[0,1]&\\\ &x&\mapsto&\frac{1}{x}&\end{array}$ We show that $(\mathbf{H},\mu_{H})$ is a fuzzy hyper BCK-algebra. Let $x,y\in H.$ 1. (i) If $x\leq y$, then $x*y=[1,x]$; i.e for all $t\in x*y,1\leq t\leq x\leq y$ and so $\frac{1}{y}\leq\frac{1}{x}\leq\frac{1}{t}.$ So, $\mu_{H}(t)\geq\frac{1}{y}=\min\\{\frac{1}{y},\frac{1}{x}\\}=\min\\{\mu_{H}(x),\mu_{H}(y)\\}$. Then $\inf\\{x*y\\}\geq\min\\{\mu_{H}(x),\mu_{H}(y)\\}$ 2. (ii) If $x>y\neq 1$ then $x*y=(1,y]$. For all $t\in H\cap x*y,\frac{1}{x}\leq\frac{1}{y}\leq\frac{1}{t}\leq 1.$ therefore, $\mu_{H}(t)=\frac{1}{t}\geq\frac{1}{x}=\min\\{\mu_{H}(x),\mu_{H}(y)\\}$ for all $t\in x*y$. Then $\in\\{\mu_{H}(x*y)\\}\geq\min\\{\mu_{H}(x),\mu_{H}(y)\\}.$ 3. (iii) If $y=1,x*y=\\{x\\}$, hence $\mu_{H}(x*y)=\\{\mu_{H}(x)\\}=\\{\frac{1}{x}\\}$. $y=1$imply $y\leq x$ and $\frac{1}{x}\leq\frac{1}{y}$ for all $x\in H$; i.e; $\min\\{\mu_{H}(x),\mu_{H}(y)\\}=\frac{1}{x}$. Then $\mu_{H}(x*y)=\\{\frac{1}{x}\\}.$ Thus $\inf\\{\mu_{H}(x*y)\\}=\frac{1}{x}\geq\min\\{(\mu_{H}(x),\mu_{H}(y))\\}$ ###### Proposition 2. The fuzzy hyperBCK-algebras and homomorphisms of fuzzy hyperBCK-algebras form a category. Proof. The proof is easy. ###### Notes 1. In the following we let $\mathcal{H}$ the category of hyperBCK-algebras; $\mathbb{F}_{\mathcal{H}}$ the category of fuzzy hyperBCK-algebras; $\mathbb{H}$ the fuzzy hyper BCK-algebra $(\mathbf{H},\mu_{H})$ For any fuzzy hyper BCK-algebra $\mathbb{H}$, we associate for all $\alpha\in[0,1]$ the set $H_{\alpha}:=\\{x\in H,\mu_{H}(x)\geq\alpha\\}$ ###### Lemma 3. Let $\mathbb{H}$ a fuzzy hyper BCK-algebra. For all $\alpha\in[0,1],O\in H_{\alpha}$ and for all $x,y\in H,x*y\subseteq H_{\alpha}$ Proof. By lemma 1, for all $x\in H,\mu_{H}(x)\leq\mu_{H}(0).$ Then for all $x\in H_{\alpha},\mu_{H}(O)\geq\mu_{H}(x)>\alpha$ i.e $O\in H_{\alpha}$. Let $x,y\in H_{\alpha}$; for all $t\in x*y,$ $\mu_{H}(t)\geq\inf\\{\mu_{H}(x*y)\geq\min\\{\mu_{H}(x),\mu_{H}(y)\\}\\}\geq\alpha$ then $t\in H_{\alpha}$. therefore, $x*y\subseteq H_{\alpha}$ ###### Definition 8. Let $(H,*,O)$ be an hyper BCK-algebra. An hyper BCK-subalgebra of $H$ is a non empty subset $S$ of $H$ such that $O\in S$ and $S$ is hyper BCK-algebra with respect to the hyper operation $"*"$ on $H$ ###### Proposition 3. Let $(H,*,O)$ be an hyper BCK-algebra. A non empty subset $S$ of $H$ is hyper BCK-subalgebra of $H$ iff for all $x,y\in S,x*y\in S$ Proof. The proof is easy. ###### Definition 9. A fuzzy hyper BCK-subalgebra of $\mathbb{H}$ is an hyper BCK-subalgebra $S$ of $\mathbf{H}$ with the restriction $\mu_{S}$ of $\mu_{H}$ on $S.$ ###### Proposition 4. For all $\alpha\in[0,1],$ $(H_{\alpha},\mu_{H})$ is fuzzy hyper BCK-subalgebra of $\mathbb{H}$ Proof. By lemma 3, $H_{\alpha}$ is hyper BCK-subalgebra of $\mathbf{H}$ and $\inf\\{\mu_{H}(x*y)\\}\geq\min\\{\mu_{H}(x),\mu_{H}(y)\\}$ ###### Definition 10. Let $\mathbb{H}$ by an fuzzy hyper BCK-algebra. The fuzzy-hyperBCK-subalgebra $\mathbf{H}_{\alpha}:=(H_{\alpha};\mu_{H})$ is calling hyper $\alpha$-cut of $\mathbb{H}$ ###### Proposition 5. Let $\mathbb{H}$ be fuzzy hyper BCK-algebra. A hyper BCK-subalgebra $S$ of $\mathbf{H}$ is fuzzy hyper BCK-subalgebra iff $S$ is hyper $\alpha$-cut of $H.$ Proof. By prosition 4, any hyper $\alpha$-cut is fuzzy hyper BCK-subalgebra. Conversely, let $S$ be fuzzy hyper BCK-subalgebra of $\mathbb{H}$. Then $\mu_{H}(S)$ is subset of $[0,1].$ If $0\in\mu_{H}(S),$ then $S=H_{0}=\mathbb{H}.$ If $0<\inf(\mu_{H}(S)),$ then $S=H_{\inf(\mu_{H}(S))}.$ ###### Proposition 6. Let $\mathbb{H}$ and $\mathbb{F}$ be two fuzzy hyper BCK algebras. An $\mathcal{H}$-morphism $f:H\longrightarrow F$ is $\mathbb{F}_{\mathcal{H}}$-morphism iff for all $\alpha\in[0,1],f(H_{\alpha})\subseteq F_{\alpha}.$ Proof. Suppose that $f(H_{\alpha})\subseteq F_{\alpha}$ for all $\alpha\in[0,1]$ Let $x\in[0,1]$ we need $\mu_{H}(x)\leq\mu_{F}(f(x))$. Let $\alpha=\mu_{H}(x);\quad x\in H_{\alpha}$ and $f(x)\in f(H_{\alpha})\subseteq F_{\alpha}$. Then $\mu_{F}(f(x))>\alpha=\mu_{H}(x).$ whence for all $x\in H,\mu_{F}(f(x))\geq\mu_{H}(x)$. Conversely, suppose that $f:\mathbb{H}\longrightarrow\mathbb{F}$ is $\mathbb{F}_{\mathcal{H}}$-morphism. For all $x\in H_{\alpha}$ for some $\alpha\in[0,1]$, $\mu_{F}(f(x))\geq\mu_{H}(x)\geq\alpha$i.e; $f(x)\in[0,1]$. Then $f(H_{\alpha})\subseteq F_{\alpha}$ for all $\alpha\in[0,1]$. ###### Proposition 7. A $\mathbb{F}_{\mathcal{H}}$-morphism $f:\mathbb{H}\longrightarrow\mathbb{F}$ is $\mathbb{F}_{\mathcal{H}}$-iso iff it is both $\mathcal{H}$-iso and $\mu_{H}=\mu_{F}f.$ Proof. Suppose that $f$ is $\mathcal{H}$-iso and $\mu_{H}=\mu_{F}f$. there is $g\in Hom_{\mathcal{H}}(\mathbf{F},\mathbf{H})$; $g\circ f=Id_{H}$ and $g\circ g=Id_{F}.$ Then, for all $x\in F,\mu_{H}(g(x))=\mu_{F}(f(g(x)))\mu_{F}(x).$ And then, $g\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H}).$ Conversely, Suppose that $f$ is $\mathbb{F}_{\mathcal{H}}$-iso. There is $g\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H});g\circ f=Id_{F}$ and $f\circ g=Id_{H}.$ Since $f\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H}),\mu_{H}\leq\mu_{F}f.$ Since $g\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{H},\mathbb{H}),\mu_{F}\leq\mu_{H}g.$ $x\in H$ imply $f(x)\in F$. Then $\mu_{F}(f(x))\leq\mu_{H}(g(f(x)))=\mu_{H}(x)$ i.e; $\mu_{F}f\leq\mu_{H}.$ therefore, $\mu_{F}f=\mu_{H}$ ###### Proposition 8. Let $f\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H}).$ $f$ is $\mathbb{H}_{\mathcal{H}}$-mono iff $f$ is $\mathcal{H}$-mono Proof. Suppose that $f$ is $\mathbb{H}_{\mathcal{H}}$-mono. For all $h,g\in Hom_{\mathcal{H}}(\mathbf{K},\mathbf{H})$, such that $fh=fg,$ we define $\mu_{K}=\min(\mu_{H}(h(x);\mu_{H}(g(x))))$ for all $x\in K.$ 1. a) We show that $(K;\mu_{K})$ is fuzzy hyper BCK-algebra. $\begin{array}[]{lllccccllll}\inf(\mu_{K}(x*y))&=&\inf\\{\mu_{H}(h(x*y);\mu_{H}(g(x*y)))\\}\\\ &=&\inf\\{\mu_{H}(h(x)*h(y);\mu_{H}(g(x)*g(y)))\\}\\\ &=&\min\\{\inf\\{\mu_{H}(h(x)*h(y)\\};\inf\\{\mu_{H}(g(x)*g(y)))\\}\\}\\\ &\geq&\min\\{\min\\{\mu_{H}(h(x);\mu_{H}(h(y)\\};\min\\{\mu_{H}(g(x);g(y))\\}\\}\\\ &\geq&\min\\{\min\\{\mu_{K}(x);\mu_{K}(y)\\}\\}\\\ &\geq&\min\\{\mu_{K}(x);\mu_{K}(y)\\}\par\par\end{array}$ Then, for all $x,y\in K,\inf(\mu_{K}(x*y))\geq\min\\{\mu_{K}(x),\mu_{K}(y)\\}$. therefore, $(K;\mu_{K})$ is fuzzy hyper BCK-algebra. 2. b) We show that $h$ and $g$ are $\mathbb{F}_{\mathcal{H}}$-homomorphism. For all $x\in K,$ $\mu_{K}(x)=\min\\{\mu_{H}(h(x),\mu_{H}(g(x)))\\}.$ Then $\mu_{K}(x)\leq\mu_{H}(g(x))$ and $\mu_{K}(x)\leq\mu_{H}(h(x))$. therefore, $h$ and $g$ are $\mathbb{F}_{\mathcal{}}H$-morphism. Since $f$ is $\mathbb{F}_{\mathcal{H}}$-mono and $h,g\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{F},\mathbb{H}),$ $fh=fg$ imply $f=g$ Conversely, if $f$ is $\mathbb{F}_{\mathcal{H}}$-mono, it is $\mathcal{H}$-mono. ###### Lemma 4. The pair $\mathbb{O}=(\\{O\\},\mu_{o})$ where $\begin{array}[]{lcccll}\mu_{o}:&\\{o\\}&\longrightarrow&[0,1]\\\ &o&\mapsto&0\end{array}$ is fuzzy hyper BCK-algebra Proof. Easy ###### Lemma 5. $\mathbb{O}$ is final objet of $\mathbb{F}_{\mathcal{H}}$ ###### Proposition 9. The category $\mathbb{F}_{\mathcal{H}}$ has products. Proof. Let $(\mathbb{H}_{i};\mu_{H_{i}})_{i\in I}$ a family of fuzzy hyper BCK-algebras. Denote $\mathbf{H}=\underset{i\in I}{\prod}H_{i}$ the $\mathcal{H}$-product of $(H_{i})_{i\in I}$ with the projection morphisms $p_{i}:\mathbf{H}\longrightarrow H_{i}$. Consider the following map $\mu_{H}:H\longrightarrow[0,1]$ define by: $\mu_{H}(x)=\underset{i\in I}{\bigwedge}\mu_{H_{i}}p_{i}(x)$ for all $x\in H$ * a) We show that the pair $(\mathbf{H};\mu_{H})$ is fuzzy hyper BCK-algebra. For all $x,y\in H,p_{i}(x*y)=p_{i}(x)*p_{i}(y)$ for all $i\mathbb{N}I.$ Then $\begin{array}[]{lcllc}\inf(\mu_{H_{i}}p_{i}(x*y))&=&\inf(\mu_{H_{i}}(p_{i}(x)*p_{i}(y))\\\ &\geq&\min\\{\mu_{H_{i}}(p_{i}(x));\mu_{H_{i}}(p_{i}(y))\\}\end{array}$ for all $i\in I$. Then, $\begin{array}[]{lllcc}\inf(\underset{i\in I}{\bigwedge}\mu_{H_{i}}p_{i}(x*y))&\geq&\underset{i\in I}{\bigwedge}\inf\\{\mu_{H_{i}}(p_{i}(x)*p_{i}(y)\\}\\\ &\geq&\underset{i\in I}{\bigwedge}\min\\{\mu_{H_{i}}(p_{i}(x));\mu_{H_{i}}(p_{i}(y))\\}\\\ &\geq&\min\\{\underset{i\in I}{\bigwedge}\mu_{H_{i}}p_{i}(x);\mu_{H_{i}}p_{i}(y)\\}\\\ &\geq&\min\\{\mu_{H}(x),\mu_{H}(y)\\}\end{array}.$ * b) For all $i\in I,x\in H;\mu_{H_{i}}p_{i}(x)\geq(\underset{i\in I}{\bigwedge}\mu_{H_{i}}p_{i})(x)$. Then each $p_{i}$ is $\mathbb{F}_{\mathcal{H}}$-morphism. * c) If $q_{j}:\mathbb{F}\longrightarrow\mathbb{H}_{j}$ is family of $\mathbb{F}_{\mathcal{H}}$-morphism, there is unique $\mathcal{H}$-morphism $\varphi:\mathbf{F}\longrightarrow\mathbf{H}$ such that the following diagram commute. $\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{j}}$$\textstyle{H_{j}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\scriptstyle{q_{j}}$ i.e $p_{j}\varphi=q_{j}$ for all $j\in I$ For all $x\in F$, $\mu_{F}(x)\leq\mu_{H_{i}}q_{i}(x)$. $\begin{array}[]{lllcc}\texttt{Then}&\mu_{F}(x)&\leq\mu_{H_{i}}p_{i}\varphi(x)\quad\texttt{for all }\quad x\in F,i\in I\\\ \texttt{i.e}&\mu_{F}(x)&\leq\underset{i\in I}{\bigwedge}\mu_{H_{i}}p_{i}\varphi(x)\\\ &&\leq(\underset{i\in I}{\bigwedge}\mu_{H_{i}}p_{i})\varphi(x)\\\ &&\leq\mu_{H}(\varphi(x))\quad\texttt{for all}\quad x\in F.\\\ &\texttt{Then}&\mu_{F}\leq\mu_{H}\varphi\leq\end{array}$ Then $\varphi$ is $\mathbb{F}_{\mathcal{H}}$-morphism. ###### Proposition 10. $\mathbb{F}_{\mathcal{H}}$ have equalizers. Proof. Let $f,g\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{H},\mathbb{F}),K:=\\{x\in H,f(x)=g(x)\\}$. It is prove in [1] that $K$ is hyper BCK-subalgebra of $H$. It is clear that $(K,\mu_{H})$ is fuzzy hyper BCK-algebra. Let $i:K\longrightarrow H$ the inclusion map. $i\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{K},\mathbb{F}).$ For all $x\in K,fi(x)=f(x)=g(x)=gi(x).$ Let $h\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{L},\mathbb{F})$ such that $fh=gh,$ for all $x\in L,f(h(x))=g(h(x))$. Then $Imh\subseteq L$. Define $\delta:L\longrightarrow K$ by $\delta(x)=h(x)$ for all $x\in L.$ $\delta\in Hom_{\mathbb{F}_{\mathcal{H}}}(\mathbb{\mathbb{H}},\mathbb{F})$ and $i\delta=h.$ So, the following diagram commute. | ---|--- $\textstyle{\mathbb{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathbb{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{\mathbb{F}}$$\textstyle{\mathbb{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{h}$ Since $i$ is monic, $\delta$ is unique $\mathbb{F}_{\mathcal{H}}$-morphism such that the above diagram commute. therefore, $\mathbb{F}_{\mathcal{H}}$ have equalizers. ###### Proposition 11. $\mathbb{F}_{\mathcal{H}}$ is complet. Proof. By proposition 9, each family of objets of $\mathbb{F}_{\mathcal{H}}$ has product. By proposition 10, each pair of parallel arrows has an equalizer. Then $\mathbb{F}_{\mathcal{H}}$ is complet. ###### Corollary 1. $\mathbb{F}_{\mathcal{H}}$ has pulbacks Proof. By propositions 9 and 10, $\mathbb{F}_{\mathcal{H}}$ has equalizers and products. therefore, $\mathbb{F}_{\mathcal{H}}$ has pulbacks. ###### Proposition 12. $\mathcal{\mathbb{F}_{H}}$ have coequalizers Proof. Let $f,g\in Hom_{\mathcal{\mathbb{F}_{H}}}(\mathbb{H},\mathbb{K})$. Let $\sum_{fg}=\\{\theta,\theta\quad\texttt{regular congruence relation on}\quad\mathbf{K}\quad\texttt{such that}\quad f(a)\theta g(a)\forall a\in H\\}$ $\sum_{fg}\neq\phi$ because $K\times K\in\sum_{fg}$ Let $\rho=\underset{\theta\in\sum_{fg}}{\bigcap}\theta$. Then, $\rho$ is regular congruence relation. Define on $K/\rho$ the following hyper operation $[x]_{\rho}*[y]_{\rho}=[x*y]_{{\rho}}.$ $(K/\rho;*;[0]_{\rho})$ is an objet of $\mathcal{H}$ (see $[\ref{1}]$). Define on $K/\rho$ the following map $\begin{array}[]{lllcc}\mu_{K/\rho}:&K/\rho&\longrightarrow&[0,1]\\\ {}&\mu_{K/\rho}([x]_{\rho})&\longmapsto&\underset{a\in[x]_{\rho}}{\bigvee}\mu_{K}(a)\end{array}$ * a) We show that $(K/\rho,\mu_{K/\rho})$ is objet of $\mathbb{F}_{\mathcal{H}}$. If $x,y\in K$ such that $[x]_{\rho}=[y]_{\rho}$. Then $\underset{a\in[x]_{\rho}}{\bigvee}\mu_{K}(a)=\underset{a\in[y]_{\theta}}{\bigvee}\mu_{K}(a)$ $\forall x\in K,\,\mu_{K}(x)\leq\underset{a\in[x]_{\rho}}{\bigvee}\mu_{K}(a)$. Then $\mu_{K}\leq\mu_{K/\rho}([x]_{\rho})=\mu_{K/\rho}(\pi(x))$ Then, the canonical projection $\pi$ is an $\mathcal{\mathbb{F}_{H}}-morphism$ Since for all $x\in H,f(x)\rho g(x)$, then $[f(x)]_{\rho}=[g(x)]_{\rho}$. therefore, $(\pi\circ f)(x)=(\pi\circ g)(x).$ Then, $\pi\circ f=\pi\circ g.$ * b) Universal property of coequalizer. Let $\varphi:\mathbb{K}\longrightarrow\mathbb{L}$ and $\mathbb{F}_{\mathcal{H}}$-morphism such that $\varphi\circ f=\varphi\circ g$. Define the following mapping. $\begin{array}[]{lllcc}\psi:&K/\rho&\longrightarrow&L\\\ {}&[x]_{\rho}&\longmapsto&\varphi(x)\end{array}$ * c) We prove that $\psi$ is well define. If $[x]_{\rho}=[y]_{\rho}$ then, for all $a\in H,\varphi(f(a))=\varphi(g(a))$ imply $f(a)R_{\varphi}g(a)$ because $R_{\varphi}$ is regular congruence on $K$. Then $R_{\varphi}\in\sum_{f,g}.$ The minimality of $\rho$ on $\sum_{f,g}$ imply $\rho\subseteq R_{\varphi}$. therefore, $[x]_{\rho}=[y]_{\rho}$ imply $x\rho y$. Then $xR_{\varphi}y$. i.e $\varphi(x)=\varphi(y)$ And then, $\psi([x]_{\rho})=\psi([y]_{\rho})$ therefore, $\psi$ is well define. For all $x\in K,\mu_{L}(\psi(\pi)(x))=\mu_{L}(\varphi(x))\geq\mu_{K}(x),\forall x\in K$ then for all $a\in[x]_{\rho}$. $\mu_{L}(\varphi(a))\geq\mu_{K}(a).$ By the minimality of $\rho$, $[a]_{\rho}=[x]_{\rho}$ imply $a\rho x$ then $aR_{\varphi}x$ i.e $\varphi(a)=\varphi(x)$ (because $\rho\subseteq R_{\varphi}$). Then $\underset{a\in[x]_{\rho}}{\bigvee}\tilde{L}(\varphi(a))=\underset{a\in[x]_{\rho}}{\bigvee}\tilde{L}(\varphi(x))=\tilde{L}(\varphi(x))$ therefore $\tilde{L}(\varphi(x))\geq\underset{a\in[x]_{\rho}}{\bigvee}(a)=\tilde{K}/\rho([x]_{\rho})$ i.e $\tilde{L}(\psi([x]_{\rho}))\geq\tilde{k}/\rho([x]_{\rho})\forall x\in H$. It is clean that $\psi(\pi(x))=\psi([x]_{\rho})=\varphi(x),\forall x\in H$ i.e $\psi\circ\pi=\varphi$ This prove the commutativity of the following diagram: $\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$.$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\varphi}$$\textstyle{K/\rho\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{L}$ The unicity of $\psi$ is thus to the fat that $\pi$ is epimorphism. Then, $\mathcal{F}_{\mathcal{H}}$ have coequalizer. ## ACKNOWLEDGEMENTS ## References * [1] H. Harizavi,J. Macdonald AND A. Borzooei _Category of bck-algebras_ :Scientiae Mathematicae Japonicae Online, e-2006, 529-537. * [2] R. A. Borzooei H.Harizavi._Regular Congruence Relations on hyper BCK-algebras_ : vol.61, No.1(2005),83-97 * [3] Eun Hwan ROH, B. Davvaz And Kyung Ho Kim _T-fuzzy subhypernea-rings of hypernear-ring_ , Scientiae Mathematicae Japonicae Online, e-2005,19-29 . * [4] Carol L. Walker, _Category of fuzzy sets_. * [5] R.A. Borzoei and M.M. Zahedi, _Positive implicative hyperK-ideals_ , Scientiae Mathematicae Japonicae Online, Vol. 4,(2001), 381-389. * [6] C. LELE and M. Salissou, _Discussiones Mathematicae, general Algebra and Application_ 26: 111-135(2006).
arxiv-papers
2011-01-13T01:09:39
2024-09-04T02:49:16.399475
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Joseph Dongho", "submitter": "Joseph Dongho", "url": "https://arxiv.org/abs/1101.2471" }
1101.2513
# Pattern fluctuations in transitional plane Couette Flow Joran Rolland, Paul Manneville Laboratoire d’Hydrodynamique de l’École Polytechnique, 91128 Palaiseau, France ###### Abstract In wide enough systems, plane Couette flow, the flow established between two parallel plates translating in opposite directions, displays alternatively turbulent and laminar oblique bands in a given range of Reynolds numbers $R$. We show that in periodic domains that contain a few bands, for given values of $R$ and size, the orientation and the wavelength of this pattern can fluctuate in time. A procedure is defined to detect well-oriented episodes and to determine the statistics of their lifetimes. The latter turn out to be distributed according to exponentially decreasing laws. This statistics is interpreted in terms of an activated process described by a Langevin equation whose deterministic part is a standard Landau model for two interacting complex amplitudes whereas the noise arises from the turbulent background. ## 1 Introduction The main features of the transition to turbulence are well understood in systems prone to a linear instability like convection where chaos emerges at the end of an instability cascade. A much wilder transition is observed in wall-bounded shear flows for which the laminar and turbulent regimes are both possible states at intermediate values of the Reynolds number $R$, the natural control parameter, whereas no linear instability mechanism is effective. A direct transition can take then place via the coexistence of laminar and turbulent domains in physical space. Two emblematic cases are the pipe flow and plane Couette flow (PCF), the simple shear flow developing between two parallel plates translating in opposite directions. Both of them are stable against infinitesimal perturbations for all values of $R$ and become turbulent only provided sufficiently strong perturbations are present. In both cases, strong hysteresis is observed and, upon decreasing $R$, the turbulent state can be maintained down to a value $R_{\rm g}$. Above $R_{\rm g}$, turbulence remains localised in space, in the form of turbulent puffs in pipe flow and turbulent patches in PCF. A striking property of PCF or counter-rotating cylindrical Couette flow (CCF) is the spatial organisation of turbulence in alternatively turbulent and laminar oblique bands that takes place in large enough systems in a specific range of Reynolds numbers [1, Ch.7]. This regime was studied in depth at Saclay by Prigent et al. [2]. It can be obtained by decreasing the Reynolds number continuously from featureless turbulence below $R_{\rm t}$, the Reynolds number above which the flow is uniformly turbulent, or triggered from laminar flow by finite amplitude perturbations above $R_{\rm g}$, the Reynolds number below which laminar flow is expected to prevail in the long time limit. A similar situation is observed in pipe flow but things are complicated by the global downstream advection so that the existence of a threshold $R_{\rm t}$ above which turbulence is uniform is still a debated matter. In contrast for PCF, the pattern is essentially time-independent and can be characterised by two wavelengths $\lambda_{x}$ and $\lambda_{z}$ in the streamwise and spanwise direction, $x$ and $z$ respectively,111In the case of CCF, the pattern is time-independent in a frame that rotates at the mean angular velocity and the axial (azimuthal) direction corresponds to the spanwise (streamwise) direction. or equivalently by a wavevector $\mathbf{k}=(k_{x},k_{z})$ with $k_{x,z}=2\pi/\lambda_{x,z}$. From symmetry considerations, two orientations are possible, corresponding to two possible combinations $(k_{x},\pm k_{z})$. Whereas a single orientation is present sufficiently far from $R_{\rm t}$ so that either mode $(k_{x},+k_{z})$ or mode $(k_{x},-k_{z})$ is selected, patches of one or the other orientation have been reported to fluctuate in space and time when $R$ approaches $R_{\rm t}$ from below [2, Figs. 2 & 3]. The main features of the bifurcation diagram could then be accounted for at a phenomenological level by an approach in terms of Ginzburg–Landau equations subjected to random noise featuring the small-scale turbulent background. This patterning was reproduced by Duguet et al. [3] using fully resolved numerical simulations in an extended system of size comparable with that of the Saclay apparatus but the computational load was so heavy that a statistical study of the upper transitional range was inconceivable. Earlier, Barkley & Tuckerman [4] also succeeded in obtaining the bands by means of fully resolved simulations with less computational burden but using narrow elongated domains aligned with the pattern’s wavevector. By construction, the fluctuating domain regime could not be obtained, whereas a re-entrant featureless turbulence regime, called ‘intermittent’ was obtained closer to $R_{\rm t}$. In our previous work on this problem, we first showed that full numerical resolution was not necessary to obtain realistic patterning but that a good account of the long range streamwise correlation of velocity fluctuations was essential [5]. This next incited us to consider reduced-resolution simulations in systems of sizes sufficient to contain at least an elementary cell $(\lambda_{x},\lambda_{z})$ of the pattern [6], thus avoiding the orientation constraint inherent in the Barkley–Tuckerman approach. Here, we expand our previous work to focus on pattern fluctuations in the upper part of the PCF’s bifurcation diagram when $R$ approaches $R_{\rm t}$ from below, taking the best possible use of the inescapable resolution lowering to perform long duration simulations, so as to obtain meaningful statistics about the dynamics of this regime. Systems considered in our numerical experiment, to be described in §2, produce patterns with a few wavelengths. In the neighbourhood of $R_{\rm t}$, fluctuations manifest themselves as orientation changes in time instead of the spatiotemporal evolution of well-ordered patches. It turns out that episodes of well-formed pattern between two orientation changes can be identified reliably, so that the lifetimes of such episodes can be measured and their average determined as a function of $R$. The Langevin approach initiated by Prigent et al. in [2] was resumed in [6] as providing an appropriate framework to interpret our numerical results. Orientation fluctuations were taken into account but their detailed statistical properties left aside, which are the subject of the present paper. In the context of pattern formation, the Langevin/Fokker–Planck approach has a long history, dating back to the 1970’s when it was applied to convecting systems [7]. Noise of thermal origin is however extremely weak so that the region of parameter space where the system is sensitive to this noise is exceedingly narrow [8] and nontrivial effects can be observed only in very specific conditions [9]. When applying the approach to the description of the bifurcation from featureless turbulence to pattern in shear flows, Prigent et al. [2] implicitly took for granted that the noise intensity was an adjustable parameter linked to the turbulent background at $R>R_{\rm t}$. Here we extend the analysis started in [6] within this conceptual framework, the subject of §3, and analyse simulation results presented in §2.4 in the light of this theory. We conclude in §4 by discussing how well this approach is suited to describe mode competition and intermittent re-entrance of featureless turbulence [4, 6] and, more generally, how the noisy temporal dynamics of coherent modes can hint at the spatio-temporal nature of transitional wall- bounded flows and explain the exponentially decreasing probability distributions of residence times or decay times often observed in this field [12]. ## 2 Conditions of the numerical experiment ### 2.1 Numerical procedure Direct numerical simulation (DNS) of the incompressible Navier–Stokes equations in the geometry of PCF are performed using Gibson’s open source code ChannelFlow [10] that assumes no-slip boundary conditions at the plates driving the flow and in-plane periodic boundary conditions. The parallel plates producing the shear are placed at a distance $2h$ from each other in the wall-normal direction $y$, they move at speeds $\pm U$ in the streamwise direction $x$, $z$ labelling the spanwise direction. The length unit is $h$, the velocity unit $U$, the time unit $h/U$, and the Reynolds number $R=Uh/\nu$, where $\nu$ is the kinematic viscosity of the fluid. The problem is completely specified when the in-plane dimensions $L_{x}$ and $L_{z}$ of the set-up are chosen. The perturbation to the laminar flow $\mathbf{U}=y\,\mathbf{\hat{x}}$ is noted $\mathbf{u}$, so that $\mathbf{u}^{2}$ is the local Euclidian distance to the base flow squared. Periodic in-plane boundary conditions allow the definition of the wave-vectors $k_{x,z}=2\pi n_{x,z}/L_{x,z}$, where the wavenumbers $n_{x,z}$ are integers. Without loss of generality, we can assume $n_{x}\geq 0$. The resolution of the simulation is fixed by the number $N_{y}$ of Chebyshev polynomials used to represent the wall-normal dependence, and the numbers $N_{x,z}$ of collocation points used to evaluate the nonlinear terms in pseudo-spectral scheme of integration of the Navier–Stokes equations. The number of Fourier modes involved in the simulation is then $2N_{x,z}/3$, owing to the 3/2-rule applied to de-aliase the velocity field. The computational load necessary to obtain meaningful results in sufficiently wide domains with fully resolved simulations is unrealistically heavy. Accordingly, we take advantage of our previous work devoted to the validation of systematic under- resolution as a modelling strategy [5]. In that work, we showed that qualitatively excellent and quantitatively acceptable results could be obtained by taking $N_{y}=15$ and $N_{x,z}=8L_{x,z}/3$. The price to be paid for the resolution lowering was apparently just a downward shift of the range $[R_{\rm g},\,R_{\rm t}]$ in which the bands are obtained, but everything else was preserved, including wavelengths. Of course, as far as resolution is concerned, the finest is the best on a strictly quantitative basis but we do not expect that the observed trends and our conclusions be sensitive to our rules to fix $N_{y}$ and $N_{x,z}$. ### 2.2 Orientation fluctuations. In this article we consider domains able to contain pattern with one or two elementary cells, i.e. $L_{x,z}=|n_{x,z}|\lambda_{x,z}$ where $n_{x}=1$ or 2 and $n_{z}=\pm 1$ or $\pm 2$. According to [2], in PCF wavelength $\lambda_{x}$ is found to be approximately equal to $110$ over the whole range $[R_{\rm g},\,R_{\rm t}]$, while wavelength $\lambda_{z}$ varies as a function of $R$ in the range $[40,\,85]$ These observations serve us to fix the size of the systems that we are going to consider below. As shown in [6], the specificity of such systems is to convert the spatio-temporal evolution of fluctuating domains observed in the neighbourhood of $R_{\rm t}$ into the temporal evolution of coherent patterns characterised by the amplitudes of the corresponding fundamental Fourier modes; possible orientation changes are associated with changes of sign of the spanwise wavenumbers. Close enough to $R_{\rm t}$, there is also some probability that featureless turbulence, the state that prevails for $R>R_{\rm t}$, be observed transiently, which is akin to the intermittent regime identified in [4]. In contrast, in the lowest part of the transitional range, close to $R_{\rm g}$, the orientation remain frozen as expected for well-formed steady oblique bands. We first illustrate this phenomenon using snapshots of $\mathbf{u}^{2}$ in Figures 1 and 2. Figure 1: Snapshots of $\mathbf{u}^{2}$ in the plane $y=-0.57$ for $R=315$ in a system of size $L_{x}\times L_{z}=128\times 84$. From left to right: one band pure state with each of the two possible orientations ($n_{x}=1$, $n_{z}=+1$) or ($n_{x}=1$, $n_{z}=-1$), two band pure state ($n_{x}=1$, $n_{z}=+2$), and mixed or defective pattern. Deep blue corresponds to laminar flow. Figure 2: Snapshots of $\mathbf{u}^{2}$ in the plane $y=-0.57$ for $R=290$ in a system of size $L_{x}\times L_{z}=170\times 48$. From left to right: ($n_{x}=1$, $n_{z}=-1$), ($n_{x}=1$, $n_{z}=+1$), and ($n_{x}=2$, $n_{z}=+1$). The left and centre panels of Fig. 1 display well-oriented patterns or ‘pure states’ showing the organised cohabitation of laminar and turbulent flow; an example of defective pattern or ‘mixed state’ without much spatial organisation is shown in the right panel. (Orientation defects between well- oriented domains require wider systems to be clearly identified as such.) Figure 2 similarly displays snapshots of $\mathbf{u}^{2}$ obtained in a narrower but longer system. Typically, during long-lasting simulations at given $L_{x,z}$ and $R$, the flow displays a pure pattern for some time, then experience a brief defective stage, and next recovers a pure state, possibly with different orientation or/and wavelength, and so on. The spatial organisation of the pattern is detected via the Fourier transform of the perturbation velocity field $\hat{\mathbf{u}}$. It turns out that most of the information about the modulation is encoded in the amplitude of the dominant wavenumber [2, 11, 6]. We consider time series of $m^{2}(n_{x},n_{z},t)=\frac{1}{2}\int_{-1}^{+1}|\hat{u}_{x}(n_{x},y,n_{z},t)|^{2}\,{\rm d}y\,,$ (1) which thus characterises a flow pattern with wavelengths $(\lambda_{x},\lambda_{z})=(L_{x}/n_{x},L_{z}/|n_{z}|)$ and orientation given by the sign of $n_{z}$. In the present study, we focus on the amplitude of the turbulence modulation in the flow and not on its phase, i.e. on the position of the pattern in the system, which was shown to be a random function of time [6]. An example of such time series is displayed in Fig. 3. Figure 3: Time series of $m^{2}(t)$ for several wave numbers $n_{z}=\pm 1,\pm 2$ for $R=315$ in a system of size $L_{x}\times L_{z}=128\times 84$. A pure pattern stage corresponds to a single $m(n_{x},n_{z})$ fluctuating around a non zero value, the other $m(n_{x}^{\prime},n_{z}^{\prime})$ remaining negligible. For instance the pattern keeps wave-number $n_{z}=+2$ from $t=3\,10^{3}$ to $t=10^{4}$. The defective stage corresponds to $m(n_{x},n_{z})$ decaying to zero while another one $m(n_{x}^{\prime},n_{z}^{\prime})$ grows. Wavenumbers $(n_{x}^{\prime},n_{z}^{\prime})$ may be different from $(n_{x},n_{z})$, in which case there is an effective change of the orientation if $|n_{z}^{\prime}|=|n_{z}|$ or a change of wavelength (sometimes combined with orientation changes) if $|n_{z}^{\prime}|\neq|n_{z}|$. In Fig. 3, a change of orientation takes place at time $t=4\,10^{4}$ ($n_{z}=-1\to+1$), a change of wavelength at time $t=1.7\,10^{4}$ ($n_{z}=-2\to+1$), the pattern with $n_{z}=+1$ growing back from a defective stage at time $t=4.3\,10^{4}$. Most of the time there is no ambiguity about the value of $n$ involved so that we shall use simplified notations, i.e. just $m$ or $m_{\pm}$ instead of $m(n_{x},\pm n_{z})$, as often as possible. Except very close to $R_{\rm t}$, pure state intermissions last long and defective episodes are short, so that series of lifetime $T_{i}$ of well- oriented lapses can be defined from recording simulations of duration sufficient to make reliable statistics. ### 2.3 Lifetime computations Orientation and wavelength fluctuations are best characterised by lifetimes distributions. Beforehand, we have to define a systematic method to detect the beginning and the end of pure pattern episodes from the $m^{2}$ time series. This is done by using two thresholds: one, $s_{1}$, for the start of a pure pattern episode and the other, $s_{2}$, for its termination, see Fig. 4 (top). Figure 4: Time series of $m^{2}$ for $n_{z}=\pm 1$ at $R=330$ (top) and $R=345$ (bottom). The two horizontal lines in the top panel locate threshold $s_{1}$ (full line) and $s_{2}$ (dashed line). Close to $R_{\rm t}$, bottom panel, orientation fluctuations are short-lived and much smaller, rendering the detection of well-oriented episodes more difficult. The fast growth of $m^{2}$ makes it easy to choose $s_{1}$ and the results are not much sensitive to its exact value. In contrast, detecting the decay is more problematic. This will be discussed in detail after the presentation of a typical result obtained by assuming that the difficulty has been properly resolved. For practical reasons, we use a byproduct of the cumulated probability density function (PDF) $Q$: $Q(T)=\frac{\\#\\{T_{i}\geq T\\}}{\\#\\{T_{i}\\}}=1-\frac{\\#\\{T_{i}\leq T\\}}{\\#\\{T_{i}\\}}\,.$ Empirical distributions obtained in an experiment with $L_{x}=110$ and $L_{z}=32$ for $R=330$ are displayed in Fig. 5. They Figure 5: Logarithm of $Q$ (right) for $L_{x}\times L_{z}=128\times 84$, $R=315$, computed with $s_{1}=0.001$, $s_{2}=0.005$, for both wave numbers $|n_{z}|=1$ and $|n_{z}|=2$. We have about 40 events for $|n_{z}|=1$ and about 20 events for $|n_{z}|=2$. are obtained from the time series, a small part of which is shown in Fig. 4, distinguishing $n_{z}=\pm 1$ from $n_{z}=\pm 2$. Since, for symmetry reasons, the two orientations are supposed to have identical distributions, we sum over the $\pm$ in each case. The semi-logarithmic coordinates used to represent $Q(T)$ suggest exponentially decreasing variations, which makes orientation changes look like deriving from a Poisson process. Assuming that they are indeed in the form $\exp(-T/\langle T\rangle)$, we can obtain the mean lifetime $\langle T\rangle$ from the plain arithmetic average of the liftetime series. $\langle T\rangle$ can also be obtained by fitting the empirical cumulated distribution against an exponential law or its logarithm against a linear law. In addition to raw data, Fig. 5 displays the second kind of fits for $|n_{z}|=1$ and $2$. These three different estimates are close to each other provided that the lifetime series comprise sufficiently large numbers of events. An average of these three values will be used to define the mean lifetime and the corresponding unbiased standard deviation will give an estimate of the “error” for each lifetime series. Let us now come to the problem of the sensitivity of $Q(T)$ to the value of the thresholds $s_{1}$ and $s_{2}$ used to determine the lifetimes of the pure pattern episodes. In Figure 6 (left) the mean lifetime $\langle T\rangle$ displays a clear plateau as a functions of $s_{1}$. The width of this plateau does not depend on $s_{2}$ though its value depends on it. Figure 6: Mean lifetimes functions of $s_{1}$ given $s_{2}$ (left) and of $s_{2}$ given $s_{1}$ (right). $R=315$ and system size $L_{x}\times L_{z}=128\times 84$, $|n_{z}|=1$. The existence of this plateau is easily seen to be related to the fast growth of $m$ when the pattern sets in: $m$ always goes through most of the values corresponding to the plateau in a very short time. In practice, for $10^{-3}\leq s_{1}\leq 1.5\,10^{-3}$ the very same episodes are detected whatever the precise value of $s_{1}$. That the plateau value still depends on $s_{2}$ just expresses that the duration of the detected episodes are modified in the same way due to changes in the detection of their termination. Of course, when $s_{1}$ is taken too large, some less-well ordered episodes escape detection or are detected too late, which artificially decreases the mean. On the other hand, if $s_{1}$ is taken too small, the “signal” gets lost in the “noise”: a large number of brief noisy excursions are detected as relevant ordered episodes, again decreasing the mean. The variation of the mean lifetime with $s_{2}$ is completely different as seen in Fig. 6 (right). Here, $\langle T\rangle$ varies roughly linearly with $s_{2}$ in a wide interval above the noise level ($\sim 3\,10^{-4}$, see Fig. 4): $\langle T\rangle(s_{2})\simeq a(1-bs_{2})\,.$ Coefficient $b=1000\pm 100$ does not vary significantly over the cases that we have considered. This dependence fully explains the change of plateau value in plots of $\langle T\rangle$ as a function of $s_{1}$. Coefficient $a$, corresponding to $\langle T\rangle$ extrapolated toward $s_{2}=0$ however still depend on $R$ and the geometry. Henceforth, we define this extrapolated value as the relevant average lifetime $\langle T\rangle$, which will be supported by the theoretical considerations to be developed in the next section. The observed dependence of $\langle T\rangle$ on $s_{2}$ can be explained by the fact that the decay of a pure pattern is much more gradual than its growth, which causes significant differences when the duration of an episode is measured, leading to a decrease of $\langle T\rangle$ as $s_{2}$ increases since the termination of the episode is detected earlier. A second reason why the mean lifetime increases as $s_{2}$ decreases arises from the fact that some excursions are not counted as decay events. In physical space, this corresponds to an irregular and slow disorganisation of turbulence, contrasting with the fast installation of the pattern. In fact $\langle T\rangle$ cannot be obtained otherwise than by extrapolation of threshold $s_{2}$ to zero, as will be discussed in §3.3. ### 2.4 DNS results The two systems sizes, $L_{x}\times L_{z}=128\times 64$ and $110\times 32$, already considered in our previous work [5, 6] are studied here over the whole range of Reynolds numbers where the pattern exists at the chosen numerical resolution, $R\in[R_{\rm g},\,R_{\rm t}]=[275,345]$. Orientation fluctuations are systematically found close enough to $R_{\rm t}$, see Cases 1 & 2 below. In addition, wavelength fluctuations can take place when the size of the system is too far away from resonating with the pattern’s elementary cell $\lambda_{x}^{\rm opt}\times\lambda_{z}^{\rm opt}$, where ‘opt’ means ‘optimal’, in a sense to be defined below in §3.1. Orientation and wavelength fluctuations are observed at $R=315$ for $L_{x}=128$, $L_{z}=84$ and $90$, and for $L_{x}=110$, $L_{z}=84$, meaning that both $|n_{z}|=1$ and $|n_{z}|=2$ are competitive for $L_{z}=84$ or $L_{z}=90$. In contrast, lifetimes of single mode patterns are extremely long for $L_{z}<84$ and $L_{z}>90$, meaning that $L_{z}<84$ is optimal for $|n_{z}|=1$ and $L_{z}>90$ is optimal pour $|n_{z}|=2$. Orientation and wavelength fluctuations are similarly present in several other circumstances, at lower Reynolds number $R=272$ and $R=275$ for $L_{x}\times L_{z}=110\times 32$, as well as at $R=290$ for $L_{z}=48$ and $L_{x}=80$ or at $R=330$ for $L_{x}=90$, $140$, and $150$. #### Case 1: $L_{x}=128$, $L_{z}=84$, $R=315$, wavelength fluctuations. Several experiments under the same protocol have been performed, using different initial conditions. Integration times ranged from $5\,10^{4}$ to $10^{5}$ $h/U$. A large enough ensemble of lifetimes has been sampled, both for $|n_{z}|=1$ and $|n_{z}|=2$, allowing us to compute the corresponding order parameters $M$ – the conditional time averages of $m(t)$ as defined in (1) – with sufficient accuracy. Snapshots corresponding to this aspect ratio are displayed in Fig. 1, a typical part of the corresponding time series is shown in Fig. 3. For $|n_{z}|=1$ and $|n_{z}|=2$, we obtain $M_{1}=0.033\pm 0.001$ and $M_{2}=0.038\pm 0.001$, respectively. From the lifetime distributions in Fig 5, we get $\tau_{1}=8100\pm 200$ and $\tau_{2}=3800\pm 100$. The fact that $M_{1}<M_{2}$ is not surprising and is understood in term of optimal wavelength (§3.1, $\lambda^{\rm opt}_{z}\simeq 39$ at $R=315$ [6]). The reason why one has $\tau_{1}>\tau_{2}$ is however not clear. #### Case 2: $L_{x}=110$, $L_{z}=32$, variable $R$, orientation fluctuations. A thorough account of the behaviour of $M$ and the re-entrance featureless turbulence has been given in [6]. Here, lifetimes are computed for Reynolds number ranging from $R=325$ to $R=340$. Below $R=325$, the lifetimes are so long that a small number of events is observed despite the length of time series used ($>2\,10^{5}$), which forbids the determination of $\tau$ as a meaningful average (Fig. 10). Above $R=340$, a clear separation of time scales is lacking, which now forbids the definition of lifetimes of individual events, compare the two panels in Fig. 4. Figure 7 displays the variation of the average lifetime $\tau$ with $R$, showing that it increases by a factor of 10 as $R$ decreases from $R=340$, which is somewhat below $R_{\rm t}=355$, down to $R=325$ below which it is too long to be measured reliably. “Error bars” suggested by up and down triangles in Fig. 7 correspond to the unbiased standard deviation of the three estimates for $\tau$ mentioned earlier. Figure 7: Mean lifetime $\tau$ as a function of $R$ for $L_{x}\times L_{z}=110\times 32$ (log scale). ## 3 Conceptual framework and application to DNS results ### 3.1 The Landau–Langevin model Prigent et al. proposed to consider the turbulent bands as resulting from a conventional pattern formation problem described at lowest order, from symmetry arguments, by two coupled cubic Ginzburg–Landau equations, one for each band orientation, further subjected to noise featuring the turbulent background above $R_{\rm t}$. The slowly varying part of the velocity field component away from the laminar profile can be written as $u_{x}=A_{+}(\tilde{x},\tilde{z},\tilde{t})e^{ik_{x}^{\rm c}x+ik_{z}^{\rm c}z}+A_{-}(\tilde{x},\tilde{z},\tilde{t})e^{ik_{x}^{\rm c}x-k_{z}^{\rm c}z}+cc\,,$ where $A_{\pm}\in\mathbb{C}$ are the amplitude fields accounting for the two modulation waves, and $\tilde{x}$, $\tilde{z}$ and $\tilde{t}$ are slow variables [1]. Then, following this approach, we assume $\tau_{0}\partial_{\tilde{t}}A_{\pm}=(\epsilon+\xi_{x}^{2}\partial_{\tilde{x}\tilde{x}}^{2}+\xi_{z}^{2}\partial_{\tilde{z}\tilde{z}}^{2})A_{\pm}-g_{1}|A_{\pm}|^{2}A_{\pm}-g_{2}|A_{\mp}|^{2}A_{\pm}+\alpha\zeta_{\pm}\,,$ (2) the quantity $\epsilon=(R_{\rm t}-R)/R_{\rm t}$ measures the relative distance to the threshold222The existence of a well defined threshold in this system is attested by the behaviour of the turbulent fraction and spatially averaged kinetic energy which display a marked change of slope at $R_{\rm t}$[6] $R_{\rm t}$, $\tau_{0}$ is the ‘natural’ time scale for pattern formation, $\xi_{x,z}$ are streamwise and spanwise correlation lengths, $g_{1}$ and $g_{2}$ are the self-coupling and cross-coupling nonlinear coefficients, and $\alpha$ the strength of the noise $\zeta_{\pm}$ supposed to be a centred Gaussian process with unit variance. The strength $\alpha$ of the noise is expected to grow smoothly with $R$, regardless of the existence of the pattern since the local intensity of the turbulence is empirically not directly correlated to the amplitude and phase of the modulation $A_{\pm}$. The tilde variables describe the long-wave modulations to an ideal pattern with critical wavelengths $\lambda_{x,z}^{\rm c}$ to which correspond critical wavevectors $k_{x,z}^{\rm c}=2\pi/\lambda_{x,z}^{\rm c}$, the term critical referring to the most unstable wave vector near $R=R_{\rm t}$. The systems that we consider have periodic boundary conditions placed at distances $L_{x,z}$. Fourier analysis then leads to characterise the pattern by wavevectors ${\bf k}=(k_{x},k_{z})$, with $k_{x,z}=2\pi n_{x,z}/L_{x,z}$. It is assumed that the wave numbers obtained during a given experiment will be the integers that will be as close as possible of $n_{x,z}^{\rm c}=L_{x,z}/\lambda_{x,z}^{\rm c}$. Furthermore, our systems can accommodate a small number of cells of size $(\lambda_{x},\lambda_{z})$ so their modes are well isolated [1, Ch.4]. Assuming that a single pair $(n_{x},\pm n_{z})$ is involved, the partial differential equation (2) is turned into an ordinary differential equation for $A(n_{x},\pm n_{z})$ simply denoted $A_{\pm}\equiv A_{\pm}^{\rm r}+iA_{\pm}^{\rm i}$, close enough to $R_{\rm t}$[6]: $\tau_{0}\mbox{$\frac{\rm d}{{\rm d}t}$}A_{\pm}=\tilde{\epsilon}A_{\pm}-g_{1}|A_{\pm}|^{2}A_{\pm}-g_{2}|A_{\mp}|^{2}A_{\pm}+\alpha\zeta_{\pm}\,,$ (3) where $\tilde{\epsilon}=\epsilon-\xi_{x}^{2}\delta k_{x}^{2}-\xi_{z}^{2}\delta k_{z}^{2}$ controlling the linear stability of these modes, is evaluated for $\delta k_{x,z}=k_{x,z}-k_{x,z}^{\rm c}$ with the relevant $k_{x,z}=2\pi n_{x,z}/L_{x,z}$, as well as the nonlinear coefficients $g_{1,2}$ ($\in\mathbb{R}$ because the pattern does not drift, at least in the absence of noise). Coefficient $\alpha$ is the effective strength of the noise affecting the mode that we consider. Equation (3) can be written as deriving from a potential: $\tau_{0}\mbox{$\frac{\rm d}{{\rm d}t}$}A_{\pm}^{\rm r,i}=-\frac{\partial\mathcal{V}}{\partial A_{\pm}^{\rm r,i}}+\alpha\zeta_{\pm}\,,$ with $\mathcal{V}=-\mbox{$\frac{1}{2}$}\tilde{\epsilon}\left(|A_{+}|^{2}+|A_{-}|^{2}\right)+\mbox{$\frac{1}{4}$}g_{1}\left(|A_{+}|^{4}+|A_{-}|^{4}\right)+\mbox{$\frac{1}{2}$}g_{2}|A_{+}|^{2}|A_{-}|^{2}\,.$ (4) Usually, when making use of phenomenological equations such as (2), one relies on values of critical wavevectors $k^{\rm c}$ that are computed once for all from some linear stability theory and further introduced in the perturbation expansions solving the nonlinear wavelength selection problem beyond the threshold [13]. Here the theory is not developed enough to have such a definition and such an evaluation of nonlinearly selected ‘optimal’ wavevectors far enough from threshold. Accordingly, in (3) we introduce values of $\tilde{\epsilon}$ that do not make reference to some explicit computation involving measured values of $\epsilon$ and $\xi_{x,z}$ but values that are just estimates consistent with the empirically determined optimal wavelengths. In the same way, we keep the cubic Landau expressions (3), neglecting higher order terms that would introduce too many little-constrained parameters, without deeper insight into the problem. The stable fixed points of the deterministic part of (3) were shown to correspond to the permanent state of the pattern and the additive noise term seen to account for fluctuations quite well by solving the corresponding Fokker–Planck equation [6]. The stationary probability distribution for the moduli $|A_{\pm}|=A_{\pm}^{\rm m}$ was obtained in the form: $\Pi(A_{+}^{\rm m},A_{-}^{\rm m})=Z^{-1}A_{+}^{\rm m}A_{-}^{\rm m}\exp(-2\mathcal{V}/\alpha^{2})\,,\qquad Z=\int A_{+}^{\rm m}A_{-}^{\rm m}\exp(-2\mathcal{V}/\alpha^{2})\,{\rm d}A_{+}^{\rm m}{\rm d}A_{-}^{\rm m}\,.$ (5) The time behaviour of $A_{\pm}^{\rm m}$ is easily discussed by considering the shape of $\mathcal{V}$ within the stochastic process framework. Two limiting cases can be identified, depending on whether $\tilde{\epsilon}$ is $\mathcal{O}(1)$ or $\ll 1$. In the first case, excursions from the neighbourhood of the minima of $\mathcal{V}$ are rare; the lifetime of an ordered episode can be defined as the average time necessary for the system to go from the neighbourhood of a minimum to the potential’s saddle. It is expected to increase with the height of the potential barrier, i.e. as parameter $\tilde{\epsilon}$ grows, and to fall off as $\alpha$ increases. The lack of symmetry between the growth and the decay of a pattern has then a clear explanation when $\tilde{\epsilon}$ is large: The growth corresponds to the system falling from the neighbourhood of the saddle into one of the wells; even in the presence of noise, this evolution is fast and mostly deterministic. In contrast, the decay corresponds to the system slowly climbing toward the saddle against the deterministic flow, driven by the sole effect of noise. In the opposite limit, when $\tilde{\epsilon}$ approaches zero, the definition of a lifetime no longer makes sense since the characteristic times for growth and decay become of the same order of magnitude. ### 3.2 Orientation lifetimes from the model Orientation changes and associated lifetimes are analysed in terms of first passage time and escape from metastable states [14]. The distribution of lifetimes is anticipated to be Poissonian as expected from a jump process controlled by an activation “energy”. In a simplified one-dimensional version of potential $\cal V$ [14, Ch.11,§2,6–7], if the well is deep enough, within a parabolic approximation the mean escape time, the average time necessary to go from a well to another, is given by $\tau/\tau_{0}=\frac{2\pi}{\sqrt{\mathcal{V}_{\rm w}^{\prime\prime}\,|\mathcal{V}_{\rm s}^{\prime\prime}|}}\exp\left(2\frac{\mathcal{V}_{\rm s}-\mathcal{V}_{\rm w}}{\alpha^{2}}\right)\,,$ (6) where ‘w’ stands for ‘well’ and ‘s’ for ‘saddle’; $\mathcal{V}_{\rm w,s}$ are the values of the potentials at the corresponding points and $\mathcal{V}_{\rm w,s}^{\prime\prime}$ the values of the second order derivatives of the potential with respect to the variable at these points. The derivation of this formula shows that $\tau$ is dominated by the time spent around the saddle. In our two dimensional system with potential (4), at lowest order in $\alpha$ the coordinates of the well and saddle points are: $(A_{\pm}^{\rm w},A_{\mp}^{\rm w})=\left(\sqrt{\tilde{\epsilon}/g_{1}},\,0\right)\qquad\mbox{and}\qquad(A_{+}^{\rm s},A_{-}^{\rm s})=\left(\sqrt{\tilde{\epsilon}/(g_{1}+g_{2})},\,\sqrt{\epsilon/(g_{1}+g_{2})}\right)$ and the corresponding values of the potential: $\mathcal{V}_{\rm w}=-\tilde{\epsilon}^{2}/4g_{1}\qquad\mbox{and}\qquad\mathcal{V}_{\rm s}=-\tilde{\epsilon}^{2}/2(g_{1}+g_{2})\,.$ The second derivatives have to be replaced by the eigenvalues of the Hessian matrix of $\cal V$ computed at these points: $H_{\rm w}=\left(\begin{matrix}2\tilde{\epsilon}&0\\\ 0&\tilde{\epsilon}(g_{2}/g_{1}-1)\end{matrix}\right)\qquad\mbox{and}\qquad H_{\rm s}=\frac{2\tilde{\epsilon}}{g_{2}+g_{1}}\left(\begin{matrix}g_{1}&g_{2}\\\ g_{2}&g_{1}\end{matrix}\right)\,.$ At point ‘w’, $H_{\rm w}$ is diagonal and the eigen-direction pointing to point ‘s’ has eigen-value $\tilde{\epsilon}(g_{2}/g_{1}-1)$. At point ‘s’, $H_{\rm s}$ is diagonal in the basis $\\{(1,1),(1,-1)\\}$ and has eigenvalues $\\{2\tilde{\epsilon},\,2\tilde{\epsilon}(g_{1}-g_{2})/(g_{1}+g_{2})\\}$. The unstable eigen-direction correspond to the second one which is negative ($g_{2}>g_{1}$). Inserting these values in (6), we obtain: $\tau/\tau_{0}=\frac{2\pi\sqrt{2}}{\tilde{\epsilon}}\frac{\sqrt{g_{1}(g_{1}+g_{2})}}{g_{2}-g_{1}}\exp\left(\frac{\tilde{\epsilon}^{2}}{2\alpha^{2}g_{1}}\,\frac{g_{2}-g_{1}}{g_{1}+g_{2}}\right)\,.$ (7) In its exponential factor, this formula points out an “energy” scale $\tilde{\epsilon}^{2}/g_{1}$ to be compared to the characteristic noise energy $\alpha^{2}$ which play the role of the Boltzmann energy in thermal problems. It also shows that, especially when $g_{2}$ is larger but comparable to $g_{1}$, the noise energy has to remain small enough because the parabolic approximation which underlies the formula assumes sufficiently deep wells. The main difference between the one and two dimensions cases are in the shape of the “energy landscape”, corrections are therefore expected to be multiplicative and not depend on the value of $\tilde{\epsilon}$. ### 3.3 Simulation of the model For the deterministic part of equation (3) we use a simple first-order implicit Euler algorithm, while the additive noise $\alpha\zeta(t)$ is treated as a Gaussian random variable with standard deviation $\alpha\sqrt{dt}$ at each time step. The model is integrated over a range of $\tilde{\epsilon}$, given $g_{1}$, $g_{2}$ (several values), and $\alpha$ (assumed constant). The time series of $|A_{\pm}|^{2}$ displayed in Fig. 8 are indeed reminiscent of those obtained by numerical integration of Navier–Stokes equations (Figs. 3 and 4): pure states at the bottom of the wells correspond to $|A_{-}|^{2}$ fluctuating around 0 and $|A_{+}|^{2}$ away from 0, or the reverse. Figure 8: Typical time series from model (3), for a set of parameters corresponding to the Navier–Stokes DNS, $\tilde{\epsilon}=0.05$, $g_{1}=60$, $g_{2}=120$, $\alpha=0.002$ [6]. Large excursions can lead to a change of the dominant orientation. These excursions are more likely to occur when $\tilde{\epsilon}$ is decreased. Lifetimes have been computed in the same way as for Fig. 6. The dependence of the mean lifetimes on thresholds $s_{1,2}$ is displayed in Figure 9. Figure 9: Mean lifetime $\tau$, extracted from the model as a function of $s_{1}$, for $s_{2}=0.0163$, $0.0193$, and $0.0225$ (left) and as a function of $s_{2}$ for several $s_{1}$ ranging from $0.0784$ to $0.1296$ (right). $\tilde{\epsilon}=0.075$, $g_{1}=1$, $g_{2}=2$, $\alpha=0.002$. A neat plateau is obtained for $s_{1}\in[0.075,0.115]$ for different values of $s_{2}$ (Fig. 9, left), which corresponds to the trajectory getting away from the saddle. Extremes values of $s_{1}$ lead to bad estimates of $\tau$ for the same reasons as stated before. As to threshold $s_{2}$, an orientation change has taken place when the trajectory goes beyond the saddle, while a pure state corresponds to one amplitude large and the other at the noise level. We have thus to detect the change from large to small for one or the other amplitude. It is extremely difficult to detect the precise passage at the saddle, since it is dominated by the time spent in that region, contribution from the two sides of the saddle point having the same weight. On the contrary the passage from one state to another leaves no doubt as to its definition. Therefore, we prefer to compute the mean first passage time from one well to another, which is obtained in our simulation by the extrapolation at $s_{2}=0$. Approximating the curves in Fig. 9 (right) by linear functions $\tau=a(1-bs_{2})$, one finds that the slope $b$ depends on $\tilde{\epsilon}$ and $s_{1}$ only weakly; the value of $\tau$ retained is then the one given by the extrapolation $s_{2}=0$, i.e. coefficient $a$. Improving the definition of $\tau$ with approximations better than the linear one has not been found necessary. The general expression of the mean first passage time gives no hint as to the quantitative behaviour on the distance to the second well $s_{2}$, although it shows the same qualitative behaviour as seen in Figures 6 (right) and 9 (right). In Figure 10 (semilog coordinates), the lifetime $\tau$ measured in this way is compared to the asymptotic expression from the theory (7) as a function of $\tilde{\epsilon}$. Figure 10: $\log(\tau)$ as a function of $\tilde{\epsilon}$ for $g_{1}=60$, $g_{2}=120$, $\tilde{\alpha}=0.002$; the model was integrated over $10^{5}$ time units. It can be seen that the asymptotic formula is not valid for the smallest values of $\tilde{\epsilon}$, when the wells are not longer deep enough for the approximation to be valid, similarly to what is found in the DNS close to $R_{\rm t}$. The values given by this formula for small values of $\tilde{\epsilon}$, especially around and below the minimum it predicts at $\tilde{\epsilon}=\alpha\sqrt{g_{1}(g_{2}+g_{1})/(g_{2}-g_{1})}$, cannot be trusted. For large $\tilde{\epsilon}$, the lifetime computed from the simulation saturates because it becomes of the order of the total integration time so that only a few events smaller than this total time can be recorded. Accordingly the long time tail of the distribution is badly sampled with an under-representation of lifetimes larger than the average expected from the theory. In Fig. 10, the numerical and the asymptotic estimates of the mean lifetimes are seen to differ by a constant of order unity, which is attributed to the one-dimensional character of the approximation. ### 3.4 Generalisation This approach can be extended to wavelength fluctuations. When the size $(L_{x},L_{z})$ of the system is such that it ‘hesitates’ between two pairs of modes $(n_{x},\pm n_{z})$ and $(n^{\prime}_{x},\pm n^{\prime}_{z})$, we introduce two supplementary amplitudes $A(n^{\prime}_{x},\pm n^{\prime}_{z})$ that we denote $B_{\pm}$ for short and, extending notations straightforwardly with primes for quantities related to $B_{\pm}$, we arrive at: $\displaystyle\tau_{0}\mbox{$\frac{\rm d}{{\rm d}t}$}A_{\pm}=\tilde{\epsilon}A_{\pm}-g_{1}|A_{\pm}|^{2}A_{\pm}-g_{2}|A_{\pm}|^{2}A_{\pm}+g_{3}(|B_{\pm}|^{2}+|B_{\mp}|^{2})A_{\pm}+\alpha\zeta_{\pm}\,,$ (8) $\displaystyle\tau_{0}\mbox{$\frac{\rm d}{{\rm d}t}$}B_{\pm}=\tilde{\epsilon}^{\prime}B_{\pm}-g^{\prime}_{1}|B_{\pm}|^{2}B_{\pm}-g^{\prime}_{2}|B_{\pm}|^{2}B_{\pm}+g^{\prime}_{3}(|A_{\pm}|^{2}+|A_{\mp}|^{2})B_{\pm}+\alpha^{\prime}\zeta^{\prime}_{\pm}\,,$ (9) where $\tilde{\epsilon}$ and $\tilde{\epsilon}^{\prime}$ as well as the nonlinear coupling constants $g_{1,2,3}$, $g^{\prime}_{1,2,3}$ and even the effective noise intensities $\alpha$, $\alpha^{\prime}$ may differ since they relate to pure patterns with different $\delta k_{x,z}=k_{x,z}-k_{x,z}^{\rm c}$. A first guess would be to assume the primed and non-primed variables equal, which would bring us immediately back to the previous approach with an effective potential, wells, saddles, and potential barriers, leading to estimates for the different lifetimes involved. It is not clear how the case of turbulence re-entrance (the intermittent regime of [4]) would fit this framework but it is well described by a PDF with three peaks [6] corresponding to a probability potential with three wells and thus hopefully amenable to a similar treatment with a similar output. These generalisations have not been worked out in detail numerically since they introduces a discouragingly large number of parameters to be fitted against the experiments and from which we would learn little, owing to their phenomenological basis. Only the case involving a single pair of modes was examined in §3.3 above, mostly in order to validate the procedure followed to determine lifetimes in §2.3. ## 4 Summary and conclusion In this paper, numerical simulations of the Navier–Stokes equation in plane Couette flow configuration have been performed in a range of Reynolds numbers where the transition to turbulence happens in the form of oblique bands. Systems with sizes fitting a few elementary cells $\lambda_{x}\times\lambda_{z}$ of the pattern have been considered. These sizes are much larger than the minimal flow unit which allows the reduction of the transition problem to a temporal process familiar to chaos theory [12]. Accordingly, the considered systems are able to display the first manifestations of a genuinely spatiotemporal dynamics via patterning. Following the patterns in time, we showed that they experience orientation and wavelength fluctuations in the upper part of the range of transitional Reynolds numbers $[R_{\rm g},R_{\rm t}]$. A systematic procedure to detect the start and the termination of well-oriented episodes was defined, leading to the observation of exponentially decreasing distributions for their lifetimes (Fig. 5). A consistent interpretation scheme was then provided by adapting the noisy Ginzburg–Landau model proposed in [2] to our case, transforming the original stochastic PDE into a Landau–Langevin stochastic ODE. Besides supporting the procedure used to determine lifetimes, the approach directly leads to the determination of probability distributions for the patterned states from the shape of the potential obtained by solving the corresponding Fokker–Planck equation, as already suggested in [2, Fig. 19]. The variation of the patterns’ mean lifetimes is thus linked to the relative distance to threshold and noise intensity through an asymptotic formula involving the “energy” barrier between wells corresponding to the different well-oriented states in competition. Ingredients in the relative distance to threshold $\tilde{\epsilon}$ which is a function of both the Reynolds number and the optimal wavelength, are amply sufficient to explain most of the dependance of the mean lifetimes as functions of $R$, $L_{x,z}$, and the spontaneous appearance of defects separating patches of well-oriented patterns close enough to $R_{\rm t}$ in larger aspect-ratio systems as illustrated in Fig. 11 and seen in the experiments [2]. Figure 11: Orientation defect spontaneously appearing in the flow for $R=340$ in a domain of size $L_{x}\times L_{z}=660\times 48$. In order to explain the occurrence of exponentially decreasing lifetime distributions, the theory of dynamical systems appeals to the sensitivity to initial conditions of trajectories visiting a homoclinic tangle [12]. Here, the modelling that fits well our observations implies that exponential distributions arise from some jump random process [14]. As soon as the size of the system is much larger that the minimal flow unit (for which the temporal behaviour inherent in low dimensional dynamical systems is relevant), a spatiotemporal perspective becomes in order, and the jumps in question can easily be associated to the local chaotic dynamics of pieces of streaks and streamwise vortices involved in the self sustaining process of turbulence [15]. This local chaotic dynamics would then be responsible for the wandering of the global system through some “energy” landscape with wells and saddles. With system sizes of the order of the elementary pattern cell $\lambda_{x}\times\lambda_{z}$, this wandering amounts to orientation and/or wavelength changes. Extending these views to larger systems would then explain the statistical properties of fluctuating laminar-turbulent patches observed in the upper transitional range close enough to $R_{\rm t}$ [2]. Whereas the origin of the noise introduced in the description is understandable from chaos at the local (microscopic) scale, it remains however to understand why the coexistence of laminar and turbulent flow takes the form of oblique bands at the global (macroscopic) scale, i.e. to justify the Ginzburg–Landau approach from the first principles rather than taking it as an educated phenomenological guess. ## References * [1] For an introductory review see, e.g. P. Manneville, Instabilities, Chaos and Turbulence, 2nd edition (Imperial College Press, 2010). * [2] A. Prigent, G. Grégoire, H. Chaté, O. Dauchot, Long-wavelength modulation of turbulent shear flow, Physica D 174 100–113 (2003). * [3] Y. Duguet, P. Schlatter, D.S. Henningson Formation of turbulent patterns near the onset of transition in plane Couette flow, J. Fluid Mech. 650, 119–129 (2010). * [4] D. Barkley, L. Tuckerman, Computational study of turbulent laminar patterns in Couette flow, Phys. Rev. Lett. 94 014502 (2005). * [5] P. Manneville, J. Rolland, On modelling transitional turbulent flows using under-resolved direct numerical simulations, Theor. Comput. Fluid Dyn. in press, DOI : 10.1007/s00162-010-0215-5. * [6] J. Rolland, P. Manneville, Ginzburg–Landau description of laminar-turbulent oblique bands in transitional plane Couette flow, Eur. Phys. J. B, submitted. * [7] R. Graham, Hydrodynamics fluctuations near the convection instability, Phys. Rev. A, 10, 1762-1784, (1974). * [8] P. Hohenberg, J. Swift, Effects of additive noise at the onset of Rayleigh-Bénard convection, Phys. Rev A, 46, 4773–4785 (1992). * [9] M. Scherer, G. Ahler, F. Hörner, I. Rehberg, Deviation from linear theory for fluctuations below the super-critical primary bifurcation to electroconvection, Phys. Rev. Lett. 85 3754–3760 (2000). * [10] J. Gibson, http://www.channelflow.org/. * [11] D. Barkley, O. Dauchot, L. Tuckerman, Statistical analysis of the transition to turbulent-laminar banded patterns in plane Couette flow, Journal of Physics, Conference Series 137 (2008) 012029. * [12] B. Eckhardt, H. Faisst, A. Schmiegel, T.M. Schneider, Dynamical systems and the transition to turbulence in linearly stable shear flows, Phil. Trans. R. Soc. A 366 1297–1315 (2008). * [13] M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys 65 (1993) 851–1112 * [14] N.G Van Kampen, Stochastic processes in physics and chemistry, North-Holland (1990). * [15] F. Waleffe, On a self-sustaining process in shear flows, Phys. Fluids 9 883–900 (1997).
arxiv-papers
2011-01-13T09:36:24
2024-09-04T02:49:16.406874
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Joran Rolland, Paul Manneville", "submitter": "Joran Rolland", "url": "https://arxiv.org/abs/1101.2513" }
1101.2616
CECS-PHY-10/15 Accelerating black hole in 2+1 dimensions and 3+1 black (st)ring Marco Astorino***marco.astorino@gmail.com Instituto de Física, Pontificia Universidad Católica de Valparaíso and Centro de Estudios Científicos (CECS), Valdivia, Chile Abstract A C-metric type solution for general relativity with cosmological constant is presented in 2+1 dimensions. It is interpreted as a three-dimensional black hole accelerated by a strut. Positive values of the cosmological constant are admissible too. Some embeddings of this metric in the 3+1 space-time are considered: accelerating BTZ black string and a black ring where the gravitational force is sustained by the acceleration. ## 1 Introduction Beside the fact that general relativity in three space-time dimensions is trivial, because the lack of dynamical degrees of freedom, still it admits black holes solutions. In fact the static and rotating black hole in 2+1 dimension is very well known as remarkably discovered by Bañados, Teitelboim and Zanelli (BTZ) in [1]. Also well known is the charged and the electro- rotating flavours [2]. Taking inspiration from the four dimensional C-metric (for references see [3], [4] and [5]), we are here interested to apply acceleration to the three dimensional black hole (section 2). This is done with the motivation that such acceleration, and the physical object that produce it, could play an important role in 3+1 general relativity solutions with non-trivial topology, as shown in section 3. Our supposition is based on analogy with the five dimensional case, as explained in [6]: to forge a black ring one has to bend a string built with the cross product of a line and a Schwarzschild black hole, and then balance the gravitational self attraction which makes the string self collide. If one would do so in four dimensions, the natural candidate to start with is the cross product of a line and the three dimensional black hole, thus the cosmological constant is necessary. Cosmological constant is also prominent in introducing a length scale to obtain different scales for the radius and the thickness of the ring. In [6] the absence of black (st)rings in vacuum 4D general relativity is attributed to the lack of such a scale. ## 2 Accelerating black hole in 2+1 dimensions We begin considering the Einstein-Hilbert action for standard general relativity, with generic cosmological constant $\Lambda$, in three dimensions111We have set, for convenience, the gravitational constant $G=1/8$: $I[g_{\mu\nu}]=\frac{1}{2\pi}\int d^{3}x\sqrt{-g}(R-2\Lambda)$ (2.1) Extremization of the action with respect the metric $g_{\mu\nu}$ yields the Einstein field equations: $R_{\mu\nu}-\frac{1}{2}R\ g_{\mu\nu}+\Lambda g_{\mu\nu}=0$ (2.2) Being inspired by the four dimensional C-metric [5], we start proposing a similar ansatz but in $2+1$ dimensions: $ds^{2}=\frac{1}{(1+\alpha r\cos\theta)^{2}}\left[-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}d\theta^{2}}{g(\theta)}\right]$ (2.3) a general solution for (2.2) in terms of the unknown functions $f(r)$ and $g(\theta)$ can be find: $f(r)=c_{0}+c_{1}\ r+c_{2}\ r^{2}\ \ ,\qquad g(\theta)=\frac{c_{0}\alpha^{2}\cos^{2}(\theta)-c_{1}\alpha\cos(\theta)+c_{2}+\Lambda}{\alpha^{2}(\cos^{2}(\theta)-1)}$ (2.4) Where $c_{0},c_{1},c_{2}$ are arbitrary integration constant. This metric describes locally a constant curvature space-time: $R^{\mu\nu}_{\ \ \rho\sigma}=\Lambda\ (\delta^{\mu}_{\ \rho}\delta^{\nu}_{\ \sigma}-\delta^{\mu}_{\ \sigma}\delta^{\nu}_{\ \rho})$. In order to have a significant $\alpha\rightarrow 0$ limit we select a particular metric choosing, from (2.4), the following integration constants: $c_{0}=1-m\quad,\qquad c_{1}=0\quad,\qquad c_{2}=\alpha^{2}(m-1)-\Lambda$ After rescaling the $\theta$ coordinate one obtains the solution: $\displaystyle ds^{2}=\frac{1}{\left[1+\alpha r\cos\left(\theta\sqrt{1-m}\right)\right]^{2}}\bigg{\\{}$ $\displaystyle-\Big{[}1-m+r^{2}\big{[}\alpha^{2}(m-1)-\Lambda\big{]}\Big{]}dt^{2}$ (2.5) $\displaystyle+\frac{dr^{2}}{1-m+r^{2}\big{[}\alpha^{2}(m-1)-\Lambda\big{]}}+r^{2}d\theta^{2}\bigg{\\}}$ The coordinates $(t,r,\theta)$ are chosen to be polar, so their range is $-\infty<t<\infty$, $r\geq 0$, $-\pi\leq\theta\leq\pi$; since $\theta$ is an angular coordinate, the points $\theta=-\pi$ and $\theta=\pi$ can be considered identified. Other choises for the coordinate ranges and identification are possible and give rise to different spacetime geometries; our choice is motivated by the will to model an accelerating 2+1 black hole. In fact when the parameter $\alpha$ is imposed to be null one has: $ds^{2}=-\big{(}1-m-\Lambda\ r^{2}\big{)}dt^{2}+\frac{dr^{2}}{1-m-\Lambda\ r^{2}}+r^{2}d\theta^{2}$ (2.6) which is precisely the static BTZ black hole metric in the gauge where the ground state is the (A)dS space-time. We prefer this form of the metric respect to the one of [1] because when the mass222Computed respect to the (A)dS background, as done for instance in [7] $m\rightarrow 0$ one does recover from (2.6), not just an asymptotically (A)dS space-time as in [1], but exactly the standard vacuum (A)dS space-time. Similarly this is what happens in 3+1 (or higher) dimension, for instance to the Schwarzschild-(A)dS black hole. The fact that in this gauge when $m\in[0,1)$ naked singularity occurs (while black holes for $m\in[1,\infty)$) it’s a typical feature of Chern- Simons gravity or odd-dimensional Lovelock theories, such is three-dimensional general relativity.333Recently $m\in[0,1)$ states of the black hole spectrum were physically dignified as topological defects in [8], representing particles in $AdS_{3}$ (see also [9]) . The metric (2.5) describes, for $m\geq 1$ an accelerating black hole whose event horizon is located at: $r_{h}=\sqrt{\frac{m-1}{\alpha^{2}(m-1)-\Lambda}}$ A physical interpretation to the $\alpha$ parameter can be given, following the argument of [3] and [4]. Usually, as for the 3+1 C-metric, one considers the weak field limit of the solution (2.5), that is $m\rightarrow 0$: $ds^{2}=\frac{1}{\big{[}1+\alpha r\cos(\theta)\big{]}^{2}}\left\\{-\big{[}1-r^{2}(\alpha^{2}+\Lambda)\big{]}dt^{2}+\frac{dr^{2}}{1-r^{2}(\alpha^{2}+\Lambda)}+r^{2}d\theta^{2}\right\\}$ (2.7) The weak field limit is usefull because, in this case, black holes can be considered as test particles and cease to deform the spacetime and inertial frames around them.444A metric similar to (2.7) is studied by [10] in the case of null cosmological constant. The 3D timelike worldlines $x^{\mu}(\lambda)$ of an observer with $r=\bar{r}=constant$ and $\theta=0$ can be obtained by the property $u_{\mu}u^{\mu}=-1$ of the 3-velocity defined by $u^{\mu}=dx^{\mu}/d\lambda$: $x^{\mu}(\lambda)=\left[\frac{1+\alpha\bar{r}\cos(\theta)}{\sqrt{1-\bar{r}^{2}(\alpha^{2}+\Lambda)}}\lambda,\bar{r},0\right]\ \ \ ;$ where $\lambda$ is the proper time of the observer. Then the magnitude $a$ of the 3-acceleration, $a^{\mu}=(\nabla_{\nu}u^{\mu})u^{\nu}$, for this kind of observer results $|a|=\sqrt{a_{\mu}a^{\mu}}\ \Big{|}_{\bar{r}=0}=\alpha$ (2.8) Since $a_{\mu}u^{\mu}=0$, the value $|a|$ is also the magnitude of the 2-acceleration in the rest frame of the observer. From eq. (2.8) we achieve the conclusion that the origin of the metric (2.7), $\bar{r}=0$ is being accelerated with an uniform acceleration whose value is precisely given by the constant $\alpha$, so (2.7) is nothing but the accelerating (A)dS space-time. Outside of the weak field limit similar results can also be obtained: $a_{\mu}a^{\mu}\ |_{\bar{r}=0}=\alpha^{2}\ (1-m)$ for $m\neq 1$, while $a_{\mu}a^{\mu}\ |_{\bar{r}=0}=-\Lambda$ for $m=1$. Thus now we can interpret the BTZ metric (2.6) as the non-accelerating limit of the (2.5). Thanks to this limit the parameter $m$ which appears in (2.5) can be interpreted as the mass parameter of our solution. Of course in case of non null acceleration this does not coincide with the mass of the metric (2.5), but is somehow related to the mass. Actually is not clear how to compute energy in this class of accelerating space-time, neither in standard 3+1 dimensions, because of the non trivial asymptotic. So usually in the literature (see [5]) the mass value is estimated by thermodynamical argument as follows: The first law of black hole thermodynamics states that $dM=TdS=\frac{k}{8\pi G}dA$; while the surface gravity $k$ is defined by $k^{2}=-\frac{1}{2}\nabla^{\mu}\chi^{\nu}\nabla_{\mu}\chi_{\nu}$ where $\chi^{\mu}$ is the killing vector $\partial_{t}$: $k=r_{h}\left[(m-1)\alpha^{2}-\Lambda\right]\bigg{|}_{m=1+\frac{\Lambda r_{h}^{2}}{r_{h}^{2}\alpha^{2}-1}}=\frac{\Lambda r_{h}}{r_{h}^{2}\alpha^{2}-1}$ The area of the accelerating black hole horizon $A$ is given by: $A=\int_{-\pi}^{\pi}\sqrt{g_{\theta\theta}}\>\bigg{|}_{\\!\\!\begin{array}[]{l}\scriptscriptstyle r=r_{h}\\\\[-3.61664pt] \scriptscriptstyle t=\hbox{\tiny const.}\end{array}}\\!\\!\\!\\!\ d\theta\ =\ \frac{4}{\sqrt{\Lambda}}\arctan\left[\sqrt{\frac{\alpha r_{h}-1}{\alpha r_{h}+1}}\tanh\left(\frac{\pi}{2}\sqrt{\frac{\Lambda r_{h}^{2}}{r_{h}^{2}\alpha^{2}-1}}\right)\right]$ (2.9) So the mass estimation is, up to a integration constant: $M=\int dM=\int\frac{2\sqrt{\Lambda}}{\pi(\alpha^{2}r_{h}^{2}-1)^{3/2}}\frac{\alpha r_{h}\sinh\left(\pi\sqrt{\frac{\Lambda r_{h}^{2}}{\alpha^{2}r_{h}^{2}-1}}\right)-\pi\sqrt{\frac{\Lambda r_{h}^{2}}{\alpha^{2}r_{h}^{2}-1}}}{1+\alpha r_{h}\cosh\left(\pi\sqrt{\frac{\Lambda r_{h}^{2}}{\alpha^{2}r_{h}^{2}-1}}\right)}dr_{h}$ For small values of the acceleration parameter $\alpha<<1$, thus, in practice we restrict only to the $\Lambda<0$ sector, this integral can be evaluated: $M=-\Lambda r_{h}^{2}+2\alpha\left[\frac{r_{h}}{\pi^{2}}\cos\left(\pi r_{h}\sqrt{\Lambda}\right)+\frac{\Lambda\pi^{2}r_{h}^{2}-1}{\sqrt{\Lambda}\pi^{3}}\sin\left(\pi r_{h}\sqrt{\Lambda}\right)\right]+O(\alpha^{2})$ Evidently in case of null acceleration this results is coherent with the BTZ one: $M[r_{h}]=m[r_{h}]+k_{0}$. Note that the integration constant $k_{0}$ is background dependent, $k_{0}=0$ for the AdS background. The main difference of this three dimensional accelerating black hole, compared with the four dimensional C-metric, is the absence of a pure acceleration horizon. In fact the effect of the acceleration merges with the cosmological constant pressure to give an unique event horizon. This happens because also the standard three dimensional event horizon is just that of an accelerated observer, until one identifies $\theta$ as an angular coordinate [1]. The peculiar combination between $\Lambda$ and $\alpha$ has the remarkable consequence that also positive values for the cosmological constant are admissible in order to get a black hole configuration. This is something unexpected since even a no-go theorem (see [11]) is found to justify the lack of a 2+1 black hole, in standard general relativity, for positive cosmological constant. In our case the no-go theorem is circumvented in one of the hypothesis of regularity of the horizon: the horizon of the accelerating solution (2.5) is continuous but not smooth as required in [11]. It is worth to note that the Riemannian curvature tensor remains locally constant everywhere, but in fact in $\theta=\pm\pi$ the metric is not differentiable because of an angular singularity, a common feature which characterises this kind of accelerating black holes. Here the singularity is not a conical one due to a standard deficit angle as the four dimensional case, this may appear as another difference. But it is just because otherwise in one dimension less there would be no room for a strut or a string. Figure 1: Polar plot ($\bar{r},\theta$) of $\bar{r}=$constant radial curves embedded in the plane $\mathbb{E}^{2}$. When $\bar{r}=r_{h}$ they illustrate horizon deformations for various values of the mass: $m=0$, $0<m<1$, $m=1$ and $m>1$. In figure 1 are polar plotted the shapes of the horizon, for various values of the mass parameter $m$, as embedded in the two dimensional euclidean plane $\mathbb{E}^{2}$. Note that the sharp vertex occurring for $0<m<1$ and $m>1$ is an artifact of the embedding, in the real curved space-time there are no sharp vertexes. In fact the two vectors normal to the horizon in $\theta=\pi$ and $\theta=-\pi$, denoted by $n^{\mu}_{(\pm)}$, are parallels: $n^{\mu}_{(+)}\ n_{(-)\mu}=1$. Nevertheless the angular singularity in the conformal factor persist as pointed out by the undifferenciability of the metric in $\theta=\pm\pi$ (for $m\neq 0,1$). It is usually associated to the presence of a semi-infinite cosmic string, or a strut, which is pulling the black hole along the $\theta=\pi$ axis. In this picture the tension of the string $\tau$ is responsible for the acceleration of the 2+1 dimensional black hole. Thanks to the Israel junction condition it is possible to compute the strength of the strut’s force, geometrically due by a jump in the extrinsic curvature $\mathcal{K}_{\mu\nu}$: $\big{[}\mathcal{K}_{\mu\nu}\big{]}^{\theta=\pi}_{\theta=-\pi}-h_{\mu\nu}\big{[}\mathcal{K}\big{]}^{\theta=\pi}_{\theta=-\pi}=-\pi h_{\mu\nu}\tau$ (2.10) where $h_{\mu\nu}$ is the induced metric on the $\theta=\pm\pi$ surface, $\mathcal{K}=\mathcal{K}_{\mu\nu}h^{\mu\nu}$ and the tension is $\tau=-2\alpha\ \sqrt{m-1}\ \sinh(\pi\sqrt{m-1}).$ The negativity of $\tau$ indicate that we are dealing with a strut that is pushing the black hole, rather than a pulling string. As can be read from the metric (2.5) or perceived from figure 1, the strut tension $\tau$ vanishes for $m=0,1$, where the metric acquire regularity. The metric (2.5) is supported by the surface stress-energy tensor that can be extract from (2.10) and, for sake of precision, it should be better added in the equation of motion (2.2). The casual structure and the Carter-Penrose diagram is similar to the BTZ one, although now the casual singularity in $r=0$ has an acceleration $\alpha$. A rotating and accelerating metric can be obtained by an improper boost in the ($t,\theta$) plane of (2.5) as explained in [2]. Since in three dimensions there is no room for rotating around the strut axis as in four dimensions, the only way to pursue rotation is have the strut rotate with the black hole. As expected that metric reduces to the (2.5) when the rotation is null, while it reduces to the rotating BTZ metric when the acceleration parameter vanish. Since the metric (2.5) remains finite for $r\rightarrow\infty$ one may also think to extend the spacetime also in the negative $r$ sector in order to reach the conformal infinity for $r=-1/(\alpha\cosh(\theta\sqrt{m-1}))$. This can be done in general for $\Lambda>0$. While for $\Lambda<0$, depending also on the reciprocal values of the parameters $\Lambda,m$ and the angular direction $\theta$, one may encounter another killing horizon corresponding to the negative root of the $f(r)$ function before the conformal infinity. But here we are not interested in that. Instead we are more interested on the possible embeddings of this metric in the $3+1$ dimensional space time, as presented in the next section. ## 3 Embeddings in 3+1 ### 3.1 Accelerating BTZ black string The most direct embedding in 3+1 dimensions of the 2+1 solution (2.5) is obtained in the spirit of [12]. A string like object can be written by a warped product of the three dimensional metric and a line element $dz$: $\displaystyle ds^{2}=\frac{\cos^{2}(z)}{\left[1+\alpha r\cos\left(\theta\sqrt{1-m}\right)\right]^{2}}\bigg{\\{}$ $\displaystyle-\left[1-m+r^{2}\left(\alpha^{2}(m-1)-\frac{\Lambda}{3}\right)\right]dt^{2}+$ (3.1) $\displaystyle+\frac{dr^{2}}{1-m+r^{2}\left(\alpha^{2}(m-1)-\frac{\Lambda}{3}\right)}+r^{2}d\theta^{2}\bigg{\\}}\ +\ \frac{3}{\Lambda}\ dz^{2}$ To preserve the correct metric signature $(-,+,+,+)$ the z coordinate have to be rotated to the imaginary plane when dealing with negative cosmological constant: $z\rightsquigarrow iz$. The standard BTZ black string of [12] is precisely recovered from (3.1) in the limit of $\alpha=0$ (of course in that case just negative cosmological constant can be considered). But the periodic dependence on the $z$ coordinate, for positive $\Lambda$ (and $\alpha\neq 0$), suggests that in this case the string singularity can be though of closed type. Thus the horizon’s topology has a toroidal geometric structure, but with a couple of points in $z=\pm\pi/2$ where the horizon throat shrinks to zero555Alternatively is possible to achieve the spherical topology, for $\Lambda>0$, considering $-\pi/2<z<\pi$; the casual singularity would be a $\pi$-length segment along the $z$-axes.. Anyway many other features are shared between the static and accelerating black string. For instance the space-time curvature still remains trivially constant as the 2+1-dimensional case: $R^{\mu\nu}_{\ \ \rho\sigma}=\Lambda/3\ (\delta^{\mu}_{\ \rho}\delta^{\nu}_{\ \sigma}-\delta^{\mu}_{\ \sigma}\delta^{\nu}_{\ \rho})$. In this case the acceleration is provided by a two space dimensions membrane, which results a semi-infinite plane, at least for negative $\Lambda$. ### 3.2 Black Ring Our main interest is in topological non-trivial solution, so we consider an ansatz with toroidal base manifold, finally. Not the one of constant curvature obtained by identification of a flat rectangle edges, but rather a doughnut embedded in the full 3+1 space-time, whose metric and curvature is giving, thinking $r$ and $t$ constant, as follows: $ds^{2}=r^{2}d\theta^{2}+\left[R_{0}+r\cos\left(\theta\sqrt{1-m}\right)\right]^{2}d\phi^{2}\ ,\qquad R^{\theta\phi}_{\ \ \theta\phi}=\frac{(1-m)\cos\left(\theta\sqrt{1-m}\right)}{r\left[R_{0}+r\cos\left(\theta\sqrt{1-m}\right)\right]}$ When $m\neq 0\neq 1$, but still $\theta=\pm\pi$ identified, the circular section of the torus is deformed, respect to the smooth $m=0,m=1$ cases, in a drop or a cardioid shaped section (see figure 1). Using this base manifold with generic $m$ and the acceleration conformal factor of the previous section 2 one can obtain a ring-like solution of Einstein equations (2.2): $\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{\left[1+\alpha r\cos\left(\theta\sqrt{1-m}\right)\right]^{2}}\bigg{\\{}-\left[1-m+r^{2}\left[\alpha^{2}(m-1)-\frac{\Lambda}{3}\right]\right]dt^{2}+$ $\displaystyle+$ $\displaystyle\frac{dr^{2}}{1-m+r^{2}\left[\alpha^{2}(m-1)-\frac{\Lambda}{3}\right]}+r^{2}d\theta^{2}+\big{[}R_{0}+r\cos(\theta\sqrt{1-m})\big{]}^{2}d\phi^{2}\bigg{\\}}$ where the bigger radius of the torus $R_{0}$ is related to the acceleration and mass parameter in this way: $R_{0}=\frac{\alpha(m-1)}{\alpha^{2}(m-1)-\Lambda/3}=\alpha r_{0}^{2}$ It is evident that the acceleration plays a fundamental role in the topological structure of the solution: when the acceleration parameter decreases also the radius of the torus $R_{0}$ shrinks until it vanishes for $\alpha=0$; in the latter case local spherical symmetry in the base manifold is achieved (while globally one has a portion of $(A)dS_{4}$). So, from a physical point of view, the acceleration sustains and balances the gravitational attraction of the ring singularity, providing an equilibrium configuration (whose stability is unclear). Figure 2: “Accelerating” black ring horizon embedded in $E^{3}$ Considering a $\phi$-constant slice of (3.2) one has exactly the accelerating black hole in 2+1 of section 2, modulo the dimensional rescaling of the cosmological constant. The topology of the horizon, located at $r_{0}$, can be confirmed with the help of the Gauss-Bonnet theorem. It is simple to compute the Euler characteristic when the mass parameter gives smooth 2-surface for constant time and radius, that is $m=0$ and $m=1$. For different values of $m$ one has to take into account the correction to the Gauss-Bonnet theorem due to the sharp edge, which is given by the Gibbons-Hawking term. This involves the jump on the trace of the extrinsic curvature $\mathcal{\bar{K}}$ and the induced metric $\bar{h}$ on the circular sharp edge. Consider the surface $\mathcal{S}$ described by the two dimensional metric obtained by (3.2) fixing $r=\bar{r}=$const and $t=$const, whose embedding in the three dimensional euclidean space $E^{3}$ is portrayed in figure 2: $d\bar{s}^{2}=\frac{1}{\left[1+\alpha\bar{r}\cos\left(\theta\sqrt{1-m}\right)\right]^{2}}\left\\{\bar{r}^{2}d\theta^{2}+\big{[}R_{0}+\bar{r}\cos(\theta\sqrt{1-m})\big{]}^{2}d\phi^{2}\right\\}$ Its Euler characteristic is null: $\displaystyle\chi(\mathcal{S})$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi}\left(\int_{\mathcal{S}}\sqrt{\bar{g}}\ \bar{R}\ d\theta\ d\phi\ +2\int_{-\pi}^{\pi}\left[\sqrt{\bar{h}}\ \bar{\mathcal{K}}\right]^{\theta=\pi}_{\theta=-\pi}\ d\phi\right)=$ $\displaystyle=$ $\displaystyle\frac{\alpha R_{0}-1}{4\pi}\sqrt{1-m}\int_{-\pi}^{\pi}d\phi\int_{-\pi}^{\pi}\frac{\cos(\theta\sqrt{1-m})+\bar{r}\alpha}{[1+\alpha\bar{r}\cos(\theta\sqrt{1-m})]^{2}}d\theta$ $\displaystyle-$ $\displaystyle\frac{\alpha R_{0}-1}{2\pi}\int_{-\pi}^{\pi}\frac{2\sin(\pi\sqrt{1-m})}{1+\alpha\bar{r}\cos(\pi\sqrt{1-m})}\ d\phi\ \ =\ 0$ so, since $\chi(\mathcal{S})=2-2g$, the genus of $\mathcal{S}$ is 1: toroidal topology $S^{1}\times S^{1}$. Observe that the irregularity on the horizon can be cast also in the external part of the torus, as happens in the 5D static ring [13]. The causal singularity is located along $r=0$, forming a circle in the $\phi$ direction. It’s a naked singularity with a cosmological horizon for $m\in(0,1)$ while a black hole type for $m>1$. Coordinate $r$ is not a standard polar radius but rather the distance from the ring singularity. Of course in order to have a proper black ring torus horizon the parameters of the metric are somewhat constrained to assure that the ring radius is larger than the ring thickness, $R_{0}\geq r_{0}$: $0\leq\Lambda\leq 3(m-1)\alpha^{2}$. Note that in the null cosmological constant case we can have just a plump horn ring with $R_{0}=r_{0}$, as expected in [6], because a lack of the lengh scale furnished by $\Lambda$. It’s not anymore possible tune the acceleration parameter $\alpha$ to make one radius arbitrary larger than the other one. While in the other limiting case, that is $\Lambda=3(m-1)\alpha^{2}$, $R_{0}$ grows to infinity. To have a well behaved coordinate’s set and an Hausdorff manifold, the range of $r$ have to be restricted when $|\theta|>\pi/2$ to $r\geq-R_{0}/\cos(\theta\sqrt{1-m})$. That fact is maybe clearer in toroidal coordinates (3.3) but, on the other hand, this coordinates patch makes the physical interpretation of the metric (3.2) more opaque. Anyway to get the the metric (3.2) in the usual ring coordinate just rename $y=-1/\alpha r$, $x=\cos(\theta\sqrt{1-m})$ and rescale time: $\displaystyle ds^{2}=\frac{1}{\alpha^{2}(y-x)^{2}}\bigg{\\{}$ $\displaystyle-$ $\displaystyle\left[(y^{2}-1)(1-m)-\frac{\Lambda}{3\alpha^{2}}\right]dt^{2}+\frac{dy^{2}}{(y^{2}-1)(1-m)-\frac{\Lambda}{3\alpha^{2}}}+$ (3.3) $\displaystyle+$ $\displaystyle\frac{dx^{2}}{(1-x^{2})(1-m)}+\left(\alpha R_{0}\ y-x\right)^{2}d\phi^{2}\bigg{\\}}$ This form of the solution may be of some utility for those people interested in finding a (A)dS ring five or in higher dimensions. Overall this metric (3.2) or (3.3) is again locally (A)dS again despite the fact that the base manifold is not of constant curvature. However note that this behaviour is conceptually different from the topology of the accelerating black string (3.1), where the toroidal topology is archived by means of the identification on the fourth coordinate, when $\Lambda>0$. That identification forced a topology change in the whole universe, which is not here the case. ## 4 Comments and Conclusions In this paper a C-metric type solution for 2+1 general relativity with cosmological constant is presented. It is analysed following the standard four-dimensional techniques: in the weak field approximation the extra parameter $\alpha$ is found to be the acceleration of an observer in the origin of coordinates. When $\alpha$ parameter vanishes the usual static BTZ black hole is recovered. Thus the metric (2.5) is interpreted as an accelerating three-dimensional black hole, for certain range of the mass parameters: $m>1$. The acceleration is provided by a one-space-dimensional semi-infinite strut whose tension is proportional to the jump in the extrinsic curvature. For $0\leq m<1$ the same solution represents an accelerating naked singularity with a cosmological horizon pulled by a string. A remarkable fact is that black hole configurations are admissible even for positive cosmological constant (whenever is smaller than a certain amount of acceleration: $\Lambda<\alpha^{2}(m-1)$). This because the pushing effect of the strut arithmetically adds to the acceleration provided by the cosmological constant. Moreover a couple of embeddings of this metric in the 3+1 dimension are considered. First an accelerating (by a two space dimension membrane) black string, whose horizon topology depends on the sign of the cosmological constant: cylindrical or toroidal for negative or positive $\Lambda$ respectively. Again the standard BTZ black string is retrieved when $\alpha=0$. Lastly it is proposed a ring singularity covered by a toroidal horizon where the gravitational force is balanced by the acceleration supplied by a disk (or puncured plane in case of external irregularity) that, in fact, sustains the ring. Each $\phi$-constant slice represents an accelerating black hole of one dimension less. Up to the author knowledge this is, though not regular, the first analytical black ring solution in four dimensions and also the first not asymptotically flat in any dimensions. For future perspective would be very interesting counterweight the gravitational attraction, instead of the acceleration only, by a smoother centrifugal effect due to the angular momentum. This may provide a regular horizon such as been done in five dimension by Emparan and Reall passing from an irregular static ring in [13] to a regular rotating one in [14]. The case of null acceleration implies only negative cosmological constant, so the Hawking topology censorship theorem can be avoided. Nevertheless the presence of the cosmological constant preclude the use of standard generating solutions techniques. ## Acknowledgements I would like to thank Eloy Ayon Beato, Fabrizio Canfora, Fiorenza de Micheli, Dietmar Klemm, Julio Oliva, Souya Ray, David Tempo, Steve Willison and Jorge Zanelli for fruitful discussions. I’m deeply indebted to Hideki Maeda for his continuous encouragement, suggestions and comments, without his help this work just wouldn’t have risen. This work has been partially funded by the Fondecyt grant 1100755 and by the Conicyt grant “Southern Theoretical Physics Laboratory” ACT-91. The Centro de Estudios Científicos (CECS) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt. CECS is also supported by a group of private companies which at present includes Antofagasta Minerals, Arauco, Empresas CMPC, Indura, Naviera Ultragas and Telefónica del Sur. ## References * [1] M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, “Geometry of the (2+1) black hole,” Phys. Rev. D 48 (1993) 1506 [arXiv:gr-qc/9302012] * [2] C. Martinez, C. Teitelboim and J. Zanelli, “Charged rotating black hole in three spacetime dimensions,” Phys. Rev. D 61, 104013 (2000) [arXiv:hep-th/9912259]. * [3] J. Podolsky and J. B. Griffiths, “Uniformly accelerating black holes in a de Sitter universe,” Phys. Rev. D 63 (2001) 024006 [arXiv:gr-qc/0010109]. J. Podolsky, “Accelerating black holes in anti-de Sitter universe,” Czech. J. Phys. 52 (2002) 1 [arXiv:gr-qc/0202033]. * [4] O. J. C. Dias and J. P. S. Lemos, “Pair of accelerated black holes in anti-de Sitter background: The AdS C-metric,” Phys. Rev. D 67 (2003) 064001 [arXiv:hep-th/0210065]. * [5] J. B. Griffiths, P. Krtous and J. Podolsky, “Interpreting the C-metric,” Class. Quant. Grav. 23 (2006) 6745 [arXiv:gr-qc/0609056]. * [6] R. Emparan and H. S. Reall, “Black Holes in Higher Dimensions,” Living Rev. Rel. 11 (2008) 6 [arXiv:0801.3471 [hep-th]] * [7] S. Deser and B. Tekin, “Energy in generic higher curvature gravity theories,” Phys. Rev. D 67 (2003) 084009 [arXiv:hep-th/0212292]. * [8] O. Miskovic and J. Zanelli, “On the negative spectrum of the 2+1 black hole,” Phys. Rev. D 79 (2009) 105011 [arXiv:0904.0475 [hep-th]]. * [9] A. R. Steif,“Time-Symmetric Initial Data for Multi-Body Solutions in Three Dimensions,” Phys. Rev. D 53 (1996) 5527 [arXiv:gr-qc/9511053]. * [10] M. M. Anber, “AdS4/CFT3+Gravity for Accelerating Conical Singularities,” JHEP 0811 (2008) 026 [arXiv:0809.2789 [hep-th]] * [11] D. Ida, “No black hole theorem in three-dimensional gravity,” Phys. Rev. Lett. 85 (2000) 3758 [arXiv:gr-qc/0005129]. * [12] R. Emparan, G. T. Horowitz and R. C. Myers, “Exact description of black holes on branes. II: Comparison with BTZ black holes and black strings,” JHEP 0001 (2000) 021 [arXiv:hep-th/9912135]. * [13] R. Emparan and H. S. Reall, “Generalized Weyl solutions,” Phys. Rev. D 65 (2002) 084025 [arXiv:hep-th/0110258]. * [14] R. Emparan and H. S. Reall, “A rotating black ring in five dimensions,” Phys. Rev. Lett. 88 (2002) 101101 [arXiv:hep-th/0110260].
arxiv-papers
2011-01-13T17:36:22
2024-09-04T02:49:16.418108
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Marco Astorino", "submitter": "Marco Astorino", "url": "https://arxiv.org/abs/1101.2616" }
1101.2618
UNIVERSITÀ DEGLI STUDI DELL’INSUBRIA Dipartimento di scienze MM. FF. NN. Como Anno accademico 2008 - 2009 Sessione di laurea del 30 Settembre 2009 Tesi di Matematica: Potenziali di Evans su varietà paraboliche Autore: Daniele Valtorta, matricola 617528 e-mail: danielevaltorta@gmail.com Relatore: Alberto Giulio Setti Co-relatore: Stefano Pigola ### 0.1 Riassunto La tesi si occupa di una particolare caratterizzazione delle varietà paraboliche, in particolare una varietà Riemanniana $R$ è parabolica se e solo se ammette potenziali di Evans, funzioni armoniche proprie definite fuori da un compatto. Il principale riferimento bibliografico è il libro di Sario e Nakai [SN], libro nel quale gli autori dimostrano l’esistenza di potenziali di Evans su superfici riemanniane. La generalizzazione al caso di varietà di dimensione qualsiasi presenta alcune differenze tecniche non irrilevanti, ad esempio leggermente diverse sono le tecniche utilizzate nel paragrafo 4.4.3 relativo al principio dell’energia. Si ringrazia il professor Wolfhard Hansen (University of Bielefeld) per i suggerimenti gentilmente forniti riguardo alla teoria del potenziale, che gioca un ruolo fondamentale in queste dimostrazioni. La tesi si articola in una parte introduttiva in cui vengono ripresi alcuni concetti di geometria Riemanniana e di teoria delle funzioni armoniche. Speciale attenzione è stata posta sulla soluzione del problema di Dirichlet su varietà, in particolare è stata trovata una dimostrazione geometrica della solubilità del problema di Dirichlet per domini particolari, dimostrazione riportata nella sezione 1.9. In seguito vengono definite e studiate l’algebra di Royden $\mathbb{M}(R)$ e la compattificazione di Royden $R^{*}$ di una varietà Riemanniana $R$, strumento essenziale per la dimostrazione di esistenza dei potenziali di Evans. Essendo $R^{*}$ uno spazio topologico non primo numerabile, un capitolo della tesi (il capitolo 2) è dedicato allo studio di filtri e ultrafiltri, concetti utili a descrivere questa topologia e soprattutto a fornire esempi espliciti di elementi in $R^{*}$ non banali (vedi paragrafo 3.3.2). Importanti strumenti tecnici introdotti nella tesi sono le formule di Green e il principio di Dirichlet, relazioni che sono state dimostrate con ipotesi poco restrittive sulla regolarità degli insiemi e delle funzioni in gioco (vedi paragrafo 3.2.4). Anche il principio del massimo ha un ruolo essenziale, sia nelle sue forme “standard”, sia nelle sue versioni più sofisticate presentate nel paragrafo 3.3.5. Nel capitolo 4 viene introdotta la capacità di un insieme, e questo concetto è utilizzato per la caratterizzazione delle varietà paraboliche, seguendo il percorso tracciato nella monografia [G2]. Sono introdotti anche il diametro transfinito e la costante di Tchebycheff, e viene dimostrato che il diametro transfinito di ogni compatto contenuto nel bordo irregolare di una varietà parabolica è infinito. Questa dimostrazione è basata fortemente sulla teoria del potenziale, teoria che tratta delle proprietà di misure di Borel regolari, della loro energia e delle proprietà dei relativi potenziali di Green. La non finitezza del diametro transfinito è essenziale per la costruzione di funzioni armoniche che tendono a infinito sul bordo irregolare di una varietà iperbolica, e grazie al legame tra queste varietà e le varietà paraboliche esposto nel paragrafo 4.4.6, otteniamo la dimostrazione (costruttiva) dell’esistenza dei potenziali di Evans, riportata nel teorema 4.76. In tutte queste dimostrazioni la linearità è essenziale, in particolare si sfrutta il fatto che la somma di due funzioni armoniche è armonica. Un possibile sviluppo futuro di questa tesi è cercare di caratterizzare le varietà $p$-paraboliche con l’esistenza di potenziali di Evans $p$-armonici, che richiederebbe di adattare le tecniche utilizzate in questo lavoro al caso non lineare. ###### Contents 1. 0.1 Riassunto 2. 1 Richiami di matematica 1. 1.1 Geometria Riemanniana 1. 1.1.1 Gradiente e laplaciano di una funzione 2. 1.1.2 Punti critici, valori critici e teorema di Sard 3. 1.1.3 Coordinate polari geodetiche e modelli 4. 1.1.4 Integrazione su varietà e formula di Green 5. 1.1.5 Coordinate di Fermi e applicazioni 6. 1.1.6 Esaustioni regolari 7. 1.1.7 Insiemi chiusi e funzioni lisce 2. 1.2 Funzioni assolutamente continue 3. 1.3 Spazi L2 di forme 4. 1.4 Derivazione sotto al segno d’integrale 5. 1.5 Convoluzioni 6. 1.6 Duali di spazi di Banach 7. 1.7 Funzioni armoniche 1. 1.7.1 Principio del massimo 2. 1.7.2 Stime sul gradiente 3. 1.7.3 Disuguaglianza, funzione e principio di Harnack 4. 1.7.4 Funzioni di Green 5. 1.7.5 Singolarità di funzioni armoniche 6. 1.7.6 Principio di Dirichlet 7. 1.7.7 Funzioni super e subarmoniche 8. 1.8 Algebre di Banach e caratteri 9. 1.9 Problema di Dirichlet 1. 1.9.1 Metodo di Perron 2. 1.9.2 Domini con bordo liscio 3. 1.9.3 Altri domini regolari 4. 1.9.4 Regolarità sul bordo 3. 2 Ultrafiltri e funzionali lineari moltiplicativi 1. 2.1 Filtri, ultrafiltri e proprietà 2. 2.2 Applicazioni: caratteri sulle successioni limitate 4. 3 Algebra e compattificazione di Royden 1. 3.1 Funzioni di Tonelli 1. 3.1.1 Definizioni e proprietà fondamentali 2. 3.1.2 Operazioni con le funzioni di Tonelli 2. 3.2 Algebra di Royden 1. 3.2.1 Definizione 2. 3.2.2 Topologie sull’algebra di Royden 3. 3.2.3 Densità di funzioni lisce 4. 3.2.4 Formule di Green e principio di Dirichlet 5. 3.2.5 Ideali dell’algebra di Royden 3. 3.3 Compattificazione di Royden 1. 3.3.1 Definizione 2. 3.3.2 Esempi non banali di caratteri 3. 3.3.3 Caratterizzazione del bordo 4. 3.3.4 Bordo armonico e decomposizione 5. 3.3.5 Principio del massimo 5. 4 Varietà paraboliche e iperboliche 1. 4.1 Capacità 2. 4.2 Bordo Armonico 3. 4.3 Funzioni di Green 1. 4.3.1 Funzioni di Green sulla compattificazione di Royden 4. 4.4 Potenziali di Evans 1. 4.4.1 Diametro transfinito 2. 4.4.2 Stime per il diametro transfinito 3. 4.4.3 Pricipio dell’energia 4. 4.4.4 Il diametro transfinito è infinito 5. 4.4.5 Funzioni armoniche che tendono a infinito sul bordo di R 6. 4.4.6 Varietà iperboliche irregolari 7. 4.4.7 Potenziali di Evans su varietà paraboliche 6. A Glossario 7. Bibliography ## Chapter 1 Richiami di matematica ### 1.1 Geometria Riemanniana Lo scopo di questo capitolo è passare brevemente in rassegna alcune definizioni e risultati che riguardano le varietà Riemanniane. Nel farlo considereremo solo funzioni lisce e varietà $C^{\infty}$. Verranno dati per scontati i concetti di varietà differenziale e di metrica riemanniana, come referenza su questi argomenti consigliamo [C1], [P1] e [G1]. Ricordiamo che una varietà riemanniana $M$ di dimensione $m$ può essere dotata di una metrica (cioè una funzione $g:T(M)\times T(M)\to\mathbb{R}$ simmetrica definita positiva e liscia), e indicheremo con $g_{ij}$ la matrice definita da $g_{ij}(p)=g(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial x^{j}})$, con $\left|g\right|$ il suo determinante e con $g^{ij}$ la sua inversa. #### 1.1.1 Gradiente e laplaciano di una funzione Data una varietà Riemanniana $(M,g)$, è possibile associare a ogni funzione differenziabile $f:M\to\mathbb{R}$ il suo differenziale $df:T(M)\to T(\mathbb{R})$, che in carte locali assume la forma $\displaystyle df|_{x}(v)=\left.\frac{\partial\tilde{f}}{\partial x^{i}}\right|_{x}v^{i}$ dove $\tilde{f}$ è la rappresentazione locale della funzione $f$ rispetto a una qualsiasi carta, e $v^{i}$ le componenti del vettore $v$ nella stessa carta. Il gradiente della funzione $f$ è il duale del suo differenziale, nel senso che $\nabla f$ è l’unico elemento di $T(M)$ tale che per ogni campo vettoriale $V\in T(M)$: $\displaystyle g_{ij}(\nabla f)^{i}v^{j}=\left\langle\nabla f\middle|V\right\rangle=D_{V}(f)\equiv V(f)=\sum_{i=1}^{m}v^{i}\frac{\partial f}{\partial x^{i}}$ da questa relazione risulta che il gradiente in coordinate locali assume la forma $\displaystyle(\nabla f)^{i}=g^{ij}\frac{\partial f}{\partial x^{j}}$ (1.1) Oltre al gradiente, per una funzione reale possiamo definire anche l’operatore di Laplace-Beltrami (o laplaciano) $\Delta:C^{m+2}(M,\mathbb{R})\to C^{m}(M,\mathbb{R})$. La forma locale di questo operatore (che è l’unico aspetto di interesse in questa tesi) è la seguente: $\displaystyle\Delta f=\frac{1}{\sqrt{g}}\frac{\partial}{\partial x^{i}}\left(g^{ij}\sqrt{g}\frac{\partial f}{\partial x^{j}}\right)$ (1.2) dove si sottointende la somma degli indici ripetuti (convenzione di Einstein). È utile definire anche l’operatore divergenza. Dato un campo vettoriale $V$ liscio, la sua divergenza è una funzione reale e in coordinate locali questa funzione è descritta da: $\displaystyle div(V)=\frac{1}{\sqrt{g}}\frac{\partial}{\partial x^{i}}\left(\sqrt{g}V^{i}\right)$ dalla definizione osserviamo subito che $\displaystyle\Delta f=div(\nabla f)$ e per un campo vettoriale $V$ e una funzione $f$ qualsiasi vale che: $\displaystyle div(fV)=\left\langle\nabla f\middle|V\right\rangle+f\Delta V$ A questo punto ha senso parlare di funzioni armoniche, che per definizione sono le funzioni per le quali $\Delta f=0$. Per approfondimenti sul laplaciano consigliamo il libro [P1] (nell’esercizio 10 cap. 2.8 pag 57 si trova la maggior parte delle informazioni necessarie per questo lavoro). #### 1.1.2 Punti critici, valori critici e teorema di Sard In questa sezione riportiamo brevemente le definizioni di punto critico, valore critico e il teorema di Sard. Per approfondimenti e chiarimenti rimandiamo a [H3]. Tutte le definizioni e i teoremi sono riportati con la generalità sufficiente agli scopi di questo lavoro, in modo da non appesantire la tesi con dettagli eccessivi, quindi in particolare tutte le varietà differenziali saranno varietà lisce. La prima domanda che ci poniamo è se le funzioni $f:M\to\mathbb{R}$ possono essere utili per definire in qualche senso una sottovarietà di $M$. ###### Definizione 1.1. Data una varietà differenziale $M$ e una funzione $f\in C^{1}(M,\mathbb{R})$, si dice che $p\in M$ è un punto critico se $df|_{p}=0$ (o equivalentemente $\nabla f|_{p}=0$). Si dice invece che $x\in\mathbb{R}$ è un valore critico per $f$ se $f^{-1}(x)$ contiene almeno un punto critico. Al contrario si dice che $p\in M$ è un valore regolare per $f$ se non è un punto critico, cioè se $df|_{p}\neq 0$, mentre $x\in\mathbb{R}$ è un valore regolare per $f$ se $f^{-1}(p)$ non contiene punti critici, cioè contiene solo punti regolari 111nel caso banale $f^{-1}(x)=\emptyset$, si dice che $x$ è un valore regolare per $f$. Indichiamo $\mathcal{C}(f)\subset\mathbb{R}$ l’insieme dei valori critici di $f:M\to\mathbb{R}$. ###### Definizione 1.2. Data una funzione $f:M\to\mathbb{R}$, per ogni valore $c\in\mathbb{R}$, definiamo insieme di sottolivello di $f$ rispetto a $c$ l’insieme $f^{-1}(-\infty,c]$, insieme di sopralivello di $f$ rispetto a $c$ l’insieme $f^{-1}[c,\infty)$ e insieme di livello di $f$ rispetto a $c$ l’insieme $f^{-1}(c)$. Osserviamo subito che il bordo degli insiemi di sotto/sopra livello è contenuto nel relativo insieme di livello, cioè $\partial(f^{-1}(-\infty,c])\subset f^{-1}(c)$, e che se $c$ è un valore regolare per $f$, questa inclusione si trasforma in uguaglianza, cioè $\partial(f^{-1}(-\infty,c])=f^{-1}(c)$. Ha senso chiedersi quando gli insiemi di sottolivello e di livello sono sottovarietà di $M$. ###### Proposizione 1.3. Se $c\in\mathbb{R}$ è un valore regolare di $f\in C^{r}(M,\mathbb{R})$ 222$r\geq 1$, allora $f^{-1}(c)$ è una sottovarietà $m-1$ dimensionale di $M$ di regolarità almeno $C^{r}$, e quindi $f^{-1}(-\infty,c]$ è una sottovarietà con bordo. Questa proposizione corrisponde al teorema 3.2 cap. 1.3 pag 22 di [H3], e rimandiamo a questo libro per la dimostrazione. Ora possiamo chiederci “quanti” siano i valori regolari di una funzione indipendentemente dalla funzione data. La risposta è contenuta nel teorema di Sard. Prima di enunciarlo definiamo gli insiemi di misura nulla su una varietà. Consigliamo il paragrafo 3.1 pag 68 di [H3] per approfondimenti. ###### Definizione 1.4. Un insieme $A\subset\mathbb{R}^{m}$ si dice essere di misura nulla se e solo se per ogni $\epsilon>0$ esiste un’insieme al più numerabile di cubi $C_{n}$ in $\mathbb{R}^{m}$ 333cioè di insiemi della forma $\prod_{i=1}^{m}[a_{i},b_{i}]$ che hanno misura $\lambda(\prod_{i=1}^{m}[a_{i},b_{i}])=\prod_{i=1}^{m}(b_{i}-a_{i})$ tali che 1. 1. $A\subset\bigcup_{n}C_{n}$ 2. 2. $\sum_{n=1}^{\infty}\lambda(C_{n})<\epsilon$ Ricordiamo che questa definizione è equivalente alla richiesta che la misura di Lebesgue di $A$ sia nulla. Valgono le seguenti proprietà ###### Proposizione 1.5. Per gli insiemi di misura nulla vale che 1. 1. L’unione numerabile di insiemi di misura nulla ha misura nulla 2. 2. Gli insiemi aperti non vuoti non hanno misura nulla, e il complementare di insiemi di misura nulla è denso 3. 3. I sottoinsiemi di insiemi di misura nulla hanno misura nulla 4. 4. L’immagine attraverso una funzione localmente Lipschitziana di un insieme di misura nulla ha misura nulla Grazie alla proprietà (1) ha senso definire ###### Definizione 1.6. Un insieme $A\subset M$ si dice essere di misura nulla se per ogni carta locale $(U,\phi)$ di $M$, l’insieme $\phi(A)$ ha misura nulla. Osserviamo che per paracompattezza di $M$, esiste sempre un atlante al più numerabile $\\{(U_{n},\phi_{n})\\}$ di $M$ 444anzi se la dimensione di $M$ è $m$, esiste sempre un atlante formato da al più $m+1$ carte, vedi problema 2.8 pag 21 di [M3], quindi dato che $A=\cup_{n}(A\cap U_{n})$, ha senso la definizione di insieme di misura nulla e non dipende dall’atlante scelto. Il teorema di Sard (o meglio una sua versione non molto generale) garantisce che ###### Teorema 1.7 (Teorema di Sard). Data $M$ varietà differenziale $m-$dimensionale, sia $r\geq\max\\{1,m-1\\}$. Allora se $f\in C^{r}(M,\mathbb{R})$, l’insieme dei valori critici $\mathcal{C}(f)$ ha misura nulla in $\mathbb{R}$, quindi il suo complementare è denso. #### 1.1.3 Coordinate polari geodetiche e modelli Un sistema di coordinate che utilizzeremo spesso sulle varietà sono le coordinate polari geodetiche. In questa sezione ci occupiamo di dare una breve carrellata sulle coordinate polari e sulle varietà modello, ovvero varietà sfericamente simmetriche. Come referenze per questa sezione consigliamo il capitolo 3 di [C1] e il capitolo 3 di [G2]. In tutta la sezione lavoreremo con una varietà riemannana $R$ di dimensione $m$. Fissato un punto $p\in R$, esiste sempre un intorno normale $U$ di $p$, un intorno cioè dove la mappa esponenziale $\exp:TR\to R$ è un diffeomorfismo. Visto che $TR$ è isomorfo a $\mathbb{R}^{m}$, possiamo definire su $TR\setminus\\{0\\}$ le classiche coordinate polari $m$-dimensionali 555ogni punto può essere individuato dalla distanza dall’origine e da una coordinata su $S^{m-1}$ che rappresenta la direzione del punto e grazie alla mappa esponenziale possiamo portare queste coordinate sulla varietà. Sull’aperto $U\setminus\\{p\\}$ chiamiamo le coordinate definite da $\displaystyle\phi(q)=\\{r(\exp^{-1}(q)),\theta(\exp^{-1}(q))\\}=(r,\theta)$ coordinate polari geodetiche. In questo sistema di coordinate la metrica assume la forma: $\displaystyle g=\begin{bmatrix}1&0\\\ 0&A(q)\end{bmatrix}$ (1.3) oppure utilizzando una notazione più famigliare alla geometria riemanniana: $\displaystyle ds^{2}=dr^{2}+A(r,\theta)_{ij}d\theta^{i}d\theta^{j}$ dove $A(q)$ è una matrice definita positiva 666vedi equazione 3.1 di [G2] oppure pagina 136 di [P1]. In coordinate polari inoltre l’operatore laplaciano assume la forma: $\displaystyle\Delta(f)=\frac{1}{\sqrt{\left|A\right|}}\frac{\partial}{\partial r}\left(\sqrt{\left|A\right|}\frac{\partial f}{\partial r}\right)+\Delta_{S}(f)=\frac{\partial^{2}f}{\partial r^{2}}+\frac{1}{2}\frac{\partial\log(\left|A\right|)}{\partial r}\frac{\partial f}{\partial r}+\Delta_{S}(f)$ (1.4) Dove $\Delta_{S}$ indica il laplaciano sulla sottovarietà $r=$costante 777vedi equazione 3.4 di [G2]. Si nota quindi che per le funzioni “radiali”, ossia quelle funzioni che dipendono solo da $r$, il laplaciano assume una forma abbastanza semplificata. Grazie alla semplice forma di $g$ in coordinate polari anche il gradiente di una funzione radiale è molto semplice da calcolare, infatti grazie alla 1.1 notiamo che per le funzioni radiali $f$: $\displaystyle(\nabla f)^{1}=\frac{\partial f}{\partial r}$ (1.5) mentre tutte le altre componenti del gradiente sono nulle. Queste coordinate risultano particolarmente adatte per descrivere le varietà modello. Queste varietà sono varietà sfericamente simmetriche rispetto a rotazioni attorno a un punto fisso. ###### Definizione 1.8. Definiamo $R$ una varietà con polo $o$, se la mappa esponenziale $\exp|_{o}$ è un diffeomorfismo globale tra $\mathbb{R}^{m}$ e $R$. Inoltre se la metrica di questa varietà rispetto alle coordinate polari geodetiche in $o$ è della forma $\displaystyle ds^{2}=dr^{2}+\sigma(r)^{2}d\theta^{2}$ dove $d\theta^{2}$ è la metrica euclidea standard di $S^{m-1}$, definiamo questa una varietà modello La definizione trova la sua giustificazione in questa osservazione. Se consideriamo una rotazione $\rho:S^{m-1}\to S^{m-1}$, possiamo definire una funzione $\displaystyle\psi_{\rho}:R\to R\ \ \psi_{\rho}(r,\theta)=\psi_{\rho}(r,\rho(\theta))$ questa funzione (che possiamo considerare a tutti gli effetti una rotazione su $R$), ha la proprietà di tenere la metrica invariata, cioè $\displaystyle\psi_{\rho}^{*}(g)=g$ dove $\psi_{\rho}^{*}$ è l’operatore di push-forward, definito da: $\displaystyle\psi_{\rho}^{*}(g)(v_{1},v_{2})=g(d\psi_{\rho}(v_{1}),d\psi_{\rho}(v_{2}))$ #### 1.1.4 Integrazione su varietà e formula di Green In questa sezione riportiamo brevemente la definizione di integrazione su varietà riemanniane e la formula di Green per domini regolari. Rimandiamo al testo [M2] per approfondimenti. Ai nostri scopi interessa solo ricordare brevemente la definizione di integrale di una funzione su una varietà e di una 1-forma su un bordo. Data una funzione reale $f:R\to\mathbb{R}$, per definire il suo integrale utilizziamo l’integrale di Lebesgue su $\mathbb{R}^{m}$. Non introduciamo la teoria dell’operatore duale di Hodge, ma ne utilizziamo comunque il simbolo ($\ast$) definendolo quando necessario. ###### Definizione 1.9. Sia $f:R\to\mathbb{R}$ una funzione continua a supporto compatto $supp(f)\Subset U$ con $(U,\phi)$ carta locale. Allora definiamo $\displaystyle\int_{R}fdV=\int_{\phi(U)}\tilde{f}(x)\sqrt{\left|g\right|}dx^{1}\cdots dx^{m}=\int_{\phi(U)}f\phi^{-1}(x)\sqrt{\left|g\right|}dx^{1}\cdots dx^{m}$ Osserviamo che la presenza di $\sqrt{\left|g\right|}$ garantisce che la definizione non dipenda dalla carta locale scelta. Data una funzione $f:R\to\mathbb{R}$ senza ulteriori condizioni sul supporto, definiamo: $\displaystyle\int_{R}fdV=\sum_{n=1}^{\infty}\int_{R}f\lambda_{n}dV$ dove $\\{\lambda_{n}\\}$ è una partizione dell’unità di $R$ subordinata a un ricoprimento di carte locali qualsiasi. ###### Definizione 1.10. Date due funzioni $f,h:R\to\mathbb{R}$, e un dominio regolare $\Omega\subset R$ 888con regolare si intende un dominio aperto relativamente compatto con $\partial\Omega$ bordo liscio (almeno a pezzi), definiamo $\displaystyle\int_{\partial\Omega}h\ast df=\int_{\phi(\partial\Omega)}h(x)g^{im}(x)\frac{\partial f}{\partial x^{i}}(x)\sqrt{\left|g\right|}dx^{1}\cdots dx^{m-1}$ quando $f$ ha supporto contenuto in una carta $(U,\phi)$ regolare per $\partial\Omega$, cioè una carta in cui $\phi(\partial\Omega)\subset\\{x^{m}=0\\}$. Se $f$ non ha queste caratteristiche, l’integrale si ottiene come sopra grazie alle partizioni dell’unità. Osserviamo che la notazione $\ast df$ indica il duale di Hodge della forma $df$. In questa tesi però questo operatore verrà usato solo in casi simili a quello appena descritto, quindi per brevità tralasciamo la sua definizione e lo utilizziamo solo per comodità di scrittura. Grazie a quanto appena descritto possiamo enunciare la prima identità di Green per varietà Riemanniane: ###### Proposizione 1.11 (Prima identità di Green). Dato un dominio regolare $\Omega\subset R$, e due funzioni lisce $f,h:R\to\mathbb{R}$, si ha che: $\displaystyle\int_{\Omega}\left\langle\nabla f\middle|\nabla h\right\rangle dV+\int_{\Omega}f\Delta hdV=\int_{\partial\Omega}f\ast dh$ In seguito rilasseremo le ipotesi sulla regolarità di $f$ in questa identità. La dimostrazione di questa identità è una facile conseguenza del teorema di Stokes (consigliamo come referenza il capitolo 7 di [M2]). #### 1.1.5 Coordinate di Fermi e applicazioni In questo paragrafo descriveremo un sistema di coordinate locali intorno a una sottovarietà regolare particolarmente utile in quanto per molti aspetti molto simile alle coordinate euclidee di $\mathbb{R}^{n}$. Il riferimento principale in questa sezione è il libro [G1], in particolare il II capitolo, dal quale riporteremo alcuni risultati senza dimostrazione. Data una varietà riemanniana $R$ di dimensione $m$ e una sua sottovarietà regolare $S$ di dimensione $s$, cerchiamo un sistema di coordinate locali $(x_{1},\cdots,x_{m})$ in un intorno $U$ di $x_{0}\in S$ tali che ogni punto $p\in U\cap S$ abbia $x^{m-s+1}(p)=\cdots=x^{m}(p)=0$ e tali che la metrica in queste coordinate assuma la forma: $\displaystyle g(p)=\begin{bmatrix}A(p)&0\\\ 0&B(p)\end{bmatrix}$ dove $A$ è una matrice $(m-s)\times(m-s)$ e $B$ è una matrice $s\times s$. In realtà per gli scopi di questo lavoro siamo interessati solo al caso $s=m-1$, quindi per comodità di notazione tratteremo solo questo caso, anche se i risultati di questo paragrafo possono facilmente essere estesi a sottovarietà di dimensioni qualsiasi. Per comodità, indicheremo con $p,\ q$ i punti sulla varietà riemanniana, con $(x^{1},\cdots,x^{m-1})\equiv\vec{x}$ le prime $m-1$ funzioni coordinate sulla varietà e con $y$ l’ultima coordinata. ###### Proposizione 1.12. Data una sottovarietà $S$ di codimensione $1$ in $R$, per ogni punto $p_{0}\in S$, esiste un suo intorno $U$ e delle coordinate locali tali che: 1. 1. $y(U\cap S)=0$ 2. 2. la metrica assume la forma $\displaystyle g(p)=\begin{bmatrix}A(p)&0\\\ 0&1\end{bmatrix}$ dove $A$ è una matrice $(m-1)\times(m-1)$. Inoltre è possibile scegliere le coordinate in modo che su $S$ le funzioni $x^{1}\cdots x^{m-1}$ siano coordinate qualsiasi relative alla sottovarietà $S$. ###### Proof. Il sistema di coordinate cercato prende il nome di coordinate di Fermi per la sottovarietà $S$. L’esistenza di queste coordinate è dimostrata in [G1]. In particolare a pagina 17 si trova la definizione di coordinate di Fermi, e grazie al lemma 2.3 di pagina 18 e al corollario 2.14 di pagina 31 999il lemma di Gauss generalizzato, si dimostra la proprietà (2). ∎ Introduciamo ora gli intorni geodeticamente convessi su una varietà riemanniana. ###### Definizione 1.13. Un insieme aperto $U\subset R$ si dice geodeticamente convesso se e solo se per ogni coppia di punti $(p,q)\subset U\times U$ la geodetica che minimizza la distanza tra questi due punti esiste unica e la sua traccia è contenuta in $U$. L’esempio più classico di intorno geodeticamente convesso sono le bolle in $\mathbb{R}^{n}$. L’esistenza di intorni geodeticamente convessi è garantita da questa proposizione, di cui non riportiamo la dimostrazione. ###### Proposizione 1.14. Dato un aperto $W\in R$ e $x\in W$, esiste sempre un intorno aperto geodeticamente convesso $U$ tale che $x\in U\subset W$. ###### Proof. La dimostrazione di questa proposizione può essere trovata su [C2], proposizione 10.5.4 pagina 334. ∎ Ora siamo pronti per dimostrare un lemma molto tecnico che servirà in seguito per dimostrare una proprietà dei potenziali di Green. ###### Proposizione 1.15. Dato $K$ insieme aperto relativamente compatto in $R$ con bordo liscio, definiamo per ogni $q\in R$: $\displaystyle\pi(q)\ \ \ t.c.\ \ \ d(\pi(q),q)=\inf\\{d(q,p)\ t.c.\ p\in K\\}$ Dimostriamo che per ogni $p_{0}\in\partial K$, esiste un intorno $V$ di $p_{0}$ dove $\pi$ è una funzione continua ben definita, e per ogni $\epsilon>0$, esiste un intorno $U_{\epsilon}$ di $p_{0}$ per il quale per ogni $p\in K$ e per ogni $q\in U_{\epsilon}$: $\displaystyle d(p,q)\leq(1+\epsilon)d(p,\pi(q))$ ###### Proof. Consideriamo un intorno $V_{1}$ di $p_{0}$ dove siano definite le coordinate di Fermi per la sottovarietà regolare $\partial K$, e in particolare consideriamo un sistema di coordinate per cui la metrica nel punto $p_{0}$ assume la forma euclidea standard, cioè $\displaystyle g_{ij}(p_{0})=\delta_{ij}$ Definiamo $K_{1}=K\cap\overline{V}_{1}$ e $K_{2}\equiv K\cap V_{1}^{C}$. L’insieme definito da $\displaystyle A=\\{p\in R\ t.c.\ d(p,K_{1})<d(p,K_{2})$ è un’insieme aperto per la continuità della funzione distanza, non vuoto perché $p_{0}\in A$. Se $p\in A$, sicuramente il punto (o i punti) $\pi(p)$ sono da cercare sono nell’insieme $K_{1}$. Consideriamo un cilindro $V$ rispetto all’ultima coordinata 101010quindi un insieme della forma $B\times(-\epsilon,\epsilon)$ con $B$ aperto contenente la proiezione $p_{0}$ contenuto nell’aperto $A\cap V_{1}$. Data la particolare forma delle coordinate di Fermi, è facile dimostrare che se $q\in V$, allora il punto $\pi(q)$ è il punto di coordinate $\displaystyle y(\pi(q))=0,\ \ x^{i}(\pi(q))=x^{i}(q)\ \ \forall i=1\cdots m-1$ Infatti supponiamo per assurdo che $y(\pi(q))\neq 0$ 111111dove l’indice $1$ può essere sostituito da qualsiasi altro indice, e sia $\gamma$ la geodetica che unisce $\gamma(0)=\pi(q)$ e $\gamma(1)=q$. La curva: $\displaystyle\tilde{\gamma}(t)=(x^{1}(q),\cdots,x^{m-1}(q),y(\gamma(t)))$ è una curva che unisce i punti $(x^{1}(q),\cdots,m^{m-1}(q),0)$ con $q$, e la sua lunghezza è $\displaystyle L(\tilde{\gamma})=\int_{0}^{1}g_{ij}\dot{\tilde{\gamma}}^{i}(t),\dot{\tilde{\gamma}}^{j}(t)dt=\int_{0}^{1}(\dot{\gamma}^{m}(t))^{2}dt<\int_{0}^{1}g_{ij}\dot{\gamma}^{i}(t),\dot{\gamma}^{j}(t)dt=L(\gamma)$ Per dimostrare la seconda parte della proposizione, ragioniamo sulle prime $m-1$ righe e colonne della matrice $g_{ij}$, cioè sulla matrice $A_{ij}$ per $i,j=1\cdots m-1$. Possiamo sempre scrivere che per ogni punto di $V$: $\displaystyle A_{ij}(p)=A_{ij}(p_{0})+\alpha_{ij}(p)=\delta_{ij}+\alpha_{ij}(p)$ Per continuità della metrica, per ogni $\epsilon>0$, esiste un intorno $U^{\prime}_{\epsilon}(p_{0})$ contenuto in $V$ per il quale il modulo di tutti gli autovalori di $\alpha_{ij}(p)$ è minore di $\epsilon$. Sia $U^{\prime\prime}_{\epsilon}(p_{0})$ un cilindro rispetto all’ultima coordinata della varietà contenuto in $U^{\prime}_{\epsilon}$, e $U_{\epsilon}$ un intorno di $p_{0}$ geodeticamente convesso contenuto in $U^{\prime\prime}_{\epsilon}$. Siano $p\in U_{\epsilon}\cap K$ e $q\in U_{\epsilon}$. Se $q\in K$, non c’è niente da dimostrare in quanto $q=\pi(q)$, quindi consideriamo solo il caso $q\not\in K$. Sia $\gamma$ una geodetica normalizzata (cioè con $g_{ij}(\gamma(t))\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)=1$) che congiunge $p$ a $q$. Questa geodetica è necessariamente contenuta in $U_{\epsilon}$. Se chiamiamo $l=d(p,q)$, sappiamo che $\displaystyle l=\int_{0}^{l}\sqrt{g_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)}dt=\int_{0}^{l}\sqrt{A_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}^{m}(t)\right|^{2}}dt$ Sia $\eta(t)$ una curva che unisce i punti $p$ e $\pi(q)$ descritta da: $\displaystyle\eta(t)=\left(\gamma^{1}(t),\cdots,\gamma^{m-1}(t),\frac{y(p)}{y(p)-y(q)}(\gamma^{m}(t)-y(q))\right)$ questa curva è contenuta nell’insieme $U^{\prime\prime}_{\epsilon}$ e anche se non è necessariamente una geodetica, vale che: $\displaystyle d(\pi(q),p)\leq\int_{0}^{l}\sqrt{g_{ij}|_{\eta(t)}\dot{\eta}^{i}(t)\dot{\eta}^{j}(t)}dt=$ $\displaystyle=\int_{0}^{l}\sqrt{A_{ij}|_{\eta(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\kappa^{2}\left|\dot{\gamma}(t)\right|^{2}}dt\leq\int_{0}^{l}\sqrt{A_{ij}|_{\eta(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}(t)\right|^{2}}dt$ dove $\kappa=\left|\frac{y(p)}{y(p)-y(q)}\right|<1$ grazie al fatto che $p\in K$, quindi $y(p)\leq 0$, e $q\not\in K$, quindi $y(q)>0$. Consideriamo inoltre che, pochè $\gamma$ è normalizzata: $\displaystyle A_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)\leq 1$ Se indichiamo $\left\|\dot{\gamma}(t)\right\|_{m-1}=\sum_{i,j\leq m-1}\delta_{ij}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)$ ricaviamo che per ogni $t$: $\displaystyle\delta_{ij}\dot{\gamma}^{i}(t)\dot{\gamma}^{i}(t)+\alpha|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)\leq 1$ $\displaystyle\left\|\dot{\gamma}(t)\right\|_{m-1}\leq 1-\alpha|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)\leq 1+\epsilon\left\|\dot{\gamma}(t)\right\|_{m-1}$ $\displaystyle\left\|\dot{\gamma}(t)\right\|_{m-1}\leq\frac{1}{1-\epsilon}$ dove abbiamo utilizzato il fatto che il modulo di tutti gli autovalori di tutte le matrici $\alpha_{ij}$ in $U^{\prime}_{\epsilon}$ è minore di $\epsilon$. Osserviamo che questa stima è indipendente dalla scelta di $p$ e $q$. Grazie alla definizione delle matrici $\alpha_{ij}$ abbiamo anche che: $\displaystyle A_{ij}|_{\eta(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)=[A_{ij}|_{\gamma(t)}-\alpha_{ij}|_{\gamma(t)}+\alpha_{ij}|_{\eta(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)$ quindi: $\displaystyle[A_{ij}|_{\eta(t)}-A_{ij}|_{\gamma(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)=[-\alpha_{ij}|_{\gamma(t)}+\alpha_{ij}|_{\eta(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)$ $\displaystyle\left|[A_{ij}|_{\eta(t)}-A_{ij}|_{\gamma(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)\right|\leq 2\epsilon\left\|\dot{\gamma}(t)\right\|_{m-1}\leq\frac{2\epsilon}{1-\epsilon}$ Grazie a quest’ultima disuguaglianza e ricordando che $\sqrt{a+b}\leq\sqrt{a}+\sqrt{\left|b\right|}$ per ogni coppia di numeri reali tali che $a\geq 0,\ a+b\geq 0$, possiamo concludere che: $\displaystyle d(\pi(q),p)\leq\int_{0}^{l}\sqrt{A_{ij}|_{\eta(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}(t)\right|^{2}}dt=$ $\displaystyle=\int_{0}^{l}\sqrt{A_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}(t)\right|^{2}+[A_{ij}|_{\eta(t)}-A_{ij}|_{\gamma(t)}]\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)}dt\leq$ $\displaystyle\leq\int_{0}^{l}\sqrt{A_{ij}|_{\gamma(t)}\dot{\gamma}^{i}(t)\dot{\gamma}^{j}(t)+\left|\dot{\gamma}(t)\right|^{2}}dt+\int_{0}^{l}\sqrt{\frac{2\epsilon}{1-\epsilon}}dt\leq l\left(1+\sqrt{\frac{2\epsilon}{1-\epsilon}}\right)$ ricordiamo che $l=d(p,q)$. La tesi segue dal fatto che $\displaystyle f(\epsilon)=\sqrt{\frac{2\epsilon}{1-\epsilon}}$ è una funzione continua per $0\leq\epsilon<1$ e $f(0)=0$. Osserviamo che $\epsilon$ non è stata scelta in funzione di $p$ e $q$, ma solo in funzione dell’intorno $U_{\epsilon}$, quindi la stima vale per ogni coppia di punti $p$ e $q$ in $U_{\epsilon}(p_{0})$. ∎ Osserviamo che questa proposizione vale anche in un altro caso, nel caso in cui $K$ sia una sottovarietà di $R$ con bordo regolare. La dimostrazione appena conclusa non tiene conto della possibilità che $K$ abbia un bordo regolare, ma può essere adattata facilmente. Basta considerare che se $\pi(q)\in\partial K$ (bordo inteso come bordo della sottovarietà regolare), allora la geodetica che unisce $p$ e $\pi(q)$ è di certo più corta della geodetica che unisce $p$ alla proiezione di $q$ sul piano contenente la parte di bordo considerata. Riportiamo quindi la proposizione lasciando i dettagli della dimostrazione al lettore. ###### Proposizione 1.16. Data $K$ sottovarietà regolare di $R$ possibilmente con bordo liscio, definiamo per ogni $q\in R$: $\displaystyle\pi(q)\ \ \ t.c.\ \ \ d(\pi(q),q)=\inf\\{d(q,p)\ t.c.\ p\in K\\}$ Per ogni $p_{0}\in K$, esiste un intorno $V$ di $p_{0}$ dove $\pi$ è una funzione continua ben definita, e per ogni $\epsilon>0$, esiste un intorno $U_{\epsilon}$ di $p_{0}$ per il quale per ogni $p\in K$ e per ogni $q\in U_{\epsilon}$: $\displaystyle d(p,q)\leq(1+\epsilon)d(p,\pi(q))$ #### 1.1.6 Esaustioni regolari In questa sezione ci occupiamo di dimostrare l’esistenza di esaustioni regolari per una qualsiasi varietà differenziale liscia connessa $M$ 121212in realtà come si può dedurre dalle dimostrazioni, un’esaustione esiste per ogni varietà $M$, e per ogni varietà esiste un’esaustione regolare quanto lo è la varietà. Per prima cosa dimostriamo l’esistenza di esaustioni non regolari, poi sfruttiamo la densità delle funzioni lisce nelle funzioni continue su una varietà per ottenere l’esistenza di esaustioni regolari. Come risulta evidente dalle definizioni, ha senso procurarsi un’esaustione di $M$ solo nel caso $M$ non compatta. Oltre che ad esaustioni regolari, avremo bisogno anche di esaustioni che si comportano bene in relazione ad una sottovarietà di $R$. In particolare, vogliamo dimostrare che per ogni sottovarietà regolare con bordo $S\subset R$ ($dim(S)=dim(R)$), esiste un’esausione regolare $K_{n}$ tale che $C=\partial K_{n}\cap S$ è ancora una sottovarietà regolare di codimensione $1$ con bordo. Ovviamente cominciamo con le definizioni del caso. ###### Definizione 1.17. Data una varietà differenziale non compatta $M$, diciamo che $\\{K_{n}\\}$ è un’esaustione di $M$ se $K_{n}$ sono insiemi aperti connessi relativamente compatti tali che 1. 1. $\overline{K_{n}}\Subset K_{n+1}$ 2. 2. $M=\bigcup_{n}K_{n}$ Spesso nella definzione si confondono $K_{n}$ e $\overline{K_{n}}$. Diciamo che questa esaustione è regolare se i bordi $\partial K_{n}$ sono $C^{\infty}$, o equivalentemente è una sottovarietà di $M$. Dimostriamo per prima cosa l’esistenza di un’esaustione. ###### Proposizione 1.18. Ogni varietà differenziale connessa $M$ ammette un’esaustione $K_{n}$. ###### Proof. Per ogni punto $p\in M$ consideriamo un intorno aperto connesso relativamente compatto $U(p)$ 131313che esiste poiché $M$ è localmente euclidea. Al variare di $p\in M$, $U(p)$ è un ricoprimento aperto di $M$, quindi poiché $M$ è II numerabile, esiste un sottoricoprimento numerabile $U(p_{n})\equiv U_{n}$. Costruiamo per induzione l’esaustione $K_{n}$. Per prima cosa rinumeriamo gli insiemi $U_{n}$ in modo che per ogni $N$, $S_{N}\equiv\bigcup_{n=1}^{N}U_{n}$ sia un insieme connesso. Questo è sicuramente vero se $N=1$. Supponiamo che sia vero per $N$. Dato che $M=\cup_{n}U_{n}$, se non esistesse nessun $\bar{n}\geq N$ tale che $\displaystyle S_{N}\cap U_{\bar{n}}\neq\emptyset$ allora $M$ sarebbe disconnessa perché $S_{N}\bigcap\left(\bigcup_{n=N+1}^{\infty}U_{n}\right)=\emptyset$. Rinominando $\bar{n}=N+1$, si ottiene che $S_{N+1}$ è l’unione di insiemi connessi a intersezione non vuota, quindi è connesso. Consideriamo ora $K_{1}=U_{1}$, e costruiamo per induzione una successione a valori interi strettamente crescente $k(n)$ tale che 1. 1. $k(1)=1$ 2. 2. per ogni $n$, $\bigcup_{j=1}^{k(n)}\overline{U_{j}}\Subset\bigcup_{j=1}^{k(n+1)}U_{j}$ Se definiamo $K_{n}\equiv\bigcup_{j=1}^{k(n)}U_{j}$, il gioco è fatto. Questo è possibile perché $\overline{K_{n}}=\bigcup_{j=1}^{k(n)}\overline{U_{j}}$ è un insieme compatto (unione finita di compatti), ricoperto dagli aperti $U_{j}$, quindi esiste un sottoricoprimento finito di $U_{j}$ che ricopre $K_{n}$. Se chiamiamo $k(n+1)$ l’indice massimo di questo sottoricoprimento, la successione $k(n)$ ha le proprietà desiderate. ∎ Grazie all’esistenza di questa esaustione, possiamo costruire una funzione di esaustione $f:M\to\mathbb{R}$ in questo modo: ###### Proposizione 1.19. Per ogni varietà $M$, esiste una funzione continua positiva propria $f:M\to\mathbb{R}$ 141414ricordiamo che una funzione è detta propria se la controimmagine di un qualsiasi compatto è compatta. ###### Proof. Definiamo $f(\overline{K_{1}})=1$, e $f(\partial K_{n})=n$. Per ogni insieme $\overline{K_{n+1}}\setminus K_{n}$ ($n=1,2,\cdots$) estendiamo $f$ a una funzione continua $n\leq f\leq n+1$ grazie al lemma di Urysohn 151515vedi teorema 4.1 pag 146 di [D]. Ricordiamo che ogni spazio metrizzabile (e quindi ogni varietà o ogni suo sottoinsieme) è normale. In questo modo otteniamo una funzione continua $f:M\to\mathbb{R}$ tale che $\overline{K_{n}}\Subset f^{-1}(-\infty,n]\Subset K_{n+1}$, il che garantisce che $f$ sia propria. ∎ Osserviamo che ci sono altri modi di definire una funzione continua positiva propria su $M$. Ad esempio se $M$ è riemanniana geodeticamente completa (e per ogni varietà è possibile definire una metrica $g$ che la renda completa), la funzione distanza da un punto soddisfa queste caratteristiche. Trovata questa funzione, siamo in grado di dimostrare che ###### Proposizione 1.20. Ogni varietà liscia $M$ ammette un’esaustione regolare $C_{n}$. ###### Proof. Data un’esaustione $K_{n}$ e definita $f$ come nella proposizione precedente, sia $h\in C^{\infty}(M,\mathbb{R})$ tale che $\left\|h-f\right\|_{\infty}<1/4$ 161616questo è possibile per densità delle funzioni lisce nello spazio delle funzioni continue con la norma uniforme, vedi teorema 2.6 pag. 49 di [H3], oppure è possibile dimostrare questo fatto adattando la dimostrazione della proposizione 3.27 a pagina 3.27. Per ogni $n$ consideriamo l’insieme $\Delta_{n}=(n+1/4,n+3/4)$ aperto in $\mathbb{R}$. Grazie al teorema di Sard, sappiamo che esiste un numero reale $x_{n}\in\Delta_{n}$ che sia valore regolare di $h$. Grazie alle proposizioni di questa sezione, questo garantisce che l’insieme $A_{n}\equiv h^{-1}(-\infty,x_{n})$ sia un aperto di $M$ con bordo $\partial A_{n}=h^{-1}(x_{n})$ sottovarietà regolare di $M$. Per definizione di $h$, si verifica facilmente che $K_{n}\subset A_{n}\subset K_{n+1}$, quindi $A_{n}$ è relativamente compatto, però non è detto che $A_{n}$ sia connesso. Per questo motivo definiamo $C_{n}$ la componente connessa di $A_{n}$ che contiene $K_{n}$. Poiché $\partial C_{n}$ è una componente connessa di $\partial A_{n}$, mantiene le sue proprietà di regolarità. Infine, dato che per ogni $n$ $K_{n}\subset C_{n}\subset K_{n+1}$, vale che 1. 1. $\overline{C}_{n}\Subset C_{n+1}$ 2. 2. $M=\bigcup_{n}C_{n}$ ∎ Oltre che ad esaustioni regolari, avremo bisogno anche di esaustioni che si comportano bene in relazione ad una sottovarietà di $R$. In particolare, vogliamo dimostrare che per ogni sottovarietà regolare con bordo $S\subset R$ ($dim(S)=dim(R)$), esiste un’esausione regolare $K_{n}$ tale che $C=\partial K_{n}\cap S$ è ancora una sottovarietà regolare di codimensione $1$ con bordo. A questo scopo, assumiamo che $S$ sia chiusa ma non compatta (se $S$ compatta, $S\subset K_{n}^{\circ}$ definitivamente). Dato che $C^{\circ}=\partial K_{n}\cap S^{\circ}$ è di certo una sottovarietà regolare, resta da dimostrare solo che anche il bordo $\partial C=\partial K_{n}\cap\partial S$ è regolare. Riportiamo ora un risultato riguardo alla regolarità dell’intersezione tra due sottovarietà regolari. ###### Definizione 1.21. Data $R$ varietà riemanniana e $S,M$ sue sottovarietà regolari, diciamo che $M$ è trasversale rispetto a $S$ se e solo se per ogni $x\in S\cap M$: $\displaystyle T_{x}(M)+T_{x}(S)=T_{x}(R)$ ###### Proposizione 1.22. Data $R$ varietà riemanniana e date $S,M$ sue sottovarietà regolari senza bordo, se $S$ ed $M$ sono trasversali, allora $S\cap M$ è ancora una sottovarietà regolare di $R$ con: $\displaystyle codim(S\cap M)=codim(S)+codim(M)$ dove $codim(S)$ indica la codimensione della sottovarietà, cioè $codim(S)\equiv dim(R)-dim(S)$ ###### Proof. La dimostrazione di questo risultato ed alcuni approfondimenti sulla teoria della trasversalità possono essere trovati su [GP]. In particolare questo teorema è il teorema pag. 30 del cap. 1. ∎ Con questo risultato possiamo dimostrare che: ###### Proposizione 1.23. Data una sottovarietà regolare $S\subset R$ della stessa dimensione di $R$, chiusa e con bordo regolare $\partial S$, esiste un’esaustione regolare $K_{n}$ tale che $\partial K_{n}\cap S$ è una sottovarietà regolare di codimensione $1$ con bordo. ###### Proof. Come affermato sopra, è sufficiente dimostrare che $\partial K_{n}\cap\partial S$ sia una sottovarietà regolare di codimensione $2$ (è il bordo della sottovarietà $\partial K_{m}\cap S$). Consideriamo una funzione di esaustione liscia positiva $h$ come nella proposizione 1.20. La funzione $\bar{h}\equiv h|_{\partial S}$ è anch’essa una funzione liscia propria e positiva. Sia $\\{A_{n},\phi_{n}\\}$ una successione di aperti relativamente compatti di $R$ su cui siano definite le coordinate di Fermi $\phi_{n}$ relative a $\partial S$ tali che $\displaystyle\partial S\subset\bigcup_{n}A_{n}$ Chiamiamo le funzioni coordinate di Fermi $(x_{1},\cdots,x_{m-1},y)$ e definiamo delle funzioni $f_{n}$ in modo che $f_{n}:A_{n}\to\mathbb{R}$, $f_{n}|_{\partial S}=\bar{h}$ e che rispetto alle carte $\phi_{n}$ si abbia: $\displaystyle\tilde{f_{n}}(x,y)\equiv f_{n}\circ\phi_{n}^{-1}(x,y)=f_{n}(x,0)=\bar{h}(x)$ Cioè definiamo $f_{n}$ in modo che siano costanti sulle geodetiche perpendicolari a $\partial S$. Consideriamo ora il ricoprimento di aperti di $R$ dato da $\\{A_{n},(\partial S)^{C}\\}$, e sia $\\{\lambda_{n},\lambda\\}$ una partizione dell’unità subordinata a questo ricoprimento. Definiamo: $\displaystyle F(p)\equiv\sum_{n}f_{n}(p)\lambda_{n}(p)+h(p)\lambda(p)$ Osserviamo subito che $F\in C^{\infty}(R,\mathbb{R}^{+})$. Inoltre vale che per ogni punto $p\in\partial S$, $\nabla(F)|_{p}\in T_{p}(S)$. Infatti, sia $p\in A_{n}\cap S$, indichiamo le sue coordinate rispetto alla carta $\phi_{n}$ come $p=(\bar{x},0)$ 171717$\bar{x}$ rappresenta il vettore delle prime $m-1$ coordinate : $\displaystyle\left.\frac{\partial F}{\partial y}\right|_{p}=\frac{\partial}{\partial t}F(\bar{x},t)=\frac{\partial}{\partial t}\left(\sum_{n}f_{n}(\bar{x},t)\lambda_{n}(\bar{x},t)\right)=$ $\displaystyle=\frac{\partial}{\partial t}\left(\sum_{n}h(\bar{x})\lambda_{n}(\bar{x},t)\right)=h(\bar{x})\frac{\partial}{\partial t}\sum_{n}\lambda_{n}(\bar{x},t)=0$ dove abbiamo sfruttato il fatto che in un intorno di $\partial S$, la funzione $\lambda=0$, quindi $\sum_{n}\lambda_{n}(\bar{x},t)=1$. Questo garantische che l’ultima componente del differenziale sia nulla, e data la particolare forma della metrica, anche l’ultima componente di $\nabla(F)|_{p}$ è nulla. Dato che in questa carta l’insieme $\partial S$ è il piano avente ultima coordinata nulla, questo dimostra che $\nabla(F)|_{p}\in T_{p}(\partial S)$. Consideriamo ora un valore regolare $c\in\mathbb{R}$ di $F$ 181818grazie al teorema di Sard, i valori regolari sono densi in $\mathbb{R}$. Allora l’insieme $F^{-1}(c)$ è una sottovarietà regolare $F_{c}$ di $R$. Ricordiamo che per queste sottovarietà: $\displaystyle T_{p}(F_{c})=\\{\nabla(F)|_{p}\\}^{\perp}$ Dato che $\nabla(F)|_{p}\in T_{p}(\partial S)$: $\displaystyle T_{p}(F_{c})+T_{p}(\partial S)\supset\\{\nabla(F)|_{p}\\}^{\perp}+\\{\nabla(F)|_{p}\\}=T_{p}(R)$ quindi le sottovarietà $F_{c}$ e $\partial S$ sono trasversali, e la loro intersezione è regolare. Se verifichiamo anche che la funzione $F$ è una funzione propria, allora possiamo scegliere una qualsiasi successione $c_{n}\nearrow\infty$ di valori regolari di $F$ e definire $\displaystyle K_{n}\equiv F^{-1}[0,c_{n}]$ per ottenere la tesi. Per dimostrare che $F$ è propria, dimostriamo che per ogni $x\in R^{+}$, esiste un compatto $C_{x}$ tale che $\displaystyle p\not\in C_{x}\ \Rightarrow\ F(p)>x$ Sia $K=h^{-1}[0,x]$, insieme compatto per il fatto che $h$ è propria. Per locale finitezza delle partizioni dell’unità, esiste un numero finito di indici $n$ per il quale i supporti di $\lambda_{n}$ intersecano l’insieme $K$. Indichiamo con $I_{x}$ l’insieme finito di questi indici e consideriamo l’insieme compatto: $\displaystyle C_{x}=K\bigcup_{n\in I_{x}}\overline{A_{n}}$ Supponiamo che $p\not\in C_{x}$. Allora $h(p)>x$, e anche $f_{n}(p)=h(\pi(p))>x$, dove $\pi(p)$ indica la proiezione di $p$ su $\partial S$, cioè il punto che nelle coordinate di Fermi è caratterizzato da: $\displaystyle\bar{x}(\pi(p))=\bar{x}(p),\ \ y(\pi(p))=0$ Infatti, $p\not\in C_{x}\ \Rightarrow\ \pi(p)\not\in K$. Supponiamo per assurdo il contrario. Dato che deve esistere $\bar{n}$ per cui $p\in A_{\bar{n}}$, e dato che $\pi(p)\in A_{\bar{n}}$ per costruzione delle coordinate di Fermi, allora se $\pi(p)\in K$ necessariamente $\bar{n}\in I_{x}$, assurdo poiché abbiamo assunto $p\not\in C_{x}$. Ricordando la definizione di $F$, è immediato verificare che $F(p)>x$. ∎ Ripetendo questa costruzione per una successione di sottovarietà $\\{S_{n}\\}$ con bordi disgiunti possiamo ottenere: ###### Proposizione 1.24. Sia $\\{S_{n}\\}$ una successione di sottovarietà di $R$ della sua stessa dimensione con bordi $\partial S_{n}$ lisci disgiunti tra loro. Allora esiste un’esaustione regolare $K_{m}$ tale che per ogni $n$, $m$ l’insieme $\partial K_{m}\cap S_{n}$ è una sottovarietà di $R$ $m-1$ dimensionale con bordo liscio. ###### Proof. L’idea della dimostrazione è la stessa della dimostrazione precedente. Scegliamo per ogni sottovarietà $\partial S_{n}$ un insieme numerabile di suoi intorni $\\{I_{n}^{m}\\}_{m=1}^{\infty}$ che ammettono coordinate di Fermi, restringendoli in modo che questi intorni siano disgiunti dagli altri bordi $\partial S_{k}$ (con $k\neq n$). Fissiamo inoltre una partizione dell’unità subordinata al ricoprimento di aperti $\left\\{\\{I_{n}^{m}\\}_{n,m=1}^{\infty};\ \bigcap_{n}\partial S_{n}^{C}\right\\}$ Ripetendo una costruzione del tutto analoga alla precedente otteniamo la tesi. ∎ #### 1.1.7 Insiemi chiusi e funzioni lisce Lo scopo di questo paragrafo è ottenere l’osservazione 1.27, che sarà utile nel seguito del lavoro. ###### Lemma 1.25. Dato un insieme chiuso $C\subset\mathbb{R}^{n}$, esiste una funzione $f:\mathbb{R}^{n}\to\mathbb{R}$ liscia positiva tale che $C=f^{-1}(0)$. ###### Proof. Grazie al fatto che $\mathbb{R}^{n}$ è uno spazio metrico, è facile trovare una funzione $g:\mathbb{R}^{n}\to\mathbb{R}$ continua per cui $C=g^{-1}(0)$, basta considerare infatti la funzione $\displaystyle g(x)=d(x,C)\equiv\inf_{y\in C}d(x,y)$ ha le caratteristiche cercate. Consideriamo ora gli insiemi $\displaystyle C_{\epsilon}\equiv\\{x\in\mathbb{R}^{n}\ t.c.\ d(x,C)\leq\epsilon\\}$ data la continuità della funzione $d(\cdot,C)$, tutti questi insiemi sono insiemi chiusi, quindi esistono funzioni $g_{\epsilon}:\mathbb{R}^{n}\to\mathbb{R}$ tali che $g_{\epsilon}^{-1}(0)=C_{\epsilon}$. Notiamo che per ogni $y\in C$, $B_{\epsilon}(y)\subset C_{\epsilon}$, quindi in particolare $g(B_{\epsilon(y)})=0$ per ogni $y\in C$, e che $\displaystyle C=\bigcap_{\epsilon>0}C_{\epsilon}=\bigcap_{n=1}^{\infty}C_{1/n}$ Grazie al lemma 1.40, esiste un nucleo di convoluzione $\Theta_{\epsilon}$ con supporto compatto contenuto in $B_{\epsilon}(0)$, e grazie al lemma 1.43, la funzione $\displaystyle g^{\prime}_{\epsilon}\equiv g_{\epsilon}\ast\Theta_{\epsilon}$ è una funzione liscia da $\mathbb{R}^{n}$ a $\mathbb{R}$ per la quale $g^{\prime}_{\epsilon}(y)=0$ per ogni $y\in C$ e $g^{\prime}_{\epsilon}(x)>0$ per ogni $x\in\mathbb{R}^{n}$ tale che $d(x,C)>\epsilon$. Definiamo la funzione $\displaystyle f(x)\equiv\sum_{n=1}^{\infty}\frac{g^{\prime}_{1/n}(x)}{A_{n}}\frac{1}{2^{n}}$ dove $A_{n}=\max_{i=1}^{n}\left\|D^{i}g^{\prime}_{1/n}\right\|_{\infty,B_{n}(0)}$. Grazie a questa definizione è facile verificare che la serie che definisce $f(x)$ converge localmente uniformemente in $\mathbb{R}^{n}$ assieme alla serie delle derivate di qualunque ordine, quindi la funzione $f$ è una funzione liscia. Ovviamente $f$ è positiva, e dato che per ogni $n$, $g^{\prime}_{1/n}(y)=0$ per ogni $y\in C$, anche $f(y)=0$ per ogni $y\in C$. Se $y\not\in C$, grazie al fatto che $C$ è chiuso, esiste $\bar{n}$ sufficientemente grande per cui $y\not\in C_{1/\bar{n}}$, quindi dalle considerazioni precedenti abbiamo che $g^{\prime}_{1/\bar{n}}(y)>0$, quindi anche $f(y)>0$. Da cui la tesi. ∎ ###### Osservazione 1.26. Utilizzando le partizioni dell’unità, è possibile estendere questo lemma a una varietà differenziale $M$ qualsiasi 191919se vogliamo $f$ liscia, la varietà $M$ deve avere una struttura $C^{\infty}$. ###### Proof. Sia $M$ una varietà differenziale qualsiasi e sia $\lambda_{n}$ una partizione dell’unità di $M$ subordinata a un ricoprimento di carte locali $(U_{n},\phi_{n})$. Sia $f_{n}$ la funzione descritta nel lemma precedente relativa all’insieme chiuso in $\mathbb{R}^{m}$ $\phi_{n}(C\cap\leavevmode\nobreak\ supp(\lambda_{n}))$. Allora la funzione $\displaystyle f(p)\equiv\sum_{n}f_{n}\circ\phi_{n}(p)\cdot\lambda_{n}(p)$ ha le caratteristiche cercate. Infatti dato che la somma è localmente finita, $f$ è una funzione liscia. La positività di $f$ è ovvia conseguenza della positività delle funzioni $f_{n}$ e $\lambda_{n}$; se $p\in C$, allora $f_{n}(\phi_{n}(p))\cdot\lambda_{n}(p)=0$, mentre se $p\not\in C$, esiste $\bar{n}$ tale che $\lambda_{\bar{n}}(p)>0$ 202020quindi in particolare $p\in supp(\lambda_{\bar{n}})$ e quindi $f_{n}(\phi_{n}(p))\cdot\lambda_{\bar{n}}(p)>0$. ∎ ###### Osservazione 1.27. Grazie alla precedente osservazione e al teorema di Sard, possiamo concludere che per ogni insieme $C$ chiuso in una varietà $R$, esiste una successione di insiemi aperti con bordo liscio $A_{n}\subset A_{n-1}$ in $R$ tali che $C=\cap_{n}A_{n}$. Sia infatti $f$ una funzione con le caratteristiche appena descritte. Consideriamo una successione di valori regolari $a_{n}$ di $f$ tali che $a_{n}\searrow 0$, allora è facile verificare che gli insiemi $\displaystyle A_{n}=f^{-1}[0,a_{n})$ soddisfano le proprietà richieste. Osserviamo inoltre che se $C$ è compatto, gli insiemi possono essere scelti relativamente compatti (lasciamo al lettore i facili dettagli di questa dimostrazione). ###### Osservazione 1.28. Sia $\Omega$ un aperto di $R$. Allora esiste una successione di aperti $A_{n}$ con bordo liscio tali che: $\displaystyle A_{n}\subset\Omega\ \ \ A_{n}\subset A_{n+1}\ \ \ \bigcup_{n}A_{n}=\Omega$ Se $\Omega$ è relativamente compatto, anche gli insiemi $A_{n}$ lo sono. ###### Proof. Sia $f$ una funzione liscia positiva tale che $f^{-1}(0)=\Omega^{C}$, e sia $\epsilon_{n}\searrow 0$ una successione di valori regolari per la funzione $f$. È facile verificare che gli insiemi $\displaystyle A_{n}\equiv f^{-1}(\epsilon_{n},\infty)$ hanno le caratteristiche desiderate. ∎ ### 1.2 Funzioni assolutamente continue In questa sezione riportiamo alcuni risultati sulle funzioni assolutamente continue senza dimostrazione. Come referenza consigliamo il paragrafo 5.4 di [R1]. ###### Definizione 1.29. Una funzione $f:U\to\mathbb{R}$ (dove $U\subset\mathbb{R}$ è un aperto) è detta assolutamente continua se per ogni $\epsilon>0$, esiste $\delta>0$ tale che per ogni scelta di $\\{x_{i},x_{i}^{\prime}\\}\subset U$ 212121in questa definizione è equivalente chiedere che $i$ vari su un insieme finito o numerabile di indici tali che $x_{i}^{\prime}>x_{i}>x_{i-1}^{\prime}$ e tali che $\sum_{i}(x_{i}^{\prime}-x_{i})<\delta$, allora: $\displaystyle\sum_{i}\left|f(x_{i}^{\prime})-f(x_{i})\right|<\epsilon$ ###### Proposizione 1.30. Se una funzione $f$ è assolutamente continua, allora la sua derivata esiste finita quasi ovunque ed è Lebesgue integrabile sui compatti. Inoltre $f$ è l’integrale della sua derivata se e solo se è assolutamente continua. Tutte le funzioni di Lipschitz sono assolutamente continue. ###### Proposizione 1.31. Date due funzioni assolutamente continue limitate $f$ e $g$, il loro prodotto è assolutamente continuo. ###### Proof. La dimostrazione segue facilmente dalla disuguaglianza: $\displaystyle\left|f(x)g(x)-f(y)g(y)\right|\leq\left|f(x)\cdot(g(x)-g(y))\right|+\left|g(y)\cdot(f(x)-f(y))\right|\leq$ $\displaystyle\leq\max\\{\left\|f\right\|_{\infty},\left\|g\right\|_{\infty}\\}\cdot(\left|f(x)-f(y)\right|+\left|g(x)-g(y)\right|)$ ∎ ### 1.3 Spazi $\mathcal{L}^{2}$ di forme In teoria dell’intergrazione sono famosi gli spazi $L^{p}$, spazi di funzioni misurabili il cui modulo elevato alla $p$ è integrabile sullo spazio. Per un’introduzione su questa teoria consigliamo [R4] (capitolo 3). I risultati che ci interessano sono comunque riportati in questa breve rassegna: ###### Definizione 1.32. Dato uno spazio di misura $(X,\mu)$ con $\mu$ misura positiva, definiamo per $1\leq p<\infty$ $L^{p}(X,\mu)$ come lo spazio delle funzioni $f:X\to\mathbb{R}$ (o anche $f:X\to\mathbb{C}$) misurabili 222222rispetto alla misura $\mu$ e all’algebra di Borel su $\mathbb{R}$ tali che $\displaystyle\left\|f\right\|_{p}^{p}\equiv\int_{X}\left|f\right|^{p}d\mu<\infty$ Se introduciamo la relazione di equivalenza $f\sim g$ se $f=g$ quasi ovunque, lo spazio quoziente $L^{p}/\sim$ con la norma $\left\|\cdot\right\|_{p}$, $L^{p}$ è uno spazio di Banach, e per $p=2$ uno spazio di Hilbert. ###### Proposizione 1.33. Se $f_{n}\to f$ rispetto a una qualsiasi norma $\left\|\cdot\right\|_{p}$, allora esiste una sottosuccesione $f_{n_{k}}$ che converge puntualmente quasi ovunque a $f$. Questo implica anche che se $f_{n}$ converge uniformemente a $f$ e $f_{n}$ converge in norma $p$ ad $h$, allora $f=h$ quasi ovunque. ###### Proposizione 1.34. Data una varietà Riemanniana $R$, gli spazi $L^{p}(R)$ sono separabili per $1\leq p<\infty$. ###### Proof. Grazie al teorema 3.14 pag 68 di [R4], sappiamo che l’insieme delle funzioni continue a supporto compatto in $R$ (che indicheremo $C_{C}(R)$ è denso nello spazio $L^{p}(R)$. Sia $K_{n}$ un’esaustione di $R$, allora: $C_{C}(R)=\bigcup_{n}C(K_{n})$ dato che $C(K_{n})$ dotato della norma del sup è separabile, e dato che la norma del sup è più forte della norma $L^{p}$ su insiemi compatti, allora $C(K_{n})$ è separabile anche nella norma $L^{p}$. Siano $D_{n}$ insiemi numerabili densi in $C(K_{n})$, e sia $D=\cup_{n}D_{n}$. Allora $D$ è numerabile e denso in $L^{p}(R)$, infatti data $f\in L^{p}(R)$ e dato $\epsilon>0$, esiste $g\in C_{C}(R)$ tale che $\left\|g-f\right\|_{p}<\epsilon$. Sia $K_{n}$ tale che $supp(g)\subset K_{n}$, allora esiste $h\in D_{n}\subset D$ tale che $\left\|g-h\right\|_{p}<\epsilon$, quindi: $\displaystyle\left\|f-h\right\|_{p}\leq\left\|f-g\right\|_{p}+\left\|g-h\right\|_{p}\leq 2\epsilon$ data l’arbitrarietà di $\epsilon$, si ottiene la tesi. ∎ Una facile generalizzazione di questi spazi sono gli spazi $L^{p}$ per forme su varietà Riemanniane. La teoria di questi spazi è del tutto analoga agli spazi $L^{p}$ di funzioni, quindi anche in questo caso non riporteremo tutte le dimostrazioni. Un riferimento su questo argomento può essere [S1], capitolo 7 232323in realtà questo libro si occupa solo di superfici riemanniane, ma in questo frangente non c’è nessuna sostanziale differenza con una varietà di dimensione generica. ###### Definizione 1.35. Data una varietà Riemanniana $(R,g)$, definiamo lo spazio $\mathcal{L}^{2}(R)$ come lo spazio delle 1-forme a quadrato integrabile su $R$. Sia cioè $\alpha\in T^{*}(R)$ una 1-forma (in coordinate locali $\alpha=\sum_{i=1}^{m}\alpha_{i}(x)dx^{i}$), $\alpha\in\mathcal{L}^{2}(R)$ se e solo se $\alpha_{i}$ sono tutte misurabili e $\int_{R}\left|\alpha\right|^{2}dV<\infty$ Ricordiamo che l’integrale su una varietà è definito tramite partizioni dell’unità. Sia $\\{\lambda_{n}\\}$ una partizione dell’unità di $R$ subordinata a un ricoprimento di aperti coordinati. Per definizione $\displaystyle\int_{R}fdV=\sum_{n=1}^{\infty}\int_{supp(\lambda_{n})}\lambda_{n}\cdot f\ dV=\sum_{n=1}^{\infty}\int_{\phi_{n}(supp(\lambda_{n}))}\tilde{\lambda}_{n}(x)\tilde{f}(x)\ \sqrt{\left|g\right|}dx^{1}\dots dx^{m}$ Ricoriamo anche che $\left|\tilde{\alpha}(x)\right|^{2}=g_{ij}(x)\tilde{\alpha}^{i}(x)\tilde{\alpha}^{j}(x)$, dove le funzioni con la tilde indicano le rappresentazioni in coordinate delle relative funzioni. Anche per questo spazio valgono gli analoghi delle proposizioni 1.33 e 1.34: ###### Proposizione 1.36. Lo spazio $\mathcal{L}^{2}(R)$ è uno spazio di Hilbert, e se $\alpha_{n}\to\alpha$ in norma, allora esiste una sottosuccesione $\alpha_{n_{k}}$ che converge puntualmente quasi ovunque ad $\alpha$. Inoltre $\mathcal{L}^{2}(R)$ è separabile. Anche se in questa tesi non toccheremo l’argomento, osserviamo che $\left|\alpha\right|^{2}dV=\alpha\ast\alpha$ dove $\ast$ indica l’operatore duale di Hodge. Osserviamo che l’integrale di Dirichlet di una funzione $\displaystyle\int_{R}\left|\nabla f\right|^{2}dV=\int_{R}\left|df\right|^{2}dV$ è la norma nello spazio $\mathcal{L}^{2}(R)$ della 1-forma $df$. ### 1.4 Derivazione sotto al segno d’integrale In questa sezione ripotiamo due lemmi che consentono sotto certe ipotesi di scambiare derivata e integrale. Il secondo lemma è una generalizzazione del primo, anche se più difficile da dimostrare. ###### Lemma 1.37 (Derivazione sotto al segno di integrale). Sia $U\subset\mathbb{R}$ un aperto e $(\Omega,\mu)$ uno spazio di misura. Supponiamo che $f:U\times\Omega\to\mathbb{R}$ abbia le proprietà: 1. 1. $f(x,\omega)$ è Lebesgue-integrabile per ogni $x\in U$ 2. 2. quasi ovunque rispetto a $\mu$ la funzione $\frac{\partial f(x,\omega)}{\partial x}$ esiste per ogni $x\in U$ 3. 3. esiste una funzione $g\in L^{1}(\Omega,\mu)$ tale che $\left|\frac{\partial f(x,\omega)}{\partial x}\right|\leq g(\omega)$ per ogni $x\in U$ Allora vale che: $\displaystyle\frac{d}{dx}\int_{\Omega}f(x,\omega)d\mu=\int_{\Omega}\frac{\partial}{\partial x}f(x,\omega)d\mu$ ###### Proof. La dimostrazione è una semplice applicazione del teorema di convergenza dominata (vedi teorema 1.34 pag. 26 di [R4]). ∎ Prima di enunciare il secondo lemma, riportiamo una proposizione che sarà utile per la sua dimostrazione ###### Proposizione 1.38. Se $f\in L^{1}(X,\mu)$, dove $(X,\mu)$ è uno spazio di misura positiva qualsiasi, allora per ogni $\epsilon>0$ esiste $\delta>0$ tale che $\displaystyle\mu(E)<\delta\Rightarrow\int_{E}\left|f\right|d\mu<\epsilon$ dove $E$ è un qualsiasi sottoinsieme misurabile di $(X,\mu)$. ###### Proof. Osserviamo che questo è il testo dell’esercizio 12 pag 32 di [R4]. Supponiamo per assurdo che esista $\epsilon>0$ tale che per ogni $n\in\mathbb{N}$ esiste $E_{n}$ con $\mu(E_{n})<1/n$ tale che $\int_{E_{n}}\left|f\right|\geq\epsilon$. Consideriamo la successione $\left|f\right|\chi_{E_{n}}$. Questa successione converge in misura a $0$, quindi per il teorema di convergenza dominata $\int_{E_{n}}\left|f\right|d\mu=\int_{X}f\chi_{E_{n}}d\mu\to 0$, assurdo. ∎ ###### Lemma 1.39 (Derivazione sotto al segno di integrale). Sia $U\subset\mathbb{R}$ un aperto e $(\Omega,\mu)$ uno spazio di misura. Supponiamo che $f:U\times\Omega\to\mathbb{R}$ abbia le proprietà: 1. 1. $f(x,\omega)$ è misurabile su $U\times\Omega$, e per quasi ogni $x\in U$ è integrabile su $\Omega$ 2. 2. quasi ovunque rispetto a $\mu$ la funzione $f(x,\omega)$ è assolutamente continua in $x$ 3. 3. $\displaystyle\int_{U}dx\int_{\Omega}d\mu\left|\frac{\partial f}{\partial x}(x,\omega)\right|<\infty$ Allora $F(x)\equiv\int_{\Omega}f(x,\omega)d\mu$ è assolutamente continua in $x$ e per quasi ogni $x$: $\displaystyle\frac{d}{dx}\int_{\Omega}f(x,\omega)d\mu=\int_{\Omega}\frac{\partial}{\partial x}f(x,\omega)d\mu$ ###### Proof. Per prima cosa dimostriamo che $F(x)$ è assolutamente continua. Grazie alla proposizione precedente (la 1.38), per ogni $\epsilon>0$ esiste $\delta>0$ per cui se $\lambda(E)<\delta$ 242424$\lambda$ indica la misura di Lebesgue su $\mathbb{R}$, allora $\displaystyle\int_{E}dx\int_{\Omega}d\mu\left|\frac{\partial f}{\partial x}(x,\omega)\right|<\epsilon$ Siano $\\{a^{i},b^{i}\\}$ tali che $b_{i}>a_{i}>b_{i-1}$ e $\sum_{i}(b_{i}-a_{i})<\delta$, e chiamiamo $E=\cup_{i}(a^{i},b^{i})$. Allora $\displaystyle\sum_{i}\left|F(b^{i})-F(a^{i})\right|=\sum_{i}\left|\int_{\Omega}f(b^{i},\omega)-f(a^{i},\omega)d\mu\right|=$ $\displaystyle=\sum_{i}\left|\int_{\Omega}\int_{a^{i}}^{b^{i}}\frac{\partial f}{\partial x}(x,\omega)\ dx\ d\mu\right|\leq\int_{E}dx\int_{\Omega}d\mu\left|\frac{\partial f}{\partial x}(x,\omega)\right|<\epsilon$ quindi $F$ è assolutamente continua in $x$. Questo significa che esiste quasi ovunque $F^{\prime}(x)\equiv\partial F(x)/\partial x$ e che $F(b)-F(a)=\int_{a}^{b}F^{\prime}(t)dt$ ogni volta che $[a,b]\subset U$. Ma $\displaystyle F(b)-F(a)=\int_{\Omega}f(b,\omega)-f(a,\omega)\ d\mu=\int_{\Omega}\int_{a}^{b}\frac{\partial f}{\partial x}(x,\omega)\ dx\ d\mu=$ $\displaystyle=\int_{a}^{b}\int_{\Omega}\frac{\partial f}{\partial x}(x,\omega)\ d\mu\ dx$ dove l’ultimo passaggio è giustificato dal teorema di Fubini (vedi teorema 7.8 pag 140 di [R4]). Ma allora si ha che per ogni $[a,b]\subset U$: $\displaystyle\int_{a}^{b}\left(F^{\prime}(x)-\int_{\Omega}\frac{\partial f}{\partial x}(x,\omega)\ d\mu\right)dx=0$ quindi grazie a una proprietà nota degli integrali 252525vedi teorema 1.39 (b) pag 29 di [R4] $\displaystyle F^{\prime}(x)=\int_{\Omega}\frac{\partial f}{\partial x}(x,\omega)\ d\mu$ dove l’uguaglianza è intesa quasi ovunque in $x$ in ogni componente connessa dell’insieme $U$, quindi in tutto $U$. ∎ ### 1.5 Convoluzioni In questa sezione introdurremo la convoluzione tra funzioni reali (non nella forma più generale possibile, ma nella forma utile ai nostri scopi), e esploreremo alcune tecniche di regolarizzazione di funzioni tramite convoluzione. Per prima cosa dimostriamo l’esistenza dei nuclei di convoluzione. ###### Lemma 1.40. Per ogni $\alpha>0$ esiste una funzione positiva $\Theta_{\alpha}\in C^{\infty}(\mathbb{R}^{m},\mathbb{R})$ con $supp(\Theta_{\alpha})\Subset\overline{B_{\alpha}(0)}$ e $\int_{\mbox{\footnotesize{$\mathbb{R}$}}^{m}}\Theta_{\alpha}(x)dx=1$. Chiamiamo queste funzioni nuclei di convoluzione. ###### Proof. È sufficiente trovare una funzione con le caratteristiche descritte per $\alpha=1$. Infatti è facile verificare che la funzione $\Theta_{\alpha}(x)\equiv\frac{1}{\alpha^{m}}\Theta\left(\frac{x}{\alpha}\right)$ verifica tutte le richieste. Per trovare la funzione $\Theta_{1}$ basta considerare la funzione $\displaystyle\tilde{\Theta}_{1}(x)\equiv\begin{cases}exp\left(\frac{1}{\left\|x\right\|^{2}-1}\right)&se\left\|x\right\|\leq 1\\\ 0&se\left\|x\right\|\geq 1\end{cases}$ $\displaystyle\Theta_{1}(x)\equiv\frac{\tilde{\Theta}_{1}(x)}{\int_{\mbox{\footnotesize{$\mathbb{R}$}}^{m}}\tilde{\Theta}_{1}(x)dx}$ oppure è possibile sfruttare l’esistenza delle partizioni dell’unità. Trovata una funzione a supporto compatto $0\leq\lambda(x)\leq 1$ (non identicamente nulla) con le tecniche descritte qui sopra la si può traslare e scalare in modo da ottenere la funzione desiderata. ∎ Passiamo a definire la convoluzione, operazione che si rivelerà molto utile per approssimare funzioni abbastanza generiche con una funzioni lisce. La dimostrazione di queste (e altre) proprietà può essere trovata su [R4] al §7.13, oppure su [F1] al §8.2. ###### Proposizione 1.41. Disuguaglianza di Young: Date $f\in L_{1}(\mathbb{R}^{m})$ e $g\in L_{p}(\mathbb{R}^{m})$ ($1\leq p\leq\infty$), si definisce: $f*g(x)=\int_{\mbox{\footnotesize{$\mathbb{R}$}}}\ f(x-y)g(y)dy$ Questa definizione ha senso solo quasi ovunque e vale che: $\left\|f*g\right\|_{p}\leq\left\|f\right\|_{1}\cdot\left\|g\right\|_{p}$ (1.6) ###### Proposizione 1.42. Date $f\in L_{p}(\mathbb{R}^{m})$ e $g\in L_{q}(\mathbb{R}^{m})$ con $\frac{1}{p}+\frac{1}{q}=1,\ 1\leq p\leq\infty$, ha senso definire $\forall x$: $f*g(x)=\int_{\mbox{\footnotesize{$\mathbb{R}$}}}\ f(x-y)g(y)dy$ e vale che $f*g$ è uniformemente continua su $\mathbb{R}^{m}$ e: $\left\|f*g\right\|_{\infty}\leq\left\|f\right\|_{p}\cdot\left\|g\right\|_{q}$ Osserviamo subito che $\ast$ è un’operazione commutativa, cioè $f\ast g=g\ast f$. La convoluzione può essere utilizzata per regolarizzare una funzione, cioè per trovare una funzione liscia vicina a piacere alla funzione data. I dettagli sono nei lemmi seguenti. Per comodità di notazione, ricordiamo brevemente la definizione di multiindice. Un vettore di numeri interi non negativi $\vec{k}=(k_{1},\cdots k_{m})$ è detto multiindice. La sua lunghezza è per definizione $\left|\vec{k}\right|=\sum_{i=1}^{m}k_{i}$, inoltre con il simbolo $D^{\vec{k}}f$ intendiamo: $\displaystyle D^{\vec{k}}f\equiv\frac{\partial^{\left|\vec{k}\right|}f}{\partial x_{1}^{k_{1}}\cdots\partial x_{m}^{k_{m}}}$ ###### Lemma 1.43. Sia $\Theta\in C^{r}(\mathbb{R}^{m},\mathbb{R})$ con $supp(\Theta)\Subset\overline{B_{\alpha}(0)}$ e $\int\Theta(x)dx=1$ con $0\leq r\leq\infty$, sia $f\in L^{1}(\mathbb{R}^{m})$. Allora si ha che: 1. 1. $(\Theta\ast f)\in C^{r}(\mathbb{R}^{m},\mathbb{R})$ 2. 2. Per ogni $\left|\vec{k}\right|\leq r$, $D^{\vec{k}}(\Theta\ast f)|_{x}=(D^{\vec{k}}(\Theta)\ast f)|_{x}$ Inoltre se $f\in C^{s}(\mathbb{R}^{m},\mathbb{R})$ ($0\leq s\leq\infty$) con $supp(f)=C$, allora si ha anche che: 1. 1. $(\Theta\ast f)\in C^{s}(\mathbb{R}^{m},\mathbb{R})$ 2. 2. Per ogni $\left|\vec{k}\right|\leq s$, $D^{\vec{k}}(\Theta\ast f)|_{x}=(\Theta\ast(D^{\vec{k}}f))|_{x}$ 3. 3. $supp(\Theta\ast f)\subset C+\alpha=\\{x\in\mathbb{R}^{m}\ t.c.\ d(x,C)\leq\alpha\\}$ ###### Proof. La dimostrazione è una semplice applicazione del lemma 1.37. ∎ Nel seguito avremo bisogno di una versione più raffinata di questo lemma, in particolare con richieste meno stringenti sulla regolarità di $f$. A questo scopo dimostriamo che: ###### Lemma 1.44. Sia $f\in C(\mathbb{R}^{m})$ con $supp(f)=K\Subset\prod_{i=1}^{m}(a_{i},b_{i})\equiv U$. Sia inoltre $f(\bar{x}^{1},\cdots,x^{i},\cdots,\bar{x}^{m})$ assolutamente continua rispetto a $x^{i}$ quasi ovunque rispetto a $(\bar{x}^{1},\cdots,\bar{x}^{i-1},\bar{x}^{i+1},\cdots,\bar{x}^{m})$, con $\partial f/\partial x^{i}\in L^{1}(\mathbb{R}^{m})$. Allora se $\Theta$ è un nucleo di convoluzione: $\displaystyle\frac{\partial}{\partial x^{i}}(\Theta\ast f)=\Theta\ast\frac{\partial f}{\partial x^{i}}$ ###### Proof. Questo lemma è conseguenza del lemma 1.39, ma per completezza riportiamo anche una dimostrazione più “elementare”. Per comodità di notazione indicheremo $\displaystyle\frac{\partial}{\partial x^{i}}\equiv\partial_{i}$ Osserviamo che quasi ovunque rispetto alle $\bar{x}$, $f(\bar{x},x^{i})=\int_{-\infty}^{x^{i}}\partial_{i}f(\bar{x},t)dt$. Consideriamo $f^{\prime}_{n}$ successione di funzioni continue a supporto compatto $f^{\prime}_{n}:\mathbb{R}^{m}\to\mathbb{R}$ tali che $\left\|f^{\prime}_{n}-\partial_{i}f\right\|_{L^{1}}\to 0$ 262626questa successione esiste per densità di $C_{C}(\mathbb{R}^{m})$ in $L^{1}(\mathbb{R}^{m})$, vedi teorema 3.14 pag 68 di [R4]. Definiamo $\displaystyle f_{n}(\bar{x},x^{i})=\int_{-\infty}^{x^{i}}f^{\prime}_{n}(\bar{x},t)dt$ in questo modo la successione $f_{n}$ converge in norma $L_{1}$ a $f$, infatti: $\displaystyle\left\|f_{n}-f\right\|_{L^{1}}\leq\int_{\mbox{\footnotesize{$\mathbb{R}$}}^{m}}\int_{\infty}^{x^{i}}\left|f^{\prime}_{n}(\bar{x},x^{i})-\partial_{i}f(\bar{x},x^{i})\right|dt\ d\bar{x}dx^{i}\leq$ $\displaystyle\leq(b_{i}-a_{i})\left\|f^{\prime}_{n}-\partial_{i}f\right\|_{L^{1}}\to 0$ Grazie ai lemmi precedenti osserviamo che $\displaystyle\partial_{i}(\Theta\ast f_{n})=(\partial_{i}\Theta)\ast f_{n}\to(\partial_{i}\Theta)\ast f=\partial_{i}(\Theta\ast f)$ e anche $\displaystyle\partial_{i}(\Theta\ast f_{n})=\Theta\ast(\partial_{i}f_{n})=\Theta\ast f^{\prime}_{n}=\to\Theta\ast(\partial_{i}f)$ dove la convergenza nelle ultime due uguaglianze è intesa nel senso di $L^{1}$. Per unicità del limite, $\partial_{i}(\Theta\ast f)=\Theta\ast(\partial_{i}f)$ nel senso di $L^{1}$, quindi quasi ovunque. Ma le funzioni $\Theta\ast(\partial_{i}f)$ e $\partial_{i}(\Theta\ast f)$ sono funzioni lisce, quindi uguaglianza quasi ovunque implica uguaglianza ovunque. ∎ Ora analizziamo le proprietà di approssimazione della convoluzione. A questo scopo utilizzeremo la notazione $\displaystyle\left\|f\right\|_{\infty}\equiv\max_{x\in\mathbb{R}^{m}}\left|f(x)\right|$ ###### Lemma 1.45. Sia $f\in C^{r}(\mathbb{R}^{m},\mathbb{R})$ con $supp(f)=K$ compatto e sia $\vec{k}$ un multiindice di lunghezza $\left|\vec{k}\right|\leq r$. Dato $\epsilon>0$, esiste $\alpha>0$ tale che per ogni $\Theta$ con supporto in $\overline{B_{\alpha}(0)}$ e $\int\Theta(x)dx=1$ si ha che $\left\|D^{\vec{k}}(\Theta\ast f)-D^{\vec{k}}(f)\right\|_{\infty}<\epsilon$. ###### Proof. Per uniforme continuità di $D^{\vec{k}}(f)$ su $K$, vale che dato $\epsilon>0$, esiste $\alpha>0$ tale che $\left\|x-y\right\|\leq\alpha\Rightarrow\left|D^{\vec{k}}f|_{y}-D^{\vec{k}}f|_{x}\right|<\epsilon$. Dato che: $\displaystyle D^{\vec{k}}(\Theta\ast f)|_{x}-D^{\vec{k}}(f)|_{x}=\int_{B_{\alpha}(0)}{\Theta(y)}(D^{\vec{k}}(f)|_{x-y}-D^{\vec{k}}(f)|_{x})dy$ si ottiene: $\displaystyle\left|D^{\vec{k}}(\Theta\ast f)|_{x}-D^{\vec{k}}(f)|_{x}\right|\leq$ $\displaystyle\leq\int_{B_{\alpha}(0)}\left|{\Theta(y)}\right|\left|(D^{\vec{k}}(f)|_{x-y}-D^{\vec{k}}(f)|_{x})\right|dy\leq\epsilon\int_{B_{\alpha}(0)}\left|{\Theta(y)}\right|=\epsilon$ ∎ Questo lemma garantisce che ogni funzione a supporto compatto può essere approssimata uniformemente a ogni suo ordine di regolarità con una funzione liscia. ### 1.6 Duali di spazi di Banach In questa sezione ricordiamo brevemente alcuni risultati riguardo agli spazi duali degli spazi di Banach, in particolare la definizione di topologia debole-* e il teorema di Banach-Alaoglu. Per approfondimenti rimandiamo ai capitoli 3 e 4 di [R2]. ###### Definizione 1.46. Dato uno spazio di Banach (reale) $(X,\left\|\cdot\right\|)$, si definisce $X^{*}$ il suo spazio duale, cioè: $\displaystyle X^{*}=\\{\phi:X\to\mathbb{R}\ \ t.c.\ \phi\ lineare\ e\ continuo\\}$ Possiamo rendere questo spazio uno spazio di Banach con la norma: $\displaystyle\left\|\phi\right\|^{*}\equiv\sup_{x\neq 0}\frac{\left|\phi(x)\right|}{\left\|x\right\|}=\sup_{\left\|x\right\|\leq 1}\left|\phi(x)\right|=\sup_{\left\|x\right\|=1}\left|\phi(x)\right|$ Notiamo che per un funzionale lineare, essere continuo, essere continuo nel punto 0 ed essere limitato (cioè avere $\left\|\phi\right\|^{*}<\infty$) sono proprietà equivalenti. Sullo spazio $X^{*}$ però è possibile definire anche una topologia vettoriale più debole della topologia indotta da questa norma, la topologia debole-*. ###### Definizione 1.47. Sullo spazio $X^{*}$ definiamo $\tau^{*}$ la topologia debole-*, una topologia invariante per traslazioni tale che una base di intorni del punto $0$ è costituita dagli insiemi $\displaystyle V(0,\epsilon,x_{1},\cdots,x_{n})=\\{\phi\ t.c.\ \left|\phi(x_{i})\right|<\epsilon\ \forall\ 1\leq i\leq n\\}$ al variare di $\epsilon$ e di $x_{1},\cdots,x_{n}\in X$ 272727al variare degli elementi e del numero degli elementi, che può essere un numero finito qualsiasi. Questa topologia rende $X^{*}$ uno spazio vettoriale topologico. Ricordiamo che per gli spazi di Hilbert reali esiste un’isometria lineare tra lo spazio $H$ e il suo duale $H^{*}$, isometria data dal teorema di rappresentazione di Riesz (vedi ad esempio teorema 3.4 di [C3]). Per la topologia debole-* vale un teorema molto famoso, il teorema di Banach Alaoglu (vedi teorema 3.15 pag 68 di [R2]). In questa rassegna riportiamo una versione del teorema adatta ai nostri scopi ###### Teorema 1.48 (Teorema di Banach-Alaoglu). Nello spazio $(X^{*},\tau^{*})$, l’insieme $B^{*}=\\{\left\|\phi\right\|^{*}\leq 1\\}$ è compatto. Inoltre se $X$ è separabile, allora $B^{*}$ è anche sequenzialmente compatto. ###### Proof. La compattezza di $B^{*}$ è dimostrata nel teorema 3.15 pag 68 di [R2], per quanto riguarda la compattezza per successioni, il teorema 3.17 a pagina 70 di [R2] garantisce che se $X$ è separabile, allora gli insiemi compatti di $X^{*}$ sono metrizzabili, quindi anche sequenzialmente compatti. ∎ ### 1.7 Funzioni armoniche In questa sezione riportiamo alcuni risultati riguardo alle funzioni armoniche su varietà riemanniane. In tutta la sezione $R$ sarà una varietà Riemanniana liscia senza bordo di dimensione $m$. #### 1.7.1 Principio del massimo In questo paragrafo riportiamo alcuni risultati sugli operatori ellittici, in particolare il principio del massimo. ###### Definizione 1.49. Un operatore differenziale lineare del secondo ordine $D:C^{2}(\Omega,\mathbb{R})\to C(\Omega,\mathbb{R})$ dove $\Omega$ è un aperto in $\mathbb{R}^{m}$ è detto ellittico se ha la forma $\displaystyle D(f)=a^{ij}(x)\partial_{i}\partial_{j}f+b^{i}(x)\partial_{i}f+c(x)f$ dove le funzioni $a$ e $b$ sono lisce in $\Omega$ e la matrice $a^{ij}$ è definita positiva (quindi ha tutti gli autovalori strettamente maggiori di $0$). L’operatore $D$ è detto strettamente ellittico se esiste un numero positivo $\lambda>0$ tale che per ogni vettore $v$ e per ogni $x\in\Omega$ $\displaystyle a^{ij}(x)v_{i}v_{j}\geq\lambda\sum_{i}\left|v_{i}\right|^{2}$ o equivalentemente se l’autovalore minimo della matrice $a^{ij}$ è limitato dal basso sull’insieme $\Omega$. L’operatore $D$ è detto uniformemente ellittico su $\Omega$ se il rapporto tra l’autovalore massimo e l’autovalore minimo di $a^{ij}(x)$ è limitato (indipendentemente da $x$). La definizione di operatore ellittico può essere estesa facilmente a operatori su varietà Riemanniane chiedendo semplicemente che ogni loro rappresentazione locale abbia le caratteristiche descritte sopra. La teoria di questi operatori viene sviluppata in maniera esaustiva su [GT], testo dal quale estraiamo solo i risultati che serviranno in seguito per questa tesi. Una proprietà interessante di questi operatori è che se i coefficienti $a^{ij},b^{i},c$ sono funzioni lisce e una funzione $C^{2}$ soddisfa $Du=f$ con $f\in C^{\infty}(\Omega)$, allora automaticamente la funzione $u\in C^{\infty}(\Omega)$. Anzi si può dimostrare che questo continua a valere anche se $Du=f$ solo nel senso delle distrubuzioni. Un’altra proprietà che sfrutteremo molto in questo lavoro è il principio del massimo, che permette di controllare il valore di una funzione con i suoi valori al bordo dell’insieme di definizione. ###### Proposizione 1.50 (Principio del massimo). Sia $D$ un operatore ellittico in un dominio relativamente compatto $\Omega$. Se la funzione $u:\Omega\to\mathbb{R}$ soddisfa: 1. 1. $u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$ 2. 2. $Du=0$ allora il massimo e il minimo di $u$ su $\overline{\Omega}$ sono raggiunti sul bordo $\partial\Omega$. Cioè: $\displaystyle\sup_{x\in\Omega}u(x)=\sup_{x\in\partial\Omega}u(x)\ \ \ \inf_{x\in\Omega}u(x)=\inf_{x\in\partial\Omega}u(x)$ Inoltre se $u$ non è continua su $\overline{\Omega}$, la conclusione può essere sostituita da: $\displaystyle\sup_{x\in\Omega}u(x)=\limsup_{x\to\partial\Omega}u(x)\ \ \ \inf_{x\in\Omega}u(x)=\liminf_{x\to\partial\Omega}u(x)$ dove con $\limsup_{x\to\partial\Omega}u(x)$ intendiamo il limite di $\sup_{x\in K_{n}^{C}}u(x)$ quando $K_{n}$ è un’esaustione di $\Omega$. Rimandiamo al teorema 3.1 pagina 31 di [GT] per la dimostrazione di questo teorema. Vale un principio simile anche se $\Omega$ non è relativamente compatto, infatti: ###### Proposizione 1.51. Sia $D$ un operatore ellittico in un dominio $\Omega$, e sia $u$ una funzione $u\in C^{2}(\Omega)\cap C^{0}(\overline{\Omega})$. Allora $\displaystyle\sup_{x\in\Omega}u(x)\leq\limsup_{x\to\infty}u(x)\ \ \ \inf_{x\in\Omega}u(x)\geq\liminf_{x\to\infty}u(x)$ dove con $\limsup_{x\to\infty}u(x)$ intendiamo il limite di $\sup_{x\in K_{n}^{C}}u(x)$ quando $K_{n}$ è un’esaustione di $\Omega$. ###### Proof. Supponiamo per assurdo che $\sup_{x\in\Omega}u(x)>\limsup_{x\to\infty}u(x)\equiv L$, e consideriamo l’insieme $A=u^{-1}(L,\infty)$, aperto non vuoto per ipotesi. Dato che $\limsup_{x\to\infty}u(x)=L$, $A$ è relativamente compatto, e applicando il precedente principio a questo insieme si ottiene che $u\equiv L$ in $A$, assurdo. ∎ Una forma leggermente più forte del principio del massimo è la seguente: ###### Proposizione 1.52 (Principio del massimo). Sia $D$ un operatore uniformemente ellittico con $c=0$. Se una funzione $u$ soddisfa $Du=0$ assume il suo massimo in un punto interno a $\Omega$, allora è constante Rimandiamo al teorema 3.5 pagina 34 di [GT] per la dimostrazione di questo teorema. Il principio del massimo può essere espresso anche in forme più sofisticate, alcune delle quali verranno esposte nel seguito della tesi. Osserviamo subito che questi principi possono essere applicati nel caso dell’operatore laplaciano su varietà Riemanniane. Infatti la rappresentazione locale del laplaciano (vedi 1.2) dimostra che questo è un operatore ellittico 282828ricordiamo che per definizione di metrica la matrice $g^{ij}$ è definita postiva sulla varietà $R$, quindi vale il corollario: ###### Proposizione 1.53. Sia $u$ una funzione armonica in $\Omega$ dominio relativamente compatto in $R$ e continua fino al bordo. Allora il massimo e il minimo della funzione sono assunti sul bordo $\partial\Omega$. Inoltre è facile verificare che su ogni insieme compatto $K\Subset R$ l’operatore $\Delta$ è uniformemente ellittico, quindi: ###### Proposizione 1.54. Sia $u$ una funzione armonica in $\Omega$ dominio relativamente compatto in $R$. Se $u$ assume il suo massimo in un punto interno a $\Omega$, allora $u$ è costante su $\Omega$. Ovviamente qualunque funzione armonica su tutta la varietà $R$ è armonica su ogni dominio relativamente compatto, quindi per una funzione di questo tipo che non sia costante vale che: $\displaystyle\inf_{x\in\partial\Omega}u(x)<u(p)<\sup_{x\in\partial\Omega}u(x)$ per ogni punto $p\in\Omega$. #### 1.7.2 Stime sul gradiente Uno strumento importantissimo nello studio delle funzioni armoniche sono le stime sul gradiente, cioé stime sul modulo del gradiente di una funzione armonica positiva. Riportiamo solo il risultato, la cui dimostrazione può essere trovata su [SY] 292929teorema 3.1 pagina 17 ###### Proposizione 1.55. Sia $R$ una varietà Riemanniana completa con $dim(R)\equiv m\geq 2$, e sia $B_{2r}(x_{0})$ la bolla geodetica di raggio $2r$ centrata in $x_{0}$. Supponiamo che $u$ sia una funzione armonica positiva su $B_{2r}(x_{0})$, e sia $Ric(M)\geq-(m-1)K$ su $B_{2r}$ 303030essendo la curvatura di Ricci una funzione continua su $R$, su ogni insieme compatto assume un minimo finito, quindi la costante $K$ si può sempre trovare dove $K\geq 0$ è una costante. Allora: $\displaystyle\frac{\left|\nabla u\right|}{u}\leq C_{m}\frac{1+r\sqrt{K}}{r}$ sull’insieme $B_{r}(x_{0})$, dove $C_{m}$ è una costante che dipende solo dalla dimensione $m$ della varietà. Osserviamo che sul testo [SY], il teorema è enunciato in una forma diversa, si richiede infatti che il limite inferiore sulla curvatura valga su tutta la varietà $R$. Dalla dimostrazione però è evidente che questa ipotesi può essere rilassata. #### 1.7.3 Disuguaglianza, funzione e principio di Harnack Un’altra proprietà che vale per le funzioni armoniche su varietà è la disuguaglianza di Harnack: ###### Proposizione 1.56 (Disuguaglianza di Harnack). Dato un dominio $\Omega$ e un insieme $K\Subset\Omega$, esiste una costante $\Lambda$ 313131chiamata costante di Harnack che dipende solo da $\Omega$ e $K$ tale che per ogni funzione armonica positiva $u$ su $\Omega$ vale che: $\displaystyle\sup_{x\in K}u(x)\leq\Lambda\inf_{x\in K}u(x)$ Questa disuguaglianza è una conseguenza del corollario 8.21 pagina 189 di [GT]. Grazie a questa disuguaglianza siamo in grado di provare il principio di Harnack, che riguarda successioni di funzioni armoniche positive su varietà ###### Proposizione 1.57 (Principio di Harnack). Sia $u_{m}$ una successione crescente di funzioni armoniche positive su $\Omega$ dominio in $R$. Allora o $u_{m}$ diverge localmente uniformemente, o converge localmente uniformemente in $\Omega$. Se la successione di funzioni $u_{m}$ è uniformemente limitata (non necessariamente positiva o crescente), allora esiste una sua sottosuccessione che converge localmente uniformemente su $\Omega$. Questo principio di può trovare su [ABR] pagina 49 323232in realtà il testo [ABR] tratta il caso del laplaciano standard in $\mathbb{R}^{m}$, ma per molte dimostrazioni le tecniche usate si basano su proprietà (come la disuguaglianza di Harnack) che valgono anche nel caso di laplaciano su varietà, quindi possono essere facilmente estese. Questo principio è molto significativo anche perché vale che ###### Proposizione 1.58. Se una successione di funzioni armoniche $u_{n}$ converge localmente uniformemente a una funzione $u$, allora $u$ è armonica. Oltre alla disuguaglianza di Harnack e alla relativa costante, possiamo definire una funzione di Harnack in questo modo: ###### Definizione 1.59. Su una varietà Riemanniana $R$ dato un aperto $\Omega$, possiamo definire una funzione $k:\Omega\times\Omega\to[1,\infty)$ nel seguente modo: $\displaystyle k(x,y)\equiv\sup\\{c\ t.c.\ c^{-1}u(x)\leq u(y)\leq cu(x)\ \ \forall\ u\in HP(\Omega)\\}$ dove $HP(\Omega)$ è l’insieme delle funzioni armoniche positive su $\Omega$. Chiamiamo questa funzione funzione di Harnack. Grazie all’esistenza della costante di Harnack $\Lambda(K,\Omega)$, sappiamo che la funzione $k$ è ben definita su $\Omega$, infatti dati due punti $x,y\in\Omega$, se consideriamo un compatto $K$ che li contiene, abbiamo che: $\displaystyle k(x,y)\leq\Lambda(K)$ però vale anche un’altra importante proprietà di questa funzione: ###### Proposizione 1.60. Per ogni $\Omega$ dominio in $R$, e per ogni $x\in\Omega$, vale che: $\displaystyle\lim_{y\to x,\ y\in\Omega}k(x,y)=1$ ###### Proof. Questa dimostrazione è un’applicazione delle stime sul gradiente 1.55. Consideriamo una bolla geodetica $B_{2r}(x)\subset\Omega$, e sia $K$ la costante descritta nella proposizione 1.55. Sia $y\in B_{r}(x)$ tale che $d(x,y)\equiv d$, sia $\gamma:[0,d]\to R$ la geodetica che unisce $x$ e $y$ e per una qualsiasi funzione $u$ armonica positiva su $\Omega$ definiamo la funzione $\displaystyle\phi:[0,d]\to R\ \ \ \ \ \phi(t)\equiv\log\circ u\circ\gamma(t)=\log(u(\gamma(t)))$ Sappiamo che $\displaystyle\phi(d)-\phi(0)=\log\left(\frac{u(y)}{u(x)}\right)$ e grazie alle stime sul gradiente possiamo osservare che: $\displaystyle\left|\phi(d)-\phi(0)\right|=\left|\int_{0}^{d}\frac{d\phi}{dt}(s)ds\right|\leq\int_{0}^{d}\left|\frac{d\phi}{dt}(s)\right|ds=\int_{0}^{d}\left|\left\langle\nabla\log(u)\middle|\frac{d\gamma}{dt}\right\rangle\right|ds=$ $\displaystyle=\int_{0}^{d}\frac{\left|\nabla u\right|}{u}ds\leq C_{m}\frac{1+r\sqrt{K}}{r}d$ Questo significa che per qualunque funzione armonica positiva $u$: $\displaystyle\frac{u(y)}{u(x)}\leq\exp\left(C_{m}\frac{1+r\sqrt{K}}{r}d(x,y)\right)$ e anche: $\displaystyle\frac{u(x)}{u(y)}\leq\exp\left(C_{m}\frac{1+r\sqrt{K}}{r}d(x,y)\right)$ quindi questa quantità tende a $1$ se $d(x,y)\to 0$. ∎ #### 1.7.4 Funzioni di Green ###### Definizione 1.61. Una funzione di Green per un insieme aperto $\Omega\subset R$ rispetto a un punto $p$ è una funzione $G\in C^{\infty}(\Omega\times\Omega\setminus D;\mathbb{R})$, dove $D=\\{(p,p)\ t.c.\ p\in\Omega\\}$ è la diagonale di $\Omega\times\Omega$, tale che: 1. 1. $G$ è strettamente positiva su $\Omega$ 2. 2. $G$ è simmetrica, cioè $G(p,q)=G(q,p)$ 3. 3. Fissato $p\in\Omega$, la funzione $G(p,q)$ è armonica rispetto a $q$ sull’insieme $\Omega\setminus\\{p\\}$ e superarmonica su tutto $\Omega$. 4. 4. $G$ soddisfa la condizione di Dirichlet al bordo, cioè per ogni $p\in\Omega$, $G(p,q)=0$ per ogni $q\in\partial\Omega$ 333333il valore di $G$ sul bordo è inteso come il limite per $q_{n}\to q$ dove $q_{n}\in\Omega$ 5. 5. $G$ è soluzione fondamentale dell’operatore $\Delta$, cioé per ogni funzione liscia $f$ a supporto compatto in $\Omega$: $\displaystyle\Delta_{x}\int_{\Omega}G(x,y)f(y)dy=\int_{\Omega}G(x,y)\Delta_{y}(f)(y)dy=-f(x)$ questo significa che nel senso delle distribuzioni $\Delta_{y}G(x,y)=-\delta_{x}$ 6. 6. Il flusso di $G_{\Omega}(\ast,p)$ attraverso il bordo di un’insieme regolare $K\Subset\Omega$ con $p\not\in\partial K$ vale: $\displaystyle\int_{\partial K}\ast dG(\cdot,p)=\begin{cases}-1&se\ p\in K\\\ 0&se\ p\not\in K\end{cases}$ 7. 7. La funzione $G$ ha un comportamento asintotico della forma: $\displaystyle G(x,y)\sim C(m)\begin{cases}-log(d(x,y))&m=2\\\ d(x,y)^{m-2}&m\geq 3\end{cases}$ quando $d(x,y)\to 0$. La costante $C(m)$ dipende solo dalla dimensione della varietà e può essere determinata sfruttando la condizione (5). Segnaliamo che per domini $\Omega$ con bordo liscio esiste unica la funzione di Green associata a questo dominio. Un riferimento per questa proposizione è [PSR] a pagina 165. ###### Proposizione 1.62. Dato un insieme aperto relativamente compatto $\Omega$ con bordo liscio, esiste unica la funzione di Green $G_{\Omega}$ Osserviamo che grazie al principio del massimo possiamo dimostrare che ###### Proposizione 1.63. Sia $\Omega$ un dominio relativamente compatto dal bordo liscio in $R$ e sia $G(\cdot,p)$ la funzione di Green relativa a $\Omega$. Se $K$ è un dominio relativamente compatto tale che $\displaystyle p\in K\Subset\Omega$ Allora la funzione $G(\cdot,p)$ rispetto all’insieme $\overline{\Omega}\setminus K$ assume il suo massimo su $\partial K$. Inoltre se $p\in K\Subset K^{\prime}\Subset\Omega$, $G$ assume il suo massimo rispetto a $\overline{K^{\prime}}\setminus K$ su $\partial K$. ###### Proof. La dimostrazione segue dal principio del massimo applicato all’insieme $\Omega\setminus K$. Essendo $G(\cdot,p)|_{\partial\Omega}=0$ per definizione, ed essendo $\Delta$ un operatore uniformemente ellittico su $\overline{\Omega}\setminus K$ 343434grazie alla compattezza di questo insieme, vale il principio 1.52, e quindi $\displaystyle G(\cdot,p)|_{\Omega\setminus\overline{K}}<\max_{x\in\partial K}G(x,p)$ (1.7) Questo dimostra anche che sull’insieme compatto $\partial K^{\prime}\Subset\Omega\setminus\overline{K}$ il massimo è strettamente minore del massimo di $G$ su $\partial K$. ∎ Se $R$ non è compatta, ha senso chiedersi se esiste una funzione con proprietà simili a quelle descritte definita su tutta la varietà. La risposta a questa domanda è legata alla parabolicità della varietà $R$ che introdurremo in seguito, e si trova nella sezione 4.3 #### 1.7.5 Singolarità di funzioni armoniche Grazie al principio del massimo e alle funzioni di Green siamo in grado di caratterizzare le singolarità delle funzioni armoniche positive. ###### Proposizione 1.64. Dato un dominio aperto relativamente compatto $\Omega\subset R$ con bordo liscio e una funzione $v\in H(\Omega\setminus\\{x_{0}\\})\cap C^{0}(\overline{\Omega}\setminus\\{x_{0}\\})$ positiva 353535in realtà è sufficiente che sia limitata dal basso, detta $G_{0}$ la funzione di Green $G_{\Omega}(\cdot,x_{0})$, se $\displaystyle\lim_{d(x,x_{0})\to 0}\frac{v(x)}{G_{0}(x)}=0$ allora la funzione $v$ è estendibile a una funzione armonica su tutta $\Omega$. ###### Proof. Sia $\Omega^{\prime}$ un dominio aperto relativamente compatto con bordo liscio tale che $\displaystyle x_{0}\in\Omega^{\prime}\subset\overline{\Omega^{\prime}}\Subset\Omega$ Sia $[v]$ la soluzione del problema di Dirichlet su $\Omega^{\prime}$ con valore al bordo $v|_{\partial\Omega^{\prime}}$. L’obiettivo della dimostrazione è mostrare che $v=[v]$ su $\Omega_{0}\equiv\Omega^{\prime}\setminus\\{x_{0}\\}$. A questo scopo chiamiamo $\delta=v-[v]$ la funzione differenza. Evidentemente $\displaystyle\delta\in H(\Omega_{0})\cap C^{0}({\overline{\Omega}^{\prime}\setminus\\{x_{0}\\}})\ \ \delta|_{\partial\Omega^{\prime}}=0$ Consideriamo la funzione $\phi(x)=\frac{\delta(x)}{G_{0}(x)}$ definita su $\Omega_{0}$. Questa funzione è identicamente nulla sul bordo di $\Omega^{\prime}$ e per $x$ che tende a $x_{0}$ il suo limite vale $0$ 363636grazie al fatto che $G_{0}$ tende a infinito in $x_{0}$, $[v]$ è limitata e questa proprietà vale per il rapporto $v/G_{0}$. Inoltre questa funzione soddisfa: $\displaystyle\nabla\left(\frac{\delta}{G_{0}}\right)=\frac{\nabla\delta}{G_{0}}-\frac{\delta\nabla G_{0}}{G_{0}^{2}}$ $\displaystyle\Delta(\phi)=div(\nabla(\phi))=div\left(\frac{\nabla\delta}{G_{0}}-\frac{\delta\nabla G_{0}}{G_{0}^{2}}\right)=$ $\displaystyle=\frac{\Delta\delta}{G_{0}}-2\frac{\left\langle\nabla\delta\middle|\nabla G_{0}\right\rangle}{G_{0}^{2}}-\delta\frac{\Delta G_{0}}{G_{0}^{2}}+2\frac{\delta\left\|\nabla G_{0}\right\|^{2}}{G_{0}^{3}}=-2\frac{\left\langle\nabla\delta\middle|\nabla G_{0}\right\rangle}{G_{0}^{2}}+2\frac{\delta\left\|\nabla G_{0}\right\|^{2}}{G_{0}^{3}}=$ $\displaystyle-\frac{2}{G_{0}}\left(\left\langle\frac{\nabla\delta}{G_{0}}\middle|\nabla G_{0}\right\rangle-\left\langle\frac{\delta\nabla G_{0}}{G_{0}^{2}}\middle|\nabla G_{0}\right\rangle\right)=-2\left\langle\nabla\phi\middle|\frac{\nabla G_{0}}{G_{0}}\right\rangle$ quindi: $\displaystyle\Delta(\phi)+2\left\langle\nabla\phi\middle|\frac{\nabla G_{0}}{G_{0}}\right\rangle=0$ Dato che l’operatore $\displaystyle D(\cdot)=\Delta(\cdot)+2\left\langle\nabla\cdot\middle|\frac{\nabla G_{0}}{G_{0}}\right\rangle$ è un’operatore ellittico su $\Omega_{0}$ (i coefficienti sono funzioni lisce e $\Delta$ è ellittico), allora grazie al principio del massimo 1.50 otteniamo che $\phi\equiv 0$ su tutto l’insieme $\Omega_{0}$, da cui la tesi. ∎ Grazie a questa proposizione siamo in grado di dimostrare il corollario: ###### Proposizione 1.65. Dato un dominio aperto relativamente compatto $\Omega\subset R$ con bordo liscio e una funzione $v\in H(\Omega\setminus\\{x_{0}\\})\cap C^{0}(\overline{\Omega}\setminus\\{x_{0}\\})$ positiva 373737in realtà è sufficiente che sia limitata dal basso, detta $G_{0}$ la funzione di Green $G_{\Omega}(\cdot,x_{0})$, se $\displaystyle\lim_{d(x,x_{0})\to 0}\frac{v(x)}{G_{0}(x)}=c$ allora esiste una funzione armonica $\delta$ su $\Omega$ continua fino al bordo tale che $\displaystyle v(x)=\delta(x)+cG_{0}(x)$ ###### Proof. Basta applicare la proposizione precedente alla funzione $v(x)-cG_{0}(x)$ ∎ #### 1.7.6 Principio di Dirichlet In questo paragrafo ci occupiamo di introdurre il principio di Dirichlet nella sua forma più standard. Questo principio afferma che le funzioni armoniche sono le funzioni che hanno integrale di Dirichlet minimo in una certa famiglia di funzioni. È possibile rilassare le ipotesi di regolarità su queste funzioni, come riportato nella sezione 3.2.4. Data una funzione su una varietà $f:R\to\mathbb{R}$ e un insieme misurabile $\Omega$, possiamo definire il suo integrale di Dirichlet come: $\displaystyle D_{\Omega}(f)\equiv\int_{\Omega}\left|\nabla f\right|^{2}dV$ e in maniera simile definiamo anche $\displaystyle D_{\Omega}(f,h)\equiv\int_{\Omega}\left\langle\nabla f\middle|\nabla g\right\rangle dV$ Il principio di Dirichlet 383838o meglio una versione del principio, in seguito dimostreremo versioni con ipotesi meno restrittive sulla regolarità delle funzioni in gioco afferma che: ###### Proposizione 1.66 (Principio di Dirichlet). Dato un dominio regolare 393939un insieme aperto relativamente compatto con bordo liscio a tratti $\Omega$ e una funzione continua $h:\partial\Omega\to\mathbb{R}$, per ogni funzione $f:\overline{\Omega}\to\mathbb{R}$ liscia tale che $f=h$ su $\partial\Omega$: $\displaystyle D_{\Omega}(f)=D_{\Omega}(u)+D_{\Omega}(f-u)$ dove $u$ è l’unica soluzione del problema di Dirichlet che ha $h$ come valore al bordo 404040cioè $u$ è l’unica funzione continua in $\overline{\Omega}$, tale che $\Delta u=0$ in $\Omega$ e $u=h$ su $\partial\Omega$. In particolare $u$ ha integrale di Dirichlet minimo tra tutte le funzioni $f$. La dimostrazione di questo principio si può trovare su [S2], sezione 7.1 pagina 173. #### 1.7.7 Funzioni super e subarmoniche Oltre alle funzioni armoniche, è possibile definire altre due categorie di funzioni legate all’armonicità: le funzioni superarmoniche e le funzioni subarmoniche. Prima di definire queste funzioni, ricordiamo alcune definizioni preliminari. ###### Definizione 1.67. Una funzione $f:R\to\mathbb{R}\cup\\{+\infty\\}$ si dice semicontinua inferiormente se vale una delle seguenti proprietà equivalenti: 1. 1. per ogni $x\in R$, $f(x)\leq\liminf_{y\to x}f(y)$ 2. 2. per ogni $a\in R$, $\\{x\in\mathbb{R}\ t.c.\ f(x)>a\\}$ è un’insieme aperto 3. 3. per ogni $a\in R$, $\\{x\in\mathbb{R}\ t.c.\ f(x)\leq a\\}$ è un’insieme chiuso $f$ si dice semicontinua superiormente se e solo se $-f$ è semicontinua inferiormente, o equivalentemente se e solo se $f:R\to\mathbb{R}\cup\\{-\infty\\}$ e: 1. 1. per ogni $x\in R$, $f(x)\geq\limsup_{y\to x}f(y)$ 2. 2. per ogni $a\in R$, $\\{x\in\mathbb{R}\ t.c.\ f(x)<a\\}$ è un’insieme aperto 3. 3. per ogni $a\in R$, $\\{x\in\mathbb{R}\ t.c.\ f(x)\geq a\\}$ è un’insieme chiuso Ricordiamo che: ###### Proposizione 1.68. Una successione crescente di funzioni $f_{n}$ continue converge a una funzione semicontinua inferiormente. Una successione decrescente di funzioni continue converge a una funzione semicontinua superiormente. Il minimo e il massimo tra due (o tra un numero finito) di funzioni semicontinue inferiormente o superiormente è ancora una funzione semicontinua inferiormente o superiormente. Il seguente lemma sarà utile per confrontare funzioni sub e superarmoniche. La sua formulazione può essere data con ipotesi meno restrittive, ma per gli scopi della tesi è sufficiente assumere di lavorare su spazi metrici. Questo lemma è tratto del lemma 4.3 pag 171 di [D]. ###### Lemma 1.69. Sia $X$ uno spazio metrico, e siano $G:X\to\mathbb{R}$ semicontinua superiormente, $g:X\to\mathbb{R}$ semicontinua inferiormente. Se $G(x)<g(x)$ per ogni $x\in X$, allora esiste una funzione continua $\phi:X\to\mathbb{R}$ tale che per ogni $x\in X$: $\displaystyle G(x)<\phi(x)<g(x)$ ###### Proof. Costruiamo la funzione $\phi$ grazie alle partizioni dell’unità. Per ogni numero razionale positivo $r>0$, sia $\displaystyle U_{r}=\\{x\in X\ t.c.\ G(x)<r\\}\cap\\{x\in X\ t.c.\ g(x)>r\\}$ per semicontinuità tutti questi insiemi sono aperti, inoltre evidentemente formano un ricoprimento dell’insieme $X$. Dato che $X$ è uno spazio metrico, ogni ricoprimento aperto ammette una partizione dell’unità subordinata a tale ricoprimento. Siano $\lambda_{r}$ le funzioni di questa partizione, definiamo: $\displaystyle\phi(x)=\sum_{r}r\lambda_{r}$ Dato che la somma è localmente finita, la funzione $\phi:X\to\mathbb{R}$ è continua su $X$. Inoltre, dato che per ogni $r$ $supp(\lambda_{r})\subset U_{r}$, si ha che per ogni $x\in X$: $\displaystyle G(x)=G(x)\sum_{r}\lambda_{r}<\phi(x)<g(x)\sum_{n}\lambda_{r}=g(x)$ ∎ Passiamo ora a trattare le funzioni sub e superarmoniche. Intuitivamente, una funzione superarmonica è una funzione che confrontata con una funzione armonica è maggiore di questa funzione. Se la funzione $v$ è continua, si dice che è superarmonica se e solo se per ogni compatto con bordo liscio $K$, la funzione armonica determinata da $v|_{\partial K}$ è minore della funzione $v$ su tutto $K$. È però possibile rilassare l’ipotesi sulla continuità di $v$. Comunque vogliamo che abbia senso confrontare la funzione $v$ con una funzione armonica $f$ su $K$ tale che $f|_{\partial K}\leq v_{\partial K}$. Se vogliamo che $f\leq v$ su tutto $K$, è necessario chiedere che la funzione $v$ sia semicontinua inferiormente. Quindi definiamo: ###### Definizione 1.70. Una funzione $v:R\to\mathbb{R}\cup\\{+\infty\\}$ si dice superarmonica se e solo se è una funzione semicontinua inferiormente e se per ogni insieme compatto $K\Subset R$ con bordo liscio e per ogni funzione $f$ armonica in $K^{\circ}$, continua su $K$ e tale che $f|_{\partial K}\leq v|_{\partial K}$, allora $f\leq v$ su tutto l’insieme $K$. Una funzione $v:R\to\mathbb{R}\cup\\{-\infty\\}$ si dice subarmonica se e solo se $-v$ è superarmonica, quindi se e solo se è una funzione semicontinua superiormente e se per ogni insieme compatto $K\Subset R$ con bordo liscio e per ogni funzione $f$ armonica in $K^{\circ}$, continua su $K$ e tale che $f|_{\partial K}\geq v|_{\partial K}$, allora $f\geq v$ su tutto l’insieme $K$. Osserviamo subito che la richiesta che $K$ abbia bordo liscio può essere rilassata. ###### Proposizione 1.71. Sia $\Omega$ un dominio aperto relativamente compatto in $R$. Se $v$ è superarmonica su $\overline{\Omega}$ e se $f\in H(\Omega)\cap(C(\overline{\Omega}))$ è tale che: $v|_{\partial\Omega}\geq f|_{\partial\Omega}$ allora $v\geq f$ su tutto l’insieme $\Omega$. ###### Proof. Sia $\epsilon>0$. Data la semicontinuità inferiore di $v-f$, l’insieme $\Omega_{\epsilon}\equiv(v-f)^{-1}(-\epsilon,\infty)=\\{x\ t.c.\ v(x)>f(x)-\epsilon\\}$ è aperto, e per ipotesi contiene $\partial\Omega$. Grazie all’osservazione 1.28, esiste una successione di aperti relativamente compatti con bordo liscio $K_{n}$ tali che $\displaystyle K_{n}\Subset\Omega\ \ \ K_{n}\Subset K_{n+1}\ \ \ \bigcup_{n}K_{n}=\Omega$ È facile verificare che $\Omega_{\epsilon}^{C}\subset K_{n}$ definitivamente. Infatti $\Omega_{\epsilon}^{C}$ è un compatto di $R$ ed è ricoperto dagli aperti $\\{K_{n}\\}$. Dato che $f\in H(\overline{K}_{n})$ e dato che definitivamente in $n$, $v|_{\partial K_{n}}<f|_{\partial K_{n}}-\epsilon$ poiché il bordo di $K_{n}$ è liscio, per la definizione di superarmonicità si ha che definitivamente in $n$: $\displaystyle v|_{K_{n}}\geq f|_{K_{n}}-\epsilon\ \ \Rightarrow\ \ v|_{\Omega}\geq f|_{\Omega}-\epsilon$ per l’arbitrarietà di $\epsilon$ si ottiene la tesi. ∎ Le funzioni sub e superarmoniche si possono confrontare tra loro, in particolare si ha che: ###### Proposizione 1.72. Sia $K$ un compatto con bordo liscio, e siano $u$ subarmonica su $K$ e $v$ superarmonica su $K$, allora se $u|_{\partial K}\leq v|_{\partial K}$ la disuguaglianza vale su tutto l’insieme $K$. ###### Proof. Grazie al lemma 1.69, sappiamo che per ogni $\epsilon>0$ esiste una funzione continua $\phi:\partial K\to\mathbb{R}$ tale che $\displaystyle u|_{\partial K}<\phi_{\epsilon}<v|_{\partial K}+\epsilon$ Sia $\Phi_{\epsilon}$ la soluzione del problema di Dirichlet su $K$ con condizioni al bordo $\phi_{\epsilon}$. Allora per definizione di superarmonicità, sappiamo che: $\displaystyle u\leq\Phi_{\epsilon}\leq v+\epsilon$ su tutto l’insime $K$. Data l’arbitrarietà di $\epsilon$, otteniamo la tesi. ∎ ###### Osservazione 1.73. Con una tecnica analoga a quella utilizzata per la dimostrazione della proposizione 1.71, si può dimostrare che 1.72 vale anche se si toglie l’ipotesi di liscezza del bordo di $K$. Osserviamo che condizione necessaria e sufficiente affinché una funzione $v$ sia armonica su $R$ è che sia contemporaneamente sub e superarmonica. Una proprietà elementare delle funzioni sub e superarmoniche è che: ###### Proposizione 1.74. Il minimo in una famiglia finita di funzioni superarmoniche è superarmonico, e il massimo in una famiglia finita di funzioni subarmoniche è subarmonico. Lo spazio delle funzioni sub e superarmoniche è un cono in uno spazio vettoriale, più precisamente: ###### Proposizione 1.75. Combinazioni lineari a coefficienti positivi di funzioni superarmoniche sono superarmoniche. ###### Proof. Se $f$ è superarmonica, è ovvio che per ogni $t\geq 0$, anche $tf$ è una funzione superarmonica. Resta da dimostrare che la somma mantiene la proprietà di superarmonicità. È facile dimostrare che la somma di una funzione armonica e una superarmonica è superarmonica, e analogamente la somma di una funzione armonica e una subarmonica è subarmonica. Consideriamo ora due funzioni $v_{1}$ e $v_{2}$ entrambe superarmoniche. Sia $K$ un compatto con bordo liscio in $R$ e sia $u\in H(K^{\circ})\cap C(K)$ tale che: $\displaystyle u|_{\partial K}\leq(v_{1}+v_{2})|_{\partial K}=v_{1}|_{\partial K}+v_{2}|_{\partial K}$ dato che la funzione $u-v_{1}$ è subarmonica mentre $v_{2}$ è superarmonica, grazie alla proposizione 1.72 sappiamo che $u-v_{1}\leq v_{2}$ su tutto l’insieme $K$, da cui la tesi. ∎ Evidentemente vale un’affermazione analoga per le funzioni subarmoniche. Riportiamo ora due proposizioni che saranno utili in seguito per dimostrare la superarmonicità dei potenziali di Green. ###### Proposizione 1.76. Data una funzione $f:R\times R\to\mathbb{R}$ continua per cui per ogni $y\in R$, $f(\cdot,y)$ è superarmonica in $R$, e data una misura di Borel positiva a supporto compatto $K$ con $\mu(K)=\mu(R)<\infty$, la funzione $\displaystyle F(x)\equiv\int_{R}f(x,y)d\mu(y)$ è una funzione superarmonica. ###### Proof. Grazie al teorema di convergenza dominata, è facile dimostrare che la funzione $F$ è continua su $R$, quindi in particolare semicontinua inferiormente. Consideriamo ora una successione con indice $n$ di partizioni di $K$ costituite da insiemi $E_{k}^{(n)}$ tali che il diametro di ogni $E_{k}^{(n)}$ sia minore di $1/n$, e per ogni $n$ e $k$ sia $y_{k}^{(n)}$ un punto qualsiasi dell’insieme $E_{k}^{(n)}$. La successione di funzioni $\displaystyle F^{n}(x)=\int_{K}f(x,y_{k}^{(n)})\chi(E_{k}^{(n)})(y)d\mu(y)=\sum_{k}f(x,y_{k}^{(n)})\mu(E_{k}^{(n)})$ è una successione di funzioni superarmoniche grazie alla proposizione 1.75, inoltre converge localmente uniformemente alla funzione $F(x)$. Sia infatti $C$ un qualsiasi insieme compatto in $R$. L’insieme $C\times K$ è compatto in $R\times R$, e quindi la funzione $f(x,y)$ è uniformemente continua su questo insieme. Questo significa che per ogni $\epsilon>0$, esiste $\delta>0$ tale che uniformemente in $x$ si ha: $\displaystyle d(y_{1},y_{2})<\delta\ \Rightarrow\ \left|f(x,y_{1})-f(x,y_{2})\right|<\epsilon$ Allora per ogni $\epsilon>0$, se scegliamo $n$ in modo che $1/n\leq\delta$, otteniamo: $\displaystyle\left|F(x)-F^{n}(x)\right|\leq\int_{K}\left|f(x,y)-\sum_{k}f(x,y_{k}^{(n)})\chi(E_{k}^{(n)})(y)\right|d\mu(y)\leq\mu(K)\epsilon$ Ora osserviamo che una successione di funzioni superarmoniche che converge localmente uniformemente ha limite superarmonico. Infatti sia $C$ un compatto con bordo liscio in $R$ e $u$ una funzione armonica sulla parte interna di $C$ continua fino al bordo tale che $\displaystyle u|_{\partial C}(x)\leq F|_{\partial C}(x)$ Allora per ogni $\epsilon>0$, esiste un $N$ tale che per ogni $n\geq N$: $\displaystyle F^{n}|_{\partial C}\geq F|_{\partial C}-\epsilon\geq u|_{\partial C}-\epsilon$ Per superarmonicità di $F_{n}$, vale che su tutto $C$: $\displaystyle F^{n}(x)\geq u(x)-\epsilon$ Passando al limite su $n$ e grazie all’arbitrarietà di $\epsilon$, otteniamo la tesi. ∎ ###### Proposizione 1.77. Una successione crescente di funzioni continue superarmoniche ha limite superarmonico. ###### Proof. Sia $f_{n}(x)$ una successione crescente di funzioni continue superarmoniche, e sia $f(x)$ il suo limite 414141automaticamente $f$ è una funzione semicontinua inferiormente. Per dimostrare la superarmonicità di $f$, consideriamo $K$ un compatto con bordo liscio in $R$ e una funzione $u$ armonica sull’interno di $K$ continua fino al bordo con $u|_{\partial K}\leq f|_{\partial K}$. Consideriamo la successione di funzioni $\displaystyle g_{n}(x)=\min\\{f_{n}|_{\partial K},u|_{\partial K}\\}$ questa successione è una successione di funzioni continue, crescenti, e ha come limite la funzione continua $u|_{\partial K}$. Grazie al teorema di Dini (riportato dopo questa dimostrazione, teorema 1.78) la convergenza è uniforme, quindi per ogni $\epsilon>0$, esiste $n$ tale che $\displaystyle g_{n}|_{\partial K}\geq u|_{\partial K}-\epsilon\ \Rightarrow\ f_{n}|_{\partial K}\geq u|_{\partial K}-\epsilon$ data la superarmonicità di $f_{n}$, si ha che $\displaystyle f_{n}(x)\geq u(x)-\epsilon$ per ogni $x\in K$. Per monotonia si ottiene che per ogni $x\in K$: $\displaystyle f(x)\geq u(x)-\epsilon$ data l’arbitrarietà di $\epsilon$, si ottiene la tesi. ∎ Riportiamo ora il teorema di Dini. Per ulteriori approfondimenti su questo teorema rimandiamo al testo [L] (il teorema di Dini è il teorema 1.3 pag 381). ###### Teorema 1.78 (Teorema di Dini). Sia $X$ uno spazio topologico compatto, e sia $f_{n}$ una successione crescente di funzioni $f_{n}:X\to\mathbb{R}$ continue. Se $f_{n}$ converge puntualmente a una funzione $f:X\to\mathbb{R}$ continua, allora la convergenza è uniforme. ###### Proof. Fissato $\epsilon>0$, consideriamo $X_{n}$ gli insiemi aperti $\displaystyle X_{n}\equiv\\{x\in X\ t.c.\ f(x)-f_{n}(x)<\epsilon\\}$ Dato che $f_{n}$ è una successione crescente, $X_{n}\subset X_{n+1}$, e vista la convergenza puntuale di $f$, sappiamo che $X=\cup_{n}X_{n}$. Per compattezza di $X$, esiste un indice $\bar{n}$ per cui $X=X_{\bar{n}}$, cioè per ogni $x\in X$ e per ogni $n\geq\bar{n}$ si ha che $\displaystyle f_{n}(x)>f(x)-\epsilon$ quindi la convergenza è uniforme. ∎ Concludiamo questo paragrafo con una proposizione che sarà spesso utilizzata nel seguito. ###### Proposizione 1.79. Dato $K$ compatto con bordo liscio in $R$, se $f$ è armonica in $R\setminus K$, costante sull’insieme $K$, minore o uguale alla costante fuori da $K$ e continua su $R$, allora è superarmonica. ###### Proof. Senza perdita di generalità, supponiamo che $f\equiv 1$ sull’insieme $K$. Osserviamo che se esiste $v$ superarmonica su $R$ tale che $v|_{R\setminus K}=f|_{R\setminus K}$, questa proposizione è conseguenza del fatto che il minimo tra funzioni superarmoniche è superarmonico. Non è necessario però richiedere che esista una tale funzione. Sia $C$ un insieme compatto con bordo liscio in $R$, e $u$ una funzione armonica sulla parte interna di $C$ continua fino al bordo la cui restrizione al bordo sia minore o uguale alla restrizione di $f$. Se $C\subset R\setminus K$ o $C\subset K$ non c’è niente da dimostrare. Negli altri casi, sia $C_{1}\equiv C\cap K$ e $C_{2}\equiv C\cap K^{C}$. Grazie al principio del massimo, sappiamo che $u\leq 1$ su tutto l’insieme $C$, quindi in particolare su $C_{1}$. Sempre con il principio del massimo (la forma descritta in 1.53), confrontando $u$ e $f$ sull’insieme $\overline{C_{2}}$ abbiamo la tesi. ∎ ### 1.8 Algebre di Banach e caratteri Questa sezione è dedicata a una breve rassegna sulle algebre di Banach e alcuni risultati che saranno utili nello svolgimento della tesi. Per approfondimenti sull’argomento consigliamo il testo [R2], in particolare il capitolo 10. Iniziamo con il ricordare alcune definizioni di base. ###### Definizione 1.80. Un’algebra associativa è uno spazio vettoriale su un campo $\mathbb{K}$ (che d’ora in avanti noi assumeremo sempre essere $\mathbb{R}$) con un’operazione di moltiplicazione associativa e distributiva rispetto alla somma. In simboli, uno spazio vettoriale $V$ è un’algebra associativa se è definita una funzione $\cdot:V\times V\to V$ tale che: 1. 1. $(x\cdot y)\cdot z=x\cdot(y\cdot z)$ 2. 2. $(x+y)\cdot z=x\cdot z+y\cdot z$ 3. 3. $z\cdot(x+y)=z\cdot z+z\cdot y$ 4. 4. $a(x\cdot y)=(ax)\cdot y=x\cdot(ay)$ Per ogni $x,y,z\in V$ e per ogni $a\in\mathbb{K}$. Nel seguito il simbolo di moltiplicazione $\cdot$ sarà sottointeso quando questo non causerà confusione. Un’algebra di Banach è semplicemente un’algebra associativa dotata di una norma compatibile con le operazioni di somma e moltiplicazione e che sia completa rispetto a questa norma. ###### Definizione 1.81. Un’algebra di Banach è un’algebra associativa su cui è definita un’operazione $\left\|\cdot\right\|:V\to\mathbb{R}^{+}$ tale che $\left\|x\cdot y\right\|\leq\left\|x\right\|\left\|y\right\|$ e che lo spazio normato $(V,\left\|\cdot\right\|)$ sia completo (o di Banach). Si dice che l’algebra di Banach $A$ sia dotata di unità se esiste un elemento $e$ tale che per ogni $x\in A$, $ex=xe=x$ e anche $\left\|e\right\|=1$. È facile dimostrare che se esiste, l’elemento $e$ è unico. D’ora in avanti ci occuperemo solo di algebre di Banach con unità sul campo dei numeri reali. ##### Elementi invertibili Nelle algebre di Banach con unità (che indicheremo con $A$ in tutta la sezione), ha senso parlare di “elementi invertibili”. Si dice invertibile un elemento $x$ se esiste $x^{-1}$ tale che $xx^{-1}=x^{-1}x=e$. Grazie all’associatività è facile dimostrare che gli elementi invertibili sono chiusi rispetto alla moltiplicazione, e ovviamente $e$ è un elemento invertibile, il cui inverso è sè stesso. È facile dimostrare che una categoria particolare di elementi di $A$ è sempre invertibile: ###### Proposizione 1.82. Se $\left\|x\right\|<1$, allora $(e-x)$ è invertibile in $A$. ###### Proof. Lo scopo di questa dimostrazione è provare che $\displaystyle(e-x)^{-1}=\sum_{i=0}^{\infty}x^{i}$ dove per convenzione $x^{0}=e$. Per prima cosa, grazie al fatto che $\left\|x\right\|<1$, la serie converge totalmente, quindi converge essendo $A$ uno spazio di Banach. Inoltre: $\displaystyle\left(\sum_{i=0}^{\infty}x^{i}\right)\cdot(e-x)=\left(\lim_{n\to\infty}\sum_{i=0}^{n}x^{i}\right)\cdot(e-x)=\lim_{n\to\infty}\left[\left(\sum_{i=0}^{n}x^{i}\right)\cdot(e-x)\right]=$ $\displaystyle=\lim_{n\to\infty}\left[\sum_{i=0}^{n}x^{i}-\sum_{i=1}^{n+1}x^{i})\right]=e-\lim_{n\to\infty}x^{n+1}=e$ e un ragionamento del tutto analogo vale per $(e-x)\cdot\left(\sum_{i=0}^{\infty}x^{i}\right)$. ∎ ##### Caratteri Introduciamo ora una classe particolare di funzionali lineari sulle algebre di Banach, i caratteri, o funzionali moltiplicativi. Anche in questa sezione riporteremo solo i risultati che interessano agli scopi della tesi, e per ulteriori approfondimenti rimandiamo a [R2]. ###### Definizione 1.83. Un funzionale lineare $\phi:A\to\mathbb{R}$ si dice moltiplicativo se conserva la moltiplicazione, cioè se $\forall x,y\in A$ $\displaystyle\phi(x\cdot y)=\phi(x)\phi(y)$ Le proprietà che risultano evidenti dalla definizione sono che: 1. 1. $\phi(e)=1$ 2. 2. per ogni elemento invertibile $x$, $\phi(x)\neq 0$ 3. 3. $\phi(x^{-1})=\left(\phi(x)\right)^{-1}$ Grazie all’ultima proprietà e alla proposizione 1.82 possiamo dimostrare che qualunque carattere è necessariamente continuo e ha norma 1 424242ricordiamo che la norma di un funzionale può essere definita da $\left\|\phi\right\|\equiv\sup_{\left\|x\right\|\leq 1}(\left|\phi(x)\right|)$, un funzionale è continuo se e solo se ha norma finita. ###### Proposizione 1.84. Ogni carattere $\phi$ ha norma 1. ###### Proof. Dal fatto che $\phi(e)=1$ si ricava facilmente che $\left\|\phi\right\|\geq 1$. Consideriamo ora un qualunque $x$ con norma minore o uguale a 1, e un qualunque numero reale $\lambda$ con $\left|\lambda\right|>1$. Dalla proposizione 1.82 segue che $e-\lambda^{-1}x$ è un elemento invertibile, quindi $\displaystyle\phi(e-\lambda^{-1}x)\neq 0\Longleftrightarrow 1-\frac{\phi(x)}{\lambda}\neq 0\Longleftrightarrow\phi(x)\neq\lambda$ Questa considerazione è valida per qualunque numero $\left|\lambda\right|>1$, e quindi si ricava che $\sup_{\left\|x\right\|\leq 1}\\{\left|\phi(x)\right|\\}\leq 1$, da cui la tesi. ∎ È utile osservare che: ###### Proposizione 1.85. L’insieme $\mathcal{C}$ dei funzionali lineari moltiplicativi è compatto rispetto alla topologia debole-*. ###### Proof. Grazie al teorema 1.48, è sufficiente dimostare che $\mathcal{C}$ è chiuso nella topologia debole-*434343ricordiamo che questo insieme è limitato in norma.. A questo scopo, consideriamo $\phi\in\overline{\mathcal{C}}$, e verifichiamo se $\phi(xy)=\phi(x)\phi(y)$. Dato che $\phi\in\overline{\mathcal{C}}$, per ogni $\epsilon>0$, esiste $\tilde{\phi}\in\mathcal{C}$ tale che: $\displaystyle\tilde{\phi}\in V(\phi,\epsilon,x,y,xy)\equiv\\{\psi\in A^{*}\ t.c.\ \left|\psi(t)-\phi(t)\right|<\epsilon\ \ t=x,y,xy\\}$ quindi per ogni $\epsilon>0$: $\displaystyle\left|\phi(xy)-\phi(x)\phi(y)\right|\leq$ $\displaystyle\leq\left|\phi(xy)-\tilde{\phi}(xy)\right|+\left|\tilde{\phi}(xy)-\tilde{\phi}(x)\tilde{\phi}(y)\right|+\left|\tilde{\phi}(x)\tilde{\phi}(y)-\phi(x)\phi(y)\right|\leq$ $\displaystyle\leq 2\epsilon+\left|\tilde{\phi}(x)\tilde{\phi}(y)-\tilde{\phi}(x)\phi(y)\right|+\left|\tilde{\phi}(x)\phi(y)-\phi(x)\phi(y)\right|\leq$ $\displaystyle\leq 2\epsilon+(\left|\tilde{\phi}(x)\right|+\left|\phi(y)\right|)\epsilon\leq 2\epsilon+(\left|\phi(x)\right|+\left|\phi(y)\right|+\epsilon)\epsilon$ Data l’arbitrarietà di $\epsilon$, si ottiene la tesi. ∎ ### 1.9 Problema di Dirichlet Lo scopo di questa sezione è illustrare alcuni risultati per risolvere il problema di Dirichlet su domini limitati su varietà riemanniane. Daremo la definizione di dominio regolare per il problema di Dirichlet e utilizzeremo il metodo di Perron e le barriere per caratterizzare questi domini, in seguito dimostreremo che alcuni domini particolari sulle varietà sono regolari, in particolare i domini limitati con bordo liscio e i domini della forma $\Omega\setminus K$, dove $\Omega$ è un dominio limitato con bordo liscio e $K$ una sottovarietà di codimensione $1$ con bordo liscio contenuta in $\Omega$. Il lemma 1.89 (che citeremo senza dimostrazione) sarà lo strumento principale per i nostri scopi, in quanto lega la solubilità del problema di Dirichlet per un operatore ellittico abbastanza generico al problema di Dirichlet del più noto e studiato laplaciano standard in $\mathbb{R}^{n}$. In tutta la sezione $\Omega$ indicherà un dominio (cioè un insieme aperto connesso relativamente compatto) in $R$ varietà riemanniana o in $\mathbb{R}^{n}$, mentre $L:C^{2}(R,\mathbb{R})\to C(R,\mathbb{R})$ indicherà un operatore differenziale del II ordine strettamente ellittico (il laplaciano su varietà riemanniane soddisfa queste ipotesi come illustrato nella sezione 1.7.1). Inoltre considereremo solo funzioni sub e superarmoniche continue fino alla chiusura dell’insieme di definizione. Prima di cominciare diamo una definizione preliminare. ###### Definizione 1.86. Un dominio $\Omega\subset R$ ha bordo $C^{k}$ con $0\leq k\leq\infty$ se e solo se per ogni punto $p\in\partial\Omega$, esiste un intorno $V(p)$ e una funzione $f\in C^{k}(V,\mathbb{R})$ tale che: $\displaystyle\Omega\cap V=f^{-1}(\infty,0)\ \ \ \partial\Omega\cap V=f^{-1}(0)$ Inoltre $0$ deve essere un valore regolare di $f$. Osserviamo che non è sufficiente per un dominio in $R$ avere il bordo costutuito da sottovarietà regolari per essere considerato di bordo liscio. Ad esempio consideriamo l’insieme $A=B(0,2)\setminus\\{(x,0)\ -1\leq x\leq 1\\}\subset\mathbb{R}^{2}$. Il bordo di questo dominio è costituito dal bordo della bolla e dal segmento $\\{(x,0)\ -1\leq x\leq 1\\}$, e sebbene per ogni punto del segmento $A=\\{(x,0)\ -1<x<1\\}$ esiste un intorno $V$ e una funzione $f:V\to\mathbb{R}$ tale che $A\cap V=f^{-1}(0)$, non è possibile fare in modo che $\Omega\cap V=f^{-1}(\infty,0)$. #### 1.9.1 Metodo di Perron Per prima cosa definiamo il problema di Dirichlet. ###### Definizione 1.87. Dati $\Omega\in R$ e $L$ con le caratteristiche descritte appena sopra, $\phi:\partial\Omega\to\mathbb{R}$ funzione continua, diciamo che $\Phi:\overline{\Omega}\to\mathbb{R}$ è soluzione del problema di Dirichlet se: $\displaystyle\Phi\in C^{2}(\Omega)\cap C(\overline{\Omega})\ \ \ e\ \ \ L(\Phi)=0\ \ su\ \ \Omega\ \ \ e\ \ \ \Phi|_{\partial\Omega}=\phi$ (1.8) La solubilità del problema di Dirichlet è fortemente legata alla regolarità del dominio $\Omega$. Ad esempio, se $\Omega=B(0,1)\setminus\\{0\\}\subset\mathbb{R}^{n}$ 444444$B(0,1)$ indica la bolla di raggio $1$ in $\mathbb{R}^{n}$, non esistono soluzioni del problema di Dirichlet “classico” (cioè con $L=\Delta$) uguali a $0$ sul bordo di $B(0,1)$ e diverse da $0$ ma limitate nell’origine (vedi esercizi 15 e 16 cap 3 pag 57 di [ABR], e per un altro esempio vedi es 25 cap 2 pag 54 dello stesso libro). ###### Definizione 1.88. Un dominio $\Omega\in R$ si dice essere regolare rispetto al problema di Dirichlet se il più generico di questi problemi ha un’unica soluzione. Il problema di Dirichlet per domini in $\mathbb{R}^{n}$ rispetto all’operatore laplaciano standard è un argomento molto studiato in matematica e con molti risultati nella letteratura. Il seguente lemma (che riportiamo senza dimostrazione) lega la solubilità del problema di Dirichlet per un operatore ellittico abbastanza generico a quella del laplaciano standard, il che semplifica molto il problema. ###### Lemma 1.89. Dato un dominio $\Omega\in\mathbb{R}^{n}$ con $\Omega\subset\overline{\Omega}\Subset\Omega_{1}$ e dato $L$ operatore ellittico della forma: $\displaystyle L=a^{ij}(x)\partial_{i}\partial_{j}+b^{i}(x)\partial_{i}$ con coefficienti $a^{ij}$ e $b^{i}$ localmente lipschitziani in $\Omega_{1}$ e per il quale esiste una funzione di Green su $\Omega_{1}$, allora il generico problema di Dirichlet su $\Omega$ è risolubile se e solo se lo è anche il generico problema di Dirichlet legato all’operatore laplaciano. ###### Proof. Questo lemma è il corollario al teorema 36.3 in [H2] (ultimo risultato presente nell’articolo). ∎ Se consideriamo $\Omega\subset R$ dominio contenuto in una carta locale, il laplaciano sulla varietà $R$ soddisfa tutte le ipotesi del teorema. Questo implica in particolare che: ###### Proposizione 1.90. Per ogni dominio $\Omega\subset R$ e per ogni $p\in\Omega$, esiste un intorno aperto $V(p)\subset\overline{V(p)}\Subset\Omega$ tale che il generico problema di Dirichlet rispetto al laplaciano su varietà è risolubile su $V(p)$. ###### Proof. La dimostrazione è un semplice corollario del teorema precedente. Per il laplaciano standard, il problema di Dirichlet su ogni bolla $B(x_{0},r)\subset\mathbb{R}^{n}$ è risolubile (vedi ad esempio teorema 1.17 pag 13 di [ABR]), quindi se consideriamo un qualsiasi insieme $A$ aperto intorno di $p$ la cui chiusura è contenuta in $\Omega\cap U$ dove $(U,\phi)$ è una carta locale di $R$ e tale che $\phi(A)=B(x_{0},r)$, il problema di Dirichlet relativo al laplaciano su varietà è risolubile su $A$. ∎ Per caratterizzare alcuni dei domini su cui è sempre possibile risolvere il problema di Dirichlet, utilizzeremo il metodo di Perron. ###### Definizione 1.91. Dati $\Omega$, $L$, $\phi$ come sopra, indichiamo con $P[\phi]$ la funzione $P[\phi]:\overline{\Omega}\to\mathbb{R}$ definita da $\displaystyle P[\phi](x)=\sup_{u\in S_{\phi}}u(x)$ dove $S_{\phi}$ è l’insieme delle funzioni subarmoniche $u:\overline{\Omega}\to\mathbb{R}$ continue su $\overline{\Omega}$ tali che $\displaystyle u|_{\partial\Omega}\leq\phi$ Osserviamo che per la compattezza di $\partial\Omega$, la funzione $\phi$ ha minimo $m$ e massimo $M$ finiti, e dato che le tutte le funzioni costanti soddisfano $\Delta_{R}(c)=0$, l’insieme $S_{\phi}$ non è vuoto in quanto contiene la funzione costante uguale a $m$, e tutte le funzioni in $S_{\phi}$, in quanto funzioni subarmoniche sono limitate da $M$, quindi $P[\phi](x)\leq M$ per ogni $x\in\overline{\Omega}$. Il metodo di Perron è illustrato ad esempio nel paragrafo 11.3 pag 226 di [ABR] o nel paragrafo 2.8 pag 23 di [GT] per risolvere il problema di Dirichlet per il laplaciano standard su domini qualsiasi. I cardini essenziali di questo metodo però sono il principio del massimo e la possibilità di risolvere il problema di Dirichlet sulla bolla. Come abbiamo visto, questi principi valgono anche per il laplaciano su varietà, non è quindi difficile immaginare che anche per questo problema il metodo di Perron fornisca una soluzione adeguata. Verifichiamo ora sotto quali condizioni $P[\phi]$ risolve 1.8. Per prima cosa dimostriamo che indipendentemente dal dominio $\Omega$, $P[\phi]$ è una funzione armonica. ###### Proposizione 1.92. $P[\phi]$ è armonica sull’insieme $\Omega$. ###### Proof. Per dimostrare che la funzione $P[\phi]$ è armonica, dimostriamo che per ogni punto $p\in\Omega$, esiste un intorno $V=V(p)$ su cui la funzione è armonica. Consideriamo a questo scopo $V(p)$ un aperto con chiusura contenuta in $\Omega\cap U$, dove $U$ è un intorno coordinato di $R$ tale che la rappresentazione in carte locali di $V(p)$ sia una bolla. Grazie a 1.89, sappiamo che è possibile risolvere il problema di Dirichlet relativo a $\Delta_{R}$ in carte locali su $V(p)$. Per ogni funzione continua $v:\Omega\to\mathbb{R}$, possiamo definire il suo sollevamento armonico come la funzione data da: $\displaystyle\bar{v}(x)=\begin{cases}v(x)&se\ x\in V^{C}\\\ D[v](x)&se\ x\in\overline{V}\end{cases}$ dove $D[v]$ è la soluzione del problema di Dirichlet relativo a $\Delta_{R}$ su $V(p)$ con $v|_{\partial V}$ come condizione al bordo. Osserviamo che questa funzione è continua su $\Omega$ e se $v$ è subarmonica, allora $\bar{v}$ mantiene questa proprietà in quanto massimo di due funzioni subarmoniche. Sia ora $u_{k}$ una successione di funzioni in $S_{\phi}$ tali che $u_{k}(p)\to P[\phi](p)$. Consideriamo $\bar{u}_{k}$ la successione dei sollevamenti armonici di queste funzioni relativamente all’intorno $V(p)$. Dato che $\bar{u}_{k}(p)\geq u_{k}(p)$ e che $\bar{u}_{k}\in S_{\phi}$, si ha che $\bar{u}_{k}(p)\to P[\phi](p)$. Data l’uniforme limitatezza delle funzioni $\bar{u}_{k}$, per il principio di Harnack 1.57 esiste una sottosuccessione che per comodità continueremo a indicare con lo stesso indice che converge localmente uniformemente su $V(p)$ a una funzione armonica. Sia $u=\lim_{k}\bar{u}_{k}$, vogliamo dimostrare che $u|_{V(p)}=P[\phi]|_{V(p)}$. È facile osservare che $u\leq P[\phi]$ su tutto $\Omega$, infatti ogni funzione $\bar{u}_{k}\in S_{\phi}$. Inoltre come osservato in precedenza $u(p)=P[\phi](p)$. Supponiamo per assurdo che esista $q\in V(p)$ tale che $u(q)<P[\phi](q)$. Allora per definizione per ogni $\epsilon>0$, esiste una funzione $\tilde{w}\in S_{\phi}$ tale che $\displaystyle u(q)<\tilde{w}(q)\leq P[\phi](q)\ \ \ \tilde{w}(q)>P[\phi](q)-\epsilon$ Se definiamo $w_{k}$ come il sollevamento armonico della funzione $\max\\{\tilde{w},u_{k}\\}$ rispetto all’insieme $V(p)$, otteniamo come prima una successione di funzioni armoniche limitate che, a patto di passare a una sottosuccessione, converge localmente uniformemente a una funzione $w$ armonica su $V(p)$. Poiché tutte le funzioni $w_{k}\in S_{\phi}$, sappiamo che $w\leq P[\phi]$, e quindi in particolare $w(p)\leq P[\phi](p)$. Inoltre per costruzione $u_{k}\leq w_{k}$, quindi passando al limite otteniamo che $u\leq w$ sull’insieme $V(p)$. Ma dato che $P[\phi](p)=u(p)\leq w(p)\leq P[\phi](p)$, abbiamo che $w(p)=u(p)$, cioè la funzione armonica $u-w$ assume il suo massimo in $V(p)$ in un punto interno all’insieme, quindi per il principio del massimo 1.52 $u-w=0$. Dato che per costruzione $w(q)>P[\phi](q)-\epsilon$, abbiamo che: $\displaystyle u(q)>P[\phi](q)-\epsilon$ e l’assurdo segue dall’arbitrarietà di $\epsilon$. ∎ Resta da verificare se la funzione $P[\phi]$ è continua su $\overline{\Omega}$ e se $P[\phi]|_{\partial\Omega}=\phi$. Prima di dare condizioni per verificare queste proprietà, sottolineamo che il problema di Dirichlet è risolvibile se e solo se valgono queste proprietà per $P[\phi]$. ###### Proposizione 1.93. Il problema di Dirichlet 1.8 è risolubile se e solo se $P[\phi]$ risulta essere continua su $\overline{\Omega}$ e $P[\phi]|_{\partial\Omega}=\phi$. ###### Proof. Se le condizioni su $P[\phi]$ sono verificate, automaticamente $P[\phi]$ è l’unica soluzione del problema di Dirichlet. Al contrario, supponiamo che esista $\Phi$ soluzione del problema. Allora $\Phi\in S_{\phi}$, anzi $\Phi=P[\phi]$. Infatti la funzione $\Phi$ è armonica, quindi maggiora tutte le funzioni in $S_{\phi}$. ∎ Come accennato in precedenza, non sempre il problema di Dirichlet è risolubile. Data l’ultima equivalenza questo implica che sebbene $P[\phi]$ sia sempre una funzione armonica, non sempre soddisfa tutte le condizioni del problema di Dirichlet. Un modo per verificare quando il problema è risolubile è il criterio delle barriere. ###### Definizione 1.94. Dato un punto $p\in\partial\Omega$, diciamo che una funzione $S:\overline{\Omega}\to\mathbb{R}$ è una barriera per il punto $p$ se $S$ è una funzione superarmonica in $\Omega$, continua in $\overline{\Omega}$ tale che $\displaystyle S(p)=0\ \ \ S|_{\overline{\Omega}\setminus\\{p\\}}>0$ Se la funzione $\beta$ ha queste caratteristiche ma è definita solamente su $\overline{\Omega}\cap V(p)$, dove $V(p)$ è un intorno qualsiasi del punto $p$, diciamo che $\beta$ è una barriera locale. Per prima cosa osserviamo che l’esistenza di una barriera “globale” è equivalente all’esistenza di una barriera locale, infatti: ###### Proposizione 1.95. Se esiste una barriera locale $\beta$ per un punto $p\in\partial\Omega$, allora esiste anche una barriera globale per $p$. ###### Proof. La dimostrazione è relativamente facile. Sia $V(p)$ un intorno aperto tale che $\beta$ sia definita su $\overline{\Omega}\cap V(p)$. Sia $W\subset\overline{W}\subset V$ un secondo intorno di $p$. Per continuità $\beta$ assume minimo $m$ strettamente positivo sull’insieme $W^{C}$. Consideriamo la funzione $S$ definita da: $\displaystyle S(q)=\begin{cases}\min\\{\beta(q),m\\}&se\ q\in\overline{\Omega}\cap W\\\ m&se\ q\in\overline{\Omega}\cap W^{C}\end{cases}$ dato che $S$ è il minimo tra funzioni superarmoniche, è ancora una funzione superarmonica. È facile dimostrare che anche le proprietà richieste a una barriera sono verificate. ∎ L’utilità del concetto di barriera è contenuta nella seguente proposizione: ###### Proposizione 1.96. Sia $p\in\partial\Omega$. Se esiste una barriera per $p$ rispetto a $\Omega$, allora $\displaystyle P[\phi](p)=\lim_{x\to p}P[\phi](x)=\phi(p)$ ###### Proof. Poiché $\phi$ è continua sull’insieme $\partial\Omega$, per ogni $\epsilon>0$ esiste un intorno $U_{\epsilon}(p)$ tale che per ogni $x\in U_{\epsilon}\cap\partial\Omega$: $\displaystyle\phi(p)-\epsilon<\phi(x)<\phi(p)+\epsilon$ Dato che $S$ è una funzione strettamente positiva su $\partial\Omega\setminus U_{\epsilon}$, esiste una costante $c>0$ tale che $\displaystyle\phi(p)-\epsilon-cS(x)<\phi(x)<\phi(p)+\epsilon+cS(x)$ per ogni $x\in\partial\Omega\setminus U_{\epsilon}$. Data la positività di $S$, questa relazione vale su tutto l’insieme $\partial\Omega$. Dato che $S$ è superarmonica, la funzione $\phi(p)-\epsilon-cS(x)\in S_{\phi}$, quindi $\displaystyle\phi(p)-\epsilon-cS(x)\in S_{\phi}\leq P[\phi](x)$ (1.9) ora consideriamo una funzione $u\in S_{\phi}$. Vale che: $\displaystyle u|_{\partial\Omega}\leq\phi<\phi(p)+\epsilon+cS(x)\ \ \Rightarrow\ \ [u-cS]|_{\partial\Omega}\leq\phi(p)+\epsilon$ Data la subarmonicità della funzione continua $[u-cS]$, questa relazione è valida su tutto l’insieme $\overline{\Omega}$, e data l’arbitrarietà di $u\in S_{\phi}$ otteniamo che: $\displaystyle P[\phi](x)\leq\phi(p)+\epsilon+cS(x)$ (1.10) Passando al limite per $x\to p$ nelle relazioni 1.9 e 1.10, otteniamo che per ogni $\epsilon>0$: $\displaystyle\phi(p)-\epsilon\leq\liminf_{x\to p}P[\phi](x)\leq\limsup_{x\to p}P[\phi](x)\leq\phi(p)+\epsilon$ data l’arbitrarietà di $\epsilon>0$, si ottiene la tesi. ∎ Il criterio delle barriere è molto utile perchè l’esistenza delle barriere è condizione necessaria e sufficiente per risolvere il problema di Dirichlet, infatti: ###### Proposizione 1.97. Il generico problema di Dirichlet su $\Omega\subset R$ è risolubile se e solo se per ogni $p\in\partial\Omega$ esiste una barriera relativa a $p$. ###### Proof. Grazie alla proposizione precedente, se ogni punto del bordo di $\Omega$ ha una barriera, $P[\phi]$ soddisfa tutte le condizioni del problema di Dirichlet, quindi esiste unica la soluzione di tale problema. Supponiamo al contrario che il problema sia sempre risolubile. Allora fissato $p\in\partial\Omega$, sia $\phi_{p}$ una funzione continua sul bordo tale che $\phi_{p}(p)=0$, e $\phi_{p}(x)>0$ per ogni $x\neq p$ (ad esempio, $\phi_{p}(x)=d(x,p)$). Sia $\Phi_{p}$ la soluzione del relativo problema di Dirichlet. Questa funzione è una barriera per $p$, infatti è una funzione armonica (quindi anche superarmonica) in $\Omega$, continua in $\overline{\Omega}$ e uguale a $\phi_{p}$ su $\partial\Omega$, e grazie al principio del massimo, $\Phi_{p}(x)>0$ per ogni $x\in\Omega$. ∎ Come corollario immediato di questo teorema, dimostriamo che ###### Proposizione 1.98. Se $\Omega$ e $\Omega^{\prime}$ sono domini regolari rispetto a $\Delta_{R}$, allora anche la loro intersezione $\Omega^{\prime\prime}=\Omega\cap\Omega^{\prime}$ è un dominio regolare. ###### Proof. La regolarità dei domini è equivalente all’esistenza di barriere per ogni punto del bordo. Dato che $\displaystyle\partial(\Omega\cap\Omega^{\prime})\subset\partial\Omega\cup\partial\Omega^{\prime}$ ogni punto del bordo di $\Omega^{\prime\prime}$ appartiene ad almeno uno dei due bordi. Senza perdita di generalità, consideriamo $p\in(\partial\Omega^{\prime\prime}\cap\partial\Omega)$. Per regolarità di $\Omega$, esiste una barriera $B$ per $p$ rispetto a $\Omega$. La funzione $B$ è superarmonica in $\Omega$, quindi automaticamente anche in $\Omega^{\prime\prime}$, strettamente positiva su $\overline{\Omega}\setminus\\{p\\}$, quindi anche su $\overline{\Omega^{\prime\prime}}\setminus\\{p\\}$, e ovviamente $B|_{\overline{\Omega^{\prime\prime}}}$ è una funzione continua. Questo dimostra che ogni punto di $\partial{\Omega^{\prime\prime}}$ possiede una barriera, $\Omega^{\prime\prime}$ è regolare. ∎ Riassumendo, un dominio $\Omega\subset R$ è regolare rispetto al laplaciano sulla varietà $R$ se e solo se per ogni punto del bordo di $\Omega$ esiste una barriera, barriera che come abbiamo visto è un concetto locale. Se il dominio $\Omega$ è contenuto in una carta locale, allora grazie al lemma 1.89, possiamo utilizzare tutti i risultati noti per il laplaciano standard e concludere che ogni insieme regolare per il laplaciano standard è regolare anche per il laplaciano su varietà (e viceversa). Utilizzando assieme queste due tecniche, dimostreremo che i domini con bordo liscio in $R$ sono regolari, ma anche i domini della forma $\Omega=A\setminus K$, dove $A$ è un dominio con bordo liscio, e $K$ una sottovarietà con bordo regolare di $R$ di codimensione $1$ contenuta in $A$. #### 1.9.2 Domini con bordo liscio Dalla letteratura sul laplaciano standard in $\mathbb{R}^{n}$, sappiamo che una condizione geometrica semplice per garantire la regolarità di un dominio è la condizione della bolla esterna. ###### Proposizione 1.99 (Condizione della bolla esterna). Dato un dominio $\Omega\subset\mathbb{R}^{n}$, $\Omega$ è regolare per il laplaciano standard se per ogni punto di $\Omega$ esiste una bolla $B(x(p),\epsilon(p))$ tale che $\displaystyle\overline{B(x,\epsilon)}\cap\overline{\Omega}=\\{p\\}$ cioè una bolla esterna al dominio $\Omega$ la cui chiusura interseca il bordo di $\Omega$ solo nel punto $p$. ###### Proof. Grazie all’omogeneità del dominio $\mathbb{R}^{n}$ e dell’operatore $\Delta$, e soprattutto grazie al grande dettaglio con cui questo problema è stato studiato, la trattazione dell’argomento è particolarmente semplice. Per dimostrare questa proposizione, dimostriamo che per ogni punto $p$ che soddisfa la condizione della bolla esterna, esiste una barriera $B_{p}$. Ad esempio possiamo considerare la funzione: $\displaystyle B_{p}(y)=\begin{cases}\log(\left\|x(p)-y\right\|)-\log(\epsilon(p))&se\ n=2\\\ -\frac{1}{\left\|x(p)-y\right\|^{n-2}}+\frac{1}{\epsilon(p)}&se\ n\geq 3\end{cases}$ È facile dimostrare che questa funzione soddisfa tutte le proprietà richieste a una barriera per $\Omega$ (vedi ad esempio teorema 11.13 pag 229 di [ABR]). ∎ Come corollario di questa condizione, è facile dimostrare che ###### Proposizione 1.100. Qualunque dominio $\Omega\subset\mathbb{R}^{n}$ con bordo $C^{2}$ è regolare per il problema di Dirichlet. ###### Proof. È sufficiente dimostrare che ogni punto del bordo di questi domini soddisfa la condizione della bolla esterna. Per i dettagli vedi corollario 11.13 di [ABR]. ∎ La condizione della bolla esterna implica in particolare che ogni anello in $\mathbb{R}^{n}$, cioè ogni insieme della forma $\displaystyle A(x_{0},r_{1},r_{2})\equiv B(x_{0},r_{2})\setminus\overline{B(x_{0},r_{1})}=\\{x\in\mathbb{R}^{n}\ t.c.\ r_{1}<\left\|x-x_{0}\right\|<r_{2}\\}$ è un dominio regolare per il laplaciano standard, e grazie al lemma 1.89, ogni anello in un insieme coordinato 454545cioè ogni insieme $A\subset\overline{A}\Subset U\subset R$ tale che $(U,\psi)$ sia un intorno coordinato e $\psi(A)$ sia un anello in $\psi(U)$ è regolare rispetto al laplaciano su $R$. Questo ci permette di dimostrare che ###### Proposizione 1.101. Ogni dominio $\Omega\subset R$ con bordo liscio 464646è sufficiente con bordo $C^{2}$ è un dominio regolare per $\Delta_{R}$. ###### Proof. Dimostriamo che ogni punto del bordo possiede una barriera locale. Sia a questo scopo $p\in\partial\Omega$ e $(U,\psi)$ un intorno coordinato di $p$ tale che $\displaystyle U\cap\Omega=\psi^{-1}\\{(x_{1},\cdots,x_{m-1},0)\\}$ cioè sia $\psi$ una carta che manda $U\cap\Omega$ in un semispazio chiuso di $\mathbb{R}^{n}$. È facile immaginare (vedi disegno 1.1) che esiste un anello $A(x_{0},r_{1},r_{2})$ in $\psi(U)$ tale che $\overline{B(x_{0},r_{1})}\cap\psi(\Omega)=\\{\psi(p)\\}$. Figure 1.1: L’area in grigio è l’area di definizione della barriera locale $\beta$ Visto che questo insieme è regolare per il laplaciano $\Delta_{R}$, esiste una funzione $\beta$ armonica su $A$, continua su $\overline{A}$, tale che $\beta|_{\partial B(x_{0},r_{1})}=0$, $\beta|_{\partial B(x_{0},r_{2})}=1$. Grazie al principio del massimo, è facile dimostrare che questa funzione è una barriera locale per il punto $p$. ∎ #### 1.9.3 Altri domini regolari Lo scopo di questa sezione è dimostrare che ogni dominio della forma $R\supset\Omega=A\setminus K$ dove $A$ è un dominio con bordo liscio e $K$ una sottovarietà regolare di codimensione $1$ con bordo regolare contenuta in $A$, è un dominio regolare per $\Delta_{R}$. Cercheremo di tenere la dimostrazione al livello più elementare e “geometrico” possibile. Il lemma 1.89 sarà essenziale per ottenere questo risultato. Cominciamo con alcuni risultati preliminari sul laplaciano standard in $\mathbb{R}^{n}$. Nel seguito chiameremo armoniche le funzioni armoniche nel senso di $\mathbb{R}^{n}$ e considereremo quindi domini in $\mathbb{R}^{n}$ ($n\geq 2$), indicando con $B(\bar{x},r)$ la bolla euclidea aperta di centro $\bar{x}$ e raggio $r$, e con $D(\bar{x},r)$ la bolla aperta $n-1$ dimensionale centrata in $\bar{x}$ di raggio $r$, cioè l’insieme $\displaystyle D(\bar{x},r)=\\{x\in\mathbb{R}^{n}\ t.c.\ \left\|\bar{x}-x\right\|<r\ e\ x_{m}=\bar{x}_{m}\\}=B(\bar{x},r)\cap\\{x_{m}=\bar{x}_{m}\\}$ ###### Proposizione 1.102. Sull’insieme $\Omega\equiv B(\bar{x},r_{2})\setminus\overline{D(\bar{x},r_{1})}$ con $r_{2}>r_{1}$, esiste una funzione $f:\Omega\to\mathbb{R}$ tale che $\displaystyle f\in H(\Omega)\cap C(\overline{\Omega})=H(\Omega)\cap C(\overline{B(\bar{x},r)})\ \ \ f|_{\partial B(\bar{x},r)}=0,\ \ \ f|_{\overline{D(\bar{x},r)}}=1$ ###### Proof. Vista l’invarianza per riscalamento di $\mathbb{R}^{n}$ e dell’operatore laplaciano standard, possiamo assumere senza perdita di generalità $r\equiv r_{1}<r_{2}=1$. In tutta la dimostrazione, $B=B(\bar{x},1)$, $D=D(\bar{x},r)$. Grazie all’osservazione 1.27, esiste una successione di aperti relativamente compatti con bordo liscio $A_{n}$ tali che $A_{n}\subset A_{n-1}$ e $\overline{D}=\cap_{n}A_{n}$. Definitivamente in $n$, vale che $\overline{A_{n}}\subset B(0,1)$. Assumiamo senza perdita di generalità che $\overline{A_{1}}\subset B$. Per ogni $n$, l’insieme $\Omega_{n}=B\setminus\overline{A_{n}}$ è un insieme regolare per il laplaciano (ha bordo liscio), quindi per ogni $n$ esiste una funzione: $\displaystyle f_{n}\in C(\overline{B},\mathbb{R})\ \ \ f_{n}\in H(\Omega_{n})\ \ \ f_{n}|_{\partial B}=0\ \ \ f_{n}|_{\overline{A_{n}}}=1$ Grazie al principio del massimo e al fatto che $A_{n}\subset A_{n-1}$, è facile verificare che questa successione è una successione decrescente, e che per ogni $n$, $0\leq f_{n}\leq 1$. Grazie al principio di Harnack (vedi 1.57), la successione $f_{n}$ converge localmente uniformemente su $\Omega$ a una funzione $f$ armonica su $\Omega$. Dimostriamo che questa funzione $f$ è continua e che soddisfa le richieste sui valori al bordo. Per prima cosa consideriamo $\bar{y}\in D$. Per definizione di $D$, esiste un raggio $r(\bar{y})$ sufficientemente piccolo in modo che $D(\bar{y},r(\bar{y}))\subset D$ e $B(\bar{y},r(\bar{y}))\subset B$. Consideriamo la semisfera: $\displaystyle B^{+}(\bar{y},r(\bar{y}))=B(\bar{y},r(\bar{y}))\cap\\{x\in\mathbb{R}^{n}\ t.c.\ x_{m}\geq 0\\}$ Grazie alla proposizione 1.98, questo insieme è un’insieme regolare per il laplaciano. Consideriamo una qualsiasi funzione continua $h:\partial B^{+}\to\mathbb{R}$ tale che $h|_{\partial B^{+}}=0$, $h(\bar{y})=1$ e $0\leq h\leq 1$, e sia $u$ la soluzione del relativo problema di Dirichlet (cioè $u|_{\partial B^{+}}=h$). Grazie al principio del massimo, è facile concludere che per ogni funzione $f_{n}$, $f_{n}\geq u$ su $B^{+}\cap A_{n}$, quindi anche su tutta $B^{+}$. Passando al limite otteniamo che $\displaystyle f(x)\geq u(x)$ Questo implica che la funzione $f$ non può essere identicamente nulla e che: $\displaystyle\liminf_{x\to\bar{y}}f(x)\geq\lim_{x\to\bar{y}}u(x)=1$ L’arbitrarietà di $\bar{y}\in D$ garantisce che $\displaystyle\liminf_{x\to\bar{y}}f(x)\geq 1$ per ogni $\bar{y}\in D$. Dato che per costruzione $\limsup_{x\to\bar{y}}f(x)\leq 1$, otteniamo che per ogni punto $\bar{y}\in D$, $f$ è continua e $f|_{D}=1$. Rimangono da considerare i punti sull’insieme $E\equiv\overline{D}\setminus D$. Sia a questo scopo $\tilde{y}\in E$. Per costruzione di $f$: $\displaystyle 0\leq\liminf_{x\to\tilde{y}}f(x)\leq\limsup_{x\to\tilde{y}}f(x)=1$ Consideriamo un numero $0<\lambda<1$. Ricordiamo che dato un insieme $I$, si dice $\displaystyle\lambda I=\\{x\in\mathbb{R}^{n}\ t.c.\ x/\lambda\in I\\}=\\{\lambda x\ t.c.\ x\in I\\}$ Grazie all’invarianza per traslazioni di $\mathbb{R}^{n}$ e dell’operatore laplaciano standard, possiamo ridefinire per comodità il sistema di riferimento in modo che $\tilde{y}=0$. Scegliamo $\lambda$ in modo che $\partial(\lambda B)=\lambda(\partial B)\subset B\setminus\Omega$ (vedi figura 1.2). Figure 1.2: Le linee piene rappresentano il bordo degli insiemi $m-1$ dimensionali, le linee tratteggiate il bordo degli insiemi $m$ dimensionali. Il colore rosso indica gli insiemi moltiplicati per $\lambda$. La funzione $f$ è armonica su $\Omega$, quindi assume un valore minimo su $\lambda(\partial B)$. Grazie al principio del massimo 1.52, questo valore è strettamente positivo, diciamo $\epsilon>0$. Consideriamo la funzione $\displaystyle f_{\lambda}:\lambda B\to\mathbb{R}\ \ \ f_{\lambda}(x)=f(\lambda^{-1}x)$ Osserviamo subito che $\liminf_{x\to\tilde{y}}f(x)=\liminf_{x\to\tilde{y}}f_{\lambda}(x)$ (ricordiamo che abbiamo scelto un riferimento dove $\tilde{y}=0$). Definiamo inoltre $\displaystyle f_{\epsilon}(x)=\frac{f_{\lambda}(x)+\epsilon}{1+\epsilon}$ Questa funzione è armonica su $\lambda\Omega$, quindi essendo $\lambda<1$ è armonica su $\lambda B\setminus\overline{D}$. Inoltre vale che $f_{\epsilon}|_{\partial(\lambda B)}=\frac{\epsilon}{1+\epsilon}<\epsilon$, e $f_{\epsilon}\leq 1$ ovunque definita. Dato che la successione $f_{n}$ converge a $f$ monotonamente dall’alto, per ogni $n$ si ha che: $\displaystyle f_{\epsilon}|_{\partial(\lambda B)}<f_{n}|_{\partial(\lambda B)}$ Inoltre visto che $f_{\epsilon}\leq 1$ ovunque definita, grazie al principio del massimo otteniamo che per ogni $n$: $\displaystyle f_{n}>f_{\epsilon}\ \ \Rightarrow\ \ f\geq f_{\epsilon}$ Questo significa che $\displaystyle\liminf_{x\to\tilde{y}=0}f(x)\geq\liminf_{x\to\tilde{y}=0}f|_{\epsilon}(x)=\frac{\epsilon+\liminf_{x\to\tilde{y}=0}f(x)}{1+\epsilon}$ se definiamo $\liminf_{x\to\tilde{y}=0}f(x)\equiv L$, otteniamo che: $\displaystyle L+\epsilon L\geq L+\epsilon\ \ \Rightarrow\ \ L\geq 1$ Quindi: $\displaystyle 1\leq\liminf_{x\to\tilde{y}}f(x)\leq\limsup_{x\to\tilde{y}}f(y)=1$ ripetendo la costruzione per ogni $\tilde{y}\in E$, otteniamo la tesi. ∎ Osserviamo che a patto di riscalare la funzione $f$, è possibile ottenere una funzione tale che: $\displaystyle f\in H(\Omega)\cap C(\overline{\Omega})=C(\overline{B(\bar{x},r)})\ \ \ f|_{\partial B(\bar{x},r)}=a,\ \ \ f|_{\overline{D(\bar{x},r)}}=b$ per qualsiasi valore fissato di $a$ e $b\in\mathbb{R}$. Passiamo ora a dimostrare che ogni dominio $\Omega$ della forma descritta qui sopra è regolare rispetto al laplaciano standard. ###### Proposizione 1.103. L’insieme $\Omega\equiv B(\bar{x},r_{2})\setminus\overline{D(\bar{x},r_{1})}$ con $r_{2}>r_{1}$ è regolare per il laplaciano standard, cioè per ogni funzione $\phi:\partial\Omega\to\mathbb{R}$ continua, esiste una soluzione del relativo problema di Dirichlet. ###### Proof. Come sopra, possiamo assumere $\bar{x}=0$ e $r\equiv r_{1}<r_{2}=1$. Iniziamo con il considerare solo funzioni $\phi$ identicamente nulle su $\partial B$ e $0\leq\phi\leq 1$ ovunque su $\partial\Omega$, e dimostriamo che $P[\phi]$ risolve il relativo problema di Dirichlet. Sappiamo che $0\leq P[\phi]\leq 1$ su tutto $\overline{\Omega}$. Inoltre grazie all’esistenza di barriere per ogni punto di $\partial B$, sappiamo che $P[\phi]$ è continua su un intorno di $\partial B$ (a priori disgiunto da $\overline{D}$) e $P[\phi]|_{\partial B}=0$. Sia $y\in\overline{D}$. Per continuità, dato $\epsilon>0$, esiste $\delta>0$ tale che $\displaystyle\phi(y)-\epsilon<\phi|_{B(y,\delta)\cap\overline{D}}<\phi(y)+\epsilon$ Consideriamo ora un insieme $\Omega_{y}=B(x,\rho_{2})\setminus\overline{D(x,\rho_{1})}$, dove $x$, $\rho_{1}$, $\rho_{2}$ sono scelti in modo che $B(x,\rho_{2})\subset B(y,\delta)$, $\overline{D}(x,\rho_{1})\subset D$, $y\in\overline{D}(x,\rho_{1})$. Ad esempio, se $y\in D$ è sufficiente scegliere $x=y$ e $\rho_{1}<\rho_{2}<\delta$, mentre se $y\in\overline{D}\setminus D$ (cioè se $y$ sta sul bordo del disco $D$), allora si può scegliere $D(x,\rho_{1})$ in modo che sia un disco interno e tangente a $D$ nel punto $y$, sufficientemente piccolo in modo che $\overline{D(x,\rho_{1})}\subset B(y,\delta)$. Consideriamo ora due funzioni, $f_{1}$ e $f_{2}$, definite in questo modo: $\displaystyle f_{1}|_{\partial B(x,\rho_{2})}=0\ \ \ f_{1}|_{\overline{D}(x,\rho_{1})}=\phi(y)-\epsilon\ \ \ f\in H(B(x,\rho_{2})\setminus\overline{D(x,\rho_{1}}))$ $\displaystyle f_{2}|_{\partial B(x,\rho_{2})}=1\ \ \ f_{1}|_{\overline{D}(x,\rho_{1})}=\phi(y)+\epsilon\ \ \ f\in H(B(x,\rho_{2})\setminus\overline{D(x,\rho_{1}}))$ estendiamo le due funzioni a costanti su $\overline{B}\setminus B(x,\rho_{2})$, in particolare $f_{1}=0$ e $f_{2}=1$ fuori dall’insieme $B(x,\rho_{2})$. Entrambe le funzioni risultano continue su $\overline{B}$, $f_{1}$ è subarmonica su $\Omega$ (in quanto massimo tra due funzioni armoniche), mentre $f_{2}$ è superarmonica su $\Omega$ (in quanto minimo tra due funzioni armoniche). Inoltre per costruzione $f_{1}|_{\partial\Omega}\leq\phi$ e $f_{2}|_{\partial\Omega}\geq\phi$. Per definizione di $P[\phi]$, otteniamo quindi che: $\displaystyle P[\phi]\geq f_{1}$ quindi vale che: $\displaystyle\phi(y)-\epsilon=\lim_{x\to y}f_{1}(x)\leq\liminf_{x\to y}P[\phi](x)$ Inoltre, qualunque funzione $u\in S_{\phi}$ 474747vedi 1.91 per il principio del massimo è minore della funzione $f_{2}$, quindi anche $P[\phi]$ conserva questa proprietà. Da questo ricaviamo che: $\displaystyle\phi(y)+\epsilon=\lim_{x\to y}f_{2}(x)\geq\limsup_{x\to y}P[\phi](x)$ data l’arbitrarietà di $\epsilon$ e del punto $y$, possiamo concludere che $P[\phi]$ risolve il problema di Dirichlet considerato. Osserviamo che con il procedimento appena descritto permette di costruire una barriera per ogni punto di $\overline{D}$. Indatti fissato $\bar{y}\in\overline{D}$, se scegliamo $\displaystyle\phi_{\bar{y}}(x)=\begin{cases}1-d(x,\bar{y})&se\ x\in\overline{D}\\\ 0&se\ x\in\partial B\end{cases}$ grazie a quanto appena dimostrato la funzione $P[\phi]$ è armonica in $\Omega$, continua su $\overline{B}$, $P[\phi_{y}](y)=1$ e $P[\phi_{y}](x)<1$ per ogni $x\neq y$. Quindi la funzione $1-P[\phi_{y}]$ è una barriera per il punto $y$. Dato che i punti di $\partial B$ sono regolari (hanno tutti una barriera locale), abbiamo dimostrato che il dominio $\Omega$ è regolare per l’operatore laplaciano standard. ∎ Ricordiamo che grazie al lemma 1.89, i domini appena considerati sono regolari anche per l’operatore $\Delta_{R}$ (in carte locali). Passiamo ora a dimostrare la proposizione principale di questo paragrafo: ###### Proposizione 1.104. Ogni dominio su una varietà riemanniana $R$ della forma $\Omega=A\setminus K$ dove $A$ è un dominio con bordo liscio e $K$ una sottovarietà regolare di codimensione $1$ con bordo regolare contenuta in $A$, è un dominio regolare per $\Delta_{R}$. ###### Proof. Dimostriamo che per ogni punto di $\partial\Omega$ è un punto regolare per il problema di Dirichlet. Se $p\in\partial A$, $p$ è regolare grazie alla proposizione 1.101, quindi possiamo limitarci a considerare il caso $p\in K$. D’ora in avanti indicheremo con $\partial K$ il bordo di $K$ inteso come bordo della sottovarietà, non come bordo topologico, e $K^{\circ}$ sarà la parte interna sempre intesa nel senso di sottovarietà. Se $p\in K^{\circ}$, consideriamo $(V(p),\psi)$ un aperto (nella topologia di $R$) coordinato intorno di $p$ tale che $\displaystyle V\cap K=\psi^{-1}\\{(x_{1},\cdots,x_{m-1},0)\\},\ \ \ \psi(p)=0$ ad esempio possiamo considerare le coordinate di Fermi su un intorno di $p$ (vedi proposizione 1.12). Sia $r_{2}$ tale che $\overline{B(0,r_{2})}\subset\psi(V)$ e sia $r_{1}<r_{2}$. Allora l’insieme $\displaystyle S=B(0,r_{2})\setminus\overline{D(0,r_{1})}$ è un’insieme regolare per l’operatore $\Delta_{R}$, quindi esiste la soluzione del problema di Dirichlet $\displaystyle\Delta_{R}(\Phi)=0\ su\ S,\ \ \ \Phi\in C(\overline{S}),\ \ \ \Phi(x)=d(x,0)\ \ \forall x\in\partial S$ È facile verificare che la funzione $\Phi$ è una barriera locale per il punto $p$. Se $p\in\partial K$, la dimostrazione della sua regolarità è molto simile. Consideriamo $(V,\psi)$ un’aperto con coordinate di Fermi tale che: $\displaystyle V\cap K=\psi^{-1}\\{(x_{1},\cdots,x_{m-1},0)\ t.c.\ x_{m-1}\leq 0\\}\ \ \ \psi(p)=0$ cioè un sistema di coordinate che rappresenta $K$ come un semipiano in $\mathbb{R}^{n}$. Sia $r_{2}$ abbastanza piccolo da verificare $B(0,2r_{2})\subset\psi(V)$. Se $r_{1}<r_{2}$, e $\bar{x}=(0,\cdots,0,-r_{1},0)$, l’insieme $\displaystyle S=B(\bar{x},r_{2})\setminus D(\bar{x},r_{1})$ è un’insieme regolare per l’operatore $\Delta_{R}$, quindi esiste la soluzione del problema di Dirichlet: $\displaystyle\Delta_{R}(\Phi)=0\ su\ S,\ \ \ \Phi\in C(\overline{S}),\ \ \ \Phi(x)=d(x,0)\ \ \forall x\in\partial S$ Come nel caso precedente, è facile verificare che la funzione $\Phi$ è una barriera locale per il punto $p$. ∎ #### 1.9.4 Regolarità sul bordo In questo paragrafo riportiamo un lemma che garantisce sotto certe condizioni la regolarità delle soluzioni di particolari problemi di Dirichlet fino al bordo del dominio. Il lemma è basato sul teorema 17.3 pag 28 di [F2], di cui riportiamo una versione semplificata senza dimostrazione. ###### Teorema 1.105. Sia $\Omega$ un dominio con bordo liscio, e $L$ un operatore uniformemente ellittico su $\overline{\Omega}$ con coefficienti $a,b,c$ lisci. Se $u$ è soluzione del problema di Dirichlet generalizzato $\displaystyle Lu=f\ in\ \Omega\ \ \ \ u|_{\partial\Omega}=0$ con $f\in C^{\infty}(\overline{\Omega})$, allora $u\in C^{\infty}(\overline{\Omega})$. Ricordiamo che, grazie alla teoria sviluppata in [F2], sotto le ipotesi citate il problema di Dirichlet generalizzato ha sempre un’unica soluzione. Osserviamo che considerando un dominio relativamente compatto con bordo liscio $\Omega\subset R$, il laplaciano su varietà soddisfa tutte le condizioni del teorema. Grazie a questo teorema siamo in grado di dimostrare che: ###### Lemma 1.106. Siano $L$ e $\Omega$ come nel teorema precedente, sia $u\in C^{2}(\Omega)\cap C(\overline{\Omega})$ con $Lu=0$. Se esiste una funzione $h\in C^{\infty}(\overline{\Omega})$ tale che $u|_{\partial\Omega}=h|_{\partial\Omega}$ allora $u\in C^{\infty}(\overline{\Omega})$. ###### Proof. La funzione $u-h$ soddisfa il problema di Dirichlet generalizzato: $\displaystyle L(u-h)=L(u)-L(h)=-L(h)\in C^{\infty}(\overline{\Omega})\ \ \ \ \ (u-h)|_{\partial\Omega}=0$ grazie al teorema precedente, la funzione $(u-h)\in C^{\infty}(\overline{\Omega})$, quindi per differenza anche la funzione $u\in C^{\infty}(\overline{\Omega})$. ∎ Nella tesi, spesso utilizzeremo questo lemma applicato a domini $\Omega$ con bordo costituito da 2 componenti connesse $\partial\Omega_{0}$ e $\partial\Omega_{1}$ 484848ad esempio un anello in $\mathbb{R}^{2}$ ha queste caratteristiche e funzioni $u$ costanti su ogni componente 494949ad esempio $u|_{\partial\Omega_{0}}=0$ e $u|_{\partial\Omega_{1}}=1$. In questo caso sappiamo che la funzione $h$ esiste, infatti: ###### Proposizione 1.107. Sia $\Omega$ un dominio con bordo costituito da due componenti connesse $\partial\Omega_{0}$ e $\partial\Omega_{1}$. Allora esiste una funzione $h\in\overline{\Omega}$ tale che $h|_{\partial\Omega_{0}}=0$ e $h|_{\partial\Omega_{1}}=1$. ###### Proof. Siano $V_{0}$ e $V_{1}$ due intorni disgiunti di $\partial\Omega_{0}$ e $\partial\Omega_{1}$ 505050gli insiemi $\partial\Omega_{0}$ e $\partial\Omega_{1}$ sono chiusi e disgiunti in $R$ spazio metrico, e sia $\\{\lambda_{0},\lambda_{1},\lambda\\}$ uan partizione dell’unità subordinata al ricoprimento aperto $\\{V_{0},V_{1},(\partial\Omega)^{C}\\}$. Chiamiamo $f_{0}$ e $f_{1}$ due funzioni lisce con le caratteristiche descritte in 1.25, in particolare: $\partial\Omega_{0}=f_{0}^{-1}(0)\ \ \ \partial\Omega_{1}=f_{1}^{-1}(0)$ È facile verificare che la funzione: $\displaystyle h\equiv\lambda_{0}\cdot f_{0}+\lambda_{1}\cdot(1-f_{1})$ è una funzione liscia con $h|_{\partial\Omega_{0}}=0\ \ h|_{\partial\Omega_{1}}=1$. ∎ Questo dimostra ad esempio che tutte le funzioni $u\in H(\Omega),\ \ u|_{\partial\Omega_{0}}=0\ \ u|_{\partial\Omega_{1}}=1$ sono funzioni $u\in C^{\infty}(\overline{\Omega})$. ## Chapter 2 Ultrafiltri e funzionali lineari moltiplicativi In questo capitolo introduciamo gli ultrafilti. L’introduzione è tratta da [D]. Il risultato principale riguarda l’esistenza di funzionali lineari moltiplicativi sull’algebra di $l_{\infty}(\mathbb{N})$, ed è tratto dall’articolo [N]. Lo scopo di questo capitolo è di descrivere un modo alternativo alle successioni per caratterizzare le topologie che non sono sequenziali, nelle quali cioè la chiusura per successioni non coincide con la chiusura topologica. Un esempio di questi spazi topologici è lo spazio $\mathbb{M}(R)$ con la topologia $B$ descritto nella definizione 3.13. Inoltre gli ultrafiltri possono essere utilizzati per definire dei caratteri sullo spazio $l_{\infty}(\mathbb{N})$ (vedi paragrafo 2.2), e questo risultato può essere utilizzato per dare un esempio non banale di carattere sull’algebra di Royden. A questo scopo rimandiamo alla sezione 3.3.2. ### 2.1 Filtri, ultrafiltri e proprietà Per prima cosa introduciamo la definizione di filtro. ###### Definizione 2.1. Sia $(X,\tau)$ uno spazio topologico. Un filtro in questo spazio è una collezione $\mathcal{A}=\\{A_{\alpha}\ t.c.\ \alpha\in I\\}$ di sottoinsiemi di $X$ tale che: 1. 1. $\forall\alpha\in I\ A_{\alpha}\neq\emptyset$ 2. 2. $\forall\alpha,\ \beta\in I$ $\exists\gamma\in I\ t.c.\ A_{\gamma}\subset A_{\alpha}\cap A_{\beta}$ Osserviamo dalla definizione che ogni coppia di insiemi in $\mathcal{A}$ ha intersezione non vuota, e per induzione qualunque collezione finita di insiemi di $\mathcal{A}$ ha intersezione non vuota. Gli ultrafiltri generalizzano la nozione di convergenza di successioni e sono utili negli spazi non primo numerabili (come ad esempio l’algebra di Royden). A questo scopo definiamo: ###### Definizione 2.2. Dato un ultrafiltro $\mathcal{A}$ nello spazio $(X,\tau)$ diciamo che $\mathcal{A}$ converge a $x_{0}$ e scriviamo che $\mathcal{A}\to x_{0}$ se e solo se $\forall U(x_{0})$ intorno aperto in $\tau$, esiste $\alpha$ tale che $\mathcal{A}_{\alpha}\subset U$. Diciamo invece che $\mathcal{A}$ si accumula in $x_{0}$ e scriviamo $\mathcal{A}\triangleright x_{0}$ se e solo se $\forall U(x_{0})$, $\forall\alpha$, $A_{\alpha}\cap U\neq\emptyset$ Osserviamo subito che $\mathcal{A}\to x_{0}\Rightarrow\mathcal{A}\triangleright x_{0}$, e che negli spazi di Hausdorff $\mathcal{A}$ non si accumula in nessun altro punto 111questo è il teorema 3.2 a pag. 214 di [D]. Questo è evidente dal fatto che ogni coppia di insiemi in $\mathcal{A}$ ha intersezione non vuota. Inoltre dalla definizione è evidente che $\mathcal{A}\triangleright x_{0}$ se e solo se $x_{0}\in\bigcap_{\alpha}\overline{A_{\alpha}}$. Un esempio banale di filtro convergente è l’insieme degli intorni di un punto. Dato $x_{0}\in X$, indichiamo con $\mathcal{U}(x_{0})$ il filtro dei suoi intorni. È evidente che $\mathcal{U}(x_{0})\to x_{0}$. È possibile definire in alcuni casi operazioni di unione e intersezione tra filtri: ###### Definizione 2.3. Dati due filtri $\mathcal{A}$ e $\mathcal{B}$ su $X$ definiamo i filtri: 1. 1. $\mathcal{A}\cup\mathcal{B}=\\{A_{\alpha}\cup B_{\beta}\ t.c.\ \alpha\in I,\ \beta\in Y\\}$ 2. 2. se $A_{\alpha}\cap B_{\beta}\neq\emptyset$ per ogni scelta di $\alpha$ e $\beta$, allora definiamo $\mathcal{A}\cap\mathcal{B}=\\{A_{\alpha}\cap B_{\beta}\ t.c.\ \alpha\in I,\ \beta\in Y\\}$ La facile verifica che questi insiemi sono filtri è lasciata al lettore. ###### Definizione 2.4. Dati due filtri $\mathcal{A}=\\{A_{\alpha}\ t.c.\ \alpha\in I\\}$ e $\mathcal{B}=\\{B_{\beta}\ t.c.\ \beta\in Y\\}$, diciamo che $\mathcal{B}$ è subordinato ad $\mathcal{A}$ e scriviamo $\mathcal{B}\vdash\mathcal{A}$ se e solo se $\forall\alpha\ \exists\beta\ t.c.\ B_{\beta}\subset A_{\alpha}$ È facile verificare che la relazione di subordinazione è una relazione di preordine, ma non di ordine parziale. Infatti è una relazione riflessiva e transitiva, ma non è vero che $\mathcal{A}\vdash\mathcal{B}\ \wedge\ \mathcal{B}\vdash\mathcal{A}\Rightarrow\mathcal{A}=\mathcal{B}$. Un controesempio si può trovare considerando $X=\mathbb{N}$, $\mathcal{A}=\\{[2n,\infty)\ n\in\mathbb{N}\\}$ e $\mathcal{B}=\\{[n,\infty)\ n\in\mathbb{N}\\}$. Per la relazione di subordinazione valgono le seguenti proprietà: ###### Proposizione 2.5. 1. 1. $\mathcal{A}\subset\mathcal{B}\Rightarrow\mathcal{B}\vdash\mathcal{A}$ 2. 2. Se $\mathcal{B}\vdash\mathcal{A}$, allora ogni elemento di $\mathcal{B}$ ha intersezione non vuota con ogni elemento di $\mathcal{A}$ 3. 3. $\mathcal{A}\to x_{0}$ se e solo se $\mathcal{A}\vdash\mathcal{U}(x_{0})$ 4. 4. Se $\mathcal{B}\vdash\mathcal{A}$, allora $\mathcal{A}\to x_{0}\Rightarrow\mathcal{B}\to x_{0}$ e anche $\mathcal{B}\triangleright x_{0}\Rightarrow\mathcal{A}\triangleright x_{0}$ ###### Proof. La dimostrazione di questi punti è lasciata al lettore. ∎ Per le successioni negli spazi topologici vale che una successione converge a un punto $p$ se e solo se da ogni sua sottosuccessione si può estrarre una sottosottosuccessione convergente a $p$, e una successione si accumula in un punto se e solo se esiste una sua sottosuccessione convergente a quel punto. Valgono risultati analogi per i filtri: ###### Proposizione 2.6. Un filtro $\mathcal{A}$ converge a $x_{0}$ se e solo se per ogni $\mathcal{B}\vdash\mathcal{A}$, esiste $\mathcal{C}\vdash\mathcal{B}$ tale che $\mathcal{C}\to x_{0}$ ###### Proof. Dalla proposizione precedente risulta ovvia una delle due implicazioni. Per dimostrare l’altra, supponiamo che $\mathcal{A}$ non converga a $x_{0}$. Quindi esiste $U(x_{0})$ intorno di $x_{0}$ tale che $\forall\alpha,\ A_{\alpha}\cap U^{C}\neq\emptyset$. La collezione $\mathcal{B}=\\{A_{\alpha}\cap U^{C}\ t.c.\ \alpha\in I\\}$ è un filtro subordinato a $\mathcal{A}$ e per il quale $x_{0}$ non è punto di accumulazione. Allora grazie alla proposizione precedente risulta che ogni filtro subordinato a $\mathcal{B}$ non può convergere a $x_{0}$. ∎ ###### Proposizione 2.7. $\mathcal{A}\triangleright x_{0}$ se e solo se esiste $\mathcal{B}\vdash\mathcal{A}$ tale che $\mathcal{B}\to x_{0}$ ###### Proof. Se esiste $\mathcal{B}\vdash\mathcal{A}$ tale che $\mathcal{B}\to x_{0}$, è chiaro che $\mathcal{A}\triangleright x_{0}$. Supponiamo ora che $\mathcal{A}\triangleright x_{0}$. La collezione $\mathcal{B}=\mathcal{A}\cap\mathcal{U}(x_{0})$ è un filtro dato che $\forall\alpha\ \forall U(x_{0}),\ A_{\alpha}\cap U\neq\emptyset$. Inoltre è chiaro che $\mathcal{B}\vdash\mathcal{U}(x_{0})$. Allora per le proprietà della relazione di subordinazione $\mathcal{B}\to x_{0}$. ∎ I filtri si possono definire anche tramite funzioni. ###### Proposizione 2.8. Dato un filtro $\mathcal{A}$ su $X$ e una funzione $f:X\to Y$, la collezione $f(\mathcal{A})=\\{f(A_{\alpha}\ t.c.\ A_{\alpha}\in\mathcal{A}\\}$ è un filtro su $Y$ ###### Proof. Il fatto che ogni elemento di $f(\mathcal{A})$ sia non vuoto è ovvio. La seconda proprietà caratterizzante i filtri segue facilmente dalla considerazione che $f(A\cap B)\subset f(A)\cap f(B)$ ∎ Passiamo ora a definire gli ultrafiltri, o filtri massimali, e a dimostrarne l’esistenza. ###### Definizione 2.9. Un filtro $\mathcal{M}$ è detto massimale (o ultrafiltro) se non ha filtri propriamente subordinati, cioè se $\displaystyle\mathcal{A}\vdash\mathcal{M}\Rightarrow\mathcal{M}\vdash\mathcal{A}$ Non tutti gli autori condividono questa definizione. Su alcuni testi si definisce ultrafiltro un filtro che non è contenuto propriamente in nessun altro filtro. Anche la definizione di subordinazione è diversa. Si definisce $\mathcal{B}$ subordinato a $\mathcal{A}$ se $\mathcal{A}\subset\mathcal{B}$. In questo modo la relazione di subordinazione diventa una relazione d’ordine parziale. Per gli scopi di questa tesi però non c’è differenza tra le due definizioni, quindi adotteremo quella meno restrittiva e più generale di [D]. Passiamo ora a caratterizzare gli ultrafiltri e a dimostrarne l’esistenza. ###### Proposizione 2.10. Un filtro $\mathcal{M}=\\{M_{\alpha}\ t.c.\ \alpha\in I\\}$ è un ultrafiltro se e solo se per ogni $S\subset X$, uno dei due insiemi $S$ o $S^{C}$ contiene un elemento di $\mathcal{M}$. ###### Proof. Dato un ultrafiltro $\mathcal{M}$ e un insieme $S$ è chiaro che non può accadere che sia $S$ che $S^{C}$ contengano un elemento di $\mathcal{M}$, altrimenti questi due elementi avrebbero necessariamente intersezione vuota. Assumiamo allora che $\forall\alpha,\ M_{\alpha}$ non sia contenuto in $S$, allora $\forall\alpha M_{\alpha}\cap S^{C}\neq\emptyset$. Questo implica che l’insieme $\mathcal{M}_{1}=\mathcal{M}\cap S^{C}=\\{A_{\alpha}\cap S^{C}\ t.c.\ \alpha\in I\\}$ sia un filtro tale che $\mathcal{M}_{1}\vdash\mathcal{M}$. Ma allora per ipotesi $\mathcal{M}\vdash\mathcal{M}_{1}$, quindi per ogni $\alpha$ esiste $\gamma$ tale che $A_{\gamma}\subset A_{\alpha}\cap S^{C}\subset S^{C}$. L’altra implicazione si dimostra considerando un filtro $\mathcal{M}_{1}\vdash\mathcal{M}$. L’ipotesi assicura che per ogni $A^{1}_{\beta}\in\mathcal{M}_{1}$ esiste un $\mathcal{A}_{\alpha}\in\mathcal{M}$ tale che $\mathcal{A}_{\alpha}\subset A^{1}_{\beta}$ oppure $\mathcal{A}_{\alpha}\subset(A^{1}_{\beta})^{C}$. La seconda possibilità è esclusa dal fatto che $\mathcal{M}_{1}\vdash\mathcal{M}$, quindi $\mathcal{M}\vdash\mathcal{M}_{1}$, cioè $\mathcal{M}$ è massimale. ∎ Grazie a questa caratterizzazione è facile dimostrare che dato $x_{0}\in X$, l’insieme $\mathcal{M}=\\{A\subset X\ t.c.\ x_{0}\in A\\}$ è un ultrafiltro, anche se è abbastanza banale e poco pratico. Per i nostri scopi è necessario dimostrare che ogni filtro è contenuto in un ultrafiltro, e per fare questo è necessario assumere il lemma di Zorn, che ricordiamo: ###### Teorema 2.11 (Lemma di Zorn). Ogni insieme preordinato in cui ogni catena contiene un limite superiore possiede almeno un elemento massimale Solitamente il lemma di Zorn è enunciato per insiemi parzialmente ordinati e non semplicemente preordinati. Si può dimostrare però che queste due versioni del teorema sono equivalenti (a questo scopo rimandiamo a [M1], esercizio 1.16). Osserviamo che negli insiemi preordinati (con una relazione d’ordine $\geq$) un elemento $m$ è detto massimale se $a\geq m\Rightarrow m\geq a$. ###### Teorema 2.12. Dato un insieme $X$ e un filtro $\mathcal{A}$ su questo insieme, esiste un filtro massimale (ultrafiltro) $\mathcal{M}$ tale che $\mathcal{M}\vdash\mathcal{A}$. ###### Proof. Chiamiamo $\hat{B}$ la famiglia dei filtri per i quali $\mathcal{B}\vdash\mathcal{A}$, e definiamo una relazione di preordine $\geq$ su questo insieme in questo modo: $\displaystyle\mathcal{B}_{2}\geq\mathcal{B}_{1}\Longleftrightarrow\mathcal{B}_{2}\vdash\mathcal{B}_{1}$ Ricordiamo che dalla proposizione 2.5, $\mathcal{B}_{1}\subset\mathcal{B}_{2}\Rightarrow\mathcal{B}_{2}\vdash\mathcal{B}_{1}$, quindi anche $\mathcal{B}_{2}\geq\mathcal{B}_{1}$. Consideriamo una catena $\mathcal{B}_{i}$ di elementi di $\hat{B}$ 222ricordiamo che con catena in un insieme preordinato si intende un suo sottoinsieme tale che per ogni sua coppia di elementi $a$ e $b$, $a\geq b$ oppure $b\geq a$, dove è possibile che entrambe le affermazioni siano vere. L’insieme $\tilde{\mathcal{B}}=\bigcup_{i}\mathcal{B}_{i}$ è un elemento massimale per questa catena. Per prima cosa dimostriamo che è un filtro. Ovviamente ogni suo elemento è non vuoto, inotlre dati due suoi elementi $A_{1}\in\tilde{\mathcal{B}}$ e $A_{2}\in\tilde{\mathcal{B}}$, esistono $\mathcal{B}_{i_{1}}$ e $\mathcal{B}_{i_{2}}$ filtri nella catena che li contengono. Supponiamo che $\mathcal{B}_{i_{1}}\vdash\mathcal{B}_{i_{2}}$, allora esiste $A_{3}\in\mathcal{B}_{i_{1}}$ tale che $A_{3}\subset A_{2}$, e quindi poiché $\mathcal{B}_{i_{1}}$ è un filtro, esiste anche $E\in\mathcal{B}_{i_{1}}$ tale che $E\subset A_{1}\cap A_{3}\subset A_{1}\cap A_{2}$. È poi evidente che $\tilde{\mathcal{B}}\geq\mathcal{B}_{i}\geq\mathcal{A}$ per ogni $i$, quindi $\tilde{\mathcal{B}}$ è un limite superiore per la catena. Applicando il lemma di Zorn otteniamo che esiste almeno un elemento massimale $\mathcal{M}$ in $\hat{B}$. $\mathcal{M}$ è un filtro massimale perché se $\mathcal{C}\vdash\mathcal{M}$, per transitività $\mathcal{C}\vdash\mathcal{A}$, quindi $\mathcal{C}\in\hat{B}$ e quindi per massimalità $\mathcal{M}\geq\mathcal{C}$, cioè $\mathcal{M}\vdash\mathcal{C}$. ∎ Osserviamo ora due proprietà degli ultrafiltri che ci saranno utili nel seguito: ###### Proposizione 2.13. Dato un ultrafiltro $\mathcal{M}$ su $X$, $\mathcal{M}\triangleright x_{0}$ se e solo se $\mathcal{M}\to x_{0}$ ###### Proof. Questa proposizione è un corollario della definizione di ultrafiltro e della proposizione 2.7 ∎ ###### Proposizione 2.14. Data una qualunque funzione $f:X\to Y$ insiemi qualsiasi, se $\mathcal{M}$ è un ultrafiltro su $X$, allora $f(\mathcal{M})=\\{f(M_{\alpha})\ t.c.\ M_{\alpha}\in\mathcal{M}\\}$ è un ultrafiltro in $Y$. ###### Proof. $f(\mathcal{M})$ è un filtro grazie alla proposizione 2.8, quindi rimane da dimostrare solo la sua massimalità. Utilizziamo la caratterizzazione 2.10 dei filtri massimali. Sia $S\in Y$. Visto che $f^{-1}(S^{C})=(f^{-1}(S))^{C}$, esiste un elemento $M_{\alpha}$ contenuto o in $f^{-1}(S)$ o nel suo complementare. $M_{\alpha}\subset f^{-1}(S)\Rightarrow f(M_{\alpha})\subset S$ e lo stesso vale per $S^{C}$, quindi dato ogni sottoinsieme di $Y$, esiste un elemento di $f(\mathcal{M})$ contenuto o in esso o nel suo complementare. ∎ Il tipo di ultrafiltri massimali che ci interessano sono gli ultrafiltri non costanti, ovvero gli ultrafiltri che non contengono nessun elemento finito ###### Definizione 2.15. Un ultrafiltro $\mathcal{M}$ è detto non costante se e solo se non contiene nessun insieme di cardinalità finita. Per dimostrare l’esistenza di ultrafiltri non costanti sull’insieme dei numeri naturali costruiamo un filtro con queste caratteristiche e lo estendiamo a massimale grazie al teorema 2.12 333dato che il teorema fa uso del lemma di Zorn, questa dimostrazione non è costruttiva. Non si possono cioè trovare esempi descrivibili facilmente di ultrafiltri massimali non costanti con questa tecnica.. ###### Osservazione 2.16. La collezione $\mathcal{F}$ di sottoinsiemi a complementare finito è un filtro sull’insieme $\mathbb{N}$. ###### Proof. La dimostrazione è immediata ∎ Indichiamo con $\mathcal{M}_{\mathcal{F}}$ un ultrafiltro subordinato al filtro $\mathcal{F}$. Questo ultrafiltro necessariamente non conterrà nessun insieme a cardinalità finita. Sia per assurdo $S\in\mathcal{M}_{\mathcal{F}}$ insieme a cardinalità finita. $S^{C}\in\mathcal{F}$ per definizione. Ma visto che $\mathcal{M}_{\mathcal{F}}\vdash\mathcal{F}$, esiste un elemento di $\mathcal{M}_{\mathcal{F}}$ contenuto in $S^{C}$, ma allora questo elemento ha intersezione vuota con $S$, assurdo. Prima di concludere questa sezione riportiamo la relazione tra compattezza di un insieme e filtri su quell’insieme. ###### Proposizione 2.17. Per un insieme $K\subset(X,\tau)$ spazio topologico le seguenti affermazioni sono equivalenti: 1. 1. Ogni ricoprimento di aperti di $M$ ha un sottoricoprimento finito 2. 2. Per ogni famiglia di chiusi $C_{i}$ tale che $\bigcap_{i}C_{i}=\emptyset$, esiste un numero finito di indici $i_{1},\dots,i_{n}$ tali che $\bigcap_{k=1}^{n}C_{i_{k}}=\emptyset$ 3. 3. Ogni filtro su $K$ ha un punto di accumulazione 4. 4. Ogni ultrafiltro in $K$ è convergente Se una qualsiasi delle proprietà precedenti è valida, l’insieme $K$ è detto compatto. ###### Proof. (1) e (2) sono equivalenti grazie alle leggi di De Morgan, (3)$\Rightarrow$(4) grazie alla proposizione 2.13, (4)$\Rightarrow$(3) grazie al teorema 2.12 e alla proposizione 2.5. (2)$\Rightarrow$(3) poiché dato un filtro $\mathcal{A}$, ogni collezione finita di suoi elementi $A_{\alpha_{i}}$ ha intersezione non vuota, quindi anche $\bigcap_{i=1}^{n}\overline{A_{\alpha_{i}}}\neq\emptyset$. Grazie a (2) allora $\bigcap_{\alpha}\overline{A_{\alpha}}\neq\emptyset$, quindi l’insieme dei punti di accumulazione di $\mathcal{A}$ è non vuoto. (3)$\Rightarrow$(2) perché data una qualsiasi collezione di chiusi $C_{\alpha}$ tale che ogni sua sottocollezione finita abbia intersezione non vuota, la collezione dei chiusi e di tutte le possibili intersezioni finite è un filtro, che per (2) ha un punto di accumulazione, quindi $\bigcap_{\alpha}\overline{C_{\alpha}}=\bigcap_{\alpha}C_{\alpha}\neq\emptyset$. ∎ Vale anche la seguente caratterizzazione della continuità delle funzioni tramite ultrafiltri: ###### Proposizione 2.18. Una funzione $f:X\to Y$ spazi topologici è continua in $x_{0}$ se e solo se per ogni filtro $\mathcal{A}\to x_{0}$, $f(\mathcal{A})\to f(x_{0})$ ###### Proof. Supponiamo che $f$ sia continua in $x_{0}$. Chiamando $\mathcal{U}(x)$ il filtro degli intorni aperti di $x_{0}$, dalla definizione di continuità si ha che per ogni intorno aperto $W$ di $f(x_{0})$, esiste un aperto $U$ in $X$ tale che $f(U)\subset W$. Quindi per definizione $f(\mathcal{U}(x_{0}))\to f(x_{0})$. Ora, se consideriamo un filtro $\mathcal{A}\to x_{0}$ qualsiasi, dalle proposizioni precedenti risulta che $\mathcal{A}\vdash\mathcal{U}(x_{0})$, quindi anche $f(\mathcal{A})\vdash f(\mathcal{U}(x_{0}))$, quindi necessariamente $f(\mathcal{A})\to f(x_{0})$. Per dimostrare l’implicazione inversa, dato che $f(\mathcal{U}(x_{0}))\to f(x_{0})$, per definizione per ogni intorno di $W(f(x_{0}))$, esiste $U(x_{0})$ tale che $f(U(x_{0}))\subset W$, quindi $f$ è continua. ∎ Per una successione in $\mathbb{R}$ possono accadere due cose: o la successione è limitata, quindi tutta contenuta in un compatto, oppure no. Se la successione è limitata e converge, allora converge a un punto al finito di $\mathbb{R}$, altrimenti una successione illimitata può “convergere” all’infinito. Per filtri e ultrafiltri /che giocano il ruolo delle successioni “convergenti” in senso generalizzato), vale una caratterizzazione simile. ###### Definizione 2.19. Un filtro $\mathcal{A}$ si dice avere supporto in un insieme $K\subset X$ se esiste $\alpha$ tale che $A_{\alpha}\subset K$ Da questa definizione si ricava immediatamente che ###### Proposizione 2.20. Se $\mathcal{A}$ ha supporto in $K$, allora $\mathcal{A}\triangleright x_{0}\Rightarrow x_{0}\in\overline{K}$ (quindi anche $\mathcal{A}\to x_{0}\Rightarrow x_{0}\in\overline{K}$) ###### Proof. La dimostrazione è una conseguenza quasi immediata della definizione. Supponiamo per assurdo che $x_{0}\not\in\overline{K}$. Allora esiste $U$ intorno di $x_{0}$ separato da $\overline{K}$. Ma allora $A_{\alpha}\cap U=\emptyset$ (dove $\alpha$ è l’indice tale che $A_{\alpha}\subset K$), quindi $x_{0}$ non può essere punto di accumulazione per $A$. ∎ ###### Proposizione 2.21. Se $\mathcal{A}$ ha supporto in $K$, allora $\mathcal{B}\equiv\mathcal{A}\cap K$ è un filtro subordinato ad $\mathcal{A}$. Inoltre se $\mathcal{A}$ è un ultrafiltro, anche $\mathcal{B}$ è un ultrafiltro. ###### Proof. Visto che esiste $\alpha$ tale che $A_{\alpha}\subset K$, allora ogni insieme del filtro $\mathcal{A}$ ha intersezione non vuota con $K$, quindi $\mathcal{B}\equiv\mathcal{A}\cap K$ ben definisce un filtro, ovviamente subordinato ad $\mathcal{A}$. Il fatto che $\mathcal{B}$ sia un ultrafiltro se $\mathcal{A}$ lo è è una facile conseguenza della caratterizzazione 2.10 ∎ Un filtro si dice avere supporto compatto se esiste un insieme compatto che è un suo supporto. In uno spazio localmente compatto, ogni filtro convergente ha supporto compatto (basta considerare un intorno compatto del punto al quale il filtro converge). Se il filtro $\mathcal{A}$ non ha supporto compatto, vuol dire che per ogni compatto $K$ e per ogni indice $\alpha$, $A_{\alpha}\cap K^{C}\neq\emptyset$, quindi per ogni compatto $K$ il filtro $\mathcal{B}=\mathcal{A}\cap K^{C}$ è un filtro subordinato ad $\mathcal{A}$, e se $\mathcal{A}$ è un ultrafiltro, anche $\mathcal{B}$ lo è. ### 2.2 Applicazioni: caratteri sulle successioni limitate Come applicazione delle due sezioni precedenti, e soprattutto come esempio per il seguito, presentiamo l’esistenza di particolari caratteri sullo spazio $l_{\infty}(\mathbb{N})$, cioè lo spazio di Banach delle successioni limitate a valori reali dotato della norma del sup. Questo spazio diventa un’algebra di Banach se definiamo la moltiplicazione tra successioni punto per punto, cioè date due successioni $x(n)$ e $y(n)$, definiamo $(x\cdot y)(n)\equiv x(n)\cdot y(n)$ 444lasciamo le dovute verifiche al lettore. È facile osservare che la successione costante uguale a 1 è l’unità di ques’algebra. Tutti i funzionali $\phi_{n}:l_{\infty}(\mathbb{N})\to\mathbb{R}$ definiti da $\phi_{n}(x)=x(n)$ sono caratteri. Cerchiamo però di definire altri funzionali, legati solo al comportamento di ogni successione all’infinito. Questa sezione è tratta dall’articolo [N]. Per prima cosa definiamo un’operazione di limite su $l_{\infty}(\mathbb{N})$: ###### Definizione 2.22. Un’operazione di limite sullo spazio $l_{\infty}(\mathbb{N})$ è un funzionale lineare $\phi:l_{\infty}(\mathbb{N})\to\mathbb{R}$ tale che per ogni successione $\displaystyle\liminf_{n}x(n)\leq\phi(x)\leq\limsup_{n}x(n)$ Un’operazione di limite è quindi un funzionale lineare che vale $\lim_{n}x(n)$ se questo limite esiste. Grazie al teorema di Hahn-Banach è possibile dimostrare l’esistenza di alcune operazioni di limite (vedi ad esempio [N], oppure l’esercizio 4 cap.3 pag.85 di [R2]). È anche possibile dimostrare l’esistenza di operazioni di limite moltiplicative su $l_{\infty}(\mathbb{N})$, e a questo scopo utilizzeremo la teoria sviluppata sugli ultrafiltri non costanti. Prima della proposizione ricordiamo che: ###### Osservazione 2.23. Ricordiamo la definizione di somma di insiemi. Dati due sottoinsiemi $A$ e $B$ di uno spazio vettoriale $V$, $A+B=\\{a+b\ \ t.c.\ \ a\in A,\ b\in B\\}$. Dalla definizione è chiaro che $(f+g)(A)\subset f(A)+g(A)$. In modo analogo alla somma di insiemi si può definire il loro prodotto, e vale ancora che $(f\cdot g)(A)\subset f(A)\cdot g(A)$. ###### Proposizione 2.24. Sull’insieme $l_{\infty}(\mathbb{N})$ esistono operazioni di limite moltiplicative. ###### Proof. Fissiamo un ultrafiltro non costante $\mathcal{M}$ sull’insieme $\mathbb{N}$ 555l’esistenza di questo tipo di ultrafiltro è stata dimostrata nell’osservazione 2.16. Una qualunque successione $x:\mathbb{N}\to\mathbb{R}$ permette di definire un ultrafiltro $x(\mathcal{M})$ su $\mathbb{R}$, anzi visto che la successione è limitata, l’ultrafiltro sarà contenuto nell’insieme compatto $[-\left\|x\right\|,\left\|x\right\|]$, quindi grazie a 2.17 sarà convergente. Definiamo $\displaystyle\phi_{\mathcal{M}}(x)\equiv\lim_{\mathcal{M}}x\equiv\lim x(\mathcal{M})$ Per prima cosa osserviamo che se sostituiamo $\mathcal{M}$ con il filtro $\mathcal{F}$ degli insiemi a complementare finito, il limite non è sempre definito ma coincide con il limite di una successione in senso classico. Resta da dimostrare che $\phi_{\mathcal{M}}$ è lineare e moltiplicativo. A questo scopo consideriamo due successioni $x,\ y:\mathbb{N}\to\mathbb{R}$, e dimostriamo che $\phi_{\mathcal{M}}(x+y)=\phi_{\mathcal{M}}(x)+\phi_{\mathcal{M}}(y)$. Sia $\phi_{\mathcal{M}}(x)=s$ e $\phi_{\mathcal{M}}(y)=t$. Dalla definizione di limite di ultrafiltro otteniamo che per ogni $\epsilon>0$, esistono $\alpha$ e $\beta$ tali che $\displaystyle x(M_{\alpha})\subset B_{\epsilon}(s)\ \wedge\ y(M_{\beta})\subset B_{\epsilon}(t)$ dove $M_{\alpha},\ M_{\beta}\in\mathcal{M}$ e $B_{\epsilon}(s)$ indica l’insieme aperto $(s-\epsilon,s+\epsilon)$. Dato che $\mathcal{M}$ è un filtro, esiste $\gamma$ tale che $M_{\gamma}\subset M_{\alpha}\cap M_{\beta}$. Questo significa che per ogni $\epsilon>0$ $\displaystyle(x+y)(M_{\gamma})\subset x(M_{\gamma})+y(M_{\gamma})\subset B_{\epsilon}(s)+B_{\epsilon}(t)=B_{2\epsilon}(s+t)$ dove abbiamo usato l’osservazione 2.23. Questo dimostra che $\lim_{\mathcal{M}}(x+y)=\lim_{\mathcal{M}}x+\lim_{\mathcal{M}}y$ Consideriamo ora un qualunque numero $a\in\mathbb{R}$ e una successione $x\in l_{\infty}(\mathbb{N})$. La dimostrazione del fatto che $\phi_{\mathcal{M}}(ax)=a\phi_{\mathcal{M}}(x)$ è del tutto analoga a quella appena mostrata, quindi lasciamo i dettagli al lettore 666questo completa la dimostrazione che $\phi_{\mathcal{M}}$ è lineare.. La dimostrazione che $\phi_{\mathcal{M}}$ è moltiplicativo è anch’essa analoga a questa, l’unico punto un po’ più difficile è capire come caratterizzare l’insieme $B_{\epsilon}(s)\cdot B_{\epsilon}(t)$. Osserviamo che $\displaystyle\left|a-s\right|<\epsilon\ \wedge\ \left|b-t\right|<\epsilon\ \Longrightarrow$ $\displaystyle\Longrightarrow\left|ab-st\right|\leq\left|ab- at\right|+\left|at- st\right|\leq\epsilon(\left|a\right|+t)\leq\epsilon(\left|s\right|+\left|t\right|+\epsilon)$ quindi $B_{\epsilon}(s)\cdot B_{\epsilon}(t)\subset B_{[\epsilon(\left|s\right|+\left|t\right|+\epsilon)]}(st)$. Dato che $\epsilon(\left|s\right|+\left|t\right|+\epsilon)$ diventa piccolo a piacere al diminuire di $\epsilon$, seguendo un ragionamento del tutto analogo al precedente, si ottiene che per ogni $\epsilon>0$, esiste $\gamma$ tale che $\displaystyle(x\cdot y)(M_{\gamma})\subset x(M_{\gamma})\cdot y(M_{\gamma})\subset B_{\epsilon}(s+t)$ da cui per ogni ultrafiltro $\mathcal{M}$, $\phi_{\mathcal{M}}$ è lineare e moltiplicativo (quindi anche continuo) su $l_{\infty}(\mathbb{N})$. Resta da dimostrare solo che è un’operazione di limite. A questo scopo procediamo per assurdo: sia $\phi_{\mathcal{M}}(x)>\limsup_{n}x(n)$, e scegliamo $p$ in modo che $\phi_{\mathcal{M}}(x)>p>\limsup_{n}x(n)$. L’insieme $(p,\infty)$ è un intorno di $\phi_{\mathcal{M}}(x)$, quindi contiene un elemento di $x(\mathcal{M})$, quindi esiste un elemento di $\mathcal{M}$ contenuto in $x^{-1}(p,\infty)$. Dato che $p>\limsup_{n}(x)$, l’insieme $x^{-1}(p,\infty)$ ha cardinalità finita, quindi per definizione di ultrafiltro non costante è impossibile che $x^{-1}(p,\infty)$ contenga qualunque elemento di $\mathcal{M}$, da cui $\phi_{\mathcal{M}}(x)\leq\limsup_{n}x(n)$ $\forall x,\ \forall\mathcal{M}$ ultrafiltro non costante. Osservando che $\liminf(x)=-\limsup(-x)$, si ottiene immediatamente anche l’altra disuguaglianza. ∎ Quello che abbiamo ottenuto è un’operazione $\phi_{\mathcal{M}}$ che è lineare, moltilicativa, definita su ogni successione limitata e che coincide con il limite standard quando questo è definito. Il limite standard, oltre che ad essere lineare e moltiplicativo, è anche invariante per traslazioni, nel senso che $\lim_{n}x(n)=\lim_{n}x(n+1)$ quando è definito. Possiamo chiederci se vale una proprietà simile per $\phi_{\mathcal{M}}$. ###### Osservazione 2.25. Nessuna operazione di limite lineare e moltiplicativa è anche invariante per traslazioni ###### Proof. Dimostriamo questa affermazione per assurdo. Consideriamo la successione limitata $x:\mathbb{N}\to\mathbb{R}$, $x(n)\equiv(-1)^{n}$, e sia $\phi$ un’operazione di limite lineare e moltiplicativa. Dato che $\phi(x)^{2}=\phi(x\cdot x)=\phi(1)=1$, $\phi(x)=\pm 1$. Per linearità $\phi(x(n))+\phi(x(n+1))=\phi(x(n)+x(n+1))=0$. Se $\phi(x(n))=\phi(x(n+1))$, allora dall’ultima uguaglianza si otterrebbe $\phi(x)=0$, assurdo. ∎ Ricordiamo che esistono operazioni di limite lineari invarianti per traslazione (che evidentemente non possono essere moltiplicative). La loro esistenza è dimostrata ad esempio su [R2] nell’esercizio 4 cap.3 pag.85. ## Chapter 3 Algebra e compattificazione di Royden Questo capitolo segue le orme del capitolo III di [SN] e dell’articolo [CSL]. Lo scopo è quello di introdurre l’algebra di Royden su varietà Riemanniane (di dimesione finita qualsiasi) e quindi la compattificazione di Royden delle varietà. ### 3.1 Funzioni di Tonelli #### 3.1.1 Definizioni e proprietà fondamentali Per prima cosa definiamo un’insieme particolare di funzioni, le funzioni di Tonelli. Lo scopo della sezione è dimostrare che queste funzioni su una varietà constituiscono un’algebra di Banach rispetto a una particolare norma. Iniziamo con il definire queste funzioni su rettangoli in $\mathbb{R}^{m}$ ###### Definizione 3.1. Una funzione $f:\prod_{i=1}^{m}(a_{i},b_{i})\to\mathbb{R}$ si dice di Tonelli se soddisfa: 1. 1. $f$ è continua su $\prod_{i=1}^{m}(a_{i},b_{i})$ 2. 2. per ogni $i$, $f^{i}(z)=f(\bar{x}_{1},\dots,\bar{x}_{i-1},z,\bar{x}_{i+1},\dots,\bar{x}_{m})$ è assolutamente continua rispetto a $z$ per quasi ogni valore di $(\bar{x}_{1},\dots,\bar{x}_{m})$ 111rimandiamo alla sezione 1.2 per la definizione e alcune proprietà delle funzioni assolutamente continue 3. 3. Per ogni $i=1,\cdots,m$, $\frac{\partial f}{\partial x_{i}}$ è a quadrato integrabile su ogni sottoinsieme compatto di $\prod_{i=1}^{m}(a_{i},b_{i})$, cioè $\frac{\partial f}{\partial x_{i}}\in L^{2}_{loc}(\prod_{i=1}^{m}(a_{i},b_{i}))$ Notiamo che tutte le funzioni $C^{\infty}(\prod_{i=1}^{m}(a_{i},b_{i}),\mathbb{R})$ sono ovviamente funzioni di Tonelli. L’insieme delle funzioni di Tonelli è ovviamente uno spazio vettoriale, e in un certo senso è l’insieme pià piccolo di funzioni continue per cui ha senso parlare di integrale di Dirichlet. A questo proposito dimostriamo che: ###### Proposizione 3.2. Data una funzione $f:M\to\mathbb{R}$ che sia di Tonelli su un insieme compatto $K\Subset U$, dove $U=\phi^{-1}(\prod_{i=1}^{m}(a_{i},b_{i}))$ è un insieme coordinato, allora detto $\displaystyle D_{K}(f)=\int_{K}\left|\nabla(f)\right|^{2}dV=\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}\sqrt{\left|g\right|}dx^{1}\dots dx^{n}$ (3.1) si ha che $D_{K}(f)$ è finito. ###### Proof. Dalla definizione sappiamo che $\int_{\phi(K)}\sum_{i=1}^{m}\left|\frac{\partial f}{\partial x^{i}}\right|^{2}dx^{1}\dots dx^{n}<\infty$. Iniziamo la dimostrazione col notare che per ogni punto $p$ e ogni m-upla di numeri $(y_{1},\dots,y_{m})$, esiste una costante $c_{p}$ (indipendente dai numeri $y_{i}$) tale che: $\displaystyle c_{p}^{-1}\sum_{i=1}^{m}(y_{i})^{2}\leq\sum_{i,j=1}^{m}g^{ij}y_{i}y_{j}\leq c_{p}\sum_{i=1}^{m}(y_{i})^{2}$ (3.2) questo è vero grazie al fatto che $\sum_{i,j=1}^{m}g^{ij}y_{i}y_{j}$ definisce una norma su $\mathbb{R}^{m}$, e tutte le norme negli spazi finito- dimensionali sono equivalenti. Inoltre questa costante $c_{p}$ dipende con continuità da $p$ 222le funzioni $g^{ij}$ dipendono con continuità da $p$, quindi su ogni compatto $K\Subset M$, la funzione $c_{p}$ ha un massimo, che indicheremo semplicemente con $c$. Da queste considerazioni risulta che: $\displaystyle g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}\leq\sum_{i=1}^{m}c\left|\frac{\partial f}{\partial x^{i}}\right|^{2}$ (3.3) dove gli indici ripetuti si intendono sommati. A questo punto, dato che su ogni compatto anche $\sqrt{\left|g\right|}$ assume massimo finito (che indicheremo $G$), si ottiene facilmente che: $\displaystyle D_{K}(f)=\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}\sqrt{\left|g\right|}dx^{1}\dots dx^{n}\leq cG\sum_{i=1}^{m}\int_{\phi(K)}\left|\frac{\partial f}{\partial x^{i}}\right|^{2}dx^{1}\dots dx^{m}<\infty$ ∎ La definzione di funzione di Tonelli può essere estesa in maniera naturale anche a funzioni $f:M\to\mathbb{R}$ ###### Definizione 3.3. Una funzione $f:M\to\mathbb{R}$ si dice di Tonelli se è di Tonelli in ogni insieme parametrizzabile come rettangolo. La proposizione precedente assicura che su ogni compatto coordinato (contenuto in un rettangolo coordinato per la precisione), l’integrale di Dirichlet di una funzione di Tonelli è finito. La stessa cosa vale anche per un qualsiasi insieme compatto. ###### Proposizione 3.4. Per ogni compatto $K\Subset M$, $D_{K}(f)<\infty$ se $f$ è una funzione di Tonelli ###### Proof. La dimostrazione segue tecniche standard della geometria differenziale. Per ogni punto $p$ della varietà $M$ è possibile trovare un intorno compatto coordinato che descritto in carte locali abbia la forma $\prod_{i=1}^{m}[a_{i},b_{i}]$. Su ciascuno di questi insiemi l’integrale di Dirichlet di $f$ è finito grazie alla proposizione precedente. Dato che l’insieme $K$ può essere ricoperto da un numero finito di questi insiemi, anche l’integrale di Dirichlet su $K$ sarà finito. ∎ #### 3.1.2 Operazioni con le funzioni di Tonelli Oltre che somma e moltiplicazione per un numero reale, le funzioni di Tonelli ammettono anche altre operazioni tra loro. In particolare il valore assoluto di una funzione di Tonelli è di Tonelli, e quindi anche massimo e minimo tra due funzioni di Tonelli sono funzioni di Tonelli. In questo paragrafo ci occuperemo di queste operazioni. ###### Proposizione 3.5. Se $f$ è di Tonelli, anche $\left|f\right|$ è di Tonelli. Inoltre su ogni insieme misurabile $S\subset M$, $D_{S}(\left|f\right|)\leq D_{S}(f)$. ###### Proof. Consideriamo una funzione $f:\prod_{i=1}^{m}(a_{i},b_{i})\to\mathbb{R}$ di Tonelli. Ovviamente $\left|f\right|$ è una funzione continua. L’assoluta continuità delle funzioni $\left|f\right|^{i}$ è garantita dal fatto che $\left|\left|f\right|^{i}(z_{1})-\left|f\right|^{i}(z_{2})\right|\leq\left|f^{i}(z_{1})-f^{i}(z_{2})\right|$ 333vedi definizione 1.29. Rimane da verificare solo la maggiorazione degli integrali di Dirichlet. Consideriamo come primo caso $K\Subset\prod_{i=1}^{m}(a_{i},b_{i})$. Sia $E$ un insieme di misura nulla tale che tutte le derivate $\partial\left|f\right|/\partial x^{i}$ e $\partial f/\partial x^{i}$ esistano su $L\equiv E^{C}=\prod_{i=1}^{m}(a_{i},b_{i})\setminus E$. Siano ora $L_{+}=f^{-1}(0,\infty)\cap L$, $L_{-}=f^{-1}(-\infty,0)\cap L$, $L_{0}=f^{-1}(0)\cap L$. Questi insiemi sono ovviamente misurabili. Sull’insieme $L_{+}$, $\partial\left|f\right|/\partial x^{i}=\partial f/\partial x^{i}$, infatti per ogni punto $x\in L_{+}$, grazie al teorema di permanenza del segno esiste un intorno di $x$ sul quale $\left|f\right|=f$. In modo analogo, su $L_{-}$, $\partial\left|f\right|\partial x^{i}=-\partial f/\partial x^{i}$. Su $L_{0}$ invece $\partial\left|f\right|\partial x^{i}=0$. Infatti per definizione: $\displaystyle\frac{\partial\left|f\right|}{\partial x^{i}}=\lim_{h\to 0}\frac{\left|f(\bar{x}^{1},\dots,\bar{x}^{i-1},x+h,\bar{x}^{i+1},\dots\bar{x}^{m})\right|-0}{h}$ dove abbiamo sfruttato il fatto che $x\in L_{0}$. Data la definizione di $L$, questo limite esiste necessariamente. Visto che scegliendo $h>0$ il limite è $\geq 0$ e scegliendo $h<0$ succede il contrario, questo limite è necessariamente nullo. Consideriamo ora un compatto $K\Subset\prod_{i=1}^{m}(a_{i},b_{i})$. Per additività degli integrali: $\displaystyle\int_{K}\left|\nabla\left|f\right|\right|^{2}dV=\int_{K\cap L_{+}}\left|\nabla\left|f\right|\right|^{2}dV+\int_{K\cap L_{-}}\left|\nabla\left|f\right|\right|^{2}dV+\int_{K\cap L_{0}}\left|\nabla\left|f\right|\right|^{2}dV$ Visto che in coordinate $\left|\nabla f\right|^{2}=g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}$ si ottiene che $\displaystyle\int_{K\cap L_{+}}\left|\nabla\left|f\right|\right|^{2}dV=\int_{K\cap L_{+}}\left|\nabla f\right|^{2}dV$ $\displaystyle\int_{K\cap L_{-}}\left|\nabla\left|f\right|\right|^{2}dV=\int_{K\cap L_{-}}\left|\nabla f\right|^{2}dV$ $\displaystyle\int_{K\cap L_{0}}\left|\nabla\left|f\right|\right|^{2}dV=0$ da cui si ottiene $D_{K}(\left|f\right|)\leq D_{K}(f)$. Per un insieme $S\subset M$ qualsiasi, basta applicare la definizione di integrale con le partizioni dell’unità. A questo scopo sia $\\{\lambda_{n}\\}$ una partizione dell’unità di $M$ subordinata a un ricoprimento di aperti coordinati. Per definizione $\displaystyle\int_{M}\left|\nabla f\right|^{2}dV=\sum_{n=1}^{\infty}\int_{supp(\lambda_{n})}\lambda_{n}\cdot\left|\nabla f\right|^{2}dV$ con un ragionamento analogo a quello illustrato sopra, si ottiene $D_{S}(\left|f\right|)\leq D_{S}(f)$ per ogni $S\subset M$ misurabile. ∎ Grazie a questa proposizione, è immediato verificare che ###### Proposizione 3.6. Date due funzioni di Tonelli $f$ e $g$, anche $\max\\{f,g\\}$ e $\min\\{f,g\\}$ sono funzioni di Tonelli ###### Proof. La dimostrazione è immediata se si considera che $\displaystyle\max\\{f,g\\}=\frac{1}{2}(f+g)+\frac{1}{2}\left|f-g\right|\ \ \ \min\\{f,g\\}=\frac{1}{2}(f+g)-\frac{1}{2}\left|f-g\right|$ ∎ ### 3.2 Algebra di Royden In questa sezione ci occupiamo dell’algebra di Royden su una varietà Riemanniana $R$ e delle sue proprietà. #### 3.2.1 Definizione ###### Definizione 3.7. Data una varietà riemanniana $R$, definiamo l’algebra di Royden $\mathbb{M}(R)$ l’insieme delle funzioni $f:R\to\mathbb{R}$ tali che: 1. 1. $f$ è una funzione continua e limitata su $R$ 2. 2. $f$ è una funzione di Tonelli 3. 3. $D_{R}(f)=\int_{R}\left|\nabla f\right|^{2}dV<\infty$ Notiamo subito che $\mathbb{M}(R)$ è un’algebra commutativa di funzioni, cioè un insieme chiuso rispetto a somma, prodotto e prodotto per uno scalare. ###### Proposizione 3.8. $\mathbb{M}(R)$ è un’algebra commutativa di funzioni ###### Proof. La verifica che $\mathbb{M}(R)$ è chiusa rispetto a somma e prodotto per uno scalare è ovvia. Rimane solo da verificare che date due funzioni $f,\ g\in\mathbb{M}(R)$, anche il loro prodotto appartiene all’algebra. Per prima cosa, se $f$ e $h$ sono continue e limitate, $f\cdot h$ è continua e limitata da $\left\|f\right\|_{\infty}\cdot\left\|h\right\|_{\infty}$ 444dove $\left\|f\right\|_{\infty}$ è la norma del sup. Se $f(\bar{x}^{1},\dots,\bar{x}^{i-1},x,\bar{x}^{i+1},\dots,\bar{x}^{m})$ e $h(\bar{x}^{1},\dots,\bar{x}^{i-1},x,\bar{x}^{i+1},\dots,\bar{x}^{m})$ sono entrambe assolutamente continue rispetto a $x$ (eventualità che si verifica quasi ovunque rispetto alle variabili barrate), allora la proposizione 1.31 garantisce che anche il prodotto sia assolutamente continuo, quindi esiste quasi ovunque la derivata del prodotto e vale che: $\displaystyle\frac{\partial(fh)}{\partial x^{i}}=\frac{\partial f}{\partial x^{i}}h+f\frac{\partial h}{\partial x^{i}}$ dove questa uguaglianza si intende quasi ovunque, quindi praticamente sempre dal punto di vista integrale. Mancano da verificare le proprietà sull’integrale di Dirichlet. Iniziamo con il verificare che dato un compatto $K$ contenuto in un rettangolo coordinato, $D_{K}(fh)<\infty$. Questa verifica segue dalla catena di disuguaglianze: $\displaystyle D_{K}(fh)=\int_{\phi(K)}g^{ij}\left(\frac{\partial f}{\partial x^{i}}h+f\frac{\partial h}{\partial x^{i}}\right)\left(\frac{\partial f}{\partial x^{j}}h+f\frac{\partial h}{\partial x^{j}}\right)\sqrt{\left|g\right|}dx^{1}\dots dx^{m}=$ $\displaystyle=\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}h^{2}dV+\int_{\phi(K)}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}f^{2}dV+2\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}fhdV$ dove $dV=\sqrt{\left|g\right|}dx^{1}\dots dx^{m}$ per semplicità di notazione. Considerando che la forma quadratica $g^{ij}$ è sempre definita positiva, l’argomento dei primi due integrali è sempre positivo, quindi il loro modulo può facilmente essere maggiorato da $\displaystyle\left|\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}h^{2}dV+\int_{\phi(K)}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}f^{2}dV\right|\leq$ $\displaystyle\leq\left\|h\right\|_{\infty}^{2}D_{K}(f)+\left\|f\right\|_{\infty}^{2}D_{K}(h)$ Per l’ultimo integrale, applichiamo due volte la disuguaglianza di Schwartz (vedi ad esempio paragrafo 10.8 pag 210 di [R1]) in modo da ottenere: $\displaystyle\left|\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}fhdV\right|\leq\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}\int_{\phi(K)}\left|g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}\right|dV\leq$ $\displaystyle\leq\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}\int_{\phi(K)}\left|g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}\right|^{1/2}\left|g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}\right|^{1/2}dV\leq$ $\displaystyle\leq\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}\left(\int_{\phi(K)}g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}dV\right)^{1/2}\left(\int_{\phi(K)}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}dV\right)^{1/2}=$ $\displaystyle=\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}D_{K}(f)^{1/2}D_{K}(g)^{1/2}$ dove dalla prima alla seconda riga si sfrutta il fatto che $g^{ij}$ è definita positiva, quindi $g^{ij}x_{i}y_{j}$ è in prodotto scalare tra $x$ e $y$ 555quindi possiamo applicare $\left|\left\langle x\middle|y\right\rangle\right|\leq\left\|x\right\|\left\|y\right\|$, e nella terza riga sfruttiamo la disuguaglianza di Schwartz per integrali. Riassumendo, con queste disuguaglianze otteniamo che $\displaystyle D_{K}(fh)\leq\left\|h\right\|_{\infty}^{2}D_{K}(f)+2\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}(D_{K}(f)D_{K}(h))^{1/2}+\left\|f\right\|_{\infty}^{2}D_{K}(h)=$ (3.4) $\displaystyle=\left(\left\|h\right\|_{\infty}D_{K}(f)^{1/2}+\left\|f\right\|_{\infty}D_{K}(h)^{1/2}\right)^{2}$ Sfruttanto le partizioni dell’unità, con un argomento del tutto analogo a quello utilizzato nella dimostrazione di 3.5, si ottiene la stessa disuguaglianza anche per gli integrali di Dirichlet estesi a tutta $R$, quindi: $\displaystyle D_{R}(fh)\leq\left(\left\|h\right\|_{\infty}D_{R}(f)^{1/2}+\left\|f\right\|_{\infty}D_{R}(h)^{1/2}\right)^{2}$ (3.5) ∎ L’algebra $\mathbb{M}(R)$ è quindi un’algebra dotata di unità (la funzione costante uguale a 1). Ha senso chiedersi quali siano i suoi elementi invertibili. È chiaro che l’inversa di una funzione $f$ è necessariamente la funzione $f^{-1}=1/f$, che esiste ed è continua solo se $f\neq 0$ ovunque. Però questa funzione è limitata solo se $\inf\left|f\right|>0$. Questo suggerisce che ###### Proposizione 3.9. Data $f\in\mathbb{M}(R)$, $f^{-1}\in\mathbb{M}(R)$ se e solo se $\inf\left|f\right|>0$. ###### Proof. Supponiamo che $\inf\left|f\right|>0$. Allora $f^{-1}$ esiste ed è continua e limitata. Inoltre se $f$ è derivabile in $x$, lo è anche $f^{-1}$, con $\displaystyle\left.\frac{\partial f^{-1}}{\partial x^{i}}\right|_{f(x)}=\left.-\frac{1}{f^{2}}\frac{\partial f}{\partial x^{i}}\right|_{x}$ Dato che $\left|f\right|\geq\inf\left|f\right|>0$, si ottiene che $\displaystyle D_{S}(f^{-1})\leq\frac{1}{(\inf\left|f\right|)^{4}}D_{S}(f)$ per ogni sottoinsieme $S$ misurabile di $R$. Supponiamo invece che $\inf\left|f\right|=0$. Se esiste $x$ tale che $f(x)=0$, $f^{-1}$ non è definita, ma anche se la funzione non assume mai il valore $0$, $\inf\left|f\right|=0\Rightarrow f^{-1}$ non limitata. ∎ #### 3.2.2 Topologie sull’algebra di Royden In questa sezione ci occupiamo di definire alcune topologie sull’algebra di Royden $\mathbb{M}(R)$ e di caratterizzarne alcune proprietà, in particolare la completezza. Con una di queste topologie, $\mathbb{M}(R)$ diventa un’algebra di Banach. La prima topologia che introduciamo è $\tau_{C}$, la topologia della convergenza uniforme sui compatti. ###### Definizione 3.10. Data l’algebra $\mathbb{M}(R)$, definiamo una base per una topologia $\tau_{C}$ gli insiemi della forma $\displaystyle V(f,\epsilon,K)=\\{h\in\mathbb{M}(R)\ t.c.\ \ \left\|f-h\right\|_{\infty,K}\equiv\max_{p\in K}\\{\left|f(p)-h(p)\right|<\epsilon\\}\\}$ dove $K\Subset R$ qualsiasi. Lasciamo al lettore la facile verifica che questa è una base per una topologia. Si nota subito che $\tau_{C}$ è primo numerabile, infatti fissata $f$, e fissata un’esaustione $K_{n}$ di compatti in $R$, una base di intorni è ad esempio $\displaystyle V_{n}(f)=V\left(f,\frac{1}{n},K_{n}\right)$ Seguendo la traccia di [R2] (1.38 (c) a pagina 29), si ottiene che questa topologia è metrizzabile. Se $R$ è compatta, questa topologia è anche normabile, in caso contrario no poiché non è localmente limitata (vedi teorema 1.9 pag. 9 di [R2]). Una metrica per $\tau_{C}$ può essere ad esempio definita da $\displaystyle d(f,h)=\max_{n}\frac{1}{n}\frac{\left\|f-h\right\|_{n}}{1+\left\|f-h\right\|_{n}}\ \ \ \left\|f-h\right\|_{n}=\max_{p\in K_{n}}\left|f(p)-h(p)\right|$ Sia nel caso $R$ compatta che nel caso $R$ non compatta questa metrica non è completa. L’idea è che l’algebra di Royden contiene solo funzioni derivabili in qualche senso, e una topologia che non chiede alcun controllo sulle derivate non può essere completa. ###### Osservazione 3.11. La topologia $\tau_{C}$ non è completa su $\mathbb{M}(R)$. ###### Proof. Le funzioni lisce limitate a supporto compatto appartengono all’algebra di Royden. Questo insieme però è denso nell’insieme delle funzioni continue su $R$ rispetto a $\tau_{C}$, quindi come sottospazio $\mathbb{M}(R)$ non può essere chiuso, quindi $(\mathbb{M}(R),\tau_{c})$ non può essere uno spazio completo. ∎ La topologia $\tau_{C}$ essendo metrizzabile può essere caratterizzata dal comportamento delle sue successioni convergenti ###### Proposizione 3.12. Rispetto alla topologia $\tau_{C}$, $f_{n}\to f$ se e solo se $f_{n}$ converge localmente uniformemente ad $f$, cioè se e solo se $\displaystyle\forall K\Subset R\ \ \lim_{n}\max_{K}\left|f_{n}(p)-f(p)\right|=0$ In questo caso scriviamo che $\displaystyle f=C-\lim_{n}f_{n}$ (3.6) Un altro modo di definire una topologia vettoriale su $\mathbb{M}(R)$ è il seguente: ###### Definizione 3.13. Fissata $K_{n}$ un’esaustione di $R$ 666ricordiamo che esaustione significa che $K_{n}$ sono compatti, $K_{n}\Subset K_{n+1}^{\circ}$ e $\cup_{n}K_{n}=R$, definiamo una base per la topologia $\tau_{B}$ di intorni di $0$ come: $\displaystyle V(0,E_{n})=\\{f:R\to\mathbb{R}\ t.c.\ \max_{p\in K_{n}}\left|f(p)\right|<E_{n}\\}$ (3.7) dove $E_{n}$ è una qualunque successione di numeri strettamente positivi tali che $\lim_{n\to\infty}E_{n}=\infty$ La topologia su $R$ è univocamente determinata dalla sua invarianza per traslazioni. Visto che si verifica facilmente che $\displaystyle V(0,E_{n})\cap V(0,^{\prime}E_{n})=V(0,\min\\{E_{n},E^{\prime}_{n}\\})$ e che il minino di due successioni strettamente positive e tendenti a infinito mantiene queste proprietà, la definizione di base è ben posta. Osserviamo subito che la definizione non dipende dalla scelta di $K_{n}$ (cioè al variare di $K_{n}$, tutte le topologie generate sono equivalenti tra loro), e nella definizione si può fare l’ulteriore richiesta che $E_{n}$ sia una successione crescente senza perdere di generalità. È inoltre possibile descrivere in maniera alternativa questa topologia. A questo scopo, data una funzione $e:R\to\mathbb{R}$, diciamo che $\displaystyle\lim_{p\to\infty}e(p)=+\infty\ \ \Longleftrightarrow\ \ \forall N\in\mathbb{N},\ \exists K\Subset R\ t.c.\ e(p)>N\ \forall p\not\in K$ Con questa notazione la topologia $\tau_{B}$ può essere descritta da una base di intorni fatta così: $\displaystyle V(0,e)=\\{f:R\to\mathbb{R}\ t.c.\ \left|f(p)\right|<e(p)\\}$ Al variare di $e$ tra le funzioni continue positive che tendono a infinito, questa base di intorni definisce la stessa topologia definita dagli intorni in 3.7. Un altro modo differente per definire questa topologia è il seguente. Dato $N\in\mathbb{N}$, definiamo l’insieme $\displaystyle X_{N}\equiv\\{f\in\mathbb{M}(R)\ t.c.\ \left\|f\right\|_{\infty}\leq N\\}$ Dotiamo $X_{N}$ 777che ovviamente non è uno spazio vettoriale della topologia indotta da $(\mathbb{M}(R),\tau_{C})$. Diciamo che un insieme $V$ è aperto in $\tau_{B}$ se e solo se $\forall N$ l’intersezione $V\cap X_{N}$ è aperto nella topologia $(X_{N},\tau_{C})$. Lasciamo al lettore la verifica che questa topologia è una topologia vettoriale. ###### Proposizione 3.14. La topologia generata da questi aperti è la topologia $\tau_{B}$ definita sopra. ###### Proof. Consideriamo $V$ aperto intorno di $0$ secondo la vecchia definizione. Allora esiste una successione $E_{n}\to\infty$ tale che $\displaystyle V(0,E_{n})\subset V$ L’intersezione $V(0,E_{n})\cap X_{N}$ è l’insieme delle funzioni tali che: $\displaystyle V(0,E_{n})\cap X_{N}=\\{f\in X_{N}\ t.c.\ \left\|f\right\|_{\infty,\ K_{n}}\leq E_{n}\ \forall n\\}$ Consideriamo $E_{n}$ successione crescente. Dato che $E_{n}\to\infty$, esiste $\bar{n}$ tale che $E_{n}\geq N$ per ogni $n\geq\bar{n}$, questo significa che $\displaystyle\left\|f\right\|_{\infty}\leq N\ \Rightarrow\ \left\|f\right\|_{\infty,\ K_{\bar{n}}^{C}}\leq E_{n}$ per ogni $n\geq\bar{n}$. Quindi possiamo scrivere: $\displaystyle V(0,E_{n})\cap X_{N}=\\{f\in X_{N}\ t.c.\ \left\|f\right\|_{\infty,\ K_{n}}\leq E_{n}\ \forall n\leq\bar{n}\\}\supset$ $\displaystyle\supset\\{f\in X_{N}\ t.c.\ \left\|f\right\|_{\infty,\ K_{\bar{n}}}\leq E_{1}\\}$ e questo è per definizione un insieme della topologia $(X_{N},\tau_{C})$. Sia ora un insieme aperto intorno di $0$ secondo la nuova definizione, cioè tale che per ogni $N$: $\displaystyle V\cap X_{N}\in\tau_{C}$ Allora per $N=1$, esiste un insieme $K_{1}$ e un numero $E_{1}$ tale che $\displaystyle\\{f\ t.c.\ \left\|f\right\|_{\infty,\ K_{1}}\leq E_{1}\\}\subset V\cap X_{1}\subset V$ e lo stesso per ogni valore di $N$. Questo significa che esiste una successione $E_{N}$ tale che: $\displaystyle\\{f\ t.c.\ \left\|f\right\|_{\infty,\ K_{N}}\leq E_{N}\ \forall N\\}\subset V$ Supponiamo per assurdo che $E_{N}\not\to\infty$, quindi $E_{N}$ limitata da $M$. Questo è impossibile perchè altrimenti l’insieme: $\displaystyle\\{f\ t.c.\ \left\|f\right\|_{\infty,\ K_{N}}\leq E_{N}\ \forall N\\}\cap X_{M}$ non sarebbe aperto nella topologia di $(X_{M},\tau_{C})$. ∎ La topologia $\tau_{B}$ ha lo svantaggio di non essere I numerabile (se $R$ non è compatta), quindi non è una topologia metrizzabile e soprattutto non può essere descritta in maniera completa dal comportamento delle sue successioni convergenti. ###### Osservazione 3.15. $\tau_{B}$ non è I numerabile, cioè fissato un punto in $\mathbb{M}(R)$, non esiste una base numerabile di intorni del punto. ###### Proof. Supponiamo per assurdo che esista una base numerabile $V_{k}$ di intorni di $0$. Questi intorni avranno la forma $\displaystyle V_{k}\equiv V(0,E^{(k)}_{n})$ dove per ogni $k$ la successione $\\{E^{(k)}_{n}\\}$ è strettamente positiva e divergente. Costruiamo per induzione una successione strettamente crescente $k_{m}$ di interi tali che $\displaystyle\forall n\geq k_{m},\ \ E^{(m)}_{n}\geq m$ e definiamo la successione $B_{m}$ strettamente positiva e tendente a infinito come: $\displaystyle k_{m}\leq n<k_{m+1}\ \Rightarrow\ B_{n}=m/2$ i termini $B_{1},\cdots,B_{k_{1}-1}$ possono essere assegnati casualmente. Dalla definizione, otteniamo che per nessun valore di $k$ $B_{n}\leq E^{(k)}_{n}\ \forall n$, quindi non esiste $k$ per cui $V(0,E^{(k)}_{n})\subset V(0,B_{n})$, che contraddice il fatto che $\\{V(0,E^{(k)}_{n})\\}$ sia una base di intorni di $0$. ∎ Le successioni convergenti in questa topologia sono caratterizzate da ###### Proposizione 3.16. Una successione $f_{n}$ converge a $f$ rispetto alla topologia $\tau_{B}$ se e solo se: 1. 1. esiste $M$ tale che $\left\|f_{n}\right\|_{\infty}\leq M\ \forall n$ 2. 2. $f=C-\lim_{n}f_{n}$ ###### Proof. Dimostriamo prima che se valgono (1) e (2), allora la successione converge rispetto a $\tau_{B}$. Vista l’invarianza per traslazioni della topologia, è sufficiente verificare questa condizione con $f=0$. Consideriamo un qualsiasi intorno aperto di $0$ della forma $V(0,E_{n})$. Dato che $E_{n}\to\infty$, esiste un numero $N_{1}$ tale che $E_{n}>M\ \forall n\geq N_{1}$ Inoltre poiché $f_{n}$ converge localmente uniformemente a $0$, esiste un numero $N_{2}$ tale che $\displaystyle\forall n\geq N_{2},\ \max_{p\in K_{N-1}}\\{\left|f_{n}(p)\right|\\}\leq\min_{1\leq i\leq N-1}\\{E_{i}\\}$ quindi se $N=\max\\{N_{1},N_{2}\\}$, $f_{n}\in V\ \forall n\geq N$, cioè $f_{n}$ converge a $f$ rispetto a $\tau_{B}$. Per l’implicazione inversa, dato che $\tau_{C}\subset\tau_{B}$, la convergenza nella topologia $B$ implica la convergenza locale uniforme. Inoltre supponiamo per assurdo che $\left\|f_{n}\right\|_{\infty}\to\infty$, quindi esiste $\\{x_{n}\\}\subset R$ tale che $\left|f_{n}(x_{n})\right|\to\infty$. Dato che $f_{n}$ converge localmente uniformemente a $0$, necessariamente $x_{n}\to\infty$ 888cioè $x_{n}$ abbandona definitivamente ogni compatto. Fissata un’esaustione $K_{n}$ di $R$, per ogni $n$ consideriamo $\displaystyle E_{n}=\min_{x_{i}\in K_{n}}\\{\left|f_{i}(x_{i})\right|\\}/2$ Non è difficile verificare che $E_{n}\to\infty$, e che quindi a meno di un numero finito di termini 999che in questo ragionamento sono ininfluenti, $E_{n}>0$. In questo modo la successione $\\{f_{n}\\}$ non è contenuta definitivamente nell’aperto $V(0,E_{n})$, quindi non può convergere a $0$. ∎ Indipendentemente dal fatto che $\tau_{B}$ non è primo numerabile, ha senso chiedersi se le successioni bastano a descrivere la topologia di questo spazio 101010spazi topologici di questo genere si chiamano spazi di Frechet-Urysohn. Visti gli scopi della tesi, ci limitiamo a rimandare a [A] capitolo I pag 13 per ulteriori approfondimenti. La risposta a questa domanda è negativa, anzi qualunque topologia generi la convergenza $B$ è destinata a non essere una topologia “strana”. Infatti in questi spazi la chiusura per successioni non è idempotente, cioè indicando $[A]_{seq}$ la chiusura per successioni di $A$, cioè l’insieme di tutti i possibili limiti di successioni in $A$, si ha che in generale $[[A]_{seq}]_{seq}\neq[A]_{seq}$, quindi necessariamente $[A]_{seq}\neq\overline{A}$. ###### Proposizione 3.17. Data una successione $f_{n}\in\mathbb{M}(R)$, se definiamo $\displaystyle f=B-\lim_{n}f_{n}\Longleftrightarrow f=C-\lim_{n}f_{n}\ \wedge\ \left(\exists M\ t.c.\ \left\|f_{n}\right\|_{\infty}\leq M\right)$ allora se $R$ non è compatta, qualunque topologia generi questa convergenza non è di Frechet-Urysohn, quindi non è descrivibile con il comportamento delle sue successioni convergenti. ###### Proof. La dimostrazione è ispirata all’esercizio 9 cap 3 pag 87 di [R2]. Consideriamo una successione discreta $\\{x_{n}\\}\subset\mathbb{R}$, $x_{n}\to\infty$, e sia $\\{U_{n}\\}$ una collezione di intorni disgiunti di $x_{n}$. Siano $g_{n}\in\mathbb{M}(R)$ funzioni tali che $supp(g_{n})\subset U_{n}$, $0\leq g_{n}\leq 1$ e $g_{n}(x_{n})=1$. Definiamo $f_{n,m}\in\mathbb{M}(R)$ come $f_{n,m}=g_{n}+ng_{m}$, e consideriamo $\displaystyle A=\\{f_{n,m}\ n,m\in\mathbb{N}\\}$ Allora $0\not\in[A]_{seq}$, ma $0\in[[A]_{seq}]_{seq}$. Infatti qualunque successione $\\{f_{n(k),m(k)}\\}_{k=1}^{\infty}$ in $A$ che converge nel senso $B$ a $0$ necessariamente converge localmente uniformemente a $0$, quindi $n_{k}\to\infty$, ma allora anche $\left\|f_{n(k),m(k)}\right\|_{\infty}=n_{k}\to\infty$, il che significa che questa successione non può convergere a $0$. D’altro canto è facile verificare che tutte le funzioni $g_{n}\in[A]_{seq}$, basta considerare che la successione $f_{n(k),m(k)}\to g_{n}$ se $n(k)=n$, $m(k)\to\infty$. Ma dato che $g_{n}\to 0$ rispetto alla convergenza $B$, la tesi è dimostrata. ∎ Questa proposizione impone di trattare la convergenza $B$ con particolare attenzione, perché non può essere descritta semplicemente dalle sue successioni (come accade per gli spazi metrici). Un’altra topologia che possiamo definire su $R$ è la classica topologia della convergenza uniforme ovunque: ###### Definizione 3.18. La norma del sup è una norma sullo spazio $\mathbb{M}(R)$. Definiamo la topologia descritta da questa norma $\tau_{U}$. Diciamo che $\displaystyle f=U-\lim_{n}f_{n}$ (3.8) se $f_{n}$ converge ad $f$ rispetto a questa norma, quindi se $\displaystyle\lim_{n}\left\|f_{n}-f\right\|_{\infty}=\lim_{n}\sup_{R}\left|f_{n}(p)-f(p)\right|=0$ ###### Osservazione 3.19. È facile verificare che $\tau_{C}\subset\tau_{B}\subset\tau_{U}$. Le tre topologie coincidono nel caso $R$ compatta, mentre l’inclusione è stretta se $R$ non è compatta. Anche in questo caso vale un’osservazione molto simile a 3.11 (e la dimostrazione è del tutto analoga, basta sostituire lo spazio delle funzioni continue con lo spazio delle funzioni continue limitate). ###### Osservazione 3.20. La norma dell’estremo superiore non rende $\mathbb{M}(R)$ uno spazio completo. Dalla definizione di $\mathbb{M}(R)$, non è difficile immaginare che per rendere questa algebra un’algebra di Banach bisogna in qualche senso tenere in considerazione il comportamento delle derivate delle funzioni. A questo scopo introduciamo il concetto di $D-\lim$: ###### Definizione 3.21. Data una successione di funzioni $f_{n}\in\mathbb{M}(R)$, diciamo che $\displaystyle D-\lim_{n}f_{n}=f$ (3.9) se e solo se $D_{R}(f_{n}-f)\to 0$ Ricordiamo che l’integrale di Dirichlet è molto legato al concetto di norma nello spazio $\mathcal{L}^{2}(R)$, lo spazio di Hilbert delle 1-forme a quadrato integrabili su $R$ 111111a questo scopo rimandiamo alla sezione 1.3. Questi concetti di convergenza possono essere mischiati fra loro: ###### Definizione 3.22. Data una successione $f_{n}\in\mathbb{M}(R)$, diciamo che $\displaystyle f=QD-\lim_{n}f_{n}$ (3.10) se e solo se $f=D-\lim_{n}f_{n}$ e anche $f=Q-\lim_{n}f_{n}$, dove $Q$ rappresenta una convergenza qualsiasi tra $C$, $B$, $U$. I due concetti di convergenza che saranno più usati in questo lavoro sono la $BD$-convergenza e la $UD$-convergenza. La seconda convergenza è generata da una norma, che indicheremo $\left\|\cdot\right\|_{R}$, norma che rende $\mathbb{M}(R)$ un’algebra di Banach. ###### Teorema 3.23. Sull’algebra $\mathbb{M}(R)$ definiamo la norma $\displaystyle\left\|f\right\|_{R}\equiv\left\|f\right\|_{\infty}+D_{R}(f)^{1/2}$ questa norma genera la convergenza $UD$, e rispetto a questa norma $\mathbb{M}(R)$ è un’algebra di Banach commutativa con unità. ###### Proof. Il fatto che $\left\|\cdot\right\|_{R}$ sia una norma è facile conseguenza del fatto che $\left\|\cdot\right\|_{\infty}$ è una norma, e $D_{R}(\cdot)^{1/2}$ è una seminorma. Ovviamente $\left\|1\right\|=1$, e grazie alla relazione (3.5) si ottiene: $\displaystyle\left\|fh\right\|_{R}=\left\|fh\right\|_{\infty}+D_{R}(fh)^{1/2}\leq\left\|f\right\|_{\infty}\left\|h\right\|_{\infty}+\left\|h\right\|_{\infty}D_{R}(f)^{1/2}+\left\|f\right\|_{\infty}D_{R}(h)^{1/2}\leq$ $\displaystyle\leq(\left\|f\right\|_{\infty}+D_{R}(f)^{1/2})(\left\|h\right\|_{\infty}+D_{R}(h)^{1/2})=\left\|f\right\|_{R}\left\|h\right\|_{R}$ questo rende $(\mathbb{M}(R),\left\|\cdot\right\|_{R})$ un’algebra normata. Rimane da verificare la completezza. Sia a questo scopo $\\{f_{n}\\}\subset\mathbb{M}(R)$ una successione di Cauchy. Allora $\\{f_{n}\\}$ è di Cauchy uniforme, quindi esiste una funzione continua limitata sulla varietà $R$ tale che $\left\|f_{n}-f\right\|_{\infty}\to 0$. La parte più complicata è dimostrare che questa funzione è di Tonelli e che $D_{R}(f_{n}-f)\to 0$. Ricordiamo che $\mathcal{L}^{2}(R)$, lo spazio delle 1-forme su $R$ normato con l’integrale del modulo quadro della forma, è uno spazio di Hilbert (vedi sezione 1.3). L’ipotesi che $f_{n}$ sia di Cauchy rispetto a $\left\|\cdot\right\|_{R}$ implica che $df_{n}$ sia una successione di Cauchy nello spazio $\mathcal{L}^{2}(R)$, quindi esiste una 1-forma $\alpha\in\mathcal{L}^{2}(r)$ tale che $df_{n}\to\alpha$. Se dimostriamo che $f$ è di Tonelli e $df=\alpha$, abbiamo la tesi. Osserviamo che queste due affermazioni hanno carattere locale, quindi fissiamo un qualunque aperto coordinato $(U,\phi)$ con l’accortezza che $\phi$ sia definita in un intorno di $\overline{U}$ e tale che $\phi(U)=\prod_{i=1}^{m}(a_{i},b_{i})$ e dimostriamo l’uguaglianza $df=\alpha$ in questa carta locale. A questo scopo indichiamo con $\alpha_{i}(x)$ le componenti locali di $\alpha$, cioè: $\displaystyle\alpha=\alpha_{i}(x)dx^{i}$ mentre per semplicità di notazione continuiamo a indicare con $f_{n}(x)$ e $f(x)$ le rappresentazioni locali di $f_{n}$ ed $f$ rispettivamente. Dividiamo la dimostrazione in due parti: l’idea e i conti. L’idea è dimostrare $f=f^{(i)}$, dove $f^{(i)}$ sono funzioni per definizione assolutamente continue e la cui derivata parziale rispetto a $x^{i}$ è proprio $\alpha_{i}$. A questo scopo mostriamo che le funzioni $f_{n}$, che convergono uniformemente a $f$, convergono in norma $L^{2}(\phi(U))$ a $f^{(i)}$ (per tutti gli indici $1\leq i\leq m$). Quindi $f=f^{(i)}$ quasi ovunque (vedi proposizione 1.33). E questo conclude la dimostrazione. Ora passiamo a conti. Definiamo le funzioni $f^{(i)}$ come: $\displaystyle f^{(i)}(x^{1},\cdots,x^{i},\cdots,x^{m})\equiv f(x^{1},\cdots,c_{i},\cdots,x^{m})+\int_{c_{i}}^{x^{i}}\alpha_{i}(x^{1},\cdots,t,\cdots,x^{m})dt$ (3.11) dove $c_{i}=(a_{i}+b_{i})/2$ 121212in realtà per la dimostrazione va bene qualsiasi $c_{i}\in(a_{i},b_{i})$. Osserviamo che $\partial f_{n}/\partial x^{i}(x)$ converge in norma $L^{2}$ a $\alpha_{i}(x)$ quando $n$ tende a infinito. Infatti $\displaystyle\int_{\phi(U)}\left(\frac{\partial f_{n}}{\partial x^{i}}(x)-\alpha_{i}(x)\right)^{2}dx^{1}\cdots dx^{m}\leq\int_{\phi(U)}\sum_{i=1}^{m}\left(\frac{\partial f_{n}}{\partial x^{i}}(x)-\alpha_{i}(x)\right)^{2}dx^{1}\cdots dx^{m}\leq$ $\displaystyle\leq\int_{\phi(U)}Cg^{ij}\left(\frac{\partial f_{n}}{\partial x^{i}}(x)-\alpha_{i}(x)\right)\left(\frac{\partial f_{n}}{\partial x^{j}}(x)-\alpha_{j}(x)\right)dx^{1}\cdots dx^{m}\leq$ $\displaystyle\leq\frac{C}{k}\int_{\phi(U)}g^{ij}\left(\frac{\partial f_{n}}{\partial x^{i}}(x)-\alpha_{i}(x)\right)\left(\frac{\partial f_{n}}{\partial x^{j}}(x)-\alpha_{j}(x)\right)\ \sqrt{\left|g\right|}dx^{1}\cdots dx^{m}=$ $\displaystyle=\frac{C}{k}\left\|df_{n}-\alpha\right\|_{\mathcal{L}^{2}(R)}\to 0$ dove per passare dalla prima alla seconda riga abbiamo sfruttato la relazione 3.2, scegliendo come $C$ il massimo dei $c_{p}$ al variare di $p\in\phi(\overline{U})$, mentre $k$ è il minimo della funzione $\sqrt{\left|g\right|}$ sempre sullo stesso insieme compatto 131313per questo motivo è importante scegliere $\phi$ definita su un intorno di $\overline{U}$. Fissato $i$, tutte le funzioni $f_{n}(\bar{x}^{i},\cdots,x^{i},\cdots,\bar{x}^{m})$ sono contemporaneamente assolutamente continue rispetto a $x^{i}$, quasi ovunque rispetto a $(\bar{x}^{1},\cdots,\bar{x}^{i-1},\bar{x}^{i+1},\cdots,\bar{x}^{m})$ 141414ogni funzione $f_{n}$ è assolutamente continua a meno di un insieme di misura nulla di $\bar{x}$, ma visto che l’unione numerabile di insiemi di misura nulla ha misura nulla, tutte le funzioni sono assolutamente continue contemporaneamente sullo stesso insieme con complementare di misura nulla, quindi vale che $\displaystyle f_{n}(\bar{x}^{i},\cdots,x^{i},\cdots,\bar{x}^{m})=f_{n}(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})+\int_{c_{i}}^{x^{i}}\frac{\partial f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial t}dt$ (3.12) Confrontando questa relazione con la relazione 3.11 otteniamo: $\displaystyle\left|f_{n}(\bar{x}^{i},\cdots,x_{i},\cdots,\bar{x}^{m})-f^{(i)}(\bar{x}^{i},\cdots,x_{i},\cdots,\bar{x}^{m})\right|^{2}=$ $\displaystyle=\left|f_{n}(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})-f(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})+\right.$ $\displaystyle\left.+\int_{c_{i}}^{x^{i}}\frac{\partial f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial t}dt-\int_{c_{i}}^{x^{i}}\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})dt\right|^{2}\leq$ $\displaystyle\leq 2\left|f_{n}(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})-f(\bar{x}^{i},\cdots,c_{i},\cdots,\bar{x}^{m})\right|^{2}+$ $\displaystyle+2\left|\int_{c_{i}}^{x^{i}}\frac{\partial f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial t}dt-\int_{c_{i}}^{x^{i}}\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})dt\right|^{2}$ per semplicità di notazione chiamiamo $2\left|A_{n}\right|^{2}$ la prima riga dopo l’ultimo segno di disuguaglianza, e la seconda $2\left|B_{n}\right|^{2}$. Grazie alle proprietà dell’integrale e alla disuguaglianza di Schwartz otteniamo: $\displaystyle\left|B_{n}\right|^{2}\leq\left(\int_{c_{i}}^{x^{i}}\left|\frac{\partial f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial t}-\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})\right|dt\right)^{2}\leq$ $\displaystyle\leq\left(b_{i}-a_{i}\right)\int_{a_{i}}^{b^{i}}\left|\frac{\partial f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial t}-\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})\right|^{2}dt$ e questa relazione vale quasi ovunque rispetto a $(\bar{x}^{1},\cdots,\bar{x}^{i-1},\bar{x}^{i+1},\cdots,\bar{x}^{m})$. Grazie a queste disuguaglianze ora siamo in grado di dimostrare che la successione $f_{n}$ (o meglio la successione delle rappresentazioni locali di $f_{n}$) converge in norma $L^{2}(\phi(U))$ alla funzione $f^{(i)}$ (con $1\leq i\leq m$ qualsiasi). Infatti: $\displaystyle\int_{\phi(U)}\left|f_{m}(x^{1},\cdots,x^{m})-f^{(i)}(x^{1},\cdots,x^{m})\right|^{2}dx^{1}\cdots dx^{m}\leq$ $\displaystyle\leq 2\left|A_{n}\right|^{2}Vol(\phi(U))+$ $\displaystyle+2(b_{i}-a_{i})\int_{\phi(U)}\left\\{\int_{a_{i}}^{b^{i}}\left|\frac{\partial f_{n}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})}{\partial t}-\alpha_{i}(\bar{x}^{i},\cdots,t,\cdots,\bar{x}^{m})\right|^{2}dt\right\\}dx=$ $\displaystyle=2\left|A_{n}\right|^{2}Vol(\phi(U))+2(b_{i}-a_{i})^{2}\left\|\frac{\partial f_{n}}{\partial x^{i}}-\alpha_{i}\right\|^{2}_{L^{2}(\phi(U))}$ Sappiamo che $A_{n}$ tende a 0 (poichè $f_{n}$ converge uniformemente a $f$), e anche $\left\|\frac{\partial f_{n}}{\partial x^{i}}-\alpha_{i}\right\|^{2}_{L^{2}(\phi(U))}$ tende a zero come dimostrato prima. ∎ Modificando leggermente questa dimostrazione, si può ottenere che: ###### Proposizione 3.24. Se $f_{n}$ è una successione di funzioni in $\mathbb{M}(R)$ tale che $\displaystyle f=C-\lim_{n}f_{n}$ dove $f$ è una funzione (continua) limitata, e se esiste $K<\infty$ tale che per ogni $n$: $\displaystyle D_{R}(f_{n})\leq K<\infty$ allora $f\in\mathbb{M}(R)$, $D_{R}(f)\leq K$ e inoltre esiste una sottosuccessione $n_{k}$ tale che per ogni $g\in\mathbb{M}(R)$: $\displaystyle D_{R}(f-f_{n_{k}},g)\to 0$ ###### Proof. Grazie al teorema 1.48, esiste una sottosuccessione di $f_{n}$ che continueremo a indicare nello stesso modo tale che $df_{n}$ converge debolmente a $\alpha\in\mathcal{L}^{2}(R)$. Consideriamo un rettangolo coordinato $T$ qualsiasi e sia $\phi\in C^{\infty}_{0}(T)$ lo spazio delle funzioni lisce a supporto compatto in $T$. Grazie a un’integrazione per parti otteniamo che: $\displaystyle\int_{T}f_{n}\frac{\partial\phi}{\partial x^{i}}dV=-\int_{T}\phi\frac{\partial f_{n}}{\partial x^{i}}dV$ Ora poiché $df_{n}\to\alpha$ debolmente, si ha che: $\displaystyle\int_{T}\phi\frac{\partial f_{n}}{\partial x^{i}}dV\to\int_{T}\phi\alpha_{i}dV$ e dato che $f=C-\lim_{n}f_{n}$ si ha che: $\displaystyle\int_{T}f_{n}\frac{\partial\phi}{\partial x^{i}}dV\to\int_{T}f\frac{\partial\phi}{\partial x^{i}}dV$ Da queste consideraziono otteniamo che: $\displaystyle\int_{T}f\frac{\partial\phi}{\partial x^{i}}dV=\int_{T}\phi\alpha_{i}dV$ Questo significa che le derivate distribuzionali di $f$ rispetto a $x^{i}$ coincidono con $\alpha_{i}$, e dato che $\alpha\in\mathcal{L}^{2}(R)$, grazie al lemma 3.25 (che riportiamo alla fine di questa proposizione), $f$ è di Tonelli con $df=\alpha$ nel senso standard e quindi $D_{R}(f)<\infty$, il che dimostra che $f\in\mathbb{M}(R)$. Il fatto che $\displaystyle D_{R}(f-f_{n},g)\to 0$ è conseguenza diretta del fatto che $df_{n}$ converge debolmente nel senso di $\mathcal{L}^{2}(R)$ a $df$. Inoltre dalla teoria della convergenza debole, sappiamo che $\left\|df\right\|_{L_{2}}\leq\limsup_{n}\left\|df_{n}\right\|_{L_{2}}$ quindi ad esempio $\displaystyle\left\|df_{n}\right\|_{L_{2}}\equiv D_{R}(f_{n})\leq k\ \ \forall n\ \Rightarrow\ \left\|df\right\|_{L_{2}}\equiv D_{R}(f)\leq k$ ∎ ###### Lemma 3.25. Sia $f:T\to\mathbb{R}$ una funzione $f\in L^{2}(T)$ 151515osserviamo che tutte le funzioni continue sono a quadrato integrabile su insiemi relativamente compatti di misura finita, quindi sui rettangoli coordinati $T$ rispetto alla forma volume tale che per tutti gli indici $i=1,\cdots,m$: $\displaystyle\frac{\partial f}{\partial x_{i}}\in L^{2}(T)$ dove le derivate sono intese in senso distrubuzionale. Allora $f$ è assolutamente continua rispetto a quasi tutti i segmenti paralleli agli assi coordinati in $T$, e le sue derivate standard coincidono quasi ovunque con le derivate distribuzionali. Quindi se $f$ è continua, è una funzione di Tonelli. ###### Proof. Dato che questo teorema riguarda la teoria delle distribuzioni e gli spazi di Sobolev, che sono argomenti marginali in questa tesi, per la dimostrazione rimandiamo il lettore a [Z] (teorema 2.1.4 pag. 44), oppure a [MZ] (teorema 1.41 pag. 22). ∎ ###### Osservazione 3.26. Nella dimostrazione, il fatto che $f$ sia limitata è utile esclusivamente per dimostrare che $f\in\mathbb{M}(R)$. Quindi se $f$ non è limitata, valgono tutte le conclusioni del teorema a meno dell’appartenenza all’algebra di Royden. #### 3.2.3 Densità di funzioni lisce In questo paragrafo dimostreremo la densità delle funzioni lisce nello spazio $\mathbb{M}(R)$, e utilizzeremo questo risultato per dimostrare formule di Green generalizzate. ###### Proposizione 3.27. Sia $f$ una funzione di Tonelli su $R$. Per ogni $\epsilon>0$, esiste $f_{\epsilon}\in C^{\infty}(R,\mathbb{R})$ tale che $\left\|f_{\epsilon}-f\right\|_{R}<\epsilon$ 161616in questa proposizione non è richiesto che $f$ sia limitata, e che il suo integrale di Dirichlet $D_{R}(f)<\infty$, quindi potrebbe non avere senso $\left\|f\right\|_{R}$. Inoltre se $f$ ha supporto contenuto in un aperto $U\subset R$, anche $f_{\epsilon}$ può essere scelta con supporto contenuto in $U$. ###### Proof. Come spesso accade in questi casi, la dimostrazione si divide in due parti, una “locale” e una “globale”. Nella prima parte dimostreremo il risultato per funzioni $f$ a supporto compatto in una carta locale, poi generalizzeremo il risultato utilizzando le partizioni dell’unità. Sia quindi $f$ come nelle ipotesi e anche a supporto compatto in un intorno coordinato $(U,\phi)$ di $R$, e chiamiamo $\tilde{f}$ la rappresentazione di $f$ in questa carta locale. Consideriamo una successione $\Theta_{n}:\mathbb{R}^{m}\to\mathbb{R}$ di nuclei di convoluzione con supporto contenuto in $B_{1/n}(0)$. Grazie al lemma 1.45 la successione $\tilde{f}_{n}\equiv\Theta_{n}\ast\tilde{f}$ (che ha supporto definitivamente contenuto in $\phi(U)$) converge nella norma del sup a $\tilde{f}$. Inoltre grazie al lemma 1.44, si ha che: $\displaystyle\frac{\partial\tilde{f}_{n}}{\partial x^{i}}=\Theta_{n}\ast\frac{\partial\tilde{f}}{\partial x^{i}}$ Passiamo ora a considerare $D_{R}(f_{n}-f)$. Grazie al fatto che esiste un insieme compatto $K$ tale che $supp(\tilde{f}_{n})\subset K\Subset U$ definitivamente rispetto a $n$, e grazie alla relazione 3.3, otteniamo che: $\displaystyle g^{ij}\left(\frac{\partial\tilde{f}_{n}}{\partial x^{i}}(x)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)\left(\frac{\partial\tilde{f}_{n}}{\partial x^{j}}(x)-\frac{\partial\tilde{f}}{\partial x^{j}}(x)\right)\leq c\sum_{i=1}^{m}\left(\frac{\partial\tilde{f}_{n}}{\partial x^{i}}(x)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}=$ $\displaystyle=c\sum_{i=1}^{m}\left(\int_{B_{n}}\Theta_{n}(y)\left(\frac{\partial\tilde{f}}{\partial x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)dV(y)\right)^{2}\leq$ $\displaystyle\leq c\sum_{i=1}^{m}\int_{B_{n}}\Theta_{n}(y)\left(\frac{\partial\tilde{f}}{\partial x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}dV(y)$ dove l’ultimo passaggio è giustificato dalla disuguaglianza di Jensen 171717infatti $\int\Theta_{n}(y)dV(y)=1$. Per riferimenti sulla disuguaglianza di Jensen, vedi teorema 3.3 pag 61 di [R4]. Queste considerazioni (assieme al teorema di Fubuni 181818vedi teorema 7.8 pag 140 di [R4]) permettono di concludere: $\displaystyle D_{R}(f_{n}-f)=\int_{K}g^{ij}\left(\frac{\partial\tilde{f}_{n}}{\partial x^{i}}(x)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)\left(\frac{\partial\tilde{f}_{n}}{\partial x^{j}}(x)-\frac{\partial\tilde{f}}{\partial x^{j}}(x)\right)dV(x)\leq$ $\displaystyle\leq c^{\prime}\int_{K}dV(x)\sum_{i=1}^{m}\int_{B_{n}}dV(y)\Theta_{n}(y)\left(\frac{\partial\tilde{f}}{\partial x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}=$ $\displaystyle=c^{\prime}\sum_{i=1}^{m}\int_{B_{n}}dV(y)\Theta_{n}(y)\int_{K}dV(x)\left(\frac{\partial\tilde{f}}{\partial x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}$ dove $c^{\prime}=\max_{x\in K}\\{c\sqrt{\left|g\right|(x)}\\}$. Chiamiamo $\displaystyle h(y)\equiv\int_{K}dV(x)\left(\frac{\partial\tilde{f}}{\partial x^{i}}(x-y)-\frac{\partial\tilde{f}}{\partial x^{i}}(x)\right)^{2}$ Grazie alla continuità dell’operatore traslazione in $L^{2}(R)$ 191919vedi teorema 9.5 pag 183 di [R2], e dato che $f$ è di Tonelli, possiamo concludere che: $\displaystyle\lim_{y\to 0}h(y)=0$ Per definizione questo significa che $\displaystyle\lim_{n\to\infty}\max_{y\in B_{n}(0)}\left|h(y)\right|=0$ E quindi anche: $\displaystyle\lim_{n\to\infty}\left|\int_{B_{n}}dV(y)\Theta_{n}(y)h(y)\right|\leq\lim_{n\to\infty}\max_{y\in B_{n}(0)}\left|h(y)\right|\int_{B_{n}}dV(y)\Theta_{n}(y)=0$ Questo conclude la dimostrazione che $D_{R}(f_{n}-f)\to 0$. Consideriamo ora una funzione $f$ di Tonelli qualsiasi, $\epsilon>0$, e sia $\lambda_{n}:R\to\mathbb{R}$ una partizione dell’unità liscia subordinata a aperti coordinati e a supporto compatto. Allora per ogni $n$, $f\cdot\lambda_{n}$ è una funzione del tipo descritto sopra. Scegliamo per ogni $n$ una funzione liscia $f_{n}$ 202020se $supp(f)\subset U$, allora $supp(f\cdot\lambda_{n})\Subset U$, quindi possiamo scegliere $supp(f_{n})\Subset U$ tale che $\left\|f_{n}-f\cdot\lambda_{n}\right\|_{R}<\epsilon/2^{n}$. Allora la funzione $f_{\epsilon}\equiv\sum_{n=1}^{\infty}f_{n}$ soddisfa le richieste fatte. $f_{\epsilon}$ è una funzione liscia per locale finitezza della partizione dell’unità $\lambda_{n}$, inoltre: $\displaystyle\left\|f_{\epsilon}-f\right\|_{R}=\left\|\sum_{n=1}^{\infty}(f_{n}-f\cdot\lambda_{n})\right\|_{R}\leq\sum_{n=1}^{\infty}\left\|f_{n}-f\cdot\lambda_{n}\right\|_{R}<\epsilon$ Inoltre se $supp(f_{n})\Subset U$ per ogni $n$, allora anche $supp(f)\subset U$. Questo significa che se $f$ ha supporto compatto, visto che ogni compatto in $R$ è contenuto in un aperto relativamente compatto, allora $f_{\epsilon}$ può essere scelta a supporto compatto. ∎ Vale anche una proposizione più forte rispetto a questa. Possiamo infatti chiedere che la funzione $f_{\epsilon}$ sia uguale ad $f$ su un insieme chiuso, ovviamente a patto di rilassare le ipotesi di regolarità sulla funzione $f_{\epsilon}$. ###### Proposizione 3.28. Sia $f$ una funzione di Tonelli su $R$ e $A\subset R$ aperto con bordo $\partial A$ regolare. Per ogni $\epsilon>0$, esiste $f_{\epsilon}\in C^{\infty}(A,\mathbb{R})$ di Tonelli su tutta la varietà tale che $\left\|f_{\epsilon}-f\right\|_{R}<\epsilon$ e $f_{\epsilon}=f$ sull’insieme $A^{C}$. ###### Proof. Riportiamo solo il filo conduttore della dimostrazione, lasciando alcuni dettagli al lettore. Questa dimostrazione è ispirata dal lemma 2.8 pagina 50 di [H3]. Sia $K_{n}$ un ricoprimento di aperti relativamente compatti localmente finiti in $A$ 212121con localmente finito si intende un ricoprimento tale che per ogni $p\in A$ esiste un intorno che interseca solo un numero finito di insiemi $K_{n}$, e sia $\epsilon_{n}$ una successione di numeri positivi che tende a 0 tale che $\epsilon_{n}\leq\epsilon$. Lo scopo è riuscire a creare una funzione $h_{\epsilon}\in C^{\infty}(A,\mathbb{R})$ tale che $\displaystyle\left\|h_{\epsilon}-f\right\|_{K_{n}}\equiv\left\|h_{\epsilon}-f\right\|_{\infty,\ K_{n}}+D_{K_{n}}(h_{\epsilon}-f)^{1/2}<\epsilon_{n}$ (3.13) In questo modo se estendiamo la definizione di $h_{\epsilon}$ come $\displaystyle f_{\epsilon}(p)=\begin{cases}h_{\epsilon}(p)&p\in A\\\ f(p)&p\in A^{C}\end{cases}$ questa funzione risulta di Tonelli su tutta $R$. Infatti $f_{\epsilon}$ è ovviamente continua e di Tonelli su tutti i punti di $R$ tranne che sul bordo $\partial A$. Consideriamo $p\in\partial A$, e consideriamo una successione $p_{k}\in A$, $p_{k}\to p$. Visto che $p_{k}$ converge al bordo di $A$, necessariamente $p_{k}\in K_{n(k)}$ dove $n(k)$ tende a infinito quando $k$ tende a infinito. Questo significa che $\displaystyle\left|f_{\epsilon}(p_{k})-f(p)\right|\leq\left|f_{\epsilon}(p_{k})-f(p_{k})\right|+\left|f(p_{k})-f(p)\right|\leq\epsilon_{n(k)}+\left|f(p_{k})-f(p)\right|$ se $k$ tende a infinito, $\epsilon_{n(k)}$ tende a zero per ipotesi, mentre il secondo termine tende a zero per continuità di $f$. Questo prova che $f_{\epsilon}$ è continua. Consideriamo un punto $p\in\partial A$ e una parametrizzazione locale $\phi(q)=(x^{1},\cdots x^{m})$ attorno a $p$ tale che $\partial A=\\{x^{m}=0\\}$. Allora la funzione $f_{\epsilon}$ è di certo assolutamente contiua quasi ovunque nell’insieme $x^{m}\neq 0$, quindi quasi ovunque. L’integrale di Dirichlet di $f_{\epsilon}$ è finito su ogni compatto che ha intersezione vuota con il bordo perché $f_{\epsilon}$ su questo insieme è una funzione liscia o una funzione di Tonelli per ipotesi. Se il compatto interseca il bordo di $A$, allora poiché $D_{R}(f-f_{\epsilon})$ è finito, necessariamente l’integrale di Dirichlet di $f_{\epsilon}$ su questo compatto è finita. Resta da dimostrare che è possibile scegliere una funzione $h_{\epsilon}$ che soddisfi le condizioni 3.13. La tecnica che utilizzeremo è un’adattamento della dimostrazione della proposizione 3.27. Scegliamo una partizione dell’unità $\\{\lambda_{n}\\}$ di $A$ subordinata al ricoprimento $K_{n}$. Per ogni $n$, definiamo $\alpha_{m}\equiv\frac{\epsilon_{m}}{\\#n\ t.c.\ supp(\lambda_{n})\cap K_{m}\neq\emptyset}$ $\delta_{n}\equiv\min\\{\alpha_{m}\ t.c.\ K_{m}\cap supp(\lambda_{n})\neq\emptyset\\}$ Per locale finitezza del ricoprimento $\\{K_{m}\\}$ tutte queste quantità sono ben definite e strettamente positive. Ora grazie alle tecniche esposte nella dimostrazione della proposizione 3.27, scegliamo per ogni $n$ una funzione liscia $f_{n}$ con supporto in $K_{n}$ tale che: $\displaystyle\left\|f_{n}-f\lambda_{n}\right\|_{R}<\delta_{n}$ e definiamo $h_{\epsilon}\equiv\sum_{n}f_{n}$. Per locale finitezza della partizione dell’unità questa serie è ben definita e $h_{\epsilon}\in C^{\infty}(A,\mathbb{R})$, inoltre: $\displaystyle\left\|h_{\epsilon}-f\right\|_{K_{m}}=\left\|\sum_{n}(f_{n}-f\lambda_{n})\right\|_{K_{m}}<\epsilon_{m}$ da cui la tesi. ∎ #### 3.2.4 Formule di Green e principio di Dirichlet Ricordando la notazione definita nella sezione 1.1.4, grazie ai risultati di densità appena descritti possiamo dimostrare che: ###### Proposizione 3.29. Se $f$ è di Tonelli con $D_{R}(f)<\infty$, e $u\in H(\Omega)\cap C^{\infty}(\overline{\Omega})$ 222222$H(\Omega)$ è l’insieme delle funzioni armoniche su $\Omega$. $C^{\infty}(\overline{\Omega})$ è l’insieme delle funzioni che possono essere estese a funzioni lisce in un intorno di $\overline{\Omega}$, dove $\Omega$ è un dominio regolare, abbiamo che: $\displaystyle D_{\Omega}(f,u)\equiv\int_{\Omega}\left\langle\nabla f\middle|\nabla u\right\rangle dV=\int_{\partial\Omega}f\ast du$ ###### Proof. Consideriamo una successione $f_{n}\in C^{\infty}(R,\mathbb{R})$ che converge in norma $\left\|\cdot\right\|_{R}$ a $f$. Allora grazie alla proposizione 1.11, abbiamo che per ogni $m$: $\displaystyle D_{\Omega}(f_{m},u)\equiv\int_{\Omega}\left\langle\nabla f_{m}\middle|\nabla u\right\rangle dV=\int_{\partial\Omega}f_{m}\ast du$ Poiché $D_{R}(f_{m}-f)\to 0$ e anche $\left\|f_{m}-f\right\|_{\infty}\to 0$, si ha che: $\displaystyle\left|D_{\Omega}(f_{m},u)-D_{\Omega}(f,u)\right|\equiv\left|\int_{\Omega}\left\langle\nabla(f_{m}-f)\middle|\nabla u\right\rangle dV\right|\leq D_{\Omega}(u)D_{\Omega}(f_{m}-f)\to 0$ $\displaystyle\left|\int_{\partial\Omega}f_{m}\ast du-\int_{\partial\Omega}f_{m}\ast du\right|\leq\int_{\partial\Omega}\left|f_{m}-f\right|\ast du\to 0$ da cui la tesi. ∎ Possiamo anche rilassare le ipotesi su $u$, e ottenere: ###### Proposizione 3.30. Sia $\Omega\subset R$ un dominio regolare, $f$ una funzione di Tonelli con $D_{R}(f)<\infty$, e $u\in H(\Omega)$ con $D_{\Omega}(u)<\infty$. Siano inoltre $\gamma_{1}$ e $\gamma_{2}$ due insiemi connessi disgiunti tali che $\gamma_{1}\cup\gamma_{2}=\partial\Omega$. Se $f=0$ su $\gamma_{1}$, e $u\in C^{\infty}(\gamma_{2})$, allora: $\displaystyle D_{\Omega}(f,u)\equiv\int_{\Omega}\left\langle\nabla f\middle|\nabla u\right\rangle dV=\int_{\gamma_{2}}f\ast du$ (3.14) ###### Proof. Per cominciare dimostriamo questa proposizione in un caso particolare. Sia $g$ una funzione di Tonelli con integrale di Dirichlet finito su $\Omega$, $g\geq 0$ su $\Omega$, $g|_{\gamma_{1}}=0$ e $g|_{\gamma_{2}}\geq\delta>0$. Consideriamo ora una successione di insiemi $\Omega_{n}$ con bordo liscio tali che $\Omega_{n}\subset\Omega$, $\partial\Omega_{n}=\gamma_{2}\cup\beta_{n}$ e $\cup_{n}\Omega_{n}=\Omega$ 232323è possibile costruire una successione con tali caratteristiche ad esempio considerando una funzione liscia $h:\overline{\Omega}\to\mathbb{R}$ tale che $h(\gamma_{1})=0$, $h(\gamma_{2})=1$ e $0<h(x)<1$ per ogni $x\in\Omega$. Scegliendo una successione decrescente $r_{n}\searrow 0$ di valori regolari per $h$, gli insiemi $\Omega_{n}\equiv h^{-1}(r_{n},1)$ hanno le caratteristiche cercate e definiamo le funzioni $\displaystyle g_{c}(x)\equiv\max\\{g(x)-c,0\\}$ per $0<c<\delta$. Per costruzione, queste funzioni coincidono con $g(x)-c$ su un intorno di $\gamma_{2}$ e sono tutte nulle in un intorno di $\gamma_{1}$, quindi fissato $c$, $g_{c}$ si annulla identicamente su $\beta_{n}$ definitivamente in $n$. Grazie a queste considerazioni, e grazie alla proposizione 3.29, sappiamo che esiste $n$ per cui: $\displaystyle D_{\Omega}(g_{c},u)=D_{\Omega_{n}}(g_{c},u)=\int_{\beta_{n}}g_{c}\ast du+\int_{\gamma_{2}}g_{c}\ast du=\int_{\gamma_{2}}g\ast du-c\int_{\gamma_{2}}\ast du$ Facendo tendere $c$ a $0$, la seconda parte della disuguaglianza tende a $\int_{\gamma_{2}}g\ast du$, mentre la prima tende a $D_{\Omega}(g,u)$. Infatti se chiamiamo $A_{c}=\\{x\in\Omega\ t.c.\ g(x)\leq\leavevmode\nobreak\ c\\}$ $\displaystyle\left|D_{\Omega}(g_{c},u)-D_{\Omega}(g,u)\right|=\left|\int_{\Omega}\left\langle\nabla(g_{c}-g)\middle|\nabla u\right\rangle dV\right|=\left|\int_{A_{c}}\left\langle\nabla g\middle|\nabla u\right\rangle dV\right|\leq$ $\displaystyle\leq\int_{A_{c}}\left|\nabla g\right|^{2}dV\int_{A_{c}}\left|\nabla u\right|^{2}dV\to 0$ e dato che entrambi gli integrali di Dirichlet estesi a tutta $\Omega$ sono finiti, se l’insieme la misura dell’insieme di integrazione tende a zero, anche l’integrale tende a 0. Questo dimostra la tesi su $g|_{\gamma_{2}}\geq\delta<0$. Se consideriamo una funzione $f\geq 0$ qualsiasi, la tesi si ottiene per linearità, infatti basta scegliere una funzione $g$ con le caratteristiche descritte sopra e applicare quanto appena per ottenere: $\displaystyle D_{\Omega}(f,u)=D_{\Omega}((f+g)-g,u)=D_{\Omega}(f+g,u)-D_{\Omega}(g,u)=$ $\displaystyle=\int_{\gamma_{2}}(f+g)\ast du-\int_{\gamma_{2}}g\ast du$ Se la funzione $f$ non è positiva, la tesi si ottiene applicando il ragionamento appena esposto alle funzioni $f^{+}$ e $f^{-}$, in particolare: $\displaystyle D_{\Omega}(f,u)=D_{\Omega}(f^{+}-f^{-},u)=\int_{\gamma_{2}}(f^{+}-f^{-})\ast du=\int_{\gamma_{2}}f\ast du$ ∎ Ora siamo in grado di dimostrare una generalizzazione del principio di Dirichlet esposto in 1.66. ###### Proposizione 3.31. Sia $f:R\to\mathbb{R}$ una funzione di Tonelli con $D_{R}(f)<\infty$, $u\in H(\Omega)\cap C(\overline{\Omega})$, dove $\Omega$ è un dominio regolare in $R$ varietà riemanniana. Se $f|_{\partial\Omega}=u|_{\partial\Omega}$, allora $D_{\Omega}(u)<\infty$ e: $\displaystyle D_{\Omega}(f)=D_{\Omega}(u)+D_{\Omega}(f-u)$ ###### Proof. La tesi di questa proposizione è esattamente identica alla tesi della proposizione 1.66, la differenza è nelle richieste di regolarità su $f$, che in questo caso sono molto meno stringenti. Come negli esempi sopra, dimostreremo questo teorema approssimando $f$ con funzioni lisce. Sia $f_{m}$ una successione di funzioni tali che $\left\|f_{m}-f\right\|_{R}\to 0$, e siano $u_{m}$ le soluzioni del problema di Dirichlet su $\Omega$ con $f_{m}|_{\partial\Omega}$ come dato al bordo. La proposizione 1.66 afferma che: $\displaystyle D_{\Omega}(f_{m})=D_{\Omega}(u_{m})+D_{\Omega}(f_{m}-u_{m})$ Grazie al principio del massimo otteniamo anche che $\left\|u_{m}-u\right\|_{\infty}\leq\left\|f_{m}|_{\partial\Omega}-f|_{\partial\Omega}\right\|_{\infty}\to 0$ quindi $u=U-\lim_{m}u_{m}$. Inoltre: $\displaystyle D_{\Omega}(u_{m}-u_{k})=D_{\Omega}(f_{m}-f_{k})-D_{\Omega}(f_{m}-f_{k}+u_{k}-u_{m})\leq D_{\Omega}(f_{m}-f_{k})$ e dato che $f_{m}\to f$ e che: $\displaystyle D_{\Omega}(f_{m}-f_{k})^{1/2}\leq D_{\Omega}(f_{m}-f)^{1/2}+D_{\Omega}(f_{k}-f)^{1/2}$ allora la successione $\\{u_{m}\\}$ è di Cauchy rispetto alla norma $\left\|\cdot\right\|_{R}$, quindi per completezza di $\mathbb{M}(R)$, $u_{m}$ converge e necessariamente converge a $u$. Questo implica che $D_{\Omega}(u)<\infty$. Osserviamo che: $\displaystyle D_{\Omega}(f)=\int_{\Omega}\left|\nabla f\right|^{2}dV=\int_{\Omega}\left|\nabla(f-u)+u\right|^{2}dV=$ $\displaystyle=\int_{\Omega}\left|\nabla(f-u)\right|^{2}dV+\int_{\Omega}\left|\nabla u\right|^{2}dV+2\int_{\Omega}\left\langle\nabla(f-u)\middle|\nabla u\right\rangle dV=$ $\displaystyle=D_{\Omega}((f-u))+D_{\Omega}(u)+2D_{\Omega}((f-u),u)$ Applicando il lemma 3.30 con $\gamma_{1}=\partial\Omega$ otteniamo che: $\displaystyle D_{\Omega}((f-u),u)=\int_{\Omega}\left\langle\nabla(f-u)\middle|\nabla u\right\rangle dV=0$ dato che per ipotesi $(f-u)|_{\partial\Omega}=0$. ∎ In realtà le ipotesi sulla regolarità del bordo di $\Omega$ non sono necessarie, infatti vale la proposizione ###### Proposizione 3.32. Sia $f:R\to\mathbb{R}$ una funzione di Tonelli con $D_{R}(f)<\infty$, $u\in\leavevmode\nobreak\ H(\Omega)\cap C(\overline{\Omega})$, con $\Omega$ è un aperto relativamente compatto in $R$ varietà riemanniana. Se $f|_{\partial\Omega}=u|_{\partial\Omega}$, allora $D_{\Omega}(u)<\infty$ e: $\displaystyle D_{\Omega}(f)=D_{\Omega}(u)+D_{\Omega}(f-u)$ ###### Proof. Per comodità, estendiamo l’insieme di definizione di $u$ a tutto $R$, ponendo $u|_{\Omega^{C}}=f|_{\Omega^{C}}$. In questo modo $u$ ha tutte le proprietà di una funzione in $\mathbb{M}(R)$, tranne il fatto di avere integrale di Dirichlet finito 242424proprietà che seguirà dalla dimostrazione. Sia $\Omega_{n}$ un’esaustione regolare per $\Omega$, definiamo le funzioni: $\displaystyle u_{n}:R\to\mathbb{R}\ \ u_{n}\in H(\Omega_{n})\cap C(R)\ \ \ \ u_{n}|_{\partial\Omega_{n}}=f|_{\partial\Omega_{n}}\ \ \ \ u_{n}|_{\Omega_{n}^{C}}=f|_{\Omega_{n}^{C}}$ Dalla proposizione precedente è evidente $\forall n$ $D_{\Omega}(u_{n})<\infty$, quindi $u_{n}\in\mathbb{M}(R)$, inoltre per ogni $m>n$ vale che: $\displaystyle D_{\Omega_{m}}(u_{n})=D_{\Omega_{m}}(u_{m})+D_{\Omega_{m}}(u_{n}-u_{m})$ quindi considerando che $u_{n}|_{\Omega_{m}^{C}}=u_{n}|_{\Omega_{m}^{C}}=f|_{\Omega_{m}^{C}}$, si ha che $D_{\Omega_{m}^{C}}(u_{n})=D_{\Omega_{m}^{C}}(u_{m})$ e $D_{\Omega_{m}^{C}}(u_{n}-u_{m})=0$, quindi: $\displaystyle\underbrace{D_{\Omega_{m}}(u_{n})+D_{\Omega_{m}^{C}}(u_{n})}_{D_{R}(u_{n})}=\underbrace{D_{\Omega_{m}}(u_{m})+D_{\Omega_{m}^{C}}(u_{m})}_{D_{R}(u_{m})}+\underbrace{D_{\Omega_{m}}(u_{n}-u_{m})+D_{\Omega_{m}^{C}}(u_{n}-u_{m})}_{D_{R}(u_{n}-u_{m})}$ Grazie a un ragionamento simile a quello riportato nella dimostrazione del teorema 3.53, si ha che la successione $\\{u_{n}\\}$ è $D$-cauchy. Inoltre grazie al principio del massimo si può dimostrare che $u_{n}$ converge uniformemente a $u$ su $\Omega$, infatti $\displaystyle\left|u(x)-u_{n}(x)\right|\begin{cases}=0&\text{se }x\in\Omega^{C}\\\ =\left|f(x)-u(x)\right|&\text{se }x\in\Omega\setminus\Omega_{n}\\\ \leq\max_{x\in\partial\Omega_{n}}\\{\left|f(x)-u(x)\right|\\}&\text{se }x\in\Omega_{n}\end{cases}$ dove l’ultima riga segue dal principio del massimo. Da questa relazione ricaviamo la stima: $\displaystyle\left\|u-u_{n}\right\|_{\infty}=\max_{x\in\Omega\setminus\Omega_{n}}\\{\left|u(x)-f(x)\right|\\}$ dato che $\Omega$ è compatto e $f|_{\partial\Omega}=u|_{\partial\Omega}$, la parte sinistra dell’ultima uguaglianza tende a $0$ quando $n$ tende a infinito, da cui $u$ è il limite uniforme di $u_{n}$. Dato che la successione $\\{u_{n}\\}$ è di Cauchy rispetto alla metrica $CD$, converge in questa metrica, e per unicità del limite converge a $u$, in particolare: $\displaystyle\lim_{n\to\infty}D_{R}(u-u_{n})=0$ Per dimostrare la seconda parte della proposizione, applichiamo ancora una volta la proposizione precedente, ottenendo che: $\displaystyle D_{\Omega_{n}}(f)=D_{\Omega_{n}}(u_{n})+D_{\Omega_{n}}(f-u_{n})$ e quindi anche $\displaystyle D_{\Omega}(f)=D_{\Omega}(u_{n})+D_{\Omega}(f-u_{n})$ La tesi si ottiene passando al limite per $n$ che tende a infinito. ∎ #### 3.2.5 Ideali dell’algebra di Royden In questo paragrafo parleremo di alcuni ideali dell’algebra di Royden. Ricordiamo la definizione di ideale di un’algebra (commutativa) ###### Definizione 3.33. Data un’algebra $A$, un suo sottoinsieme $I\subset A$ è detto ideale se: 1. 1. $I$ è un sottospazio vettoriale di $A$ 2. 2. per ogni $x\in A$ e $y\in I$, $xy\in I$ Un esempio di ideale su $\mathbb{M}(R)$ è l’insieme delle funzioni che si annullano in un punto $p$. Questo ideale è per altro anche il nucleo del carattere $\tau(p)$. Gli ideali a cui saremo interessati in questa sezione però sono: $\mathbb{M}_{0}(R)$ e $\mathbb{M}_{\Delta}(R)$, dove: $\displaystyle\mathbb{M}_{0}(R)\equiv\\{f\in\mathbb{M}(R)\ t.c.\ supp(f)\ compatto\\}$ mentre $\mathbb{M}_{\Delta}(R)$ è l’insieme di tutte le funzioni che sono $BD-$limiti di successioni di funzioni in $\mathbb{M}_{0}(R)$ 252525quindi $M_{\Delta}(R)$ è la chiusura sequenziale dell’insieme $M_{0}(R)$. È abbastanza facile verificare che $\mathbb{M}_{0}(R)$ è un ideale di $\mathbb{M}(R)$, meno banale è la verifica che anche $\mathbb{M}_{\Delta}(R)$ è un ideale. ###### Proposizione 3.34. $\mathbb{M}_{\Delta}(R)$ è un ideale di $\mathbb{M}(R)$. ###### Proof. Per dimostrare questa affermazione consideriamo una funzione $f\in\mathbb{M}_{\Delta}(R)$ e $h\in\mathbb{M}(R)$. Allora esiste una successione $f_{n}\in\mathbb{M}_{0}(R)$ tale che $f=BD-\lim_{n}f_{n}$ Ovviamente la successione $\\{hf_{n}\\}\subset\mathbb{M}_{0}(R)$. Dimostriamo che la successione $hf_{n}$ converge nella topologia $BD$ a $hf$, in questo modo otteniamo la tesi. Poiché $h$ è continua e limitata, la successione $hf_{n}$ converge localmente uniformemente a $hf$ ed è uniformemente limitata (quindi $hf=B-\lim_{n}hf_{n}$). Resta da dimostrare che $hf=D-\lim_{n}hf_{n}$. A questo scopo osserviamo che: $\displaystyle D_{R}(hf-hf_{n})=\int_{R}g^{ij}\left(\frac{\partial(hf- hf_{n})}{\partial x^{i}}\right)\left(\frac{\partial(hf-hf_{n})}{\partial x^{j}}\right)dV=$ $\displaystyle=\int_{R}g^{ij}\left(\frac{\partial h}{\partial x^{i}}f+h\frac{\partial f}{\partial x^{i}}-\frac{\partial h}{\partial x^{i}}f_{n}-h\frac{\partial f_{n}}{\partial x^{i}}\right)\left(\frac{\partial h}{\partial x^{j}}f+h\frac{\partial f}{\partial x^{j}}-\frac{\partial h}{\partial x^{j}}f_{n}-h\frac{\partial f_{n}}{\partial x^{j}}\right)dV=$ $\displaystyle\int_{R}g^{ij}\left(\frac{\partial h}{\partial x^{i}}(f-f_{n})+h\left(\frac{\partial f}{\partial x^{i}}-\frac{\partial f_{n}}{\partial x^{i}}\right)\right)\left(\frac{\partial h}{\partial x^{j}}(f-f_{n})+h\left(\frac{\partial f}{\partial x^{j}}-\frac{\partial f_{n}}{\partial x^{j}}\right)\right)dV\leq$ $\displaystyle\leq 2\int_{R}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}(f-f_{m})^{2}dV+2\int_{R}g^{ij}\left(\frac{\partial f}{\partial x^{i}}-\frac{\partial f_{n}}{\partial x^{i}}\right)\left(\frac{\partial f}{\partial x^{j}}-\frac{\partial f_{n}}{\partial x^{j}}\right)h^{2}dV$ dove l’ultima disuguaglianza segue dal fatto che per ogni norma $\left\|A+B\right\|^{2}\leq 2\left(\left\|A\right\|^{2}+\left\|B\right\|^{2}\right)$ applicando questa disuguaglianza alla norma $\left\|A\right\|=g^{ij}A_{i}A_{j}$ si ottiene il risultato. Ora consideriamo che per ogni $K\Subset R$: $\displaystyle\int_{R}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}(f-f_{m})^{2}dV=$ $\displaystyle=\int_{K}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}(f-f_{m})^{2}dV+\int_{R\setminus K}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}(f-f_{m})^{2}dV$ e poiché $f_{n}$ converge localmente uniformemente a $f$, il primo integrale converge a $0$ quando $m$ tende a infinito, mentre il secondo è limitato in modulo da: $\displaystyle\left|\int_{R\setminus K}g^{ij}\frac{\partial h}{\partial x^{i}}\frac{\partial h}{\partial x^{j}}(f-f_{m})^{2}dV\right|\leq ND_{R\setminus K}(h)$ dove $N$ è tale che $\left|f_{n}(x)-f(x)\right|^{2}\leq N\ \forall n\ \forall x$ 262626poiché $f$ è limitata e $f_{n}$ è uniformemente limitata, questo numero esiste. Inoltre abbiamo che: $\displaystyle\left|\int_{R}g^{ij}\left(\frac{\partial f}{\partial x^{i}}-\frac{\partial f_{n}}{\partial x^{i}}\right)\left(\frac{\partial f}{\partial x^{j}}-\frac{\partial f_{n}}{\partial x^{j}}\right)h^{2}dV\right|\leq$ $\displaystyle\leq M\int_{R}g^{ij}\left(\frac{\partial f}{\partial x^{i}}-\frac{\partial f_{n}}{\partial x^{i}}\right)\left(\frac{\partial f}{\partial x^{j}}-\frac{\partial f_{n}}{\partial x^{j}}\right)dV=MD_{R}(f_{n}-f)$ dove $M=\left\|g\right\|_{\infty}^{2}$. Dato che $D_{R}(f_{n}-f)\to 0$, otteniamo che: $\displaystyle\limsup_{n\to\infty}D_{R}(hf-hf_{n})\leq 2ND_{R\setminus K}(h)$ data l’arbitrarietà di $K$ e dato che $D_{R}(h)<\infty$, otteniamo che: $\displaystyle\lim_{n\to\infty}D_{R}(hf-hf_{n})=0$ Quindi $BD-\lim_{n}gf_{n}=gf$, il che dimostra che $gf\in\mathbb{M}_{\Delta}(R)$. ∎ Vale una proposizione simile a 3.24 per lo spazio $\mathbb{M}_{\Delta}(R)$: ###### Proposizione 3.35. Sia $f_{n}$ una successione in $\mathbb{M}_{\Delta}(R)$ tale che $\displaystyle f=C-\lim_{n}f_{n}$ con $f$ funzione (continua) limitata e $\displaystyle D_{R}(f_{n})\leq K$ dove $K$ non dipende da $n$. Allora $f\in\mathbb{M}_{\Delta}(R)$. ###### Proof. Grazie al teorema 3.24, $f\in\mathbb{M}(R)$ e esiste una sottosuccessione $f_{n_{k}}$ (che per comodità continueremo a indicare con $f_{n}$) tale che per ogni $g\in\mathbb{M}(R)$ $\displaystyle D_{R}(f-f_{n};g)\to 0$ Sia $R_{n}$ un’esaustione regolare di $R$, e definiamo le funzioni $\phi_{n}$ in modo che: 1. 1. $\phi_{n}=0$ su $R\setminus\overline{R_{2}}$ 2. 2. $\phi_{n}=f_{n}$ su $R_{1}$ 3. 3. $\phi_{n}\in H(R_{2}\setminus\overline{R_{1}})$ e sia $\phi$ definita in maniera analoga per $f$. Dato che per ogni $n$, $\phi_{n}\in\mathbb{M}_{0}(R)$, al posto che $f$ e $f_{n}$, nella dimostrazione possiamo considerare le funzioni $f-\phi$ e $f_{n}-\phi_{n}$, cioè possiamo assumere per ipotesi che $f_{n}=f=0$ sull’insieme $R_{1}$. Sia $u_{m}$ una successione di funzioni definite da: 1. 1. $u_{m}=0$ su $R_{1}$ 2. 2. $u_{m}=f$ su $R_{m}\setminus R_{1}$ 3. 3. $u_{m}\in H(R_{m}\setminus\overline{R_{1}})$ Chiaramente $u_{m}\in\mathbb{M}(R)$, e per il principio del massimo $\left\|u_{m}\right\|_{\infty}\leq\left\|f\right\|_{\infty}$. Poiché $u_{m}=u_{n}=f$ su $\partial R_{p}$ e $u_{p}$ è armonica su $R_{p}$, grazie al principio di Dirichlet (vedi 3.31) otteniamo che se $p>m$: $\displaystyle D_{R}(u_{p}-u_{m})=D_{R}(u_{m})-D_{R}(u_{p})$ questo dimostra che la successione $D_{R}(u_{m})$ è decrescente al crescere di $m$ 272727infatti se $p>m$, $D_{R}(u_{m})-D_{R}(u_{p})\geq 0$, e ovviamente limitata dal basso da $0$, quindi è convergente, e quindi se $m$ è sufficientemente grande e $p>m$, $D_{R}(u_{m}-u_{p})$ è piccolo a piacere, cioè $\\{u_{m}\\}$ è $D$-Cauchy. Grazie al principio 1.57, esiste una sottosuccessione di $u_{m}$ (che continueremo a indicare nello stesso modo) tale che: $\displaystyle u=B-\lim_{n}u_{n}\ \ \Rightarrow\ \ u=BD-\lim_{n}u_{n}$ allora $u\in\mathbb{M}(R)$ con $u(R_{1})=0$ e $u\in H(R\setminus R_{1})$. Dato che $f-u=BD-\lim_{n}(f-u_{n})$, e per costruzione $(f-u_{n})\in\mathbb{M}_{0}(R)$, $f-u\in\mathbb{M}_{\Delta}(R)$. Grazie alla formula di Green 3.30, si ha che: $\displaystyle D_{R}(f-u_{n};u)=0$ poiché $f-u_{n}=0$ sull’insieme $R\setminus R_{n}$, e anche su $R_{1}$ (quindi anche sui relativi bordi). Quindi anche $D_{R}(f-u;u)=\lim_{n}D_{R}(f-u_{n};u)=0$. Quindi otteniamo che: $\displaystyle D_{R}(f-u;u)=0\ \ \Rightarrow\ \ D_{R}(u)\equiv D_{R}(u;u)=D_{R}(f;u)=\lim_{n}D_{R}(f_{n};u)$ Dato che $f_{n}\in\mathbb{M}_{\Delta}(R)$ per ipotesi, per ogni $n$ esiste una successione $h_{k}$ di funzioni in $\mathbb{M}_{0}(R)$ tali che $f_{n}=BD-\lim_{k}h_{k}$. Questo permette di osservare che $\displaystyle D_{R}(f_{n};u)=\lim_{k}D_{R}(h_{k};u)=\lim_{k}\int_{\partial R_{1}}h_{k}\ast du$ dove l’ultima uguaglianza è conseguenza di 3.30. Dato che $\left\|h_{k}\right\|_{\infty,\partial R_{1}}\to 0$ (poiché la funzione $f_{n}=0$ su $\partial R_{1}$), l’ultimo limite è $0$, quindi $D_{R}(u)=0$, il che significa che $u\equiv 0$, e quindi $f=f-u\in\mathbb{M}_{\Delta}(R)$, come volevasi dimostrare. ∎ Come corollario a questa proposizione, osserviamo che: ###### Osservazione 3.36. Se $\\{f_{n}\\}\subset\mathbb{M}_{\Delta}(R)$ e $f=BD-\lim_{n}f_{n}$, allora $f\in\mathbb{M}_{\Delta}(R)$. ### 3.3 Compattificazione di Royden Ora siamo pronti per introdurre il concetto di compattificazione di Royden. Data una varietà riemanniana $R$, la sua compattificazione di Royden $R^{*}$ è uno spazio compatto che in qualche senso contiene $R$. La compattificazione di Royden è l’iniseme dei caratteri sull’algebra $\mathbb{M}(R)$, quindi l’integrale di Dirichlet delle funzioni gioca un ruolo fondamentale nella costruzione di questa compattificazione. Questo fa intuire che proprietà come parabolicità e iperbolicità vengano in qualche modo riflesse nelle proprietà di $R^{*}$. #### 3.3.1 Definizione ###### Definizione 3.37. Data una varietà riemanniana $(R,g)$, si definisce compattificazione di Royden uno spazio $R^{*}$ tale che: 1. 1. $R^{*}$ è uno spazio compatto di Hausdorff 2. 2. $R$ è un sottoinsieme aperto e denso di $R^{*}$ 3. 3. ogni funzione in $\mathbb{M}(R)$ può essere estesa per continuità a una funzione definita su $R^{*}$ 4. 4. l’insieme $\overline{\mathbb{M}}(R)$ delle funzioni che sono estensione delle funzioni in $\mathbb{M}(R)$ separa i punti di $R^{*}$ Osserviamo subito che se $R$ è compatta, necessariamente $R^{*}=R$. ###### Teorema 3.38. Per ogni varietà riemanniana $R$ esiste $R^{*}$ ed è unica a meno di omeomorfismi che mantengaono fissi i punti di $R$. Spezziamo la dimostrazione di questo teorema in alcuni lemmi per facilitarne la lettura, e riportiamo nell proposizione 3.44 lo schema riassuntivo della dimostrazione. Sia $R^{*}$ l’insime dei funzionali lineari moltiplicativi su $\mathbb{M}(R)$ dotato della topologia debole-* rispetto a $\mathbb{M}(R)$. Ovviamente questo spazio è uno spazio di Hausdorff 282828sia $p\neq q$, allora per definizione esiste $f\in\mathbb{M}(R)$ tale che $\left|p(f)-q(f)\right|=\delta>0$. Quindi gli intorni $V(p)=\\{h\in R^{*}\ t.c.\ \left|h(f)-p(f)\right|<\delta/3\\}$ e $W(q)=\\{h\in R^{*}\ t.c.\ \left|h(f)-q(f)\right|<\delta/3\\}$ sono intorni disgiunti dei due caratteri $p$ e $q$., e grazie alla proposizione 1.85 è anche uno spazio compatto. $R^{*}$ è il candidato a compattificazione di Royden di $R$. Per ogni punto $p\in R$, il funzionale $x_{p}(f)\equiv f(p)$ appartiene di certo all’algebra di Royden. Definiamo la funzione $\tau:R\to R^{*}$ come $\displaystyle\tau(p)=x_{p}\ \Longleftrightarrow\ \tau(p)(f)=f(p)$ ###### Lemma 3.39. La funzione $\tau$ è un’omeomorfismo sulla sua immagine. ###### Proof. Infatti sia $p\neq q$, necessariamente $\tau(p)\neq\tau(q)$. Infatti poiché $R$ è una varietà riemanniana, per ogni coppia di punti disgiunti esiste una funzione $f$ liscia a supporto compatto (quindi $f\in\mathbb{M}(R)$) tale che $f(p)\neq f(q)$ 292929questa funzione si può costruire ad esempio sfruttando le partizioni dell’unità, quindi $\tau(p)(f)\neq\tau(q)(f)$. Inoltre $\tau$ è continua. Consideriamo un qualunque aperto di $R^{*}$. Per definizione di topologia debole-*, questo aperto contiene un’intersezione finita di insiemi della forma $\displaystyle V(q,f,\epsilon)=\\{p\in R^{*}\ t.c.\ \left|p(f)-q(f)\right|<\epsilon\\}$ La controimmagine $\tau^{-1}(V)$ è aperta poiché $f$ è continua. Inoltre la mappa $\tau$ è anche una mappa aperta sulla sua immagine. Consideriamo a questo scopo un punto $p\in R$ e un suo intorno aperto $V$ qualsiasi, e dimostriamo che $\tau(p)$ ha un intorno aperto contenuto in $\tau(V)\cap\tau(R)$. Sia a questo proposito $\psi$ una funzione liscia tale che $\psi(p)=0$ e $1-\psi$ abbia supporto compatto contenuto in $V$ (questo garantisce che $\psi\in\mathbb{M}(R)$). L’insieme $\displaystyle A=\\{q\in\tau(R)\ t.c.\ \left|\psi(q)-\psi(p)\right|<1/2\\}$ è un aperto su $\tau(R)$ ed è ovviamente contenuto in $\tau(V)$. ∎ Per quanto riguarda l’estendibilità delle funzioni $f\in\mathbb{M}(R)$, osserviamo che: ###### Lemma 3.40. Tutte le funzioni $f\in\mathbb{M}(R)$ possono essere estese per continuità a $R^{*}$. ###### Proof. Definiamo $\bar{f}$ l’estensione di $f$ a $R^{*}$, e definiamo in maniera naturale $\displaystyle\bar{f}(x)\equiv x(f)\ \ \forall x\in R^{*}$ Se $p\in R$, allora $\bar{f}(p)=\tau(p)(f)=f(p)$, il che dimostra che $\bar{f}$ è un’estensione di $f$. La continuità di $\bar{f}$ segue dalla definizione di topologia debole-*, infatti: $\displaystyle\\{y\in R^{*}\ t.c.\ \left|\bar{f}(x)-\bar{f}(y)\right|<\epsilon\\}=\\{y\in R^{*}\ t.c.\ \left|x(f)-y(f)\right|<\epsilon\\}$ dove l’ultimo insieme è aperto in $R^{*}$ per definizione. ∎ Per dimostrare la densità di $R$ in $R^{*}$, utilizzeremo quest’altro lemma: ###### Lemma 3.41. Lo spazio $\overline{\mathbb{M}(R)}=\\{\bar{f}\ t.c.\ f\in\mathbb{M}(R)\\}$ è denso rispetto alla norma del sup nell’algebra delle funzioni continue da $R^{*}$ a $\mathbb{R}$, $C(R^{*})$. ###### Proof. Questa dimostrazione è una facile conseguenza del teorema di Stone-Weierstrass (vedi ad esempio teorema 7.31 pag 162 di [R3]). L’algebra $\overline{\mathbb{M}}(R)$ infatti separa i punti su $R^{*}$ e contiene la fuzione costante uguale a 1 che non si annulla mai (quindi l’algebra non si annulla in nessun punto di $R^{*}$). ∎ ###### Lemma 3.42. L’insieme $\tau(R)$ è denso in $R^{*}$. ###### Proof. Dimostriamo questa affermazione per assurdo: sia $\tilde{x}\in R^{*}\setminus\overline{\tau(R)}$. Allora per il lemma di Uryson (vedi teorema 2.12 pag 39 di [R4]) esiste una funzione continua $h:R^{*}\to\mathbb{R}$ tale che $h(\tilde{x})=0$ e $h(\tau(R))=1$. Grazie al lemma precedente, esiste una funzione $\bar{f}\in\overline{\mathbb{M}(R)}$ tale che $\bar{f}(\tilde{x})=0$ e $\bar{f}(\tau(R))>1/3$ 303030infatti grazie al lemma precedente esiste una funzione $\bar{h}\in\overline{\mathbb{M}}(R)$ tale che $\bar{h}(\tilde{x})<1/3$ e $\bar{h}(\tau/R))>2/3$. La funzione $\bar{f}=\bar{h}-\bar{h}(\tilde{x})$ ha le caratteristiche richieste.Allora la sua restrizione $f:R\to\mathbb{R}$ ha estremo inferiore positivo, quindi è invertibile in $\mathbb{M}(R)$. Questo significa che esiste $g$ tale che $(fg)(p)=1\ \forall p\in R$. Ma per definizione di carattere $\overline{(fg)}(x)=x(fg)=1\ \forall x\in R^{*}$, quindi anche $\overline{(fg)}(\tilde{x})=\bar{f}(\tilde{x})\bar{g}(\tilde{x})=1$ 313131dato che $x(fg)=x(f)x(g)$, $\overline{(fg)}=\bar{f}\bar{g}$, impossibile se $\bar{f}(\tilde{x})=0$ ∎ In seguito a questi lemmi siamo pronti a dimostrare che ###### Proposizione 3.43. $R^{*}$ l’insieme dei caratteri su $\mathbb{M}(R)$ con la topologia debole ereditata da questo spazio è una compattificazione di Royden per $R$. ###### Proof. Il fatto che $R^{*}$ sia compatto e di Hausdorff è stato dimostrato a pagina 3.38. Che $R$ sia denso in $R^{*}$ è il contenuto dei lemmi 3.39 e 3.42. Inoltre $R$ è aperto come sottoinsieme di $R^{*}$ grazie alla sua locale compattezza. Infatti sia $p\in R$ qualsiasi, e $U$ un suo intorno aperto relativamente compatto nella topologia di $R$, allora esiste un aperto $A\in R^{*}$ tale che $\tau(U)=A\cap R$. Ma allora $\tau(U)=A$, infatti se per assurdo non fosse così, $A\setminus\tau(\overline{U})$ sarebbe ancora un insieme aperto per compattezza di $\tau(\overline{U})$, e quindi se fosse non vuoto avrebbe intersezione non vuota con $R$ per densità, ma questo è impossibile, quindi $A\subset\tau(\overline{U})$, e cioè $A=\tau(U)$. Ogni funzione in $\mathbb{M}(R)$ può essere estesa per continuità a una funzione definita su $R^{*}$ grazie al lemma 3.40, e infine l’insieme $\overline{\mathbb{M}}(R)$ delle funzioni che sono estensione delle funzioni in $\mathbb{M}(R)$ separa i punti di $R^{*}$ per definizione di carattere, oppure come conseguenza del lemma 3.41. ∎ Resta da verificare l’unicità di $R^{*}$ a meno di omeomorfismi che tengano fissi gli elementi di $R$. ###### Proposizione 3.44. Sia $X$ uno spazio con le proprietà (1),(2),(3),(4) di 3.38. Allora la mappa $\sigma:X\to R^{*}$ definita da $\displaystyle\sigma(p)(f)=f(p)$ dove $f\in\overline{\mathbb{M}}(R)$, è un omeomorfismo che tiene fissi gli elementi di $R$. ###### Proof. È ovvio che $\sigma$ tiene fissi gli elementi di $R$, e grazie a un ragionamento molto simile a quello del lemma 3.39, $\sigma$ è un omeomorfismo sulla sua immagine. Essendo però $X$ compatto, $R$ denso in $R^{*}$, e anche $R^{*}$ compatto, allora necessariamente $\sigma(X)=R^{*}$, quindi $\sigma$ è anche suriettivo. ∎ D’ora in avanti per comodità di notazione confonderemo la scrittura $f$ e $\bar{f}$, cioè indicheremo una funzione $f:R\to\mathbb{R}$ e la sua estensione a tutta la compattificazione di Royden nello stesso modo. #### 3.3.2 Esempi non banali di caratteri Come preannunciato nel capitolo 2, utilizziamo gli ultrafiltri e in particolare i risultati ottenuti nella sezione 2.2 per descrivere alcuni caratteri non banali sull’algebra di Royden (quindi punti di $R^{*}$). Dalla definizione di carattere, ci si aspetta che sull’algebra di Royden esistano dei caratteri che in qualche senso rappresentino il limite della funzione in una certa direzione. Ad esempio se consideriamo una successione $x_{n}\in R$, l’operazione di limite lungo questa successione è un’operazione lineare e moltiplicativa su $\mathbb{M}(R)$ quando definita, nel senso che date due funzioni $f$ e $g$ tali che esista il limite $\lim_{n}f(x_{n})$ e $\lim_{n}g(x_{n})$, allora vale che: $\displaystyle\lim_{n}(f\cdot g)(x_{n})=\lim_{n}f(x_{n})\cdot\lim_{n}g(x_{n})$ data una funzione in $\mathbb{M}(R)$, non sempre è garantito che su ogni successione che tende a infinito esista il limite di $f(x_{n})$. Per aggirare questo problema, utilizziamo i limiti lungo ultrafiltri. ###### Osservazione 3.45. Con la tecnica mostrata nella sezione 2.2 è possibile definire caratteri anche sulle algebre di Royden. Basta considerare una qualsiasi successione $\\{x_{n}\\}$ in $R$ e un qualsiasi ultrafiltro $\mathcal{M}$ su $\mathbb{N}$ e definire $\displaystyle\phi(f)=\lim_{\mathcal{M}}f(x_{n})$ Se l’ultrafiltro è non costante, allora $\phi$ è un’operazione di limite, quindi: $\displaystyle\liminf_{n}f(x_{n})\leq\phi(f)\leq\limsup_{n}f(x_{n})$ Questo garantisce che l’operazione descritta sia un’estensione dell’operazione di limite standard, nel senso che se esiste $\lim_{n}f(x_{n})$, allora $\phi(f)=\lim_{n}f(x_{n})$, e anche che questo carattere non sia un carattere banale, non sia cioè un carattere immagine attraverso $\tau$ di un punto di $R$. Infatti è sempre possibile creare una funzione in $\mathbb{M}(R)$ che valga $1$ su un qualsiasi punto fissato $\bar{x}$ e che valga $O$ definitivamente su una successione $x_{n}$ che tende a infinito 323232quindi $\phi(f)=0$, mentre $\bar{x}(f)\equiv f(\bar{x})=1$. La compattificazione di Royden quindi contiene tutti i caratteri che rendono conto del comportamento di una funzione al “limite” lungo una successione. In realtà non è necessario passare attraverso le successioni, con un ragionamento molto simile a quello sviluppato nella sezione precendente, si osserva che ###### Proposizione 3.46. Ogni ultrafiltro $\mathcal{M}$ su $R$ definisce un carattere sull’algebra di Royden, quindi un elemento di $R^{*}$ ###### Proof. La dimostrazione è la generalizzazione della dimostrazione di 2.24. Consideriamo una funzione continua limitata $f:R\to\mathbb{R}$. La collezione $f(\mathcal{M})$ è un ultrafiltro in $\mathbb{R}$, anzi un ultrafiltro in $[\inf_{R}(f),\sup_{R}(f)]$, insieme compatto. Quindi per ogni funzione possiamo definire $\displaystyle\phi_{\mathcal{M}}(f)=\lim f(\mathcal{M})$ e con argomenti del tutto analoghi alla dimostrazione 2.24 ottenere che $\phi$ è un carattere su $M(R)$. ∎ Quello che manca in questo caso rispetto a sopra è la proprietà che vale per gli ultrafiltri non costanti $\displaystyle\liminf_{n}x(n)\leq\phi(x)\leq\limsup_{n}x(n)$ quindi una specie di controllo del carattere con il comportamento di $f$ all’infinito. Ovviamente non ha senso la definizione standard di $\liminf$ e $\limsup$ per una funzione che come dominio ha un’insieme con la potenza del continuo, ma comunque anche in questo caso esiste una proprietà simile, che verrà esplorata nella sezione successiva. #### 3.3.3 Caratterizzazione del bordo In questa sezione ci occupiamo di due caratterizzazioni del bordo di $R^{*}$, una di natura funzionale e una di natura topologica. ###### Definizione 3.47. Indichiamo con $\Gamma=R^{*}\setminus R$ il bordo di $R^{*}$ Ovviamente $\Gamma$ è un’insieme compatto in $R^{*}$. Per prima cosa dimostreremo che per un carattere $p\in\Gamma$ $p(f)=p((1-\lambda)f)$ per ogni funzione $\lambda$ a supporto compatto in $R$, cioè il valore di un carattere in $\Gamma$ dipende solo dal comportamento “all’infinito” della funzione a cui è applicato. ###### Proposizione 3.48. $p\in\Gamma$ se e solo se $p(f)=0$ $\forall f\in\mathbb{M}_{0}(R)$, o equivalentemente se per ogni funzione $f\in\mathbb{M}(R)$ e $\forall\lambda\in C^{\infty}(R,\mathbb{R})$ a supporto compatto $\displaystyle p(\lambda f)=0\ \ \Longleftrightarrow\ \ p(f)=p((1-\lambda)f)$ ###### Proof. Questa dimostrazione è un’adattamento dell’esempio 11.13 (a) pag 283 di [R2]. Per prima cosa notiamo che vale una ovvia dicotomia per gli elementi di $R^{*}$. Consideriamo un’esaustione $K_{n}$ di $R$, e consideriamo una successione di funzioni di cut-off $\lambda_{n}$ tali che $\lambda_{n}(K_{n})=1$ e $supp(\lambda_{n})\subset K_{n+1}$. Allora $\forall p\in R^{*}$ vale che $\displaystyle\forall n\ p(\lambda_{n})=0\ \vee\ \exists n\ t.c.\ p(\lambda_{n})\neq 0$ Nel primo caso, consideriamo $\lambda$ una funzione a supporto compatto. Allora esiste $n$ tale che $supp(\lambda)\subset K_{n}$, quindi $\lambda\lambda_{n}f=\lambda f$ da cui: $\displaystyle p(\lambda f)=p(\lambda_{n}\lambda f)=p(\lambda_{n})p(\lambda f)=0$ Dimostriamo che nel secondo caso necessariamente $p=\tau(x)$ per qualche $x\in K_{n+1}$. Per prima cosa osserviamo che nel secondo caso $\forall f\in\mathbb{M}(R)$ $\displaystyle p(f)=p(f\lambda_{n})/p(\lambda_{n})$ (3.15) Supponiamo per assurdo che questo non sia vero, cioè che per ogni $x$ in $K_{n+1}$, $p\neq\tau(x)$. Questo significa che esiste $f\in\mathbb{M}(R)$ tale che $\displaystyle x(f)=f(x)\neq p(f)\ \Rightarrow\ p(f-p(f)1)=0\neq x(f-p(f)1)=f(x)-p(f)$ cioè per ogni punto $x$ in $K_{n+1}$, esiste $f_{x}$ tale che $p(f_{x})=0$ e $f_{x}(x)\neq 0$. Visto che tutte le funzioni $f_{x}$ sono continue, per ogni $x$ esiste un intorno $U_{x}$ in cui $f_{x}\neq 0$. Quindi per compattezza di $K_{n+1}$ esiste un numero finito di punti $\\{x_{k}\\}\subset K_{n+1}$ tali che $\cup_{k}U_{k}\supset K_{n+1}$. Consideriamo la funzione $\displaystyle F=\sum_{k}f_{k}^{2}$ questa funzione è strettamente positiva su tutto l’insieme $K_{n+1}$, quindi anche su un suo intorno compatto $A$. Sia $\psi$ una funzione di cut-off con $supp(\psi)\Subset A$ e $\psi(K_{n+1})=1$. La funzione $\displaystyle\tilde{F}=F\cdot\psi+(1-\psi)$ è per costruzione una funzione strettamente positiva su tutta la varietà $R$ e $\tilde{F}(x)=1\ \forall x\not\in A$ Quindi $\inf_{R}(\tilde{F})>0$ e grazie alla proposizione 3.9 è un’elemento invertibile di $\mathbb{M}(R)$. Questo implica che necessariamente $p(\tilde{F})\neq 0$, ma ciò nonostante: $\displaystyle p(\tilde{F})=p(\lambda_{n}\tilde{F})/p(\lambda_{n})=p(\lambda_{n}F)/p(\lambda_{n})=p(F)=\sum_{k}[p(f_{k})]^{2}=0$ dove abbiamo sfruttato la relazione 3.15 e la definizione delle varie funzioni in gioco. ∎ Passiamo ora alla caratterizzazione topologica del bordo $\Gamma$. ###### Proposizione 3.49. $p\in\Gamma$ se e solo se $p$ non è un insieme $G_{\delta}$, cioè se e solo se $p$ non è intersezione numerabile di aperti. ###### Proof. Supponiamo per assurdo che $p$ sia un $G_{\delta}$, quindi siano $V_{n}$ tali che $\overline{V_{n+1}}\subset V_{n}$ e sia $\\{K_{n}\\}$ un’esaustione di $R$. Dato che $p\in\Gamma$, $W_{n}\equiv V_{n}\cap K_{n}^{C}$ sono ancora intorni di $p$, e vale ancora che $\overline{W_{n+1}}\subset W_{n}$. Per ogni aperto $W_{n}\setminus\overline{W_{n+1}}$, scegliamo una funzione $w_{n}$ con supporto in questo aperto, con massimo $1$ e di integrale di Dirichlet $D_{R}(w_{n})\leq 2^{-n}$ 333333possibile grazie alla proposizione 4.10. Allora la funzione $\displaystyle f=\sum_{n=1}^{\infty}w_{n}$ converge localmente uniformemente a una funzione continua su $R$ con integrale di Dirichlet finito, quindi è estendibile con contintuità a $R^{*}$. Ma visto che per ogni intorno $W_{n}$ di $p$ la funzione oscilla tra $0$ e $1$, non è estendibile con continuità in $p$, contraddizione. ∎ Questo prova che la topologia di $R^{*}$ non è I numerabile, quindi neanche metrizzabile. #### 3.3.4 Bordo armonico e decomposizione In questa sezione definiamo il bordo armonico di $R^{*}$, concetto che sarà utile per dimostrare una versione del principio del massimo. ###### Definizione 3.50. Definiamo $\Delta$ in bordo armonico di $R^{*}$ come: $\displaystyle\Delta=\\{p\in R^{*}\ t.c.\ \forall f\in\mathbb{M}_{\Delta}(R)\ f(p)=0\\}=\bigcap_{f\in\mathbb{M}_{\Delta}(R)}f^{-1}(0)$ Dalla definizione risulta evidente che $\Delta$ è un insieme chiuso, e che $\Delta\cap R=\emptyset$, cioè $\Delta\subset\Gamma$. Infatti per ogni punto di $R$ è sempre possibile trovare una funzione liscia a supporto compatto che vale 1 sul punto. Il seguente lemma dimostra l’esistena di particolari funzioni in $\mathbb{M}_{\Delta}(R)$. ###### Lemma 3.51. Sia $K$ un insieme compatto in $R^{*}$ tale che $K\cap\Delta=\emptyset$. Allora esiste una funzione $f\in\mathbb{M}_{\Delta}(R)$ identicamente uguale a $1$ su $K$. ###### Proof. Dato che $K\cap\Delta=\emptyset$, per ogni punto $p\in K$ esiste una funzione $f_{p}\in\mathbb{M}_{\Delta}(R)$ tale che $f_{p}(p)>1$. Per compattezza di $K$, esiste un numero finito di punti $p_{1},\cdots,p_{n}$ tali che $\displaystyle\bigcup_{i=1}^{n}f_{p_{n}}^{-1}(1,\infty)\supset K$ Quindi la funzione $\displaystyle f_{1}(x)\equiv\sum_{i=1}^{n}f^{2}_{p_{n}}(x)$ è una funzione appartenente a $\mathbb{M}_{\Delta}(R)$ (che ricordiamo essere un’ideale), e maggiore di $1$ sull’insieme $K$. La funzione $g(x)=\max\\{1/2,f_{1}(x)\\}$ è una funzione $g\in\mathbb{M}(R)$ invertibile grazie al lemma 3.9 e identicamente uguale a $f_{1}$ su $K$, quindi grazie al fatto che $\mathbb{M}_{\Delta}(R)$ è un’ideale di $\mathbb{M}(R)$, si ha che la funzione $\displaystyle f=f_{1}\cdot g^{-1}$ soddisfa la tesi del lemma. ∎ Come corollario, osserviamo che: ###### Osservazione 3.52. Se e solo se $\Delta=\emptyset$, allora $\mathbb{M}_{\Delta}(R)=\mathbb{M}(R)$. ###### Proof. L’implicazione da destra a sinistra è evidente dalla definizione dell’insime $\Delta$. Per quanto riguarda l’altra implicazione, se $\Delta=\emptyset$, grazie alla proposizione precedente la funzione costante uguale a $1$ su tutto $R^{*}$ appartiene a $\mathbb{M}_{\Delta}(R)$, da cui segue la tesi. ∎ Il seguente teorema permette di scrivere ogni funzione $f\in\mathbb{M}(R)$ come la somma di due funzioni, una armonica e l’altra in $\mathbb{M}_{\Delta}(R)$. Questa decomposizione sarà utile sia in questa sezione per dimostrare una forma del principio del massimo che nel seguito. ###### Teorema 3.53. Sia $f\in\mathbb{M}(R)$. Allora esistono una funzione $u\in HDB(R)$ 343434$HDB(R)$ è l’insime delle funzioni armoniche su $R$, limitate e con integrale di Dirichlet finito che indicheremo per comodità $u=\pi(f)$ e una funzione $h\in\mathbb{M}_{\Delta}(R)$ tali che 1. 1. $f=\pi(f)+h$ 2. 2. se $\Delta\neq\emptyset$, la decomposizione è unica, altrimenti unica a meno di costanti 3. 3. se $f\geq 0$, allora anche $\pi(f)\geq 0$ 4. 4. $\left\|f\right\|_{\infty}\geq\left\|\pi(f)\right\|_{\infty}$ 5. 5. per ogni funzione $\phi\in\mathbb{M}_{\Delta}(R)$, $D_{R}(\phi,\pi(f))=0$, quindi $\displaystyle D_{R}(f)=D_{R}(\pi(f))+D_{R}(h)$ ###### Proof. La dimostrazione di questo teorema è costruttiva. Sia $K_{n}$ un’esaustione di $R$ con domini regolari. Definiamo $u_{n}\in\mathbb{M}(R)$ come $u_{n}=f$ sull’insieme $R\setminus K_{n}$, e $u_{n}\in H(K_{n})$. Grazie al principio del massimo, notiamo subito che $\left\|u_{n}\right\|_{\infty}\leq\left\|f\right\|_{\infty}$ e che se $f\geq 0$, anche $u_{n}\geq 0$. Visto che la successione $u_{n}$ è uniformemente limitata, grazie al principio di Harnack 1.57 esiste una sua sottosuccessione (che continueremo a indicare con $u_{n}$) che converge localmente uniformemente a una funzione armonica $u$ definita su $R$. Inoltre, grazie all’identità di Green 3.30, se $k>n$: $\displaystyle D_{K_{k}}(u_{k}-u_{n},u_{k})=D_{R}(u_{n}-u_{k},u_{k})=0$ e quindi: $\displaystyle D_{R}(u_{n})=D_{R}(u_{k})+D_{R}(u_{n}-u_{k})+2D_{R}(u_{k},u_{n}-u_{k})=D_{R}(u_{k})+D_{R}(u_{n}-u_{k})$ in particolare, se $k>n$, $D_{R}(u_{k})\leq D_{R}(u_{n})$, quindi la successione $D_{R}(u_{n})$ è decrescente e converge a un valore $\geq 0$. Grazie all’ultima uguaglianza, questo significa anche che $D_{R}(u_{n}-u_{k})$ è piccolo a piacere se $n$ e $k$ sono sufficientemente grandi, cioè la successione $\\{u_{n}\\}$ è $D-$Cauchy. Vista la completezza $BD$ dello spazio $\mathbb{M}(R)$, la funzione $u\in\mathbb{M}(R)$, quindi $u\in HDB(R)$ Consideriamo ora la successione $h_{n}=f-u_{n}$. Per le proprietà di $u_{n}$, $h_{n}\in\mathbb{M}_{0}(R)$, e $BD-\lim_{n}h_{n}=f-u=h$, quindi chiaramente $h\in\mathbb{M}_{\Delta}(R)$. Rimanne da dimostrare l’unicità della decomposizione. Supponiamo che esistano due decomposizioni distinte per $f$: $\displaystyle f=u+h=\bar{u}+\bar{h}$ Definiamo $v=u-\bar{u}=\bar{h}-h$. Necessariamente $v\in HDB(R)\cap\mathbb{M}_{\Delta}(R)$. Quindi esiste una successione $v_{n}\in\mathbb{M}_{0}(R)$ tale che $v=BD-\lim_{n}v_{n}$, quindi: $\displaystyle D_{R}(v)=\lim_{n}D_{R}(v_{n},v)=\lim_{n}D_{K_{k}}(v_{n},v)=\lim_{n}\int_{\partial K_{k}}v_{n}\ast dv=0$ Dove $K_{k}$ è scelto in modo che $supp(v_{n})\subset K_{k}$. Questo dimostra che $v$ ha derivata nulla, quindi è costante su $R$, e se $\Delta\neq\emptyset$, allora $v=0$ dato che $v(\Delta)=0$. Il punto (5) è una facile applicazione della formula di Green 3.29. Infatti se $\phi\in\mathbb{M}_{\Delta}(R)$, esiste una successione $\phi_{n}\in\mathbb{M}_{0}(R)$ tale che $\phi=BD-\lim_{n}\phi_{n}$, quindi: $\displaystyle D_{R}(\phi,u)=\lim_{n}D_{R}(\phi_{n},u)$ Per ogni $n$, sia $K$ un compatto dal bordo liscio in $R$ tale che $supp(\phi_{n})\subset K$, allora per 3.29. $\displaystyle D_{R}(\phi_{n},u)=D_{K}(\phi_{n},u)=0$ poiché $\phi_{n}=0$ su $\partial K$. ∎ Dalla dimostrazione è facile ricavare questo corollario: ###### Proposizione 3.54. Data $f\in\mathbb{M}(R)$, se esiste una funzione subarmonica $v$ definita su $R$ tale che $v\leq f$, allora $\displaystyle v\leq\pi(f)$ allo stesso modo, se esiste $v$ superarmonica tale che $v\geq f$ allora $\displaystyle v\geq\pi(f)$ Questa proposizione può essere migliorata, infatti nella composizione si può chiedere che $\pi(f)=f$ su un predeterminato insieme compatto $K$ con bordo regolare a tratti. ###### Teorema 3.55. Sia $f\in\mathbb{M}(R)$ e sia $K$ un compatto non vuoto in $R$ con bordo regolare, allora esistono uniche una funzione $u\in HBD(R\setminus K)\cap\mathbb{M}(R)$ e $g\in\mathbb{M}_{\Delta}(R)$, $g=0$ su $K$, tali che: 1. 1. $f=u+g$ 2. 2. $D_{R}(u,\phi)=0$ per ogni $\phi\in\mathbb{M}_{\Delta}(R)$ e $\phi=0$ su $K$ 3. 3. $D_{R}(f)=D_{R}(u)+D_{R}(g)$ 4. 4. se $v$ è superarmonica su $R\setminus K$ e su questo insieme $v\geq f$, allora $v\geq u$ su $R\setminus K$ 5. 5. $\left\|u\right\|_{\infty,R\setminus K}\leq\left\|f\right\|_{\infty,R\setminus K}$ ###### Proof. La dimostrazione è del tutto analoga alla dimostrazione precedente, quindi lasciamo i dettagli al lettore. ∎ ###### Osservazione 3.56. Nelle ipotesi del teorema precedente, denotiamo la funzione $u$ con $\pi_{K}(f)$. Osserviamo che se $K\subset C$: $\displaystyle\pi_{K}(\pi_{C}(f))=\pi_{K}(f)$ ###### Proof. La dimostrazione è una semplice applicazione del teorema precedente. Infatti sappiamo che: $\displaystyle f=\pi_{K}(f)+g_{K}$ $\displaystyle f=\pi_{C}(f)+g_{C}$ $\displaystyle\pi_{C}(f)=\pi_{K}(f)+g_{K}-g_{C}\equiv\pi_{K}(\pi_{C}(f))+g$ dato che $g|_{K}\equiv(g_{K}-g_{C})|_{K}=0$ e che $g\in\mathbb{M}_{\Delta}(R)$, per unicità della decomposizione abbiamo la tesi. ∎ #### 3.3.5 Principio del massimo Ora siamo pronti a presentare questa versione del principio del massimo. ###### Teorema 3.57 (Principio del massimo). Data una funzione $u\in HBD(R)$, si ha che: $\displaystyle\min_{p\in\Delta}u(p)=\inf_{p\in R^{*}}u(p)\ \ \ \max_{p\in\Delta}u(p)=\sup_{p\in R^{*}}u(p)$ In particolare se $\Delta=\emptyset$, allora $u\in HBD(R)\Rightarrow u=cost.$ ###### Proof. Se $\Delta=\emptyset$ allora $\mathbb{M}(R)=\mathbb{M}_{\Delta}(R)$ (vedi osservazione 3.52), e si può dimostrare che $u=cost$ con un ragionamento del tutto simile a quello utilizzato nel teorema 3.53 per dimostrare che $v=cost$. Supponiamo quindi che $\Delta\neq\emptyset$. Per dimostrare la tesi dimostriamo che $u(\Delta)\leq 0$ implica $u(R)\leq 0$. Grazie alla densità di $R$ in $R^{*}$, l’ultima affermazione implica che $u(R^{*})\leq 0$, e se questo è vero, la tesi si ottiene considerando la funzione $u-\leavevmode\nobreak\ \max_{p\in\Delta}u(p)$, oppure la funzione $-u+\min_{p\in\Delta}u(p)$. Supponiamo quindi che $u(\Delta)\leq 0$. Fissato $\epsilon>0$, definiamo l’insieme $\displaystyle A=\\{p\in R^{*}\ t.c.\ u(p)\geq\epsilon\\}$ Dato che ovviamente $A\cap\Delta=\emptyset$, per ogni $p$ 353535e per definizione di $\Delta$ esiste una funzione $f_{p}\in\mathbb{M}_{\Delta}(R)$ tale che $f_{p}(p)\geq 2$. Visto che l’algebra di Royden è chiusa rispetto all’operazione di massimo, possiamo fare in modo che $f_{p}\geq 0$ su $R^{*}$ 363636basta considerare $\max\\{f_{p},0\\}$. Allora il ricoprimento $\displaystyle U_{p}=\\{q\in R^{*}\ t.c.\ f_{p}(q)>1\\}$ è un ricoprimento aperto di $A$, e quindi per compattezza esiste un numero finito di punti $p_{1},\cdots,p_{n}$ tali che $U_{p_{i}}$ ricopre $A$. Questo significa che la funzione $\displaystyle f(q)\equiv\sum_{i=1}^{n}f_{p_{n}}(q)$ è una funzione $f\in\mathbb{M}_{\Delta}(R)$ tale che $f|_{A}\geq 1$. Visto che per ipotesi la funzione $u$ è limitata da sopra, esiste un numero $M$ tale che $u-Mf\leq 0$ sull’insieme $A$, e quindi per definizione di $A$, $u-Mf-\epsilon\leq 0$ su tutta $R^{*}$. Grazie al teorema di decomposizione appena dimostrato possiamo concludere che esistono (uniche visto che $\Delta\neq\emptyset$) due funzioni $v\in HBD(R)$ e $h\in\mathbb{M}_{\Delta}(R)$ tali che $\displaystyle u-Mf-\epsilon=v+h$ e per la proposizione 3.54 $v\leq 0$ su $R^{*}$. Per unicità, $v=u-\epsilon$, e per arbitrarietà di $\epsilon$, $u\leq 0$ su $R^{*}$. ∎ La dimostrazione di questo teorema può essere facilmente adattata per ottenere: ###### Teorema 3.58. Data una funzione $u\in HBD(R\setminus K)$ con $K$ compatto in $R$, si ha che: $\displaystyle\min_{p\in(\Delta\cup\partial K)}u(p)=\inf_{p\in R^{*}\setminus K}u(p)\ \ \ \max_{p\in(\Delta\cup\partial K)}u(p)=\sup_{p\in R^{*}\setminus K}u(p)$ Come corollario a questo teorema possiamo facilmente dimostrare che ###### Proposizione 3.59. Data una funzione subarmonica $u\in\mathbb{M}(R)$ limitata, allora: $\displaystyle\max_{p\in\Delta}u(p)=\sup_{p\in R^{*}}u(p)$ Data una funzione superarmonica $v\in\mathbb{M}(R)$: $\displaystyle\min_{p\in\Delta}v(p)=\inf_{p\in R^{*}}v(p)$ ###### Proof. Dimostriamo solo il primo caso. La dimostrazione è identica al caso armonico, ma una volta trovata la decomposizione di $u-Mf-\epsilon=v+h$, non possiamo concludere che $v=u-\epsilon$, ma grazie alla proposizione 3.54, possiamo concludere che $u\leq v$. ∎ Un altro corollario di questo principio è il seguente: ###### Proposizione 3.60. Se $u\in H(R\setminus K)$ con $K$ insieme compatto 373737possibilmente vuoto è una funzione limitata e $u\in\mathbb{M}_{\Delta}(R)$, allora $u=BD-\lim_{n}u_{n}$, dove $u_{n}$ sono le funzioni definite da: $\displaystyle u_{n}(x)=\begin{cases}u(x)&se\ x\in K\\\ 0&se\ x\in K_{n}^{C}\\\ u_{n}\in H(K_{n}\setminus K)\end{cases}$ dove $K_{n}$ è un’esaustione regolare di $R$ con $K\subset K_{1}$. Questo implica che $\forall x\in R\setminus K$: $\displaystyle\min\\{0,\min u|_{\partial K}\\}\leq u(x)\leq\max\\{0,\max u|_{\partial K}\\}$ ###### Proof. Grazie alle considerazioni fatte in precedenza, sappiamo che esiste il limite $\displaystyle v=BD-\lim_{n}u_{n}$ la funzione $u-v$ è una funzione armonica su $R\setminus K$ con $(u-v)|_{K}=0$ e $(u-v)|_{\Delta}=0$, quindi grazie al principio del massimo appena dimostrato, $u=v$. L’ultima considerazione segue dal principio del massimo applicato alle funzioni $u_{n}$ sull’insieme $K_{n}\setminus K$. Per le funzioni $u_{n}$ infatti massimo e minimo sono assunti su $\partial(K_{n}\setminus K)=\partial K\cup\partial K_{n}$. Passando al limite si ottiene la tesi. ∎ La costruzione della compattificazione di Royden trova una buona giustificazione nel principio appena dimostrato. Su insiemi compatti con bordo, uno strumento fondamentale per lo studio delle funzioni armoniche è il principio del massimo che garantisce che una funzione armonica assume il suo massimo in un punto del bordo. Questo principio non è (ovviamente) applicabile se studiamo funzioni armoniche su varietà non compatte senza bordo. La compattificazione di Royden è il tentativo di “rendere compatta” una varietà in modo da poter applicare ancora i principi del massimo. Come ci si può aspettare, un insieme compatto con bordo e la compattificazione di Royden non si comportano esattamente nello stesso modo. Nelle ipotesi di quest’ultimo principio infatti compare l’ipotesi che $u\in HBD(R)$, cioè che l’integrale di Dirichlet della funzione sia finito, però questo principio dice anche dove esattamente cercare il massimo delle funzioni armoniche. Non in un punto qualsiasi del bordo come nel caso di insiemi compatti, ma è sufficiente controllare il comportamento di $f$ sul bordo armonico della varietà. Concludiamo questo capitolo con una caratterizzazione di $\mathbb{M}_{\Delta}(R)$. Dalla definizione 3.50, il bordo armonico è l’insieme dove tutte le funzioni in $\mathbb{M}_{\Delta}(R)$ sono nulle. Vale anche una sorta di viceversa, nel senso che: ###### Proposizione 3.61. $\displaystyle\mathbb{M}_{\Delta}(R)=\\{f\in\mathbb{M}(R)\ \ t.c.\ f(\Delta)=0\\}$ ###### Proof. La dimostrazione segue facilmente dalla decomposizione descritta nella proposizione 3.53 e dal principio del massimo 3.57. Se $\Delta=\emptyset$, $\mathbb{M}_{\Delta}(R)=\mathbb{M}(R)$ come dall’osservazione 3.52. Supponiamo quindi che $\Delta\neq\ \emptyset$. Se $f\in\mathbb{M}_{\Delta}(R)$, è ovvio dalla definizione che $f(\Delta)=0$. Supponiamo quindi che $f(\Delta)=0$. Allora, grazie al teorema 3.53, esistono due funzioni $u\in HBD(R)$ e $h\in\mathbb{M}_{\Delta}(R)$ tali che $\displaystyle f=u+h$ Essendo $h(\Delta)=f(\Delta)=0$, si ha che $u(\Delta)=0$, quindi per il principio del massimo 3.57, $u$ è identicamente nulla, da cui la tesi. ∎ ## Chapter 4 Varietà paraboliche e iperboliche In questo capitolo tratteremo proprietà e caratterizzazioni di varietà paraboliche e iperboliche. Il risultato principale è l’esistenza dei potenziali di Evans sulle varietà paraboliche. In tutto il capitolo $R$ sarà una varietà riemanniana sensa bordo di dimesione $m$. ### 4.1 Capacità In questa sezione introduciamo la capacità di una coppia di insiemi e definiamo di conseguenza le varietà paraboliche. I risultati principali sono tratti da [G2] e [PSR] capitolo 7. ###### Definizione 4.1. Dati due insiemi $K$ compatto e $K\subset\Omega$ aperto paracompatto, definiamo la capacità di $K$ rispetto a $\Omega$: $\displaystyle\text{Cap}(K,\Omega)=\inf{\int_{R}(\nabla f)^{2}dV}$ dove l’inf è preso su tutte le funzioni di Tonelli 111in realtà l’insieme di funzioni considerate può essere più generale senza cambiare il valore della capacità, vedi [G2] tali che $\displaystyle f(K)=1\ \ supp(f)\subset\overline{\Omega}\ \ 0\leq f\leq 1$ Data questa definizione, osserviamo subito che la capacità di una coppia di insiemi aumenta se rimpiccioliamo $\Omega$ e se ingrandiamo $K$, cioè se $K\subset K^{\prime}$ e $\Omega\subset\Omega^{\prime}$: $\displaystyle\text{Cap}(K,\Omega^{\prime})\leq\text{Cap}(K,\Omega)\leq\text{Cap}(K^{\prime},\Omega)\ ;\ \text{Cap}(K,\Omega^{\prime})\leq\text{Cap}(K^{\prime},\Omega^{\prime})\leq\text{Cap}(K^{\prime},\Omega^{\prime})$ Visti gli scopi della tesi, d’ora in avanti considereremo solo insiemi $K$ e $\Omega$ con bordo regolare. La prima domanda data questa definizione è se l’inf è raggiunto, e in tale caso da quale funzione. La risposta dipende dalla regolarità dei bordi di $K$ e $\Omega$, in particolare se entrambi i bordi sono lisci grazie al principio di Dirichlet possiamo dimostrare che: ###### Proposizione 4.2. Il valore della capacità di una coppia di insiemi $K\Subset\Omega$ con bordi lisci è equivalente all’integrale di Dirichlet della funzione soluzione del problema $\displaystyle\ \begin{cases}\Delta u=0\ \ in\ \Omega\setminus K\\\ u|_{K}=1\\\ u|_{\Omega^{C}}=0\end{cases}$ (4.1) Chiamiamo la funzione $u$ potenziale di capacità della coppia $K$, $\Omega$. ###### Proof. Osserviamo che se i bordi degli insiemi sono lisci, è possibile risolvere il problema di Dirichlet che definisce $u$. La dimostrazione è una diretta conseguenza del principio di Dirichlet riportato nella proposizione 3.31. ∎ Nella seguente proposizione osserviamo che se è possibile risolvere il problema di Dirichlet 4.1, anche se $\Omega$ e $K$ non hanno bordo liscio $\text{Cap}(K,\Omega)=D_{\Omega}(u)$. ###### Proposizione 4.3. Siano $K$ e $\Omega$ tali che $\Omega\setminus K$ sia regolare per il problema di Dirichlet 222cioè tale che il problema di Dirichlet 4.1 abbia soluzione, vedi sezione 1.9 per condizioni sulla regolarità degli insiemi. Se esiste una funzione di Tonelli $f$ tale che $f|_{K}=1$, $f|_{\Omega^{C}}=0$ e $D_{R}(f)<\infty$, allora $\text{Cap}(K,\Omega)=D_{R}(u)<\infty$. ###### Proof. La dimostrazione è una diretta conseguenza del principio di Dirichlet riportato nella proposizione 3.32. ∎ ###### Proposizione 4.4. Nella definizione di capacità, se $\Omega$ e $K$ hanno bordi regolari, allora possiamo sostituire l’insieme delle funzioni di Tonelli con l’insieme delle funzioni $f$ lisce in $\Omega\setminus K$ uguali a $1$ su $K$ e nulle su $\Omega^{C}$. ###### Proof. Questo risultato segue dalla densità delle funzioni lisce in $\Omega\setminus K$ e uguali a $1$ su $K$ e nulle in $\Omega^{C}$ illustrato nella proposizione 3.28. ∎ Possiamo ulteriormente caratterizzare la capacità di una coppia di insiemi grazie alle formule di Green. ###### Proposizione 4.5. Sia $u$ il potenziale di capacità di $(K,\Omega)$. Allora si ha che: $\displaystyle\text{Cap}(K,\Omega)=\int_{R}\left|\nabla u\right|^{2}dV=-\int_{\partial K}\ast du=-\int_{\partial\Omega}\ast du$ ###### Proof. La dimostrazione della prima uguaglianza è una diretta conseguenza della formula di Green 3.29. La seconda segue dalla considerazione che $u$ è armonica in $\Omega\setminus K$, quindi grazie alla proposizione 1.11 $\displaystyle 0=\int_{\Omega\setminus K}\Delta u\ dV=\int_{\partial(\Omega\setminus K)}\ast du=\int_{\partial\Omega}\ast du+\int_{\partial K}\ast du$ ∎ Oltre alla capacità di una coppia di insiemi, è possibile definire la capacità di un insieme compatto come: ###### Definizione 4.6. Dato un compatto $K\Subset R$ regolare per il problema di Dirichlet, definiamo $\displaystyle\text{Cap}(K)=\lim_{n}\text{Cap}(K,E_{n})$ dove $E_{n}$ è una qualsiasi esaustione regolare di $R$ 333per l’esistenza di queste esaustioni, vedi 1.20. Questa definizione è ben posta, nel senso che non dipende dalla scelta dell’esaustione. ###### Proof. Date due esaustioni regolari $E_{n}$ e $C_{n}$, definiamo $\displaystyle e_{n}\equiv\text{Cap}(K,E_{n})\to e\ \ c_{n}\equiv\text{Cap}(K,C_{n})\to c$ le due successioni sono monotone decrescenti grazie alle considerazioni fatte prima. Inoltre per la compattezza di ogni $E_{n}$, esiste un intero $k$ tale che $E_{n}\Subset C_{k}$. Questo implica che $e_{n}\geq c_{k}\geq c$. Quindi anche $e\geq c$. Scambiando i ruoli di $E_{n}$ e $C_{n}$ si ottiene $e=c$. ∎ Possiamo definire un potenziale di capacità anche per un singolo insieme compatto. ###### Proposizione 4.7. Dato un insieme aperto relativamente compatto regolare per il problema di Dirichlet $K$ e una qualsiasi esaustione regolare $K_{n}$ con $K\subset K_{1}^{\circ}$ di $R$, sia $u_{n}$ il potenziale di capacità di $(K,K_{n})$, allora la funzione $\displaystyle u=\lim_{n}u_{n}$ è il potenziale di capacità per $K$, nel senso che $u$ è una funzione armonica su $R\setminus K$, $u=1$ su $K$ tale che $\displaystyle\text{Cap}(K)=D_{R}(u)$ ###### Proof. La successione $u_{n}$ è una successione di funzioni armoniche su $K\setminus K_{n}$ strettamente positive su $K_{n}^{\circ}$ grazie alla proposizione 1.54. Dato che tutte queste funzioni sono uniformemente limitate da $1$, grazie alle proposizioni 1.57 e 1.58 $u_{n}$ converge localmente uniformemente a una funzione $u$ armonica in $R\setminus K$. Sempre grazie alla proposizione 1.58 si ha che $\displaystyle\lim_{n}D_{R}(u_{n})=D_{R}(u)$ come volevasi dimostrare. ∎ Osserviamo che possiamo estendere la nozione di capacità e di potenziale di capacità anche a insiemi più generali di quelli utilizzati fino ad ora. In particolare, dato $C\subset R$ compatto, se è possibile risolvere il problema di Dirichlet $\displaystyle u|_{K}=1\ \ \ u|_{\partial E_{n}}=0\ \ \ u\in H(E_{n}\setminus K)$ dove $E_{n}$ è un’esaustione regolare di $R$ con $K\subset E_{1}$, allora ha senso parlare di capacità di $K$ e di potenziale di capacità di $K$. In particolare: ###### Osservazione 4.8. Se $K$ è una sottovarietà regolare compatta di $R$ di codimensione $1$ possibilmente con bordo, allora ha senso definire la sua capacità e il suo potenziale di capacità. ###### Proof. Questo risultato segue dalle considerazioni della sezione 1.9.3. ∎ Prima di proseguire, riportiamo un’applicazione di quanto appena descritto. In particolare dimostriamo che in ogni insieme aperto, esiste sempre una coppia $(K,\Omega)$ contenuta in questo insieme con capacità grande a piacere e piccola a piacere (quindi una funzione armonica con integrale di Dirichlet grande a piacere e piccolo a piacere). ###### Proposizione 4.9. Dato un qualsiasi aperto $A\subset R$, esistono $K\Subset\Omega\subset A$ tali che $\text{Cap}(K,\Omega)$ è grande a piacere. ###### Proof. La dimostrazione di questa proposizione è costruttiva. Consideriamo un punto $p\in A$, allora esiste un intorno normale $B(p)\subset A$, intorno che possiamo dotare delle coordinate geodetiche $(r,\theta)$ (fuori da $p$) 444vedi sezione 1.1.3. Consideriamo $\displaystyle K=\\{p\in B\ t.c.\ r(p)\leq R_{1}\\}\ \ \Omega=\\{p\in B\ t.c.\ r(p)<R_{2}\\}$ dove $R_{1}<R_{2}$ e $R_{2}$ è tale che $\Omega\subset B(p)$. Consideriamo una qualsiasi funzione di Tonelli $0\leq f\leq 1$ tale che $f(K)=1$ e $f=0$ fuori da $\Omega$. Per comodità di notazione, utiliziamo lo stesso simbolo $f$ con la sua rappresentazione in coordinate geodetiche. Abbiamo che $\displaystyle\int_{R}\left|\nabla f\right|^{2}\ dV=\int_{\Omega\setminus K}\left|\nabla f\right|^{2}\ dV=\int_{\Omega\setminus K}g^{ij}\frac{\partial f}{\partial x^{i}}\frac{\partial f}{\partial x^{j}}\ \sqrt{\left|g\right|}drd\theta^{1}\cdots d\theta^{m-1}$ dove $r=x^{1}$ e $\theta$ rappresenta tutte le altre coordinate. Grazie alla particolare forma che la metrica assume in coordinate polari geodetiche (vedi 1.3) si ha che: $\displaystyle\int_{\Omega\setminus K}\left|\nabla f\right|^{2}\ dV\geq\int_{\Omega\setminus K}\left(\frac{\partial f}{\partial r}\right)^{2}\ \sqrt{\left|g\right|}drd\theta^{1}\cdots d\theta^{m-1}\geq$ $\displaystyle\geq c\omega_{m-1}\int_{R_{1}}^{R_{2}}\left(\frac{\partial f}{\partial r}\right)^{2}dr$ dove $c$ è un limite inferiore positivo per $\sqrt{\left|g\right|}$ su $B(p)\setminus K$. Grazie alla disuguaglianza di Schwartz, otteniamo che: $\displaystyle\frac{1}{R_{2}-R_{1}}=\frac{1}{R_{2}-R_{1}}\left(\int_{R_{1}}^{R_{2}}\frac{\partial f}{\partial r}dr\right)^{2}\leq\int_{R_{1}}^{R_{2}}\left(\frac{\partial f}{\partial r}\right)^{2}dr$ il che dimostra che a patto di scegliere $R_{2}-R_{1}$ sufficientemente piccolo, la capacità dell’anello è abbastanza grande. Osserviamo che in questa dimostrazione è essenziale trovare il limite inferiore per $\sqrt{\left|g\right|}$ 555che ad esempio nel caso di $\mathbb{R}^{m}$ vale $\omega_{m-1}r^{m-1}$, dove $\omega_{m-1}$ è l’area della sfera $m-1$ dimensionale rispetto alla metrica euclidea standard, quindi è essenziale che $R_{1}$ non tenda a $0$ quando scegliamo $R_{2}-R_{1}$ piccolo. In pratica per ottenere la tesi, fissato $R_{1}$ e trovata la costante $c$, scegliamo $R_{2}$ in modo che $R_{2}-R_{1}$ sia sufficientemente piccolo. ∎ ###### Proposizione 4.10. Dato un qualsiasi aperto $A\subset R$, esistono $K\Subset\Omega\subset A$ tali che $\text{Cap}(K,\Omega)$ è piccola a piacere. ###### Proof. Procediamo in maniera del tutto analoga a sopra, solo che in questo caso al posto di considerare una funzione di Tonelli qualsiasi, ne consideriamo una particolare. Infatti se vogliamo dimostrare che la capacità di un’insieme è piccola, basta trovare una funzione che renda piccolo l’integrale di Dirichlet. Sia come sopra $B(p)$ un intorno normale contenuto in $A$, e siano $\displaystyle K=\\{p\in B\ t.c.\ r(p)\leq R_{1}\\}\ \ \Omega=\\{p\in B\ t.c.\ r(p)<R_{2}\\}$ dove $R_{1}<R_{2}$ e $R_{2}$ è tale che $\Omega\subset B(p)$. Consideriamo la funzione $\displaystyle f(p)=\begin{cases}1&se\ p\in K\\\ 0&se\ p\in\Omega^{C}\\\ \frac{\log(r(p)/R_{2})}{\log(R_{1}/R_{2})}&se\ p\in\Omega\setminus K\end{cases}$ Dato che $f$ dipende solo da $r(p)$, il suo integrale di Dirichlet vale: $\displaystyle\int_{R}\left|\nabla f\right|^{2}dV=\int_{\Omega\setminus K}\left(\frac{\partial f}{\partial r}\right)^{2}\sqrt{\left|g\right|}drd\theta^{1}\cdots d\theta^{m-1}=$ $\displaystyle=\int_{\Omega\setminus K}\frac{1}{\log^{2}(R_{1}/R_{2})}\frac{1}{r^{2}}\sqrt{\left|g\right|}drd\theta^{1}\cdots d\theta^{m-1}$ Notiamo che l’integrale $\int_{r=k}\sqrt{\left|g\right|}d\theta^{1}\cdots d\theta^{m-1}$ è la superficie della sfera di raggio $k$, quindi per le proprietà della metrica, se $k$ è tale che $B_{k}\subset B(p)$, allora esiste una costante $C$ tale che: $\displaystyle\int_{r=k}\sqrt{\left|g\right|}d\theta^{1}\cdots d\theta^{m-1}\leq Ck^{m-1}$ dove $m$ rappresenta la dimensione della varietà. Questa stima permette di concludere che: $\displaystyle\int_{R}\left|\nabla f\right|^{2}dV\leq\frac{C}{\log^{2}(R_{1}/R_{2})}\int_{R_{1}}^{R_{2}}r^{m-3}dr$ Se $m=2$, si ottiene: $\displaystyle\int_{R}\left|\nabla f\right|^{2}dV\leq\frac{C}{\log(R_{2}/R_{1})}$ quindi fissato $R_{2}$, è possibile scegliere $R_{1}>0$ in modo che questa quantità sia piccola a piacere. In caso $m\geq 3$ invece si ottiene: $\displaystyle\int_{R}\left|\nabla f\right|^{2}dV\leq\frac{C}{\log^{2}(R_{1}/R_{2})}(R_{2}^{m-2}-R_{1}^{m-2})$ Anche in questo caso, fissato $R_{2}$ è sempre possibile scegliere $R_{1}$ in modo che $D_{R}(f)$ sia piccolo a piacere. ∎ La capacità di una coppia di insiemi è legata al comportamento della funzione di Green su quell’insieme, in particolare si ha che 666proposizione 4.1 pag 154 di [G2]: ###### Proposizione 4.11. Siano $K,\ \Omega$ insiemi compatti dal bordo liscio in $R$ con $K\Subset\Omega^{\circ}$, e sia $p\in K$. Allora: $\displaystyle\min_{x\in\partial K}G_{\Omega}(p,x)\leq(\text{Cap}(K,\Omega))^{-1}\leq\max_{x\in\partial K}G_{\Omega}(p,x)$ dove $G_{\Omega}$ indica la funzione di Green relativa al dominio $\Omega$ 777vedi proposizione 1.62. ###### Proof. Siano $\displaystyle b\equiv\min_{x\in\partial K}G_{\Omega}(p,x)\ \ \ a\equiv\max_{x\in\partial K}G_{\Omega}(p,x)$ e definiamo per ogni $c>0$ l’insieme compatto $\displaystyle F_{c}\equiv\\{x\in\Omega\ t.c.\ G_{\Omega}(p,x)\geq c\\}$ Per prima cosa osserviamo che $F_{a}\subset K\subset F_{b}$. Questo è conseguenza del principio del massimo, infatti la funzione $G_{\Omega}(\cdot,p)$ è armonica su $\Omega\setminus K$, quindi assume il suo massimo su $\partial K$, cioè $G_{\Omega}(x,p)<a$ se $x\in\Omega\setminus K$. Inoltre dato che $G_{\Omega}$ è superarmonica su $K^{\circ}$, si ha che $G_{\Omega}(x,p)>b$ se $x\in K$. Grazie alla “monotonia” della capacità, si ha che: $\displaystyle\text{Cap}(F_{a},\Omega)\leq\text{Cap}(K,\Omega)\leq\text{Cap}(F_{b},\Omega)$ la tesi segue dalla considerazione che $\text{Cap}(F_{c},\Omega)=1/c$. Per il teorema si Sard, quasi ogni $c$ è un valore regolare per $G_{\Omega}(\cdot,p)$, quindi $F_{c}$ ha bordo regolare. Per questi valori di $c$, la funzione $G_{\Omega}/c$ (estesa costante uguale a $1$ dentro $F_{c}$ e uguale a $0$ fuori da $\Omega$) è il potenziale di capacità per la coppia $(F_{c},\Omega)$, quindi grazie alla proposizione 4.5: $\displaystyle\text{Cap}(F_{c},\Omega)=-\frac{1}{c}\int_{\partial F_{c}}\ast dG_{\Omega}(\cdot,p)=\frac{1}{c}$ Se $c$ non è un valore regolare di $G_{\Omega}(\cdot,p)$, allora esiste $c_{n}\nearrow c$ e $c_{n}^{\prime}\searrow c$ successioni di valori regolari di $G_{\Omega}(\cdot,p)$. Dalla monotonia della capacità si ottiene che per ogni $n$: $\displaystyle\frac{1}{c_{n}}\leq\text{Cap}(F_{c},\Omega)\leq\frac{1}{c_{n}^{\prime}}$ da cui $Cap(F_{c},\Omega)=1/c$. ∎ Grazie alla definizione di capacità appena data possiamo dividere in 2 categorie le varietà Riemanniane, quelle per cui ogni insieme aperto non vuoto relativamente compatto 888noi considereremo solo compatti con bordo liscio per comodità ha capacità positiva, e quelle per cui ogni insieme compatto ha capacità nulla. In questa sezione daremo delle caratterizzazioni equivalenti di queste proprietà. ###### Definizione 4.12. Una varietà Riemanniana $R$ si dice parabolica se ogni insieme aperto non vuoto relativamente compatto con bordo liscio ha capacità nulla, in caso contrario la varietà si definisce iperbolica Per prima cosa osserviamo che nella definizione non è necessario chiedere che ogni insieme aperto relativamente compatto abbia capacità nulla, basta che un solo aperto relativamente compatto possieda questa proprietà e automaticamente tutti gli aperti non vuoti relativamente compatti hanno capacità nulla. Questo implica anche che se un solo aperto relativamente compatto ha capacità positiva, tuttigli aperti relativamente compatti hanno capacità postiva. ###### Proposizione 4.13. Sia $R$ una varietà Riemanniana. Se un insieme aperto non vuoto relativamente compatto con bordo liscio $K\Subset R$ ha capacità nulla, allora tutti gli aperti non vuoti relativamente compatti in $R$ hanno capacità nulla. ###### Proof. Sia $K$ un aperto relativamente compatto con bordo liscio di capacità nulla. Allora tutti gli insiemi compatti contenuti in $K$ hanno capacità nulla grazie alla “monotonia” della capacità. Consideriamo un aperto relativamente compatto con bordo liscio $C\supset K$ e chiamiamo $u$ il potenziale di capacità di $K$ e $u^{\prime}$ il potenziale di capacità di $C$. Sia $K_{n}$ un’esaustione regolare di $R$ con $C\subset K_{1}^{\circ}$ e sia $u_{n}$ il potenziale di capacità di $(K,K_{n})$, $u_{n}^{\prime}$ il potenziale di capacità di $(C,K_{n})$. Grazie al principio del massimo applicato all’insieme $K_{n}\setminus C$ (vedi 1.50), si ha che $u_{n}\leq u_{n}^{\prime}$, quindi $u_{n}\leq u^{\prime}$, disuguaglianza valida per ogni $n$. Passando al limite si ottiene che $u\leq u^{\prime}$. Se $K$ ha capacità nulla, il suo potenziale di capacità è la funzione costante uguale a $1$, quindi necessariamente anche $u^{\prime}$ è costante uguale a $1$, cioè la capacità di $C$ è nulla. Se $C\cap K\neq\emptyset$, allora considerando una bolla $B\subset C\cap K$, $B$ ha capacità nulla poiché contenuta in $K$, e quindi $C$ ha capacità nulla poiché contiene $B$. Infine se $C\cap K=\emptyset$, allora $K\subset C\cup K$, quindi grazie a quanto appena dimostrato $C\cup K$ ha capacità e anche $C\subset C\cup K$ ha capacità nulla. ∎ Osserviamo da questa definizione che modificando la metrica di $R$ su un insieme compatto, la varietà rimane parabolica o non parabolica come la varietà non modificata. ###### Proposizione 4.14. Sia $(R,g)$ una varietà riemanniana, e sia $(R,g^{\prime})$ un’altra varietà con $g=g^{\prime}$ fuori da un compatto $K$. Allora $(R,g)$ è parabolica se e solo se $(R,g^{\prime})$ lo è. ###### Proof. Consideriamo un compatto $C$ che contenga $K$ nella sua parte interna. La capacità di questo compatto nelle due varietà è identica, perché su $C^{C}$ le due metriche coincidono, da cui la tesi. ∎ ### 4.2 Bordo Armonico La prima caratterizzazione che diamo della parabolicità riguarda il bordo armonico di $R^{*}$. ###### Proposizione 4.15. La varietà $R$ è parabolica se e solo se $1\in\mathbb{M}_{\Delta}(R)$. ###### Proof. Una parte della dimostrazione è semplice. Se la capacità di un insieme compatto $K$ è nulla, questo implica che il suo potenziale armonico $u$ è costante uguale a $1$. Dato che $u$ per definizione è il limite locale uniforme del potenziale armonico $u_{n}$ della coppia $(K,K_{n})$, e dato che $u_{n}$ è limitata da $1$, si ha che $u=B-\lim_{n}u_{n}$. La successione $u_{n}$ inoltre è anche di Cauchy rispetto alla seminorma $D$, infatti se $m>n$: $\displaystyle D_{R}(u_{n}-u_{m})=D_{R}(u_{n})-2D_{R}(u_{n};u_{m})+D_{R}(u_{m})$ Poiché la funzione di Tonelli $u_{m}-u_{n}$ è nulla sul bordo di $K_{m}\setminus K$, grazie alla formula di Green 3.30 osserviamo che $\displaystyle D_{R}(u_{m}-u_{n};u_{m})=D_{K_{m}\setminus K}(u_{m}-u_{n};u_{m})=0$ Questo ci permette di concludere che: $\displaystyle 2D_{R}(u_{m})-2D_{R}(u_{m};u_{n})=0$ da cui $\displaystyle D_{R}(u_{n}-u_{m})=D_{R}(u_{n})-D_{R}(u_{m})$ poiché la successione $D_{R}(u_{n})$ è decrescente e converge a $\text{Cap}(K)$, abbiamo dimostrato che $\\{u_{n}\\}$ è $D-$Cauchy. Quindi $1=BD-\lim_{n}u_{n}$, e poiché $u_{n}\in\mathbb{M}_{0}(R)$, $1\in\mathbb{M}_{\Delta}(R)$. Per dimostrare l’implicazione inversa, sia $u$ il potenziale di capacità di un compatto $K$ e sia $f_{n}$ una successione in $\mathbb{M}_{0}(R)$ tale che: $\displaystyle 1=BD-\lim_{n}f_{n}$ Chiaramente: $\displaystyle u=B-\lim_{n}uf_{n}$ Per dimostrare che $u=D-\lim_{n}uf_{n}$ consideriamo un compatto qualsiasi $C\Subset R$, e osserviamo che: $\displaystyle D_{R}(uf_{n}-u)=\int_{R}\left|\nabla(u(f_{n}-1))\right|^{2}dV\leq$ $\displaystyle\leq 2\int_{R}\left|u\right|^{2}\left|\nabla(f_{n}-1)\right|^{2}dV+2\int_{R}\left|\nabla u\right|^{2}\left|f_{n}-1\right|^{2}dV$ il primo termine della somma converge a $0$ per ipotesi, mentre per il secondo termine osserviamo che: $\displaystyle\int_{R}\left|\nabla u\right|^{2}\left|f_{n}-1\right|^{2}dV=\int_{C}\left|\nabla u\right|^{2}\left|f_{n}-1\right|^{2}dV+\int_{R\setminus C}\left|\nabla u\right|^{2}\left|f_{n}-1\right|^{2}dV$ dove $C$ è un compatto qualsiasi. Dato che $u\in\mathbb{M}(R)$ (quindi ha integrale di Dirichlet finito) e $f_{n}$ converge localmente uniformemente a $1$, il primo addendo tende a zero indipendentemente dal compatto scelto. Detto $b=\limsup_{n}\left|f_{n}-1\right|^{2}$ 999sicuramente finito grazie all’uniforme limitatezza della successione $\\{f_{n}\\}$, abbiamo che per ogni $C\Subset R$: $\displaystyle\limsup_{n}D_{R}(uf_{n}-u)\leq 2b\int_{R\setminus C}\left|\nabla u\right|^{2}dV$ vista l’arbitrarietà di $C$, e dato che $\int_{R}\left|\nabla u\right|^{2}dV<\infty$, possiamo concludere che $\displaystyle\lim_{n}D_{R}(uf_{n}-u)=0$ Sia $F_{n}$ un compatto dal bordo regolare che contiene $supp(f_{n})$, allora grazie alla formula di Green 3.30 applicata all’insieme $A_{n}=F_{n}\setminus K$ , osserviamo che $\displaystyle D_{R}((1-u)f_{n},1-u)=D_{A_{n}}((1-u)f_{n},1-u)=0$ poiché la funzione di Tonelli $(1-u)f_{n}$ è nulla sul bordo di $A_{n}$. Grazie alle ultime due considerazioni possiamo concludere che: $\displaystyle D_{R}(u)=D_{R}(1-u)\equiv D_{R}(1-u;1-u)=\lim_{n}D_{R}((1-u)f_{n},1-u)=0$ quindi la capacità di $K$ è nulla, il che dimostra la parabolicità di $R$. ∎ Come corollario di questa proposizione osserviamo che: ###### Proposizione 4.16. Una varietà $R$ è parabolica se e solo se 1. 1. $1\in\mathbb{M}_{\Delta}(R)$ 2. 2. $\mathbb{M}_{\Delta}(R)=\mathbb{M}(R)$ 3. 3. $\Delta=\emptyset$ ###### Proof. Il punto (1) è il contenuto della proposizione precedente. Dato che $\mathbb{M}_{\Delta}(R)$ è un’ideale di $\mathbb{M}(R)$, se $1\in\mathbb{M}_{\Delta}(R)$ necessariamente $\mathbb{M}(R)=\mathbb{M}_{\Delta}(R)$ e viceversa. L’equivalenza tra (2) e (3) è il contenuto dell’osservazione 3.52. ∎ Osserviamo che grazie ad una forma del principio del massimo (la forma riportata in 3.57), una varietà parabolica non ammette funzioni armoniche limitate con integrale di Dirichlet finito che non siano costanti. Il viceversa non è vero, ad esempio $\mathbb{R}^{n}$ con la metrica euclidea standard non ammette funzioni armoniche limitate non costanti pur non essendo una varietà parabolica (per $n\geq 3$) 101010vedi teorema 2.1 pag. 31 di [ABR]. Possiamo migliorare questa osservazione, infatti non è necessario chiedere che la funzione armonica abbia integrale di Dirichlet finito, e neanche che sia limitata, basta una delle due condizioni per dimostrare che la funzione è costante. ###### Proposizione 4.17. Una varietà Riemanniana $R$ parabolica non ammette funzioni armoniche limitate non costanti. ###### Proof. Sia $u$ una funzione subarmonica limitata dall’alto su $R$. A meno di una traslazione, possiamo supporre $u\geq 0$. Sia $a=\sup_{x\in R}u(x)<\infty$. Consideriamo un’esaustione regolare $K_{n}$ $n=0,1,\cdots$, e siano $(1-\omega_{n})$ i potenziali armonici della coppia $(K_{0},K_{n})$, e sia $b=\max_{x\in K_{0}}u(x)$. Allora grazie al principio del massimo applicato a $K_{n}\setminus K_{0}$ abbiamo che: $\displaystyle u(x)\leq b+a\omega_{n}$ per ogni $n$. Passando al limite, visto che $R$ è parabolica, otteniamo che: $\displaystyle u(x)\leq b+a0=b$ Questo è valido per ogni $K_{0}$ dominio relativamente compatto con bordo liscio. Se consideriamo l’insieme delle bolle coordinate centrate in un punto qualsiasi $x_{0}$, otteniamo quindi che $u(x)\leq u(x_{0})$. Ripetendo il ragionamento con $-u$, otteniamo che $u(x)=u(x_{0})$ per ogni $x\in R$. ∎ ###### Proposizione 4.18. Una varietà parabolica $R$ non ammette funzioni armoniche con integrale di Dirichlet finito che non siano costanti. ###### Proof. Sia $u$ una funzione con integrale di Dirichlet $D_{R}(u)<\infty$. Dato che $R$ è parabolica, $1\in\mathbb{M}_{\Delta}(R)$ 111111vedi 4.15, quindi esiste una successione di funzioni $\phi_{n}\in\mathbb{M}_{0}(R)$ tale che $1=BD-\lim_{n}\phi_{n}$. Consideriamo la funzione $\displaystyle u_{m}(z)\equiv\min\\{\max\\{u(z),-m\\}m\\}$ cioè $u_{m}$ è la funzione $u$ troncata in $[-m,m]$. Grazie alla formula di Green 3.29 se consideriamo un insieme compatto dal bordo liscio $K$ tale che $supp(\phi_{n})\subset K$, otteniamo che: $\displaystyle D_{R}(\phi_{n}u_{m},u)=D_{K}(\phi_{n}u_{m},u)=0$ poiché la funzione $\phi_{n}u_{m}|_{\partial K}=0$. Dato che $1=BD-\lim_{n}\phi_{n}$, $u_{m}=BD-\lim_{n}\phi_{n}u_{m}$. Infatti: $\displaystyle D_{R}(\phi_{n}u_{m}-u_{m})=\int_{R}\left|\nabla{(\phi_{n}-1)u_{m}}\right|^{2}dV=\int_{R}\left|\nabla(\phi_{n}-1)u_{m}+(\phi_{n}-1)\nabla u_{m}\right|^{2}dV$ dato che $u_{m}$ è limitata e ha integrale di Dirichlet finito, si ottiene facilmente che il limite per $n\to\infty$ di questa quantità è $0$. Questo dimostra che $u_{m}=D-\lim_{n}\phi_{n}u_{m}$, il fatto che $u_{m}=B-\lim_{n}\phi_{n}u_{m}$ è quasi scontato. Dato che per ogni $n$, $D_{R}(\phi_{n}u_{m},u)=0$, si ha che per ogni $m$: $\displaystyle D_{R}(u_{m},u)=0$ È facile verificare che $u=CD-\lim_{m}u_{m}$, quindi abbiamo che: $\displaystyle D_{R}(u)\equiv D_{R}(u,u)=\lim_{m}D_{R}(u_{m},u)=0$ da cui $u$ è costante. ∎ ### 4.3 Funzioni di Green Un’altra caratterizzazione delle varietà paraboliche riguarda l’esistenza di funzioni di Green definite su tutta la varietà. ###### Proposizione 4.19. Consideriamo un’esaustione regolare $K_{n}$ di $R$, e siano $G_{n}\equiv G_{K_{n}}$ le funzioni di Green 121212estese a $0$ fuori da $K_{n}$ rispetto a questi compatti. La successione di funzioni $\displaystyle G_{n}(\cdot,p)$ con $p\in K_{1}$ fissato converge a una funzione armonica positiva $G(\cdot,p)$ su $R\setminus\\{p\\}$ se e solo se la varietà $R$ non è parabolica. ###### Proof. Per $m>n$, sia $\displaystyle\delta_{n}^{m}(p)\equiv G_{m}(\cdot,p)-G_{n}(\cdot,p)$ dove $\delta_{n}^{m}$ è definita sull’insieme $K_{n}$. Grazie al fatto che per $d(x,p)\to 0$ le due funzioni $G_{n}$ e $G_{m}$ hanno lo stesso comportamento asintotico e grazie alla proposizione 1.65 $\delta_{n}^{m}$ è una funzione armonica su tutto $K_{n}$ per ogni $m$. Osserviamo che per ogni $n$, $\delta_{n}^{n+1}$ è una funzione strettamente positiva su $K_{n}$, infatti su $\partial K_{n}$ $G_{n}(\cdot,p)=0$, mentre per il principio del massimo $G_{n+1}(\cdot,p)>0$, quindi $\delta_{n}^{n+1}|_{\partial K_{n}}>0$, e sempre per il principio del massimo $\delta_{n}^{n+1}>0$. Dato che $\displaystyle\delta_{n}^{m}=\sum_{i=n+1}^{m}\delta_{n}^{i}$ otteniamo che al variare di $m$, $\delta_{n}^{m}$ è una successione di funzioni armoniche positive crescenti. Per il principio di Harnack (vedi 1.57), la successione $\delta_{n}^{m}$ al variare di $m$ converge localmente uniformemente (in $K_{n}$) o diverge localmente uniformemente. È evidente che il comportamento delle successioni $\delta_{n}^{m}$ è indipendente dal parametro $n$, infatti sull’intersezione dei vari insiemi di definizione vale che $\displaystyle\delta_{n}^{m}=\delta_{k}^{m}-\delta_{n}^{k}$ quindi se e solo se al variare di $m$ $\delta_{n}^{m}$ è limitata, anche $\delta_{k}^{m}$ lo è. Consideriamo $K\subset K_{1}$ un compatto contenente $p$ come punto interno, allora dalla proposizione 4.11, sappiamo che per ogni $m$: $\displaystyle\min_{x\in\partial K}G_{m}(x,p)\leq\text{Cap}(K,K_{m})^{-1}\leq\max_{x\in\partial K}G_{m}(x,p)$ facendo tendere $m$ a infinito, osserviamo che se $\text{Cap}(K)=\lim_{m}\text{Cap}(K,K_{m})>0$, allora necessariamente $G_{m}|_{\partial K}(\cdot,p)$ è limitata, altrimenti tende a infinito. Dato che al variare di $m$ $\delta_{n}^{m}$ è limitata se e solo se $G_{m}(\cdot,p)|_{\partial K}$ lo è, allora indipendentemente da $n$, $\delta_{n}^{m}$ converge su $K_{n}$ a una funzione armonica se e solo se $R$ è non parabolica. Quindi la successione $G_{m}(\cdot,p)$, che sul compatto $K_{n}$ è uguale a $G_{n}(\cdot,p)+\delta_{n}^{m}(\cdot)$ 131313se $m>n$, converge localmente uniformemente in $R$ 141414anche se le funzioni $G_{m}$ non sono definite in $p$, la loro differenza può essere estesa in $p$, e in questo senso diciamo che la convergenza è uniforme anche su $p$ a una funzione armonica se e solo se $\delta_{n}^{m}$ converge, quindi se e solo se $R$ non è parabolica. ∎ Da questa dimostrazione deduciamo che se $\Omega\Subset\Omega^{\prime}$, allora $G_{\Omega}(\cdot,p)\leq G_{\Omega^{\prime}}(\cdot,p)$, quindi che la funzione $G(\cdot,p)$ ottenuta come limite di $G_{n}$ è indipendente dalla scelta dell’esaustione $K_{n}$. ###### Proposizione 4.20. La funzione $G(\cdot,p)=\lim_{n}G_{n}(\cdot,p)$ è indipendente dalla scelta dell’esaustione $K_{n}$ utilizzata per definire $G_{n}$. ###### Proof. Siano $K_{n}$ e $K_{m}^{\prime}$ due esaustioni regolari di $R$ con $p\in K_{1}\cap K_{1}^{\prime}$, e siano $G_{n}(\cdot,p)$ e $G_{m}^{\prime}(\cdot,p)$ i relativi nuclei di Green, dove $\displaystyle G(\cdot,p)\equiv\lim_{n}G_{n}(\cdot,p)\ \ \ G^{\prime}(\cdot,p)\equiv\lim_{m}G_{m}^{\prime}(\cdot,p)$ Per ogni $n$, esiste $\bar{m}$ tale che $K_{n}\Subset K^{\prime}_{\bar{m}}$, allora grazie all’osservazione precedente: $\displaystyle G_{n}(\cdot,p)\leq G_{\bar{m}}^{\prime}(\cdot,p)\leq G^{\prime}(\cdot,p)$ quindi passando al limite su $n$, $G(\cdot,p)\leq G^{\prime}(\cdot,p)$, e poiché i ruoli di $G_{n}$ e $G_{m}^{\prime}$ sono simmetrici, vale anche il viceversa, quindi $\displaystyle G(\cdot,p)=G^{\prime}(\cdot,p)$ ∎ Osserviamo che anche per le funzioni di Green definite su tutta $R$ vale una proprietà analoga a 1.63: ###### Proposizione 4.21. Sia $G(\cdot,p)$ la funzione di Green ottenuta con il metodo di esaustione, allora se $K$ e $K^{\prime}$ sono domini relativamente compatti con $p\in K\Subset K^{\prime}$, vale che: $\displaystyle G|_{K^{\prime}\setminus\overline{K}}(\cdot,p)\leq\max_{x\in\partial K}G(x,p)\ \ \ \ \ \ G(\cdot,p)|_{R\setminus\overline{K}}\leq\max_{x\in\partial K}G(x,p)$ ###### Proof. Sia $K_{n}$ un’esaustione regolare di $R$ con $p\in K_{1}$ e siano $G_{n}(\cdot,p)$ le relative funzioni di Green. Se $n$ è abbastanza grande per cui $K^{\prime}\Subset K_{n}$, allora vale che: $\displaystyle G_{n}|_{K^{\prime}\setminus\overline{K}}(\cdot,p)<\max_{x\in\partial K}G_{n}(x,p)$ poiché questa relazione vale definitivamente, passando al limite su $n$ otteniamo che: $\displaystyle G|_{K^{\prime}\setminus\overline{K}}(\cdot,p)\leq\max_{x\in\partial K}G(x,p)$ Data l’indipendenza da $K^{\prime}$ di questa proprietà, possiamo concludere che: $\displaystyle G(\cdot,p)|_{R\setminus\overline{K}}\leq\max_{x\in\partial K}G(x,p)$ come volevasi dimostrare. ∎ Grazie alle tecniche usate, possiamo facilmente dimostrare anche alcune proprietà della funzione $G(x,p)=\lim_{n}G_{n}(x,p)$ che corrispondono alle proprietà delle funzioni $G_{n}$. ###### Proposizione 4.22. Per la funzione $G(x,p)=\lim_{n}G_{n}(x,p)$ vale che: 1. 1. per ogni dominio relativamente compatto con bordo liscio $\Omega$ contenente $p$, vale che $G(\cdot,p)-G_{\Omega}(\cdot,p)$ è una funzione estendibile a una funzione armonica su tutto $\Omega$. 2. 2. $G$ è strettamente positiva su $\Omega$ 3. 3. $G$ è simmetrica, cioè $G(p,q)=G(q,p)$ 4. 4. Fissato $q\in\Omega$, la funzione $G(q,p)$ è armonica rispetto a $p$ sull’insieme $\Omega\setminus\\{q\\}$ e superarmonica su tutto $\Omega$. 5. 5. $G$ è soluzione fondamentale dell’operatore $\Delta$, cioé per ogni funzione liscia $f$ a supporto compatto in $R$: $\displaystyle\Delta_{x}\int_{R}G(x,y)f(y)dy=\int_{R}G(x,y)\Delta_{y}(f)(y)dy=-f(x)$ questo significa che nel senso delle distribuzioni $\Delta_{y}G(x,y)=-\delta_{x}$ 6. 6. Il flusso di $G_{\Omega}(\ast,p)$ attraverso il bordo di un’insieme regolare $K\Subset\Omega$ con $p\not\in\partial K$ vale: $\displaystyle\int_{\partial K}\ast dG(\cdot,p)=\begin{cases}-1&se\ p\in K\\\ 0&se\ p\not\in K\end{cases}$ 7. 7. La funzione $G$ ha un comportamento asintotico della forma: $\displaystyle G(x,y)\sim C(m)\begin{cases}-log(d(x,y))&m=2\\\ d(x,y)^{m-2}&m\geq 3\end{cases}$ quando $d(x,y)\to 0$. La costante $C(m)$ dipende solo dalla dimensione della varietà e può essere determinata sfruttando la condizione (5). ###### Proof. In tutta la dimostrazione supponiamo che $G$ esista, cioè che $R$ non sia parabolica. Il punto (1) è il punto chiave per dimostrare tutte le altre proprietà. Dalla costruzione nella proposizione precedente sappiamo che per ogni $n$ sull’insieme $K_{n}$ la funzione $G(\cdot,p)-G_{n}(\cdot,p)=\lim_{m}\delta_{n}^{m}\equiv\delta_{n}$ è una funzione armonica su $K_{n}$. Consideriamo una funzione di Green $G_{\Omega}(\cdot,p)$ con $\Omega$ qualsiasi, e sia $\Omega\Subset K_{n}$. Dalla proposizione 1.65 sappiamo che $G_{n}(\cdot,p)-G_{\Omega}(\cdot,p)$ è estendibile a una funzione armonica $\phi$ definita su tutto $\Omega$. Quindi: $\displaystyle G(\cdot,p)-G_{\Omega}(\cdot,p)=G(\cdot,p)-G_{n}(\cdot,p)+G_{n}(\cdot,p)-G_{\Omega}(\cdot,p)=\delta_{n}(\cdot)-\phi(\cdot)$ è una funzione armonica su $\Omega$. Questo implica in particolare che fissato $q\in\Omega$, $G(q,p)$ si può scrivere come la somma tra la funzione $G_{\Omega}(q,p)$ e una funzione armonica, il che dimostra anche il punto (4). La positività e la simmetria di $G$ sono una ovvia conseguenza del fatto che queste proprietà valgono anche per tutte le $G_{n}$. Per dimostrare il punto (5), consideriamo una funzione $f\in C^{\infty}_{C}(R)$. Sia $\Omega$ tale che $supp(f)\subset\Omega$, allora grazie al punto (1) sappiamo che $G(x,y)=G_{\Omega}(x,y)+\phi(x,y)$ con $\phi(\cdot,y)$ armonica in $\Omega$ per $y$ fissato 151515e viceversa per simmetria, quindi: $\displaystyle\Delta_{x}\int_{R}G(x,y)f(y)dy=\Delta_{x}\int_{\Omega}G_{\Omega}(x,y)f(y)dy+\Delta_{x}\int_{\Omega}\phi(x,y)f(y)dy=-f(x)$ Grazie al fatto che $\Delta_{x}\phi(x,y)=0$ e al teorema di derivazione sotto al segno d’integrale. Allo stesso modo: $\displaystyle\int_{R}G(x,y)\Delta_{y}f(y)dy=\int_{\Omega}G_{\Omega}(x,y)\Delta_{y}f(y)dy+\int_{\Omega}\phi(x,y)\Delta_{y}f(y)dy$ grazie a un’integrazione per parti su $\Omega$ (il termine al bordo è nullo grazie al fatto che $supp(f)\Subset\Omega$): $\displaystyle\int_{\Omega}\phi(x,y)\Delta_{y}f(y)dy=-\int_{\Omega}\Delta_{y}\phi(x,y)\Delta_{y}f(y)dy=0$ otteniamo che: $\displaystyle\int_{R}G(x,y)\Delta_{y}f(y)dy=\int_{\Omega}G_{\Omega}(x,y)\Delta_{y}f(y)dy=-f(x)$ In maniera del tutto analoga, si dimostra il punto (6), basta considerare il fatto che $\ast dG=\ast dG_{\Omega}+\ast d\phi$ e il flusso di una funzione armonica attraverso il bordo di un compatto regolare è nullo grazie alla formula di green 1.11. Il punto (7) è conseguenza del fatto che la funzione armonica $\phi$ è limitata. ∎ Osserviamo che anche nel caso di varietà paraboliche è possibile definire delle funzioni di Green, solo che in questo caso le funzioni non sono positive 161616e neanche limitate dal basso. Dall’articolo [LT] riportiamo il teorema: ###### Teorema 4.23. Sia $R$ una varietà riemanniana completa non compatta e senza bordo. Allora esiste un nucleo di Green simmetrico $G(x,y)$ liscio sull’insieme $R\times\leavevmode\nobreak\ R\setminus\leavevmode\nobreak\ D$ dove $D=\\{(x,x)\ t.c.\ x\in R\\}$ è la diagonale di $R\times R$. In particolare $G$ soddisfa: $\displaystyle\Delta_{x}\int_{R}G(x,y)f(y)dy=\int_{R}G(x,y)\Delta_{y}(f)(y)dy=-f(x)$ per ogni funzione $f$ liscia a supporto compatto in $R$. #### 4.3.1 Funzioni di Green sulla compattificazione di Royden In questa sezione dimostriamo che per una varietà iperbolica le funzioni di Green si possono estendere alla compattificazione di Royden $R^{*}$. Utilizzeremo i risultati ottenuti nella sezione 4.3. ###### Proposizione 4.24. Per ogni $c>0$, la funzione $\displaystyle G_{c}(x,p)\equiv G(x,p)\curlywedge c\equiv\min\\{G(x,p);c\\}$ appartiene a $\mathbb{M}_{\Delta}(R)$, inoltre $\displaystyle D_{R}(G(x,p)\curlywedge c)\leq c$ ###### Proof. Data un’esaustione regolare $K_{n}$ sia $G_{n}(x,p)$ la funzione di Green relativa a $K_{n}$ e $G(x,p)$ il suo limite. L’obiettivo della dimostrazione è mostrare che per ogni $c>0$ 171717questo è vero anche per $c\leq 0$, ma è privo di senso essendo $G(x,p)>0$ su $R$ la successione $G_{n}(\cdot,p)\curlywedge c$ converge localmente uniformemente a $G_{c}(\cdot,p)$ 181818questa considerazione è quasi ovvia e ha integrale di Dirichlet limitato, quindi dato che ovviamente $G_{n}(x,p)\curlywedge c\in\mathbb{M}_{0}(R)\subset\mathbb{M}_{\Delta}(R)$, grazie al teorema 3.24 $G_{c}\in\leavevmode\nobreak\ \mathbb{M}_{\Delta}(R)$. Per dimostrare che $D_{R}(G_{n}(x,p)\curlywedge c)<M(c)$, per prima cosa osserviamo che se $c^{\prime}\leq c$ $\displaystyle D_{R}(G_{n}(x,p)\curlywedge c^{\prime})=\int_{G_{n}(x,p)<c^{\prime}}\left|\nabla(G_{n}(x,p))\right|^{2}dV\leq$ $\displaystyle\leq\int_{G_{n}(x,p)<c}\left|\nabla(G_{n}(x,p))\right|^{2}dV=D_{R}(G_{n}(x,p\curlywedge c)$ quindi basta dimostrare che $D_{R}(G_{n}(x,p)\curlywedge c)<M(c)$ è valida per un insieme illimitato di $c$. Consideriamo un insieme aperto relativamente compatto $p\in\Omega$. Grazie a 4.21, $G(x,p)$ ristretta a $R\setminus\Omega$ assume il suo massimo $M$ in $\partial\Omega$. Allora l’insieme $U_{c}\equiv\\{x\ t.c.\ G(x,p)>c\\}$ è contenuto in $\Omega$ se $c>M$, e per monotonia anche gli insiemi $\displaystyle U_{c}^{n}\equiv\\{x\ t.c.\ G_{n}(x,p)>c\\}$ sono contenuti in $\Omega$ se $c>M$, quindi sono relativamente compatti. All’interno di $(M,\infty)$, consideriamo i valori di $c$ che sono valori regolari per tutte le funzioni $G_{n}(\cdot,p)$. Per il teorema di Sard 191919e per il fatto che l’unione numerabile di insiemi di misura nulla ha misura nulla questi valori sono densi in $(M,\infty)$. D’ora in avanti consideriamo solo $c$ con queste caratteristiche. Osserviamo ora che per ogni $n$, l’integrale: $\displaystyle D_{R}(G_{n}(x,p)\curlywedge c)=\int_{K_{n}\cap(U_{c}^{n})^{C}}\left|\nabla(G_{n}(x,p))\right|^{2}dV=$ $\displaystyle\int_{\partial(K_{n}\cap(U_{c}^{n})^{C})}G_{n}\ast dG_{n}=-c\int_{\partial U_{c}^{n}}\ast dG_{n}=c$ grazie alle proprietà di $G_{n}$ descritte in 1.7.4. Questo e il teorema 3.24 danno la tesi. Inoltre osserviamo che dall’ultima considerazione possiamo ottenere che: $\displaystyle D_{R}(G(x,p)\curlywedge c)\leq c$ per ogni $c>0$. ∎ Grazie al fatto che $G_{c}(x,p)\in\mathbb{M}_{\Delta}(R)$ per ogni $c$, possiamo estendere la definizione del nucleo di Green a $R\times R^{*}$. Sia a questo scopo $U_{c}\equiv\\{x\in R\ t.c.\ G(x,z)>c\\}$, e sia $c$ tale che $U_{c}$ è relativamente compatto in $R$. Allora grazie all’osservazione 3.48, per ogni funzione liscia a supporto compatto $\lambda$, se $p\in\Gamma$ e $z\in R$ fisso: $\displaystyle p(G_{c}(\cdot,z))=p(G_{c}(\cdot,z)(1-\lambda))$ se scegliamo $\lambda$ in modo che $supp(\lambda)\supset U_{c}$, questa relazione mostra che per ogni $c^{\prime}>c$ si ha che: $\displaystyle p(G_{c}(\cdot,z))=p(G_{c^{\prime}}(\cdot,z))$ quindi il valore che assume $G_{c}(\cdot,z)$ sul punto $p\in\Gamma$ è indipendente dalla scelta di $c$, e quindi ha senso definire: $\displaystyle G(p,z)=\lim_{q\to p}G(q,z)$ Da questa definizione segue che: ###### Proposizione 4.25. La funzione $G^{\prime}(z,p)$ con $(z,p)\in R\times R^{*}$ è una armonica in $R\setminus\\{p\\}$, continua su $R\times R^{*}\setminus D$, $G^{\prime}_{c}(\cdot,p)\equiv\min\\{G^{\prime}(\cdot,p),c\\}\in\mathbb{M}_{\Delta}(R)$ per ogni $c>0$, e $D_{R}(G_{c}^{\prime}(\cdot,p))\leq c$. ###### Proof. È necessario dimostrare la continuità della funzione $G^{\prime}(\cdot,\cdot)$ solo sui punti $R\times\Gamma$, dato che $G^{\prime}$ è estensione di $G$ che è continua su $R\times R\setminus D$. Consideriamo quindi una coppia $(z_{0},p_{0})\in R\times\Gamma$, $\epsilon>0$, e siano $U$ e $V$ due intorni di $z_{0}$ e $p_{0}$ aperti in $R^{*}$ e a chiusura disgiunta. Fissato $z_{0}$, la funzione $G^{\prime}(z_{0},\cdot)$ è continua su $R^{*}\setminus\\{z_{0}\\}$, questo significa che possiamo scegliere $V$ in modo che per ogni $p\in W$: $\displaystyle\left|G^{\prime}(z_{0},p)-G^{\prime}(z_{0},p_{0})\right|\leq\epsilon$ Sia $s=\sup_{p\in R\setminus U}{G^{\prime}(z_{0},p)}=\sup_{p\in R^{*}\setminus U}{G^{\prime}(z_{0},p)}<\infty$ 202020l’ultima uguaglianza segue dal fatto che $R$ è denso in $R^{*}$. Consideriamo la funzione di Harnack definita sull’aperto $U\times U$. Allora vale che: $\displaystyle k(z,z_{0})^{-1}G^{\prime}(z_{0},p)\leq G^{\prime}(z,p)\leq k(z_{0},z)G^{\prime}(z_{0},p)$ per ogni punto $p\in W\cap R$. Data la proposizione 1.60, esiste un intorno $U^{\prime}\subset U$ tale che $k(U^{\prime},z_{0})\subset[1,1+\epsilon/s)$. Allora per $z\in U^{\prime}$ e per ogni $p\in W\cap R$, si ha che: $\displaystyle\left|G^{\prime}(z,p)-G^{\prime}(z_{0},p)\right|\leq\epsilon$ Sempre per continuità su $R^{*}$ di $G^{\prime}(z_{0},\cdot)$, si ha che in realtà questa relazione vale per ogni $p\in W$. Riassumendo otteniamo che se $(z,p)\in U^{\prime}\times W$, si ha che: $\displaystyle\left|G^{\prime}(z_{0},p_{0})-G^{\prime}(z,p)\right|\leq\left|G^{\prime}(z_{0},p_{0})-G^{\prime}(z_{0},p)\right|+\left|G^{\prime}(z_{0},p)-G^{\prime}(z,p)\right|\leq 2\epsilon$ data l’arbitrarietà di $\epsilon$ si ottiene la tesi. La funzione $G^{\prime}(\cdot,p)$ è il limite di funzioni armoniche positive, quindi grazie al principio di Harnack (vedi 1.57) è una funzione armonica. Sappiamo che per ogni $p\in R$: $\displaystyle D_{R}(G(x,p)\curlywedge c)\leq c$ Sia $p_{0}\in\Gamma$. Dato che $G^{\prime}(\cdot,p_{0})\curlywedge c=C-\lim_{p\in R,\ p\to p_{0}}G(\cdot,p)\curlywedge c$, al teorema 3.24, abbiamo che $G^{\prime}(\cdot,p_{0})\curlywedge c\in\mathbb{M}_{\Delta}(R)$ e anche $D_{R}(G^{\prime}(\cdot,p_{0})\curlywedge c)\leq c$. ∎ Grazie alle proprietà fino a qui dimostrate, possiamo definire $G^{*}(\cdot,\cdot)$ l’estensione di $G^{\prime}(\cdot,\cdot)$ a tutto $R^{*}\times R^{*}\setminus D$. Infatti, essendo $G^{\prime}(\cdot,p)\curlywedge c\in\mathbb{M}_{\Delta}(R)\subset\mathbb{M}(R)$ per ogni $c>0$, ha senso definire: $\displaystyle G^{*}(z_{0},p_{0})\curlywedge c=\lim_{z\in R,\ z\to z_{0}}G^{\prime}(z,p_{0})\curlywedge c\equiv\lim_{z\in R,\ z\to z_{0}}\lim_{p\in R,\ p\to p_{0}}G(z,p)\curlywedge c$ per ogni $c>0$, quindi: ###### Definizione 4.26. Definiamo $G^{*}(\cdot,\cdot):R^{*}\times R^{*}\setminus D\to[0,\infty)$ con il doppio limite: $\displaystyle G^{*}(z_{0},p_{0})=\lim_{z\in R,\ z\to z_{0}}G^{\prime}(z,p_{0})\equiv\lim_{z\in R,\ z\to z_{0}}\lim_{p\in R,\ p\to p_{0}}G(z,p)$ Definiamo inoltre il bordo essenziale irregolare di $R^{*}$ come: ###### Definizione 4.27. L’insieme dei punti in $\Gamma$ $\displaystyle\Xi\equiv\\{p\in\Gamma\ t.c.\ G(z,p)>0\ z\in R\\}$ è detto bordo essenziale irregolare di $R$. Riassumendo le proprietà fin qui dimostrate, per la funzione $G^{*}$ valgono le seguenti proprietà: ###### Proposizione 4.28. Sia $R$ una varietà non parabolica senza bordo, allora per la funzione $G^{*}(\cdot,\cdot)$, l’estensione del nucleo di Green alla compattificazione $R^{*}$, valgono le proprietà: 1. 1. $G^{*}(x,y)=G(x,y)\ \forall(x,y)\in R$ 2. 2. $G^{*}(\cdot,p)$ è continua sull’insieme $R^{*}\setminus\\{p\\}$ 3. 3. $G^{*}(\cdot,p)$ è una funzione armonica su $R\setminus\\{p\\}$ 4. 4. $D_{R}(G^{*}(\cdot,p)\curlywedge c)\leq c$ per ogni $c>0$ 5. 5. $G^{*}(\cdot,\cdot)|_{(\Delta\times R^{*})}=0$ 6. 6. $G^{*}(\cdot,\cdot)|_{(R\cup\Xi)\times(R\cup\Xi)}>0$ ###### Proof. (1) e (2) e (3) seguono dal fatto che $G^{*}$ è un’estensione di $G^{\prime}$ che è un’estensione di $G$, e visto che sia $G$ che $G^{\prime}$ sono in $\mathbb{M}(R)$, l’estensione è continua su tutta $R^{*}$ per definizione della compattificazione di Royden. Grazie alla 4.25, possiamo dedurre (4) e (5), infatti dato che $G^{\prime}\in\mathbb{M}_{\Delta}(R)$, $D_{R}(G^{\prime}\curlywedge c)\leq c$ e grazie alla proposizione 3.35, possiamo dedurre che $G^{*}\curlywedge c\in\mathbb{M}_{\Delta}(R)$, quindi, $G^{*}(\cdot,\cdot)|_{\Delta\times R^{*}}=0$. Rimane da dimostrare (6). Come spesso accade, questa dimostrazione è un’applicazione del principio del massimo. Sappiamo che $G^{*}(x,y)>0$ se $(x,y)\in R\times R$, e anche se $(x,y)\in R\times\Xi$, resta il caso $(x,y)\in\Xi\times\Xi$. La funzione $G^{*}(\cdot,y)$ è armonica positiva su $R$ (come dimostrato sopra). Quindi se consideriamo un aperto $U\in R$ e un punto $p\in U$, esiste un numero positivo $a$ tale che $aG^{*}(\cdot,y)-G(\cdot,p)|_{\partial U}>0$. Consideriamo una successione $G_{n}(\cdot,p)$ di funzioni di Green relative a un’esaustione regolare $R_{n}$. Allora definitivamente $aG^{*}(\cdot,y)-G_{n}(\cdot,p)|_{\partial U}>0$ e anche $aG^{*}(\cdot,y)-G_{n}(\cdot,p)|_{\partial R_{n}}>0$. Quindi per il principio del massimo $aG^{*}(\cdot,y)-G_{n}(\cdot,p)|_{R_{n}\setminus U}\geq 0$. Allora questa disuguaglianza vale anche per il limite su $n$, cioè $\displaystyle aG^{*}(\cdot,y)-G(\cdot,p)|_{\partial U}\geq 0$ dato che $G(x,p)>0$ per definizione di $\Xi$, si ha la tesi. ∎ D’ora in avanti confonderemo la notazione di $G,\ G^{\prime},\ G^{*}$ quando non ci sia rischio di confusione. ### 4.4 Potenziali di Evans In questa sezione diamo un’altra caratterizzazione della parabolicità di una varietà $R$ attraverso l’esistenza di particolari funzioni armoniche, i potenziali di Evans. ###### Definizione 4.29. Dato un compatto $K\Subset R$, un potenziale di Evans rispetto a questo compatto è una funzione armonica $f:(R\setminus K)\to\mathbb{R}$ tale che $\displaystyle\lim_{x\to\infty}f(x)=\infty\ \ \ f|_{\partial K}=0$ L’ultimo limite può essere inteso equivalentemente in 2 sensi: per ogni successione $x_{n}\to\infty$, $f(x_{n})\to\infty$, oppure per ogni $N>0$, esiste $K\subset K_{N}\Subset R$ tale che $f(K_{N}^{C})\subset(N,+\infty)$. Se la funzione $f$ è solo superarmonica in $R\setminus K$, viede definita potenziale di Evans superarmonico. I potenziali di Evans sono caratteristici delle varietà paraboliche, nel senso che una varietà è parabolica se e solo se per ogni compatto $K$ esiste un potenziale di Evans relativo a questo compatto. La dimostrazione di un’implicazione è quasi immediata, mentre l’altra implicazione è il contenuto fondamentale di questa tesi. Osserviamo che per caratterizazione delle varietà paraboliche, è sufficiente chiedere che il potenziale di Evans sia superarmonico. ###### Proposizione 4.30. Se esiste un compatto $K$ che ammette un potenziale di Evans superarmonico, allora la varietà è parabolica ###### Proof. Sia $K$ un compatto e $f$ il relativo potenziale di Evans superarmonico. Consideriamo un compatto $C$ con $K\Subset C^{\circ}$ e bordo liscio. La funzione $f$ continua a essere superarmonica positiva su $C^{C}$ e tende a infinito. Inoltre grazie al principio del massimo 1.52 $f|_{\partial C}>0$. Dimostriamo che il potenziale armonico $u$ di $C$ è costante uguale a $1$. Sia $b=1-u$ funzione armonica definita su $C^{C}$ 212121estendibile per continuità a $0$ su $C$, e sia $C_{k}$ un’esaustione regolare di $R$ con $C_{1}=C$. Per ogni $N>1$, esiste $\bar{k}$ tale che $f(C_{\bar{k}}^{C})\subset(N,\infty)$. Dato che la funzione $b$ è necessariamente $\leq 1$ 222222indipendendemente dal fatto che $R$ sia parabolica, e che $Nb|_{\partial C}<f|_{\partial C}$, per il principio del massimo abbiamo che per ogni $k>\bar{k}$, $Nb\leq f$ sull’insieme $C^{k}\setminus C$, quindi su tutto l’insieme $R\setminus C$. Data l’arbitrarietà di $N$, otteniamo che per ogni $N>1$ $\displaystyle Nb\leq f\ \ \Rightarrow\ \ b\leq\frac{f}{N}$ quindi necessariamente $b=0$, cioè $u=1$, come volevasi dimostrare. ∎ I paragrafi seguenti si occupano di dimostrare l’implicazione inversa. #### 4.4.1 Diametro transfinito In questo paragrafo definiamo due strumenti che saranno utili per costruire particolari funzioni armoniche, tra cui i potenziali di Evans. D’ora in avanti assumiamo che $R$ sia una varietà riemanniana non parabolica. ###### Definizione 4.31. Detto $G^{*}(\cdot,\cdot)$ il nucleo di Green su $R^{*}$, e dato $X\subset R^{*}$ definiamo: $\displaystyle\binom{n}{2}\rho_{n}(X)\equiv\inf_{p_{1},\cdots,p_{n}\in X}\sum_{i<j}^{1\cdots n}G(p_{i},p_{j})$ (4.2) inoltre per convenzione $\rho(\emptyset)=\infty$ ###### Proposizione 4.32. Fissato l’insieme $X$, $\rho_{n}(X)$ è una successione crescente al crescere di $n$. ###### Proof. La dimostrazione è un semplice esercizio di algebra, non riguarda le caratteristiche di $R$. Siano $p_{1},\cdots,p_{n}$ punti qualsiasi in $X$, e consideriamo per ogni $1\leq k\leq n+1$: $\displaystyle\sum_{i<j}^{1\cdots n+1}G(p_{i},p_{j})=\sum_{i=1}^{k-1}G(p_{i},p_{k})+\sum_{j=k+1}^{n+1}G(p_{k},p_{j})+\sum_{i<j;\ i,j\neq k}^{1\cdots n}G(p_{i},p_{j})$ e quindi dalla definizione di $\rho_{n}(X)$: $\displaystyle\sum_{i<j}^{1\cdots n+1}G(p_{i},p_{j})\geq\sum_{i=1}^{k}G(p_{i},p_{k})+\sum_{j=k+1}^{n+1}G(p_{k},p_{j})+\binom{n}{2}\rho_{n}(X)$ Sommando tutte queste disuguaglianze al variare di $k$ tra $1$ e $n+1$, otteniamo che: $\displaystyle(n+1)\sum_{i<j}^{1\cdots n+1}G(p_{i},p_{j})\geq 2\sum_{i<j}^{1\cdots n+1}G(p_{i},p_{j})+(n+1)\binom{n}{2}\rho_{n}(X)$ cioè: $\displaystyle(n-1)\sum_{i<j}^{1\cdots n+1}G(p_{i},p_{j})\geq(n+1)\binom{n}{2}\rho_{n}(X)$ vista l’arbitrarietà dei punti considerati, possiamo concludere che: $\displaystyle(n-1)\binom{n+1}{2}\rho_{n+1}(X)\geq(n+1)\binom{n}{2}\rho_{n}(X)$ considerando che $\displaystyle(n-1)\binom{n+1}{2}=(n+1)\binom{n}{2}$ otteniamo la tesi, cioè $\displaystyle\rho_{n+1}(X)\geq\rho_{n}(X)$ ∎ Questa proposizione ci permette di definire il limite al variare di $n$ di $\rho_{n}(X)$, in particolare: ###### Definizione 4.33. Dato $X\in R^{*}$, definiamo il diametro transfinito di $X$: $\displaystyle\rho(X)\equiv\lim_{n}\rho_{n}(X)$ (4.3) Assieme al diametro transfinito, definiamo la costante di Tchebycheff, e esploriamo alcuni legami tra i due concetti. ###### Definizione 4.34. Dato $X\subset R^{*}$, definiamo: $\displaystyle n\tau_{n}(X)=\sup_{p_{1},\cdots,p_{n}\in X}\left(\inf_{p\in X}\sum_{i=1}^{n}G(p,p_{i})\right)$ e per convenzione $\tau_{n}(\emptyset)=\infty$ Anche in questo caso vale una relazione che ci consente di definire il limite di $\tau_{n}$. Infatti osserviamo che: $\displaystyle\sum_{i=1}^{n+m}G(p,p_{i})=\sum_{i=1}^{n}G(p,p_{i})+\sum_{i=n+1}^{n+m}G(p,p_{i})$ quindi applicando l’$\inf$ a entrambi i membri otteniamo: $\displaystyle\inf_{p\in X}\sum_{i=1}^{n+m}G(p,p_{i})\geq\inf_{p\in X}\sum_{i=1}^{n}G(p,p_{i})+\inf_{p\in X}\sum_{i=n+1}^{n+m}G(p,p_{i})$ cioè per definizione: $\displaystyle(n+m)\tau_{n+m}(X)\geq n\tau_{n}(X)+m\tau_{m}(X)$ (4.4) se consideriamo $n=m$ otteniamo il caso particolare: $\displaystyle(qm)\tau_{qm}(X)\geq q\tau_{m}(X)\Rightarrow\tau_{qm}(X)\geq\tau_{m}(X)$ (4.5) Grazie a questa relazione possiamo dimostrare che: ###### Proposizione 4.35. Sia $\alpha\equiv\sup_{n}\tau_{n}(X)$, allora $\lim_{n}\tau_{n}(X)=\alpha$. ###### Proof. La dimostrazione segue dalla relazione 4.4. Infatti scegliamo un qualunque $\beta<\alpha$. Per definizione di $\sup$, esiste un $m$ tale che $\tau_{m}(X)>\beta$. È un fatto noto che ogni numero $n$ può essere scritto in maniera univoca come: $\displaystyle n=qm+r$ dove $q,r\in\mathbb{N}$ e $0\leq r\leq m-1$. Quindi per $\tau_{n}(X)$ vale che: $\displaystyle n\tau_{n}(X)=(qm+r)\tau_{qm+r}(X)\geq qm\tau_{qm}(X)+r\tau_{r}(X)\geq qm\tau_{m}(X)$ dove abbiamo utilizzato le relazioni 4.4 ed 4.5. Quindi possiamo concludere che $\displaystyle\tau_{n}(X)\geq\frac{qm}{qm+r}\tau_{m}(X)>\frac{qm}{qm+r}\beta$ Se $n$ tende a infinito, $q$ tente a infinito, mentre $0\leq r\leq n-1$ continua a valere, quindi: $\displaystyle\liminf_{n}\tau_{n}(X)\geq\liminf_{n}\frac{qm}{qm+r}\beta=\beta$ Cioè per ogni $\beta<\alpha=\sup_{n}\tau_{n}(X)$ abbiamo che: $\displaystyle\alpha\geq\limsup_{n}\tau_{n}(X)\geq\liminf_{n}\tau_{n}(X)\geq\beta$ data l’arbitrarietà di $\beta$, otteniamo la tesi. ∎ ###### Definizione 4.36. Dato $X\subset R$ definiamo la sua costante di Tchebycheff il limite: $\displaystyle\tau(X)\equiv\lim_{n\to\infty}\tau_{n}(X)$ Il diametro transfinito di un insieme e la costante di Tchebycheff misurano in qualche senso la grandezza di un insieme. Più il nucleo di Green $G^{*}$ ha un valore alto sui punti di $X$, più queste due costanti hanno valore alto, ed è facile osservare che per insiemi di cardinalità finita $\tau(X)=\rho(X)=\infty$, e che se $X^{\prime}\subset X$, allora $\tau(X^{\prime})\geq\tau(X)$. Di seguito riportiamo un’altra proprietà di queste due costanti. ###### Proposizione 4.37. Per ogni $X\subset R^{*}$ si ha che: $\displaystyle\tau(X)\geq\rho(X)$ (4.6) ###### Proof. Assumiamo che $X$ abbia cardinalità infinita 232323altrimenti abbiamo visto che $\tau(X)=\rho(X)=\infty$. Per $n>1$ sia $r\equiv\frac{1}{n-1}$. È possibile scegliere $n$ punti $p_{1},\cdots,p_{n}$ tali che per ogni $i=1,\cdots,n-1$ abbiamo che: $\displaystyle\sum_{j=n+1-i}^{n}G(p_{n-i};p_{j})\leq\inf_{p\in X}\sum_{j=n+1-i}^{n}G(p,p_{j})+r$ (4.7) Dimostriamo questa affermazione per induzione su $i$. Per $i=1$ l’affermazione segue direttamente dalla definizione di $\inf$. Infatti se scegliamo $p_{n}$ arbitrariamente e $p_{n-1}$ in modo che $\displaystyle G(p_{n-1},p_{n})\leq\inf_{p\in X}G(p,p_{n})+r$ Supponiamo che la tesi sia vera per $1\leq i<n-1$ 242424quindi supponiamo di aver determinato $p_{n},\cdots,p_{n-i+1}$ e consideriamo la funzione $\displaystyle f(p)=\sum_{j=n-i+1}^{n}G(p,p_{i})$ osserviamo che se $p=p_{k}$ per qualche $k=n-i+1,\cdots n$, allora $f(p)$ vale infinito. La funzione $f$ comunque è positiva, quindi possiamo trovare un punto $p_{n-i}$ tale che: $\displaystyle f(p_{n-i})\leq\inf_{p\in X}f(p)+r$ quindi otteniamo che esistono punti $p_{1},\cdots,p_{n}\in X$ tali che $\displaystyle\sum_{j=n+1-i}^{n}G(p_{n-i};p_{j})\leq\inf_{p\in X}\sum_{j=n+1-i}^{n}G(p;p_{j})\leq i\tau_{i}(X)+r$ sommando queste disuguaglianze per $i=1,\cdots,n-1$ otteniamo che $\displaystyle\sum_{i<j}^{1\cdots n}G(p_{i};p_{j})\leq\sum_{i=1}^{n-1}i\tau_{i}(X)+1$ e dalla definizione di $\rho_{n}$ abbiamo: $\displaystyle\binom{n}{2}\rho_{n}(X)\leq\sum_{i=1}^{n-1}i\tau_{i}(X)+1$ quindi: $\displaystyle\rho_{n}(X)\leq\binom{n}{2}^{-1}\left(\sum_{i=1}^{n-1}i\tau_{i}(X)+1\right)$ Passando al limite otteniamo la tesi, cioè: $\displaystyle\rho(X)=\lim_{n}\rho_{n}(X)\leq\lim_{n}\binom{n}{2}^{-1}\left(\sum_{i=1}^{n-1}i\tau_{i}(X)+1\right)=\tau(X)$ dove l’ultimo passaggio è giustificato nel seguente lemma ∎ ###### Lemma 4.38. Sia $a_{n}$ tale che $\lim_{n}a_{n}=a$, allora: $\displaystyle\lim_{n}\sum_{k=1}^{n-1}\frac{ka_{k}}{\binom{n}{2}}=a$ ###### Proof. Per definizione di limite, per ogni $\epsilon>0$, esiste $N$ tale che $a_{k}>a-\epsilon$ per ogni $k>N$, quindi per ogni $\epsilon>0$ e per ogni $n>N+1$ (quindi definitivamente): $\displaystyle\binom{n}{2}^{-1}\sum_{k=1}^{n-1}ka_{k}=\binom{n}{2}^{-1}\left(\sum_{k=1}^{N}ka_{k}+\sum_{k=N+1}^{n-1}ka_{k}\right)>$ $\displaystyle>\binom{n}{2}^{-1}\left(\sum_{k=1}^{N}ka_{k}\right)+(a-\epsilon)\binom{n}{2}^{-1}\left(\sum_{k=N+1}^{n-1}k\right)$ applicando il $\liminf_{n}$, e osservando che $\binom{n}{2}\to\infty$, $\sum_{k=1}^{N}ka_{k}$ è costante al variare di $n$ e $\sum_{k=N+1}^{n-1}k=\binom{n}{2}-\binom{N+1}{2}$ otteniamo: $\displaystyle\liminf_{n}\binom{n}{2}^{-1}\sum_{k=1}^{n-1}ka_{k}\geq a-\epsilon$ con un ragionamento del tutto analogo si ottiene anche: $\displaystyle\limsup_{n}\binom{n}{2}^{-1}\sum_{k=1}^{n-1}ka_{k}<a+\epsilon$ e per l’arbitrarietà di $\epsilon$, otteniamo la tesi. ∎ #### 4.4.2 Stime per il diametro transfinito Lo scopo di questo paragrafo è ottenere la stima riportata nella proposizione 4.40, una stima tecnica che servirà nei paragrafi successivi a dimostrare che per ogni $\Xi^{\prime}\Subset\Xi$, $\rho(\Xi^{\prime})=\tau(\Xi^{\prime})=\infty$. L’affermazione è vuotamente vera se $R$ è una varietà iperbolica regolare (cioè se $\Xi=\emptyset$), quindi in tutta la sezione assumeremo che $R$ sia una varietà iperbolica irregolare. A questo scopo, introduciamo un insieme di funzioni ausiliarie che serviranno a stimare il diametro di $\Xi_{n}$. Fissato un punto $z_{0}\in R$, sia $r_{n}$ una successione di numeri reali positivi tali che $\displaystyle r_{n}>r_{n+1}\ \ \ \lim_{n}r_{n}=0$ inoltre chiediamo che $U_{n}\equiv\\{z\in R\ t.c.\ G(z,z_{0})>r_{n}\\}$ non sia relativamente compatto e che $r_{n}$ sia un valore regolare di $G(\cdot,z_{0})$ 252525in modo che $U_{n}$ sia un insieme con bordo regolare. In questo modo possiamo descrivere l’insieme $\Xi$ come unione di $\Xi_{n}$, dove $\displaystyle\Xi_{n}=\overline{U_{n}}\cap\Gamma=\\{z\in\Gamma\ t.c.\ G(z,z_{0})\geq r_{n}\\}$ Osserviamo che l’insieme $U_{n}$ è necessariamente connesso, infatti se avesse una componente connessa non contenente $z_{0}$, per il principio del massimo e per il fatto che $G(\cdot,z_{0})|_{\Delta}=0$, allora $G(\cdot,z_{0})$ sarebbe necessariamente minore di $r_{n}$ su questa componente (assurdo per definizione di $U_{n}$. Grazie alla proposizione 1.24, possiamo scegliere un’esaustione regolare $K_{m}$ di $R$ in modo che gli insieme $\displaystyle F_{nm}\equiv\overline{U_{n}}\cap\partial K_{m}$ siano per ogni $n$ e $m$ sottovarietà regolari di codimensione $1$ con bordo liscio. Fissato un indice $m$, esiste una successione di insiemi $C_{p}$ aperti relativamente compatti in $R$ con bordo liscio tale che $\overline{C_{p+1}}\subset C_{p}$ e $K_{m}=\cap_{p}C_{p}$. Questa affermazione può essere dimostrata con argomentazioni simili a quelle riportate nella proposizione 1.20. La successione $C_{p}$ è utile per dimostrare che: ###### Lemma 4.39. Con le notazioni introdotte qui sopra, esiste una funzione $w_{n,m,p}$ tale che: 1. 1. $w_{n,m,p}\in H(U_{n+1}\setminus\overline{K_{m}})$ 2. 2. $w_{n,m,p}\geq 0$ 3. 3. $w_{n,m,p}|_{\partial U_{n+1}\setminus C_{p}}=0$ 4. 4. $w_{n,m,p}|_{\partial U_{n+1}\setminus K_{m}}\leq 1$ 5. 5. $w_{n,m,p}|_{F_{n+1.m}}=1$ 6. 6. $w_{n,m,p}|_{\Xi_{n}}\geq\sigma_{n}$ 7. 7. $D_{R}(w)<\infty$ dove $\sigma_{n}$ è un numero strettamente positivo indipendente da $m$ e $p$. ###### Proof. La dimostrazione di questo lemma è molto tecnica. La sua utilità sarà illustrata nella proposizione seguente. Costruiamo le funzioni $w_{n,m,p}$ su $U_{n+1}\setminus K_{m}$ per esaustione, dimostriamo che è possibile estendere queste funzioni a una funzioni in $\mathbb{M}(R)$ (quindi ha senso parlare di $w_{n,m,p}|_{\Xi_{n}}$) e dimostriamo l’ultima disuguaglianza confrontando queste funzioni con funzioni di Green opportunamente modificate. In tutta la dimostrazione considereremo fissati i valori di $n$, $m$, $p$, quindi $w_{n,m,p}\equiv w$, $U_{n+1}\equiv U$, $K_{m}\equiv K$ e $C_{p}\equiv C$. Fissiamo una funzione liscia a supporto compatto $\lambda:R\to[0,1]$ tale che: $\displaystyle\lambda|_{(\partial K)\cap U}=1,\ \ \ \text{supp}(\lambda)\subset C$ Per $k>m$, sia $u_{k}$ la soluzione del problema di Dirichlet $\displaystyle u_{k}\in H(U\cap(K_{k}\setminus\overline{K})),\ \ \ u_{k}|_{\partial K\cup[(\partial U)\cap K_{k}\setminus K]}=\lambda|_{\partial K\cup[(\partial U)\cap K_{k}\setminus K]},\ \ \ u_{k}|_{\partial K_{k}}=0$ Per il principio del massimo, la successione $u_{k}$ è una successione crescente e limitata da $1$, quindi grazie al principio di Harnack, $u_{k}$ ammette come limite una funzione armonica su $U\setminus\overline{K}$, in particolare: $\displaystyle\lim_{k}u_{k}=w$ Osserviamo che grazie al principio di Dirichlet 3.32, $D_{R}(u_{k})\leq D_{R}(\lambda)<\infty$ per ogni $k$, e inoltre se $i>k$ $\displaystyle D_{R}(u_{k})=D_{R}(u_{i})+D_{R}(u_{k}-u_{i})$ da cui con un ragionamento simile a quello riportato nella dimostrazione del teorema 3.53, otteniamo che la successione $\\{u_{k}\\}$ è $D$-Cauchy in $\mathbb{M}(R)$, quindi per completezza $u_{k}$ ammette limite $CD$, e per unicità del limite $\displaystyle CD-\lim_{k}u_{k}=w$ da cui $D_{R}(w)<\infty$, quindi (7) è dimostrata. Per quanto riguarda le altre proprietà, (2), (3), (4) e (5) seguono direttamente dal fatto che tutte le funzioni $u_{k}$ soddisfano queste proprietà. Per dare senso alla richiesta $w|_{\Xi_{n}}\geq\sigma_{n}$, estendiamo la funzione $w$ a una funzione $\tilde{w}$ definita su tutta $R$ in questo modo: $\displaystyle\tilde{w}(x)=\begin{cases}w(x)&\text{se }x\in U\setminus K\\\ \lambda(x)&\text{se }x\in(U\setminus K)^{C}\end{cases}$ Osserviamo che $\Xi_{n}\subset\overline{U}$, quindi il valore di $\tilde{w}|_{\Xi_{n}}$ è indipendente dall’estensione di $w$ che si sceglie. È facile osservare che $\tilde{w}$ è continua e di Tonelli grazie a un ragionamento simile a quello riportato in 3.28, e il suo integrale di Dirichlet è finito poiché gli integrali di Dirichlet di $\lambda$ e $w$ lo sono. Questo dimostra che ha senso parlare di $w|_{\Xi_{n}}$ grazie al fatto che $w$ può essere estesa a tutto $R^{*}$ e che $w$ è definita su $U_{n+1}\setminus K_{m}$, quindi in un intorno di $\Xi_{n}$. Per dimostrare (5), consideriamo la funzione $\displaystyle h(x)=\frac{G(x,z_{0})-r_{n+1}}{b-r_{n+1}}$ dove $b$ è un numero positivo sufficientemente grande da rendere l’insieme $\displaystyle B\equiv\\{z\in R\ \ r.c.\ \ G(z,z_{0})>b\\}$ contenuto nell’insieme $K_{1}\cap U_{n+1}$. In questo modo $h(x)\leq 1$ sull’insieme $U\setminus K$. Confrontiamo ora le funzioni $w$ e $h$ sull’insieme $U\setminus\overline{K}$. Osserviamo prima di tutto che queste sono entrambe funzioni armoniche su $U\setminus\overline{K}$ e continue fino al bordo di questo insieme. Per definizione di $U\equiv U_{n+1}$, la funzione $h$ è negativa su $\partial U$, e come osservato in precedenza $h\leq 1=w$ sull’insieme $\partial K\cap U$. Inoltre entrambe le funzioni sono nulle sull’insieme $\Delta\cap U$, quindi grazie al principio del massimo 3.58, $w\geq h$ su tutto l’insieme di definizione, quindi in particolare anche su $\Xi_{n}$. Dato che $h|_{\Xi_{n}}\geq\frac{r_{n}-r_{n+1}}{b-r_{n+1}}\equiv\sigma_{n}$, si ha la tesi. ∎ Grazie alla funzione $w_{nm}$ possiamo ottenere una stima su $\rho(\Xi_{n})$, infatti vale che: ###### Proposizione 4.40. Secondo le notazioni fino a qui introdotte: $\displaystyle\rho(\Xi_{n})\geq\sigma_{n}^{2}\rho(F_{n+1,m})$ (4.8) per ogni $m$. ###### Proof. In tutta la dimostrazione, sottointendiamo che le sommatorie che scorrono su nessun indice sono nulle, nel senso che ad esempio: $\displaystyle\sum_{i=1}^{0}a_{i}\equiv 0$ Assumiamo inoltre che $\Xi_{n}$ abbia cardinalità infinita, in caso contrario la tesi è ovvia essendo $\rho(A)=\infty$ per ogni insieme $A$ di cardinalità finita. Sia $k\geq 4$ un intero fissato e $p_{1},\cdots,p_{k}$ punti arbitrari in $\Xi_{n}$. Ci prefiggiamo di trovare $k$ punti $z_{1},\cdots,z_{k}\in F_{n+1,m}$ tali che $\displaystyle\sigma^{2}_{n}\sum_{i<j}^{1\cdots t}G(z_{i},z_{j})+\sigma_{n}\sum_{i=1}^{t}\sum_{j=t+1}^{k}G(z_{i},p_{j})+\sum_{i<j}^{t+1,\cdots,k}G(p_{i},p_{j})\leq\sum_{i<j}^{1,\cdots,k}G(p_{i},p_{j})$ (4.9) per ogni $t=1,\cdots,k$. Una volta dimostrato questo otteniamo in particolare per $t=k$ che: $\displaystyle\sigma^{2}_{n}\sum_{i<j}^{1\cdots k}G(z_{i},z_{j})\leq\sum_{i<j}^{1,\cdots,k}G(p_{i},p_{j})$ e per definizione di $\rho(F_{n+1,m})$ questo implica che: $\displaystyle\sigma^{2}_{n}\binom{k}{2}\rho_{k}(F_{n+1,m})\leq\sum_{i<j}^{1,\cdots,k}G(p_{i},p_{j})$ inoltre data l’arbitrarietà della scelta dei punti $p_{1},\cdots,p_{k}$, si ha che: $\displaystyle\sigma^{2}_{n}\rho_{k}(F_{n+1,m})\leq\binom{k}{2}\rho_{k}(\Xi_{n})$ passando al limite per $k$ che tende a infinito, si ottiene la tesi. Resta da dimostrare la parte tecnica della prova, cioè la relazione 4.9. A questo scopo utilizzeremo l’induzione sull’indice $1\leq t\leq k$. Supponiamo di aver trovato dei punti $z_{1},\cdots,z_{h-1}$ con $1\leq h\leq k-1$ per cui vale 4.9 262626ovviamente l’ipotesi di induzione garantisce che questa relazione valga solo per $1\leq t\leq h-1$, perché per indici più grandi le equazioni non hanno senso non avendo determinato tutti punti $z_{i}$. Definiamo $\displaystyle u_{h}(z)\equiv\sum_{j=h+1}^{k}G(z,p_{j})+\sigma_{n}\sum_{i=1}^{h-1}G(z_{i},z)$ Dato che $u_{h}$ è continua e positiva su $R\setminus\\{z_{1},\cdots,z_{h-1}\\}$, ammette un minimo positivo su $F_{n+1,m}$ assunto in $z_{h}$. Osserviamo che per ogni $\delta>0$: $\displaystyle u_{h}(z)-(u_{h}(z_{h})-\delta)>\delta$ sull’insieme $\partial K_{m}\cap U_{n+1}$. Questo implica che esiste un intorno di questo insieme su cui questa funzione è strettamente positiva. Denotiamo questo intorno con il simbolo $A$. Per definizione della successione $C_{p}$, esiste un indice $p$ per il quale l’insieme $(C_{p}\setminus K_{m})\cap U_{n+1}\subset A$. Consideriamo quindi la funzione $w_{n,m,p}$ definita nel lemma precedente. Sappiamo che $\displaystyle\phi_{h}^{\delta}(z)\equiv u_{h}(z)-(u_{h}(z_{h})-\delta)w_{n,m,p}\geq 0$ sull’insieme $\partial(U_{n+1}\setminus K_{m})$, quindi per il principio del massimo questa funzione è positiva su tutto l’insieme $U_{n+1}\setminus K_{m}$, e in particolare sull’insieme $\Xi_{n}$, cioé: $\displaystyle u_{h}(z)\geq(u_{h}(z_{h})-\delta)\sigma_{n}$ per ogni $z\in\Xi_{n}$. Data l’arbitrarietà di $\delta$, otteniamo che $\displaystyle u_{h}(z)\geq u_{h}(z_{h})\sigma_{n}$ per ogni $z\in\Xi_{n}$. Applicando le definizioni otteniamo che: $\displaystyle\sum_{j=h+1}^{k}G(p_{h},p_{j})+\sigma_{n}\sum_{i=1}^{h-1}G(z_{i},p_{h})\geq\sigma_{n}\sum_{j=h+1}^{k}G(z_{h},p_{j})+\sigma_{n}^{2}\sum_{i=1}^{h-1}G(z_{i},z_{h})$ (4.10) questa relazione per $h=1$ è la dimostrazione di 4.9 per $t=1$. Per gli altri valori di $t$, dall’ipotesi induttiva sappiamo che per $t=h-1$: $\displaystyle\sum_{i<j}^{1\cdots,k}G(p_{i},p_{j})\geq\sigma^{2}_{n}\sum_{i<j}^{1\cdots h-1}G(z_{i},z_{j})+\sigma_{n}\sum_{i=1}^{h-1}\sum_{j=h}^{k}G(z_{i},p_{j})+\sum_{i<j}^{h,\cdots,k}G(p_{i},p_{j})=$ $\displaystyle=\sigma^{2}_{n}\sum_{i<j}^{1\cdots h-1}G(z_{i},z_{j})+\sigma_{n}\sum_{i=1}^{h-1}\sum_{j=h+1}^{k}G(z_{i},p_{j})+$ $\displaystyle+\sum_{i<j}^{h+1,\cdots,k}G(p_{i},p_{j})+\left\\{\sigma_{n}\sum_{i=1}^{h-1}G(z_{i},p_{h})+\sum_{j=h+1}^{k}G(p_{h},p_{j})\right\\}$ applicando la relazione 4.10 nelle parentesi graffe, otteniamo la 4.9 per $t=h$, il che completa il ragionamento di induzione. ∎ #### 4.4.3 Pricipio dell’energia Anche questo paragrafo, come il precedente, riporta alcuni risultati tecnici utili per stimare il diametro transfinito degli insiemi compatti contenuti in $\Xi$. In particolare introduciamo l’energia e il potenziale di Green relativo a una misura, e troveremo come queste nozioni sono legate alla capacità di un insieme. Alcuni risultati della teoria del potenziale sono stati cortesemente segnalati dal professor. Wolfhard Hansen (University of Bielefeld), che ringraziamo. In tutto il paragrafo, $K$ indicherà una sottovarietà regolare $m-1$ dimensionale di $R$ possibilmente con bordo liscio. ###### Definizione 4.41. Sull’insieme delle misure di Borel regolari con supporto in $K$ definiamo il prodotto 272727il cui risultato potrebbe anche essere $\infty$: $\displaystyle\left\langle\mu\middle|\nu\right\rangle_{B}\equiv\int_{K}d\mu(x)\ \int_{K}\ d\nu(y)G(x,y)$ la simmetria di questo prodotto segue dalla simmetria di $G(x,y)$ rispetto a $x$ e $y$. Chiamiamo $\left\langle\mu\middle|\nu\right\rangle_{B}$ l’energia relativa tra le misure $\mu$ e $\nu$, e definiamo energia di $\mu$ la quantità $\left\langle\mu\middle|\mu\right\rangle_{B}$. ###### Proposizione 4.42. $\left\langle\mu\middle|\mu\right\rangle_{B}=0$ se e solo se $\mu=0$· ###### Proof. Supponiamo per assurdo che $\mu\neq 0$. Allora la funzione $\displaystyle f(x)\equiv\int G(x,y)d\mu(y)$ è strettamente positiva per ogni $x$, infatti su $K$, $G(x,\cdot)$ assume un minimo positivo $\lambda>0$, e quindi: $\displaystyle f(x)\geq\lambda\mu(K)$ con un ragionamento analogo, si ottiene che anche $\int f(x)d\mu(x)=\int d\mu(x)\int d\mu(y)G(x.y)>0$ ∎ ###### Definizione 4.43. Dato un insieme $K$ con le caratteristiche descritte all’inizio del paragrafo, definiamo l’energia minima di $K$ la quantità $\displaystyle\epsilon(K)\equiv\inf_{\mu\in m_{K}}\left\langle\mu\middle|\mu\right\rangle_{B}$ dove $m_{K}$ è l’insieme delle misure di Borel regolari positive unitarie con supporto su $K$ (cioè $\mu(K)=1$). Per convenzione $\epsilon(\emptyset)=\infty$, e osserviamo immediatamente che per definizione di $\inf$ se $K\subset K^{\prime}$, $\epsilon(K)\geq\epsilon(K^{\prime})$. ###### Osservazione 4.44. Per gli insiemi $K$ con le caratteristiche descritte all’inizio del paragrafo, $\epsilon(K)<\infty$. ###### Proof. Per dimostrare questa affermazione, basta trovare una misura $\mu$ di Borel regolare finita 282828il fatto che $\mu(R)\neq 1$ non è importante, infatti è sufficiente riscalare la misura per la quale $\left\langle\mu\middle|\mu\right\rangle_{B}<\infty$. Sia $(U,\phi)$ un insieme coordinato che interseca $K$ per il quale $\phi(K)=(x_{1},\cdots,x_{m-1},0)$ e sia $B$ una bolla di raggio $\bar{r}$ in $\mathbb{R}^{m-1}$ contenuta in $\phi(K)$. Consideriamo la misura di superficie data dalla metrica riemanniana su questo insieme, misura che ha la forma $\displaystyle dV=\sqrt{\left|g\right|}d\lambda$ dove $\lambda$ è la misura di Legesue standard. Grazie alle proprietà del nucleo di Green, sappiamo che sull’insieme $B$ $G$ in coordinate locali si può scrivere come $\displaystyle G(x,y)=f(x,y)+C(m)\gamma(x,y)$ dove $f(x,y)$ è una funzione continua nelle due variabili $x$ e $y$, $C(m)$ una costante che dipende solo dalla dimensione $m$ e $\gamma(x,y)$ è una funzione che dipende dalla dimensione $m$ della varietà $R$, in particolare: $\displaystyle\gamma(x,y)=\begin{cases}-\log(d(x,y))&m=2\\\ d(x,y)^{m-2}&m\geq 3\end{cases}$ Per limitatezza di $f$ sull’insieme $B\times B$, è ovvio che: $\displaystyle\int_{B}f(x,y)dV(x)dV(y)=\int_{B}f(x,y)\sqrt{\left|g\right|}d\lambda(x)\sqrt{\left|g\right|}d\lambda(y)<\infty$ Consideriamo quindi l’integrale $\displaystyle\int_{B}\gamma(x,y)dV(x)dV(y)=\int\gamma(x,y)\sqrt{\left|g\right|}d\lambda(x)\sqrt{\left|g\right|}d\lambda(y)\leq M^{2}\int_{B}\gamma(x,y)d\lambda(x)d\lambda(y)$ dove $M$ è un limite superiore per la funzione $\sqrt{\left|g\right|}$ sull’insieme relativamente compatto $B$. La funzione $\displaystyle\bar{\gamma}(x)\equiv\int_{B}\gamma(x,y)d\lambda(y)$ è una funzione limitata in $x$, infatti scelto $x\in B$, detto $2B(x)$ la bolla di raggio $2\bar{r}$ centrata in $x$, si ha che: $\displaystyle\bar{\gamma}(x)=\int_{B}\gamma(x,y)d\lambda(y)\leq\int_{2B(x)}\gamma(x,y)d\lambda(y)$ per $m=2$, si ha che: $\displaystyle\int_{2B(x)}-\log(\left|x-y\right|)d\lambda(y)=\int_{-\bar{r}}^{\bar{r}}-\log(r)dr=2\bar{r}(\log(\bar{r})-1)<\infty$ mentre per $m\geq 3$, passando alle coordinate polari centrate in $x$, si ha che $\displaystyle\int_{2B(x)}\frac{1}{\left|x-y\right|^{m-2}}d\lambda(y)=\int_{S_{m-2}}d\theta\int_{0}^{\bar{r}}dr\ r^{m-2}\frac{1}{r^{m-2}}=\omega_{m-2}\bar{r}$ dove $\omega_{m}$ è la superficie della sfera $m$ dimensionale. Essendo la funzione $\bar{\gamma}$ limitata, abbiamo che: $\displaystyle\int_{B\times B}\gamma(x,y)d\lambda(x)d\lambda(y)=\int_{B}\bar{\gamma}(x)d\lambda(x)\leq\lambda(B)\sup_{B}(\bar{\gamma}(x))<\infty$ ∎ ###### Definizione 4.45. Per una misura $\mu\in m_{K}$, definiamo il relativo potenziale di Green la funzione: $\displaystyle G_{\mu}(x)\equiv\int_{K}G(x,y)d\mu(y)$ Per prima cosa esploriamo le proprietà di $G_{\mu}$ per una qualsiasi misura di Borel positiva regolare a supporto compatto $S\in R$. ###### Proposizione 4.46. $G_{\mu}\in H(R\setminus S)$, $G_{\mu}$ è positiva, semicontinua inferiormente, e può essere estesa a tutto $R^{*}$ con $G_{\mu}|_{\Delta}=0$. Inoltre $G_{\mu}$ è superarmonica su $R$. ###### Proof. La positività di $G_{\mu}$ è un’ovvia conseguenza della positività della misura e del nucleo di Green. Per dimostrare che $G_{\mu}\in H(R\setminus S)$, sfruttiamo il teorema di derivazione sotto al segno di integrale 1.37. Sia $x\in R\setminus S$. Allora esiste un intorno compatto $V$ di $x$ disgiunto da $S$, e per le proprietà di $G(\cdot,\cdot)$, sappiamo che $\displaystyle G(x,y)\leq\sup_{z\in\partial V}G(x,z)<\infty$ per ogni $y\in S$. Grazie al fatto che $G\in C^{\infty}(R\times R\setminus D)$, sappiamo che tutte le derivate $\partial_{i}G(x,y)$ e $\partial_{i}\partial_{j}G(x,y)$ sono uniformemente limitate se $(x,y)\in V\times S$, quindi possiamo applicare il teorema 1.37 e ottenere che: $\displaystyle\Delta G_{\nu}(x)=g^{ij}(x)\partial_{i}\partial_{j}\int_{S}G(x,y)d\mu(y)=$ $\displaystyle=g^{ij}(x)\partial_{i}\int_{S}\partial_{j}G(x,y)d\mu(y)=\int_{S}g^{ij}(x)\partial_{i}\partial_{j}G(x,y)d\mu(y)=0$ Con un raginamento del tutto analogo possiamo dimostrare che la funzione $G_{\mu}$ è continua sull’insieme $R\setminus S$. Il fatto che $G_{\mu}$ è semicontinua inferiormente su $R$ segue dalla considerazione che grazie al teorema di convergenza monotona: $\displaystyle G_{\mu}(x)=\int_{S}G(x,y)d\mu(y)=\lim_{n}\int_{S}(G(x,y)\curlywedge n)d\mu(y)\equiv\lim_{n}G_{\mu}^{n}(x)$ La successione $G_{\mu}^{n}$ è una successione crescente di funzioni continue 292929la continuità segue dalla continuità della funzione $G(x,y)\curlywedge n$ e dal teorema di convergenza dominata, quindi $G_{\mu}(x)$ è necessariamente una funzione semicontinua inferiormente. Sempre grazie al teorema di convergenza monotona, sappiamo che: $\displaystyle G_{\mu}(x)\curlywedge c=\int_{S}(G(x,y)\curlywedge c)d\mu(y)$ Queste funzioni appartengono tutte a $\mathbb{M}(R)$, infatti sono continue, di Tonelli, e: $\displaystyle D_{R}(G_{\mu}(x)\curlywedge c)=\int_{R}dx\left|\nabla\int_{S}(G(x,y)\curlywedge c)d\mu(y)\right|^{2}\leq$ $\displaystyle\leq\int_{R}dx\int_{S}\left|\nabla(G(x,y)\curlywedge c)\right|^{2}d\mu(y)=\int_{S}d\mu(y)\int_{R}dx\left|\nabla(G(x,y)\curlywedge c)\right|^{2}\leq$ $\displaystyle\leq\int_{S}d\mu(y)c=c$ dove abbiamo sfruttato il fatto che $\mu(R)=1$ 303030quindi $\left|\int_{S}f(y)d\mu(y)\right|^{2}\leq\int_{S}\left|f(x)\right|^{2}d\mu(y)$ e il teorema di Fubuni per lo scambio di integrali quando l’integrando è positivo. Sempre per il teorema di convergenza monotona, detto $G_{n}(x,y)$ i nuclei di Green rispetto a un’esaustione regolare di $R$, sappiamo che: $\displaystyle G_{\mu}(x)\curlywedge c=\lim_{n}\int_{S}(G_{n}(x,y)\curlywedge c)d\mu(y)$ Poiché tutte le funzioni $G_{n}(x,y)\curlywedge c$ appartengono a $\mathbb{M}_{\Delta}(R)$ e $\displaystyle D_{R}(G_{n}(x,y)\curlywedge c)\leq c$ (4.11) grazie al teorema 3.35, $G_{\mu}(x)\curlywedge c\in\mathbb{M}_{\Delta}(R)$, quindi $G_{\mu}|_{\Delta}=0$. La superarmonicità di $G_{\mu}$ segue dalle proposizioni 1.76 e 1.77. Infatti sappiamo che: $\displaystyle G_{\mu}(x)=\int_{K}G(x,y)d\mu(y)=\lim_{n}\int_{K}(G(x,y)\curlywedge n)d\mu(y)$ Le funzioni $\int_{K}(G(x,y)\curlywedge n)d\mu(y)$ sono superarmoniche grazie alla proposizione 3.54, e anche il loro limite è superarmonico grazie a 1.77. ∎ ###### Proposizione 4.47. Se $\epsilon(K)<\infty$, esiste una misura $\nu\in m_{K}$ tale che: $\displaystyle\epsilon(K)=\left\langle\nu\middle|\nu\right\rangle_{B}$ Cioè esiste una misura che realizza il minimo dell’energia. ###### Proof. Dalla definizione di $\inf$, esiste una successione $\nu_{n}\in m_{K}$ tale che: $\displaystyle\lim_{n}\left\langle\nu_{n}\middle|\nu_{n}\right\rangle_{B}=\epsilon(K)$ Dalla teoria della misura e degli spazi di Banach, sappiamo che esiste una sottosuccessione di $\\{\nu_{n}\\}$ (che per comodità continueremo a indicare con lo stesso indice) tale che $\nu_{n}$ converge debolmente a $\nu\in m_{K}$, cioè per ogni funzione $f\in C(K)$: $\displaystyle\lim_{n}\int fd\nu_{n}=\int fd\nu$ Ora consideriamo le funzioni $\displaystyle\phi_{n}^{c}(x)\equiv\int_{K}(G(x,y)\curlywedge c)d\nu_{n}(y)\ \ \ \ \ \ \phi^{c}(x)\equiv\int_{K}(G(x,y)\curlywedge c)d\nu(y)$ Ricordiamo che per convergenza monotona: $\displaystyle\int_{K}G(x,y)d\nu(y)=\lim_{c\to\infty}\phi^{c}(x)$ Osserviamo che la successione $\\{\phi_{n}^{c}\\}$ converge uniformemente su $K$ a $\phi^{c}$, infatti grazie alla definizione di $\nu$, c’è convergenza puntuale. Inoltre: $\displaystyle\left|\phi_{n}^{c}(x)\right|=\int_{K}(G(x,y)\curlywedge c)d\nu_{n}(y)\leq c\nu_{n}(K)\leq cM$ $\displaystyle\left|\phi_{n}^{c}(x_{1})-\phi_{n}^{c}(x_{2})\right|\leq\int_{K}\left|(G(x_{1},y)\curlywedge c)-G(x_{2},y)\curlywedge c\right|d\nu_{n}(y)$ Data la continuità uniforme della funzione $G(\cdot,\cdot)\curlywedge c$ sull’insieme $K\times K$, si ha che per ogni $\epsilon>0$, esiste $\delta$ per cui $d(x_{1},x_{2})<\delta\ \ \Rightarrow\ \ \left|G(x_{1},y)\curlywedge c-G(x_{2},y)\curlywedge c\right|<\epsilon$ quindi se $d(x_{1},x_{2})<\delta$: $\displaystyle\left|\phi_{n}^{c}(x_{1})-\phi_{n}^{c}(x_{2})\right|\leq\epsilon\nu_{n}(K)\leq\epsilon M$ Queste osservazioni dimostrano l’uniforme limitatezza e l’equicontinuità della successione $\phi_{n}^{c}$ sull’insieme $K$, quindi grazie al teorema di Ascoli-Arzelà (vedi ad esempio appendice A5 pag 394 di [R2]) per ogni sottosuccessione di $\\{\phi_{n}^{c}\\}$, esiste una sua sottosottosuccessione convergente uniformemente su $K$. Dato che $\\{\phi_{n}^{c}\\}$ converge puntualmente a $\phi^{c}$, allora la convergenza è uniforme su $K$. Questo implica che: $\displaystyle\left|\int_{K}\phi_{n}^{c}d\nu_{n}-\int_{K}\phi^{c}d\nu\right|\leq\left|\int_{K}(\phi_{n}^{c}-\phi^{c})d\nu_{n}\right|+\left|\int_{K}\phi^{c}d\nu_{n}-\int_{K}\phi^{c}d\nu\right|\leq$ (4.12) $\displaystyle\leq\left\|\phi_{n}^{c}-\phi^{c}\right\|_{\infty,K}\nu_{n}(K)+\left|\int_{K}\phi^{c}d\nu_{n}-\int_{K}\phi^{c}d\nu\right|\to 0$ Da queste considerazioni otteniamo che: $\displaystyle\left\langle\nu\middle|\nu\right\rangle_{B}=\lim_{c\to\infty}\int_{K}d\nu(x)\int_{K}d\nu(y)(G(x,y)\curlywedge c)=\lim_{c\to\infty}\lim_{n\to\infty}\int_{K}\phi_{n}^{c}d\nu_{n}=$ $\displaystyle\lim_{c\to\infty}\liminf_{n\to\infty}\int_{K}\phi_{n}^{c}d\nu_{n}\leq\liminf_{n\to\infty}\lim_{c\to\infty}\int_{K}\phi_{n}^{c}d\nu_{n}=\liminf_{n\to\infty}\left\langle\nu_{n}\middle|\nu_{n}\right\rangle$ dove il penultimo passaggio, lo scambio tra limite e liminf, è giustificato nel lemma seguente (lemma 4.48). Questa disuguaglianza ci permette di concludere che: $\displaystyle\epsilon(K)\leq\left\langle\nu\middle|\nu\right\rangle_{B}\leq\liminf_{n}\left\langle\nu_{n}\middle|\nu_{n}\right\rangle_{B}=\epsilon(K)$ da cui segue la tesi. ∎ ###### Lemma 4.48. Sia $a_{n,m}$ una successione con due indici crescente in $m$ per ogni $n$ fissato. Allora: $\displaystyle\lim_{m\to\infty}\liminf_{n\to\infty}a_{n,m}\leq\liminf_{n\to\infty}\lim_{m\to\infty}a_{n,m}$ ###### Proof. Sia per definizione $\displaystyle L\equiv\lim_{m\to\infty}\liminf_{n\to\infty}a_{n,m}\ \ \ \ \ \ \ b_{m}\equiv\liminf_{n\to\infty}a_{n,m}$ Dalla definizione di limite, sappiamo che per ogni $\epsilon>0$, esiste $M$ tale che per ogni $m\geq M$, $b_{m}>L-\epsilon$, quindi dalla definizione di $\liminf$ sappiamo che fissato $\bar{m}$, esiste $N(\bar{m})$ tale che per ogni $n\geq N(\bar{m})$: $\displaystyle a_{n,\bar{m}}>L-\epsilon$ Data la monotonia di $a_{n,m}$ rispetto all’indice $m$, sappiamo che per ogni $m\geq\bar{m}$ e per ogni $n\geq N(\bar{m})$, $a_{n,m}>L-\epsilon$, quindi passando al limite in $m$: $\displaystyle\lim_{m}a_{n,m}\geq L-\epsilon$ quindi passando al $\liminf$: $\displaystyle\liminf_{n}\lim_{m}a_{n,m}\geq L-\epsilon$ la tesi si ottiene per arbitrarietà di $\epsilon$. ∎ Ora introduciamo una particolare misura positiva di Borel regolare sull’insieme $K$, ragionando con le funzioni armoniche. Sia $K$ una sottovarietà regolare compatta di codimensione $1$ (possibilmente con bordo liscio). $R\setminus K$ avrà $n$ componenti connesse $R_{1},\cdots,R_{n}$ 313131ad esempio, se $K$ è una sfera, $R\setminus K$ avrà due componenti connesse, se invece $K$ è una parte di piano, $R\setminus K$ è connesso, inoltre ovviamente $K=\cup\partial R_{i}$. Su ogni insieme $R_{i}$ possiamo risolvere un problema di Dirichlet, nel senso che data una funzione $f$ continua su $\partial R_{i}$, esiste una funzione armonica in $R_{i}$ che indicheremo $H^{i}(f)$ che ristretta al bordo di $R_{i}$ uguagli $f$. Questa caratterizzazione però è sufficiente solo se $R_{i}$ è relativamente compatto. In caso contrario indichiamo con $H^{i}(f)$ la funzione armonica ottenuta per esaustione di $R_{i}$. In dettaglio, sia $C_{n}$ un’esaustione regolare di $R$, e sia $H^{i}_{n}(f)$ la funzione $\displaystyle H^{i}_{n}(f)\in H(R_{i}\cap C_{n}),\ \ H^{i}_{n}(f)|_{\partial R_{i}\cap C_{n}}=f,\ \ H^{i}_{n}(f)|_{\partial C_{n}\cap R_{i}}=0$ grazie al principio del massimo si ottiene che $H^{i}_{n}(f)$ è una successione crescente in $n$ di funzioni armoniche, e poiché sempre per il principio del massimo $\displaystyle\left\|H^{i}_{n}(f)\right\|_{\infty}\leq\left\|f\right\|_{\infty}$ grazie al principio di Harnack $H^{i}_{n}(f)$ converge a una funzione $H^{i}(f)$ armonica su $R_{i}$ e continua su $\overline{R_{i}}$. Ora se scegliamo un punto qualsiasi $z_{0}\in R^{i}$, per quanto appena visto possiamo definire un funzionale lineare continuo positivo 323232per definizione, $\phi$ è positivo se $\phi(f)\geq 0$ per ogni $f\geq 0$ sullo spazio delle funzioni continue in $K$ in questo modo: $\displaystyle\phi_{z_{0}}(f)\equiv H^{i}(f)(z_{0})$ dalla teoria 333333vedi teorema di rappresentazione di Riestz, ad esempio su [R4], teorema 2.14 pag 40 sappiamo che esiste unica una misura di Borel positiva regolare $\xi_{z_{0}}$ con supporto su $\partial R^{i}$ 343434quindi con supporto in $K$ a patto di estenderla a nulla su $K\setminus\partial R^{i}$ tale che: $\displaystyle\phi_{z_{0}}(f)\equiv H^{i}(f)(z_{0})=\int_{K}fd\xi_{z_{0}}$ per ogni funzione $f$ continua su $K$. ###### Definizione 4.49. La misura $\xi_{z_{0}}$ appena caratterizzata è detta la misura armonica dell’insieme $K$ rispetto a $z_{0}$. Nel seguente lemma ci occupiamo di un aspetto tecnico legato a questa misura, in particolare a cosa succede su un insieme $F\subset K$ di misura nulla rispetto a una di queste $\xi_{z_{0}}$. ###### Lemma 4.50. Se $\xi_{z_{0}}(F)=0$, esiste una funzione armonica positiva $h:R^{i}\to[0,\infty)$ tale che per ogni $x_{0}\in F$: $\displaystyle\lim_{x\to x_{0}}h(x)=\infty$ Come corollario di questo lemma, possiamo dimostrare che la proprietà $\xi_{z_{0}}(F)=0$ NON dipende dalla scelta di $z_{0}$ (se $z_{0}$ viene scelto tra gli elementi dello stesso insieme $R^{i}$). ###### Proof. Dato che $\xi_{z_{0}}\equiv\xi_{0}$ è una misura regolare, esiste una successione di insiemi $\\{F_{n}\\}$ aperti nella topologia di $K$ tale che $F\subset F_{n}\subset\overline{F_{n}}\subset F_{n+1}\ \ \ \ \bigcap_{n}F_{n}=F\ \ \ \ \xi_{0}(F_{n})\leq 2^{-n}$ Costruiamo le funzioni $q_{n}:K\to\mathbb{R}$ con queste proprietà: 1. 1. $q_{n}\in C(K,[0,1])$ per ogni $n$ 2. 2. $supp(q_{n})\subset F_{n}$, e $q_{n}|_{F_{n+1}}=1$ Sia $s_{n}\equiv\sum_{i=1}^{n}q_{i}$. Visto quanto abbiamo appena osservato, possiamo costruire la successione di funzioni armoniche positive $\displaystyle h_{n}\equiv H^{i}(s_{n})$ Per definizione di $s_{n}$, sappiamo che $h_{n}|_{F}=s_{n}|_{F}=n$, inoltre la successione di funzioni armoniche positive crescenti $h_{n}$ è tale che: $\displaystyle h_{n}(z_{0})=H^{i}(s_{n})(z_{0})=\int_{K}s_{n}d\xi_{z_{0}}=\sum_{j=1}^{n}\int_{K}q_{j}d\xi_{z_{0}}\leq\sum_{i=j}^{n}\int_{K}\chi_{F_{j}}d\xi_{z_{0}}\leq\sum_{j=1}^{n}2^{-j}\leq 1$ quindi per il principio di Harnack 353535vedi 1.57 la successione $h_{n}$ converge a una funzione armonica $h$ su $R^{i}$ semicontinua inferiormente su $\overline{R^{i}}$ 363636dato che la successione $h_{n}$ è una successione crescente di funzioni continue su $\overline{R^{i}}$ tale che $h(z)=\infty$ per ogni $z\in F$. Consideriamo ora un punto $z_{1}\in R^{i}$, e sia $\xi_{z_{1}}$ la relativa misura armonica. Per la successione $h_{n}$ vale che; $\displaystyle h_{n}(z_{1})=\int_{K}s_{n}d\xi_{z_{1}}$ quindi passando al limite su $n$ otteniamo che: $\displaystyle\lim_{n}\int_{K}s_{n}d\xi_{z_{1}}=h(z_{1})<\infty$ se per assurdo $\xi_{z_{1}}(F)>0$, allora necessariamente $\displaystyle\int_{K}s_{n}d\xi_{z_{1}}\geq n\xi_{z_{1}}(F)\to\infty$ che contraddice quanto appena scoperto, quindi se $\xi_{z_{0}}(F)=0$, allora per ogni $z_{1}\in R^{i}$ si ha che $\xi_{z_{1}}(F)=0$. Scambiando i ruoli di $z_{0}$ e $z_{1}$ si ottiene che per un insieme $F\subset K$, il fatto di avere misura armonica $\xi$ nulla è indipendente dalla scelta del punto $z_{0}\in R^{i}$. ∎ La misura armonica è legata all’energia dal fatto che ###### Proposizione 4.51. Sia $z_{0}\not\in K$ e $\xi_{z_{0}}=\xi$ la misura armonica di $K$ relativa a $z_{0}$. Allora se un insieme di Borel $F\subset K$ ha misura armonica non nulla ($\xi(F)\neq 0$), allora ha energia finita. ###### Proof. Consideriamo la misura di Borel $\xi_{1}\equiv\xi|_{F}$. Il potenziale di Green di questa misura è caratterizzato dal fatto che per ogni $x\not\in K$: $\displaystyle G_{\xi_{1}}(x)=\int_{K}G(x,y)d\xi_{1}(y)\leq\int_{K}G(x,y)d\xi(y)=H(G(x,\cdot)|_{z_{0}}\leq G(x,z_{0})$ infatti per superarmonicità, la funzione $G(x,\cdot)$ è maggiore della funzione armonica che assume valore $G(x,\cdot)$ sull’insieme $K$. Data la continutà di $G(\cdot,z_{0})$ su $R\setminus\\{z_{0}\\}$ e data la semicontinuità inferiore di $G_{\xi_{1}}$, la disuguaglianza continua a valere anche se $x\in K$, dimostrando che: $\displaystyle G_{\xi_{1}}(x)\leq G(z_{0},x)\leq M$ dove $M$ è un maggiorante per la funzione continua $G(z_{0},\cdot)|_{K}$, quindi: $\displaystyle\left\langle\xi_{1}\middle|\xi_{1}\right\rangle_{B}\leq M\xi_{1}(K)<\infty$ ∎ Con la seguente proposizione ci occupiamo di alcune proprietà fini del potenziale $G_{\nu}$, dove $\nu$ è la misura che minimizza l’energia su $K$. La proposizione e la relativa dimostrazione sono tratte dal paragrafo 9G di [SN] e dal teorema III.12 di [T]. ###### Proposizione 4.52. Sia $K$ un’insieme con le caratteristiche descritte all’inizio del paragrafo, sia $\xi$ la misura armonica di $K$ rispetto a un punto qualsiasi $z_{0}\not\in K$ e sia $\nu\in m_{K}$ una misura tale che $\displaystyle E\equiv\epsilon(K)=\left\langle\nu\middle|\nu\right\rangle_{B}$ Allora: 1. 1. $G_{\nu}(x)\leq E$ sul supporto $S_{\nu}$ della misura $\nu$ 2. 2. $G_{\nu}(x)\geq E$ a meno di un insieme di misura armonica $\xi$ nulla ###### Proof. Sia $A_{\delta}=\\{x\in K\ t.c.\ G_{\nu}(x)>E+2\delta\\}$, dove $\delta>0$. Data la semicontinuità inferiore di $G_{\nu}$, $A_{\delta}$ sono insiemi aperti per ogni $\delta$, quindi Borel misurabili. Supponiamo per assurdo che $\nu(A_{\delta})>0$ per qualche $\delta>0$. Per la regolarità della misura $\nu$, esiste un insieme $C\subset A_{\delta}$ compatto tale che $\nu(C)\neq 0$. Dato che $\displaystyle E=\epsilon(K)=\left\langle\nu\middle|\nu\right\rangle_{B}=\int_{S_{\nu}}G_{\nu}(x)d\nu(x)$ esiste un punto $x_{0}\in S_{\nu}$ tale $G_{\nu}(x_{0})\leq E+\delta$, inoltre l’insieme $B=\\{x\in K\ t.c.\ G_{\nu}(x)\leq E+\delta\\}$ è chiuso e $\nu(B_{\delta})\neq 0$. Consideriamo la misura di Borel $\displaystyle\sigma(\cdot)=-\frac{\nu(\cdot\cap C)}{\nu(C)}+\frac{\nu(\cdot\cap B)}{\nu(B)}$ Dato che $\displaystyle 0\leq\left\langle\nu(\cdot\cap C)\middle|\nu(\cdot\cap C)\right\rangle_{B}=\int_{K}G_{\nu(\cdot\cap C)}d\nu{\cdot\cap C}\leq\int_{K}G_{\nu}d\nu(\cdot\cap C)=$ $\displaystyle=\int_{K}G_{\nu(\cdot\cap C)}d\nu\leq\int_{K}G_{\nu}d\nu=\left\langle\nu\middle|\nu\right\rangle_{B}<\infty$ e con un ragionamento analogo anche $\left\langle\nu(\cdot\cap B)\middle|\nu(\cdot\cap B)\right\rangle_{B}<\infty$, e dato che gli insiemi $B$ e $C$ sono compatti disgiunti, allora $\displaystyle 0\leq\left\langle\nu(\cdot\cap B)\middle|\nu(\cdot\cap C)\right\rangle_{B}=\int_{K}G_{\nu(\cdot\cap B)}d\nu(\cdot\cap C)\leq M\nu(C)$ dove $M$ è un maggiorante per la funzione $G_{\nu(\cdot\cap B)}$ continua sul compatto $C$,. Inoltre notiamo che $\left\langle\sigma\middle|\sigma\right\rangle_{B}\neq\infty$, infatti: $\displaystyle\left\langle\sigma\middle|\sigma\right\rangle_{B}=\left\langle-\frac{\nu(\cdot\cap C)}{\nu(C)}+\frac{\nu(\cdot\cap B)}{\nu(B)}\middle|-\frac{\nu(\cdot\cap C)}{\nu(C)}+\frac{\nu(\cdot\cap B)}{\nu(B)}\right\rangle_{B}=$ $\displaystyle=\frac{1}{\nu(C)^{2}}\left\langle\nu(\cdot\cap C)\middle|\nu(\cdot\cap C)\right\rangle_{B}-\frac{2}{\nu(C)\nu(B)}\left\langle\nu(\cdot\cap C)\middle|\nu(\cdot\cap B)\right\rangle_{B}+$ $\displaystyle+\frac{1}{\nu(B)^{2}}\left\langle\nu(\cdot\cap B)\middle|\nu(\cdot\cap B)\right\rangle_{B}$ Consideriamo ora per $0\leq\eta<\nu(C)$ la misura $\nu_{\eta}=\nu+\eta\sigma$. È facile verificare che $\nu_{\eta}$ è una misura di Borel regolare positiva e $\nu_{\eta}(K)=1$, quindi $\nu_{\eta}\in m_{K}$. D’altronde osserviamo che: $\displaystyle\left\langle\nu_{\eta}\middle|\nu_{\eta}\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B}=\left\langle\nu+\eta\sigma\middle|\nu+\eta\sigma\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B}=2\eta\left\langle\nu\middle|\sigma\right\rangle_{B}+\eta^{2}\left\langle\sigma\middle|\sigma\right\rangle_{B}$ Inoltre: $\displaystyle\left\langle\nu\middle|\sigma\right\rangle_{B}=-\frac{1}{\nu(C)}\int_{C}G_{\nu}d\nu+\frac{1}{\nu(B)}\int_{B}G_{\nu}d\nu\leq-E-2\delta+E+\delta<-\delta<0$ Quindi per $\eta$ sufficientemente piccolo, $\left\langle\nu_{\eta}\middle|\nu_{\eta}\right\rangle_{B}<\left\langle\nu\middle|\nu\right\rangle_{B}$, contraddicendo la definizione di $\nu$. La seconda parte della proposizione si dimostra in maniera analoga alla prima. Supponiamo per assurdo che esista $\delta>0$ tale che l’insieme $A_{\delta}=\\{x\in K\ t.c.\ G_{\nu}(x)\leq E-2\delta\\}$ abbia misura armonica positiva. Grazie alla proposizione 4.51, l’energia di $A$ è finita, quindi esiste una misura positiva unitaria $\sigma_{1}$ con supporto in $A$ tale che $\left\langle\sigma_{1}\middle|\sigma_{1}\right\rangle_{B}<\infty$. Sia $B=\\{x\in Kt.c.\ G_{\nu}(x)>E-\delta\\}$. Dato che $\displaystyle\int_{K}G_{\nu}(x)d\nu(x)=E$ $\nu(B)\neq 0$. Se definiamo $\sigma(\cdot)=\sigma_{1}(\cdot)-\frac{\nu(\cdot\cap B)}{\nu(B)}$, analogamente a quanto dimostrato sopra, si dimostra che $\sigma(K)=\sigma(R)=0$ e che per ogni $0\leq\eta<\nu(B)$, la misura $\nu+\eta\sigma\in m_{K}$. La contraddizione segue dal fatto che: $\displaystyle\left\langle\nu_{\eta}\middle|\nu_{\eta}\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B}=2\eta\left\langle\sigma\middle|\nu\right\rangle_{B}+\eta^{2}\left\langle\sigma\middle|\sigma\right\rangle_{B}$ $\displaystyle\left\langle\sigma\middle|\nu\right\rangle_{B}=\int_{K}G_{\nu}(x)d\sigma=$ $\displaystyle=\int_{A_{\delta}}G_{\nu}(x)d\sigma_{1}-\frac{1}{\nu(B)}\int_{B}G_{\nu}(x)d\nu\leq E-2\delta-E+\delta\leq-\delta<0$ ∎ Nella seguente proposizione dimostriamo una proprietà di continuità del potenziale di Green $G_{\mu}$ rispetto a una qualsiasi $\mu\in m_{K}$. ###### Proposizione 4.53. Sia $\mu$ una misura di Borel positiva finita e $G_{\mu}$ il relativo potenziale di Green. Se $G_{\mu}$ ristretto al supporto $S_{\mu}$ della misura $\mu$ è continuo, allora $G_{\mu}$ è una funzione continua su tutta la varietà $R$. ###### Proof. Dato che $G_{\mu}\in H(R\setminus S_{\mu})$, è sufficiente dimostrare che $G_{\mu}$ è continua sull’insieme $S_{\mu}$, quindi che per ogni $x_{0}\in S_{\mu}$: $\displaystyle\lim_{x\to x_{0}}G_{\mu}(x)=G_{\mu}(x_{0})$ Sappiamo dalle proposizioni precedenti che $G_{\mu}$ è semicontinua inferiormente, quindi basta dimostrare che: $\displaystyle\limsup_{x\to x_{0}}G_{\mu}(x)\leq G_{\mu}(x_{0})$ (4.13) inoltre possiamo assumere che $G_{\mu}(x_{0})<\infty$, quindi $\mu(\\{x_{0}\\})=0$. Consideriamo un intorno coordinato di $U$ di $x_{0}$. Su questo intorno possiamo scrivere: $\displaystyle G(x,y)=f(x,y)+C(m)h(x,y)$ dove $f(x,y)$ è una funzione continua in entrambe le sue variabili, $C(m)$ una costante che dipende solo da $m=dim(R)$, mentre $\displaystyle h(x,y)=\begin{cases}-\log(d(x,y))\ &\ se\ m=2\\\ d(x,y)^{-m+2}\ &\ se\ m\geq 3\end{cases}$ Decomponendo la misura $\mu$ in $\mu_{0}(\cdot)\equiv\mu(\cdot\cap U)$ e $\mu^{\prime}\equiv\mu-\mu_{0}$ possiamo scrivere il potenziale di Green come: $\displaystyle G_{\mu}(x)=G_{\mu_{0}}(x)+G_{\mu^{\prime}}(x)$ e dato che $G_{\mu^{\prime}}(x)$ è una funzione continua in $x_{0}$, basta dimostrare 4.13 per il solo potenziale $G_{\mu_{0}}$. Inoltre $\displaystyle G_{\mu_{0}}(x)=\int_{K}f(x,y)d\mu_{0}(y)+C(m)\int_{K}h(x,y)d\mu_{0}(y)$ e dato che la misura $\mu_{0}$ è finita, il primo integrale è una funzione continua della variabile $x$, quindi ancora basta dimostrare 4.13 per la funzione $H(x)\equiv\int_{K}h(x,y)d\mu_{0}(y)$ Poiché $\mu(\\{x_{0}\\})=0$, e dato che $\mu$ è una misura regolare, per ogni $\epsilon>0$, esiste un aperto $U_{\epsilon}\subset U$ contenente $x_{0}$ tale che $\mu(U_{\epsilon})=\mu_{0}(U_{\epsilon})<\epsilon$, quindi vale che: $\displaystyle H(x)=\int_{U_{\epsilon}}h(x,y)d\mu_{0}(y)+\int_{U\setminus U_{\epsilon}}h(x,y)d\mu_{0}(y)$ Per ogni $x\in U$, definiamo $\pi(x)\in S_{\mu}$ un punto tale che $d(x,\pi(x))=\min_{y\in S_{\mu}}d(x,y)$. Osserviamo che in generale il punto $\pi(x)$ non è unico, ma comunque $\lim_{x\to x_{0}}\pi(x)=x_{0}$ se $x_{0}\in S_{\mu}$, inoltre vale che per ogni $y\in S_{\mu}$: $\displaystyle d(y,\pi(x))\leq d(x,y)+d(x,\pi(x))\leq 2d(x,y)$ quindi: $\displaystyle h(x,y)\leq\begin{cases}\log(2)+h(\pi(x),y)&se\ m=2\\\ 2^{m-2}h(\pi(x),y)&se\ m\geq 3\end{cases}$ Dividiamo la dimostrazione in due casi: se $n=2$: $\displaystyle H(x)\leq\int_{U_{\epsilon}}h(\pi(x),y)d\mu_{\epsilon}+\epsilon\log(2)+\int_{U\setminus U_{\epsilon}}h(x,y)d\mu_{0}(y)=$ $\displaystyle=H(\pi(x))+\epsilon\log(2)+\int_{U\setminus U_{\epsilon}}(h(x,y)-h(\pi(x),y)d\mu_{0}(y)$ Applicando il $\limsup_{x\to x_{0}}$ a entrambi i membri e tenedo conto dell’ipotesi di continuità di $G_{\mu}$ (quindi di $H$) ristretta all’insieme $S_{\mu}$ otteniamo: $\displaystyle\limsup_{x\to x_{0}}H(x)\leq H(x_{0})+\epsilon\log(2)+\limsup_{x\to x_{0}}\int_{U\setminus U_{\epsilon}}(h(x,y)-h(\pi(x),y)d\mu_{0}(y)$ Per convergenza dominata, l’ultimo addendo è nullo, e data l’arbitrarietà di $\epsilon$, otteniamo la tesi. In maniera analoga, se $m\geq 3$: $\displaystyle H(x)\leq\int_{U_{\epsilon}}h(\pi(x),y)d\mu_{\epsilon}+\alpha_{m}\int_{U_{\epsilon}}h(\pi(x),y)d\mu(y)+\int_{U\setminus U_{\epsilon}}h(x,y)d\mu_{0}(y)$ dove $\alpha_{m}=2^{m-2}-1$. Applicando il $\limsup_{x\to x_{0}}$ a entrambi i membri e tenendo conto della continuità della funzione $H$ ristretta all’insieme $S_{\mu}$, otteniamo: $\displaystyle\limsup_{x\to x_{0}}H(x)\leq H(x_{0})+\alpha_{m}\int_{U_{\epsilon}}h(x_{0},y)d\mu(y)+$ $\displaystyle+\limsup_{x\to x_{0}}\int_{U\setminus U_{\epsilon}}(h(x,y)-h(\pi(x),y)d\mu_{0}(y)$ Come nel caso bidimensionale, l’ultimo limite è nullo per convergenza dominata 373737infatti dato che $x\to x_{0}$, $x$ e $\pi(x)$ appartengono definitivamente a un intorno compatto di $x_{0}$ contenuto in $U_{\epsilon}$, quindi se $y\in U\setminus U_{\epsilon}$, sia $h(x,y)$ che $h(\pi(x),y)$ sono uniformemente limitate da una costante, mentre dato che $\int_{U}h(x_{0},y)d\mu(y)<\infty$, grazie a una nota proprietà degli integrali se $\epsilon$ è sufficientemente piccolo $\int_{U_{\epsilon}}h(x_{0},y)d\mu(y)$ può essere reso piccolo a piacere. Quindi ancora una volta otteniamo che: $\displaystyle\limsup_{x\to x_{0}}H(x)\leq H(x_{0})$ da cui la tesi. ∎ ###### Osservazione 4.54. Osserviamo che nel caso $m=2$, la relazione 4.13 segue dal “principio di Frostman” (vedi ad esempio la sezione 9E pag 320 di [SN], o il teorema III.1 di [T]) e non è necessario assumere che $G_{\mu}$ sia continuo quando ristretto all’insieme $S_{\mu}$. In dimensione maggiore, però, la dimostrazione di questo principio non è facilmente estendibile, per questa ragione riportiamo solo la versione generale della proposizione (che comunque è sufficiente per i nostri scopi). Questa proposizione e l’armonicità di $G_{\mu}$ su $R\setminus S_{\mu}$ ci permettono di dimostrare che: ###### Proposizione 4.55. Se $G_{\mu}$ è continua quando ristretta all’insieme $S_{\mu}$, allora il suo massimo è raggiunto su $S_{\mu}$, quindi se $G_{\mu}$ è continua quando ristretta a $S_{\mu}$ e $G_{\mu}(x)\leq c$ per ogni $x\in S_{\mu}$, allora $G_{\mu}(x)\leq c$ per ogni $x\in R$. ###### Proof. La dimostrazione è una semplice applicazione del teorema 3.58 (ricordiamo che $G_{\mu}|_{\Delta}=0$). ∎ Questa proposizione può essere migliorata, nel senso che non è necessario chiedere la continuità di $G_{\mu}$ sull’insieme $S_{\mu}$. A questo scopo riportiamo i seguenti lemmi, tratto dai teoremi 3.6.2 e 3.6.3 di [H1], e cortesemente segnalati dal professor. Wolfhard Hansen (University of Bielefeld): ###### Lemma 4.56. Data una misura positiva regolare finita $\mu$ con supporto compatto $S_{\mu}$ tale che $G_{\mu}<\infty$ su $S_{\mu}$, per ogni $\epsilon>0$, esiste un insieme compatto $C\subset S_{\mu}$ tale che, detta $\mu|(\cdot)C\equiv\mu(\cdot\cap C)$, $G_{\mu|C}$ è continua su $C$ e $\mu(S_{\mu}\setminus C)<\epsilon$. ###### Proof. Grazie al teorema di Lusin (vedi ad esempio teorema 2.23 pag 53 di [R4]), per ogni $\epsilon>0$ esiste un compatto $C\subset S_{\mu}$ tale che $G_{\mu}|_{C}$ è una funzione continua. Dato che: $\displaystyle G_{\mu|C}=G_{\mu}-G_{\mu-\mu|C}$ e dato che su $C$ $G_{\mu}$ è continua, $G_{\mu|C}$ è semicontinua superiormente su $C$, e quindi continua su $C$ e automaticamente continua su tutto $R$ grazie alla proposizione 4.53. ∎ ###### Lemma 4.57. Se $G_{\mu}<\infty$ su $S_{\mu}$, allora esiste una successione di misure di Borel regolari $\mu_{n}$ tali che $\\{G_{\mu_{n}}\\}$ è una successione crescente di funzioni continue su $R$ e: $\displaystyle G_{\mu}(x)=\lim_{n}G_{\mu_{n}}(x)$ ###### Proof. Scegliamo per induzione una successione di insiemi compatti $C_{n}$ tali che $C_{n}\subset C_{n+1}$, $\mu(S_{\mu}\setminus C_{n})\leq 2^{-n}$ e $G_{\mu}$ continua sull’insieme $C_{n}$, e consideriamo $\mu_{n}=\mu|C_{n}$. Dal lemma precedente, sappiamo che $G_{\mu_{n}}$ è una funzione continua, e per costruzione delle misure $\mu_{n}$, ovviamente $\\{G_{\mu_{n}}(x)\\}$ è una successione crescente $\forall x\in R$. Inoltre: $\displaystyle G_{\mu_{n}}(x)=\int_{K}G(x,y)\chi_{C_{n}}(y)d\mu(y)$ e quindi per convergenza monotona, $\lim_{n}G_{\mu_{n}}(x)=G_{\mu}(x)$. ∎ Ora siamo pronti per dimostrare che: ###### Proposizione 4.58. Se $G_{\mu}(x)\leq c$ per ogni $x\in S_{\mu}$, allora $G_{\mu}(x)\leq c$ per ogni $x\in R$. ###### Proof. Sia $\\{\mu_{n}\\}$ una successione di misure con le caratteristiche descritte nell’ultimo lemma, allora $G_{\mu_{n}}(x)\leq G_{\mu}(x)\leq c$ per ogni $x\in S_{\mu}$, e dato che $G_{\mu_{n}}$ sono funzioni continue, grazie alla proposizione 4.55, sappiamo che per ogni $x$ per ogni $n$: $G_{\mu_{n}}(x)\leq c$ Passando al limite su $n$ otteniamo la tesi. ∎ Grazie a questa proposizione, possiamo dimostrare questo teorema che garantisce lega l’energia di un insieme alla sua capacità: ###### Teorema 4.59. Sia $K$ una sottovarietà regolare di codimensione $1$ di $R$ possibilmente con bordo liscio. Esiste una misura di Borel regolare unitaria 383838cioè $\nu(R)=1$ $\nu$ con supporto in $K$ tale che: $\displaystyle\epsilon(K)=\left\langle\nu\middle|\nu\right\rangle_{B}<\infty$ Inoltre detto $u$ il potenziale armonico dell’insieme $K$, per il potenziale di Green relativo a $\nu$ vale che: 1. 1. $G_{\nu}\in H(R\setminus K)$ 2. 2. $G_{\nu}(x)=\epsilon(K)u(x)$ per ogni $x\in R$ 3. 3. $G_{\nu}\in\mathbb{M}_{\Delta}(R)$ 4. 4. $D_{R}(G_{\nu})=\epsilon(K)$ 5. 5. $S_{\nu}=K$ Inoltre la misura $\nu$ è unica. ###### Proof. Sia $u$ il potenziale di capacità dell’insieme $K$ (vedi osservazione 4.8). Sia $\nu$ la misura che risolve il problema 4.47. Sappiamo che il potenziale $G_{\nu}$ soddisfa $G_{\nu}|_{S_{\nu}}\leq\epsilon(K)u|_{S_{\nu}}=\epsilon(K)$, e grazie alla proposizione 4.58, sappiamo anche che $G|_{\nu}\leq\epsilon(K)$ su tutta la varietà $R$. Dato che entrambe le funzioni $G_{\nu}$ e $u$ sono armoniche in $R\setminus K$, su annullano su $\Delta$ e $G_{\nu}|_{K}\leq\epsilon(K)=\epsilon(K)u|_{K}$, grazie al principio 3.58, possiamo concludere che $\displaystyle G_{\nu}(x)\leq\epsilon(K)u(x)\ \ \ \forall x\in R$ Per dimostrare questa disuguaglianza con il verso opposto, sappiamo che $G_{\nu}(x)\geq\epsilon(K)u(x)$ per $x\in K\setminus F$ dove $F$ è un’insieme di misura armonica $\xi$ nulla, quindi grazie al lemma 4.50 esiste una funzione $w$ armonica su $R\setminus K$ che tende a infinito nei punti di $F$. Allora per ogni $\epsilon>0$, la funzione $\displaystyle G_{\nu}(x)+\epsilon w(x)\geq\epsilon(K)u(x)$ per ogni $x\in R\setminus K$ grazie al teorema 3.58. Data l’arbitrarietà di $\epsilon$, possiamo concludere che $\displaystyle G_{\nu}(x)=\epsilon(K)u(x)\ \ \ \forall x\in R\setminus K$ Dato che $u$ è continua su $R$ e $G_{\nu}$ è superarmonica, quindi semicontinua inferiormente, l’uguaglianza vale su tutto $R$. Sia infatti $x_{0}\in K$. Allora esiste una successione $\\{x_{n}\\}\subset R\setminus K$ che converge a $x_{0}$ e per la quale: $\displaystyle G_{\nu}(x_{0})\geq\liminf_{n}G_{\nu}(x_{n})\geq\lim_{n}\epsilon(K)u(x_{n})=\epsilon(K)$ dato che $G_{\nu}\leq\epsilon(K)$, necessariamente $G_{\nu}(x_{0})=\epsilon(K)=\epsilon(K)u(x_{0})$. Per quanto riguarda la proprietà (4), osserviamo che per ogni aperto relativamente compatto con bordo liscio $C$ tale che $K\subset C$: $\displaystyle D_{R}(u)=-\int_{\partial C}\ast du$ Consideriamo infatti una successione di aperti con bordo liscio $\\{A_{n}\\}$ tali che $A_{n}\subset A_{n-1}$ e $K=\cap_{n}A_{n}$ 393939un esempio di insiemi con queste caratteristiche è descritto nell’osservazione 1.27. Allora sappiamo che: $\displaystyle D_{R}(u)=\lim_{n}D_{R\setminus A_{n}}(u)=\lim_{n}\int_{\partial(R\setminus A_{n})}u\ast du=-\lim_{n}\int_{\partial A_{n}}u\ast du$ Grazie al fatto che $u$ è continua, è anche uniformemente continua su ogni insieme compatto, quindi ad esempio sull’insieme $\overline{A_{1}}$. Quindi per ogni $\epsilon>0$, esiste $\delta$ tale che $\displaystyle d(x,y)<\delta\ \ \Rightarrow\ \ \left|u(x)-u(y)\right|<\epsilon$ In particolare, se $d(x,K)<\delta$, otteniamo che $0\leq 1-u(x)\leq\epsilon$. Dato che $\cap_{n}A_{n}=K$, $A_{n}$ è contenuto definitivamente nell’aperto $\displaystyle K+\delta\equiv\\{x\in R\ t.c.\ d(x,K)<\delta\\}$ quindi definitivamente in $n$ vale che: $\displaystyle\int_{\partial A_{n}}(1-u)\ast du=\int_{\partial A_{n}}(1-u)(\ast du)^{+}-\int_{\partial A_{n}}(1-u)(\ast du)^{-}\leq$ $\displaystyle\leq\epsilon\int_{\partial A_{n}}(\ast du)^{+}\leq\epsilon\int_{\partial A_{n}}\ast du$ $\displaystyle\int_{\partial A_{n}}(1-u)\ast du=\int_{\partial A_{n}}(1-u)(\ast du)^{+}-\int_{\partial A_{n}}(1-u)(\ast du)^{-}\geq$ $\displaystyle\geq-\epsilon\int_{\partial A_{n}}(\ast du)^{-}\geq-\epsilon\int_{\partial A_{n}}\ast du$ cioé: $\displaystyle-\epsilon\int_{\partial A_{n}}\ast du\leq\int_{\partial A_{n}}(1-u)\ast du\leq\epsilon\int_{\partial A_{n}}\ast du$ dato che $u\in H(\overline{A_{n}}\setminus A_{m})$: $\int_{\partial A_{n}}\ast du=\int_{\partial A_{m}}\ast du$ per ogni $n$ e $m$, quindi possiamo concludere che $\displaystyle\lim_{n}\int_{\partial A_{n}}(1-u)\ast du=0\ \ \Rightarrow$ $\displaystyle\Rightarrow\ \ -D_{R}(u)=\lim_{n}\int_{\partial A_{n}}u\ast du=\lim_{n}\int_{\partial A_{n}}\ast du=\int_{\partial C}\ast du$ Passiamo ora a considerare $D_{R}(G_{\mu})$ $\displaystyle- D_{R}(G_{\nu})=-D_{R}(\epsilon(K)u)=\epsilon(K)^{2}\int_{\partial C}\ast du=\epsilon(K)\int_{\partial C}\ast d(G_{\nu})=$ $\displaystyle=\epsilon(K)\int_{\partial C}\ast d\left(\int_{K}G(x,y)d\mu(y)\right)=\epsilon(K)\int_{\partial C}\int_{K}\ast dG(\cdot,y)d\mu(y)=$ $\displaystyle=\epsilon(K)\int_{K}d\mu(y)\int_{C}\ast dG(\cdot,y)=-\epsilon(K)$ dove abbiamo sfruttato il fatto che $G(x,y)\in C^{\infty}(K\times C)$ per scambiare tra loro i segni di integrali e derivate, e la relazione (6) di 1.61. Resta da dimostrare la proprietà (5). A questo scopo notiamo che per il principio del massimo forte, una funzione armonica su $R$ non può assumere il suo massimo in un punto interno del dominio, quindi se $x\in K\setminus S_{\nu}$, $G_{\nu}(x)<\max_{x\in R}G_{\nu}(x)=\epsilon(K)$. Ma dato che $G_{\nu}(x)=\epsilon(K)u(x)$, e dato che il potenziale di capacità $u$ è identicamente uguale a $1$ su $S_{\nu}$, necessariamente l’insieme $K\setminus S_{\nu}$ è vuoto. Una dimostrazione alternativa di questa affermazione si può ottenere sfruttando una tecnica del tutto analoga a quella utilizzata nella dimostrazione della proposizione 4.52. Supponiamo per assurdo che $S_{\nu}\not=K$. Allora, dato che $S_{\nu}$ è chiuso, sulla sottovarietà $K$ esiste una bolla chiusa $m-1$ dimensionale contenuta in $K$ ma disgiunta da $S_{\nu}$. Grazie all’osservazione 4.44, esiste una misura $\mu\in m_{K}$ con supporto contenuto nella bolla tale che $\left\langle\mu\middle|\mu\right\rangle_{B}<\infty$. Per ogni $0\leq\lambda\leq 1$, consideriamo la misura: $\displaystyle\nu_{\lambda}\equiv(1-\lambda)\nu+\lambda\mu$ È facile verificare che $\nu_{\lambda}\in m_{K}$ per ogni $0\leq\lambda\leq 1$ inoltre: $\displaystyle\left\langle\nu_{\lambda}\middle|\nu-\lambda\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B}=$ $\displaystyle=[(1-\lambda)^{2}-1]\left\langle\nu\middle|\nu\right\rangle_{B}+\lambda^{2}\left\langle\mu\middle|\mu\right\rangle_{B}+2\lambda(1-\lambda)\left\langle\nu\middle|\mu\right\rangle_{B}=$ $\displaystyle=\lambda^{2}(\left\langle\nu\middle|\nu\right\rangle_{B}+\left\langle\mu\middle|\mu\right\rangle_{B}-2\left\langle\nu\middle|\mu\right\rangle_{B})+2\lambda(\left\langle\nu\middle|\mu\right\rangle_{B}-\left\langle\nu\middle|\nu\right\rangle_{B})$ Dato che $\displaystyle\left\langle\nu\middle|\mu\right\rangle_{B}=\int_{K}G_{\nu}(x)d\mu\leq M<\epsilon(K)<\infty$ dove $M$ è il massimo della funzione $G_{\nu}$ sull’insieme $S_{\mu}$, che è strettamente minore di $\epsilon(K)$ grazie al principio del massimo, $\left\langle\nu\middle|\mu\right\rangle_{B}-\left\langle\mu\middle|\mu\right\rangle_{B}<0$, quindi esiste un valore di $\lambda$ sufficientemente piccolo per il quale $\displaystyle\left\langle\nu_{\lambda}\middle|\nu_{\lambda}\right\rangle_{B}<\left\langle\nu\middle|\nu\right\rangle_{B}$ contraddicendo l’ipotesi di minimalità di $\nu$. Concludiamo con la dimostrazione dell’unicità della misura $\nu$. Supponiamo per assurdo che esista un’altra misura di minimo $\bar{\nu}$. Vale comunque che $G_{\nu}(x)=\epsilon(K)u(x)=G_{\bar{\nu}}(x)$. Grazie al punto (5) della proposizione 4.28, per ogni funzione liscia a supporto compatto in $R$, abbiamo che: $\displaystyle\int_{R}f(y)d\nu(y)=\int_{R}\left(\int_{R}G(x,y)\Delta f(x)d\lambda(x)\right)d\nu(y)$ dato che $\displaystyle\int_{R}\left|\Delta f(x)\right|\int_{R}G(x,y)d\nu(y)d\lambda(x)\leq M\epsilon(K)\lambda(S)<\infty$ dove $M$ è un maggiorante di $\left|\Delta f(x)\right|$ e $S$ il supporto della funzione $f$, grazie al teorema di Fubini possiamo scambiare l’ordine di integrazione e ottenere: $\displaystyle\int_{R}f(y)d\nu(y)=\int_{R}\Delta f(x)G_{\nu}(x)d\lambda(x)=\int_{R}f(y)d\bar{\nu}(y)$ Per densità delle funzioni lisce a supporto compatto nelle funzioni continue a supporto compatto, necessariamente $\nu=\bar{\nu}$. ∎ ###### Osservazione 4.60. Osserviamo subito che nelle ipotesi del teorema precedente, otteniamo che la capacità di $K$ è precisamente l’inverso di $\epsilon(K)$, infatti: $\displaystyle Cap(K)=D_{R}(u)=\frac{D_{R}(G_{\mu})}{\epsilon(K)^{2}}=\epsilon(K)^{-1}$ Con la seguente stima leghiamo l’energia al diametro transfinito di un insieme. ###### Proposizione 4.61. Sia $K\subset R$. Allora $\rho(K)\geq\epsilon(K)$. ###### Proof. Grazie alla definizione di $\rho_{n}(K)$, per ogni $n$ possiamo scegliere $n$ punti $p_{1}\cdots,p_{n}$ in $K$ tali che $\displaystyle\binom{n}{2}\rho_{n}(K)\geq\sum_{i<j}^{1,\cdots,n}G(p_{i},p_{j})-\frac{1}{n}$ (4.14) Sia $\mu_{n}$ la misura data da $\displaystyle\mu_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{p_{i}}$ dove $\delta_{x}$ è la misura che associa $1$ agli insiemi contenenti $x$ e $0$ agli altri. Quindi $\mu_{n}\in m_{K}$. Allora esiste una sottosuccessione di $\\{\mu_{n}\\}$ (che per comodità continueremo a indicare con lo stesso simbolo) che converge debolmente nel senso della misura a una misura $\mu$, cioè esiste una misura $\mu$ tale che per ogni funzione $\phi$ continua su $K$: $\displaystyle\lim_{n}\int_{K}\phi d\mu_{n}=\int_{K}\phi d\mu$ quindi la misura $\mu$ è di Borel, positiva e unitaria. Definiamo le funzioni: $\displaystyle\phi_{n}^{c}(x)\equiv\int_{K}(G(x,y)\curlywedge c)d\mu_{n}(y)\ \ \ \ \ \ \phi^{c}(x)\equiv\int_{K}(G(x,y)\curlywedge c)d\mu(y)$ e seguendo lo stesso ragionamento della dimostrazione della proposizione 4.47, otteniamo la validità dell’equazione 4.12, cioè otteniamo: $\displaystyle\lim_{n}\int_{K}d\mu_{n}(x)\int_{K}d\mu_{n}(y)(G(x,y)\curlywedge c)=\int_{K}d\mu(x)\int_{K}d\mu(y)(G(x,y)\curlywedge c)$ Moltiplicando ambo i membri di 4.14 per $\binom{n}{2}^{-1}$ otteniamo: $\displaystyle\rho_{n}(K)+\frac{1}{n}\binom{n}{2}^{-1}\geq\binom{n}{2}^{-1}\sum_{i<j}^{1,\cdots,n}G(p_{i},p_{j})\geq\frac{2}{n^{2}}\sum_{i<j}^{1,\cdots,n}(G(p_{i},p_{j})\curlywedge c)$ dove abbiamo sfruttato il fatto che $\binom{n}{2}\leq\frac{n^{2}}{2}$. Ora dalla definizione di $\mu_{n}$ otteniamo che: $\displaystyle\sum_{i,j}^{1,\cdots,n}(G(p_{i},p_{j})\curlywedge c)=n^{2}\int_{K}d\mu_{n}\int_{K}d\mu_{n}(G(x,y)\curlywedge c)$ considerando che la funzione $G$ è simmetrica nei suoi due argomenti e che: $\displaystyle\sum_{i=j=1}^{n}(G(p_{i},p_{j})\curlywedge c)=\sum_{i=1}^{n}(G(p_{i},p_{i})\curlywedge c)=nc$ otteniamo: $\displaystyle\sum_{i<j}^{1,\cdots,n}(G(p_{i},p_{j})\curlywedge c)=\frac{n^{2}}{2}\int_{K}d\mu_{n}\int_{K}d\mu_{n}(G(x,y)\curlywedge c)-\frac{nc}{2}$ quindi: $\displaystyle\rho_{n}(K)+\frac{1}{n}\binom{n}{2}^{-1}\geq\int_{K}d\mu_{n}\int_{K}d\mu_{n}(G(x,y)\curlywedge c)-\frac{c}{2n}$ facendo tendere $n$ a infinito, otteniamo: $\displaystyle\rho(K)\geq\lim_{n}\int_{K}d\mu_{n}\int_{K}d\mu_{n}(G(x,y)\curlywedge c)=\int_{K}d\mu\int_{K}d\mu(G(x,y)\curlywedge c)$ grazie all’arbitrarietà del parametro $c$, concludiamo: $\displaystyle\rho(K)\geq\int_{K}d\mu\int_{K}d\mu G(x,y)\geq\epsilon(K)$ come volevasi dimostrare. ∎ L’ultima proposizione “tecnica” della dimostrazione è: ###### Proposizione 4.62. La capacità di $F_{n+1,m}$ tende a $0$ se $m$ tende a infinito. ###### Proof. Ricordiamo che la capacità di $F_{n+1,m}$ (insieme che in questa dimostrazione indicheremo per comodità con $F_{m}$) equivale per definizione all’integrale di Dirichlet della funzione $u_{n+1,m}$ (che indicheremo per semplicità $u_{m}$), funzione armonica su $R\setminus F_{m}$, $u_{m}\in\mathbb{M}_{\Delta}(R)$ e $u_{m}|_{\partial F_{m}}=1$. Grazie al lemma 3.51, sappiamo che esiste una funzione $f_{m}\in\mathbb{M}_{\Delta}(R)$, $f_{m}=1$ su $\displaystyle E_{n+1,m}\equiv\overline{U_{n+1}\cap K_{m}^{C}}$ Infatti per definizione: $\displaystyle U_{n}\equiv\\{z\in R\ t.c.\ G(z,z_{0})>r_{n}\\}$ e per 4.28, $G(\cdot,z_{0})|_{\Delta}=0$, cioè $U_{n}\cap\Delta=\emptyset\ \forall n$. Possiamo scegliere questa funzione armonica su $R\setminus E_{n+1,m}$, infatti definiamo $W_{n+1,m,p}$ con $p>m$ l’insieme $\displaystyle W_{n+1,m,p}\equiv U_{n+1}\cap(\overline{K_{p}}\setminus K_{m})$ e chiamiamo $\pi_{m,p}$ l’operatore definito in 3.55 rispetto all’insieme $W_{n+1,m,p}$. La successione $\\{w_{m,p}\equiv\pi_{m,p}(f_{m})\\}$ è una successione di funzioni armoniche su $R\setminus E_{n+1,m}$, appartenenti all’insieme $\mathbb{M}_{\Delta}(R)$. Grazie al punto (2) del teorema 3.55, si ha che: $\displaystyle D(w_{m,p+q}-w_{m,p},w_{m,p})=0\ \ \Rightarrow\ \ D(w_{m,p+q}-w_{m,p})=D(w_{m,p})-D(w_{m,p+q})$ Con un ragionamento analogo a quello esposto in 3.35, otteniamo che la successione $\\{w_{m.p}\\}$ è D-cauchy rispetto all’indice $p$. Osserviamo che la successione $\\{w_{m,p}\\}$ è una successione crescente in $p$, infatti dato che $w_{m,p}=1$ su $W_{n+1,m,p}$ e $w_{m,p}|_{\Delta}=0$, grazie all’osservazione 3.60 $w_{m,p}\leq 1$ su $R$, quindi in particolare su $W_{n+1,m,p+1}$ si ha che $w_{m,p}\leq w_{m,p+1}$. Sempre grazie alla proposizione 3.60, otteniamo che questa disuguaglianza è valida su tutto $R$. Quindi la successione $w_{m,p}$ converge monotonamente rispetto a $p$ a una funzione $w_{m}$ armonica su $R\setminus E_{n+1,m}$ per il principio di Harnack e identicamente uguale a $1$ su $E_{n+1,m}$ per costruzione. È facile verificare che la convergenza è locale uniforme su $R$. Dimostriamo che: $\displaystyle 0=BD-\lim_{m}w_{m}$ Il punto (2) di 3.55 garantisce che per ogni $p$: $\displaystyle D_{R}(w_{m+q,\ p}-w_{m},w_{m+q,\ p})=0$ infatti $w_{m+q,\ p}-w_{m}=0$ su $W_{n+1,m,p}$ e questa funzione è in $\mathbb{M}_{\Delta}(R)$. Passando al limite su $p$, otteniamo che: $\displaystyle 0=D_{R}(w_{m+q}-w_{m},w_{m+q})=D_{R}(w_{m+q})-D_{R}(w_{m+q},w_{m})$ da cui: $\displaystyle D_{R}(w_{m+q}-w_{m})=D_{R}(w_{m})-D_{R}(w_{m+q})$ quindi grazie a un ragionamento simile a quello riportato in 3.35, otteniamo che la successione $\\{w_{m}\\}$ è D-cauchy. La successione $w_{n}$ inoltre è una successione decrescente di funzioni. Infatti per ogni $p$ abbiamo che $w_{m+1,p}\leq 1$ sull’insieme $E_{n+1,m}$ grazie alla proposizione 3.60, e quindi $w_{m+1,p}\leq w_{m}$ sull’insieme $E_{n+1,m}$. La disuguaglianza vale su tutta la varietà $R$ (quindi per continuità anche su tutta $R^{*}$) grazie al teorema 3.58. Questo garantisce che esiste il limite $\displaystyle w=BD-\lim_{m}w_{m}$ Grazie al principio di Harnack, questa funzione è armonica su tutta $R$, e per l’osservazione 3.36 $w\in\mathbb{M}_{\Delta}(R)$. Grazie al principio del massimo 3.57, otteniamo che $w=0$, quindi: $\displaystyle\lim_{m}D_{R}(w_{m})=0$ Questo risultato è utile in quanto $\displaystyle D_{R}(u_{m})\leq D_{R}(w_{m})$ dove questa osservazione segue dal teorema 3.55. Consideriamo infatti l’insieme $F_{n+1,m}$ come l’insieme $K$ del teorema. La proiezione della funzione $w_{m}$ su questo insieme è la funzione $u_{m}$ dato che $(w_{m}-u_{m})|_{F_{n+1,m}}=0$ e questa funzione è in $\mathbb{M}_{\Delta}(R)$. Quindi grazie al punto (3) del teorema abbiamo che: $\displaystyle D_{R}(w_{m})=D_{R}(u_{m})+D_{R}(w_{m}-u_{m})$ da cui la tesi. ∎ #### 4.4.4 Il diametro transfinito $\rho(\Xi_{n})=\infty$ Mettendo assieme le varie proposizioni e lemmi visti fino ad ora, siamo pronti per dimostrare che: ###### Proposizione 4.63. Per ogni $n$, $\tau(\Xi_{n})=\rho(\Xi_{n})=\infty$. Questo implica che ogni compatto contenuto nel bordo irregolare $\Xi$ ha diametro transfinito infinito. ###### Proof. Grazie alla proposizione 4.37 sappiamo che $\displaystyle\tau(X)\geq\rho(X)$ per ogni insieme $X\subset R^{*}$. Quindi basta dimostrare che $\rho(\Xi_{n})=\infty$. La relazione 4.8 combinata con la proposizione 4.61 ci permette di scrivere: $\displaystyle\rho(\Xi_{n})\geq\sigma_{n}^{2}\rho(F_{n+1,m})\geq\sigma_{n}^{2}\epsilon(F_{n+1,m})$ (4.15) Questo assicura che se $\lim_{m}\epsilon(F_{n+1,m})=\infty$ otteniamo la tesi. Per dimostrare questa uguaglianza, ricordiamo che $F_{n+1,m}$ è una sottovarietà di codimensione $1$ di $R$ con bordo liscio, quindi grazie a 4.60: $\displaystyle\epsilon(F_{n+1,m})=Cap(F_{n+1,m})^{-1}$ e grazie alla proposizione 4.62, che assicura che: $\displaystyle\lim_{m}Cap(F_{n+1,m})=0$ otteniamo la tesi. ∎ #### 4.4.5 Funzioni armoniche che tendono a infinito sul bordo di R Utilizzando in fatto che $\tau(\Xi_{n})=\infty$ per ogni $n$, in questa sezione costruiremo una funzione armonica su $R$ che converge a infinito su $\Xi$ (in un senso che analizzeremo meglio in seguito). ###### Proposizione 4.64. Esiste una funzione armonica $E:R\to\mathbb{R}$ tale che $E(p)=\infty$ per ogni $p\in\Xi$. ###### Proof. La dimostrazione è costruttiva. Dato che $\tau(\Xi_{n})=\infty$ per ogni $n$, per definizione di $\tau$ 404040vedi 4.34 esiste una successione di interi $n_{k}$ tale che $\displaystyle\tau_{n_{k}}(\Xi_{n})\geq 2^{k}$ cioè sempre per definizione di $\tau_{n}$, esistono $n_{k}$ punti $p_{1},\cdots,p_{n_{k}}\in\Xi_{n}$ tali che $\displaystyle\inf_{p\in\Xi_{n}}\sum_{i=1}^{n_{k}}G(p,p_{i})>2^{k}n_{k}$ definiamo la funzione $\displaystyle E_{n,k}(x)\equiv\sum_{i=1}^{n_{k}}\frac{1}{2^{k}n_{k}}G(x,p_{n_{k}})$ Osserviamo che $E_{n,k}(p)>1$ per ogni $p\in\Xi_{n}$. Se definiamo $\displaystyle E_{n}(x)\equiv\sum_{k=1}^{\infty}E_{n,k}(x)$ osserviamo che in ogni punto di $R$ la somma converge. Infatti per come è definita $E_{n,k}$, $E_{n}$ è una combinazione convessa delle funzioni $G(x,p_{n,k})$, quindi una serie di funzioni armoniche positive. Per il principio di Harnack, questa successione o converge localmente uniformemente in $R$ o diverge in tutti i punti di $R$. Consideriamo $z_{0}\in R$ un punto qualsiasi. Sia $V$ un suo intorno relativamente compatto. Sappiamo che $\displaystyle\sup_{x\in R^{*}\setminus V}G(x,z_{0})\leq\sup_{x\in\partial V}G(x,z_{0})<\infty$ questo vuol dire che la successione $G(z_{0},p_{k})$ è uniformemente limitata, quindi qualunque sua combinazione convessa converge 414141e dal principio di Harnack converge localmente uniformemente a una funzione armonica. Un modo meno banale ma più veloce di dimostrare la stessa cosa, è osservare che $G(z_{0},\cdot)$ è una funzione continua su $\Gamma$ insieme compatto, quindi assume massimo finito su questo insieme. Inoltre osserviamo che per ogni $p\in\Xi_{n}$, $E_{n}(p)=\infty$. Per definire la funzione $E$ ripetiamo un ragionamento simile a quello fin qui esposto, in particolare definiamo: $\displaystyle E(x)\equiv\sum_{n=1}^{\infty}\frac{1}{2^{n}}E_{n}(x)$ anche in questo caso $E$ è una combinazione convessa di funzioni della forma $G(\cdot,p)$ con $p\in\Xi$ 424242$p$ può stare in un $\Xi_{n}$ qualsiasi, quindi in generale $p\in\Xi$, cioè esiste una successione di numeri positivi $t_{k}$ con somma $\sum_{k=1}^{\infty}t_{k}=1$ e una successione di punti $p_{k}$ in $\Xi$ tale che: $\displaystyle E(x)=\sum_{k=1}^{\infty}t_{k}G(x,p_{k})$ (4.16) Grazie al principio di Harnack, $E(x)$ è una funzione armonica su $R$ e per ogni $p\in\Xi$, $E(p)=\infty$. ∎ Il fatto che $E(p)=\infty$ per ogni $p\in\Xi$ è assolutamente inutile se non consideriamo che: ###### Osservazione 4.65. La funzione $E$ è semicontinua inferiormente su $R^{*}$, nel senso che $\displaystyle\liminf_{p\to p_{0}}E(p)\geq E(p_{0})$ per ogni $p$ in $R^{*}$. Ricordiamo la definizione di $\liminf$ in una topologia non I numerabile: $\displaystyle\liminf_{p\to p_{0}}f(p)=L\ \Longleftrightarrow\ \forall\epsilon>0,\ \exists V(p_{0})\ t.c.\ E(p)>L-\epsilon\ \forall p\in V(p_{0})\ \wedge$ $\displaystyle\wedge\ \forall\epsilon>0,\ \forall V(p_{0}),\ \exists p\in V(p_{0})\ t.c.\ E(p)<L+\epsilon$ ###### Proof. Se $p\in R\cup\Delta$, la dimostrazione è ovvia essendo $E$ continua su $R$, positiva ovunque e uguale a $0$ su $\Delta$ 434343infatti tutte le funzioni $G(\cdot,p)$ si annullano sul bordo armonico $\Delta$, vedi 4.28. Data la positività delle funzioni di Green, abbiamo anche che: $\displaystyle\inf_{q\in V\setminus\\{q_{0}\\}}\sum_{i=1}^{\infty}\alpha_{i}G(q,p_{i})\geq\inf_{q\in V\setminus\\{q_{0}\\}}\sum_{i=1}^{N}\alpha_{i}G(q,p_{i})$ per ogni scelta di $\alpha_{i}\geq 0$ e per ogni interno $N$. Dato che le funzioni di Green sono semicontinue inferiormente 444444infatti se $q_{0}\neq p_{i}$, questo segue dalla continuità della funzione $G$, altrimenti dal fatto che $\lim_{V\to x}G(y,x)=\infty$, vedi sezione 1.7.4, cioè: $\displaystyle\lim_{V\to q_{0}}\inf_{V\setminus\\{q_{0}\\}}G(q,p_{i})=G(q_{0},p_{i})$ possiamo osservare che: $\displaystyle\lim_{V\to q_{0}}\inf_{q\in V\setminus\\{q_{0}\\}}\sum_{i=1}^{\infty}\alpha_{i}G(q,p_{i})\geq\lim_{V\to q_{0}}\inf_{q\in V\setminus\\{q_{0}\\}}\sum_{i=1}^{N}\alpha_{i}G(q,p_{i})=\sum_{i=1}^{N}\alpha_{i}G(q,p_{i})$ visto che la relazione vale per ogni $n$, si ha la tesi. ∎ Questa osservazione garantisce che per $p\to p_{0},\ p\in R$, $E(p)\to\infty$. Riassumendo, abbiamo dimostrato l’esistenza di una funzione $E:R^{*}\to\mathbb{R}\cup\\{\infty\\}$ continua su $R$ e semicontinua inferiormente su $R^{*}$, tale che $E(p)=\infty$ per ogni $p\in\Xi$. Ora ci concentriamo sul dimostrare altre proprietà di questa funzione, come ad esempio caratterizzare l’integrale di Dirichlet $D_{R}(E\curlywedge c)$ per $c>0$ 454545come avevamo fatto per le funzion di Green $G^{*}(\cdot,p)$. ###### Proposizione 4.66. Per ogni $c>0$ vale che $\displaystyle D_{R}(E(\cdot)\curlywedge c)\leq c$ ###### Proof. Siano $t_{k}$ e $p_{k}$ tali che valga la relazione 4.16. Definiamo la successione di funzioni $\displaystyle\psi_{n}(p)\equiv\sum_{k=1}^{n}t_{k}G(p,p_{k})$ cioè la successione delle somme parziali che definiscono $E(\cdot)$. Grazie alle considerazioni precedenti, sappiamo che $\psi_{n}$ converge localmente uniformemente a $E$, e se dimostriamo che: $\displaystyle D_{R}(\psi_{n}\curlywedge c)\leq c$ allora grazie al teorema 3.24 464646e all’osservazione 3.26 abbiamo la tesi. Resta da dimostrare la parte tecnica del teorema, cioè l’ultima relazione. Fissiamo $n$ e definiamo per comodità $\psi_{n}(p)\equiv\psi(p)$ 474747fino a quando non ci può essere confusione con l’indice $n$. Sia $\alpha$ un valore regolare per la funzione $\psi$. Definiamo la quantità: $\displaystyle L(\alpha)\equiv\sum_{j=1}^{n}\int_{\psi=\alpha}\left|\ast dG(\cdot,p_{j})\right|=\sum_{j=1}^{m}\int_{\phi(\partial\Omega)}\left|\frac{\partial G(\cdot,p_{j})}{\partial x}(y)\right|\sqrt{\left|g\right|}dy^{1}\cdots dy^{m-1}$ dove $(x,y^{1}\cdots,y^{m-1})$ sono coordinate per la sottovarietà $\psi=\alpha$ dove $\\{\psi=\alpha\\}=\\{y^{m}=0\\}$ e dove la metrica assume la forma particolare 484848cioè per coordinatizzare un intorno di $\\{\psi=\alpha\\}$ scegliamo come coordinate $x$ e altre tutte ortogonali a $x$, possibile grazie al fatto che $\alpha$ è un valore regolare per la funzione $x$ $\displaystyle g=\begin{bmatrix}1&0\\\ 0&A(y)\end{bmatrix}$ Ricordando che per ogni $n$-upla di numeri positivi $\displaystyle\left(\sum_{i=1}^{n}a_{i}\right)^{2}\leq n\sum_{i=1}^{n}a_{i}^{2}$ e la disuguaglianza di Schwartz, cioè: $\displaystyle\left(\int\left|f\right|gdx\right)^{2}\leq\int f^{2}dx\int g^{2}dx$ abbiamo che: $\displaystyle L(\alpha)^{2}\leq$ $\displaystyle\leq n\sum_{j=1}^{m}\int_{\phi(\partial\Omega)}\left(\frac{\partial G(\cdot,p_{j})}{\partial x}(y)\right)^{2}\sqrt{\left|g\right|}dy^{1}\cdots dy^{m-1}\int_{x=\alpha}\sqrt{\left|g\right|}dy^{1}\cdots dy^{m-1}\leq$ $\displaystyle\leq n\sum_{j=1}^{n}\int_{\psi=\alpha}\left|\nabla G(\cdot,p_{j})\right|^{2}\sqrt{\left|g\right|}dy^{1}\cdots dy^{m-1}\int_{\psi=\alpha}\ast dx$ Dimostreremo nel lemma seguente (lemma 4.67 e osservazione seguente) che vale la proprietà: $\displaystyle D_{R}(\psi\curlywedge\alpha)=\alpha\int_{\psi=\alpha}\ast dx$ (4.17) Consideriamo due numeri reali $c,\ c^{\prime}$ con $0<c<\alpha<c^{\prime}$ dove $c$ e $c^{\prime}$ sono valori regolari della funzione $\psi$. Allora vale che: $\displaystyle\int_{\psi=\alpha}\ast dx=\alpha D_{R}(\psi\curlywedge\alpha)\leq c^{\prime}D_{R}(\psi\curlywedge c^{\prime})$ quindi $\displaystyle L(\alpha)\leq nc^{\prime}D_{R}(\psi\curlywedge c^{\prime})\sum_{j=1}^{n}\int_{\psi=\alpha}\left|\nabla G(\cdot,p_{j})\right|^{2}\sqrt{\left|g\right|}dy^{1}\cdots dy^{m-1}$ Inoltre dato che l’insieme dei valori non regolari per $\psi$ ha misura nulla (vedi teorema di Sard), vale che: $\displaystyle\int_{c}^{c^{\prime}}d\alpha\sum_{j=1}^{n}\int_{\psi=\alpha}\left|\nabla G(\cdot,p_{j})\right|^{2}\sqrt{\left|g\right|}dy^{1}\cdots dy^{m-1}=\sum_{j=1}^{n}\int_{c<\psi<c^{\prime}}\left|\nabla G(\cdot,p_{j})\right|^{2}dV\leq$ $\displaystyle\leq\sum_{j=1}^{n}\int_{\psi<c^{\prime}}\left|\nabla G(\cdot,p_{j})\right|^{2}dV$ Per definizione della funzione $\psi$, se $\psi(z)<c^{\prime}$, allora necessariamente per ogni $j=1\cdots n$ $\displaystyle G(z,p_{j})<\frac{c^{\prime}}{t_{j}}$ quindi grazie alla proprietà (5) di 4.28, si la stima sull’integrale di Dirichlet: $\displaystyle\int_{\psi<c^{\prime}}\left|\nabla G(\cdot,p_{j})\right|^{2}dV\leq D_{R}(G(\cdot,p_{j})\curlywedge c^{\prime}t_{j}^{-1})\leq\frac{c^{\prime}}{t_{j}}$ Da questa stima, ricordando che $\psi\curlywedge c\in\mathbb{M}_{\Delta}(R)$, possiamo ottenere che: $\displaystyle\int_{c<\psi<c^{\prime}}L(\alpha)^{2}d\alpha\leq nc^{\prime}D_{R}(\psi\curlywedge c^{\prime})\sum_{j=1}^{n}\frac{c^{\prime}}{t_{j}}<\infty$ Il ragionamento fino a qui discusso ha l’utilità di dimostrare (grazie al teorema di Fubini) che a meno di un insieme di $\alpha$ di misura nulla rispetto alla misura di Lebesgue su $\mathbb{R}$, vale che: $\displaystyle\int_{\psi=\alpha}\left|\ast dG(\cdot,p_{j})\right|<\infty$ D’ora in avanti considereremo valori di $\alpha$ regolari per la funzione $\psi$ e per i quali vale la relazione appena scritta per ogni $j=1\cdots,n$ 494949grazie alle considerazioni fatte fino ad ora, risulta evidente che questo insieme abbia complementare di misura nulla in $\mathbb{R}$, quindi sia denso in $\mathbb{R}$. Grazie alla relazione 4.17 e alla definizione della funzione $\psi$, sappiamo che: $\displaystyle D_{R}(\psi\curlywedge\alpha)=\alpha\int_{\psi=\alpha}\ast dx=\alpha\sum_{k=1}^{n}t_{k}\int_{\psi_{\alpha}}\ast dG(\cdot,p_{k})$ (4.18) Sia $c_{k}>\alpha/t_{k}$ un valore regolare per la funzione $G(\cdot,p_{k})$. Grazie alla positività di tutte le funzioni $G(\cdot,p_{k})$, abbiamo che: $\displaystyle K\equiv\\{G(\cdot,p_{k})\geq c_{k}\\}\subset K^{\prime}\equiv\\{\psi(\cdot)\geq\alpha\\}$ applicando ancora il lemma seguente (lemma 4.67) alla funzione $(G(\cdot,p_{k})\curlywedge c_{k})/c_{k})$, otteniamo con l’aiuto della proposizione 4.28 $\displaystyle\int_{\psi=\alpha}\ast d(c_{k}^{-1}G(\cdot,p_{k}))=\int_{\partial K^{\prime}}\ast d(c_{k}^{-1}G(\cdot,p_{k}))=\int_{\partial K}\ast d(c_{k}^{-1}G(\cdot,p_{k}))=$ $\displaystyle=\int_{\partial K}\ast d(c_{k}^{-1}(G(\cdot,p_{k})\curlywedge c_{k}))=D_{R}(c_{k}^{-1}(G(\cdot,p_{k})\curlywedge c_{k}))\leq c_{k}^{-2}c_{k}$ e quindi: $\displaystyle\int_{\psi=\alpha}\ast dG(\cdot,p_{k})\leq 1$ da cui utilizzando 4.18: $\displaystyle D_{R}(\psi\curlywedge\alpha)\leq\alpha\sum_{i=k1}^{n}t_{k}\int_{\psi_{\alpha}}\leq\alpha$ Se $\alpha$ non è un valore regolare per $\psi$, comunque esiste una successione $\alpha_{n}$ che converge dall’alto a $\alpha$ di valori regolari di $\psi$, quindi: $\displaystyle D_{R}(\psi\curlywedge\alpha)\leq D_{R}(\psi\curlywedge\alpha_{n})\leq\alpha_{n}$ dato che la relazione vale $\forall n$, abbiamo la tesi. ∎ ###### Lemma 4.67. Sia $K$ un’insieme compatto in $R^{*}$ il cui bordo in $R$ sia liscio liscio e tale che $K\cap\Delta=\emptyset$, e sia $u$ una funzione limitata tale che $u\in\mathbb{M}_{\Delta}(R)$, $u|_{K}=1$, $u\in H(R\setminus K)$. Allora si ha che: 1. 1. $D_{R}(u)=-\int_{\partial K}\ast du$ 2. 2. se $K^{\prime}$ è un insieme compatto in $R^{*}$ con bordo liscio, disgiunto da $\Delta$ e la cui parte interna contiene $K$, allora se $\int_{\partial(K^{\prime}\setminus K)}\left|\ast du\right|<\infty$: $\displaystyle D_{R}(u)=-\int_{\partial K}\ast du=-\int_{\partial K^{\prime}}\ast du$ ###### Proof. Sia $R_{n}$ un’esaustione regolare di $R$, definiamo $K_{n}\equiv K\cap\overline{R_{n}}$. Con le notazioni dell’osservazione 3.56, abbiamo che: $\displaystyle\pi_{K_{n}}(\pi_{K_{n+p}}(u))=\pi_{K_{n}}(u)$ per ogni $n$ e $p$ naturali. Grazie al punto (2) del teorema 3.55, si ha che: $\displaystyle D(\pi_{K_{n+p}}(u)-\pi_{K_{n}}(u),\pi_{K_{n}}(u))=0\ \ \Rightarrow$ $\displaystyle\Rightarrow\ \ D(\pi_{K_{n+p}}(u)-\pi_{K_{n}}(u))=D(\pi_{K_{n+p}}(u))-D(\pi_{K_{n}}(u))$ Con un ragionamento analogo a quello esposto in 3.35, otteniamo che la successione $\\{\pi_{K_{n}}(u)\\}$ è D-cauchy. Osserviamo che la successione $\pi_{K_{n}}(u)$ è una successione crescente in $n$, infatti dato che $\pi_{K_{n}}(u)=1$ su $K_{n}$ e $\pi_{K_{n}}(u)|_{\Delta}=0$, grazie all’osservazione 3.60 $\pi_{K_{n}}(u)\leq 1$ su $R$, quindi in particolare su $K_{n+1}$ si ha che $\pi_{K_{n}}(u)\leq\pi_{K_{n+1}}(u)$. Sempre grazie alla proposizione 3.60, otteniamo che questa disuguaglianza è valida su tutto $R$. Quindi la successione $\pi_{K_{n}}(u)$ converge monotonamente a una funzione $f$ armonica su $R\setminus K$ per il principio di Harnack e identicamente uguale a $1$ su $K$ per costruzione. È facile verificare che la convergenza è locale uniforme su $R$. Dato che $(f-u)|_{K}=0$ e $(f-u)|_{\Delta}=0$, per la proposizione 3.60, $f=u$ su $R$, quindi $\displaystyle u=BD-\lim_{n}\pi_{K_{n}}(u)$ Fissato $n$, per ogni $m>n$, definiamo la funzione $v^{(n)}_{m}$ tale che: 1. 1. $v^{(n)}_{m}|_{K_{n}}=1$ 2. 2. $v^{(n)}_{m}|_{R\setminus R_{m}}=0$ 3. 3. $v^{(n)}_{m}\in H(R_{m}\setminus K_{n})$ dalla dimostrazione del teorema 3.53 505050o meglio dal suo adattamento alla dimostrazione di 3.55, si ha che $\displaystyle\pi_{K_{n}}(u)=BD-\lim_{m}v^{(n)}_{m}$ Applicando la formula di Green 3.30 all’insieme $R_{m}\setminus K$ otteniamo che: $\displaystyle D_{R}(v^{(n)}_{m},u)=D_{R_{m}\setminus K}(v^{(n)}_{m},u)=-\int_{\partial K}v_{m}\ast du$ dove il segno $-$ viene dall’orientazione di $\partial K$, contraria a quella di $\partial(R_{m}\setminus K)$. Grazie al fatto che $u|_{K}=1$ e $u|_{R\setminus K}\leq 1$, possiamo dedurre che $\ast du\leq 0$ su $\partial K$, e dato che $v^{(n)}_{m}$ è una successione crescente in $m$ 515151semplice applicazione del principio del massimo, grazie al teorema di convergenza monotona possiamo concludere che: $\displaystyle D_{R}(\pi_{K_{n}}(u),u)=\lim_{n}D_{R}(v^{(n)}_{m},u)=\lim_{n}-\int_{\partial K}v^{(n)}_{m}\ast du=-\int_{\partial K}\pi_{K_{n}}(u)\ast du$ La successione $\pi_{K_{n}}(u)$ è crescente sull’insieme $\partial K_{n}$ e in particolare tende a $1$ su questo insieme, quindi sempre per convergenza monotona: $\displaystyle D_{R}(u)\equiv D_{R}(u,u)=\lim_{n}D_{R}(\pi_{K_{n}}(u),u)=\lim_{n}-\int_{\partial K}\pi_{K_{n}}(u)\ast du=-\int_{\partial K}1\ast du$ Per dimostrare il punto (3), sfruttiamo il lemma 3.51. Sia $f\in\mathbb{M}_{\Delta}(R)$ tale che $f_{K^{\prime}}=1$. Allora per definizione di $\mathbb{M}_{\Delta}(R)$, esiste una successione di funzioni $f_{n}\in\mathbb{M}_{0}(R)$ tale che $f=BD-\lim_{n}f_{n}$. Quindi vale che: $\displaystyle 0=D_{K^{\prime}\setminus K}(1,u)=D_{K^{\prime}\setminus K}(f,u)=\lim_{n}D_{K^{\prime}\setminus K}(f_{n},u)$ Sia $S_{n}\equiv supp(f_{n})$ 525252compatto in $R$ per definizione di $\mathbb{M}_{0}(R)$, allora: $\displaystyle D_{K^{\prime}\setminus K}(f_{n},u)=D_{(K^{\prime}\setminus K)\cap S_{n}}(f_{n},u)$ applicando la formula di Green 3.30 otteniamo: $\displaystyle D_{(K^{\prime}\setminus K)\cap S}(f_{n},u)=\int_{\partial(K^{\prime}\setminus K)}f_{n}\ast du$ Per ipotesi, sappiamo che $\int_{\partial(K^{\prime}\setminus K)}\left|\ast du\right|<\infty$, quindi per uniforme limitatezza della successione $f_{n}$ vale anche che $\int_{\partial(K^{\prime}\setminus K)}\left|f_{n}\right|\left|\ast du\right|<M<\infty$ dove $M$ è indipendente da $n$. Quindi possiamo applicare il teorema di convergenza dominata e ottenere che: $\displaystyle\lim_{n}\int_{\partial(K^{\prime}\setminus K)}f_{n}\ast du=\int_{\partial(K^{\prime}\setminus K)}\ast du$ Riassumendo, abbiamo ottenuto: $\displaystyle 0=\int_{\partial(K^{\prime}\setminus K)}\ast du=\int_{\partial K^{\prime}}\ast du-\int_{\partial K}\ast du$ ∎ Riassumendo abbiamo dimostrato l’esistenza di funzioni con queste caratteristiche: ###### Proposizione 4.68. Data una varietà $R$ iperbolica irregolare ($\Xi(R)\neq\emptyset$), esiste una funzione armonica $E:R\to[0,\infty)$ tale che: 1. 1. $E\in H(R)$ 2. 2. $E(\cdot)$ è la combinazione convessa di funzioni $G(p_{i},\cdot)$ dove $p_{i}\in\Xi$ 3. 3. $E(z)=\infty$ se $z\in\Xi$, $E(z)=0$ se $z\in\Delta$ 4. 4. $E(z)$ è semicontinua inferiormente su $R^{*}$ 5. 5. $D_{R}(E(z)\curlywedge c)\leq c$ per ogni $c>0$ #### 4.4.6 Varietà iperboliche irregolari Data una varietà iperbolica $R$, sappiamo che esiste una funzione di Green $G(\cdot,\cdot):R^{*}\times R^{*}\to\mathbb{R}\cup\\{\infty\\}$ con le proprietà descritte nella proposizione 4.28. L’insieme $\Xi$ è definito come il sottoinsieme di $\Gamma$ tale che $\displaystyle\Xi\equiv\\{p\ t.c.\ G(z,p)>0\ z\in R\\}$ dove la definizione non dipende dalla scelta del punto $z$. In funzione del fatto che $\Xi$ sia vuoto o meno definiamo una varietà iperbolica regolare o irregolare ###### Definizione 4.69. Data una varietà iperbolica $R$, diciamo che è regolare se $\Xi=\emptyset$, altrimento diciamo che è irregolare. Una facile caratterizzazione delle varietà regolari è la seguente: ###### Proposizione 4.70. Sia $G(z,\cdot)$ un nucleo di Green su $R$ iperbolica. $R$ è regolare se e solo se per ogni $c>0$ l’insieme $\displaystyle A_{c}\equiv\\{p\in R\ t.c.\ G(z.p)\geq c\\}$ è compatto in $R$. ###### Proof. La dimostrazione di questa affermazione è abbastanza immediata. Supponiamo che gli insiemi $A_{c}$ siano compatti. Allora per ogni $p\in\Gamma$, sappiamo che: $\displaystyle G(z,p)=p(G(z,\cdot))=p(G(z,\cdot)(1-\lambda_{c}))$ dove $\lambda_{c}$ è una funzione liscia a supporto compatto tale che $\lambda_{c}=1$ sull’insieme $A_{c}$. Grazie all’osservazione 3.48 vale la seconda uguaglianza scritta in formula, e inoltre grazie all’osservazione 4.21 possiamo concludere che la funzione $G(z,\cdot)(1-\lambda_{c})$ è compresa tra $0$ e $c$. Quindi per continuità, anche: $\displaystyle 0\leq p(G(z,\cdot)(1-\lambda_{c}))=G(z,p)\leq c$ data l’arbitrarietà di $c>0$, abbiamo che se $A_{c}$ è compatto per ogni $c$, allora fissato $z$, $G(z,p)=0$ per ogni $p\in\Gamma$, quindi $\Xi=\emptyset$, cioè la varietà iperbolica è regolare. Supponiamo ora che esiste un insieme $A_{c}$ non compatto, allora esiste una successione $p_{n}\in R$, $p_{n}\to\infty$ tale che $\displaystyle G(z,p_{n})\geq c\ \ \ \forall n$ Consideriamo un qualunque carattere $p$ su $R$ descritto da un ultrafiltro su $z_{n}$. Dall’ultima equazione sappiamo che $\displaystyle G(z,p)\geq c$ da cui $p\in\Xi\neq\emptyset$. ∎ ###### Osservazione 4.71. Per le proprietà della funzione $G$, il fatto che per ogni $c>0$ $A_{c}$ sia compatto, è equivalente a chiedere che per ogni $c$ l’insieme $\partial A_{c}$ sia compatto. Per le varietà iperboliche irregolari, definiamo l’insieme delle successioni irregolari, cioé ###### Definizione 4.72. Data una varietà iperbolica $R$, definiamo: $\displaystyle\Sigma(R)=\\{\\{z_{n}\\}\subset R\ t.c\ z_{n}\to\infty\ \ e\ \liminf_{n}G(z_{0},z_{n})>0\\}$ Grazie a un ragionamento simile a quello riportato nella dimostrazione di 4.28, si ottiene che questa condizione è indipendente dalla scelta di $z_{0}$, quindi $\Sigma(R)$ è ben definito. Inoltre dalla caratterizzazione appena dimostrata, è immediato verificare che $R$ è irregolare se e solo se $\Sigma(R)\neq\emptyset$. Possiamo riformulare la proposizione 4.68 con questa nuova terminologia. In particolare: ###### Proposizione 4.73. Sia $R$ una varietà iperbolica irregolare, e sia $E$ la funzione descritta in 4.68. Allora per ogni successione $\\{z_{n}\\}\in\Sigma(R)$: $\displaystyle\lim_{n\to\infty}E(z_{n})=\infty$ ###### Proof. Questa osservazione segue dalla semicontinuità di $E$ su $\Xi$. Per prima cosa osserviamo che la proprietà appena enunciata è equivalente a: $\displaystyle\lim_{n\to\infty}\inf_{z\in V(z_{0},a,n)}E(z)=\infty$ (4.19) per ogni $a>0$, $a$ tale che $G(z_{0},\cdot)\geq a$ non sia un insieme compatto in $R$, dove $V(z_{0},a,n)\equiv\\{z\in R\ t.c.\ G(z_{0},z)>a\\}\setminus K_{n}$ e $K_{n}$ è un’esaustione di $R$ 535353dalle considerazioni precedenti, sappiamo che questa proprietà è indipendente dalla scelta di $z_{0}\in R$. Infatti, se questo è vero e consideriamo una successione $\\{z_{m}\\}\in\Sigma(R)$, dato che $\liminf_{m}G(z_{0},z_{m})=\lambda>0$ $z_{m}$ appartiene definitivamente a tutti gli insiemi $V(z_{0},\lambda/2,n)$, e per $m$ che tende a infinito, esiste una successsione $n_{m}$ che tende a infinito tale che $z_{n}\in V(z_{0},\lambda/2,n_{m})$. Per dimostrare l’altra implicazione, supponiamo che esista $a>0$ tale che: $\displaystyle\lim_{n\to\infty}\inf_{z\in V(z_{0},a,n)}E(z)=M<\infty$ questo significa che per ogni $n$ esiste un $z_{n}\in V(z_{0},a,n)$ tale che $E(z_{n})\leq M+1$. Per costruzione, la successione $z_{n}$ tende a infinito, e $G(z_{0},z_{n})\geq a$, quindi $\\{z_{n}\\}\in\Sigma(R)$. Questo completa la dimostrazione dell’equivalenza tra le due proprietà. Dato che: $\displaystyle\bigcap_{n}V(z_{0},a,n)=\Xi_{a}\equiv\\{p\in\Gamma\ t.c.\ G(z_{0},p)>a\\}\subset\Xi$ allora l’equazione 4.19 è vera perché $E$ è semicontinua inferiormente sull’insieme $R\cup\Xi$ e $E(\Xi)=\infty$. ∎ #### 4.4.7 Potenziali di Evans su varietà paraboliche Osserviamo che tutti gli spazi $R^{n}$ con $n\geq 3$ dotati della metrica euclidea standard sono iperbolici regolari, e gli stessi spazi privati di un punto qualsiasi sono iperbolici irregolari. Consideriamo ad esempio l’insieme $\mathbb{R}^{3}\setminus\\{(0,0,1)\\}$. Sappiamo che la funzione di Green su $\mathbb{R}^{3}$ è proporzionale all’inverso della distanza: $\displaystyle G(x,y)=C(3)\frac{1}{d(x,y)}$ Si vede facilmente che la stessa funzione ristretta all’insieme $\mathbb{R}^{3}\setminus\\{(0,0,1)\\}$ è ancora un nucleo di Green. Fissiamo il punto $x=(0,0,0)$ e consideriamo la funzione $G(0,y)$. L’insieme $\\{y\in\mathbb{R}^{3}\setminus\\{(0,0,1)\ t.c.\ G(0,y)\geq C(3)/2\\}\\}$ NON è un insieme compatto in $\mathbb{R}^{3}\setminus\\{(0,0,1)\\}$, quindi questa varietà è iperbolica irregolare. Altri esempi di varietà iperboliche irregolari possono essere costruiti togliendo un insieme compatto da una varietà riemanniana parabolica. Illustriamo questo risultato in due proposizioni. ###### Proposizione 4.74. Sia $R$ una varietà Riemanniana e $K$ un compatto che sia chiusura della sua parte interna in $R$. Allora la varietà $R\setminus K$ dotata della metrica di sottospazio è una varietà iperbolica. ###### Proof. Consideriamo un compatto $C$ in $R^{\prime}\equiv R\setminus K$ con bordo regolare, e dimostriamo che la capacità di questo compatto è necessariamente diversa da $0$. A questo scopo consideriamo un altro insieme compatto con bordo regolare $K^{\prime}$ tale che: $\displaystyle K\Subset K^{\prime\circ}\subset K^{\prime}\ \ \ K^{\prime}\cap C=\emptyset$ Per ora trattiamo il caso di $K$ compatto con bordo regolare. Sia per definizione $w$ la funzione armonica in $K^{\prime}\setminus K$ tale che: $\displaystyle w|_{\partial K}=0\ \ \ w|_{\partial K^{\prime}}=1$ Consideriamo un’esaustione regolare $V_{n}$ di $R^{\prime}$ tale che $C^{\prime}\Subset V_{1}$, e sia $v_{n}$ il potenziale armonico della coppia $(C,V_{n})$. Sappiamo che la capacità di $C$ è per definizione l’integrale di Dirichlet del limite di $v_{n}$, ed è nulla se e solo se $v=1$ costantemente. Applicando il principio del massimo sull’insieme $V_{n}\cap K^{\prime}$, otteniamo che per ogni $n$, $v_{n}\leq w$. Infatti sul bordo di $K^{\prime}$, $w=1$ mentre $v_{n}\leq 1$, e sul bordo di $V_{n}$, $v_{n}=0$ mentre $w\geq 0$. Questo garantisce che $\displaystyle v\leq w\ \ \ su\ \ K^{\prime}\setminus K$ quindi $v$ non può essere costante uguale a $1$. Se $K$ non ha bordo liscio, è sufficiente ripetere la costruzione di $w$ con un insieme $K^{\prime\prime}\Subset K$ dal bordo liscio. Tutte le considerazioni fatte si applicano a questo caso senza complicazioni. ∎ ###### Proposizione 4.75. Sia $R$ una varietà parabolica e sia $K$ un insieme compatto che sia chiusura della sua parte interna e abbia bordo regolare. Allora la varietà $R^{\prime}\equiv R\setminus K$ è iperbolica irregolare. Inoltre tutte le successioni $\\{z_{n}\\}\subset R^{\prime}\subset R$ che tendono a infinito in $R$ sono irregolari, appartengono all’insieme $\Sigma(R^{\prime})$, e dato $z_{0}\in R^{\prime}$, la funzione $G(z_{0},\cdot)$ può essere estesa per continuità anche a $\partial K$ e su questo insieme è nulla. ###### Proof. Dalla proposizione precedente sappiamo che $R^{\prime}$ è una varietà iperbolica, quindi ammette nucleo di Green positivo. Sia $z_{0}\in R^{\prime}$ qualsiasi, e consideriamo un compatto con bordo regolare $K^{\prime}\Subset R$ 545454questo insieme NON è compatto in $R^{\prime}$ tale che $K\subset K^{\prime\circ}$ e $z_{0}\in K^{\prime\circ}$. Sia inoltre $K_{n}$ un’esaustione regolare di $R$ con $K^{\prime}\subset K_{1}$. Osserviamo che il nucleo di Green $G(z_{0},\cdot)$ su $R^{\prime}$ è una funzione armonica sull’insieme $R\setminus K^{\prime}$, e su $\partial K^{\prime}$ assume minimo strettamente positivo $\lambda$ 555555$\lambda>0$ perchè il principio del massimo assicura che $G(z_{0},\cdot)$ non può assumere il suo minimo in un punto interno all’insieme di definizione. Allora grazie al principio del massimo possiamo confrontare $G(z_{0},\cdot)$ con $v_{n}$, i potenziali armonici della coppia $(K^{\prime},K_{n})$. Per ogni $n$ si ha infatti che sull’insieme $R\setminus K^{\prime}$: $\displaystyle G(z_{0},\cdot)\geq\lambda v_{n}(\cdot)$ passando al limite e ricordando che $R$ è parabolica (quindi $\lim_{n}v_{n}=1$) otteniamo che su $R\setminus K^{\prime}$: $\displaystyle G(z_{0},\cdot)\geq\lambda$ Questo dimostra anche che qualunque successione tendente a infinito in $R$ è in $\Sigma(R^{\prime})$. Per dimostrare l’ultimo punto, consideriamo il massimo $\Lambda$ della funzione $G(\cdot,z_{0})$ sul bordo di un dominio compatto 565656rispetto alla topologia di $R$ con bordo liscio $C$ tale che $K\subset C$ e $z_{0}\not\in C$. Sia $v$ il potenziale di capacità della coppia $(K,C)$. Grazie al principio del massimo, $G(z_{0},\cdot)\leq\Lambda(1-v(\cdot))$, infatti $G(z_{0},\cdot)$ è costruita come limite crescente di nuclei di Green $G_{n}$ su domini compatti, ed è facile verificare che $\displaystyle G_{n}(z_{0},\cdot)\leq\Lambda(1-v(\cdot))$ passando al limite su $n$ si ottiene la disuguaglianza desiderata. Questo dimostra che $G(z_{0},\cdot)$ tende a zero se l’argomento tende a un punto qualsiasi di $\partial K$. Osserviamo anche che grazie a questa proprietà, tutte le successioni che tendono a $\partial K$ 575757con questo si intendono tutte le successioni di elementi di $R^{\prime}$ tali che per ogni intorno $U$ di $\partial K$, intorno rispetto alla topologia di $R$, esiste $N$ tale che $x_{n}\in U$ per ogni $n\geq N$. Ad esempio tutte le successioni che convergono rispetto alla topologia di $R$ a un punto di $\partial K$ non appartengono a $\Sigma(R^{\prime})$. ∎ Per dimostrare l’esistenza di potenziali di Evans relativi a un qualsiasi dominio compatto con bordo liscio $K$ in $R$ varietà parabolica, sfruttiamo quest’ultima proposizione, l’esistenza delle funzioni $E$ descritte in 4.68 e la proposizione 4.73. ###### Teorema 4.76. Data una varietà parabolica $R$ e un dominio compatto con bordo liscio $K$, esiste un potenziale di Evans su $R$ rispetto a $K$, cioè una funzione $E:R\setminus K^{\circ}\to[0,\infty)$ tale che: 1. 1. $E\in H(R\setminus K)$ 2. 2. $E(z)=0$ se $z\in\partial K$ 3. 3. $E(z_{n})\to\infty$ per ogni successione $z_{n}$ che tende a infinito in $R$. 4. 4. $D_{R}(E\curlywedge c)\leq c$ per ogni $c>0$ ###### Proof. La dimostrazione è semplicemente una raccolta di risultati precedentemente ottenuti. Grazie a quanto appena dimostrato, $R\setminus K$ è una varietà iperbolica irregolare, quindi considerando la funzione $E$ costruita nella proposizione 4.68 e grazie alla proposizione 4.73, otteniamo che $E$ soddisfa (1), (3) e (4). Il punto (2) si ricava considerando che $E$ è una combinazione convessa di nuclei di Green sulla varietà $R\setminus K$, che grazie alla proposizione 4.75 si annullano sull’insieme $\partial K$. ∎ ## Appendix A Glossario $A^{C}$ | complementare dell’insieme $A$ ---|--- $\overline{A}$ | chiusura topologica dell’insieme $A$ d$\lambda$, d$\lambda^{m}$ | misura di Lebesgue su $\mathbb{R}$ o su $\mathbb{R}^{m}$ $(R,g)$ | varietà Riemanniana $R$ con tensore metrico $g$ $\Omega$ | dominio (insieme aperto connesso) in $R$ $\sqrt{\left|g\right|}$ | radice quadrata del determinante di $g$ $dV$ | $dV=\sqrt{\left|g\right|}dx^{1}\cdots dx^{n}$ elemento di volume su $(R,g)$ $L^{2}(\Omega)$ | spazio delle funzioni a quadrato integrabile su $\Omega$ $\mathcal{L}^{2}(\Omega)$ | spazio delle $1$-forme a quadrato integrabile su $\Omega$ | (vedi sezione 1.3) $\left\|f\right\|_{\infty}$ | norma del sup della funzione $f$ $D_{\Omega}(f)$ | integrale di Dirichlet della funzione $f$ sull’insieme $\Omega$. $\left\|f\right\|_{R}$ | norma di $f$ nell’algebra di Royden $\mathbb{M}(R)$ | (vedi teorema 3.23) $\mathbb{M}(R)$ | Algebra di Royden su $R$ (vedi definizione 3.7) $\mathbb{M}_{0}(R)$ | Insieme delle funzioni a supporto compatto in $\mathbb{M}(R)$ $\mathbb{M}_{\Delta}(R)$ | Completamento di $\mathbb{M}_{0}(R)$ nella topologia $BD$ | (vedi paragrafo 3.2.5) $R^{*}$ | Compattificazione di Royden di $R$ (vedi definizione 3.37) $\Gamma$ | $\Gamma=R^{*}\setminus R$ è il bordo di $R^{*}$ (vedi definizione 3.47) $\Delta$ | bordo armonico di $R$ (vedi definizione 3.50) $\Xi$ | bordo irregolare di $R$ (vedi definizione 4.27) $f\ast h$ | convoluzione della funzione $f$ con $h$ (vedi sezione 1.5) $\int f\ast du$ | vedi definizione 1.10 $H(\Omega)$ | spazio delle funzioni armoniche su $\Omega$ $HP(\Omega)$ | spazio delle funzioni armoniche positive su $\Omega$ $HD(\Omega)$ | spazio delle funzioni armoniche con $D_{\Omega}(u)<\infty$ $C(\Omega)=C(\Omega,\mathbb{R})$ | spazio delle funzioni continue a valori reali su $\Omega$ $C(\overline{\Omega})=C(\overline{\Omega},\mathbb{R})$ | spazio delle funzioni continue definite in un intorno di $\overline{\Omega}$ $\text{Cap}(K,\Omega)$ | capacità della coppia $(K,\Omega)$ (vedi definizione 4.1) $\text{Cap}(K)$ | capacità di $K$ (vedi definizione 4.6) ## Bibliography * [1] * [A] Arkhangel’skii, A.V.; Pontryagin, L.S. General Topology I Springer-Verlag, New York (1990) ISBN 3-540-18178-4 * [ABR] Axler, Bourdon, Ramey Harmonic Function Theory, II edition Springer Verlag http://www.axler.net/HFT.pdfhttp://www.axler.net/HFT.pdf * [C1] do Carmo Riemannian geometry Birkhäuser * [CSL] Chang, John; Sario, Leo Royden’s Algebra on Riemannian Spaces Math. Scand. 28 (1971), 139-158 http://www.mscand.dk/article.php?id=2001http://www.mscand.dk/article.php?id=2001 * [C2] Conlon, Lawrence Differentiable Manifolds, Second edition Birkhäuser * [C3] Conway, John A First Course in Functional Analysis Springer-Verlag * [D] Dugundji, James Topology Allyn and Bacon, Inc. * [F1] Folland, G. B. Real Analysis, Modern techniques and their applications (II ed.) Wiley-Interscience Publication * [F2] Friedman, Avner Partial Differential Equations Holt, Rinehart and Winston Inc. * [G1] Gray, Alfred Tubes Addison Wesley * [G2] Grigor’yan, Alexander Analytic and Geometric Background of Recurrence and non-explosion of the Brownian Motion on Riemannian Manifolds Bullettin of the american mathematical society * [GP] Guillemin, Victor; Pollack, Alan Differential Topology Prentice Hall * [GT] Gilbard, Trudinger Elliptic partial differential equations Springer * [H1] Helms Potential theory Springer * [H2] Hervé, Rose-Marie Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel http://archive.numdam.org/article/AIF_1962__12__415_0.pdfhttp://archive.numdam.org/article/AIF_1962__12__415_0.pdf * [H3] Hirsch, Morris W. Differential topology Springer-Verlag * [L] Lang, Serge Real Analysis, II ed Addison Wesley * [LT] Li, Peter; Tam, Luen-Fai Symmetric Green’s Functions on complete manifolds American Journal of Mathematics, Vol 109, n6, Dic 1987, pp. 1129-1154 http://www.jstor.org/stable/2374588?origin=JSTOR-pdfstable link Jstor * [M1] Magginson, Robert E. An Introduction to Banach Space Theory Graduate texts in mathematics disponibile in anteprima limitata su http://books.google.it/http://books.google.it/books?id=fD- GeCsqoqkC&printsec=frontcoverdq=megginson+banach * [MZ] Malỳ, Jean; Ziemer, William P. Fine Regularity of Solutions of Elliptic Partial Differential Equations American Mathematical Society disponibile in anteprima limitata su http://books.google.it/http://books.google.it/ * [M2] Munkres, James R. Analysis on Manifolds Addison-Wesley Pubblishing Company * [M3] Munkres, James R. Elementary differential topology Princeton University Press * [N] Nagy, Gabriel “Strange” Limits Notes from the Functional Analysis Course (Fall 07 - Spring 08) www.math.ksu.edu/ nagy/func-an-2007-2008/strange-limits- HB.pdfhttp://www.math.ksu.edu/ nagy/func-an-2007-2008/strange-limits-HB.pdf * [P1] Petersen, Peter Riemannian Geometry, II ed Springer * [PSR] Pigola; Setti; Rigoli Vanishing and Finiteness Results in Geometric Analysis Birkhäuser * [R1] Royden, H. L. Real Analysis The Macmillan Company * [R2] Rudin, Walter Functional Analysis, II ed. McGraw-Hill * [R3] Rudin, Walter Principles of Mathematical Analysis, III ed McGraw-Hill * [R4] Rudin, Walter Real and Complex Analysis McGraw-Hill * [SN] Sario, L.; Nakai, M. Classification theory of Riemann Surfaces Springer Verlag * [SY] Schoen, R.; Yau, S.T. Lectures on Differential Geometry International Press * [S1] Springer, George Introduction to Riemann Surfaces Addison-Wesley Pubblishing Company * [S2] Strauss, Walter A. Partial Differential Equations An Introduction John Wiley and Sons * [T] Tsuji, M. Potential theory in modern function theory (2ed) Chelsea * [Z] Ziemer, William P.Weakly differentiable functions. Sobolev spaces and functions of bounded variation Springer Verlag ### Ringraziamenti… e non solo I miei ringraziamenti vanno innanzitutto al prof. Alberto Giulio Setti e al prof. Stefano Pigola. In particolare ringrazio Alberto soprattutto per i consigli che mi ha dato e per l’estrema gentilezza, pazienza e diponibilità dimostrate, ben oltre quanto richiesto e non solo dal punto di vista accademico. Ringrazio anche il prof. Wolfhard Hansen (University of Bielefeld) per la disponibilità e per i suggerimenti dati riguardo alla teoria del potenziale. Vuoi studiare a Princeton? - … Ovviamente ringrazio tutti i miei amici dell’università, la compagnia del $4^{\circ}$ piano e della Norvegia, nanetti compresi. Non vedo l’ora del prossimo ciclo! C’è uno spiffero… Non posso dimenticare di ringraziare Madesimo, che mi ha fornito un posto idilliaco dove pensare e scrivere questa tesi, e non posso dimenticare tutti gli amici che mi hanno accompagnato in questa impresa. Ai larici a mezzogiorno? Un ringraziamento importante va a Internet e a tutti quelli che non credono nel diritto d’autore (senza i quali non avrei potuto studiare tutto quello che ho studiato), da Wikipediahttp://www.wikipedia.org a Ubuntu - Linuxhttp://www.ubuntulinux.org passando per l’indispensabile Gigapediahttp://www.gigapedia.org. A journey of a thousand sites begins with a single click. Ringrazio l’Unione Europea, per quello che è ma soprattutto per quello che potrebbe essere. Ringrazio in particolare tutti quelli che ci hanno lavorato seriamente, perché hanno aperto una strada che ora noi abbiamo la responsabilità di portare avanti e soprattutto perché ci hanno lasciato qualcosa che sprecare sarebbe una follia. European Union - United in diversity Un pensiero speciale va a Lucy, e a suo padre scomparso di recente. Cosa, cosa? Ringrazio Homer, Marge, Bart, Lisa e Maggie, che mi hanno divertito, intrattenuto, e dato da pensare. Farfalla vendetta! Ringrazio Kathryn, Seven, Benjamin, Jean-Luc e rispettive compagnie perché mi hanno fatto sognare. I look forward to it. Or should I say backward? - Don’t get started! Un ringraziamento speciale a Oriana Fallaci, e alla maga. […] Ad esempio perché si trovasse qui, perché avesse scelto un mestiere che non si addiceva al suo carattere e alla sua struttura mentale cioè il mestiere di soldato, perché con quel mestiere avesse tradito la matematica. Quanto gli mancava la matematica, quanto la rimpiangeva! Massaggia le meningi come un allenatore massaggia i muscoli di un atleta, la matematica. Le irrora di pensiero puro, le lava dai sentimenti che corrompono l’intelligenza, le porta in serre dove crescono fiori stupendi. I fiori di un’astrazione composta di concretezza, di una fantasia composta di realtà […] No, non è vero che è una scienza rigida, la matematica, una dottrina severa. È un’arte seducente, estrosa, una maga che può compiere mille incantesimi e mille prodigi. Può mettere ordine nel disordine, dare un senso alle cose prive di senso, rispondere ad ogni interrogativo. Può addirittuta fornire ciò che in sostanza cerchi: la formula della Vita. Doveva tornarci, ricominciare da capo con l’umiltà d’uno scolaro che nelle vacanze ha dimenticato la tavola pitagorica. Due per due fa quattro, quattro per quattro fa sedici, sedici per sedici fa duecentocinquantasei, e la derivata di una costante è uguale a zero. La derivata di una variabile è uguale a uno, la derivata di una potenza di una variabile… Non se ne ricordava? Sì che se ne ricordava! La derivata di una potenza di una variabile è uguale all’esponente della potenza moltiplicata per la variabile con lo stesso esponente diminuito di uno. E la derivata di una divisione? È uguale alla derivata del dividendo moltiplicato per il divisore meno la derivata del divisore moltiplicata per il dividendo, il tutto diviso il dividendo moltiplicato per sé stesso. Semplice! Bè, naturalmente trovare la formula della Vita non sarebbe stato così semplice. Trovare una formula significa risolvere un problema, e per risolvere un problema bisogna enunciarlo, e per enunciarlo bisogna partire da un presupposto… Ah perché aveva tradito la maga? Che cosa lo aveva indotto a tradirla? […] Oriana Fallaci: Insciallah. Atto primo, Capitolo primo
arxiv-papers
2011-01-13T17:38:05
2024-09-04T02:49:16.432306
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Daniele Valtorta", "submitter": "Daniele Valtorta Mr.", "url": "https://arxiv.org/abs/1101.2618" }
1101.2913
# Hypercontractivity and its Applications Punyashloka Biswal ###### Abstract Hypercontractive inequalities are a useful tool in dealing with extremal questions in the geometry of high-dimensional discrete and continuous spaces. In this survey we trace a few connections between different manifestations of hypercontractivity, and also present some relatively recent applications of these techniques in computer science. ## 1 Preliminaries and notation #### Fourier analysis on the hypercube. We define the inner product $\langle f,g\rangle=\operatorname*{\mathbb{E}}_{x}f(x)g(x)$ on functions $f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, where the expectation is taken over the uniform (counting) measure on $\\{-1,1\\}^{n}$. The multilinear polynomials $\chi_{S}(x)=\prod_{i\in S}x_{i}$ (where $S$ ranges over subsets of $[n]$) form an orthogonal basis under this inner product; they are called the Fourier basis. Thus, for any function $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$, we have $f=\sum_{S\subseteq[n]}\hat{f}(S)\chi_{S}(x)$, where the Fourier coefficients $\hat{f}(S)=\langle f,\chi_{S}\rangle$ obey Plancherel’s relation $\sum\hat{f}(S)^{2}=1$. It is easy to verify that $\operatorname*{\mathbb{E}}_{x}f(x)=\hat{f}(0)$ and $\operatorname*{Var}_{x}f(x)=\sum_{S\neq\emptyset}\hat{f}(S)^{2}$. #### Norms. For $1\leq p<\infty$, define the $\ell_{p}$ norm $\|f\|_{p}=(\operatorname*{\mathbb{E}}_{x}|f(x)|^{p})^{1/p}$. These norms are monotone in $p$: for every function $f$, $p\geq q$ implies $\|f\|_{p}\geq\|f\|_{q}$. For a linear operator $M$ carrying functions $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$ to functions $Mf=g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, we define the $p$-to-$q$ operator norm $\|M\|_{p\to q}=\sup_{f}\|Mf\|_{q}/\|f\|_{p}$. $M$ is said to be a contraction from $\ell_{p}$ to $\ell_{q}$ when $\|M\|_{p\to q}\leq 1$. Because of the monotonicity of norms, a contraction from $\ell_{p}$ to $\ell_{p}$ is automatically a contraction from $\ell_{p}$ to $\ell_{q}$ for any $q<p$. When $q>p$ and $\|M\|_{p\to q}\leq 1$, then $M$ is said to be hypercontractive. #### Convolution operators. Letting $xy$ represent the coordinatewise product of $x,y\in\\{-1,1\\}^{n}$, we define the convolution $(f*g)(x)=\operatorname*{\mathbb{E}}_{y}f(x)g(xy)$ of two functions $f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, and note that it is a linear operator $f\mapsto f*g$ for every fixed $g$. Convolution is commutative and associative, and the Fourier coefficients of a convolution satisfy the useful property $\widehat{f*g}=\hat{f}\hat{g}$. We shall be particularly interested in the convolution properties of the following functions * • The Dirac delta $\delta\colon\\{-1,1\\}^{n}\to\mathbb{R}$, given by $\delta(1,\dotsc,1)=1$ and $\delta(x)=0$ otherwise. It is the identity for convolution and has $\hat{\delta}(S)=1$ for all $S\subseteq[n]$. * • The edge functions $h_{i}\colon\\{-1,1\\}^{n}\to\mathbb{R}$ given by $h_{i}(x)=\begin{cases}\phantom{-}1/2&x=(1,\dotsc,1)\\\ -1/2&x_{i}=-1,x_{[n]\setminus\\{i\\}}=(1,\dotsc,1)\\\ \phantom{-}0&\text{otherwise.}\end{cases}$ $\hat{h}_{i}(S)$ is $1$ or $0$ according as $S$ contains or does not contain $i$, respectively. For any function $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$, $(f*h_{i})(x)=(f(x)-f(y))/2$, where $y$ is obtained from $x$ by flipping just the $i$th bit. Convolution with $h_{i}$ acts as an orthogonal projection (as we can easily see in the Fourier domain), so for any functions $f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, we have $\langle f*h_{i},g\rangle=\langle f,h_{i}*g\rangle=\langle f*h_{i},g*h_{i}\rangle$ * • The Bonami-Gross-Beckner noise functions $\operatorname{BG}_{\rho}\colon\\{-1,1\\}^{n}\to\mathbb{R}$ for $0\leq\rho\leq 1$, where $\widehat{\operatorname{BG}}_{\rho}(S)=\rho^{|S|}$ and we define $0^{0}=1$. These operators form a semigroup, because $\operatorname{BG}_{\sigma}*\operatorname{BG}_{\rho}=\operatorname{BG}_{\sigma\rho}$ and $\operatorname{BG}_{1}=\delta$. Note that $\operatorname{BG}_{\rho}(x)=\sum_{S}\rho^{|S|}\chi_{S}(x)=\prod_{i}(1+\rho x_{i})$. We define the noise operator $T_{\rho}$ acting on functions on the discrete cube by $T_{\rho}f=\operatorname{BG}_{\rho}*f$. In combinatorial terms, $(T_{\rho}f)(x)$ is the expected value of $f(y)$, where $y$ is obtained from $x$ by independently flipping each bit of $x$ with probability $1-\rho$. ###### Lemma 1. $\frac{d}{d\rho}\operatorname{BG}_{\rho}=\frac{1}{\rho}\operatorname{BG}_{\rho}*\sum h_{i}$ ###### Proof. This is easy in the Fourier basis: $\widehat{\operatorname{BG}}_{\rho}^{\prime}=(\rho^{|S|})^{\prime}=|S|\rho^{|S|-1}=\sum_{i\in[n]}\hat{h}_{i}\frac{\widehat{\operatorname{BG}}_{\rho}}{\rho}.\qed$ ## 2 The Bonami-Gross-Beckner Inequality ### 2.1 Poincaré and Log-Sobolev inequalities The Poincaré and logarithmic Sobolev inequalities both relate a function’s global non-constantness to how fast it changes “locally”. The amount of local change is quantified by the _energy_ $\operatorname{\mathbb{D}}(f,f)$, where the Dirichlet form $\operatorname{\mathbb{D}}$ is defined as $\operatorname{\mathbb{D}}(f,g)=\tfrac{1}{2}\operatorname*{\mathbb{E}}_{xy\in E}(f(x)-f(y))(g(x)-g(y))$ ($E$ is the set of pairs $x,y$ that differ in a single coordinate). In terms of the edge functions $h_{i}$, observe that $\operatorname{\mathbb{D}}(f,g)=\frac{2}{n}\sum_{i}\langle f*h_{i},g*h_{i}\rangle$. In the case of the Poincaré inequality, we measure the distance of $f$ to a constant by its variance $\operatorname*{Var}(f)=\operatorname*{\mathbb{E}}(f-\operatorname*{\mathbb{E}}f)^{2}=\operatorname*{\mathbb{E}}f^{2}-(\operatorname*{\mathbb{E}}f)^{2}$. Then the Poincaré constant (of the discrete cube) is the supremal $\lambda$ such that the inequality $\operatorname{\mathbb{D}}(f,f)\geq\lambda\operatorname*{Var}(f)$ holds for all $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$. This quantity is also the smallest nonzero eigenvalue of the Laplacian of the discrete cube, viewed as a graph (i.e., its spectral expansion). Another way of measuring the non-constantness of a function is to consider its entropy $\operatorname*{Ent}(f)=\operatorname*{\mathbb{E}}[f\log\frac{f}{\operatorname*{\mathbb{E}}f}]$ (where we assume $f\geq 0$ and use the convention that $0\log 0=0$). Note that $\operatorname*{Ent}(cf)=c\operatorname*{Ent}(f)$ for any $c\geq 0$, so the entropy is homogenous of degree $1$ in its argument. Because we are comparing the entropy with the energy (which is homogenous of degree $2$) we use the entropy of the _square_ of the function to define the Log-Sobolev constant: the largest $\alpha$ such that the inequality $\operatorname{\mathbb{D}}(f,f)\geq\alpha\operatorname*{Ent}(f^{2})$ holds for all $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$. For the discrete cube $\\{-1,1\\}^{n}$, we have $\lambda=2/n$ and $\alpha=1/n$, as we shall see below. It is interesting to ask how these quantities are related when we consider other probability spaces equipped with a suitable Dirichlet form (for example, $d$-regular graphs with $\operatorname{\mathbb{D}}(f,g)=\operatorname*{\mathbb{E}}_{xy\in E}(f(x)-f(y))(g(x)-g(y))$, where the expectation is taken over all edges). Set $f=1+\epsilon g$ for a sufficiently small $\epsilon$ and observe that $\operatorname*{Var}(f)=\epsilon^{2}\operatorname*{Var}(g)$ and $\operatorname{\mathbb{D}}(f,f)=\epsilon^{2}\operatorname{\mathbb{D}}(g,g)$, whereas $\displaystyle\operatorname*{Ent}(f^{2})$ $\displaystyle=\operatorname*{\mathbb{E}}\left[(1+\epsilon g)^{2}(2\log(1+\epsilon g)-\log\operatorname*{\mathbb{E}}[(1+\epsilon g)^{2}])\right]$ $\displaystyle=2\epsilon^{2}\operatorname*{Var}(g)+O(\epsilon^{3})$ This shows that $\alpha\leq\lambda/2$, which is tight in the case of the cube. However, for constant-degree expander families (in particular, for random $d$-regular graphs with high probability) we have [DSC96, Example 4.2] $\lambda=\Omega(1)$ but $\alpha=O(\log\log n/\log n)\ll\lambda$. ### 2.2 Hypercontractivity and the log-Sobolev inequality When $\rho\in[0,1]$, the noise operator $T_{\rho}$ is easily seen to contract $\ell_{2}$: for any $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$, we have $\|T_{\rho}f\|_{2}^{2}=\sum_{S}\rho^{|S|}\hat{f}(S)^{2}\leq\sum_{S}\hat{f}(S)^{2}=\|f\|_{2}^{2}$. Now consider its behavior from $\ell_{2}$ to $\ell_{q}$ for some $q>2$. When $\rho=1$, we have $T_{1}f=f$; in particular, for $g(x)=(1+x_{1})/2$, $\|g\|_{q}=1/2^{1/q}>1/2^{1/2}=\|g\|_{2}$. On the other hand, $T_{0}f=\operatorname*{\mathbb{E}}f$, so $\|T_{0}f\|_{q}=|\operatorname*{\mathbb{E}}f|\leq\|\operatorname*{\mathbb{E}}f^{2}\|^{1/2}$. By the intermediate value theorem, there must be some $\rho\in(0,1)$ such that $\|T_{0}\|_{2\to q}=1$. A theorem of Gross [Gro75] connects this critical $\rho$ with the Log-Sobolev constant $\alpha$ of the underlying space: ###### Theorem 2. $\|T_{\rho}f\|_{p\to q}\leq 1$ if and only if $\rho^{-2\alpha n}\geq\frac{q-1}{p-1}$. Stated differently, $\|T_{1-\epsilon}f\|_{q}\leq\|f\|_{2}$ when $q\leq(1-\epsilon)^{-2}+1\approx 2+2\epsilon$. Thus to prove hypercontractive inequalities on the discrete cube, it suffices to bound the log-Sobolev constant. We shall prove this claim for $p=2$, which turns out to imply the general version. ###### Proof of Theorem 2. We shall prove that $\|T_{\rho}f\|_{q}\leq\|f\|_{2}$ for $q=1+\rho^{-2\alpha n}$; the remainder of the theorem can be shown using similar techniques. As we observed before, this inequality is tight when $\rho=1$, so it suffices to show that $\frac{d}{d\rho}\|T_{\rho}f\|_{q}\geq 0$ for $0\leq\rho\leq 1$. For notational convenience, let $G=\|T_{\rho}f\|_{q}^{q}$. Then $\|T_{\rho}f\|_{q}^{\prime}=(G^{1/q})^{\prime}=q^{-2}G^{(1/q)-1}\left(qG^{\prime}-q^{\prime}G\log G\right).$ Now we use the fact that $G=\operatorname*{\mathbb{E}}(T_{\rho}f)^{q}$ to get $G^{\prime}=q\operatorname*{\mathbb{E}}\left[(T_{\rho}f)^{q-1}(T_{\rho}f)^{\prime}\right]+q^{\prime}\operatorname*{\mathbb{E}}\left[(T_{\rho}f)^{q}\log(T_{\rho}f)\right].$ Applying Lemma 3 and simplifying, we get $qG^{\prime}-q^{\prime}G\log G=q^{\prime}\operatorname*{Ent}\left((T_{\rho}f)^{q}\right)+\frac{nq^{2}}{2\rho}\operatorname{\mathbb{D}}\left((T_{\rho}f)^{q-1},T_{\rho}f\right).$ We use Lemma 4 to handle the second term, and plug in $q=1+\rho^{-2\alpha n}$ to get $qG^{\prime}-q^{\prime}G\log G=n\rho^{-2\alpha n-1}\bigl{[}\operatorname{\mathbb{D}}\bigl{(}(T_{\rho}f)^{q/2},(T_{\rho}f)^{q/2}\bigr{)}-\operatorname*{Ent}\left((T_{\rho}f)^{q}\right)\bigr{]},$ whose positivity we are guaranteed by the log-Sobolev inequality applied to $(T_{\rho}f)^{(q-1)/2}$. ∎ ###### Lemma 3. For any $f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, $\langle g,\frac{d}{d\rho}(T_{\rho}f)\rangle=\frac{n}{2\rho}\operatorname{\mathbb{D}}(g,T_{\rho}f)$. ###### Proof. Recalling Lemma 1 and the projection property of the $h_{i}$s, we have $\langle g,(T_{\rho}f)^{\prime}\rangle=\langle g,\operatorname{BG}_{\rho}^{\prime}*f\rangle=\biggl{<}g,\frac{1}{\rho}\operatorname{BG}_{\rho}*f*\sum_{i}h_{i}\biggr{>}=\frac{1}{\rho}\sum_{i}\langle g*h_{i},\operatorname{BG}_{\rho}*f\rangle=\frac{n}{2\rho}\operatorname{\mathbb{D}}(g,T_{\rho}f).\qed$ ###### Lemma 4. For any $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$ and $q\geq 2$, $\operatorname{\mathbb{D}}(f,f^{q-1})\geq\frac{4(q-1)}{q^{2}}\operatorname{\mathbb{D}}\left(f^{q/2},f^{q/2}\right)$. ###### Proof. It suffices to show that $(a^{q-1}-b^{q-1})(a-b)>\frac{4(q-1)}{q^{2}}(a^{q/2}-b^{q/2})^{2}$ for all $a>b\geq 0$ and $q\geq 2$. But observe that $\displaystyle\left(\int_{a}^{b}t^{q/2-1}dt\right)^{2}$ $\displaystyle=\frac{4}{q^{2}}(a^{q/2}-b^{q/2})^{2}$ $\displaystyle\int_{a}^{b}t^{q-2}dt\ \int_{a}^{b}dt$ $\displaystyle=\frac{1}{q-1}(a^{q-1}-b^{q-1})(a-b)$ and the inequality between the integrals follows from convexity. ∎ ### 2.3 Two-point inequality We begin by showing that the log-Sobolev inequality holds for the uniform distribution on the two-point space $\\{-1,1\\}$ with $\alpha=2$. Without loss of generality, consider $f(x)=1+sx$. Then $\operatorname*{Ent}(f^{2})=\tfrac{1}{2}(1+s)^{2}\log(1+s)^{2}+\tfrac{1}{2}(1-s)^{2}\log(1-s)^{2}-(1+s^{2})\log(1+s^{2})$ and $\operatorname{\mathbb{D}}(f,f)=2s^{2}$. We shall show that $\phi(s)=\operatorname{\mathbb{D}}(f,f)-\alpha\operatorname*{Ent}(f^{2})$ is non-negative for $-1\leq s\leq 1$. By symmetry it suffices to consider $s\geq 0$. But $\phi(0)=0$ and $\phi^{\prime}(s)=4s+2s\log(1+s^{2})+2(1-s)\log(1-s)-2(1+s)\log(1+s),$ which is non-negative because $\phi^{\prime}(0)=0$ and $\phi^{\prime\prime}(s)=\frac{4s^{2}}{s^{2}+1}+2\log\frac{1+s^{2}}{1-s^{2}}\geq 0.$ ### 2.4 Tensoring property ###### Theorem 5. Let $\alpha$ be the log-Sobolev constant of $\\{-1,1\\}^{n}$. Then the log- Sobolev constant of $\\{-1,1\\}^{2n}$ is $\alpha/2$. When $n$ is a power of $2$, we can conclude inductively that $\alpha=1/n$; a proof along similar lines works for arbitrary $n$ as well. ###### Proof of Theorem 5. For any $f\colon\\{-1,1\\}^{n}\times\\{-1,1\\}^{n}\to\mathbb{R}$, set $g(x)=\|f(x,\cdot)\|_{2}$. Then by the conditional entropy formula, $\operatorname*{Ent}(f^{2})\leq\operatorname*{Ent}(g^{2})+\operatorname*{\mathbb{E}}_{x}\operatorname*{Ent}_{y}(f(x,y)^{2})\leq\frac{\operatorname{\mathbb{D}}(g,g)+\operatorname*{\mathbb{E}}_{x}\operatorname{\mathbb{D}}_{y}(f(x,y),f(x,y))}{\alpha}$ and by convexity, $\displaystyle\operatorname{\mathbb{D}}(g,g)$ $\displaystyle=\tfrac{1}{2}\operatorname*{\mathbb{E}}_{x\sim x^{\prime}}(g(x)-g(x^{\prime}))^{2}\leq\tfrac{1}{2}\operatorname*{\mathbb{E}}_{x\sim x^{\prime}}\operatorname*{\mathbb{E}}_{y}\left[(f(x,y)-f(x^{\prime},y))^{2}\right]=\operatorname*{\mathbb{E}}_{y}\operatorname{\mathbb{D}}_{x}(f(x,y),f(x,y))$ where the notation $x\sim x^{\prime}$ ranges over edges of $\\{-1,1\\}^{n}$. Taken together, these give $\operatorname*{Ent}(f^{2})\leq\frac{\operatorname*{\mathbb{E}}_{x}\operatorname{\mathbb{D}}_{y}(f(x,y),f(x,y))+\operatorname*{\mathbb{E}}_{y}\operatorname{\mathbb{D}}_{x}(f(x,y),f(x,y))}{\alpha}\leq\frac{2\operatorname{\mathbb{D}}(f)}{\alpha}=\frac{\operatorname{\mathbb{D}}(f)}{\alpha/2}.$ as claimed. ∎ ### 2.5 Non-product groups Recall that we defined the Dirichlet form $\operatorname{\mathbb{D}}(f,g)=\tfrac{1}{2}\operatorname*{\mathbb{E}}_{u\sim v}(f(u)-f(v))(g(u)-g(v))$ for functions $f,g\colon\\{-1,1\\}^{n}\to\mathbb{R}$, but it makes sense for any regular graph if we sample $u,v$ uniformly from the edges. Thus, given any family of regular graphs, we can ask if they satisfy a log-Sobolev inequality of the form $\operatorname{\mathbb{D}}(f,f)\geq\alpha\operatorname*{Ent}(f)$ for all suitable $f$. It turns out that the relationship between logarithmic Sobolev inequalities and hypercontractive noise operator subgroups, as stated by Gross [Gro75], holds for a wide class of spaces, not just the hypercube $\\{-1,1\\}^{n}$. Diaconis and Saloff-Coste [DSC96] explored an intermediate between these two extremes of specialization to give improved mixing time results for Markov chains on various graphs. One of the first discrete applications of hypercontractivity was a celebrated theorem of Kahn, Kalai and Linial [KKL88] relating the maximum influence of a function on the hypercube to its variance. In Theorem 7, we discuss some recent work [OW09b] of O’Donnell and Wimmer generalizing the KKL theorem to apply to the wider class of Schreier graphs associated with group actions (defined below). An action of a group $G$ on a set $X$ is a homomorphism from $G$ to the group of bijections on $X$, and we write $x^{g}$ for the image of $x$ under the bijection for $g$. If $S$ is a set of generators for $G$, then the Schreier graph $\operatorname{Sch}(G,S,X)$ has vertex set $X$ and edges $(x,x^{g})$ for all $x\in X$ and $g\in S$. It is known that every connected regular graph of even degree can be obtained in this way [Gro77]. The definition of the Dirichlet form $\operatorname{\mathbb{D}}$ generalizes without change, but to be able to derive a log-Sobolev inequality for this space, we must define the noise operator $T_{\rho}$ in an appropriate fashion to satisfy the claim of Lemma 1: $\langle g,\frac{d}{d\rho}(T_{\rho}f)\rangle\propto\frac{1}{\rho}\operatorname{\mathbb{D}}(g,T_{\rho}f)$. ## 3 Boolean-Valued Functions ### 3.1 Influences Write $x_{-i}$ for the collection of random variables $\\{x_{1},\dotsc,x_{n}\\}\setminus\\{x_{i}\\}$. The influence of the $i$th coordinate on a function $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$ is given by $\operatorname{Inf}_{i}(f)=\operatorname*{\mathbb{E}}_{x_{-i}}\operatorname*{Var}_{x_{i}}f(x)=\operatorname*{\mathbb{E}}_{x_{-i}}\left[\operatorname*{\mathbb{E}}_{x_{i}}f(x)^{2}-(\operatorname*{\mathbb{E}}_{x_{i}}f(x))^{2}\right].$ When $f$ is Boolean-valued, this quantity is just the probability that changing $x_{i}$ changes $f(x)$. Writing $f$ in the Fourier basis, we have $\operatorname*{\mathbb{E}}_{x_{-i}}\operatorname*{\mathbb{E}}_{x_{i}}f(x)^{2}=\operatorname*{\mathbb{E}}_{x}f(x)^{2}=\sum_{S}\hat{f}(S)^{2}$ and $\operatorname*{\mathbb{E}}_{x_{-i}}(\operatorname*{\mathbb{E}}_{x_{i}}f(x))^{2}=\sum_{S\not\ni i}\hat{f}(S)^{2}$, so that $\operatorname{Inf}_{i}(f)=\sum_{S\ni i}\hat{f}(S)^{2}=\operatorname*{\mathbb{E}}(f*h_{i})^{2}$. In addition, we define the total influence $\operatorname{Inf}(f)=\sum_{i}\operatorname{Inf}_{i}(f)=\sum_{S}|S|\hat{f}(S)^{2}$. ### 3.2 Structural results Boolean functions are natural combinatorial objects, but they were first studied from an analytical viewpoint in work on voting and social choice. In this setting, a function $f\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$ is viewed as a way to combine the preferences of $n$ voters to yield the result of the election. This explains the notions of dictator or junta functions, which depend on only one or a few of their coordinates, respectively. In this context it is also natural to consider functions where no coordinate (“voter”) has a very large influence. Kahn, Kalai, and Linial [KKL88] first introduced the Fourier analysis of Boolean functions as a technique in computer science. Their theorem establishes that if a function is far from a constant (i.e., has variance at least a constant), then it must have a variable of influence $\Omega(\frac{\log n}{n})$. We state a strengthening of their original inequality due to Talagrand [Tal95]: ###### Theorem 6 ([KKL88, Tal95]). For any $f\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$, $\sum_{i}\frac{\operatorname{Inf}_{i}(f)}{\log(1/\operatorname{Inf}_{i}(f))}\geq\Omega(1)\cdot\operatorname*{Var}(f).$ We can compare this to the Poincaré inequality on the cube, which can be stated as $\sum_{i}\operatorname{Inf}_{i}(f)\geq\Omega(1)\cdot\operatorname*{Var}(f).$ (In particular, there exists a variable of influence $\Omega\bigl{(}\tfrac{1}{n}\bigr{)}\operatorname*{Var}(f)$.) The KKL theorem is a stronger result of the same form: it is a comparison between a local and a global measure of variation. The proofs of KKL and Talagrand used the hypercontractivity of the cube, but we present here a more recent proof due to Rossignol that uses the log-Sobolev inequality instead. For simplicity we’ll just show the weaker statement that the maximum influence is $\Omega\bigl{(}\tfrac{\log n}{n}\bigr{)}\operatorname*{Var}(f)$. ###### Proof. Write $f-\operatorname*{\mathbb{E}}f=f_{1}+\dotsb+f_{n}$, where $f_{j}=\sum_{S:\max S=j}\hat{f}(S)\chi_{S}$. For each $f_{j}$, the log-Sobolev inequality states that $\operatorname{\mathbb{D}}(f_{j},f_{j})\geq\alpha\operatorname*{Ent}(f_{j}^{2})=\frac{1}{n}\operatorname*{Ent}(f_{j}^{2})$. By writing $\operatorname{\mathbb{D}}(f_{j},f_{j})$ in terms of the Fourier coefficients $\hat{f}(S)$, we can check that $\operatorname{\mathbb{D}}(f,f)=\sum_{j=0}^{n}\operatorname{\mathbb{D}}(f_{j},f_{j})$, so that we can sum all these inequalities to obtain $n\operatorname{\mathbb{D}}(f,f)\geq\sum_{j}\operatorname*{Ent}(f_{j}^{2})=\underbrace{\sum_{j}\operatorname*{\mathbb{E}}\left[f_{j}^{2}\log(f_{j}^{2})\right]}_{A}+\underbrace{\sum_{j}\operatorname*{\mathbb{E}}f_{j}^{2}\log\frac{1}{\operatorname*{\mathbb{E}}f_{j}^{2}}}_{B}.$ In order to bound $B$, we begin by noting that $\operatorname*{\mathbb{E}}f_{j}^{2}=\sum_{S:\max S=j}\hat{f}(S)^{2}\leq\sum_{S\ni j}\hat{f}(S)^{2}=\operatorname*{\mathbb{E}}(f*h_{j})^{2}$ where the $h_{j}$s are the edge functions we defined earlier. Letting $M(f)=\max_{j}\operatorname*{\mathbb{E}}(f*h_{j})^{2}=\max_{j}\operatorname{Inf}_{j}(f)$, we have $B=\sum_{j}\operatorname*{\mathbb{E}}f_{j}^{2}\log\frac{1}{\operatorname*{\mathbb{E}}f_{j}^{2}}\geq\sum_{j}\operatorname*{\mathbb{E}}f_{j}^{2}\log\frac{1}{M(f)}=\operatorname*{Var}(f)\log\frac{1}{M(f)}$ where we have used the orthogonality of the $f_{j}$s and the fact that $\operatorname*{Var}(f)=\sum_{S\neq\emptyset}\hat{f}(S)^{2}$. To bound $A$, we split it up further: $A=\underbrace{\sum_{j}\operatorname*{\mathbb{E}}\left[f_{j}^{2}\log(f_{j}^{2})\cdot 1_{f_{j}^{2}\leq t}\right]}_{A_{1}}+\underbrace{\sum_{j}\operatorname*{\mathbb{E}}\left[f_{j}^{2}\log(f_{j}^{2})\cdot 1_{f_{j}^{2}>t}\right]}_{A_{2}}.$ For $0\leq t\leq 1/e^{2}$, we have that $\sqrt{t}\log\sqrt{t}$ is a nonpositive decreasing function and therefore, $A_{1}=2\sum_{j}\operatorname*{\mathbb{E}}\left[|f_{j}|\log|f_{j}|\cdot|f_{j}|1_{f_{j}^{2}\leq t}\right]\geq 2\sqrt{t}\log\sqrt{t}\sum_{j}\operatorname*{\mathbb{E}}|f_{j}\cdot 1_{f_{j}^{2}\leq t}|\geq\sqrt{t}\log t\sum_{j}\operatorname*{\mathbb{E}}|f_{j}|.$ By comparing Fourier coefficients, it is easy to verify that $f_{j}=\operatorname*{\mathbb{E}}_{x_{j+1},\dotsc,x_{n}}(f*h_{j})$. Therefore, by convexity, $\operatorname*{\mathbb{E}}|f_{j}|\leq\operatorname*{\mathbb{E}}|f*h_{j}|.$ Until now, the proof has made no use of the fact that $f$ takes on only Boolean values. Now we argue that because $f(x)\in\\{-1,1\\}$, we must have $(f*h_{j})(x)\in\\{-1,0,1\\}$, so that $\operatorname*{\mathbb{E}}|f*h_{j}|=\operatorname*{\mathbb{E}}(f*h_{j})^{2}$. Plugging this into our bound for $A_{1}$ yields $A_{1}\geq\sqrt{t}\log t\sum_{j}\operatorname*{\mathbb{E}}(f*h_{j})^{2}=\frac{n}{2}\sqrt{t}\log t\cdot\operatorname{\mathbb{D}}(f,f).$ For $A_{2}$, note that $\log(\cdot)$ is increasing, so $A_{2}\geq\log t\sum_{j}\operatorname*{\mathbb{E}}f_{{}_{j}}^{2}=\log t\operatorname*{Var}f.$ Summing all these bounds gives us $n\operatorname{\mathbb{D}}(f,f)\geq\log\frac{1}{M(f)}\operatorname*{Var}(f)+\frac{n}{2}\sqrt{t}\log t\cdot\operatorname{\mathbb{D}}(f,f)+\log t\cdot\operatorname*{Var}(f).$ By the Poincaré inequality, $\operatorname{\mathbb{D}}(f,f)\geq\frac{2}{n}\operatorname*{Var}(f)$, so we can set $t=\bigl{(}\frac{2\operatorname*{Var}(f)}{ne\operatorname{\mathbb{D}}(f,f)}\bigr{)}^{2}\leq 1/e^{2}$. With this substitution, the above inequality becomes $\frac{2}{e\sqrt{t}}\geq\log\frac{t^{1+1/e}}{M(f)}.$ Suppose $t\leq(\frac{4}{e\log n})^{2}$. Then $\operatorname{\mathbb{D}}(f,f)\geq\frac{2\operatorname*{Var}(f)}{en}\cdot\frac{e\log n}{4}=\Omega\Bigl{(}\frac{\log n}{n}\Bigr{)},$ and we know that $M(f)\geq 2\operatorname{\mathbb{D}}(f,f)$. On the other hand, if $t>(\frac{4}{e\log n})^{2}$, then $M(f)>t^{1+1/e}\exp\Bigl{(}\frac{-2}{e\sqrt{t}}\Bigr{)}=\Bigl{(}\frac{4}{e\log n}\Bigr{)}^{2+2/e}\exp\Bigl{(}\frac{-\log n}{2}\Bigr{)}\gg\frac{\log n}{n}.\qed$ We are now in a position to state the recent result of O’Donnell and Wimmer [OW09b] generalizing the KKL theorem to Schreier graphs satisfying a certain technical property. ###### Theorem 7 ([OW09b]). Let $G$ be a group acting on a set $X$, $U\subseteq X$ be a union of conjugacy classes that generates $G$, and $\alpha$ be the log-Sobolev constant of $\operatorname{Sch}(G,X,U)$. Then for any $f\colon X\to\\{-1,1\\}$, $\frac{\sum_{U}\operatorname{Inf}_{u}(f)}{\log(1/\max_{U}\operatorname{Inf}_{u}(f))}\geq\Omega(\alpha\operatorname*{Var}(f)).$ In particular, there is some $u\in U$ such that $\operatorname{Inf}_{u}(f)\geq\Omega(\alpha\log\frac{1}{\alpha})\operatorname*{Var}(f)$. For an Abelian group such as $\mathbb{Z}_{2}^{n}$ (the cube), every group element is in a conjugacy class by itself, so the extra condition on $U$ is vacuous. Using $\alpha=\Omega(\frac{1}{n})$ for the cube, we recover the original KKL theorem. O’Donnell et al. apply the generalized result to the non-Abelian group $S_{n}$ of permutations on $[n]$, generated by transpositions and acting on the family $\binom{[n]}{k}$ of $k$-subsets of $[n]$. By viewing these families as sets of $n$-bit strings, they recover a “rigidity” version of the Kruskal-Katona theorem that states (roughly) that if a subset of a layer of a cube has a small expansion to the layer above it, then it must be correlated to some dictator function. #### Coding theoretic interpretation. In the _long code_ , an integer $i\in[n]$ is encoded as the dictator function $(x_{1},\dotsc,x_{n})\mapsto x_{i}$. By using many more bits ($2^{n}$ rather than $\log n$) of redundant storage, we hope to be able to recover from corruptions in the data. The theorem tells us that as long as the corrupted version of an encoding is far from a constant function, it can be decoded to a coordinate whose influence is $\Omega(\log n)$ times the average influence. Since every coordinate’s influence is nonnegative, only $O(\log n)$ coordinates can have influence this large. Thus, we have a “small” set of candidate long codes to which we might decode the word. To complete this picture, we’d like to understand how far the word can be from functions that depend only on these coordinates; the following theorem of Friedgut, which we state without proof, furnishes this information. ###### Theorem 8 ([Fri98]). For every $f\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$ and $0<\epsilon<1$, there is a function $g\colon\\{-1,1\\}^{n}\to\\{-1,1\\}$ depending on at most $\exp\bigl{(}\frac{2+o(1)}{\epsilon n}\operatorname{Inf}(f)\bigr{)}$ variables such that $\operatorname*{\mathbb{E}}|f-g|\leq\epsilon$. ## 4 Gaussian isoperimetry and an algorithmic application Hypercontractive inequalities were first investigated in the context of Gaussian probability spaces, for their applications to quantum field theory. The following simple proof reduces the continuous Gaussian hypercontractive inequality to its discrete counterpart on the cube. ### 4.1 From the central limit theorem to Gaussian hypercontractivity ###### Theorem 9 ([Gro75]). Let $x\in\mathbb{R}$ be normally distributed, i.e., $\Pr[x\in A]=\frac{1}{\sqrt{2\pi}}\int_{A}\exp\left(-\frac{x^{2}}{2}\right)\,dx.$ Then for a smooth function $f\colon\mathbb{R}\to\mathbb{R}$, the random variable $F=f(x)$ satisfies $\operatorname{\mathbb{D}}(F,F)\geq\alpha\operatorname*{Ent}(F^{2})$ with $\alpha=1$ and $\operatorname{\mathbb{D}}(F,G)=\frac{1}{2}\left<{\frac{dF}{dx},\frac{dG}{dx}}\right>.$ ###### Proof. We shall approximate the Gaussian distribution by a weighted sum of Bernoulli variables. Let $y\in\\{-1,1\\}^{k}$ be uniformly distributed, and set $g(y)=\frac{y_{1}+\dotsb+y_{k}}{\sqrt{k}}$. By the log-Sobolev inequality applied to $f\circ g(y)$, we have $\operatorname{\mathbb{D}}\left(f\circ g(y),f\circ g(y)\right)\geq\operatorname*{Ent}(f\circ g(y)^{2})$. By the central limit theorem, the right side converges to $\operatorname*{Ent}(f(x)^{2})=\operatorname*{Ent}(F^{2})$ as $k\to\infty$, so it remains to show that the left side converges to $\operatorname{\mathbb{D}}(F,F)$ as well. Let $y|_{y_{i}=\theta}$ be the value obtained by replacing the $i$th coordinate of $y$ with the value $\theta$, and observe that $g(y|_{y_{i}=1})-g(y|_{y_{i}=-1})=2/\sqrt{k}$. Then, using the smoothness of $f$, we have $\left|\left(h_{i}*(f\circ g)\right)(y)\right|=\frac{1}{2}\left|f\circ g(y|_{y_{i}=1})-f\circ g(y|_{y_{i}=-1})\right|=\frac{1}{\sqrt{k}}\left|f^{\prime}\circ g(y)\right|+o\left(\frac{1}{\sqrt{k}}\right),$ so that $\operatorname{\mathbb{D}}\left(f\circ g(y),f\circ g(y)\right)=\frac{1}{2}\operatorname*{\mathbb{E}}_{y}\left[\sum_{i}\left(h_{i}*(f\circ g)\right)(y)^{2}\right]=\frac{1}{2}\operatorname*{\mathbb{E}}_{y}\left[f^{\prime}\circ g(y)^{2}+o(1)\right].$ The second term vanishes as $k\to\infty$, and the first term converges to $\operatorname{\mathbb{D}}(F,F)$ by the Central Limit Theorem. ∎ The tensoring property of log-Sobolev inequalities lets us extend this result to Gaussian distributions over $\mathbb{R}^{d}$. We are also interested in the corresponding noise operator $S_{\rho}$, known as the Ornstein-Uhlenbeck operator, which is given by $S_{\rho}f(x)=\operatorname*{\mathbb{E}}_{z\sim\mathcal{N}(0,1)^{d}}f(\rho x+(1-\rho^{2})^{1/2}z).$ Theorem 2 has an analog in this setting, which lets us conclude that every function $f\colon\mathbb{R}^{d}\to\mathbb{R}$ satisfies $\|S_{\rho}f\|_{q}\leq\|f\|_{p}$ where $q>p\geq 1$ and $\rho^{-2}\geq(p-1)/(q-1)$. ### 4.2 Reverse hypercontractivity and isoperimetry In 1982, Borell showed a _reversed_ inequality of a similar form when $q<p<1$: ###### Theorem 10 (Reverse hypercontractivity, [Bor82]). Fix $q<p\leq 1$ and $\rho\geq 0$ such that $\rho^{-4}\geq(p-1)/(q-1)$. Then for any positive-valued function $f\colon\mathbb{R}^{d}\to\mathbb{R}^{+}$, we have $\|S_{\rho}f\|_{q}\geq\|f\|_{p}$. Note that the expressions $\|\cdot\|_{p}$ are not norms when $p<1$; in particular, they are not convex. However, this theorem can be proved by means similar to our proof for the Gaussian log-Sobolev inequality: we start with a base result for the 2-point space, proceed by tensoring to the hypercube, and use the central limit theorem to cover Gaussian space. As an application of Borell’s result, consider the following strong isoperimetry theorem for Gaussian space (due to Sherman). ###### Theorem 11 (Gaussian isoperimetry, [She09]). Let $u,u^{\prime}\in\mathbb{R}^{d}$ be independent Gaussian random variables. Then for any set $A\subseteq\mathbb{R}^{d}$ and any $\tau>0$, we have $\Pr_{u}\left[\Pr_{u^{\prime}}[\rho u+(\sqrt{1-\rho^{2}})u^{\prime}\in A]\leq\tau\right]\leq\frac{\tau^{1-\rho}}{\mu(A)}$ ###### Proof. When $\mu(A)\leq\tau^{1-\delta}$, there is nothing to prove. Otherwise, let $f$ be the indicator function of $A$ and observe that $\Pr_{u^{\prime}}\bigl{[}\rho u+(1-\rho^{2})^{1/2}u^{\prime}\in A\bigr{]}=S_{\rho}f(u)$. Therefore, for $q=1-1/\rho<0$, we have $\displaystyle\Pr_{u}\left[\Pr_{u^{\prime}}[u^{\prime}\in A]\leq\tau\right]$ $\displaystyle=\Pr_{u}[S_{\rho}f(u)\leq\tau]$ $\displaystyle=\Pr_{u}[S_{\rho}f(u)^{q}\geq\tau^{q}]$ $\displaystyle\leq\frac{\operatorname*{\mathbb{E}}_{u}(S_{\rho}f(u))^{q}}{\tau^{q}}$ by an application of Markov’s inequality. But $\operatorname*{\mathbb{E}}_{u}(S_{\rho}f(u))^{q}$ is just $\|S_{\rho}f\|_{q}^{q}$, and we know by Borell’s theorem that $\|S_{\rho}f\|_{q}\geq\|f\|_{p}$ for $p=1-\rho$. Thus $\Pr_{u}\left[\Pr_{u^{\prime}}[u^{\prime}\in A]\leq\tau\right]\leq\frac{\|f\|_{p}^{q}}{\tau^{q}}=\frac{\mu(A)^{q/p}}{\tau^{q}}=\left(\frac{\tau^{1-\rho}}{\mu(A)}\right)^{1/\rho}\leq\frac{\tau^{1-\rho}}{\mu(A)}$ where we have used the facts that $q<0$ and $\rho\leq 1$. ∎ ### 4.3 Fast graph partitioning and the constructive Big Core Theorem #### Problem and SDP rounding algorithm. In the $c$-balanced separator problem, we are given a graph $G=(V,E)$ on $n$ vertices and asked to find the smallest set of edges such that their removal disconnects the graph into pieces of size at most $cn$. The problem is NP- hard, and the best known approximation ratio111For technical reasons, it is actually a pseudo-approximation: the algorithm’s output for $c$ is compared to the optimal value for $c^{\prime}\neq c$. is $\Theta(\sqrt{\log n})$. The first algorithm to achieve this bound was based on a semidefinite program that assigns a unit vector to each vertex and minimizes the total embedded squared length of the edges subject to the constraint that the vertices are spread out and that the squared distances between the points form a metric: minimize $\displaystyle\textstyle\sum_{i\sim j}\|x_{i}-x_{j}\|_{2}^{2}$ subject to $\displaystyle\|x_{i}\|_{2}^{2}=1$ $\displaystyle\forall i\in V$ $\displaystyle\textstyle\sum_{i,j}\|x_{i}-x_{j}\|^{2}\geq c(1-c)n$ $\displaystyle\|x_{i}-x_{j}\|_{2}^{2}+\|x_{j}-x_{k}\|_{2}^{2}\geq\|x_{i}-x_{k}\|^{2}$ $\displaystyle\forall i,j,k\in V$ To round this SDP, Arora, Rao and Vazirani [ARV09] pick a random direction $u$ and project all the points along $u$. They then define sets $A$ and $B$ consisting of points $x$ whose projections are sufficiently large, i.e., $A=\\{x\mid\langle x,u\rangle<-K\\}$ and similarly $B=\\{x\mid\langle x,u\rangle>K\\}$, where $K$ is chosen to make $A$ and $B$ have size $\Theta(n)$ with high probability. Next, they discard points $a\in A,b\in B$ such that $\|a-b\|$ is much smaller than expected for a pair whose projections are $\geq 2K$ apart. Finally, if the resulting pruned sets $A^{\prime}\subset A$ and $B^{\prime}\subset B$ are large enough, they show that greedily growing $A$ yields a good cut. #### Matchings and cores. The key step in making this argument work is to ensure that not too many pairs $(a,b)$ are removed in the pruning step. To bound the probability of this bad event, we consider the possibility that for a large fraction $\delta=\Omega(1)$ of directions $u$, there exists a matching of points $M_{u}$ such that each pair $(a,b)\in M_{u}$ is short (i.e., $\|a-b\|\leq\ell=O(1/\sqrt{\log n})$) but stretched along $u$ (i.e., $|\langle a-b,u\rangle|\geq\sigma=\Omega(1)$). Such a set of points is called a _$(\sigma,\delta,\ell)$ -core_. The big core theorem (first proved with optimal parameters by Lee [Lee05]) asserts that this situation can’t arise: for a fixed $\sigma,\delta$, and $\ell$, we must have $n\gg\exp(\sigma^{6}/\ell^{4}\log^{2}(1/\delta))$, which is a contradiction for our chosen values of $\sigma,\delta,\ell$. In order to prove the big core theorem, Lee concatenates pairs that share a point and belong in matchings for nearby directions. The existence of a long chain of such concatenations is what leads to a contradiction: if we consider the endpoints $a,b$ of a chain of length $p$, the projection $|\langle a-b,u\rangle|$ grows linearly in $p$ whereas the distance $\|a-b\|$ grows only as $\sqrt{p}$ (recall that the SDP constrained the _squared_ distances to form a metric). #### Boosting. The matching chaining argument we have just presented in its simple form doesn’t work, for the following reason. At each chaining step, the fraction of nearby directions available for our use reduces by roughly $1-\delta$ (by a union bound) so that we are rapidly left with no direction to move in. To remedy this situation, we need to boost the fraction of usable directions at each step, say from $\delta/2$ to $1-\delta/2$, so that we can carry on chaining in spite of a $1-\delta$ loss. Lee’s proof uses the standard isoperimetric inequality for the sphere to show that this boosting can be performed with no change in $\ell$ and a very small penalty in $\sigma$. In other words, we take advantage of the fact that a very small dilation of a set of constant measure (i.e., the set of available directions) has measure close to $1$. #### Faster algorithms. Lee’s big core theorem is non-constructive in the sense that it only shows the _existence_ of such a long chain of matched pairs in order to give a contradiction. While this form suffices to bound the approximation ratio of the ARV rounding scheme, other variants of their technique require a way to _efficiently sample_ long chains, not just show their existence. Sherman constructs a distribution over directions that does not depend on the point set at all, yet is guaranteed to always have a non-trivial probability of producing long chains of stretched pairs. More precisely, ###### Theorem 12 (Constructive big core [She09]). For any $1\leq R\leq\Theta(\sqrt{\log n})$, there is $P\geq\Theta(R^{2}/\log n)$ and an efficiently sampleable distribution $\mu$ over the set of sequences of $\leq P$ direction vectors (each in $\mathbb{R}^{d}$), such that: for any $(\sigma,\delta,\ell)$-core $M$, if the string of directions is sampled from $\mu$, the expected number of chains whose endpoints are $\geq P\ell$ apart is at least $\exp(-O(P^{2})n)$. We sketch some of the ideas of the proof here. Sherman constructs two sequences of Gaussian directions $u_{1},\dotsc,u_{P}$ and $w_{1},\dotsc,w_{P}$. Each $w_{i}$ is an independent Gaussian vector, whereas each $u_{i}$ for $i>1$ is a Gaussian vector $\rho$-correlated with $u_{i-1}$. Finally, the distribution $\mu$ is given by randomly shuffling together the $u_{i}$ and $w_{i}$, picking a uniformly random $R$ between $1$ and $P$, and returning the first $R$ elements of the shuffled sequence. The correlated directions $u_{i}$ correspond to the steps in which Lee’s proof chained pairs from similar directions, whereas the independent $w_{i}$ correspond to the region-growing steps necessary for boosting. By randomly interleaving these two types of moves, Sherman’s sampling algorithm can be oblivious to the actual point set it is acting on. ## 5 Complexity theoretic applications ### 5.1 Dictatorship testing with perfect completeness #### Definitions. A function $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$ is said to be $(\epsilon,\delta)$-quasirandom if $\hat{f}(S)\leq\epsilon$ whenever $|S|\leq 1/\delta$. In order to show that a given problem is hard to approximate, we often need to design a test that * • performs $q$ _queries_ on a black-box function $f$, * • accepts every dictator function with probability $\geq c$ (the _completeness_ probability), and * • accepts every $(\epsilon,\delta)$-quasirandom function with probability $\leq s$ (the _soundness_ probability). A test is said to be _adaptive_ if each query is allowed to depend on the result of the queries so far. While dictatorship tests for the $c<1$ setting have been known for over a decade (first from the work of Håstad and more recently via the Unique Games Conjecture of Khot), there were no nontrivial bounds for $c=1$ until some recent results of O’Donnell and Wu. Their analysis, which we show below, relies heavily on the hypercontractive inequality. ###### Theorem 13 ([OW09a]). For every $n>0$, there is a $3$-query non-adaptive test that accepts every dictator function $(x_{1},\dotsc,x_{n})\mapsto x_{i}$ with probability $c=1$ but accepts any $(\delta,\delta/\log(1/\delta))$-quasirandom odd function $f\colon\\{-1,1\\}^{n}\to[-1,1]$ with probability $\leq s=5/8+O(\sqrt{\delta})$. The proof uses the following strengthening of the hypercontractive inequality for restricted parameter values. ###### Lemma 14. If $0\leq\rho\leq 1$, $q\geq 1$, and $0\leq\lambda\leq 1$ satisfy $\rho^{\lambda}\leq 1/\sqrt{q-1}$, then for all $f\colon\\{-1,1\\}^{n}\to\mathbb{R}$, $\|T_{\rho}f\|_{q}\leq\|T_{\rho}f\|_{2}^{1-\lambda}\|f\|_{2}^{\lambda}$. ###### Proof. $\displaystyle\|T_{\rho}f\|_{q}^{2}$ $\displaystyle=\|T_{\rho^{\lambda}}T_{\rho^{1-\lambda}}f\|_{q}^{2}$ $\displaystyle\leq\|T_{\rho^{1-\lambda}}f\|_{2}^{2}$ $\displaystyle=\sum_{S}|\rho\hat{f}(S)|^{2(1-\lambda)}|\hat{f}(S)|^{2\lambda}$ $\displaystyle=\|T_{\rho}f\|_{2}^{2(1-\lambda)}\|f\|_{2}^{2\lambda}\qquad\qed$ ###### Proof of Theorem 13. Define the “not-two” predicate $\operatorname{\texttt{NTW}}\colon\\{-1,1\\}^{3}\to\\{-1,1\\}$ as follows: $\operatorname{\texttt{NTW}}(a,b,c)=1$ if exactly two of $a,b,c$ equal $-1$, and $\operatorname{\texttt{NTW}}(a,b,c)=-1$ otherwise. Explicitly, $\begin{array}[]{rrrrrrrrr}a&-1&-1&-1&-1&1&1&1&1\\\ b&-1&-1&1&1&-1&-1&1&1\\\ c&-1&1&-1&1&-1&1&-1&1\\\ \hline\cr\operatorname{\texttt{NTW}}(a,b,c)&-1&1&1&-1&1&-1&-1&-1\end{array}$ Let $\delta\in[0,1]$ be a parameter to be fixed later. For $i=1,\dotsc,n$, we pick bits $x_{i},y_{i},z_{i}\in\\{-1,1\\}$ as follows: * • with probability $1-\delta$: we choose $x_{i},y_{i}$ uniformly and independently, then set $z_{i}=-x_{i}y_{i}$; * • with probability $\delta$: we choose $x_{i}$ uniformly, then set $y_{i}=z_{i}=x_{i}$. Note that for $i\neq j$, $(x_{i},y_{i},z_{i})$ is independent of $(x_{j},y_{j},z_{j})$. We accept if $\operatorname{\texttt{NTW}}(f(x),f(y),f(z))=-1$. It is immediate from the construction of $x_{i},y_{i},z_{i}$ that $\operatorname{\texttt{NTW}}(x_{i},y_{i},z_{i})=-1$ for $i=1,\dotsc,n$. Therefore, if $f$ is a dictator function, it follows that $\operatorname{\texttt{NTW}}(f(x),f(y),f(z))$ must also equal $-1$. #### Soundness. It remains to analyze the test when $f$ is pseudorandom. We begin by writing $\operatorname{\texttt{NTW}}$ in the Fourier basis: $\operatorname{\texttt{NTW}}=-\frac{1}{4}\chi_{\emptyset}-\frac{1}{4}(\chi_{\\{1\\}}+\chi_{\\{2\\}}+\chi_{\\{3\\}})-\frac{1}{4}(\chi_{\\{1,2\\}}+\chi_{\\{2,3\\}}+\chi_{\\{1,3\\}})+\frac{3}{4}\chi_{\\{1,2,3\\}}$. Therefore, by symmetry, $\operatorname*{\mathbb{E}}_{x,y,z}\operatorname{\texttt{NTW}}(f(x),f(y),f(z))=-\tfrac{1}{4}-\tfrac{3}{4}\operatorname*{\mathbb{E}}_{x}f(x)-\tfrac{3}{4}\operatorname*{\mathbb{E}}_{x,y}f(x)f(y)+\tfrac{3}{4}\operatorname*{\mathbb{E}}_{x,y,z}f(x)f(y)f(z).$ We shall systematically rewrite the right-hand side in terms of the Fourier coefficients of $f$. By our assumption that $f$ is odd, we have $\hat{f}(S)=0$ whenever $S$ has even cardinality. Therefore $\operatorname*{\mathbb{E}}f(x)=\hat{f}(\emptyset)=0$. Also, $\operatorname*{\mathbb{E}}_{x,y}f(x)f(y)=\sum_{S,T}\hat{f}(S)\hat{f}(T)\operatorname*{\mathbb{E}}_{x,y}\chi_{S}(x)\chi_{T}(y).$ Consider a summand where $S\neq T$, and without loss of generality fix $i\in S\setminus T$. It is easy to see that the contributions due to $x_{i}=\pm 1$ cancel each other. Thus, the only terms that remain are of the form $S=T$, i.e., $\operatorname*{\mathbb{E}}_{x,y}f(x)f(y)=\sum_{S}\hat{f}(S)^{2}\operatorname*{\mathbb{E}}_{x,y}\chi_{S}(x)\chi_{S}(y)=\sum_{S}\hat{f}(S)^{2}\left(\operatorname*{\mathbb{E}}_{x_{i},y_{i}}x_{i}y_{i}\right)^{|S|}=\sum_{S}\hat{f}(S)^{2}\delta^{|S|},$ where we have used the fact that $\operatorname*{\mathbb{E}}(x_{i}y_{i})=(1-\delta)\cdot 0+\delta\cdot 1=\delta$. But $\hat{f}(S)$ is nonzero only for $|S|$ odd, and $\sum_{S}\hat{f}(S)^{2}=1$, so we can upper-bound the above sum by $\delta$. #### Bounding the cubic term. We proceed similarly: $\operatorname*{\mathbb{E}}_{x,y,z}f(x)f(y)f(z)=\sum_{S,T,U}\hat{f}(S)\hat{f}(T)\hat{f}(U)\operatorname*{\mathbb{E}}_{x,y,z}\chi_{S}(x)\chi_{T}(y)\chi_{U}(z).$ (1) Each of the expectations can be written as a product over coordinates $i\in[n]$ using the fact that individual coordinates of $x,y,z$ are chosen independently. When $i$ belongs to exactly one of $S,T,U$ (say $S$), then it contributes a factor $\operatorname*{\mathbb{E}}x_{i}=0$, making the product zero. Similarly, when $i$ belongs to two of the sets (say $S,T$), then the contribution is $\operatorname*{\mathbb{E}}x_{i}y_{i}=\delta$ by our earlier calculation. Finally, when $i$ belongs to all three of the sets, we have $\operatorname*{\mathbb{E}}x_{i}y_{i}z_{i}=(1-\delta)\cdot(-1)+\delta\cdot(0)=-(1-\delta)$. In light of this calculation, any triple $S,T,U$ that makes a nonzero contribution to the sum (1) must be of the form $\displaystyle S$ $\displaystyle=A\cup B\cup C$ $\displaystyle T$ $\displaystyle=A\cup C\cup D$ $\displaystyle U$ $\displaystyle=A\cup D\cup B$ for suitable sets $A,B,C,D\subseteq[n]$ where $A$ is disjoint from $B,C,D$. Also $|S|,|T|,|U|$ must be odd, from which we can show that $|A|$ must be odd. In terms of these new sets we can rewrite $\operatorname*{\mathbb{E}}_{x,y,z}f(x)f(y)f(z)=-\sum_{\begin{subarray}{c}B,C,D\text{ disj. from }A\\\ |A|\text{ odd}\end{subarray}}\hat{f}(A\cup B\cup C)\hat{f}(A\cup C\cup D)\hat{f}(A\cup D\cup B)(1-\delta)^{|A|}\delta^{|B|+|C|+|D|}.$ For a fixed $A$, define the function $g_{A}\colon\\{-1,1\\}^{[n]\setminus A}\to\mathbb{R}$ by $\hat{g}_{A}(X)=\hat{f}(A\cap X)$. Then we have $\displaystyle\operatorname*{\mathbb{E}}_{x,y,z}f(x)f(y)f(z)$ $\displaystyle=-\sum_{|A|\text{ odd}}(1-\delta)^{|A|}\sum_{\begin{subarray}{c}B,C,D\\\ \text{disj. from }A\end{subarray}}\hat{g}_{A}(B\cup C)\sqrt{\delta}^{|B\cup C|}\cdot\hat{g}_{A}(C\cup D)\sqrt{\delta}^{|C\cup D|}\cdot\hat{g}_{A}(D\cup B)\sqrt{\delta}^{|D\cup B|}$ $\displaystyle=-\sum_{|A|\text{ odd}}(1-\delta)^{|A|}\sum_{\begin{subarray}{c}B,C,D\\\ \text{disj. from }A\end{subarray}}\widehat{T_{\sqrt{\delta}}g_{A}}(B\cup C)\cdot\widehat{T_{\sqrt{\delta}}g_{A}}(C\cup D)\cdot\widehat{T_{\sqrt{\delta}}g_{A}}(D\cup B)$ $\displaystyle=-\sum_{|A|\text{ odd}}(1-\delta)^{|A|}\|T_{\sqrt{\delta}}g_{A}\|_{3}^{3}.$ Write $g_{A}(u)=\operatorname*{\mathbb{E}}_{x}g_{A}(u)+\tilde{g}_{A}(u)=\hat{f}(A)+\tilde{g}_{A}(u)$. Then, using the inequality $|a+b|^{3}\leq 4(|a|^{3}+|b|^{3})$, we have $\|T_{\sqrt{\delta}}g_{A}\|_{3}^{3}=\|\hat{f}(A)+T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}\leq 4|\hat{f}(A)|^{3}+4\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}$ and therefore, $\displaystyle\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}g_{A}\|^{3}\leq 4\sum(1-\delta)^{|A|}|\hat{f}(A)|^{3}+4\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}.$ To bound the first term, note that $\sum(1-\delta)^{|A|}|\hat{f}(A)|^{3}\leq\sum\hat{f}(A)^{2}\cdot\max\\{(1-\delta)^{|A|}|\hat{f}(A)|)\\}$. The sum of the squared Fourier coefficients is just $1$ (by Parseval’s identity) and we can use the $(\delta,\frac{\delta}{\log(1/\delta)})$-pseudorandomness property to bound the quantity in the maximum: when $|A|<\frac{1}{\delta}\log\frac{1}{\delta}$, then $|\hat{f}(A)|\leq\sqrt{\delta}$ and when $|A|\geq\frac{1}{\delta}\log\frac{1}{\delta}$ then $(1-\delta)^{|A|}\leq\delta$. Thus the entire first summand is $O(\sqrt{\delta})$. #### Hypercontractivity. It remains to bound $\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}$. Fix $\lambda=\frac{\log 2}{\log(1/\delta)}$ and apply the modified hypercontractive inequality: $\displaystyle\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{3}^{3}$ $\displaystyle\leq\sum(1-\delta)^{|A|}\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{2}^{3-3\lambda}\|\tilde{g}_{A}\|_{2}^{3\lambda}$ Now, $\|\tilde{g}_{A}\|_{2}^{3\lambda}\leq 1$ and $\|T_{\sqrt{\delta}}\tilde{g}_{A}\|_{2}^{3-3\lambda}=O(\sqrt{\delta})\sum_{\emptyset\neq B\subseteq\overline{A}}\delta^{|B|}\hat{f}(A\cup B)^{2}$. The contribution of the corresponding term to the sum we were trying to bound is $O(\sqrt{\delta})\cdot\hat{f}(A\cup B)^{2}\cdot(1-\delta)^{|A|}\delta^{|B|}$. For each choice of $A\cup B$, the $(1-\delta)^{|A|}\delta^{|B|}$ terms sum to at most one, and all the $\hat{f}(A\cup B)^{2}$ terms themselves sum to at most one. Therefore, we have bounded the entire sum by $O(\sqrt{\delta})$ as desired. ∎ ### 5.2 Integrality gap for Unique Label Cover SDP #### Problem and SDP relaxation. In the Unique Label Cover problem, we are given a label set $L$ and a weighted multigraph $G=(V,E)$ whose edges are labeled by permutations $\\{\pi_{e}\colon L\to L\\}_{e\in E}$, and are asked to find an assignment $f\colon V\to L$ of labels to edges that maximizes the fraction of edges $e\\{u,v\\}$ that are “consistent” with our labeling, i.e., $\pi_{e}(f(u))=f(v)$. If there exists a labeling that satisfies all the edges, then it is easy to find such a labeling. However, when all we can guarantee is that $99\%$ fraction of the edges can be satisfied, it is not known how to find a labeling satisfying even $1\%$ of them. At the same time, present techniques cannot show that finding a $1\%$-consistent labeling is NP-hard. One approach to solving this problem is to use an extension of the Goemans- Williamson SDP for Max-Cut, where we set up a vector $v_{i}$ for every vertex $v$ and label $i$: maximize $\displaystyle\textstyle\operatorname*{\mathbb{E}}_{e\\{u,v\\}}\sum_{i\in L}\langle u_{i},v_{\pi_{e}(i)}\rangle$ subject to $\displaystyle\langle u_{i},v_{j}\rangle\geq 0$ $\displaystyle\forall u,v\in V,\forall i,j\in L$ $\displaystyle\textstyle\sum_{i\in L}\langle v_{i},v_{i}\rangle=1$ $\displaystyle\forall v\in V$ $\displaystyle\langle\textstyle\sum_{i\in L}u_{i},\textstyle\sum_{j\in L}v_{j}\rangle=1$ $\displaystyle\forall u,v\in L$ $\displaystyle\langle v_{i},v_{j}\rangle=0$ $\displaystyle\forall v\in V,\forall i\neq j\in L$ (The expectation in the objective is over a distribution where $e\\{u,v\\}$ is picked with probability proportional to its weight.) The intent is that $\|v_{i}\|^{2}$ should be the probability that $v$ receives label $i$, and $\langle u_{i},v_{j}\rangle$ should be the corresponding joint probability. It is easy to see that this SDP is a relaxation of the original problem. #### Gap instance. In an influential paper, Khot and Vishnoi [KV05] constructed an integrality gap for this SDP: for a label set of size $2^{k}$ and an arbitrary parameter $\eta\in[0,\frac{1}{2}]$, a graph whose optimal labeling satisfies $\leq 1/2^{\eta k}$ fraction of the edges, but for which the SDP optimum is at least $1-\eta$. The hypercontractive inequality plays a central role in the soundness analysis, which we present below. Let $\tilde{V}$ be the set of all functions $f\colon\\{-1,1\\}^{k}\to\\{-1,1\\}$ and $L$ be the Fourier basis $\\{\chi_{S}\mid S\subseteq[k]\\}$; clearly, $|L|=2^{k}$. Observe that $\tilde{V}$ is an Abelian group under pointwise multiplication, and $L$ is a subgroup. We take the quotient $V=\tilde{V}/L$ to be the vertex set. Fix an arbitrary representative for each coset and write $V=\\{f_{1}L,f_{2}L,\dotsc,f_{|V|}L\\}$. We shall define a weighted edge between every pair of these representative functions, then show how to extend this definition to all pairs of functions, and finally map these edges to edges between cosets. * • The edge $\tilde{e}\\{f,g\\}$ has weight equal to $\Pr_{h,h^{\prime}}[(f,g)=(h,h^{\prime})]$, where $h,h^{\prime}\in V$ are drawn to be $\rho$-correlated on every bit with uniform marginals, where $\rho=1-2\eta$. * • With every edge $\tilde{e}\\{f_{i},f_{j}\\}$ between representative functions, we associate the identity permutation. * • A non-representative function acts as if its label is assigned according to its coset’s representative. Thus, the permutation associated with $\tilde{e}\\{f_{i}\chi_{S},f_{j}\chi_{T}\\}$ is $\chi_{U}\chi_{S}\mapsto\chi_{U}\chi_{T}$. * • In the actual graph under consideration, every edge $\tilde{e}\\{f_{i}\chi_{S},f_{j}\chi_{T}\\}$ appears as an edge $e\\{f_{i}L,f_{j}L\\}$ (with the same permutation and weight). #### Soundness analysis. Given a labeling $R\colon V\to L$ on the cosets, we consider the induced labeling $\tilde{R}\colon\tilde{V}\to L$ given by $\tilde{R}(f_{i}\chi_{S})=R(f_{i}L)\chi_{S}$. From our definitions, it is clear that the objective value attained by $\tilde{R}$ is precisely $\Pr_{h,h^{\prime}}[\tilde{R}(h)=\tilde{R}(h^{\prime})]$, where $h,h^{\prime}$ are chosen as before. Fix any label $\chi_{S}$ and consider the indicator function $\phi\colon\tilde{V}\to\\{0,1\\}$ of functions that $\tilde{R}$ labels with $\chi_{S}$. Since exactly one function in each coset gets labeled $\chi_{S}$, we know that $\operatorname*{\mathbb{E}}\phi=1/2^{k}$. Therefore, $\Pr_{h,h^{\prime}}[\tilde{R}(h)=\tilde{R}(h^{\prime})=\chi_{S}]=\operatorname*{\mathbb{E}}_{h,h^{\prime}}[\phi(h)\phi(h^{\prime})]=\langle h,T_{\rho}h\rangle=\|T_{\sqrt{\rho}}h\|^{2}_{2},$ which we can upper-bound (using hypercontractivity) by $\|h\|^{2}_{1+\rho}=1/2^{\frac{2k}{1+\rho}}\leq 1/2^{\eta k}$. ## References * [ARV09] Sanjeev Arora, Satish Rao, and Umesh V. Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2), 2009. * [Bor82] C. Borell. Positivity improving operators and hypercontractivity. Mathematische Zeitschrift, 180(3):225–234, 1982. * [DSC96] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. The Annals of Applied Probability, 6(3):695–750, 1996. * [Fri98] E. Friedgut. Boolean functions with low average sensitivity depend on few coordinates. Combinatorica, 18(1):27–35, 1998. * [Gro75] L. Gross. Logarithmic sobolev inequalities. 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arxiv-papers
2011-01-14T21:30:47
2024-09-04T02:49:16.481227
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Punyashloka Biswal", "submitter": "Punyashloka Biswal", "url": "https://arxiv.org/abs/1101.2913" }
1101.3257
# The Chromospheric Activity, Age, Metallicty and Space Motions of 36 Wide Binaries J. K. Zhao11affiliation: Florida Institute of Technology, Melbourne, USA, 32901 22affiliation: Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China , T. D. Oswalt11affiliation: Florida Institute of Technology, Melbourne, USA, 32901 , M. Rudkin11affiliation: Florida Institute of Technology, Melbourne, USA, 32901 , G. Zhao22affiliation: Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China , Y. Q. Chen22affiliation: Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China jzhao@fit.edu toswalt@fit.edu mrudkin@fit.edu gzhao@bao.ac.cn cyq@bao.ac.cn ###### Abstract We present the chromospheric activity (CA) levels, metallicities and full space motions for 41 F, G, K and M dwarf stars in 36 wide binary systems. Thirty-one of the binaries, contain a white dwarf component. In such binaries the total age can be estimated by adding the cooling age of the white dwarf to an estimate of the progenitor’s main sequence lifetime. To better understand how CA correlates to stellar age, 14 cluster member stars were also observed. Our observations demonstrate for the first time that in general CA decays with age from 50 Myr to at least 8 Gyr for stars with 1.0 $\leq$ V-I $\leq$ 2.4. However, little change occurs in CA level for stars with V-I $<$ 1.0 between 1 Gyr and 5 Gyr, consistent with the results of Pace et al. (2009). Our sample also exhibits a negative correlation between stellar age and metallicity, a positive correlation between stellar age and W space velocity component and the W velocity dispersion increases with age. Finally, the population membership of these wide binaries is examined based upon their U, V, W kinematics, metallicity and CA. We conclude that wide binaries are similar to field and cluster stars in these respects. More importantly, they span a much more continuous range in age and metallicity than is afforded by nearby clusters. activity: Stars —late-type: Stars—white dwarfs: Stars ## 1 Introduction Age is one of the most difficult to determine properties of a star. The Vogt- Russell theorem asserts that the structure of a star is uniquely determined by its mass and composition. Nucleosynthesis in the core results in changes in composition and this implies at least some measurable property(ies) of a star must vary with age. Unfortunately, these changes are subtle and difficult to measure. It is ironic that the age of the Universe (13.7 $\pm$ 0.2 Gyr; Bennett et al. 2003) is known to better precision than the age of any star other than the Sun. The present methods by which stellar ages can be estimated are seldom consistent within 50% (Soderblom 2010). Even the Sun does not reveal its age directly; this key calibration point is determined from the decay of radioisotope samples to be 4,566${}_{-1}^{+2}$ Myr (Chaussidon 2007). One of the ‘semifundamental’ methods of stellar age determination is isochrone fitting the position of a star in the Hertzsprung-Russell diagram (HRD). However, because of the degeneracy of theoretical isochrones, this technique does not work well for the vast majority of stars-those on the lower main sequence (MS). Here small errors in luminosity or metallicity translate into large errors in age. CaII H$\&$K features in the violet spectra of MS stars are one of the more well-studied indicators of CA. Early work by Wilson (1963; 1968) and Vaughan $\&$ Preston (1980) established CaII H$\&$K emission as a useful marker of CA in lower MS stars. In F to early M stars Skumanich (1972) found that CaII H$\&$K emission, magnetic field strength and rotation all decay as the inverse square root of stellar age. Mamajek $\&$ Hillenbrand (2008) and others have shown that the CA vs. age relation is much more complex than Skumanich envisioned; such factors as metallicity, photospheric contamination of CA indices and variation in CA must be considered. Clusters provide only a limited range of ages and metallicities to investigate these effects. A self-sustaining magnetic dynamo driven by rotation and convection is believed to be the source of CA in MS stars of spectral type F, G, K and early M. According to this paradigm, due to magnetic breaking, rotational velocity decreases with age, which leads to a decrease in CA as well, unless angular momentum is sustained by tidal interaction, as in the case of short-period binaries, or maintained by convection as in late M type dwarfs. Using a lower resolution analog of the Mount Wilson S index in eight clusters and the Sun, Barry, Cromwell $\&$ Hege (1987) found that the decay of CA is well-represented by an exponential over the age range from $10^{7}$ to 6 $\times$ $10^{9}$ yr. Barry (1988) adjusted the age of three clusters from Barry, Cromwell $\&$ Hege (1987). Using a color correction C${}_{cf}^{\prime}$ from Noyes et al. (1984), Barry found empirically that CA $\sim$ t1/e. In addition, he concluded that the star formation rate in the solar neighborhood has not been constant, suggesting a recent burst of star formation because of the large number of stars in his youngest age bins. Working toward a more detailed understanding of the CA vs. age relation, Soderblom, Duncan $\&$ Johnson (1991) found that, while a power law is generally the best fit to the CaII H$\&$K vs. age relation, it implies a constant star formation rate (SFR). Any different SFR causes the Skumanich relation to indicate an excess of young stars in the solar neighborhood. They also found that the calibrated cluster data presented in Barry, Cromwell $\&$ Hege (1987) were consistent with a constant SFR. Pace $\&$ Pasquini (2004) found that for several clusters older than 1 Gyr there appeared to be a constant activity level. Pace et al. (2009) believed stars change from active to inactive, crossing the activity range corresponding to the so-called ‘Vaughan-Preston gap’, on a time-scale that might be as short as 200 Myr. If true, this would bring into question whether the Skumanich relation is valid for MS stars in all age regimes. Among wide white dwarf (WD)+dM binaries Silvestri, Hawley $\&$ Oswalt (2005) used the cooling ages of WD components, plus an average estimate for MS lifetime, to explore the activity vs. age relation among lower MS stars. This study confirmed that stars later than spectral type dM3 do not exhibit a Skumanich-style CA vs. age relation. In a study of activity among unresolved WD+dM cataclysmic variable candidates Silvestri et al (2006) again used the white dwarf (WD) cooling times alone as lower age limit. Both of these studies found the general trend seen in clusters, i.e, that later dM spectral types remain active at a roughly constant level for a longer period of time than earlier spectral types, whereupon each star becomes inactive. The transition from active to inactive appears to take place over a relatively short period of time. However, in both studies some dM stars in binaries were found to exhibit activity more characteristic of brighter, bluer and more massive M dwarfs than seen in clusters. West et al. (2008) examined the age-activity relation among a sample of over 38,000 low-mass (M0-M7) stars drawn from the Sloan Digital Sky Survey (SDSS) Data Release 5 (DR5) and also found later spectral types remain active longer. It is important to note that late type stars do not seem to decline in activity monotonically, they are either ‘on’ if young, or ‘off’ if old. While the activity turnoff point on the lower MS constrains a cluster’s age it is not a useful means for estimating the age of a field star. This paper describes our analysis of a sample of common proper motion binary (CPMB) systems. Most of the systems selected have a late MS star paired with a WD component. All have relatively wide orbital separations ($<a>$ $\sim$ 103 AU; Oswalt et al. 1993). Thus, each component is assumed to have evolved independently, unaffected by mass exchange or tidal coupling that complicates the evolution of closer pairs (Greenstein 1986). It is also assumed that members of such binaries are coeval. Essentially, each may be regarded as an open cluster with only two components. Such binaries are far more numerous than clusters and span a much broader and more continuous range in age. Thus, they are potentially very valuable to investigations of phenomena that depend upon age. The total age of a CPMB can be estimated from the cooling age of the WD component added to an estimate of its progenitor’s MS lifetime. Since some CPMBs in our sample have ages well beyond the present $\sim$4 Gyr nearby cluster limit, this provides an opportunity to test the CA vs. age relation in much older MS stars. Wide binaries present another opportunity. From the spectra of the MS components, the metallicity can be measured. Presumably this is also the original metallicity of the WD progenitor. In addition, the radial velocities of the MS stars are easy to determine. Since proper motions are available, with trigonometric or photometric parallaxes, full space motions for all systems in the observed sample can be estimated. Thus, CPMBs present an opportunity to investigate the relations among age, metallicity and space motion, as well as population membership even for WDs, which often have weak or no spectral feature. In section 2 we present an overview of the observations and reductions for our sample. A discussion of the CA vs. age relation is given in section 3. Our age and metallicity relation is presented in section 4. In section 5, we describe our analysis of population membership. We conclude with a discussion of the implications of our findings in section 6. ## 2 Observations and Data Reduction Most stars chosen for this study are components of wide MS+WD pairs from the Luyten (1979) and Giclas, Burnham $\&$ Thomas (1971) proper motion catalogs chosen by Oswalt, Hintzen $\&$ Luyten (1988). A key impetus for using such pairs in this study is that the total lifetime of each pair should be approximately the age derived from measurements of the MS component. In addition the total age of a pair should be approximately the sum of the WD component’s cooling time and the MS lifetime of its progenitor. Table 1 gives the observed data for 36 wide binaries. Column 1 is a unique ID number. Columns 2 - 4 list each component’s name, right ascension, and declination (coordinates are for epoch 1950). The V magnitudes and original low-resolution spectroscopic identifications are given in columns 5 - 8. Column 9 is the observation date for the high resolution ($\sim$ 2$\rm\AA$) spectra in the present study. Columns 10 - 13 list the proper motion, direction of proper motion (measured east of north), position angle (centered on the primary measured east of north), and separation of the components (in arcseconds), respectively. Of the 36 wide binaries, 5 systems consist of two MS stars, 1 system is a triple WD+dK+dM, and the remaining 30 are MS+WD pairs. A sample of cluster MS stars of known age, such as IC 2391, IC 2602 and M67, previously studied by Patten $\&$ Simon (1993, 1996), Barrado y Navascués, Stauffer $\&$ Jayawardhana (2004) and Giampapa et al. (2006), were adopted as ‘CA standards’. These stars were routinely observed in the course of our observing program for the CPMBs. Table 2 provides the observational information for these cluster member stars. Column 1 is a unique ID. Columns 2 - 4 list the name, V magnitude and observation date. The colors V-I and B-V are given in column 5 - 6. The CA flux ratio SHK (see Section 3.1), age and cluster membership are given in columns 7 - 9. The last column provides the corresponding literature source for each star. ### 2.1 BVRI Photometry We used BVRI photometric data for our wide binaries from Smith (1997) whenever available. Photometric data for some stars that were not in this source were taken from the literature identified by the Simbad Astronomical Database (Genova 2006). Photometric colors of other stars were estimated from our spectra by empirical calibrations. For example, Fig.1 is the relation between V-I and CaI4227 flux ratio. The feature of this index is within 4211 - 4242 $\rm\AA$ and the continuum ranges within 4152 - 4182 $\rm\AA$. The filled circles are stars with known V-I color. The dotted line is a least-squares fit. The scatter $\rm\sigma_{V-I}\approx$ 0.22. We used this relation to estimate the V-I for stars with no published colors. For the stars in cluster IC 2602, IC 2391 and M67, photometric colors were taken from Barnes, Sofia $\&$ Prosser (1999 and references therein), Patten $\&$ Simon (1996) and Giampapa et al. (2006). ### 2.2 Spectroscopic Observations In the southern hemisphere, observations were conducted at Cerro Tololo Interamerican Observatory (CTIO) using the Blanco 4-meter telescope. The Ritchey-Chr$\acute{e}$tien (RC) Cassegrain spectrograph was used on two separate observing runs (February 2004 and February 2005) to obtain optical spectroscopy of CPMBs, as well as the CA standard stars. During the two observation runs, the KPGL1 grating was used to obtain spectra with a scale of 0.95 $\rm{\AA}$/pixel. A Loral 3K CCD (L3K) was used with the RC spectrograph. It is a thinned 3K$\times$1K CCD with 15 $\mu$m pixels. A spectral range of approximately 3800 $\rm{\AA}$ to 6700 $\rm{\AA}$ was achieved. Northern hemisphere observations were conducted at Kitt Peak National Observatory (KPNO) using the Mayall 4-meter telescope. The RC spectrograph, with the BL450 grating set for the 2nd order to yield a resolution of 0.70 $\rm{\AA}$/pixel, was used to obtain optical spectra during the July 2005 and November 2006 observing runs. The 2K$\times$2K T2KB CCD camera with 24 $\mu$m pixels was used to image the spectra. An 8-mm CuSO4 order-blocking filter was added to decrease 1st-order overlap at the blue end of the spectrum. A spectral range of approximately 3800 $\rm{\AA}$ to 5100 $\rm{\AA}$ was achieved. ### 2.3 Data Reduction The data were reduced with standard IRAF111IRAF is distributed by the National Optical Astronomy observatories, which are operated by the Association of Universitites for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation (http://iraf.noao.edu). reduction procedures. In all cases, program objects were reduced with calibration data (bias, flat, arc, flux standard) taken on the same night. Data were bias- subtracted and flat-fielded, and one-dimensional spectra were extracted using the standard aperture extraction method. A wavelength scale was determined for each spectrum using HeNeAr arc lamp calibrations. Flux standard stars were used to place the spectra on a calibrated flux scale. We emphasize that these are only relative fluxes, as most nights were not spectrophotometric. The radial velocity of each MS star was determined by cross-correlation between the observed spectra and a set of MS template spectra. The F, G and K template spectra were generated from a theoretical atmosphere grid (Castelli $\&$ Kurucz 2003). The dM template spectra were compiled using observed M dwarf spectra from the Sloan Digital Sky Survey (SDSS)222http://www.astro.washington.edu/slh/templates. The wavelength ranged from roughly 3900 - 9200 $\rm\AA$ (see Bochanski et al. 2007). Our typical internal measurement uncertainties in radial velocity were $\sigma\rm_{v_{r}}$ = $\pm$ 4.6 km s-1. The final radial velocities listed in column 5 of Table 3 were corrected to the heliocentric frame. ## 3 CA-Age Relation ### 3.1 Measurement of SHK The flux ratio $\displaystyle\rm{S_{HK}}$ $\displaystyle=$ $\displaystyle\alpha\rm{\frac{H+K}{R+V}}$ (1) was determined for each MS star, where H and K are the fluxes measured in 2 $\rm{\AA}$ rectangular windows centered on the line cores of CaII H$\&$K. Here R and V are the fluxes measured in 20 $\rm{\AA}$ rectangular ‘pseudocontinuum’ windows on either side. Although these are not strictly equivalent to the triangular windows Wilson (1968) used with his photomultiplier-based spectrometer, Hall, Lockwood $\&$ Skiff (2007) have shown that using 1.0 $\rm{\AA}$ rectangular H$\&$K windows produces results that are easily calibrated to the Baliunas et al. (1995) analysis of Wilson’s (1968) original survey of bright MS stars. Our choice of 2 $\rm{\AA}$ windows is set by the resolution of the CTIO and KPNO instrumentation, but it detects CaII H$\&$K emission nearly as well and allows fainter stars to be observed. Gray et al. (2003) have shown that even a resolution of $\sim$4$\rm\AA$ can produce useful measure of SHK. In our measurement, the scale $\alpha$ is 10.0, reflecting the fact that the continuum windows are 10 times wider than the H$\&$K windows. In our sample some stars were observed two or three times. In such cases, the mean of these measurements was adopted as the star’s $\rm{S_{HK}}$ and the scatter as the uncertainty. For those stars observed only once, we adopted the average uncertainty derived from those stars having more than one observation ($\pm$4.6%). The $\rm{S_{HK}}$ indices of all the MS stars are shown in column 4 of Table 3. For the purpose of this study we need only a calibration of CA vs. age on our instrumental system. Some stars were observed both at CTIO and KPNO. We found an empirical calibration: Sctio=Skpno+0.095. To remove this small instrumental effect, all the SHK measured at KPNO were transformed into the CTIO instrumental system with this relation. ### 3.2 The age determination Our sample includes 14 cluster member stars: 6 in IC2602, 6 in IC2391 and 2 in M67. The ages of IC2602 and IC 2391 are approximately the same ($\sim$ 50$\pm$5 Myr; Barrado y Navascues, Stauffer $\&$ Jayawardhana 2004). They obtained intermediate-resolution optical spectroscopy of 44 potentially very low mass members of IC2391 and derived the cluster age from a comparison of several theoretical models. The most recent age determination for M67 is 4.05$\pm$0.05 Gyr (Jorgensen $\&$ Lindegren 2005). Among our 31 systems containing WD components, the ages of 23 were determined by using computed cooling times of WD companions added to estimates of their progenitor’s MS lifetimes. The $T$eff and log $g$ of each WD companion was obtained from the literature (see Table 4). The ages of the remaining 8 wide binaries were not obtained because the $T$eff and log $g$ of WD companions could not be obtained from our spectra (3: DC type; 1: DQ; 4: low S/N) and could not be found in the literature. In cases where uncertainties were not given, we adopted 200 K and 0.05 as the average uncertainty for $T$eff and log $g$, respectively. This decision was based on the recommendation of Bergeron, Saffer $\&$ Liebert (1992) who believe the internal errors are typically 100-300 K in $T$eff and 0.02-0.06 in log $g$. From the $T$eff and log $g$ of each WD, its mass (MWD) and cooling time (tcool) were estimated from Bergeron’s cooling sequences333The cooling sequences can be downloaded from the website: http://www.astro.umontreal.ca/ bergeron/CoolingModels/.. For the pure hydrogen model atmospheres above $T$eff = 30,000 K we used the carbon- core cooling models of Wood (1995), with thick hydrogen layers of qH = MH/M∗ = 10-4. For $T$eff below 30,000 K we used cooling models similar to those described in Fontaine, Brassard $\&$ Bergeron (2001) but with carbon-oxygen cores and qH = 10-4 (see Bergeron, Leggett $\&$ Ruiz 2001). For the pure helium model atmospheres we used similar models but with qH = 10-10. MWD and tcool were then calculated by spline interpolation based on the $T$eff and log $g$. In Table 4, columns 6 and 7 list the final MWD and tcool for these 23 WDs. Although $T$eff and log $g$ are from different literature sources, the parameters are consistent for common stars in these references. Thus, there appears to be no systematic uncertainties expected among the $T$eff and log $g$ we used. Using the Initial-Final Mass Relation (IFMR; equations 2 - 3 presented in Catalán et al. 2008b), we then estimated the progenitor masses Mi of the WDs. There are two WDs whose masses are lower than 0.5M⊙ and the current IFMR does not extend to such low mass objects. From Fig. 2 in Catalán et al. (2008b), we adopted Mi $\sim$ 1.25M⊙ for MWD $<$ 0.5M⊙. Next, using the third-order polynomial of Iben $\&$ Laughlin (1989), $\displaystyle\rm{log~{}t_{evol}}$ $\displaystyle=\rm{9.921-3.6648(logM_{i})+1.9697(logM_{i})^{2}-0.9369(logM_{i})^{3}}$ (2) we determined the MS lifetime tevol (in years) corresponding to each Mi, progenitor mass of the WD (in M⊙). Columns 8 - 9 in Table 4 are the Mi and tevol we derived for 23 WDs. Finally, the total ages of these wide binaries were computed by adding tevol to the cooling ages of the WDs (tcool; column 11 in Table 3). Independent age determination for 4 pairs were found in the literature. These pairs are listed in Table 5. Column 1 gives their identifications. Columns 2-3 list the ages included in Holmberg, Nordström $\&$ Andersen (2009) and Valenti $\&$ Fischer (2005), respectively. It can be seen that our ages are younger than isochrone fitting ages in all four cases. For 40 Eri A the isochone fitting age is unreasonably large, while our age of this star is consistent with the rotation age 4.75$\pm$0.75 Gyr from Barnes (2007). For CD-37 10500, the error bar is too large for isochrone fitting age to be useful. It is very difficult to estimate the age for K and M dwarfs with the isochrone fitting method because of the narrow vertical dispersion of isochrones within the MS in an HR diagram. Small uncertainties in luminosity and metallicity cause large uncertainties in age. Using the white dwarf cooling times plus the estimated progenitor lifetimes should give a more consistent age for such stars. Note that the internal uncertainties of our age determinations are smaller than those derived from isochrone fitting. Although we have taken a similar approach, there are a few differences between our age determinations and those by Silvestri, Hawley $\&$ Oswalt (2005) and Silvestri et al. (2006). Silvestri et al. (2006) only used the WD cooling time as a lower limit to the age of each binary without considering the WD progenitor’s lifetime in evaluating the age-activity relation among close binaries. Therefore, the ages were somewhat less than actual ages. In Silvestri, Hawley $\&$ Oswalt (2005), each age was estimated from the WD’s cooling time plus an estimate of its progenitor’s lifetime. The Mi of each WD was computed from the IFMR of Weidemann(2000) using an adopted mean WD mass of 0.61 M⊙. We explicitly computed the Mi of each WD from the IFMR of Catal$\rm\acute{a}$n (2008b) using our own estimate of its current mass (Mf). In view of this, our method should provide more precise age estimates. ### 3.3 Discussion All MS stars are plotted in Fig. 2, which displays our empirical relation among log(SHK), age and V-I. Asterisks represent the young cluster stars in IC2602 and IC2391 that have the same age 50$\pm$5 Myr (Barrado y Navascues, Stauffer $\&$ Jayawardhana 2004). It is clear that log(S$\rm{}_{HK})$ tends to be larger in red stars and smaller in blue stars at the same age. Two M67 cluster member stars are plotted with squares. M67 is much older than IC2602 and IC2391 and these points clearly demonstrate the expectation that CA declines with age. The plus ( + ) signs represent field stars whose ages are between 1.0 Gyr and 5.0 Gyr. Diamonds represent stars whose ages are between 5.0 Gyr and 8.0 Gyr. Our sample includes five known flare stars (40 Eri C, LP891-13, LP888-63, G216-B14A, BD+26 730) which are displayed as filled circles. These clearly show enhanced CA for their supposed age. Fig. 2 can be divided into four arbitrary age domains, represented by dotted lines. The top domain mainly consists of young active stars and flare stars. The other three domains consist of less active stars whose ages are, respectively, 1.0 - 5.0 Gyr, 5.0 - 8.0 Gyr and $>$ 8.0 Gyr. The typical uncertainty in V-I and log(SHK) for inactive stars are displayed at the right- top of Fig. 2 as an error bar. A least-squares fit to stars in IC2602 and IC2391 (dash-dot line) illustrates that this group conforms to a consistent age in the log(S$\rm{}_{HK})$ vs. V-I plane. Stars to the right of the vertical dashed line at V-I = 2.4 are later in spectral type than $\sim$dM3. Silvestri, Hawley $\&$ Oswalt (2005) found no Skumanich-style CA vs. age relation among stars later than dM3, in accord with the expectation that a so-called turbulent dynamo drives CA in such stars (See Reid $\&$ Hawley 2000 and reference therein). Thus, stars in this region are not expected to follow the CA vs. age relation seen in earlier spectral type MS stars. Overall, it can be seen that CA generally decays with age from 50 Myr to at least 8 Gyr for stars with 1.0 $\leq$ V-I $\leq$ 2.4. However, for stars bluer than V-I $\approx$ 1.0, CA shows little variation between 1.0-5.0 Gyr, which is consistent with the results of Pace et al. (2009) who found that CA from 1.4 Gyr (NGC3680) to 4.0 Gyr (M67) remains almost constant. Also note that lines of constant age do not have the same slope. For young stars the lines of constant age have a relatively steep slope, while they appear to be much flatter for old stars. Some comments on a few individual stars in Fig. 2 are appropriate. The star labeled as 1 in Fig. 2 is G163-B9A. Its companion, G163-B9B, was identified as an sdB star by Wegner $\&$ Reid (1991) and Catalán et al. (2008a). The CA of G163-B9A is very strong. Thus, it is either a very young star or perhaps was observed during a flare event. The latter possibility was eliminated based on a detailed examination of its spectrum. Thus we conclude G163-B9A/B is probably not a physical pair. Stars labeled as 2-3 are possible halo stars; star 4 has the weakest CA. These will be discussed in Section 5. There are 5 wide binaries (CD-31 1454/LP888-25; G114-B8A/B; CD-31 7352/LP902-30; LP684-1/2 and LP387-1/2) that consist of pairs of MS stars. Only LP387-1/2 was observed at KPNO while all others were observed at CTIO. Since the two components of coeval binaries are expected to have consistent levels of CA, this provides a good reality check on our CA, V-I and age relation. Unfortunately, the spectrum of LP888-25, the companion to CD-31 1454, was unusable because of low signal-to-noise (S/N). In addition, we concluded that G114-B8A/B is not a physical binary because the components’ radial velocities are inconsistent. The solid lines in Fig. 2 connect the MS components in the other three pairs CD-31 7352/LP902-30, LP684-1/2 and LP387-1/2, that most likely are physical pairs. The components in CD-31 7352/LP902-30 and LP387-1/2 have CA consistent with the same age. Some age difference is implied in LP684-1/2 though the difference is within the uncertainty implied by the error bar in the upper right corner of Fig. 2. Clearly, CA, along with radial velocity, provides a very useful way for filtering out nonphysical pairs. ## 4 The age-[Fe/H] Relation The metallicity of each MS star was estimated by comparing the observed spectra to a set of template spectra. Initially, a library of low resolution theoretical spectra was generated using the SYNTH program, based on Kurucz’s New ODF atmospheric models (Castelli $\&$ Kurucz 2003). The atmosphere models assume local thermodynamic equilibrium (LTE). A mixing-length of l$/$Hp = 1.25 and a microturbulence $\xi$ = 2 km s-1 were adopted. The line list included the atoms and molecules from Kurucz (1993). The maximum correlation method was applied to find the closest matching theoretical spectra to each observed spectrum. Since our objects are MS stars, we adopted log $g$ = 4.5 and [Fe/H] = 0.0 as initial values. The effective $T$eff was then estimated based on the maximum correlation method. Once an estimated $T$eff was obtained, a new estimated [Fe/H] could be obtained with the same procedure. After several iterations, the best-fitting parameters stabilized and computations were terminated. The left panel of Fig. 3 displays the correlation coefficient vs. [Fe/H] in a typical fit. Each open circle represents one template. The filled circle is the maximum correlation coefficient obtained from polynomial fitting (solid line). The value of [Fe/H] at this point is regarded as our best estimate for this star’s metallicity. The diamond points mark the templates yielding the maximum correlation (‘+’ : highest template point above; ‘-’: the highest template point below). The difference in [Fe/H] between the two diamond points and the filled circle is taken as the estimated uncertainty in metallicity. The right panel of Fig. 3 compares our [Fe/H] estimates to those which could be found in the literature. The mean difference is smaller than 0.15 dex, suggesting our metallicity is basically consistent with that of other work. [Fe/H] measurements were not made for stars later than M3 because we do not have templates for the latest spectra nor could we expect the CA vs. age determination to be valid in such late type MS stars. The resulting metallicities estimated for 37 MS stars in our sample are given in column 10 of Table 3. Although one of the key consequences of the stellar evolution theory is the gradual increase in the metal content of the interstellar medium (ISM) and the progressive enrichment of subsequent stellar generations, some studies have found little, if any, indication that an age-metallicity relation (AMR) exists amongst solar neighborhood late-type stars. For example, Rocha-Pinto et al. (2000) studied the AMR using a sample of 552 late-type dwarfs. For those stars, metallicities were estimated from uvby data, and ages were calculated from their chromospheric emission levels using a metallicity-dependent CA vs. age relation developed by Rocha-Pinto $\&$ Maciel (1998). The resulting AMR was found to be a smooth function in their analysis. Feltzing, Holmberg $\&$ Hurley (2001) found that the solar neighborhood age-metallicity diagram is well populated at all ages in a sample of 5828 dwarf and sub-dwarf stars from the Hipparcos Catalogue. Bensby, Feltzing $\&$ Lundström (2004) investigated the AMR using a sample of 229 nearby thick disk stars. Their results indicate that there is indeed an AMR in the thick disk. They found that the median age decreases by about 5-7 Gyr when going from [Fe/H]$\approx$ -0.8 to [Fe/H]$\approx$ -0.1. Fig. 4 is our [Fe/H] vs. age relation for 21 stars. An asterisk represents one likely halo star (see next section). Disk stars are displayed as filled circles. The typical uncertainties in age and [Fe/H] are shown at the left- bottom of this figure. The dotted line is a least-squares fit for only disk stars, while the dashed line is the fit for all stars. The expected trend in age-metallicity is apparent: old stars tend to be of lower metallicity. Our [Fe/H] vs. age relation differs somewhat from the early work on single stars by Barry (1988; see Fig. 5 in that reference). Our Fig. 4 shows a more clear trend, quite similar in fact to the newer study of field stars by Rocha- Pinto $\&$ Maciel (2000); the slope of their relation and ours are approximately the same. We conclude that wide binaries exhibit a [Fe/H] $\sim$ age relation similar to field stars. However, at present our sample contains too few stars to support a detailed examination of the nearby star formation history. ## 5 Population Membership Column 6 of Table 3 lists the parallaxes of our wide binaries. Some trigonometric parallaxes were obtained from the Simbad Astronomical Database (Genova 2006). For 11 wide binaries lacking trigonometric parallaxes, we computed photometric parallaxes using equation 3-5 which were derived from the data in Bergeron, Leggett $\&$ Ruiz (2001). $\displaystyle\rm{\pi}$ $\displaystyle=\rm{10^{-(\frac{V-M_{v}+5}{5}})}$ (3) $\displaystyle\rm{M_{v}}$ $\displaystyle=\rm{12.2199+1.8152(V-I)+2.9704(V-I)^{2}-1.7082(V-I)^{3}}$ (4) $\displaystyle\rm{M_{v}}$ $\displaystyle=\rm{11.7099+6.6038(B-V)-3.7273(B-V)^{2}+0.8066(B-V)^{3}}$ (5) The rectangular velocity components relative to the Sun for 41 MS stars were then computed and transformed into Galactic velocity components U, V, and W, and corrected for the peculiar solar motion (U, V, W) = (-9, +12, +7) km s-1 (Wielen 1982). The UVW-velocity components are defined as a right-handed system with U positive in the direction radially outward from the Galactic center, V positive in the direction of Galactic rotation, and W positive perpendicular to the plane of the Galaxy in the direction of the north Galactic pole. The typical uncertainties in U, V and W are no more than $\sim$10 km s-1. Columns 7-9 of Table 3 list our computed U, V and W velocity components. The top panel of Fig. 5 shows contours, centered at (U, V) = (0, -220) km s-1, that represent 1$\sigma$ and 2$\sigma$ velocity ellipsoids for stars in the Galactic stellar halo as defined by Chiba $\&$ Beers (2000). Six stars lie outside the 2$\sigma$ velocity contour centered on (U, V) = (0, -35) km s-1 defined for disk stars (Chiba $\&$ Beers 2000). The bottom of Fig. 5 shows the Toomre diagram of our stars. Venn et al. (2004) suggest stars with Vtotal $>$ 180km s-1 are possible halo members. There are two stars that meet this criterion. Taking metallicity, space motion and CA into account, they have a high probability of belonging to the halo population. One is LHS300A ([Fe/H]= -0.95 $\pm$ 0.25) which is identified as a thick disk star in Monteiro et al. (2006). Considering the metallicity and space motion, we think it is a halo star. The other is CD-31 1454 ([Fe/H] = -0.48 $\pm$ 0.02) which is regarded as a halo star by Chanamé $\&$ Gould (2004). The two likely halo stars are also labeled in Fig. 2 as number 2 and 3 respectively. Their CA is weak, suggesting ages in excess of 5 Gyr. These two plausible halo stars are displayed as asterisks plus filled circles in the bottom pane of Fig. 5. The star numbered 4 in Fig. 2 is G114-B8B ([Fe/H] = -0.40 $\pm$ 0.08) which may be the oldest star in our sample. Its age appears to be older than 8 Gyr as suggested by its location in Fig. 2. The other stars are most likely members of the disk. The above analysis demonstrates the difficulty of making a population assignment on the basis of only space velocity or metallicity or CA. Ideally all three should be used. The left panel of Fig. 6 displays the computed absolute value of the W components of our stars’ space motions vs. estimated ages for 21 stars for which complete information is available. As expected, old stars tend to have larger W velocity. A weak positive correlation between the vertical velocity (W) of stars in CPMBs and age is expected based on the standard paradigm for stellar encounters in the disk. The right panel of Fig. 6 presents the dispersion in W as a function of age. It is clear that in general the W dispersion becomes larger with age from 1 Gyr to 8 Gyr. ## 6 Conclusion In this study we presented the CA levels, ages, metallicities and space motions for components of 36 wide binaries. WD components were identified in 31 wide binaries. We also observed a sample of cluster member stars with well- determined ages in order to test the expected CA vs. age relation. The ages of 23 wide binaries were derived by the cooling time of each WD companion added to the lifetime estimate of its progenitor. We first examined the relation among log (SHK), V-I and age. Our results support the expected hypothesis established among single and cluster MS stars, i.e., in general CA declines with age for stars with 1.0 $\leq$ V-I $\leq$ 2.4 from 50 Myr to at least 8 Gyr. However, for stars with V-I $<$ 1.0, the CA varies little between 1 Gyr to 5 Gyr. This is consistent with results of Pace et al. (2009) who found nearly constant CA level from 1.4 Gyr (NGC3680) to 4.0 Gyr (M67). Apparently the slope in the log(SHK) vs. V-I plane for young stars is relatively steep, while for old stars it appears to be flatter. Additional observations will be required to determine whether this slope changes monotonically or discontinuously with age. These limitations will need to be taken carefully into account by anyone attempting to use CA to determine ages for single stars. The metallicities of stars earlier in spectral type than M3 were measured by template matching. Our sample generally supports the expected paradigm, i.e. older stars tend to have lower metallicity. However, it also underscores the fact that there is much variation in metallicity at all ages, precluding its use for determining ages for single stars. Also, the AMR among wide binaries appears to be quite comparable to that found in single field stars (Rocha- Pinto $\&$ Maciel 2000). With trigonometric parallaxes from the literature and photometric parallaxes derived from B, V, R, I data, proper motions and our measured vr values, we calculated full space motions U, V and W for as many of our MS stars as possible. Our results clearly show that the W dispersion increases with age. In general, the W velocity component is also relatively larger for old stars. Using our measurements of CA and metallicity, we concluded that 2 wide binaries ( CD-31 1454/LP888-25 and LHS300A/B) are probably halo members, while the others are disk stars. This low fraction of halo members among wide binaries is consistent with the earlier results of Silvestri, Oswalt $\&$ Hawley (2002) who found only one halo binary in their sample of WD+dM pairs. We estimated the oldest disk star in our sample is G114-B8B ($>$ 8 Gyr) based on its weak CA. Five of our wide binaries consist of two MS stars. Two (G114-B8A/B; G163-B9A/B) apparently are not physical pairs, since the two companions have inconsistent CA levels, radial velocities and/or metallicities. CD-31 7352/LP902-30 and LP387-1/2 are physical pairs because the two MS components have very similar velocities, metallicty and CA level. LP684-1/2 is probably a physical pair since the radial velocity difference between two components is within the range of uncertainty and its components display comparable SHK value. In conclusion, our study affirms the assumption that wide binaries (CPMBs) share the same kinematic $\&$ spectroscopic properties as nearby single field and cluster stars. Thus they are very promising resources for studying stellar populations and age groups that are not well sampled by nearby clusters. TDO acknowledges supported from NSF grant AST-0807919 to Florida Institute of Technology. JKZ, GZ and YQC acknowledge support from NSFC grant No. 10821061, 11073026 and 11078019. We are also grateful for constructive comments by the reviewer that substantially improved our paper. ## References * Baliunas et al. (1995) Baliunas, S. L., Donahue, R. A., Soon, W. H., Horne, J. H., Frazer, J., Woodard-Eklund, L., Bradford, M., Rao, L. M., Wilson, O. C., Zhang, Q., Bennett, W., Briggs, J., Carrol, S. M., Duncan, D. K., Figueroa, D., Lanning, H. H., Misch, A., Mueller, J., Noyes, R. W., Poppe, D., Porter, A. C., Robinson, C. R., Russell, J., Shelton, J. C., Soyumber, T., Vaughan, A. 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See text for details. Figure 2: The relation among log($\rm{S_{HK}}$), V-I and age. Open circles represent CPMB stars with unknown age. Filled circles are flare stars. Stars in cluster IC2602 + IC2391 and M67 are shown as asterisks and squares respectively. Plus signs and diamonds represent stars with different ages: 1Gyr $<$ age $<$ 5Gyr and 5Gyr $<$ age $<$ 8Gyr, respectively. Four domains are divided by dotted lines. The dash-dot line is the least-squares fit of young clusters IC2602+IC2391. The vertical dashed line is V-I = 2.4. To the right of this, CA is not expected to depend on age. Solid lines connect the MS components in CD-31 7352/LP902-30, LP684-1/2 and LP387-1/2. The star indicated as number 1 is G163-B9A. Numbers 2-3 denote halo star candidates (LHS300A and CD-31 1454). Number 4 is the star with the weakest activity in our sample (G114-B8B). The typical error bar for our measurement is in the upper right corner. Separate V-I uncertainties are given for 4 stars whose V-I were estimated by the relation in Fig.1. A square with a plus sign is an outlier (G95-B5A) that seems overactive for its age domain. This star is perhaps a close binary and it warrants follow-up observations to determine the cause of its high CA. Figure 3: Left: An example of how [Fe/H] estimates were made with the maximum correlation method. The filled circle is the maximum of fitted polynomial. The [Fe/H] correlated to this maximum is regarded as our final result. + and - signs indicate our estimate of uncertainty as discussed in the text. Right: The comparison between our [Fe/H] results and those in the literature. Filled circles represent stars in Catalán et al. 2008a. Single open circles represent stars in Holmberg, Nordström $\&$ Andersen (2009). Double open circles, X signs and squares represent the clusters IC2391, IC2602 and M67, respectively. We adopted the average [Fe/H] of member stars as the final cluster [Fe/H]. The [Fe/H] of IC2391 and IC2602 are from D’Orazi $\&$ Randich (2009), while that of M67 is from Pancino et al. 2010. Diamonds and triangles represent stars in Mallik (1998) and Sousa et al. (2008), respectively. Note that many of these sources did not give uncertainties, hence no vertical error bar could be plotted. Figure 4: Our [Fe/H] vs. age relation for 21 MS stars. Typical uncertainties in age and [Fe/H] are displayed at the lower left of this figure. Filled circles represent disk stars. Only one star indicated by an asterisk (LHS300A) is a likely halo star. The dotted line is a least-squares fit for only disk stars, while the dashed line includes all stars. Figure 5: Top: UV-velocity distribution of our sample with measured vr (km s-1). The ellipsoids indicate the 1 $\sigma$ (inner) and 2 $\sigma$ (outer) contours for Galactic thick-disk and stellar halo populations, respectively. Typical error bars are given in both panels. Bottom: Toomre diagram of our stars. Dashed line is Vtotal = 180 km s-1. Figure 6: Left: age vs. absolute value of the W component of space velocity for 21 MS stars in our CPMB sample. Solid line is a least-squares fit. Typical uncertainties in age and W are displayed at the upper right of this figure. Right: W dispersion vs. age derived by binning the data in the left panel in five age group ranging from 1-8 Gyr. Table 1: Observed wide binaries ID | Name | R.A. | Decl. | | | | | UT | $\mu$ | $\theta_{\mu}$ | pos | Sep | Location ---|---|---|---|---|---|---|---|---|---|---|---|---|--- | (Star 1/Star 2) | (B1950.0) | (B1950.0) | V1 | Sp1aaSpectral types for the WD and MS stars were determined from low - resolution ($\sim$ 7 - 15 $\rm{\AA}$) spectra (Oswalt et al. 1988, 1991, 1993). | V2 | Sp2aaSpectral types for the WD and MS stars were determined from low - resolution ($\sim$ 7 - 15 $\rm{\AA}$) spectra (Oswalt et al. 1988, 1991, 1993). | (mm/yy) | (arcsec yr${}^{-1})$ | (deg) | (deg) | (arcsec) | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | 1 | CD-31 1454/LP888-25 | 03 32 36 | -31 14 00 | 11.8 | dG | 15.9 | dK | 02/05 | 0.51 | 186 | 264 | 223 | CTIO 2 | G114-B8A/B | 08 58 17 | -04 10 24 | 11.0 | dK0 | 15.9 | dG2 | 02/05 | 0.10 | 150 | 151 | 54 | CTIO 3 | BD+12 937/G102-39 | 05 51 05 | 12 23 48 | 7.6 | dF8 | 15.7 | DC | 02/04 | 0.28 | 184 | 47 | 91 | CTIO 4 | G272-B5A/B | 02 00 31 | -17 07 30 | 12.5 | dG | 15.8 | DA | 02/05 | 0.05 | 181 | 79 | 72 | CTIO 5 | BD-03 2935/LP670-9 | 10 27 24 | -03 57 00 | 11.2 | dG | 18.7 | | 02/05 | 0.18 | 103 | 139 | 35 | CTIO 6 | G163-B9A/B | 10 43 39 | -03 24 06 | 12.6 | dF9 | 15.6 | | 02/05 | 0.08 | 115 | 122 | 76 | CTIO 7 | BD+23 2539/LP378-537 | 13 04 48 | 22 43 00 | 9.8 | dK0 | 16.2 | DA | 02/04 | 0.11 | 300 | 106 | 20 | CTIO 8 | CD-25 8487/LP849-59 | 11 07 00 | -25 43 00 | 9.3 | sdM0 | 16.8 | DC | 02/04 | 0.25 | 106 | 181 | 100 | CTIO 9 | CD-28 3361/LP895-41 | 06 42 34 | -28 30 48 | 11.2 | dK | 16.8 | DA | 02/04 | 0.16 | 227 | 75 | 16 | CTIO 10 | BD-5 3450/LP674-29 | 12 09 48 | -06 05 00 | 12.0 | dK5 | 17.2 | DC | 02/05 | 0.44 | 220 | 102 | 202 | CTIO 11 | BD-18 2482/LP786-6 | 08 45 18 | -18 48 00 | 12.8 | dK3 | 15.1 | DB | 02/04 | 0.16 | 268 | 236 | 31 | CTIO 12 | 40 Eri A/B/C | 04 13 03 | -07 44 06 | 5.3 | dG | 9.5 | DA | 02/04,02/05 | 4.08 | 213 | 105 | 82 | CTIO 13 | CD-31 7352/LP902-30 | 09 28 20 | -31 53 12 | 9.3 | dK | 14.5 | dM | 02/05 | 0.34 | 348 | 206 | 12 | CTIO 14 | LP684-1/2 | 15 54 00 | -04 41 00 | 12.7 | dM | 15.5 | dM | 02/05 | 0.32 | 244 | 202 | 5 | CTIO 15 | BD-18 3019/LP791-55 | 10 43 30 | -18 50 00 | 12.9 | dM0 | 16.6 | DQ | 02/04 | 1.98 | 250 | 356 | 7.5 | CTIO 16 | LP856-54/53 | 13 48 30 | -27 19 00 | 13.9 | dM | 15.1 | DA | 02/04 | 0.24 | 166 | 233 | 9 | CTIO 17 | LP498-25/26 | 13 36 45 | 12 23 48 | 13.9 | dM | 14.5 | DB | 02/04 | 0.19 | 134 | 307 | 87 | CTIO 18 | LP672-2/1 | 11 05 30 | -04 53 00 | 12.6 | dM6 | 13.8 | DA | 02/04 | 0.44 | 184 | 160 | 279 | CTIO 19 | LP916-26/27 | 15 42 18 | -27 30 00 | 15.5 | dM | 16.3 | DB | 02/04 | 0.24 | 235 | 330 | 52 | CTIO 20 | LP891-13/12 | 04 43 18 | -27 32 00 | 15.6 | dM | 15.9 | DQ | 02/05 | 0.24 | 246 | 62 | 49 | CTIO 21 | LP783-2/3 | 07 38 02 | -17 17 24 | 12.9 | dM | 17.6 | DB | 02/04,02/05 | 1.26 | 117 | 276 | 21 | CTIO 22 | CD-37 10500/L481-60 | 15 44 12 | -37 46 00 | 6.8 | dG | 13.2 | DA | 02/05 | 0.48 | 243 | 131 | 15 | CTIO 23 | CD-59 1275/L182-61 | 06 15 36 | -59 11 24 | 7.0 | dG0 | 13.7 | DB | 02/04,02/05 | 0.33 | 190 | 302 | 41 | CTIO 24 | LP888-63/64 | 03 26 45 | -27 18 36 | 13.9 | | 15.6 | | 02/05 | 0.83 | 63 | 227 | 7 | CTIO 25 | CD-38 10983/10980 | 16 20 38 | -39 04 42 | 6.1 | dG | 10.7 | DA | 02/04,02/05 | 0.08 | 95 | 248 | 345 | CTIO 26 | LHS193A/B | 04 30 50 | -39 08 55 | 11.7 | dM | 17.7 | DB | 02/05 | 1.023 | 44.5 | | | CTIO 27 | LHS300A/B | 11 08 58 | -40 49 05 | 13.2 | dK | 17.8 | DB | 02/05 | 1.277 | 264.5 | | | CTIO 28 | LP387-2/1 | 16 44 18 | 24 06 00 | 16.8 | dG | 17.6 | DG | 07/05 | 0.11 | 163 | 296 | 37 | KPNO 29 | BD-8 0980/G156-64 | 22 53 12 | -08 05 24 | 9.0 | dG | 16.4 | DA | 07/05 | 0.59 | 91 | 168 | 43 | KPNO 30 | G171-62/G172-4 | 00 30 17 | 44 27 18 | 10.3 | dK | 16.6 | DA | 11/06 | 0.16 | 285.9 | | | KPNO 31 | BD-1 469/LP592-80 | 03 15 48 | -01 06 18 | 6.6 | dG | 17.2 | DA | 11/06 | 0.18 | 192 | 50 | 49 | KPNO 32 | G216-B14A/B | 22 58 49 | 40 40 12 | 12.0 | | 15.5 | | 11/06 | 0.07 | 215 | 261 | 23 | KPNO 33 | BD+44 1847/G116-16 | 09 11 51 | 44 15 36 | 10.2 | | 15.5 | | 11/06 | 0.28 | 174 | 95 | 1020 | KPNO 34 | G273-B1A/B | 23 50 54 | -08 21 06 | 12.0 | dG | 16.4 | DA | 11/06 | 0.12 | 75 | 210 | 36 | KPNO 35 | G95-B5A/B | 02 20 44 | 22 14 00 | 9.2 | | 15.6 | | 11/06 | 0.15 | 113 | 94 | 26 | KPNO 36 | BD+26 730/LP358-525 | 04 33 42 | 27 02 00 | 9.4 | dK | 16.3 | DA | 11/06 | 0.28 | 122 | 338 | 128 | KPNO Note. — Units of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcseconds. $V$ magnitude values (except 26 and 27) are $m_{pg}$ magnitudes from Oswalt, Hintzen $\&$ Luyten 1988. Table 2: The list of cluster member stars ID | Name | V | UT | V-I | B-V | SHK | age (Gyr) | Cluster | Refaa1: Barnes, Sofia $\&$ Prosser (1999); 2: Patten $\&$ Simon (1996); 3: Giampapa et al. (2006) ---|---|---|---|---|---|---|---|---|--- (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) 37 | [RSP95] 15 | 11.75 | 02/05 | 1.0600 | 0.9300 | 0.6819 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1 38 | [RSP95] 32 | 15.06 | 02/05 | 2.1600 | 1.6300 | 1.5799 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1 39 | [RSP95] 66 | 11.07 | 02/04 | 0.8300 | 0.6800 | 0.4241 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1 40 | [RSP95] 70 | 10.92 | 02/04 | 0.7100 | 0.6900 | 0.3553 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1 41 | [RSP95] 72 | 10.89 | 02/05 | 0.7600 | 0.6400 | 0.5110 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1 42 | [RSP95] 80 | 11.75 | 02/04 | 1.0900 | 0.9300 | 0.5738 | 0.050${}_{0.045}^{0.055}$ | IC2602 | 1 43 | VXR PSPC 12 | 11.86 | 02/04 | 0.9100 | 0.8300 | 0.5429 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2 44 | VXR PSPC 14 | 10.45 | 02/04 | 0.6900 | 0.5700 | 0.4302 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2 45 | VXR PSPC 70 | 10.85 | 02/04 | 0.7500 | 0.6400 | 0.3716 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2 46 | VXR PSPC 72 | 11.46 | 02/04 | 0.8400 | 0.7300 | 0.4984 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2 47 | VXR PSPC 76a | 12.76 | 02/04 | 1.2414 | 1.0400 | 0.8793 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2 48 | VXR PSPC 77a | 9.91 | 02/04 | 0.6000 | 0.5000 | 0.4184 | 0.050${}_{0.045}^{0.055}$ | IC2391 | 2 49 | Cl* NGC 2682 SAND 785 | 14.8 | 02/05 | 0.8315 | 0.6500 | 0.3233 | 4.000${}_{3.800}^{4.300}$ | M67 | 3 50 | Cl* NGC 2682 SAND 1477 | 14.6 | 02/05 | 0.8497 | 0.6700 | 0.2981 | 4.000${}_{3.800}^{4.300}$ | M67 | 3 Table 3: The V-I, B-V, SHK, vr, full space motions and age for 27 wide binaries ID | V-I | B-V | SHK | vr | $\pi$ | U | V | W | [Fe/H] | age ---|---|---|---|---|---|---|---|---|---|--- | | | | (km s${}^{-1})$ | (mas) | (km s-1) | (km s-1) | (km s-1) | | (Gyr) (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) 1a | 0.5210 | 0.8230 | 0.2303 | 76.1$\pm$7.3 | 5.89$\pm$2.77 | -329.0 | -242.5 | -101.4 | -0.48$\pm$0.02 | 2a | 0.6595 | 0.5660 | 0.2843 | 16.1$\pm$4.3 | 9.06$\pm$2.68 | -42.5 | -38.6 | 11.7 | -0.17$\pm$0.07 | 2b | 1.0701 | 0.9111 | 0.2413 | 114.3$\pm$2.1 | 9.06$\pm$2.68 | 10.3 | -109.3 | 54.9 | -0.40$\pm$0.08 | 3 | 0.6624 | 0.5699 | 0.2891 | 58.8$\pm$9.7 | 18.70$\pm$0.80 | 26.4 | -65.8 | -39.0 | -0.25$\pm$0.02 | 4 | 0.6800 | 0.5230 | 0.2689 | 28.3$\pm$5.4 | 62.95 | -10.3 | 16.5 | -19.1 | -0.34$\pm$0.06 | 5 | 0.6901 | 0.6762 | 0.2666 | -54.3$\pm$6.1 | 28.3 | -51.2 | 44.0 | -18.6 | 0.10$\pm$0.01 | 6 | 0.7420 | 0.5970 | 0.4582 | -20.8$\pm$6.9 | 83.3 | -18.5 | 18.6 | -7.1 | -0.25$\pm$0.15 | 7 | 0.7754 | 0.6422 | 0.3150 | -7.7$\pm$4.5 | 56.7 | 0.4 | 4.4 | 0.4 | 0.06$\pm$0.04 | 1.0${}_{0.9}^{1.2}$ 8 | 0.7892 | 0.6963 | 0.2856 | 28.5$\pm$4.9 | 26.76$\pm 1.09$ | -59.4 | -8.6 | 30.9 | -0.16$\pm$0.06 | 9 | 1.0776 | 0.9720 | 0.3454 | 27.0$\pm$1.6 | 28.17$\pm 0.06$ | -6.4 | -15.4 | -26.4 | -0.13$\pm$0.13 | 10 | 1.1181 | 0.9884 | 0.3180 | 55.9$\pm$1.6 | 22.94$\pm 1.63$ | -0.3 | -101.2 | 5.9 | -0.48$\pm$0.02 | 11 | 1.1316 | 1.0362 | 0.4074 | 69.2$\pm$5.6 | 29.49 | 35.6 | -53.3 | 3.5 | 0.19$\pm$0.09 | 2.0${}_{1.4}^{2.8}$ 12a | 1.1469 | 0.7750 | 0.3051 | -37.9$\pm$1.7 | 198.25$\pm$0.84 | -104.0 | -8.5 | -37.4 | -0.17$\pm$0.17 | 5.0${}_{4.0}^{6.1}$ 12c | 2.8297 | 1.3938 | 3.0759 | -54.0$\pm$1.7 | 198.25$\pm$0.84 | -104.0 | -8.5 | -37.4 | -0.17$\pm$0.17 | 5.0${}_{4.0}^{6.1}$ 13a | 1.1891 | 0.9822 | 0.3140 | 21.8$\pm$2.0 | 51.71$\pm$0.91 | 19.6 | -9.4 | 28.7 | -0.30$\pm$0.10 | 13b | 2.3761 | 1.4432 | 0.5585 | 33.8$\pm$3.1 | 51.71$\pm$0.91 | 21.6 | -21.6 | 31.7 | -0.22$\pm$0.08 | 14a | 1.4140 | 1.2120 | 0.4563 | 67.2$\pm$1.8 | 42.69 | -58.5 | -23.0 | 58.5 | 0.19$\pm$0.09 | 14b | 2.2270 | 0.6870 | 0.5730 | 60.3$\pm$6.9 | 42.69 | -52.8 | -23.5 | 54.5 | 0.05$\pm$0.09 | 15 | 1.8888 | 1.4484 | 0.6202 | 57.7$\pm$2.0 | 56.92 | 236.9 | -187.8 | -229.5 | -0.25$\pm$0.15 | 3.6${}_{2.7}^{5.0}$ 16 | 1.9870 | 1.4550 | 0.74958 | 14.4$\pm$4.2 | 49.63 | -28.6 | -11.1 | -4.2 | 0.15$\pm$0.05 | 1.2${}_{0.9}^{1.4}$ 17 | 2.3380 | 1.5710 | 1.0492 | 11.7$\pm$5.2 | 46.12 | -33.3 | 3.1 | 11.8 | -0.24$\pm$0.10 | 1.5${}_{0.8}^{2.9}$ 18 | 2.4790 | 1.5020 | 0.7738 | 28.6$\pm$4.2 | 57.70$\pm$14.40 | -22.9 | -39.0 | 8.8 | -0.02$\pm$0.03 | 3.5${}_{2.5}^{4.5}$ 19 | 2.8165 | 1.6318 | 0.9172 | -53.4$\pm$4.0 | 40.05 | 48.1 | -9.6 | -7.5 | | 2.2${}_{1.6}^{4.8}$ 20 | 2.8474 | 1.5111 | 2.5424 | 24.0$\pm$3.2 | 38.80 | -14.0 | 3.4 | -33.0 | | 21 | 4.2570 | 1.8760 | 1.0893 | -24.6$\pm$3.9 | 102.00$\pm$14.00 | -65.8 | -3.1 | 40.6 | | 2.4${}_{1.8}^{4.2}$ 22 | 0.8933 | 0.7180 | 0.3031 | -18.3$\pm$10.1 | 65.1$\pm$0.4 | 24.8 | -24.1 | 15.5 | 0.02$\pm$0.08 | 1.1${}_{0.9}^{1.3}$ 23 | 0.7769 | 0.5900 | 0.27162 | -41.9$\pm$4.2 | 27.5$\pm$0.5 | -63.3 | 12.9 | -17.8 | 0.03$\pm$0.07 | 1.4${}_{0.4}^{2.4}$ 24 | 1.8366 | 1.5000 | 1.2597 | 3.2$\pm$3.8 | 57.6 | 29.6 | -4.0 | 83.1 | | 3.6${}_{2.2}^{5.6}$ 25 | 0.8133 | 0.630 | 0.32504 | -2.5$\pm$4.1 | 77.69$\pm$0.86 | -9.5 | 9.7 | 2.1 | -0.10$\pm$0.12 | 1.4${}_{0.9}^{2.3}$ 26 | 1.5700 | 1.1600 | 0.3815 | 56.6$\pm$4.4 | 32.06$\pm$1.65 | 131.7 | -50.6 | 48.2 | -0.45$\pm$0.26 | 7.5${}_{6.1}^{8.7}$ 27 | 1.6900 | | 0.3835 | 145.0$\pm$1.8 | 32.3 | 100.3 | -195.3 | -46.3 | -0.95$\pm$0.25 | 7.9${}_{5.5}^{9.0}$ 28a | 1.0215 | 0.8394 | 0.2264 | -29.9$\pm$2.2 | 45.0 | -14.0 | -19.3 | 27.4 | 0.04$\pm$0.06 | 28b | 0.901 | 0.8470 | 0.2054 | -32.1$\pm$5.8 | 45.0 | -14.2 | -20.4 | 29.3 | 0.03$\pm$0.07 | 29 | 0.7100 | 0.5300 | 0.2189 | -28.1$\pm$4.1 | 28.7$\pm$1.3 | 80.4 | -38.6 | -12.3 | -0.40$\pm$0.02 | 3.5${}_{3.0}^{3.9}$ 30 | 0.8800 | 0.9800 | 0.2389 | 44.4$\pm$1.5 | 9.52$\pm$1.63 | 101.3 | -20.3 | -36.3 | -0.23$\pm$0.03 | 1.6${}_{1.0}^{2.4}$ 31 | 0.3758aaV-I is from the relation in Fig.1 | 1.0400 | 0.1585 | 33.2$\pm$1.9 | 14.68$\pm$0.96 | -19.7 | -29.4 | -49.3 | -0.16$\pm$0.04 | 2.6${}_{1.3}^{3.9}$ 32 | 2.0889aaV-I is from the relation in Fig.1 | | 1.0034 | 1.6$\pm$1.2 | | | | | -0.15$\pm$0.05 | 1.2${}_{1.0}^{1.3}$ 33 | 0.7160 | 0.6600 | 0.1820 | -61.4$\pm$12.0 | 19.36$\pm$1.30 | -57.6 | -65.3 | -27.8 | -0.40$\pm$0.10 | 1.6${}_{1.4}^{1.8}$ 34 | 0.6861aaV-I is from the relation in Fig.1 | | 0.2081 | 28.9$\pm$1.4 | | | | | -0.16$\pm$0.06 | 3.4${}_{2.7}^{4.2}$ 35 | 0.9195aaV-I is from the relation in Fig.1 | | 0.3607 | 27.3$\pm$3.6 | 21.65 | 26.4 | -11.3 | -7.7 | -0.50$\pm$0.03 | 3.4${}_{2.6}^{4.2}$ 36 | 1.52 | 1.1200 | 1.7876 | 42.7$\pm$1.3 | 56.02$\pm$1.21 | 35.8 | -11.6 | 4.4 | -0.21$\pm$0.03 | 4.5${}_{3.6}^{5.4}$ Table 4: The $T$eff, log $g$, mass, cooling time and reference of fifteen wide dwarfs ID | Name | Sp | $T$eff | log $g$ | MWD | cooling time | Mi | tevol | RefaaThis column lists the source of $T$eff, log $g$. 1: Catalán et al. 2008; 2: Voss et al. 2007; 3: Sion et al. 2009; 4: Bergeron et al. 1995; 5: Koester et al. 2009; 6: Koester et al. 2001; 7: Monteiro et al. 2006; 8: Weidemann $\&$ Koester 1984 ---|---|---|---|---|---|---|---|---|--- | | | (K) | | (M⊙) | (Gyr) | (M⊙) | (Gyr) | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) 7b | LP378-537 | DA | 10800$\pm$120 | 8.21$\pm$0.05 | 0.732$\pm$0.032 | 0.6760$\pm$0.0626 | 3.02$\pm$0.23 | 0.3${}_{0.2}^{0.5}$ | 1 11b | LP786-6 | DB | 17566$\pm$200 | 7.97$\pm$0.05 | 0.579$\pm$0.028 | 0.1182$\pm$0.0135 | 1.56$\pm$0.29 | 1.9${}_{1.3}^{2.7}$ | 2 12b | 40Eri B | DA | 16570$\pm$350 | 7.86$\pm$0.05 | 0.540$\pm$0.019 | 0.1122$\pm$0.0116 | 1.15$\pm$0.20 | 4.9${}_{3.9}^{6.0}$ | 4 15b | LP791-55 | DQ | 6190$\pm$200 | 8.09$\pm$0.05 | 0.630$\pm$0.029 | 2.8244$\pm$0.6080 | 2.09$\pm$0.31 | 0.8${}_{0.3}^{2.1}$ | 3 16b | LP856-53 | DA | 10080$\pm$200 | 8.17$\pm$0.05 | 0.705$\pm$0.032 | 0.7572$\pm$0.0911 | 2.82$\pm$0.23 | 0.4${}_{0.2}^{0.6}$ | 4 17b | LP498-26 | DB | 16779$\pm$200 | 8.00$\pm$0.05 | 0.595$\pm$0.027 | 0.1475$\pm$0.0155 | 1.73$\pm$0.28 | 1.4${}_{0.6}^{2.7}$ | 2 18b | LP672-1 | DA | 15996$\pm$11 | 7.753$\pm$0.002 | 0.486$\pm$0.001 | 0.1066$\pm$0.0004 | 1.25$\pm$0.01 | 3.4${}_{2.4}^{4.4}$ | 5 19b | LP916-27 | DB | 10826$\pm$200 | 8.00$\pm$0.05 | 0.585$\pm$0.029 | 0.5365$\pm$0.0531 | 1.62$\pm$0.30 | 1.7${}_{1.2}^{4.3}$ | 2 21b | LP783-3 | DZ | 7590$\pm$200 | 8.07$\pm$0.05 | 0.619$\pm$0.031 | 1.4854$\pm$0.1856 | 1.93$\pm$0.31 | 0.9${}_{0.3}^{2.7}$ | 3 22b | L481-60 | DA | 10613$\pm$18 | 8.12$\pm$0.03 | 0.675$\pm$0.018 | 0.6167$\pm$0.0254 | 2.56$\pm$0.19 | 0.5${}_{0.3}^{0.7}$ | 6 23b | L182-61 | DB | 16714$\pm$200 | 8.07$\pm$0.05 | 0.605$\pm$0.029 | 0.1539$\pm$0.0160 | 1.83$\pm$0.30 | 1.2${}_{0.4}^{2.2}$ | 2 24b | LP888-63 | DA | 9408$\pm$8 | 7.93$\pm$0.02 | 0.559$\pm$0.011 | 0.6515$\pm$0.0159 | 1.35$\pm$0.11 | 3.0${}_{1.6}^{5.0}$ | 3 25b | CD-38 10980 | DA | 24276$\pm$200 | 8.01$\pm$0.05 | 0.641$\pm$0.028 | 0.0279$\pm$0.0043 | 2.21$\pm$0.29 | 0.7${}_{0.3}^{1.6}$ | 6 26b | LHS193B | DA | 4394$\pm$200 | 8.10$\pm$0.05 | 0.632$\pm$0.032 | 6.6900$\pm$0.8100 | 2.11$\pm$0.33 | 0.8${}_{0.3}^{1.3}$ | 7 27b | LHS300B | DA | 4705$\pm$200 | 7.80$\pm$0.05 | 0.456$\pm$0.026 | 4.5385$\pm$0.7298 | 1.25$\pm$0.01 | 3.4${}_{2.4}^{4.4}$ | 7 29b | G156-64 | DA | 7165$\pm$165 | 8.43$\pm$0.07 | 0.869$\pm$0.046 | 3.2983$\pm$0.4292 | 4.02$\pm$0.34 | 0.1${}_{0.1}^{0.3}$ | 4 30b | G172-4 | DA | 10440$\pm$240 | 8.02$\pm$0.07 | 0.613$\pm$0.043 | 0.5566$\pm$0.0803 | 1.92$\pm$0.45 | 1.1${}_{0.3}^{1.9}$ | 8 31b | LP592-80 | DA | 7520$\pm$260 | 8.01$\pm$0.45 | 0.600$\pm$0.256 | 1.2822$\pm$0.7289 | 1.78$\pm$0.20 | 1.3${}_{0.3}^{2.0}$ | 1 32b | G216-B14B | DA | 9860$\pm$226 | 8.20$\pm$0.07 | 0.724$\pm$0.045 | 0.8443$\pm$0.1246 | 2.96$\pm$0.33 | 0.3${}_{0.2}^{0.4}$ | 4 33b | G116-16 | DA | 8750$\pm$201 | 8.29$\pm$0.07 | 0.780$\pm$0.045 | 1.3397$\pm$0.1785 | 3.37$\pm$0.33 | 0.2${}_{0.1}^{0.3}$ | 4 34b | G273-B1B | DA | 18529$\pm$37 | 7.79$\pm$0.01 | 0.509$\pm$0.004 | 0.0617$\pm$0.0136 | 0.83$\pm$0.04 | 3.4${}_{2.7}^{4.0}$ | 5 35b | G94-B5B | DA | 15630$\pm$65 | 7.89$\pm$0.01 | 0.555$\pm$0.005 | 0.1456$\pm$0.0384 | 1.31$\pm$0.05 | 3.3${}_{2.4}^{4.0}$ | 5 36b | LP358-525 | DA | 5620$\pm$110 | 8.14$\pm$0.07 | 0.673$\pm$0.044 | 4.0693$\pm$0.8147 | 2.54$\pm$0.46 | 0.5${}_{0.2}^{0.8}$ | 4 Table 5: The comparison between our ages and those from literature Name | agenaaages are from Holmberg, Nordström $\&$ Andersen (2009) | agevbbages are from Valenti $\&$ Fischer (2005) | age ---|---|---|--- | (Gyr) | (Gyr) | (Gyr) (1) | (2) | (3) | (4) 40 Eri A | $\sim$ | 12.2${}_{8.5}^{14.5}$ | 5.0${}_{4.0}^{6.1}$ CD-38 10983 | 2.5${}^{7.0}_{\sim}$ | 2.0${}_{0.4}^{3.9}$ | 1.2${}_{0.4}^{2.2}$ CD-59 1275 | 5.9${}_{5.4}^{6.6}$ | 3.7${}_{3.4}^{4.7}$ | 1.4${}_{0.4}^{2.4}$ CD-37 10500 | 7.4${}_{1.9}^{13.0}$ | 4.4${}_{1.4}^{7.0}$ | 1.1${}_{0.9}^{1.3}$
arxiv-papers
2011-01-17T16:28:58
2024-09-04T02:49:16.498241
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "J. K. Zhao, T. D. Oswalt, M. Rudkin, G. Zhao, Y. Q. Chen", "submitter": "Jingkun Zhao", "url": "https://arxiv.org/abs/1101.3257" }
1101.3260
2009 Vol. 9 No. XX, 000–000 11institutetext: Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China; gzhao@bao.ac.cn Received [year] [month] [day]; accepted [year] [month] [day] # Metallicity calibration for solar type stars based on red spectra J. K. Zhao G. Zhao Y. Q. Chen A. L. Luo ###### Abstract Based on high resolution and high signal-to-noise ratio (S/N) spectra analysis of 90 solar type stars, we have established several new metallicity calibrations in $T\rm{{}_{eff}}$ range [5600, 6500] K based on red spectra with the wavelength range of 560-880 nm. The new metallicity calibrations are applied to determine the metallicity of solar analogs selected from SDSS spectra. There is a good consistent result with the adopted value presented in SDSS-DR7 and a small scatter of 0.26 dex for stars with S/N $>$ 50 is obtained. This study provides a new reliable way to derive the metallicity for solar-like stars with low resolution spectra. In particular, our calibrations are useful for finding metal-rich stars, which are missing in SSPP. ###### keywords: techniques: radial velocities - stars: temperatures - stars: abundances ## 1 Introduction The stellar spectroscopic survey with the Large Area sky Multi-Object fiber Spectroscopic Telescope (LAMOST) will provide a huge amount of data, which can be used for the study of chemical and kinematical evolution of our Galaxy. In this respect, stellar metallicity and radial velocity, being two main parameters, can be derived from spectra. The determination of radial velocity is generally easier mainly by using either cross-correlation of the template spectra or Doppler shift through line calibration. The consistency is usually quite good depending on the quality of the spectra. The metallicity estimation from stellar spectra is based on various methods as shown in Lee et al. (2008a) (hereafter Lee08). However, for solar type stars, these values can be underestimated by up to 0.5 dex in the previous version of SSPP (SEGUE Stellar Parameter Pipeline; Lee et al. 2008a). The current version of SSPP has made great improvement, reaching about 0.1 dex in the underestimation (Lee et al. 2008b). From Fig A1 in Bond et al. (2009), the largest difference between the SDSS (Sloan Digital Sky Survey; York et al. 2001) spectroscopic metallicity values with DR6 (Data Release 6; Adelman-McCarthy et al. 2008) and DR7 (Data Release 7; Abazajian et al. 2009) is shown for solar type stars, so it is worth the effort to do more research about deriving the metallicty of those stars. In this work, we attempt to establish a new metallicity calibration for low resolution solar type stars based on the result from high resolution and high signal-to-noise ratio spectral analysis performed by Chen et al. (2000, hereafter Chen00). In comparison with the methods presented in Lee08, this work has some advantages. Firstly, the calibration is based on the real stellar (empirical) spectra and their metallicity is derived from fine analysis of high resolution and high S/N spectra. Secondly, we have used the red spectral coverage of 560-880 nm but most of the methods in Lee08 are based on blue spectra with $\lambda<600$ nm. As is well known, the advantage of the red spectra is easier to define continuum, which is not possible for blue spectra due to the heavy line blanketing at low resolution observations. In view of this advantage, we adopted the equivalent widths (EW) of individual lines in the calibration instead of the line index. As for line index, there might be different definition for different authors while EW is a fixed value. For example, CaII K line, there are K6, K12 and K18 among its definition. If we can define the continuum well, the EW is better than line index. Thirdly, we have adopted only Fe lines for metallicity calibration and avoid contributions from other elements, which do not exactly trace iron evolution at different times and different nucleosynthesis sites. In Lee08, the wavelength ranges of templates match include all lines from different elements. The KP line index is also an indicator of [Ca/H]. Although these weak Fe lines are undetectable in metal-poor stars because of the noise, we can recognize them in solar type stars where the S/N is higher than 20. Moreover, the EW of weak lines is more sensitive to abundance than that of strong lines. For high resolution spectra analysis, strong lines, e.g. Na 5895$\rm\AA$ and Na 5890$\rm\AA$ are saturated and in the growth curve the increasing EWs do not give higher abundance. In general, the EW of strong lines does not change a lot with the degradation of resolution. In view of this, it is not optimal to establish relation between abundance and EW in combination with colors using very strong line. Finally, the calibration is internally consistent, while Lee08 adopted the average of different values from various methods. In Section 2, a description of the data for the calibration is presented. The template matching analysis is described in Section 3. The detailed analysis procedure to get the calibration formula is given in Section 4. The application of the calibration to SDSS spectra is illustrated in Section 5. Finally, the conclusion is given in Section 6. ## 2 Data The real spectra are taken from Chen00 which has a resolving power of 37 000 and S/N of 150-300 obtained with the Coud$\acute{e}$ Echelle Spectrograph mounted on the 2.16 m telescope of the National Astronomical Observatories (Zhao $\&$ Li 2001). The sample has $T\rm{{}_{eff}}$, log $g$ and [Fe/H] distributions as shown in Fig. 1. It is clear that $T\rm{{}_{eff}}$ ranges between [5600, 6500], log $g$ is in [3.98, 4.43] and [Fe/H] is in [-1.04, 0.06]. Moreover, the range of b-y is within [0.28, 0.43], B-V is within [0.40, 0.67], V-I is within [0.46, 0.73] and V-K is within [0.9, 1.65]. We convolved the normalized spectra to low resolution of 2000 with a Gaussian Function. In addition, the spectra were rebinned to 1.5$\rm{\AA}/pix$ after smoothed to R$\sim$2000. Figure 1: Stellar parameter distribution of the sample in Chen00 Table 1: [Fe/H] results of template match with different wavelength ranges wavelength range(nm) | mean deviation | scatter ---|---|--- 570-653 | 0.448 | 0.175 570-684 | 0.226 | 0.275 570-700 | 0.016 | 0.230 640-670 | 0.470 | 0.280 651-662 | 0.599 | 0.390 690-713 | -0.07 | 0.590 735-756 | 0.350 | 0.290 772-810 | -0.100 | 0.260 ## 3 The template match analysis Following one method of Lee08, we have performed the template spectra match (also see Allende et al 2006, Re Fiorentin et al 2007, Zwitter et al. 2005) for the normalized low resolution spectra of 90 stars and derived the stellar temperature, gravity and metallicity. In this method, we generate a library of low resolution theoretical spectra by using the SYNTH program based on Kurucz New ODF (Castelli $\&$ Kurucz 2003) atmospheric models. The atmosphere models are under the assumption of local thermodynamic equilibrium (LTE). The mixing- length is adopted to be l$/$Hp = 1.25 and microturbulence is 1.5 km s-1. The line list, including the atoms and molecules, are all from Kurucz (1993). The molecular species include CH, CN, OH, and TiO. Solar abundances are from Asplund (2005). As for those grids, $T\rm{{}_{eff}}$ covers the range [3500-9750]K with 250K interval; log $g$ is within [1.0, 5.0] dex with 0.5 dex interval; [Fe/H] is within [-4.0, -3.0] dex with 0.25 dex interval and 0.1 dex interval in [-3.0,+0.5] dex. The minimum distance method is applied to obtain the parameters by interpolation among several of the closest theoretical spectra with the observed one. Figure 2: [Fe/H] and $T\rm{{}_{eff}}$ comparison between those of Chen00 and the results from the template match with spectral range of 570nm-750nm We have adopted different wavelength ranges in the matching procedure and obtained different results as shown in Table 1. As compared with the ‘standard’ values presented in Chen00 paper, we have found that the spectral range of 570-700 nm is the best choice with a mean deviation of 0.016 dex and scatter of 0.23 dex in [Fe/H]. The comparison of temperature and metallicity of the 570-700 nm spectral range is shown in Fig. 2. It is clear that the temperature estimation has systematical deviation with a lower value in the present work. Since the high resolution spectra in Chen00 are obtained with echelle spectrograph. The order is not wide enough to include the whole H alpha line region and the normalization is implemented order by order. Thus, the continuum is not very reliable in H alpha region, which might be the reason of systematical deviation in temperature estimation. The metallicity is quite consistent with an rms of 0.23 dex. ## 4 [Fe/H] vs. EW of the FeI line - An empirical calibration Although the template match method can be used to obtain accurate stellar parameters, it may give different results with different wavelength ranges. Hence, we will determine stellar metallicity based on the strength of some FeI lines. Figure 3: A portion of the spectra of HD94280, HD100446 and HD184601. Solid line is the high resolution spectra while thick dotted line is the low resolution spectra. The two FeI lines are those that meet the stated requirements. Table 2: The definition of FeI lines. Left | Right | Center | Element ---|---|---|--- 6060.917 | 6069.954 | 6065.492 | FeI 6217.769 | 6221.788 | 6219.292 | FeI 6391.147 | 6395.953 | 6393.605 | FeI 6398.256 | 6402.209 | 6400.232 | FeI 6675.000 | 6682.326 | 6678.256 | FeI ### 4.1 The spectral lines selection What we want is to choose some Fe lines that are not heavily blended and have better profiles in the low resolution (R$\sim$2000) spectra. At the beginning, we draw the original spectrum, then overplot the low resolution spectrum on it. After checking the lines one by one, we select five FeI lines as our indices of metallicity. The spectral lines for three stars with different metallicity values are given in Fig. 3: HD94280 has [Fe/H] of 0.06; HD100446 has -0.48; HD184601 has -0.81. In Fig. 3, the solid line is the high resolution spectrum while the thick dotted line is that of the low resolution one. In this segment of the spectrum, only two FeI lines meet our requirements since they are detectable; they have good shapes and are not seriously blended in the spectra with a resolution of R$\sim$2000\. From the top to the bottom of Fig. 3, it is obvious that the strength of FeI lines decreases. Also, for stars with [Fe/H] $>$ -0.8, the two lines can be identified. Table 2 presents the definition of five FeI line indices used for our metallicity calibration. There are three parts for each line including red, center and blue spectra. The EW of each line can be measured by using a direct integration method. Figure 4: The relation between [Fe/H] and EW for five FeI lines for the stars with [Fe/H] $>$ -0.8 Table 3: The coefficient and $\sigma$ of the fitting between [Fe/H] and EW for each FeI line Lines | a | b | $\sigma$ ---|---|---|--- FeI1 | -0.922 | 5.063 | 0.170 FeI2 | -0.866 | 6.559 | 0.143 FeI3 | -1.010 | 5.960 | 0.167 FeI4 | -0.970 | 4.958 | 0.163 FeI5 | -1.026 | 6.056 | 0.204 Table 4: The coefficient and $\sigma$ of [Fe/H] calibration based on the EW of five FeI lines and temperature Lines | a | b | c | $\sigma$ ---|---|---|---|--- FeI1 | 1.731 | 5.929 | -3.281 | 0.150 FeI2 | 2.138 | 7.726 | -3.692 | 0.112 FeI3 | 2.282 | 7.386 | -4.116 | 0.136 FeI4 | 2.793 | 6.381 | -4.702 | 0.122 FeI5 | 0.974 | 6.949 | -2.505 | 0.194 ### 4.2 The metallicity calibration based on EWs Since our spectra are normalized and there are no spectra with flux calibration, it is difficult to derive reliable temperature measurements. In order to improve the metallicity determination, we resort to using the strengths of iron lines and establish a calibration between metallicity and EWs of iron lines. The EWs are derived with Equation 1. $\displaystyle\rm{EW}$ $\displaystyle=$ $\displaystyle\int_{\lambda 1}^{\lambda 2}{\frac{f_{c}-f_{\lambda}}{f_{c}}d_{\lambda}}$ (1) In Equation1, the fλ represents the flux of wavelength $\lambda$, while fc means the continuum of wavelength $\lambda$. It is shown in Fig. 4 that there is a good correlation between [Fe/H] and EWs for the five FeI lines for stars whose [Fe/H] $>$ -0.8. The calibration (Equation 2) for each line is shown in Table 3. $\displaystyle\rm[Fe/H]$ $\displaystyle=$ $\displaystyle a+b*\rm{EW(FeI)}$ (2) As seen in Fig. 4, FeI2 and FeI4 show the best result with the lowest scatter in the relation. To show the temperature effect of EW, we divide the temperature range into three parts. The first part is the range of [5265, 5900]; the second part is [5900, 6200] and the last part is [6200, 6496]. In Fig. 5, the stars in first part are given with the sign of the dot and asterisks represent the stars in second part, while the stars in last part are plotted with diamonds. From Fig. 5, it is clear that the relation between EW and [Fe/H] changes with temperature. FeI2 and FeI4 have lower scatter than other FeI lines and this may be due to the lower sensitivity of line strength with temperature. In order to understand this issue, we added the temperature term in the calibration in Fig. 6. The coefficients and scatter of each line are given in Table 4. It is obvious that the $\sigma$ becomes small for the calibration of each line after considering the effect of temperature. Since the temperature is usually unknown in the spectra analysis, it may be good to replace the temperature term with the color index. Thus, we collected (b-y), (B-V), (V-I) and (V-K) and performed a similar calibration (Equation 3). log $g$ also will bring more or less uncertainty on metallicity determination. However, the gravity range considered in the calibration sample is narrow, hence its effect is very little and could be ignored. Table 5 is the coefficients and scatter of the calibration of [Fe/H] through EW and B-V. Table 5: The coefficient, $\sigma$ and the EW range of the fitting between [Fe/H] and the EW plus B-V for each FeI line Lines | a | b | c | $\sigma$ | EW range ---|---|---|---|---|--- FeI1 | -0.790 | 5.351 | -0.315 | 0.169 | 0.025$\sim$0.225 FeI2 | -0.290 | 8.144 | -1.342 | 0.132 | 0.006$\sim$0.158 FeI3 | -0.555 | 7.352 | -1.167 | 0.160 | 0.044$\sim$0.185 FeI4 | -0.321 | 6.622 | -1.643 | 0.151 | 0.029$\sim$0.224 FeI5 | -1.200 | 5.497 | 0.453 | 0.202 | 0.046$\sim$0.193 $\displaystyle\rm[Fe/H]$ $\displaystyle=$ $\displaystyle a+b*\rm{EW(FeI)}+c*(B-V)$ (3) Figure 5: The relation between the metallicity and EW of FeI lines in different $T\rm{{}_{eff}}$ ranges. Dots represent stars in [5265, 5900]K; asterisks are those in [5900, 6200]K; diamonds are those in [6200, 6500]K. Figure 6: The comparison between [Fe/H] in Chen00 and that of calibration from each FeI line and the temperature Figure 7: The application of our [Fe/H] calibration in Miles spectra library. The x axis is [Fe/H] from the Miles catalog and the y axis is [Fe/H] obtained by our calibration. The solid line is x=y Figure 8: Plots of (a) number vs. S/N (b) log $g$ vs. $T\rm{{}_{eff}}$ Figure 9: The spectra of 53738-2051-030. ‘a’ is the original spectrum while ‘b’ is normalized spectrum. ‘c’,‘d’,‘e’,‘f’ and ‘g’ present the spectral lines in our calibration ### 4.3 Calibration in Miles spectra library To make an external calibration, we selected 107 spectra from the Miles spectral library (Sánchez-Blázquez et al. 2006; Cenarro et al. 2007) which meet the following conditions: -0.8 $\leq$ [Fe/H] $\leq$ 0.5; 4.0 $\leq$ log $g$ $\leq$ 4.5; 5600 $\leq$ $T\rm{{}_{eff}}$ $\leq$ 6500\. Thus, it is available to estimate the metallicity of these 107 spectra using above calibrations. The resolution of Miles spectra is about 2.3$\rm\AA$ and the wavelength has already been calibrated with radial velocity. First, we do normalization for these 107 spectra. The continuum is determined by iteratively smoothing with a Gaussian profile, and then clipping off points that lie beyond 1 $\sigma$ or 4 $\sigma$ above the smooth curve. Second, the EWs of five FeI lines are measured. B-V of those 107 stars are taken from the literature identified by the Simbad Astronomical Database (Genova 2006). Finally, the metallicities are derived by EW of FeI4 line using equation 3 (the coefficients is shown in Table 5). Fig. 7 is [Fe/H] comparison between our results and those from Miles library. The mean error is about 0.08 dex, and the scatter is about 0.21 dex. [Fe/H] of Miles library is obtained by the compilation from the literature. This suggests that our metallicity is basically consistent with that of other work. ## 5 Application of these calibrations In order to check the accuracy of [Fe/H] calibration from the EW of five FeI lines, we implement this calibration to determine [Fe/H] for solar-like stars with SDSS spectra. The selection limitation is as follows: $0.4<(g-r)_{0}<0.5$, $0.10<(r-i)_{0}<0.14$, $0.02<(i-z)_{0}<0.06$, and $g_{0}<20$. The above color ranges come from the color of the Sun and its error bars. Based on this limitation, 4356 stars are extracted from the SDSS DR7 archive. Fig. 8 presents the information of this sample, from which we can see the peak of S/N is 20; the effective temperature of most stars is located in the range of [5500, 6000]K; log $g$ is in [4.0, 4.5]. By transforming equation of Bilir et al. (2005), the g-r range of Chen00 is [0.197, 0.501], so it is available to estimate the metallicity of these solar-like stars using Equation 2 and Equation 3. ### 5.1 Preprocess Our first preprocessing procedures mainly include radial velocity correction and normalization. The value of radial velocity comes from the FIT head of each spectra. The pseudocontinuum is determined with the same method illustrated in Sec. 4.3. Although the method of pseudocontinuum determination is different with that in Chen00, it is good enough for solar type stars in the red spectral region. Since the SDSS spectra are relative flux calibrated, the pseudocontinuum in red region is easier represented by a lower order polynomial. Thus, iteratively smoothing with a Gaussian profile to original spectrum will get a good continuum shape. After normalization, then the EW of the lines can be measured by the direct integration method. Fig. 9 is an example of the SDSS spectrum. ‘a’ is the original spectrum while ‘b’ is the normalized spectrum. ‘c’, ‘d’, ‘e’ and ‘f’ present the FeI lines in our calibration. It is clear that these FeI lines are detectable and show a good profile in the SDSS spectra. Figure 10: [Fe/H] comparison of the sdss solar-like stars between those from SSPP and those derived from the calibration using five FeI lines ### 5.2 Calibration After obtaining the EW of five FeI lines, [Fe/H] can be determined by our calibration formula. We do some comparison between [Fe/H] from our calibration and those from SSPP. Equation 4 is the calibration formula based on the FeI4 line. B-V can be obtained from g-r transformation (Bilir et al. 2005). Fig. 10 is [Fe/H] comparison between those from SSPP and the results from Equation 4. The comparison of stars with S/N $>$ 50 is shown in the top panel and the difference distribution is given in bottom panel. It is clear that our calibration from Equation 4 is very consistent with [Fe/H] of SSPP for those with S/N $>$ 50\. The mean difference is about 0.018 dex and the scatter is around 0.26 dex. $\displaystyle\rm[Fe/H]=-0.321+6.622*\rm{EW(FeI4)}-1.643*(B-V)$ (4) Moreover, we extract 51 stars which meet these conditions: [Fe/H]$\geq$0 from our result; [Fe/H]$<0$ from SSPP; S/N$>50$. So these 51 stars are metal rich stars if our result is right. As for these 51 stars, the temperature range is about [5650, 5865]K and the gravity range is in [4.3, 4.5] dex. To check the reliability of our result, we make a comparison between the strength of some spectral lines in these stars and those in the Sun since $T\rm{{}_{eff}}$ and log $g$ of these stars are very close to those of the Sun. If the strength is stronger than that of the Sun then this star can be regarded as a metal rich star and its [Fe/H] is larger than 0. Since the NaI and CaII line are strong in the red band, these lines, as well as two FeI lines, are selected for comparison. There are three cases: one is that the strengths of these lines are all stronger than those of the Sun (see Fig. 11); one is that the strengths of these lines are all close to those of the Sun (See Fig. 12); the others are taken as the third case (See Fig. 13). After comparison, there are 33 stars in first case, 10 stars in second case and 8 stars in third case. In view of this, the metal rich stars account for 84% of these 51 stars. So our calibration also provides a reliable way to identify metal rich stars. Figure 11: The comparison of the strength of spectral lines. The solid line is the object spectrum and the dotted line is the solar spectrum Figure 12: The comparison of the strength of spectral lines. The solid line is the object spectrum and the dotted line is the solar spectrum Figure 13: The comparison of the strength of spectral lines. The solid line is the object spectrum and the dotted line is the solar spectrum Figure 14: Left:[Fe/H] change vs. S/N for seven stars. The original spectra of these seven stars were extracted from DR7 with S/N $\sim$ 100\. By introducing Gaussian noise in original spectra, we degraded them to S/N of 50, 30, and 20, respectively. Filled squares are [Fe/H] changes of these seven stars with different S/N, while asterisks represent average [Fe/H] change for the given S/N. Right: [Fe/H] differences between our calibration results and those from SSPP vs. S/N. Asterisks represent average [Fe/H] differences and vertical lines represent the scatter of [Fe/H] differences at the given S/N. ### 5.3 The effect of S/N To investigate the S/N effect on our calibration, we selected seven spectra from DR7 with S/N (r band) $\sim$ 100\. By introducing Gaussian noise in these spectra, we degraded them to S/N of 50, 30, and 20, respectively. Then, normalization and EW measurement were carried out to all spectra. Finally, the metallicity were derived with our above calibration. The left panel of Fig. 14 presents the S/N effect on metallicity change. Filled squares represent [Fe/H] changes for these seven stars with different S/N. Asterisks represent the average [Fe/H] changes for the given S/N. It can be seen that [Fe/H] changes about 0.22 dex from S/N = 100 to 20. [Fe/H] differences between our calibration results and those of SSPP vs. S/N are shown in right panel of Fig. 14. Asterisks are average differences and vertical lines represent scatters for the given S/N for these seven stars. Average [Fe/H] differences nearly keep the same while the scatter will be smaller than 0.4 dex when S/N is higher than 30. To sum up, our metallicity determination is quite robust to reductions in S/N. ## 6 Conclusions For solar type stars, although template matching can derive reliable results with a suitable wavelength range, it is very difficult to determine the most appropriate wavelength range for matching. We selected five FeI lines from the red part of the R$\sim$2000 resolution spectra. These lines, which have a good profile, are not seriously blended and could be detectable with [Fe/H] $>$ -0.8. At the beginning, the metallicity calibrations are set up only through the EW and the scatters are from 0.14 to 0.20 dex. The dispersion becomes small after adding the temperature into the calibrations. Since the temperature is usually unknown in the spectra analysis, it may be good to replace the temperature term with the color index. In view of this, several metallicity calibrations are constructed by the EW of FeI lines and colors based on the 90 solar type stars. The dispersion of all the calibrations is smaller than 0.21 dex. Among the five FeI lines, FeI2 and FeI4 have contributed the better calibrations (Equation 5-6) which have smaller scatters (0.13 dex, 0.15 dex). $\displaystyle\rm[Fe/H]=-0.290+8.144*\rm{EW(FeI2)}-1.342*(B-V),~{}~{}~{}0.006<EW(FeI2)<0.158$ (5) $\displaystyle\rm[Fe/H]=-0.321+6.622*\rm{EW(FeI4)}-1.643*(B-V),~{}~{}~{}0.029<EW(FeI4)<0.224$ (6) Moreover, we use the calibration from the EW of FeI4 and the B-V to estimate [Fe/H] of the solar type stars in DR7. After comparing with the value from SSPP, our method gives a good consistency for S/N larger than 50. In addition, we analyze the stars for which [Fe/H] $\geq 0$ by the spectral lines comparison and found that 84% of them are reliable. Usually, [Na/Fe]=0 $\&$ [Ca/Fe]=0 for most stars with [Fe/H] $>$ -0.4 in the solar neighborhood. In view of this, Na and Ca lines are stronger in Fe-rich stars. So this provides a new formula to estimate [Fe/H] with the red band and presents a reliable way to identify metal rich stars. ###### Acknowledgements. This work is supported by the National Natural Science Foundation of China under grant Nos. 10673015, 10821061, 10973021, 11078019 and 11073026, the National Basic Research Program of China (973 program) No. 2007CB815103/815403, the Academy program No. 2006AA01A120 and the Youth Foundation of the National Astronomical Observatories of China. Many thanks to James Wicker for his help revising English grama of this paper. ## References * Abazajian (2009) Abazajian K., et al. 2009, ApJS, 182, 543 * Ademan-McCarthy (2008) Adelman-McCarthy J. K., et al., 2008, ApJS, 175, 297 * Allende (2006) Allende P. C., Beers T. C., Wilhelm R., Newberg H. J., Rockosi C. M., Yanny B., Lee Y. S., 2006, ApJ, 636,804 * Allende (2008) Allende P. C., Sivarani T., Beers T. C., Lee Y. S., Koesterke L., Shetrone M., Sneden C., Lambert D. L., Wilhelm R., Rockosi C. 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C., Sivarani T., Allende P.C., Koesterke L., Wilhelm R., Re Fiorentin P., Bailer-Jones C. A. L., Norris J. E., Rockosi C. M., Yanny B., Newberg H. J., Covey K. R.,Zhang H. T., Luo A. L., 2008, AJ, 136, 2022 * Lee (2008b) Lee Y. S., Beers T. C., Sivarani T., Johnson J. A., An D., Wilhelm R., Allende P. C., Koesterke L., Re Fiorentin P., Bailer-Jones C. A. L., Norris J. E., Yanny B., Rockosi C., Newberg H. J., Cudworth K. M., Pan K., 2008, AJ, 136, 2050 * Re Fiorentin (2007) Re Fiorentin P., Bailer-Jones C. A. L., Lee Y. S., Beers, T. C., Sivarani T., Wilhelm R., Allende P. C., Norris J. E., 2007, A&A, 467, 1373 * York (2000) York D. G., et al., AJ, 2000, 120, 1579 * Sánchez-Blázquez (2003) Sánchez-Blázquez P., Peletier R. F., Jiménez-Vicente J., Cardiel N., Cenarro A. J., Falcón-Barroso J., Gorgas J., Selam S., Vazdekis A, 2006, MNRAS, 371, 703 * Zwitter (2005) Zwitter T., Munari U., Siebert A., 2005, ESASP, 576, 623 * Zhao (2001) Zhao G., & Li H. B., 2001, Chinese J. Astron. 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arxiv-papers
2011-01-17T16:36:38
2024-09-04T02:49:16.506636
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "J. K. Zhao, G. Zhao, Y. Q. Chen, A. L. Luo", "submitter": "Jingkun Zhao", "url": "https://arxiv.org/abs/1101.3260" }
1101.3373
# Effects of interaction and polarization on spin-charge separation: A time- dependent spin-density-functional theory study Gao Xianlong gaoxl@zjnu.edu.cn Department of Physics, Zhejiang Normal University, Jinhua 321004, Zhejiang Province, China ###### Abstract We calculate the nonequilibrium dynamic evolution of a one-dimensional system of two-component fermionic atoms after a strong local quench by using a time- dependent spin-density-functional theory. The interaction quench is also considered to see its influence on the spin-charge separation. It is shown that the charge velocity is larger than the spin velocity for the system of on-site repulsive interaction (Luttinger liquid), and vise versa for the system of on-site attractive interaction (Luther-Emery liquid). We find that both the interaction quench and polarization suppress the spin-charge separation. ###### pacs: 71.15.Mb, 03.75.Ss, 71.10.Pm ## I Introduction While the nonequilibrium dynamic evolution of quantum systems has long been extensively studied, Mattis progress is hindered by the tremendous difficulties in solving the nonequilibrium quantum many-body Schrödinger equation. This situation is going to be changed due to the progress in experiments and the development in numerical methods. On the experimental side, the development in manipulating ultracold atomic gases makes it feasible to study strongly correlated systems with time-varying interactions and external potentials and in out-of-equilibrium situations. The high controllability in ultracold atomic-gases’ systems provides an ideal testbed to observe the long-time evolution of strongly correlated quantum many-body systems, and to test theoretical predictions, such as the Bloch oscillation, Bloch the absence of thermalization in nearly integrable one- dimensional (1D) Bose gases, Kinoshita and the expansion of BEC in a random disorder after switching off the trapping potential. Anderson These efforts allow us to study the nonequilibrium dynamics of strongly correlated systems from a new perspective. Numerically, many techniques have been developed, such as, the time-adaptive density-matrix renormalization group (t-DMRG), Schollwoek the time-dependent numerical renormalization group, Costi continuous-time Monte Carlo algorithm, Eckstein and time-evolving block decimation method. Vidal Time-dependent spin-density-functional theory (TDSDFT) has been proved to be a powerful numerical tool beyond the linear-response regime in studying the interplay between interaction and the time-dependent external potential. gaoprb78Liwei ; verdozzi_2007 More tests of the performance of TDSDFT will be done in this paper on the polarized system with attractive or repulsive interactions. Compared to the algorithms, such as the t-DMRG, this technique gives numerically inexpensive results for large lattice systems and long-time evolution, but with difficulties in calculating some properties, such as, the correlation functions. The 1D bosonic or fermionic systems accessible by the present ultracold experiments, moritz ; bosonic are exactly solvable in some cases Gaudin and can be used to obtain a thorough understanding of the many-body ground-state and the dynamical properties. The nonequilibrium problems in 1D system are especially remarkable in which the 1D systems are strongly interacting, weakly dissipative, and lack of thermalization. Sutherland The 1D systems, belonging to the universality class described by the Luttinger-liquid theory, have its particularity in its low-energy excitations, characterized by charged, spinless excitations and neutral, spin-carrying collective excitations. Generically, the different dynamics is determined by the velocities of the charge and spin collective excitations, which has been verified experimentally in semiconductor quantum wires by Auslaender et al.. Auslaender The possibility of studying these phenomena experimentally in 1D two-component cold Fermi gases, moritz where ”spin” and ”charge” refer, respectively, to the density difference and the total atomic mass density of the two internal atomic states, was first highlighted by Recati et al. recati_prl_2003 . The different velocities for spin and charge in the propagation of wave packets have been demonstrated by Kollath et al. kollath ; Kollath2 in a numerical t-DMRG study of the 1D Fermi-Hubbard model, by Kleine et al. kleine in a similar study of the two-component Bose-Hubbard model, and, analytically, by Kecke et al. kecke_prl_2005 for interacting fermions in a 1D harmonic trap. Exact diagonalization and quantum Monte Carlo simulations are also used in studying the spin and charge susceptibilities of the Hubbard model. Jagla ; Zacher Dynamic structure factors of the charge density and spin are analyzed for the partially spin-polarized 1D Hubbard model with strong attractive interactions using a time-dependent density-matrix renormalization method. Huse The spin-charge separation is well addressed for this system. Huse We would like to mention here that a genuine observation of spin-charge separation requires one to explore the single-particle excitation, which is studied recently in simulating the excitations created by adding or removing a single particle. Kollath2 ; Ulbricht The nonequilibrium dynamics in 1D systems has attracted a growing attention in the possible equilibrium properties after an external perturbation and the changes in physical quantities after the quench. quench ; kollath ; Karlsson The dynamic phase transition and different relaxation behavior are studied with a sudden interaction quench Eckstein ; Barmettler . The relation between the thermalization and the integrability in 1D system is well addressed. Rigol The real-time evolution for the magnetization in the 1D spin chain is also studied in great details using the t-DMRG. Langer In this paper, we study the 1D system under an instantaneous switching off a strong local potential or on-site interactions, namely, a sudden quantum quench is considered. The strong local potential creates Gaussian-shaped charge and/or spin accumulations at some position in space. After the quantum quench, the time-evolution of spin and charge densities is then calculated at later times. We tackle this problem using TDSDFT based on an adiabatic local spin density approximation (ALSDA). The contents of the paper are as follows. In Sec. II, we introduce the model: a time-dependent lattice Hamiltonian that we use to study spin-charge separation and quench dynamics. Then we briefly summarize the self-consistent lattice TDSDFT scheme that we use to deal with the time-dependent inhomogeneous system. In Sec. III, we report and discuss our main numerical results. At last, a concluding section summarizes our results. ## II Model and the method We consider a two-component repulsive/attractive Fermi gas with $N_{f}$ atoms loaded in a 1D optical lattice with $N_{s}$ lattice sites. At time $t\leq 0$, a localized spin- and charge-density perturbation is created by switching on slowly the local potential, such that the system is in the ground state of the system with the additional potential. At $t=0^{+}$, the localized potential is removed abruptly and/or the on-site interaction is switched off instantaneously. This system is modeled by a time-dependent Fermi-Hubbard Hamiltonian as follows: $\displaystyle{\hat{\cal H}}(t)$ $\displaystyle=$ $\displaystyle-\gamma\sum_{i,\sigma}({\hat{c}}^{\dagger}_{i\sigma}{\hat{c}}_{i+1\sigma}+{\rm H}.{\rm c}.)+U(t)\sum_{i}{\hat{n}}_{i\uparrow}{\hat{n}}_{i\downarrow}$ (1) $\displaystyle+$ $\displaystyle\sum_{i,\sigma}V_{i\sigma}(t){\hat{n}}_{i\sigma}~{}.$ Here $\gamma$ is the hopping parameter, ${\hat{c}}^{\dagger}_{i\sigma}$ (${\hat{c}}_{i\sigma}$) creates (annihilates) a fermion in the $i$th site ($i\in[1,N_{s}]$), $\sigma=\uparrow,\downarrow$ is a pseudospin-$1/2$ degree- of-freedom (hyperfine-state label), $U(t)$ is the time-dependent on-site Hubbard interaction of negative or attractive nature, and ${\hat{n}}_{i\sigma}={\hat{c}}^{\dagger}_{i\sigma}{\hat{c}}_{i\sigma}$. We also introduce for future purposes the local number operator ${\hat{n}}_{i}=\sum_{\sigma}{\hat{n}}_{i\sigma}$ and the local spin operator ${\hat{s}}_{i}=\sum_{\sigma}\sigma{\hat{n}}_{i\sigma}/2$. The external time-dependent potential $V_{i\sigma}(t)=V^{\rm ext}_{i\sigma}\Theta(-t)$, which simulates the spin-selective focused laser- induced potential. $\Theta(t)$ is the Heaviside step function which relates the quench dynamics to the modification of the local potential. $\Theta(-t)=0$ for $t>0$. $V^{\rm ext}_{i\sigma}$ is taken to be of the following Gaussian form: $\displaystyle V^{\rm ext}_{i\sigma}$ $\displaystyle=$ $\displaystyle W_{\sigma}\exp{\left\\{-\frac{[i-(N_{s}+1)/2]^{2}}{2\alpha^{2}}\right\\}}~{}.$ (2) Here $W_{\sigma}$ is the amplitude of the local potential. We discuss the system of conserved particle number in the canonical ensemble. The number of atoms for spin up and spin down is, $N_{\uparrow}$ and $N_{\downarrow}$, respectively. The polarization is defined as $p=(N_{\uparrow}-N_{\downarrow})/N_{f}$. The on-site interaction and $W_{\sigma}$ are scaled in units of $\gamma$ as, $u=U/\gamma$ and $w_{\sigma}=W_{\sigma}/\gamma$, respectively. A powerful theoretical tool to investigate the dynamics of many-body systems in the presence of time-dependent inhomogeneous external potentials, such as that in Eq. (1), is TDSDFT, Giuliani_and_Vignale ; marques_2006 based on the Runge-Gross theorem rgt and on the time-dependent single-particle Kohn-Sham equations. The complication of the problem is hidden in the unknown time- dependent exchange and correlation (xc) potential. Most applications of TDSDFT use the simple adiabatic local spin-density approximation for the dynamical xc potential, Giuliani_and_Vignale ; zangwill which has often been proved to be successful in studying the real-time evolution. marques_2006 In this approximation, one assumes that the time-dependent xc potential is just the static xc potential evaluated at the instantaneous density, where the xc potential is local in time and space. The static xc potential is then treated within the static local spin-density approximation. Very attractive features of the ALSDA are its extreme simplicity, the ease of implementation, and the fact that it is not restricted to mean-field approximation and small deviations from the ground-state density, i.e., to the linear response regime. The dynamics induced by the strong local perturbation discussed here cannot be dealt with the theory based on the linear response while TDSDFT is a good candidate. We here employ a lattice version of spin-density-functional theory (SDFT) and TDSDFT. gaoprb78Liwei In short, for times $t\leq 0$, the spin-resolved site- occupation profiles can be calculated by means of a static SDFT. For times $t>0$, we calculate the time evolution of spin-resolved site-occupation profiles $n_{i\sigma}(t\leq 0)$ by means of a TDSDFT scheme in which the time- dependent xc potential is determined exactly at the ALSDA level (details see, Ref. [gaoprb78Liwei, ]). The performance of this method has been tested systematically against accurate t-DMRG simulation data for the repulsive Hubbard model. gaoprb78Liwei It is found that, the simple ALSDA for the time- dependent xc potential is surprisingly accurate in describing collective density and spin dynamics in strongly correlated 1D ultracold Fermi gases in a wide range of coupling strengths and spin polarizations. The performance of TDSDFT in describing the nonequilibrium behavior of strongly correlated lattice models has also been recently addressed in Ref. [verdozzi_2007, ]. In this work, we use this method to mainly discuss the nature of the interactions on the velocities of the density and spin evolution. The spin- charge dynamics after a local quench is discussed in Luttinger liquids (for $U>0$, gapless spin and charge excitations) and in Luther-Emery liquids (for $U<0$, gapless charge and gapful spin excitations). We consider at the same time the influence of polarization on the spin-charge dynamics. For attractive interactions, we limit our discussion on the weak-interaction case because for strong attractive interactions we found our SDFT code overestimates the amplitude of the bulk atomic density waves, which will greatly influence the TDSDFT results based on that. Experimentally the strong local potential can be obtained by a blue- or red- detuned laser beam tightly focused perpendicular to the 1D atomic wires, which generates locally repulsive or attractive potentials for the atoms in the wires, corresponding to $W_{\sigma}>0$ or $W_{\sigma}<0$. In this paper, we are interested in the repulsive potential for the atoms. The charge and spin densities can be observed by using in situ sequential absorption imaging, electron beams, or noise interference, Shin which, in principle, gives an unambiguous information on the spin-charge separation. ## III Numerical results and discussion In this section, we report on the results calculated by solving the time- dependent Kohn-Sham equations. Mathematically the solution of the time- dependent Kohn-Sham equations is an initial value problem. A given set of initial orbitals calculated from the static Kohn-Sham equations is propagated forward in time. No self-consistent iterations are required as in the static case. For times $t\leq 0$, the system is in the presence of a strong local potential, which creates a strong local disturbance in ultracold gases and makes the total density and spin-density distributions in the center of the system locally different (up to a few lattice sites). We are interested in two kinds of quench dynamics. The first one is that, at time $t=0^{+}$, the local potential is quenched with the time-independent on-site interaction $U(t)=U$. The second is that, at time $t=0^{+}$, the local potential is switched off and at the same time the on-site interaction is quenched instantaneously with $U(t)=U\Theta(-t)$. After the quench, excitations are produced. We concern in this paper the subsequent real-time evolution of the spin-resolved densities after the quench, $n_{i\sigma}(t)=\langle\Psi(t)|\hat{n}_{i\sigma}|\Psi(t)\rangle$ with $|\Psi(t)\rangle$ the state of the system at time $t$. Charge density and spin density are defined accordingly as $n_{i}(t)=n_{i\uparrow}(t)+n_{i\downarrow}(t)$ and $s_{i}(t)=[n_{i\uparrow}(t)-n_{i\downarrow}(t)]/2$. If not mentioned otherwise, the numerical results presented below correspond to a system with $N_{f}=30$ atoms on $N_{s}=100$ sites, and with open (hard wall) boundary conditions imposed at the sites $i=0$ and $i=101$. The external potential is chosen to be spin dependent: $w_{\uparrow}=-1$ and $w_{\downarrow}=0$, used to form a local density and spin density occupations in the center of the system. ### III.1 $u>0$ and $p=0$ In Fig. 1, we show results for a spin-unpolarized system ($N_{\uparrow}=N_{\downarrow}=15$) with repulsive interaction of $u=2$. At $t\leq 0$, a dominant local charge- and spin-density profiles in the center of the system are generated by the strong local potential. After the quench of the local potential, the charge and spin densities evolve and split into two counterpropagating density wave packets. The propagation in time is in fact due to the nonequilibrium initial condition. The charge density evolves with a quicker velocity than the spin, which is in agreement with the general picture of spin-charge separation. giamarchi_book A qualitative analysis based on the continuity equation for the momentum density can also well explain the phenomena of spin-charge separation. gaoprl102 Figure 1: (Color online) Charge $n_{i}(t)$ and spin $s_{i}(t)$ occupations as functions of lattice site $i$ and time $t$ for $N_{s}=100$, $N_{\uparrow}=N_{\downarrow}=15$, $w_{\uparrow}=-1$, $w_{\downarrow}=0$, $\alpha=2$, and repulsive interaction of $u=+2$. Top panel: ground-state charge and spin occupations for times $t\leq 0$ (solid line) and at time $t=5~{}\hbar/\gamma$ (dashed-dotted line). Bottom panel: same as in the top panels but at time $t=10~{}\hbar/\gamma$ (solid line) and $t=20~{}\hbar/\gamma$ (dashed-dotted line). The charge and spin densities are plotted in the top and bottom of the panel, respectively. The arrows in the plot indicate the positions where the wave packets propagate. In the inset, we show the velocities of the charge $v_{c}$ (open circles) and spin $v_{s}$ (solid circles) density wave packets as a function of the amplitude of the local potential $|w_{\uparrow}|$. Both velocities are increasing functions of $|w_{\uparrow}|$. We notice a common feature in almost all the figures in this paper, that is, the spin and charge densities have an asymmetric forward-leaning shape. This is caused by a nonlinear effect, i.e., the different local velocities in the center and at the edges. Since the local velocity is proportional to the density, the higher density in the center gains larger velocity than that at the edges, which qualitatively explains why the asymmetric forward-leaning shape happens during the density propagation. For perturbations with small amplitude, the charge velocity is studied in details by t-DMRG and compared to the Bethe-ansatz results with good agreement. kollath For the strong local potential studied here, the spin and charge velocities, determined from the propagation of the maximum of the charge and spin wave packets away from the center, vary with time. We thus calculate and compare the velocities determined at the fixed time $t=10\hbar/\gamma$. In the inset of Fig. 1, we show the spin and charge velocities as a function of the amplitude of the local potential $|w_{\uparrow}|$. We find both velocities are increasing functions of $|w_{\uparrow}|$. For the charge background density ($\sim 0.3$) in Fig. 1, the charge and spin velocities by the Bethe-ansatz method are $v_{c}=1.15$ and $v_{s}=0.75$. In the limit of $w_{\uparrow}\rightarrow 0$, but $w_{\downarrow}\equiv 0$, our results give $v_{c}=1.3$ and $v_{s}=0.65$. The differences are possibly caused by the simultaneous local perturbations in the charge and spin densities used here, which break the spin-charge scenario and couple the spin and charge modes, similar to the effects caused by the finite spin polarization (see Secs. III-C and III-D). Figure 2: (Color online) Charge $n_{i}(t)$ and spin $s_{i}(t)$ occupations as functions of lattice site $i$ and time $t$ with quenches for the local potential and on-site interaction, i.e., $V_{i\sigma}(t)=V^{\rm ext}_{i\sigma}\Theta(-t)$ and $U(t)=U\Theta(-t)$. The other parameters are the same as that in Fig. 1. The static density (solid line) is shown together with two time shots for $t=5~{}\hbar/\gamma$ (dash line) and $t=10~{}\hbar/\gamma$ (dashed-dotted line). Figure 3: (Color online) 3D plots for the Charge density $n_{i}(t)$ (Top panel) and spin density $s_{i}(t)$ (Bottom panel) as functions of lattice site $i$ and time $t$ (in units of $\hbar/\gamma$) for a harmonically trapped system with $N_{s}=200$, $N_{\uparrow}=N_{\downarrow}=15$, $V_{2}/\gamma=5\times 10^{-4}$, $w_{\uparrow}=-1$, $w_{\downarrow}=0$, $\alpha=2$, and repulsive interaction of $u=2$. In Fig. 2, we study the local potential quench together with an on-site interaction quench, i.e., $V_{i\sigma}(t)=V^{\rm ext}_{i\sigma}\Theta(-t)$ and $U(t)=U\Theta(-t)$. We find that, the spin- and charge-density wave packets split and counterpropagate as usual but the phenomena of the spin-charge separation completely disappears. That is, the spin and charge densities evolve with the same velocity. From the Luttinger-liquid theory based on the bosonization method Coll or from the Bethe-ansatz solution, Schulz one can derive that the spin velocity $v_{s}$ and the charge velocity $v_{c}$ satisfy $v_{c}=v_{s}=v_{F}$ in the noninteracting limit, with $v_{F}=2\gamma\sin(\pi n/2)$ the Fermi velocity. The interaction between the different species is one of the important ingredients for the spin-charge separation, which explains the suppression of the spin-charge separation after the interaction quench. Making use of the techniques from the cold atomic gases, two different ways of quenching, used in Figs. 1 and 2, respectively, can give a clear signal that different collective spin and charge dynamics happens when starting from the same initial strong local perturbation. We would like to mention that Kollath proposed to repeat the dynamics in Fig. 1 in higher dimensions where no separation of spin and charge should be seen. Kollath2 We notice that in Fig. 2 already at short time, some density waves coming from the sharp edges begin to influence the charge- and spin-density wave packets from the center. At larger time, they will mix with the original packets. Figure 4: (Color online) Charge $n_{i}(t)$ and spin $s_{i}(t)$ occupations as functions of lattice site $i$ and time $t$ for $N_{s}=100$, $N_{\uparrow}=N_{\downarrow}=15$, $w_{\uparrow}=-1$, $w_{\downarrow}=0$, $\alpha=2$, and attractive interaction of $u=-1$. Top panel: ground-state charge and spin occupations for times $t\leq 0$ (solid line) and at time $t=5~{}\hbar/\gamma$ (dashed-dotted line). Bottom panel: same as in the top panels but at time $t=10~{}\hbar/\gamma$ (solid line) and $t=20~{}\hbar/\gamma$ (dashed-dotted line). The inset shows the velocities of the charge $v_{c}$ (open circles) and spin $v_{s}$ (solid circles) density wave packets as a function of the amplitude of the local potential $|w_{\uparrow}|$. Figure 5: (Color online) Contour plots for the Charge density $n_{i}(t)$ (Top panel) and spin density $s_{i}(t)$ (Bottom panel) as functions of lattice site $i$ and time $t$ (in units of $\hbar/\gamma$) for a harmonically trapped system with $N_{s}=200$, $N_{\uparrow}=N_{\downarrow}=15$, $V_{2}/\gamma=5\times 10^{-4}$, $w_{\uparrow}=-1$, $w_{\downarrow}=0$, $\alpha=2$, and attractive interaction of $u=-1$. In practice, an additional trapping potential is unavoidable in the present experimental set-ups. We thus present our simulations for the system in the presence of an additional weak superimposed harmonic trapping potential, namely, $V^{\rm ext}_{i\sigma}$ in Eq. (2) is changed into, $\displaystyle V^{\rm ext}_{i\sigma}=W_{\sigma}e^{-\frac{[i-(N_{s}+1)/2]^{2}}{2\alpha^{2}}}+V_{2}\left(i-\frac{N_{s}+1}{2}\right)^{2}.$ (3) Here we take $V_{2}/\gamma=5\times 10^{-4}$. The three-dimensional (3D) plots of the time evolution of the spin- and charge-density wave packets are shown in Fig. 3. From the figure, we observe that, in the presence of the harmonic potential the charge and spin wave packets are highly inhomogeneous, but the counter-propagation and the separation of the charge- and spin-density wave packets are still visible in the background of the inverted parabola. ### III.2 $u<0$ and $p=0$ In one-dimensional Hubbard model, away from half filling, the spin and charge velocities of the low-energy collective excitations satisfy, Coll ; Schulz $v_{s,c}=v_{F}\sqrt{1\mp\frac{U}{\pi v_{F}}}\,.$ This gives a qualitative explanation that for the positive-$U$ Hubbard model, the charge velocity is larger than the spin velocity, while for the negative-$U$ Hubbard model, the charge velocity is smaller than the spin velocity. In Fig. 4, the quench dynamics for the attractive Hubbard model, which belongs to the Luther-Emery universality class, illustrates that spin- wave packets evolve with a faster speed than the charge branches. In the inset of Fig. 4, we show the spin and charge velocities evaluated at $t=10\hbar/\gamma$ as a function of the amplitude of the local potential $|w_{\uparrow}|$. We notice that an abrupt change appears in the charge velocity at $|w_{\uparrow}|\approx 0.55$. For attractive interactions, Luther- Emery paring induces a prominent density wave characterized by the dip-hump structure. While the charge velocity is determined from the propagation of the maximum of the charge wave packets located at one of the humps of the density wave. The increase in the amplitude of the local potential makes the maximum of the charge wave packets move from the lattice site $i=44$ to $i=39$, which explains the discontinuity of the charge velocity for attractive interactions at $|w_{\uparrow}|\approx 0.55$. However, this discontinuous change has artifacts because the way of extracting $v_{c,s}$ used here is not an optimum one. In Fig. 5, we present the contour plots of the time evolution of the density and spin packets for the system in the presence of a harmonic trapping potential with $V_{2}/\gamma=5\times 10^{-4}$. The different evolution velocities for the charge- and spin-density wave packets are clearly visible. Figure 6: (Color online) The ground-state charge and spin occupations as functions of lattice site $i$ and time $t$ for the system of repulsive interaction of $u=2$ in the polarized case of $P=0.47$ ($N_{\uparrow}=22,N_{\downarrow}=8$). Besides the ground-state density and spin density (solid line), three time shots are shown with $t=5~{}\hbar/\gamma$ (dash line,) $t=10~{}\hbar/\gamma$ (dashed-dotted line), and $t=15~{}\hbar/\gamma$ (dotted line). Figure 7: (Color online) Same as Fig. 6 but for the polarized system of $P=0.87$ ($N_{\uparrow}=28,N_{\downarrow}=2$). Figure 8: (Color online) The ground- state charge and spin occupations as functions of lattice site $i$ and time $t$ for the system of attractive interaction of $u=-1$ in the polarized case of $p=0.47$ ($N_{\uparrow}=22,N_{\downarrow}=8$). Besides the ground-state density and spin density (solid line), three time shots are shown with $t=5~{}\hbar/\gamma$ (dash line,) $t=10~{}\hbar/\gamma$ (dashed-dotted line), and $t=15~{}\hbar/\gamma$ (dotted line). Figure 9: (Color online) Same as Fig. 8, but for the polarized system of $p=0.87$ ($N_{\uparrow}=28,N_{\downarrow}=2$). ### III.3 $u>0$ and $p\neq 0$ The spin-charge separation in a spin-polarized one-dimensional system is quite different from the fully polarized one. The spin-charge-coupled dynamics in a polarized system formulated with the first-quantized path-integral formalism and bosonization techniques provides us a new non-Tomanaga-Luttinger-liquid universality class. Akhanjee For the Luther-Emery liquid of unpolarized attractive Fermi gases, the spin and charge degrees of freedom are decoupled. In contrast, in the system with finite spin imbalance, spin-charge mixing is found based on an effective-field theory for the long-wavelength and low- energy properties. Erhai In Figs. 6 and 7, the quench dynamics for spin- and charge-density waves is shown for the system of repulsive interaction ($u=2$) with polarization of $p=0.47$ and $0.87$, respectively. For $p\geq 0.47$, there is only small difference between spin and charge velocities. In the case of a large polarization, the same propagating velocities for spin and charge are obtained. ### III.4 $u<0$ and $p\neq 0$ The quench dynamics for spin and charge density waves of the attractive case for $u=-1$ is shown in Figs. 8 and 9. We find with the increasing of the polarization, the spin-charge separation is strongly suppressed due to the interplay between charge and spin degrees of freedom. Theoretically, for the partially polarized system, the spin and charge modes are coupled. In this case, there is no strict spin-charge separation scenario, namely, the spin- charge separation breaks down. Numerically, we observe that, at small polarization the spin and charge wave packets still evolve at different velocities although they are coupled and influence each other. At large polarization, the spin-charge separation disappears and evolves at the same velocities for both the repulsive and the attractive systems we studied. ## IV Conclusions In summary, we have calculated the non-equilibrium dynamic evolution of a one- dimensional system of two-component fermionic atoms after a strong local quench with or without interaction quench by using a time-dependent density- functional theory with a suitable Bethe-ansatz based adiabatic local spin- density approximation. A test of the performance of TDSDFT is provided for the unpolarized systems with attractive or repulsive interactions in the presence of a harmonic trapping potential. Under the same local perturbation, the charge velocity is larger than the spin velocity for the system of repulsive interaction and vice versa for the attractive case, which is compatible with the low-energy collective dynamics from the Bethe-ansatz solution or the bosonization techniques. 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arxiv-papers
2011-01-18T03:01:48
2024-09-04T02:49:16.513969
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Gao Xianlong", "submitter": "Gao Xianlong", "url": "https://arxiv.org/abs/1101.3373" }
1101.3398
11institutetext: National Key Lab. of ISN, Xidian University , Xi’an 710071, P.R.China 11email: wp_ma@mail.xidian.edu.cn # New Quadriphase Sequences families with Larger Linear Span and Size Wenping Ma ###### Abstract In this paper, new families of quadriphase sequences with larger linear span and size have been proposed and studied. In particular, a new family of quadriphase sequences of period $2^{n}-1$ for a positive integer $n=em$ with an even positive factor $m$ is presented, the cross-correlation function among these sequences has been explicitly calculated. Another new family of quadriphase sequences of period $2(2^{n}-1)$ for a positive integer $n=em$ with an even positive factor $m$ is also presented, a detailed analysis of the cross-correlation function of proposed sequences has also been presented. ## 1 Introduction Family of pseudorandom sequences with low cross correlaton and large linear span has important application in code-division multiple access communications and cryptology. Quadriphase sequences are the one most often used in practice because of their easy implementation in modulators. However, up to now, only few families of optimal quadriphase sequences are found [1],[5, 6, 7, 8, 9, 10, 11, 12]. Among the known optimal quadriphase sequence families, the most famous ones are the families $\mathcal{A}$ and $\mathcal{B}$ investigated by Boztas, Hammons, and Kummar in[5]. The family $\mathcal{A}$ has period $2^{n}-1$ and family size $2^{n}+1$, while the two corresponding parameters of the family $\mathcal{B}$ are $2(2^{n}-1)$ and $2^{n-1}$, respectively. Another optimal family $\mathcal{C}$ was discussed in [10], and this family has the same correlation properties as the family $\mathcal{B}$. Families $S(t)$ were defined by Kumar et.[6], and when $t=0$ or $m$ is odd, the correlation distributions of families $S(t)$ are established by Kai-Uwe Schmidt[7]. Tang, Udaya, and Fan generalized the family $\mathcal{A}$ and proposed a new family of quadriphase sequences with low correlation in [12]. By utilizing a variation of family $\mathcal{B}$ and $\mathcal{C}$, Tang and Udaya obtained the family $\mathcal{D}$, which has period $2(2^{n}-1)$ and a larger family size $2^{n}$[9]. Recently, Wenfeng Jiang, Lei Hu, Xiaohu Tang, and Xiangyong Zeng proposed two new families $\mathcal{S}$ and $\mathcal{U}$ of quadriphase sequences with larger linear spans for a positive integer $n=em$ with an odd positive factor $m$. Both families are asymptotically optimal with respect to the Wech and Sidelnikov bounds. The family $\mathcal{S}$ has period $2^{n}-1$, family size $2^{n}+1$, and maximum correlation magnitude $2^{\frac{n}{2}}+1$. The family $\mathcal{U}$ has period $2(2^{n}-1)$, family size $2^{n}$, and maximum correlation magnitude $2^{\frac{n+1}{2}}+2$ [1]. In this paper, motivated by the constructions proposed in [1, 2, 3, 4, 6], the new families of quadriphase sequences with larger linear span and size have been presented. As a special case of the sequence families, a new family of quadriphase sequences of period $2^{n}-1$ for a positive integer $n=em$ with an even positive factor $m$ is presented, the cross-correlation function among these sequences has been explicitly calculated. Another new family of quadriphase sequences of period $2(2^{n}-1)$ for a positive integer $n=em$ with an even positive factor $m$ is also presented, a detailed analysis of the cross-correlation function of proposed sequences has been presented. The sequences have low correlations and are useful in code division multiple access communication systems and cryptography. This paper is organized as follows. Section 2 introduces the preliminaries and notations. In section 3, we give the constructions and properties of the new sequences families $\mathcal{L}$ and $\mathcal{V}$ with period $2^{n}-1$. The constructions and correlation properties of the new sequences family $\mathcal{W}$ with period $2(2^{n}-1)$ are presented in section 4. The conclusions and acknowledgement are presented in section 5 and 6 respectively. ## 2 Preliminaries ### 2.1 Basic Concepts Let $a=\\{a(t)\\}$ and $b=\\{b(t)\\}$ be two quadriphase sequences of period $L$, the correlation function $R_{a,b}(\tau)$ between them at a shift $0\leq\tau\leq L-1$ is defined by $R_{a,b}(\tau)=\sum_{t=0}^{L-1}\omega^{a(t)-b(t+\tau)}$ where $\omega^{2}=-1$. Let $\mathcal{F}$ be a family of $M$ quadriphase sequences $\mathcal{F}=\\{a_{i}=\\{a(t)\\}:1\leq i\leq M\\}.$ The maximum correlation magnitude $R_{max}$ of $\mathcal{F}$ is $R_{max}=\max\\{|R_{a_{i},a_{j}}(\tau)|:1\leq i,j\leq M,\ \ i\neq j\ or\ \tau\neq 0\\}.$ ### 2.2 Galois Ring Let $Z_{4}[x]$ be the ring of all polynomials over $Z_{4}$ . A monic polynomial $f(x)\in Z_{4}[x]$ is said to basic primitive if its projection $\overline{f(x)}$ $\overline{f(x)}=f(x)\ mod\ 2$ is primitive over $Z_{2}[x].$ Let $f(x)$ be a basic primitive polynomial of degree $n$ over $Z_{4}$, and $Z_{4}[x]/(f(x))$ denotes the ring of residue classes of polynomials over $Z_{4}$ modulo $f(x)$. It can be shown that this quotient ring is a commutative ring with identity called Galois ring, denoted as $GR(4,n)$[11]. As a multiplicative group, the units $GR^{*}(4,n)$ have the following structure: $GR^{*}(4,n)=G_{A}\otimes G_{C}$ where $G_{C}$ is a cyclic group of order $2^{n}-1$ and $G_{A}$ is an Abelian group of order $2^{n}$. Naturally, the projection map $\overline{a}$ from $Z_{4}$ to $Z_{2}$ induces a homomorphism from $GR(4,n)$ to finite field $GF(2^{n})$. Let $\beta\in GR^{*}(4,n)$ be a generator of the cyclic group $G_{C}$ , then $\alpha=\overline{\beta}$ is a primitive root of $GF(2^{n})$ with primitive polynomial $\overline{f(x)}$ over $Z_{2}$. For each element $x\in GR(4,n)$ has a unique $2-adic$ representation of the form $x=x_{0}+2x_{1},x_{0},x_{1}\in G_{C}.$ (1) Let $n=em$, The $Frobenius$ automorphism of $GR(4,n)$ over $GR(4,e)$ is given by $\sigma(x)=x_{0}^{2^{e}}+2x_{1}^{2^{e}}$ for any element $x$ expressed as (1), and the trace function $Tr_{e}^{n}$ from $GR(4,n)$ to $GR(4,e)$ is defined by $Tr^{n}_{e}(x)=x+\sigma(x)+\sigma^{2}(x)+\cdots+\sigma^{m-1}(x)$ where $\sigma^{i}(x)=\sigma^{i-1}(\sigma(x))$ for $1<i\leq m-1$. Let $GF(q)$ is the finite field with $q$ elements, $tr^{n}_{e}(x)$ is the trace function from $GF(2^{n})$ to $GF(2^{e})$, i.e., $tr_{e}^{n}(x)=x+x^{2^{e}}+\cdots=x^{2^{e(\frac{n}{e}-1)}},x\in GF(2^{n}).$ We have $\overline{Tr_{e}^{n}(x)}=tr_{e}^{n}(\overline{x})$, where $x\in GR(4,n)$. Throughout this paper, we suppose (1) $n=em$ with $e\geq 2\ and\ m\geq 2$,(2) $\lambda\in GR(4,e)$ such that $\overline{\lambda}\in GF(2^{e})\setminus\\{1,0\\}$. ### 2.3 Linear Span Let $f(x)=Tr_{1}^{n}[(1+2\alpha)x]+2\sum_{i=1}^{r}Tr_{1}^{n_{i}}(A_{i}x^{v_{i}}),\alpha\in G_{C},A_{i}\in GF(2^{n_{i}}),x\in G_{C},$ where $v_{i}$ is a coset leader of a cyclotomic coset modulo $2^{n_{i}}-1$, and $n_{i}|n$ is the size of the cyclotomic coset containing $v_{i}$. For sequence $a=\\{a_{i}\\}$ such that $a_{i}=f(\beta),i=0,1,2,\cdots$ where $\beta$ is a primitive element of $G_{C}$. Linear span of a sequence $a$ is equal to $n+\sum_{i,A_{i}\neq 0}n_{i}$ , or equivalently, the degree of the shortest linear feedback shift register that can generates $a$ [1, 4]. ## 3 New Quadriphase Sequences with Larger Size and Linear Span Define a function $P(x)$ over $GR(4,n)$ as $P(x)=\left\\{\begin{array}[]{ll}$$\sum_{j=1}^{l-1}Tr_{1}^{n}(x^{2^{ej}+1})+Tr_{1}^{le}(x^{2^{le}+1}),if\ m=2l$$,\\\ \\\ $$\sum_{j=1}^{l}Tr_{1}^{n}(x^{2^{ej}+1}),if\ m=2l+1.$$\end{array}\right.$ For any $x,y\in GR(4,n)$, it is easy to check that[1, 2, 3] $\begin{array}[]{lll}2P(x)+2P(y)+2P(x+y)=2Tr_{1}^{n}[y(x+Tr^{n}_{e}(x))].\end{array}$ (2) ###### Definition 1 Let $\rho$ be an integer such that $1\leq\rho<\displaystyle{\lfloor\frac{n}{2}\rfloor}$ , a family of quaternary sequences of period $2^{n}-1$, $\mathcal{L}=\\{s_{i}(t):0\leq t<2^{n}-1,1\leq i\leq 2^{\rho n}+1\\}$ is defined by $s_{i}(t)=\left\\{\begin{array}[]{ll}$$Tr^{n}_{1}[(1+2\lambda_{0}^{i})\beta^{t}]+2\sum_{k=1}^{\rho-1}Tr^{n}_{1}(\lambda_{k}^{i}\beta^{t(1+2^{k})})+2P(\lambda\beta^{t}),1\leq i\leq 2^{\rho n},$$\\\ \\\ $$2Tr^{n}_{1}(\beta^{t}),i=2^{\rho n}+1$$\end{array}\right.$ where $\\{(\lambda_{0}^{i},\lambda_{1}^{i},\cdots,\lambda_{\rho-1}^{i}),i=1,2,\cdots,2^{\rho n}\\}$ is an enumeration of the elements of $G_{C}\times G_{C}\times\cdots\times G_{C}$, $\beta$ is a generator element of group $G_{C}$. ###### Lemma 1 All sequence in $\mathcal{L}$ are cyclically distinct. Thus, the family size of $\mathcal{L}$ is $2^{n\rho}+1$ . ###### Proof The proof of lemma 1 is similar to the proofs of Lemma 1 and Lemma 6 in [4], we cancel the details. ### 3.1 The Correlation Function of the Sequence Family (1) Suppose $s_{i},s_{j}$,$1\leq i,j\leq 2^{\rho n}$, are two sequences, the correlation function between $s_{i}$ and $s_{j}$ is $R_{s_{i},s_{j}}(\tau)=\sum_{x\in G_{C}}\omega^{Tr^{n}_{1}[(1+2\gamma^{i}_{0}-(1+2\gamma^{j}_{0})\delta)x]+2\sum_{k=1}^{\rho-1}Tr_{1}^{n}(\eta_{k}x^{1+2^{k}})+2(P(\lambda x)+P(\lambda\delta x))}-1$ (3) where $\delta=\beta^{\tau}$, $\tau\neq 0$, $\lambda_{k}^{i}-\delta^{1+2^{k}}\lambda^{j}_{k}=\eta_{k}$,$k=1,2,\cdots,\rho-1$. $\displaystyle(R_{s_{i},s_{j}}(\tau)+1)(R_{s_{i},s_{j}}(\tau)+1)^{*}$ $\displaystyle=\sum_{s\in G_{C}}\sum_{y\in G_{C}}\omega^{Tr_{1}^{n}[(1+2\gamma_{0}^{i}-(1+2\gamma_{0}^{j})\delta)x+2\sum_{k=1}^{\rho-1}Tr_{1}^{n}(\eta_{k}x^{1+2^{k}})+2(P(\lambda x)+P(\lambda\delta x))]}$ $\displaystyle\verb+ +\cdot\omega^{-Tr_{1}^{n}[(1+2\gamma_{0}^{i}-(1+2\gamma_{0}^{j})\delta)y+2\sum_{k=1}^{\rho-1}Tr_{1}^{n}(\eta_{k}y^{1+2^{k}})+2(P(\lambda y)+P(\lambda\delta y))]}$ $\displaystyle=\sum_{x\in G_{C}}\sum_{y\in G_{C}}\omega^{Tr_{1}^{n}((1+2\gamma_{0}^{i}-(1+2\gamma_{0}^{j})\delta)(x+3y))}$ $\displaystyle\verb+ +\cdot\omega^{2[\sum_{k=1}^{\rho-1}Tr_{1}^{n}[\eta_{k}(x^{1+2^{k}}+y^{1+2^{k}})]+P(\lambda x)+P(\lambda\delta x)+P(\lambda y)+P(\lambda\delta y)]}$ $\displaystyle=\sum_{x\in G_{C}}\sum_{y\in G_{C}}\omega^{Tr_{1}^{n}((\Delta(x+3y))+2(\sum_{k=1}^{\rho-1}Tr_{1}^{n}[\eta_{k}(x^{1+2^{k}}+y^{1+2^{k}})]+P(\lambda x)+P(\lambda\delta x)+P(\lambda y)+P(\lambda\delta y))}$ $\displaystyle=\sum_{z\in G_{C}}\sum_{y\in G_{C}}\omega^{Tr_{1}^{n}(\Delta z)+2[\sum_{k=1}^{\rho-1}Tr_{1}^{n}[\eta_{k}((y+z+2\sqrt{yz})^{1+2^{k}}+y^{1+2^{k}})]+2Tr_{1}^{n}(\Delta\sqrt{yz})}$ $\displaystyle\verb+ +\cdot\omega^{2P(\lambda(y+z+2\sqrt{yz}))+2P(\lambda\delta(y+z+2\sqrt{yz}))+2P(\lambda y)+2P(\lambda\delta y)]}$ $\displaystyle=\sum_{z\in G_{C}}\omega^{\phi(z)}\sum_{y\in G_{C}}\omega^{2[Tr_{1}^{n}(y(\Delta^{2}z))+v(y,z)]}$ where $\Delta=1+2\gamma_{0}^{i}-(1+2\gamma_{0}^{j})\delta$, $x=y+z+2\sqrt{yz}$, $\phi(z)=Tr_{1}^{n}(\Delta z)+2[P(\lambda z)+2P(\lambda\delta z)]+2\sum_{k=1}^{\rho-1}Tr^{n}_{1}(\eta_{k}z^{1+2^{k}})$, $v(y,z)=\sum_{k=1}^{\rho-1}Tr_{1}^{n}[\eta_{k}((y+z+2\sqrt{yz})^{1+2^{k}}+y^{1+2^{k}}+z^{1+2^{k}})]+P(\lambda(y+z+$ $2\sqrt{yz}))+P(\lambda\delta(y+z+2\sqrt{yz}))+P(\lambda y)+P(\lambda\delta z)+P(\lambda z)+P(\lambda\delta z).$ Then, by (2), we have $2v(y,z)=2Tr_{1}^{n}[\lambda y(\lambda z+Tr^{n}_{e}(\lambda z))+\lambda\delta y(\lambda\delta z+Tr^{n}_{e}(\lambda\delta z))]+2Tr_{1}^{n}\sum_{k=1}^{\rho-1}[\eta_{k}(zy^{2^{k}}+z^{2^{k}}y)]$ $=2Tr^{n}_{1}[y(\lambda^{2}z+\lambda Tr_{e}^{n}(\lambda z)+\lambda^{2}\delta^{2}z+\lambda\delta Tr^{n}_{e}(\lambda\delta z)+\sum_{k=1}^{\rho-1}(\eta_{k}^{-2^{k}}z^{-2^{k}}+\eta_{k}z^{2^{k}}))].$ Define $L(z)=\overline{\delta}tr^{n}_{e}(\overline{\delta}z)+tr^{n}_{e}(z)+\frac{1}{\overline{\lambda}^{2}}(\overline{\lambda}^{2}+1)(\overline{\delta}^{2}+1)z+\frac{1}{\overline{\lambda}^{2}}\sum_{k=1}^{\rho-1}(\overline{\eta}^{-2^{k}}_{k}z^{-2^{k}}+\overline{\eta}_{k}z^{2^{k}}),$ (4) where $z\in GF(2^{n})$, then $L(z)$ is a linear equation over $GF(2^{n})$. For $L(z)=0$, we have to count the number of solutions in the equation $\overline{\delta}tr^{n}_{e}(\overline{\delta}z)+tr^{n}_{e}(z)+\frac{1}{\overline{\lambda}^{2}}(\overline{\lambda}^{2}+1)(\overline{\delta}^{2}+1)z+\frac{1}{\overline{\lambda}^{2}}\sum_{k=1}^{\rho-1}(\overline{\eta}^{-2^{k}}_{k}z^{-2^{k}}+\overline{\eta}_{k}z^{2^{k}})=0$ (5) for given $\eta_{i}$’s in $G_{C}$,$\lambda\in G_{C}$ such that $\overline{\lambda}\in GF(2^{e})\backslash\\{0,1\\}$, and $\overline{\delta}\in GF(2^{n})\backslash\\{0\\}$. It is easy to verify that $\frac{1}{\overline{\lambda}^{2}}(\overline{\lambda}^{2}+1)(\overline{\delta}^{2}+1)z+\frac{1}{\overline{\lambda}^{2}}\sum_{k=1}^{\rho-1}(\overline{\eta}^{-2^{k}}_{k}z^{-2^{k}}+\overline{\eta}_{k}z^{2^{k}})$ is not a constant polynomial of z, and the maximum number of solutions of equation $L(z)=0$ is at most $2^{2(\rho-1)+2e}$. Thus $|R_{s_{i},s_{j}}(\tau)+1|\leq 2^{\frac{n+2(\rho-1)+2e}{2}}.$ (6) (2) If $1\leq i\leq 2^{\rho n}$ and $j=2^{n\rho}+1$ are two sequences, then the correlation function between $s_{i}$ and $s_{j}$ is $R_{s_{i},s_{j}(\tau)}=\sum_{x\in G_{C}}\omega^{Tr_{1}^{n}[(1+2\gamma_{1}-2\delta)x]+2\sum_{k=1}^{\rho-1}Tr_{1}^{n}(\lambda_{k}^{i}x^{1+2^{k}})+2P(\lambda x)}-1,$ similar to analysis above, the equation (4) become the following equation. $L(z)=\frac{1+\bar{\lambda}^{2}}{\bar{\lambda}^{2}}z+tr^{n}_{e}(z)+\frac{1}{\overline{\lambda}^{2}}\sum_{k=1}^{\rho-1}((\overline{\lambda}_{k}^{i})^{-2^{k}}z^{-2^{k}}+\overline{\lambda}^{i}_{k}z^{2^{k}}).$ Thus $|R_{s_{i},s_{j}}(\tau)+1|\leq 2^{\frac{n+2(\rho-1)+e}{2}}.$ (7) (3) If $i=j=2^{\rho n}+1$, then $s_{i}$ is essentially a binary $m-$sequence, Then $R_{s_{i},s_{j}}(\tau)=-1$ for $\tau\neq 0$. (4) Suppose that $s_{i}$,$s_{j}$, $1\leq i,j\leq 2^{\rho n}$, are two sequences, then $R_{s_{i},s_{j}}(0)=\sum_{x\in T}\omega^{2Tr_{1}^{n}[(\gamma^{i}_{0}+\gamma^{j}_{0})x]+2\sum^{\rho-1}_{k=1}Tr_{1}^{n}(\eta_{k}x^{1+2^{k}})}-1,$ similar to the analysis above, we have $|R_{s_{i},s_{j}(0)}+1|\leq 2^{\frac{n+2(\rho-1)}{2}}.$ (8) It seems difficult to get tighter bound for inequality(6)-(8), thus we propose the following open problem. Open problem:For $n=em$ , how many solutions exist exactly for the equation (5) over finite field $GF(2^{n})$. Following the discussion above, we have the following theorem. ###### Theorem 3.1 For $n=em$ and an integer $\rho$ such that $1\leq\rho<\displaystyle{\lfloor\frac{n}{2}\rfloor}$, the proposed quadriphase family has $2^{n\rho}+1$ cyclically distinct binary sequences of period $2^{n}-1$. The maximum correlation magnitude of sequences is smaller than $1+2^{\frac{n+2(\rho-1)+2e}{2}}$. Therefore, the sequences family constitutes a $(2^{n}-1,2^{n\rho}+1,1+2^{\frac{n+2(\rho-1)+2e}{2}})$ quadriphase signal set. ### 3.2 Linear Spans of the Sequence In order to express clearly, let $s(\lambda_{0},\Lambda,t)=Tr^{n}_{1}[(1+2\lambda_{0})\beta^{t}]+2\sum_{k=1}^{\rho-1}Tr^{n}_{1}(\lambda_{k}\beta^{t(1+2^{k})})+2P(\lambda\beta^{t})$, where $\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$. We divide the set $\Delta=\\{1,2,\cdots,\rho-1\\}$ into two sets $A$ and $B$ such that $\Delta=A\bigcup B$, where $A=\\{ke+r:1\leq k\leq\displaystyle{\lfloor\frac{\rho-1}{e}\rfloor,0<r<e}\\}$, $B=\\{ke:1\leq k\leq\displaystyle{\lfloor\frac{\rho-1}{e}\rfloor}\\}$. ###### Theorem 3.2 (1)Consider a sequence represented by $s(\lambda_{0},\Lambda,t)$ where $j\ \lambda_{i}$’s with $i\in A$ in $\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$ are equal to 0 and $l\ \lambda_{i}$’s with $i\in B$ in $\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$ are equal to $\overline{\lambda}$.Let $LS_{j,l}(\rho)$ be the linear span of the sequence.Then $LS_{j,l}(\rho)=n(\frac{m-1}{2}+\rho-1-\lfloor\frac{\rho-1}{e}\rfloor+1-j-l),0\leq j\leq|A|,0\leq l\leq|B|.$ and there are $(^{\rho-1-\lfloor\frac{\rho-1}{e}\rfloor}_{\verb+ +j})(_{\verb+ +l}^{\lfloor\frac{\rho-1}{e}\rfloor})2^{n}(2^{n}-1)^{\rho-1-j-l}$ sequences having linear span $LS_{j,l}(\rho)$. (2) The linear span of the sequences $s_{2^{n\rho}+1}(t)$ is $n$. ###### Proof First, consider the linear span of sequences with $m$ is odd. A sequence constructed above has a total of $\displaystyle{\frac{m-1}{2}+\rho-1-\lfloor\frac{\rho-1}{e}\rfloor+1}$ trace terms and each trace term has the linear span of $n$. If $j\ \lambda$’s with $i\in A$ in $\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$ are equal to 0, and $l\ \lambda_{i}$’s with $i\in B$ in $\Lambda=(\lambda_{1},\cdots,\lambda_{\rho-1})$ are equal to $\overline{\lambda}^{2^{i}+1}$, it has $\displaystyle{\frac{m-1}{2}+\rho-1-\lfloor\frac{\rho-1}{e}\rfloor+1-j-l}$ nonzero trace terms and the corresponding linear span of the sequences is given by $LS_{j,l}(\rho)=n(\frac{m-1}{2}+\rho-1-\lfloor\frac{\rho-1}{e}\rfloor+1-j-l),0\leq j\leq|A|,0\leq l\leq|B|.$ Since (1) $j\ \lambda_{i}$’s with $i\in A$ are 0 and $(|A|-j)\ \lambda$ ’s are nonzero, (2) $l\ \lambda_{i}$’s with $i\in B$ are $\overline{\lambda}^{2^{i}+1}$ and $(|B|-l)\ \lambda_{i}$’s are not equal to $\overline{\lambda}^{2^{i}+1}$, (3) the number of $\lambda_{0}$ is $2^{n}$. Therefore, the number of corresponding sequences given above is $\displaystyle(_{\verb+ +j}^{\rho-1-\lfloor\frac{\rho-1}{e}\rfloor})(2^{n}-1)^{\rho-1-\lfloor\frac{\rho-1}{e}\rfloor-j}\cdot(_{\verb+ +l}^{\lfloor\frac{\rho-1}{e}\rfloor})(2^{n}-1)^{\lfloor\frac{\rho-1}{e}\rfloor-l}\cdot 2^{n}$ $\displaystyle=(_{\verb+ +j}^{\rho-1-\lfloor\frac{\rho-1}{e}\rfloor})\cdot(_{\verb+ +l}^{\lfloor\frac{\rho-1}{e}\rfloor})2^{n}(2^{n}-1)^{\rho-1-l-j}.$ Applying this result to each $j$ and each $l$, we obtain the linear span of the proposed sequence. Using a similar approach to the odd case, we see that the linear span of both sequences is same. For the sequences families above, some special conditions had already been discussed, for example, the case with $n=em$, where $m$ is an odd, and $\rho=1$ had been discussed in [1]. In the following, we will discuss another special case with $n=em$, where $m$ is an even, and $\rho=1$, we call this special sequence family as family $\mathcal{V}$. ### 3.3 Correlation Function of the Sequence family for even $m$ and $\rho=1$ If $\rho=1$, then the equation (5) becames $\overline{\delta}Tr^{n}_{e}(\overline{\delta}z)+Tr^{n}_{e}(z)+\frac{(\overline{\lambda}^{2}+1)(\overline{\delta}^{2}+1)}{\overline{\lambda}^{2}}z=0$ (9) In the following, we study the solution of the equation (9). Let $Tr_{e}^{n}(\overline{\delta}z)=a$, $Tr_{e}^{n}(z)=b$, then $z=\frac{\overline{\lambda}^{2}}{\overline{\lambda}^{2}+1}\frac{\overline{\delta}a+b}{\overline{\delta}^{2}+1}.$ By computing $Tr^{n}_{e}(z)$ and $Tr^{n}_{e}(\overline{\delta}z)$, we have $\left\\{\begin{array}[]{lll}$$aTr^{n}_{e}(\displaystyle{\frac{\overline{\delta}}{\overline{\delta}^{2}+1}})+b[Tr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})-\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}]=0$$\\\ \\\ $$\displaystyle{a[tr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})-\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}]+bTr^{n}_{e}(\frac{\overline{\delta}}{\overline{\delta}^{2}+1})=0}$$\end{array}\right.$ (10) The determinant of corresponding coefficient matrix of (10) is equal to $\displaystyle[Tr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})-\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}]^{2}+[Tr^{n}_{e}(\frac{\overline{\delta}}{1+\overline{\delta}^{2}})]^{2}$ $\displaystyle=(\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}})^{2}+[Tr^{n}_{e}(\frac{1}{1+\overline{\delta}})]^{2}$ (1) If $\displaystyle{Tr^{n}_{e}(\frac{1}{1+\overline{\delta}})\neq\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}}$, then the determinant of coefficient matrix (10) is not equal to zero, the equation (10) has unique solution $a=0$, $b=0$, then $z=0$. (2) If $\displaystyle{Tr^{n}_{e}(\frac{1}{1+\overline{\delta}})=\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}}$, then the determinant of coefficient matrix (10) is equal to zero, $\displaystyle aTr^{n}_{e}(\frac{1}{\overline{\delta}+1})+aTr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})+b[Tr^{n}_{e}(\frac{1}{\overline{\delta}^{2}+1})-\frac{\overline{\lambda}^{2}+1}{\overline{\lambda}^{2}}]=0,$ thus $a=b$, the equation (10) has $2^{e}$ solutions, then $z=\frac{\overline{\lambda}^{2}}{\overline{\lambda}^{2}+1}\frac{1}{\overline{\delta}+1}a$. $\displaystyle 2P(\overline{\lambda}z)+2P(\overline{\lambda}\overline{\delta}z)=2\sum^{l-1}_{j=1}Tr^{n}_{1}[(\overline{\lambda}z)^{2^{ej}+1}+(\overline{\lambda}\overline{\delta}z)^{2^{ej}+1}]+2Tr^{le}_{1}[(\overline{\lambda}x)^{2^{le}+1}+(\overline{\lambda}\overline{\delta}z)^{2^{el}+1}]$ $\displaystyle=2\sum^{l-1}_{j=1}Tr^{n}_{1}[(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2^{ej}+1}(\frac{1}{1+\overline{\delta}})^{2^{ej}+1}+(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2^{ej}+1}(\frac{\overline{\delta}}{1+\overline{\delta}})^{2^{ej}+1}]$ $\displaystyle\verb+ ++2Tr^{el}_{1}[(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2^{el}+1}(\frac{1}{1+\overline{\delta}})^{2^{el}+1}+(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2^{el}+1}(\frac{\overline{\delta}}{1+\overline{\delta}})^{2^{el}+1}]$ $\displaystyle=2Tr^{e}_{1}\\{(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2}[\sum^{l-1}_{j=1}Tr^{n}_{e}[(\frac{1}{1+\overline{\delta}})^{2^{ej}+1}+(\frac{\overline{\delta}}{1+\overline{\delta}})^{2^{ej}+1}]$ $\displaystyle\verb+ ++Tr^{le}_{e}[(\frac{1}{1+\overline{\delta}})^{2^{el}+1}+(\frac{\overline{\delta}}{1+\overline{\delta}})^{2^{el}+1}]]\\}$ $\displaystyle=2Tr^{e}_{1}{[(\frac{\overline{\lambda}^{3}a}{1+\overline{\lambda}^{2}})^{2}}(Tr^{le}_{e}1+Tr^{n}_{e}(\frac{1}{1+\overline{\delta}}))]$ $\displaystyle=\left\\{\begin{array}[]{ll}$$2Tr^{n}_{1}(\frac{\overline{\lambda}^{2}z}{1+\overline{\lambda}^{2}}),\ for\ odd\ l$$,\\\ \\\ $$2Tr^{n}_{1}(\frac{\overline{\lambda}^{2}z}{1+\overline{\lambda}}),\ for\ even\ l.$$\end{array}\right.$ Thus, $\phi(z)=Tr^{n}_{1}(\Delta z)+2[P(\lambda z)+P(\lambda\delta z)]$ $=\left\\{\begin{array}[]{ll}$$2Tr^{n}_{1}[(\overline{\gamma}_{0}^{i}+(\overline{\gamma}_{0}^{j}+1)\overline{\delta}+\frac{\overline{\lambda}^{2}}{1+\overline{\lambda}^{2}})z],\ \ for\ odd\ l$$,\\\ \\\ $$2Tr^{n}_{1}[(\overline{\gamma}_{0}^{i}+(\overline{\gamma}_{0}^{j}+1)\overline{\delta}+\frac{\overline{\lambda}^{2}}{1+\overline{\lambda}})z],\ for\ even\ l.$$\end{array}\right.$ Because the solutions space of equation (9) is a linear subspace, following the discussions above, we have ###### Theorem 3.3 for $m$ is even, $\rho=1$, the nontrivial correlation function of the proposed sequences family $\mathcal{V}$ takes values in $\\{-1,-1\pm 2^{\frac{n}{2}},-1\pm 2^{\frac{n}{2}}\omega,-1\pm 2^{\frac{n+e}{2}},-1\pm 2^{\frac{n+e}{2}}\omega\\}$. ## 4 Quadriphase Sequences with period $2(2^{n}-1)$ Similar to the [1, 2], for an even $m$, we propose the following sequence family, the correlation function of the sequences family is calculated. In this section, let $G=\\{\eta_{1},\eta_{2},\cdots,\eta_{2^{n-1}}\\}$ be a maximum subset of $G_{C}$ such that $2\eta_{i}\neq 2(\eta_{j}+1)$ for arbitrary $1\leq i,j\leq 2^{n-1}$. By convention, denote $\beta^{\frac{1}{2}}=\beta^{2^{n-1}}$. We present another family of quadriphase sequences with period $2(2^{n}-1)$ as follows. ###### Definition 2 A family $\mathcal{W}$ of quadriphase sequences with period $2(2^{n}-1)$ is defined as $\mathcal{W}=\\{u_{i}(t),v_{i}(t):0\leq i<2^{n-1}\\}$ is given by 1) $u_{i}(t)=$$\left\\{\begin{array}[]{lll}Tr^{n}_{1}[(1+2\eta_{i})\beta^{t_{0}}]+2P(\lambda\beta^{t_{0}}),t=2t_{0}\\\ \\\ Tr^{n}_{1}[(1+2(\eta_{i}+1))\beta^{t_{0}+\frac{1}{2}}]+2P(\lambda\beta^{t_{0}+\frac{1}{2}}),t=2t_{0}+1\end{array}\right.$ for $0\leq i<2^{n-1}$, where $\eta_{i}\in G$. 2) $v_{i}(t)=$$\left\\{\begin{array}[]{lll}Tr^{n}_{1}[(1+2\eta_{i})\beta^{t_{0}}]+2P(\lambda\beta^{t_{0}})+2,t=2t_{0}\\\ \\\ Tr^{n}_{1}[(1+2(\eta_{i}+1))\beta^{t_{0}+\frac{1}{2}}]+2P(\lambda\beta^{t_{0}+\frac{1}{2}}),t=2t_{0}+1\end{array}\right.$ for $0\leq i<2^{n-1}$, where $\eta_{i}\in G$. ###### Theorem 4.1 the correlation functions of the family $\mathcal{W}$ satisfy the following properties. 1) if $\tau=2^{n}-1$, then $R_{u_{i},u_{j}}(\tau)=-2$, $R_{v_{i},v_{j}}(\tau)=2$, $R_{u_{i},v_{j}}(\tau)=0$. 2) if $\tau=0$, then $R_{u_{i},v_{j}}(\tau)=0$ and $R_{u_{i},u_{j}}(\tau)=R_{v_{i},v_{j}}(\tau)=\left\\{\begin{array}[]{ll}2(2^{n}-1),i=j\\\ -2,i\neq j,\end{array}\right.$ 3)If $\tau=2\tau_{0}+1\neq 2^{n}-1$, then $\displaystyle a)\ R_{u_{i},u_{j}}(\tau)\ takes\ values\ in\ \\{-2,-2\pm 2^{\frac{n}{2}+1},-2\pm 2^{\frac{n+e}{2}+1}\\},$ $\displaystyle b)\ R_{v_{i},v_{j}}(\tau)\ takes\ values\ in\ \\{2,2\pm 2^{\frac{n}{2}+1},2\pm 2^{\frac{n+e}{2}+1}\\},$ $\displaystyle c)\ R_{u_{i},v_{j}}(\tau)\ takes\ values\ in\ \\{\pm 2^{\frac{n}{2}+1}\omega,\pm 2^{\frac{n+e}{2}+1}\omega\\}.$ 4)If $\tau=2\tau_{0}$ and $\tau_{0}\neq 0$, then $\displaystyle a)\ R_{u_{i},u_{j}}(\tau)\ takes\ values\ in\ \\{-2,-2\pm 2^{\frac{n}{2}+1},-2\pm 2^{\frac{n+e}{2}+1}\\},$ $\displaystyle b)\ R_{v_{i},v_{j}}(\tau)\ takes\ values\ in\ \\{-2,-2\pm 2^{\frac{n}{2}+1},-2\pm 2^{\frac{n+e}{2}+1}\\},$ $\displaystyle c)\ R_{u_{i},v_{j}}(\tau)\ takes\ values\ in\ \\{\pm 2^{\frac{n}{2}+1}\omega,\pm 2^{\frac{n+e}{2}+1}\omega\\}.$ ###### Proof In order to analysis easily, let $\varsigma(\gamma_{1},\gamma_{2},\delta)=\sum_{x\in G_{C}}\omega^{Tr^{n}_{1}[(1+2\gamma_{1}-(1+2\gamma_{2})\delta)x]+2(P(\lambda x)+P(\lambda\delta x))}.$ (11) It is easy to check that $\varsigma(\gamma_{1}+1,\gamma_{2},\delta)=\varsigma(\gamma_{1},\gamma_{2}+1,\delta)^{*}$, where $*$ denotes complex conjugate. Similar to [1], the following facts can be easily checked. 1) if $\tau=2\tau_{0}+1$, then $\displaystyle R_{u_{i},u_{j}}(\tau)=\varsigma(\eta_{i},\eta_{j}+1,\delta)+\varsigma(\eta_{i}+1,\eta_{j},\delta)-2,$ $\displaystyle R_{v_{i},v_{j}}(\tau)=-\varsigma(\eta_{i},\eta_{j}+1,\delta)-\varsigma(\eta_{i}+1,\eta_{j},\delta)+2,$ $\displaystyle R_{u_{i},v_{j}}(\tau)=\varsigma(\eta_{i},\eta_{j}+1,\delta)-\varsigma(\eta_{i}+1,\eta_{j},\delta).$ 2) if $\tau=2\tau_{0}$, then $\displaystyle R_{u_{i},u_{j}}(\tau)=\varsigma(\eta_{i},\eta_{j}+1,\delta)+\varsigma(\eta_{i}+1,\eta_{j},\delta)-2,$ $\displaystyle R_{v_{i},v_{j}}(\tau)=\varsigma(\eta_{i},\eta_{j}+1,\delta)+\varsigma(\eta_{i}+1,\eta_{j},\delta)-2,$ $\displaystyle R_{u_{i},v_{j}}(\tau)=-\varsigma(\eta_{i},\eta_{j}+1,\delta)+\varsigma(\eta_{i}+1,\eta_{j},\delta).$ Due to (3),(11) and the theorem 3, the theorem 4 is proved. Similar to the proof of the theorem 2 above, or the proof of theorem 3 and theorem 7 [1] the following theorem is obtained. ###### Theorem 4.2 the linear spans of the sequences in $\mathcal{W}$ are given as follows (1) For $u_{i}\in\mathcal{W}$, the linear span $LS(u_{i})$ of $u_{i}$ is given by $\displaystyle{LS(u_{i})=\frac{n(n+e)}{2e}}$. (2) For $v_{i}\in\mathcal{W}$, the linear span $LS(v_{i})$ of $v_{i}$ is given by $\displaystyle{LS(u_{i})=\frac{n(n+e)}{2e}+2}$. ## 5 Conclusions In this paper, we have proposed the new families of quadriphase sequences with larger linear span and size. The maximum correlation magnitude of proposed sequences family is bigger then that of the related sequence in [1], and is smaller than that of the related binary sequences family in [2, 3] with same parameters. The proposed two families of quadriphase sequences with period $2^{n}-1$ and $2(2^{n}-1)$ respectively for a positive integer $n=em$ where $m$ is an even positive can be take as an extensions of the results in [1] where $m$ is an odd positive. ## 6 Acknowledgment This work was supported by National Science Foundation of China under grant No.60773002 and 61072140, the Project sponsored by $SRF$ for $ROCS$, $SEM$, 863 Program (2007AA01Z472), and the 111 Project (B08038). ## References * [1] Jiang W.F.,Hu L.,Tang X.H.,Zeng X.Y.: New optimal quadriphase sequences with larger linear span. IEEE Trans. Inform. Theory, Vol.55, No.1, pp.458-470, Jan. 2009. * [2] Tang X.H.,Udaya P.,Fan P.Z.:Generalized binary Udaya-Siddiqi sequences. IEEE Trans. Inform. Theory, Vol.53, No.3, pp.1225-1230, Mar. 2007. * [3] KimS H.,No J.S.: New families of binary sequences with low cross correlation property.IEEE Trans. Inform. Theory, Vol.49, No.1, pp.3059-3065, Jan. 2009. * [4] Yu N.Y.,Gong G.: A new binary sequence family with low correlation and large size. IEEE Trans. Inform. Theory, Vol.52, No.4, pp.1624-1636, Mar.2006. * [5] Boztas S.,Hammons R.,Kumar P.V.: 4-phase sequences with near optimum correlation properties. IEEE Trans. Inform. Theory, Vol.14, No.3, pp.1101-1113, May 1992. * [6] Kumar P.V.,Helleseth T.,Calderbank A.R.,Hammons A.R.Jr.: Large families of quaternary sequences with low correlation. IEEE Trans. Inform. Theory, Vol.42, No.2, pp.579-592, Mar.1996. * [7] Schmidt K.-U.:$Z_{4}$-valued quadratic forms and quaternary sequence families. IEEE Trans. Inform. Theory, Vol.55, No.12, pp.5803-5810, Dec.2009. * [8] Sole P.: A quaternary cyclic code, and a family of quadriphase sequences with low correlation properties.Lecture Notes in Computer Science, Vol.388, pp.193-201,1989. * [9] Tang X.H.,Udaya P.:A note on the optimal quadriphase sequences families. IEEE Trans. Inform. Theory, Vol.53, No.1, pp433-436, Jan. 2007\. * [10] Udaya P.,Siddiqi M.U.: Optimal and suboptimal quadriphase sequences derived from maximal length sequences over $Z_{4}$.Appl. Algebra Eng. Commun.Comput., Vol.9, no.2, pp.161-191,1998. * [11] Hammons A.R.Jr.,Kumar P.V.,Calderbank A.R.,Sloane N.J.A.,Sole P.: The $Z_{4}$ linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory, Vol.40, No.2, pp.301-319, Mar. 1994\. * [12] Tang X.T.,Udaya P.,Fan P.Z.:Quadriphase sequences obtained from binary quadratic form sequences. Lecture Note in Computer Science, Vol.3486, pp.243-254, 2005.
arxiv-papers
2011-01-18T08:35:50
2024-09-04T02:49:16.519842
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Wenping Ma", "submitter": "Wenping Ma", "url": "https://arxiv.org/abs/1101.3398" }
1101.3652
# Three-dimensional topological insulators in the octahedron-decorated cubic lattice Jing-Min Hou jmhou@seu.edu.cn Wen-Xin Zhang Guo-Xiang Wang Department of Physics, Southeast University, Nanjing, 211189, China ###### Abstract We investigate a tight-binding model of the octahedron-decorated cubic lattice with spin-orbit coupling. We calculate the band structure of the lattice and evaluate the $Z_{2}$ topological indices. According to the $Z_{2}$ topological indices and the band structure, we present the phase diagrams of the lattice with different filling fractions. We find that the $(1;111)$ and $(1;000)$ strong topological insulators occur in some range of parameters at $1/6,1/2$ and $2/3$ filling fractions. Additionally, the $(0;111)$ weak topological insulator is found at $1/6$ and $2/3$ filing fractions. We analyze and discuss the characteristics of these topological insulators and their surfaces states. ###### pacs: 73.43.-f, 71.10.Fd, 73.20.-r, 72.25.-b ## I Introduction Usually, different phases of matter can be classified using Landau’s approach according to their underling symmetriesLandau . In 1980s, the discovery of the quantum Hall effect changed physicists’ viewpoint on the classification of matterKlitzing . The quantum Hall states can be classified by a topological invariant, now named the TKNN numberThouless (equivalent to the first Chern number), which is directly connected to the quantized Hall conductivity, but they have the same symmetry. Since the Hall conductivity is odd under time reversal, the topological non-trivial quantum Hall states can only occur when time reversal symmetry is broken, which is performed by a magnetic field. In 1988, Haldane also proposed a time reversal symmetry broken toy model without a magnetic field to realize quantum Hall statesHaldane . All the quantum Hall states have a gapped band structure in bulk and chiral gapless edge states that are topologically protected. Recently, the promising prospect of spintronics in technology stimulates physicists to generate spin current. Quantum spin Hall effect was proposed to create spin currentBernevig ; Kane . The quantum spin Hall states are non- trivial topological phases with time reversal symmetry, which have a bulk gap and topologically protected gapless helical edge states. For the above reason, the quantum spin Hall states also called topological insulators. Two- dimensional topological insulators are characterized by a $Z_{2}$ topological index $\nu=0,1$Kane2 . For a non-trivial topological insulator the topological index has a value $\nu=1$ while $\nu=0$ for a trivial band insulator. Therefore, a topological insulator always has a metallic boundary when placed next to a vacuum or an ordinary band insulator because topological invariants cannot change as long as a material remains insulating. The remarkable metallic boundaries of topological insulators may result in new spintronic or magnetoelectric devices and a new architecture for topological quantum bits. In quantum spin Hall phases, the spin-orbit coupling plays the role of the spin-dependent effective magnetic field. The first real material, a HgTe quantum well, supporting two-dimensional topological insulators was predicted by Bernevig, et al.Bernevig2 and experimentally conformed by König et al.Konig . Figure 1: (Color online). (a) The octahedron-decorated cubic lattice which can be obtained by replacing every lattice site of a cubic lattice with an octahedral cluster as shown in (b). (c) The three-dimensional Brillouin zone and high symmetry points. (d) The two-dimensional Brillouin zone of a slab with two $001$ surfaces. Soon after the quantum spin Hall insulator was discovered, time-reversal invariant topological insulators were generalized to three dimensionsFu ; Moore ; Roy . Three-dimensional time-reversal invariant band insulators are classified according to four $Z_{2}$ topological indices $(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$ with $\nu_{i}=0,1$Fu . In three dimensions, the time-reversal invariant band insulators can be classified into 16 phases according to the four $Z_{2}$ topological indices. A band insulator with $\nu_{0}=1$ is called a strong topological insulator(STI), a band insulator with $\nu_{0}=0$ and at least one non-zero $\nu_{i}(i=1,2,3)$ is called a weak topological insulator(WTI), while an ordinary trivial band insulator has an index $(0;000)$. For an STI phase, the surface states have an odd number of Dirac points, which are topologically protected and for a WTI or trivial band insulator phase, the surface states have an even number of Dirac points. Fu and Kane firstly predicted that Bi1-x Sbx supports a three-dimensional topological insulatorFu2 , which was conformed experimentally by Hsieh, et al. in 2008Hsieh . Later, Bi2Se3 was discovered to be a three-dimensional insulator experimentally as a second generation materialXia , which also was supported by theoretical calculationsXia ; H.Zhang . Additionally, reference H.Zhang also predicted that Bi2Te3 and Sb2Te3 are second generation materials supporting three-dimensional topological insulators. The later experimental studies on Bi2Te3Chen ; Hsieh2 ; Hsieh3 and Sb2Te3Hsieh3 identified their topological band structures. To help experimental physicists find more topological insulator materials, theoretical physicists have investigated several models that support non- trivial topological insulators. Theoretical studies have demonstrated that, within the tight-binding approximation and with the spin-orbit coupling, the honeycombKane , kagomeGuo , checkerboardSun , decorated honeycombRuegg , LiebWeeks , and square-octagonKargarian lattices support two-dimensional topological insulators and the diamondFu , pyrochloreGuo2 , and perovskiteWeeks lattices support three-dimensional topological insulators. In this paper, we shall show that a new lattice, the octahedron-decorated cubic lattice as shown in Fig.1 (a), supports three-dimensional topological insulators with the spin-orbit coupling existing. This lattice can be regarded as a three-dimensional generalization of the square-octagon latticeRuegg . We find that this model supports STI and WTI phases for $1/6$ and $2/3$ filling and STI phases for $1/2$ filling as well as ordinary band insulator and metal phases. ## II Model We consider the octahedron-decorated cubic lattice as shown in Fig.1 (a), which can be obtained by replacing every lattice site of a cubic lattice with an octahedral cluster as shown in Fig.1(b). This lattice has a unit cell with six different lattice sites as denoted in Fig.1(b) so that it contains six sublattices. Here, we assume that the distance between the centers of two nearest-neighbor octahedral clusters is $a$, which is the same with the lattice constant of all sublattices, the distance of every lattice site of an octahedral cluster from its center is $a/4$, and the distance of two nearest- neighbor lattice sites in different octahedral clusters is $a/2$. With the tight-binding approximation, we can write the second quantized Hamiltonian of the lattice as follows, $\displaystyle H_{0}=-t\sum_{\langle i,j\rangle,\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}-t_{1}\sum_{[i,j],\sigma}c_{i\sigma}^{\dagger}c_{j\sigma}$ (1) where $c_{i\sigma}$ is the annihilation operator destructing an electron with spin $\sigma$ on the site ${\bf r}_{i}$ of the octahedron-decorated cubic lattice, $\langle i,j\rangle$ represents nearest-neighbor hopping in the same octahedral cluster with amplitude $t$ and $[i,j]$ denotes nearest-neighbor hopping between two different octahedral clusters with amplitude $t_{1}$. Figure 2: (Color online). Phase diagrams of the octahedron-decorated cubic lattice for (a) $1/6$ filling, (b) $1/2$ filling, and (c) $2/3$ filling. Here, BI denotes a trivial band insulator; STI1 and STI2 denote $(1;111)$ and $(1;000)$ strong topological insulators, respectively; WTI denotes a $(0;111)$ weak topological insulator; and M denotes a metal phase. In momentum space, the Hamiltonian (1) can be represented by $H_{0}=\sum_{{\bf k}\sigma}\Psi^{\dagger}_{{\bf k}\sigma}{\cal H}^{(0)}_{\bf k}\Psi_{{\bf k}\sigma}$ with $\Psi_{{\bf k}\sigma}=(c_{1{\bf k}\sigma},c_{2{\bf k}\sigma},c_{3{\bf k}\sigma},c_{4{\bf k}\sigma},c_{5{\bf k}\sigma},c_{6{\bf k}\sigma})^{T}$, which are ordered according to the sequence denoted in Fig.1(b). Here, ${\cal H}^{(0)}_{\bf k}$ takes the following form, $\displaystyle{\cal H}_{\bf k}^{0}=$ $\displaystyle-\left(\matrix{0&t&t&t&t&t_{1}e^{ik_{z}}\cr t&0&t&t_{1}e^{ik_{x}}&t&t\cr t&t&0&t&t_{1}e^{ik_{y}}&t\cr t&t_{1}e^{-ik_{x}}&t&0&t&t\cr t&t&t_{1}e^{-ik_{y}}&t&0&t\cr t_{1}e^{-ik_{z}}&t&t&t&t&0}\right)$ (2) Since $H_{0}$ is spin-decoupling, ${\cal H}_{\bf k}^{0}$ is spin-independent, i.e. it is the same for both spin-up and spin-down electrons. Fig.1(c) shows the first Brillouin zone of the octahedron-decorated cubic lattice. The spectrum of Eq.(2) with $t_{1}=t$ is calculated and shown in Fig.3(d). The spectrum contains six bands which come from the six sites in every unit cell. A gap exists between the first and second bands. The second, third and fourth bands touch together at points $\Gamma,R$ and $M$. The third, fourth and fifth band touch at point $X$, near which a Dirac cone occurs. Five bands including the second, third, fourth, fifth and sixth bands meet at point $\Gamma$. Along the $\Gamma\rightarrow R$ line in momentum space, the second and third bands are degenerate and the fifth and sixth bands are degenerate. Now, in order to find non-trivial topological insulators in the octahedron- decorated cubic lattice, we proceed to introduce the spin-orbit interactions between next-nearest-neighbor sites as follows, $\displaystyle H_{\rm SO}=i\frac{8\lambda_{\rm SO}}{a^{2}}\sum_{\langle\langle i,j\rangle\rangle\alpha\beta}({\bf d}_{ij}^{1}\times{\bf d}_{ij}^{2})\cdot\boldmath{\mbox{$\sigma$}}_{\alpha\beta}c_{i\alpha}^{\dagger}c_{j\beta},$ where $\langle\langle i,j\rangle\rangle$ represents two next-nearest-neighbor sites $i,j$, and $\lambda_{\rm SO}$ is the amplitude of spin-orbit coupling of the two next-nearest-neighbor sites. ${\boldmath{\mbox{$\sigma$}}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the vector of Pauli spin matrices. ${\bf d}_{ij}^{1,2}$ are the two nearest neighbor bond vectors traversed between sites $i$ and $j$ with $8|{\bf d}_{ij}^{1}\times{\bf d}_{ij}^{2}|/a^{2}=1$. In momentum space, the Hamiltonian for spin-orbit coupling (II) can be expressed as $H_{\rm SO}=\sum_{\bf k}\Psi_{\bf k}^{\dagger}{\cal H}_{\bf k}^{\rm SO}\Psi_{\bf k}$ with $\Psi_{\bf k}=(c_{1{\bf k}\uparrow},c_{2{\bf k}\uparrow},$ $c_{3{\bf k}\uparrow},c_{4{\bf k}\uparrow},c_{5{\bf k}\uparrow},c_{6{\bf k}\uparrow},c_{1{\bf k}\downarrow},c_{2{\bf k}\downarrow},c_{3{\bf k}\downarrow},c_{4{\bf k}\downarrow},c_{5{\bf k}\downarrow},c_{6{\bf k}\downarrow})^{T}$. Since ${\cal H}_{\bf k}^{\rm SO}$ does not decouple for the two spin projections, it is a $12\times 12$ matrix. In momentum space, the total single particle Hamiltonian is ${\cal H}_{\bf k}={\cal H}_{\bf k}^{0}+{\cal H}_{\bf k}^{\rm SO}$. The bands and eigenstates can be obtained by exactly diagonalizing ${\cal H}_{\bf k}$. ## III Three-dimensional topological insulators Figure 3: (Color online). Band structures of the octahedron-decorated cubic lattice for various parameters $t_{1}$ and $\lambda_{\rm SO}$. Here, the horizontal axis represents the wave vectors along the path in the first Brillouin zone indicated by the red lines in Fig.1(c). (a) $t_{1}=t,\lambda_{\rm SO}=0.5t$, (b) $t_{1}=-t,\lambda_{\rm SO}=-0.5t$, (c) $t_{1}=t,\lambda_{\rm SO}=t$, (d) $t_{1}=t,\lambda_{\rm SO}=0$, (e) $t_{1}=3t,\lambda_{\rm SO}=-0.4t$, (f) $t_{1}=-3t,\lambda_{\rm SO}=0.4t$, (g) $t_{1}=3.2t,\lambda_{\rm SO}=-0.2t$, (h) $t_{1}=t,\lambda_{\rm SO}=0.2t$, (i) $t_{1}=t,\lambda_{\rm SO}=-0.2t$, (j) $t_{1}=-t,\lambda_{\rm SO}=0.2t$, (k) $t_{1}=2.5t,\lambda_{\rm SO}=-0.2t$, and (l) $t_{1}=0.5t,\lambda_{\rm SO}=0$. The classification of three-dimensional topological insulators is presented in Ref.Fu . For three-dimensional lattices there eight distinct time reversal invariant momenta (TRIM), which can be expressed in terms of primitive reciprocal lattice vectors as $\Gamma_{i=(n_{1},n_{2},n_{3})}=(n_{1}{\bf b}_{1}+n_{2}{\bf b}_{2}+n_{3}{\bf b}_{3})/2$ with $n_{j}=0,1$. Three- dimensional topological insulators can be distinguished by four $Z_{2}$ topological invariants $(\nu_{0};\nu_{1}\nu_{2}\nu_{3})$, which are defined as $(-1)^{\nu_{0}}=\prod_{n_{j}=0,1}\delta_{n_{1}n_{2}n_{3}}$ and $(-1)^{\nu_{i=1,2,3}}=\prod_{n_{j\neq i}=0,1;n_{i}=1}\delta_{n_{1}n_{2}n_{3}}$, where $\delta_{n_{1}n_{2}n_{3}}=\sqrt{\det[w(\Gamma_{n_{1}n_{2}n_{3}})]}/{\rm Pf}[w(\Gamma_{n_{1}n_{2}n_{3}})]=\pm 1$. Here the unitary matrix $w$ is defined as $w_{ij}({\bf k})=\langle u_{i}(-{\bf k})|\Theta|u_{j}({\bf k}\rangle$ with $\Theta$ being the time reversal operator and $|u_{j}({\bf k})\rangle$ being the Bloch wave functions for occupied bands. Fu and Kane have found a simple method to identify the $Z_{2}$ invariants for the system with the presence of inversion symmetryFu2 . In this case, $\delta_{n_{1}n_{2}n_{3}}$ can be calculated by $\delta_{n_{1}n_{2}n_{3}}=\prod_{m=1}^{N}\xi_{2m}(\Gamma_{n_{1}n_{2}n_{3}})$, where $N$ is the number of occupied bands and $\xi_{2m}(\Gamma_{n_{1}n_{2}n_{3}})=\pm 1$ is the parity eigenvalue of the $2m$th occupied band at $\Gamma_{n_{1}n_{2}n_{3}}$. Our model is inversion symmetric so we will adopt this method to evaluate the $Z_{2}$ invariants $\nu_{i}(i=0,1,2,3)$. We select the center of an octahedron in the lattice as the center of inversion, then the parity operator acts as ${\cal P}[\psi_{1}({\bf r}),\psi_{2}({\bf r}),\psi_{3}({\bf r}),\psi_{4}({\bf r}),\psi_{5}({\bf r}),\psi_{6}({\bf r})]^{T}=[\psi_{6}(-{\bf r}),\psi_{4}(-{\bf r}),\psi_{5}(-{\bf r}),\psi_{2}(-{\bf r}),\psi_{3}(-{\bf r}),\psi_{1}(-{\bf r})]^{T}$, where $[\psi_{1}({\bf r}),\psi_{2}({\bf r}),\psi_{3}({\bf r}),\psi_{4}({\bf r}),\psi_{5}({\bf r}),\psi_{6}({\bf r})]^{T}$ is the six-component wave function. Taking Fourier transformation, we can write the six-component wave function as $[\psi_{1}({\bf r}),\psi_{2}({\bf r}),\psi_{3}({\bf r}),\psi_{4}({\bf r}),\psi_{5}({\bf r}),\psi_{6}({\bf r})]=\sum_{\bf k}[\phi_{1}({\bf k}),\phi_{2}({\bf k}),\phi_{3}({\bf k}),\phi_{4}({\bf k}),\phi_{5}({\bf k}),\phi_{6}({\bf k})]e^{i{\bf k}\cdot{\bf r}}$ and the parity operator as ${\cal P}=\sum_{\bf k}e^{i{\bf k}\cdot{\bf r}}{\cal P}_{\bf k}e^{-i{\bf k}\cdot{\bf r}}$. Then, in momentum space, we obtain the equation ${\cal P}_{\bf k}[\phi_{1}({\bf k}),\phi_{2}({\bf k}),\phi_{3}({\bf k}),\phi_{4}({\bf k}),\phi_{5}({\bf k}),\phi_{6}({\bf k})]^{T}=[\phi_{6}(-{\bf k}),\phi_{4}(-{\bf k}),\phi_{5}(-{\bf k}),\phi_{2}(-{\bf k}),\phi_{3}(-{\bf k}),\phi_{1}(-{\bf k})]^{T}$. Considering the degree of spin, we can express the parity operator at the time reversal invariant momenta $\Gamma_{n_{1}n_{2}n_{3}}$ as follows, $\displaystyle{\cal P}_{\Gamma_{n_{1}n_{2}n_{3}}}=\left(\matrix{1&0\cr 0&1}\right)\otimes\left(\matrix{0&0&0&0&0&1\cr 0&0&0&1&0&0\cr 0&0&0&0&1&0\cr 0&1&0&0&0&0\cr 0&0&1&0&0&0\cr 1&0&0&0&0&0}\right)$ (3) where the $4\times 4$ matrix is the unit matrix in spin space. We diagonalize the total single-particle Hamiltonian ${\cal H}_{\bf k}$ and calculate the $Z_{2}$ topological invariants for different filling fractions. We find that non-trivial topological insulators exist for $1/6,1/2$ and $2/3$ filling while only metal phase occurs for $1/3$ and $5/6$ filling. Thus, we will focus on and discuss the cases with $1/6,1/2$ and $2/3$ filling fractions in the following part of the paper. We identify phases for different parameters $t_{1}$ and $\lambda_{\rm SO}$ with $1/6,1/2$ and $2/3$ filling fractions and draw phase diagrams as shown in Fig.2. Figs.2(a), 2(b) and 2(c) show the phase diagrams for $1/6,1/2$ and $2/3$ filling, respectively. For $1/6$ and $2/3$ filling, there are $(1;111)$ and $(1;000)$ STI phases, $(0;111)$ WTI phase as well as trivial band insulator and metal phases. For $1/2$ filling, there are $(1;111)$ and $(1;000)$ STI phases, trivial band insulator and metal phases except $(0;111)$ WTI phase. To clearly manifest the bulk band structure of different phases for various filling factions, we calculate the bulk energy bands for several cases with different parameters $t_{1}$ and $\lambda_{\rm SO}$, which are shown in Fig.3. In order to investigate the characteristics of surface states for various phases, we evaluate the energy bands in a slab geometry with two $001$ surfaces. The Brillouin zone of the slab is shown in Fig.1(d). The energy bands are present along lines that connect the four surface TRIM as shown in Fig.4. With the assistance of the bulk energy bands shown in Fig.3 and the two-dimensional energy bands for a slab shown in Fig.4, we will sequentially analyze various phases, identify three-dimensional topological insulators, and discuss their characteristics for $1/6,1/2$ and $2/3$ filling. Figure 4: (Color online). Band structures of a slab with two $001$ surfaces for various parameters $t_{1}$ and $\lambda_{\rm SO}$. Here, the horizontal axis represents the wave vectors along the path in the surface Brillouin zone indicated by the red lines in Fig.1(d). (a) $t_{1}=t,\lambda_{\rm SO}=0.5t$, (b) $t_{1}=-t,\lambda_{\rm SO}=-0.5t$, (c) $t_{1}=t,\lambda_{\rm SO}=t$, (d) $t_{1}=t,\lambda_{\rm SO}=0$, (e) $t_{1}=3t,\lambda_{\rm SO}=-0.4t$, (f) $t_{1}=-3t,\lambda_{\rm SO}=0.4t$, (g) $t_{1}=3.2t,\lambda_{\rm SO}=-0.2t$, (h) $t_{1}=t,\lambda_{\rm SO}=0.2t$, (i) $t_{1}=t,\lambda_{\rm SO}=-0.2t$, (j) $t_{1}=-t,\lambda_{\rm SO}=0.2t$, (k) $t_{1}=2.5t,\lambda_{\rm SO}=-0.2t$, and (l) $t_{1}=0.5t,\lambda_{\rm SO}=0$. ### III.1 $1/6$ filling Fig.2(a) shows the phase diagram of the octahedron-decorated cubic lattice for $1/6$ filling. In this case, the $(1;111)$ and $(1;000)$ STI phases are discovered. The non-trivial STI phases have a gap between the first and second bands as shown in Fig.3(a) and 3(b) corresponding to $(1;111)$ and $(1;000)$ STI phases, respectively. We note that for $1/6$ filling there is only one Dirac point on TRIM as shown in Fig.4(a) and (b), that is, only a pair of robust spin-filtered states exists. We also find a $(0;111)$ WTI phase for $1/6$ filling e.g., as shown in Fig.3(c). Fig.4(c) shows the surface states for a $(0;111)$ WTI phase that has two Dirac points between the first and second bands on TRIM. We note that trivial band insulators occur for smaller $t_{1}$ and smaller $\lambda_{\rm SO}$ parameters, which is easily understood for when $t_{1}$ and $\lambda_{\rm SO}$ approaches to zero the lattice becomes separated octahedral clusters. For a trivial band insulator there is a gap between the first and second bands as shown in Fig.3(d), but there are not surface states as shown in Fig.4(d). For a metal phase, the gap vanishes. ### III.2 $1/2$ filling Fig.2(b) shows the phase diagram of the octahedron-decorated cubic lattice for $1/2$ filling. For $1/2$ filling, $(1;111)$ and $(1;000)$ STI phases, trivial band insulators, and metal phases occur, but WTI phases are not found. Figs.3(e) and 3(f) show the band structure for $(1;111)$ and $(0;111)$ STI phases, respectively. We can find from these diagrams that a gap opens between the third and fourth bands. For STI phases, there only one Dirac point on TRIM as shown in Fig.4(e) and (f). For trivial band insulators, there is also a gap between the third and fourth bands as shown in Fig.3(g), but even number of Dirac points exist on TRIM as shown in Fig.4(g). For smaller $t_{1}$ and smaller $\lambda_{\rm SO}$, a metal phase occurs except a special point $t_{1}=0$ and $\lambda_{\rm SO}=0$. For $t_{1}=0$ and $\lambda_{\rm SO}=0$, the second, third and fourth bands are degenerate and become a flat band, which means that electrons are localized. In other works, the system with $t_{1}$ and $\lambda_{\rm SO}$ for $1/2$ filling is a trivial band insulator. However, a tiny change from $t_{1}=0$ and $\lambda_{\rm SO}=0$ for parameters $t_{1}$ and $\lambda_{\rm SO}$ makes the flat band become three dispersive bands that are crossover each other, then the lattice with three bands occupied becomes a metal. ### III.3 $2/3$ filling Fig.2(c) shows the phase diagram of the octahedron-decorated cubic lattice for $2/3$ filling. We note that, similar to $1/6$ filling, $(1;111)$ and $(1;000)$ STI phases, $(0;111)$ WTI phase, trivial band insulator, and metal phase occur in different ranges of parameters $t_{1}$ and $\lambda_{\rm SO}$. Fig.3(i) shows the band structure for $t_{1}=t,\lambda_{\rm SO}=-0.2t$ at which a $(1;111)$ STI phase occurs. We can find that a gap opens between the fourth and fifth bands as shown in Fig.3(i). There is an odd number of surface states which traverse the gap as shown in Fig.4(i). For the $(1;000)$ STI phase, the similar characteristics are exemplified in Figs.3(j) and 4(j). The $(0;111)$ WTI phase is found as well. Fig.3(k) and Fig.4(k) show the $(0;111)$ WTI phase has a gap between the fourth and fifth bands and an even number of surface states traversing the gap. For smaller $t_{1}$ and smaller $\lambda_{\rm SO}$, the system for $2/3$ filling is a trivial band insulator, which is feathered by a gap between the fourth and fifth bands combined with an even number of surface states traversing the gap as shown in Fig.3(l) and Fig.4(l), respectively. ## IV Conclusion In summary, we have shown that the octahedron-decorated cubic lattice with spin-orbit coupling supports three-dimensional topological insulators at $1/6,1/2$ and $2/3$ filling fractions. For $1/6$ and $2/3$ filling, $(1;111)$ and $(1;000)$ STI phases, $(0;111)$ WTI phase, trivial band insulator, and metal phase are found, while for $1/2$ filling, $(1;111)$ and $(1;000)$ STI phases, trivial band insulator, and metal phase occur except $(0;111)$ WTI phase. We have calculated the band structure and surface band structure for the tight-binding model of the octahedron-decorated cubic lattice with spin- orbit coupling and evaluated the $Z_{2}$ topological invariants. We have analyzed and discussed the characters of the band structures and the surface states of different phases. Although the octahedron-decorated cubic lattice we considered is a toy model, our study points out an alternative path to search for real topological materials. On the other hand, it might as well be built from optical lattices due to their diversity and controllability. ###### Acknowledgements. This work was supported by the National Natural Science Foundation of China under Grant No. 11004028 and the Science and Technology Foundation of Southeast University under Grant No. KJ2010417 ## References * (1) L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937). * (2) K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980). * (3) D. J. Thouless, M. 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Analytis, J. H. Chu, et al., Science 325, 178 (2009). * (18) D. Hsieh, Y. Xia, D. Qian, et al., Nature 460, 1101 (2009). * (19) D. Hsieh, Y. Xia, D. Qian, et al., Phys. Rev. Lett. 103 146401 (2009). * (20) H. M. Guo and M. Franz, Phys. Rev. B 80, 113102 (2009). * (21) K. Sun, H. Yao, E. Fradkin, and S. A. Kivelson, Phys. Rev. Lett. 103, 046811 (2009). * (22) A. Rüegg, J. Wen, and G. A. Fiete, Phys. Rev. B 81, 205115 (2010). * (23) C. Weeks and M. Franz, Phys. Rev. B 82, 085310 (2010). * (24) M. Kargarian and G. A. Fiete, Phys. Rev. B 82, 085106 (2010). * (25) H. M. Guo and M. Franz, Phys. Rev. Lett. 103, 206805 (2009).
arxiv-papers
2011-01-19T10:07:03
2024-09-04T02:49:16.530060
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jing-Min Hou, Wen-Xin Zhang, and Guo-Xiang Wang", "submitter": "Jing-Min Hou", "url": "https://arxiv.org/abs/1101.3652" }
1101.3766
# Quantum coherence between two atoms beyond $Q=10^{15}$ C. W. Chou chinwen@nist.gov D. B. Hume M. J. Thorpe D. J. Wineland T. Rosenband Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305 ###### Abstract We place two atoms in quantum superposition states and observe coherent phase evolution for $3.4\times 10^{15}$ cycles. Correlation signals from the two atoms yield information about their relative phase even after the probe radiation has decohered. This technique was applied to a frequency comparison of two 27Al+ ions, where a fractional uncertainty of $3.7^{+1.0}_{-0.8}\times 10^{-16}/\sqrt{\tau/s}$ was observed. Two measures of the Q-factor are reported: The Q-factor derived from quantum coherence is $3.4^{+2.4}_{-1.1}\times 10^{16}$, and the spectroscopic Q-factor for a Ramsey time of 3 s is $6.7\times 10^{15}$. As part of this experiment, we demonstrate a method to detect the individual quantum states of two Al+ ions in a Mg+-Al+-Al+ linear ion chain without spatially resolving the ions. Coherent evolution of quantum superpositions follows directly from Schrödinger’s equation, and is a hallmark of quantum mechanics. Quantum systems with a high degree of coherence are desirable for sensitive measurements and for studies in quantum control. Typically, quantum superposition states quickly decohere due to uncontrolled interactions between the system and its environment. However, through careful isolation and system preparation, quantum coherence has been observed in naturally occurring systems including photons and atoms, as well as in engineered macroscopic systems Haroche and Raimond (2006); Nat ; Leggett (2002); O’Connell et al. (2010). In order to observe the coherence time of a system, it must be compared to a reference system that is at least as coherent, a requirement that can be difficult to satisfy, particularly in systems with the highest degree of coherence. In atomic physics, quality (Q-) factors as high as $1\times 10^{14}$ to $4\times 10^{14}$ Rafac et al. (2000); Boyd et al. (2006); Chou et al. (2010a) have been observed with laser spectroscopy, where the linewidths are often limited by laser noise rather than atomic decoherence. In this report we apply a recent spectroscopic technique Chwalla et al. (2007) to directly observe atomic coherence beyond the laser limit and probe an atomic resonance with a Q-factor above $10^{15}$. Historically, Mössbauer spectroscopy with $\gamma$-rays has exhibited the highest relative coherence, as quantified by the spectroscopic Q-factor (the ratio of oscillation frequency to observed resonance linewidth). Values as high as $8.3\times 10^{14}$ are observed Potzel et al. (1976) in the 93.3 keV radioactive decay of 67Zn, limited by the nuclear lifetime of 13.4 $\mu$s. In those measurements, two separate crystals that contained 67Zn nuclei were compared. One sample provided the probe radiation, while the other served as the resonant absorber. The Mössbauer method might be extended to characterize optical transitions in atoms Dehmelt and Nagourney (1988), but here we use a method based on Ramsey spectroscopy in which the phase fluctuations of the probe source are rejected as common-mode noise Chwalla et al. (2007), enabling Ramsey times much longer than the probe coherence time. Other experiments that compare pairs of microwave Bize et al. (2000) or optical clocks Katori et al. (2010) use a related technique to reduce the Dick-effect noise Dick et al. (1987); Lodewyck et al. (2010) that can limit the stability of frequency comparisons. Figure 1: (Color online) Illustration of the protocol. (a) The detected states from the previous Ramsey experiment serve as the initial states for the the current measurement. (b) The first $\pi/2$-pulse is applied. This is accomplished by a laser beam whose axis coincides with the axis of the ion array. (c) The clock state superpositions freely evolve. (d) The spacing is adjusted at the end of the free-evolution period to vary the differential phase $\Delta\phi$. This is followed immediately by the second $\pi/2$-pulse. At the end of the sequence, the final states are detected to obtain the correlation. In the experiment reported here, atomic superposition states evolve coherently for up to 5 s at a frequency of $1.12\times 10^{15}$ Hz. Following Chwalla et al. Chwalla et al. (2007), a Ramsey pulse sequence Ramsey (1956) is simultaneously applied to two trapped 27Al+ ions, labeled $i\in\\{1,2\\}$ (see Fig. 1). The probe radiation for both ions is derived from the same source. Each ion is initialized in one of the two quantum states that make up the clock transition (clock states), which need not be the same for both ions. Immediately prior to the second $\pi/2$-pulse, a variable displacement $\mathbf{r_{i}}$ is applied to the ions. This Ramsey sequence induces a state change with probability $p_{i}=(1+\cos{\delta\phi_{i}})/2$, where $\delta\phi_{i}=\phi_{L}+\mathbf{k}\cdot\mathbf{r_{i}}-\phi_{i}$ is the difference between the phase accumulated by the laser ($\phi_{L}$+$\mathbf{k}\cdot\mathbf{r_{i}}$) and ion ($\phi_{i}$) during the free-evolution period $T$, and $\mathbf{k}$ is the laser beam wavevector, $\mathbf{k}=\hat{z}2\pi/(267\text{ nm})$. The correlation probability (the probability that both ions make a transition, or both do not make a transition) is then $P=[2+\cos{(\delta\phi_{1}-\delta\phi_{2})}+\cos{(\delta\phi_{1}+\delta\phi_{2})}]/4$. Here the relative phase, $\delta\phi_{1}-\delta\phi_{2}$, is independent of $\phi_{L}$, which is uniformly randomized over the interval $[0,2\pi)$ with a pseudo-random number generator. Without knowledge of $\phi_{L}$, the probability of correlated transitions is $P_{c}=\frac{1}{2\pi}\int^{2\pi}_{0}Pd\phi_{L}=\frac{1}{2}+\frac{C}{2}\cos{\Delta\phi},$ (1) where $\Delta\phi=\phi_{2}-\phi_{1}+\mathbf{k}\cdot(\mathbf{r_{1}-r_{2}})$ and $C\equiv P_{c,\text{ max}}-P_{c,\text{ min}}\leq\frac{1}{2}$ is the contrast. The correlation signal $P_{c}$ provides a measurement of the differential phase evolution of the two Al+ “clock” ions similar to the measurement of differential phase between source and absorber in Mössbauer spectroscopy. Its statistical properties are equivalent to that of a single-ion Ramsey experiment with reduced contrast, and the ultimate measurement uncertainty is determined by quantum projection noise Itano et al. (1993). When $|\Delta\phi|$ is kept near $\pi/2$, the statistical uncertainty of the ion- ion fractional frequency difference, or measurement instability, is $\sigma(\tau)\equiv\sigma_{\nu}/\nu=(2\pi\nu C\sqrt{T\tau})^{-1}$, where $\tau$ is the total measurement duration, $\sigma_{\nu}$ is the uncertainty in the measured frequency difference $(\phi_{2}-\phi_{1})/(2\pi T)$, and $\nu\approx 1.12$ PHz is the transition frequency. Importantly, the free- evolution period $T$ is not limited by laser phase noise. In the experiment, a linear Paul trap confines one 25Mg+ ion and two Al+ ions in an array Chou et al. (2010b); Hume et al. (2007) along the trap z-axis (Fig. 1). The motional frequencies of a single Mg+ in the trap are $\\{f_{x},f_{y},f_{z}\\}=\\{5.13,6.86,3.00\\}$ MHz. The ions are maintained in the order of Mg+-Al+-Al+ (inter-ion spacing 2.69 $\mu$m) by periodically adjusting the trap conditions and verifying via Mg+ spectroscopy the frequency of the “stretch” mode of motion, whose value is 5.1 MHz for the correct order DBH . The two states involved in the Al+ clock transition, $|\\!\\!\downarrow\rangle\equiv|^{1}S_{0}$, $m_{F}=5/2\rangle$ and $|\\!\\!\uparrow\rangle\equiv|^{3}P_{0}$, $m_{F}=5/2\rangle$, are detected with an adaptive quantum non-demolition process Hume et al. (2007). The present implementation distinguishes all four states $|\\!\\!\downarrow_{1}\downarrow_{2}\rangle$, $|\\!\\!\downarrow_{1}\uparrow_{2}\rangle$, $|\\!\\!\uparrow_{1}\downarrow_{2}\rangle$, and $|\\!\\!\uparrow_{1}\uparrow_{2}\rangle$ by observing Mg+ fluorescence after controlled interactions between the Al+ and Mg+ ions. Individual state detection relies on the two Al+ ions having different amplitudes in several motional eigenmodes, which affects the state-mapping probability onto the Mg+ ion. Information from several measurements is combined in a Bayesian process Hume et al. (2007), to determine the most likely state of the two Al+ ions with typically 99 % probability in an average of 30 detection cycles (approximately 50 ms total duration). We note that this technique allows individual state detection of two ions in the same trap, without the need for high spatial-resolution optics. The Ramsey experiments use $\pi/2$-pulse durations of 1.2 ms and are carried out for various free-evolution periods $T$. For each $T$, $\Delta\phi_{z}\equiv\mathbf{k}\cdot(\mathbf{r_{1}-r_{2}})$ is varied from 0 to beyond $2\pi$ to characterize the correlation. The duration required to shift the positions by $\mathbf{r_{i}}$ is approximately 10 ms. Figure 2 shows the correlation signals for $T$ between 0.1 and 5 s. Currently, collisions between the ions and background gas make it impractical to generate sufficient statistics for $T$ greater than 5 s. The collisions result in changes of ion order and loss of ions due to chemical reactions. Figure 2: (Color online) Correlation probabilities $P_{c}$ versus $\Delta\phi_{z}$ for various Ramsey times: (a) 0.1 s, 1500 probes; (b) 0.5 s, 600 probes;(c) 1 s, 600 probes; (d) 2 s, 360 probes; (e) 3 s, 300 probes; (f) 5 s, 100 probes. Dots: measurement outcomes; lines: maximum-likelihood fits to the fringes. The phase difference $\phi_{2}-\phi_{1}$, and thus the frequency difference, between the two Al+ ions can be determined from the phases of the $P_{c}$ fringes in Fig. 2. In the experiment, we apply a magnetic field gradient of $dB/dz=1.32\pm 0.33$ mT/m, as measured by monitoring the frequency of the $|F=3\text{, }m_{F}=-3\rangle\rightarrow|F=2\text{, }m_{F}=-2\rangle$ magnetic-field dependent transition in the 25Mg+ $3s\text{ }S_{1/2}$ ground state hyperfine manifold, when the Mg+ position along the trap axis is adjusted. This gradient induces a fractional frequency shift $(\nu_{2}-\nu_{1})/\nu=1.32\pm 0.33\times 10^{-16}$ between the $|\\!\\!\downarrow\rangle\leftrightarrow|\\!\\!\uparrow\rangle$ transitions of the two Al+ ions. The phases of the $P_{c}$ fringes, determined by maximum- likelihood fits Sivia and Skilling (2006), increase linearly with $T$, as shown in Fig. 3a. A linear fit has a slope of $0.84\pm 0.06$ rad/s, corresponding to a measured shift of $1.19\pm 0.08\times 10^{-16}$, in agreement with the shift caused by the magnetic-field gradient. All reported uncertainties represent a 68 % confidence interval. We derive the contrast $C$ from the maximum-likelihood fits to the data in Fig. 2. An exponential fit of $C$ versus $T$ yields a relative coherence time $T_{C}$ of $9.7^{+6.9}_{-3.1}$ s, corresponding to a Q-factor ($Q=\pi\nu T_{C}$ Vion et al. (2002)) of $3.4^{+2.4}_{-1.1}\times 10^{16}$. A uniform prior distribution of $T_{C}$ on the interval 0 s to 25 s is assumed. The measured coherence time is compatible with the expected result, which is given by the lifetime $T^{\prime}=20.6\pm 1.4$ s Rosenband et al. (2007) of the Al+ $|^{3}P_{0}\rangle$ state. When viewed in terms of Ramsey spectroscopy, for $T=3$ s, the full-width-at-half-maximum of the Ramsey signal corresponds to a Q-factor of $2\nu T=6.7\times 10^{15}$. Figure 3: (Color online) (a) Differential phase $\phi_{2}-\phi_{1}$ versus Ramsey time $T$. The solid line is a linear fit, with slope $0.84\pm 0.06$ rad/s. (b) Measurement uncertainty extrapolated to 1 s averaging time as a function of Ramsey time. Dots: measurement results, where the uncertainties are derived from the uncertainties in the contrast $C$; solid line: theoretical lifetime-limited instability, where only phases corresponding to $\Delta\phi\approx\pm\pi/2$ are probed; dashed line: expected experimental instability, with $\Delta\phi$ uniformly distributed over $[0,2\pi)$. The dashed line is derived from the measured coherence time of 9.7 s, and an approximate overhead of 100 ms per Ramsey measurement, which reduces the duty cycle. The current protocol could significantly reduce the total duration of future high-precision measurements with atomic clocks. Figure 3b shows the measurement uncertainties extrapolated to 1 s ($\sigma_{1s}$) versus Ramsey time $T$. The long-term statistical uncertainty is then $\sigma(\tau)=\sigma_{1s}/\sqrt{\tau/s}$ for a measurement duration $\tau$. Note that, for $T=3$ s, the frequency difference between the two Al+ ions can be determined with a fractional uncertainty $\sigma=1.1\times 10^{-17}$ in a 1126 s measurement (900 s integrated free-evolution time), which can be extrapolated to infer a relative measurement uncertainty $\sigma_{1s}=3.7\times 10^{-16}$. This result may be compared to a recent frequency difference measurement of two Al+ clocks, where 65,000 s were required to reach the same uncertainty of $1.1\times 10^{-17}$ Chou et al. (2010c). In general, the lifetime-limited contrast is $C=\frac{1}{2}\exp(-T/T^{\prime})$, yielding an instability of $\sigma(\tau)=\exp(T/T^{\prime})/(\pi\nu\sqrt{T\tau})$, which is shown for Al+ in Fig. 3b (solid line). The optimal probe time of $T=T^{\prime}/2$ yields $\sigma_{1s}=1.4\times 10^{-16}$. Although we have used the technique to measure two ions in the same trap, it may also be applied to clocks at different locations. A proposed frequency comparison of remote optical clocks is depicted in Fig. 4. Note, however, that due to the requirement that $\phi_{L}$ be the same for both clocks, this technique is limited to comparisons between clocks operating at similar frequencies. Although the clocks need not be identical, the differential phase, $\Delta\phi$, must be known well enough to make phase errors of $\pi$ unlikely. In order to retain control over the differential phase, the individual paths (paths 1 and 2 in Fig. 4) need to be phase-stabilized and the Ramsey pulses at the two locations need to be synchronized so that the two clocks experience the same laser phase noise. For ions with very long radiative lifetimes, the same technique could be used to compare two ion samples, each composed of maximally entangled states Leibfried et al. (2005); Monz et al. . Figure 4: (Color online) Proposed frequency cmparison of remote optical clocks, here based on Al+ ions. The paths 1 and 2 that direct the clock laser light to the ions need to be controlled so that they can faithfully transmit the light without introducing additional phase noise. Local frequency fluctuations, such as those caused by fluctuating magnetic fields, should be minimized. The free evolution periods need to be synchronized so that the atoms are subjected to the same phase noise in the Ramsey pulses, the effect of which cancels in the protocol. A similar approach can be taken in comparisons of two clocks composed of many unentangled atoms. The measurement protocol is again based on synchronized Ramsey pulses where the free-evolution time $T$ exceeds the laser coherence time. The two clocks (labeled $X\in\\{A,B\\}$) measure transition probabilities $p_{X}=\frac{1}{2}[1+\cos{(\phi_{X}-\phi_{L}-\theta_{X})}]$, and the quantity of interest $\delta\phi_{AB}=\phi_{A}-\phi_{B}$ is determined from $\delta\phi_{AB}=\cos^{-1}{(2p_{A}-1)}-\cos^{-1}{(2p_{B}-1)}+\theta_{A}-\theta_{B}$, where $\theta_{X}$ are the controlled laser phase offsets at the two clocks. If we consider only atomic projection noise in $p_{A}$ and $p_{B}$, this measurement has a variance of $var(\delta\phi_{AB})=\frac{1}{N_{A}}+\frac{1}{N_{B}}$, where $N_{A}$ and $N_{B}$ are the numbers of atoms in clocks $A$ and $B$. The fractional frequency stability of the clock comparison is then $\sigma_{y}(\tau)=\sqrt{var(\delta\phi_{AB})}/(2\pi\nu\sqrt{T\tau})$. A complication is introduced by the fact that $\phi_{L}$ will be initially unknown, which leads to ambiguities in the trigonometric inversions from which $\delta\phi_{AB}$ is calculated. Such ambiguities will be absent in the majority of measurements, if an approximate value of $\delta\phi_{AB}$ can be determined through prior calibrations (with $var(\delta\phi_{AB})\ll 1)$, and phase offsets $\theta_{X}$ are adjusted such that $\phi_{A}-\phi_{B}-(\theta_{A}-\theta_{B})\approx\pi/2$. After this calibration procedure, $p_{A}$ and $p_{B}$ represent approximate quadratures of the laser-atom phase difference, and for most values of $\phi_{L}$ the trigonometric inversions are unambiguous. In such a measurement the Ramsey free-evolution time is no longer constrained by laser decoherence, and the Dick effect due to the probe source is absent. Therefore, more rapid frequency comparisons of similar-frequency many-atom optical clocks should also be possible. Small values of $\sigma(\tau)$ in frequency comparisons are useful for evaluating and improving the performance of optical clocks and for metrological applications. For example, comparison of clocks in geographically distinct locations can be used to evaluate spatial and temporal variations in the geoid Chou et al. (2010a). More generally, any physical process that leads to small, constant frequency shifts in an optical clock can be studied in this way. This includes relativistic effects as well as shifts caused by electric fields, magnetic fields and atom collisions. Our observation of a Q-factor beyond $10^{15}$ and a frequency ratio measurement instability of $3.7\times 10^{-16}/\sqrt{\tau/s}$ highlights the intrinsic sensitivity of optical clocks as a metrological tool. This work is supported by ONR, AFOSR, DARPA, and IARPA. We thank D. Leibrandt and J. Sherman for comments on the manuscript. Publication of NIST, not subject to U.S. copyright. ## References * Haroche and Raimond (2006) S. Haroche and J. M. Raimond, _Exploring the quantum_ (Oxford University Press, Oxford, U.K., 2006). * (2) Quantum coherence experiments in several technologies are reviewed in R. Blatt and D. Wineland, Nature 453, 1008 (2008); I. Bloch, ibid. 453, 1016 (2008); H. J. Kimble, ibid. 453, 1023 (2008); J. Clarke and F. K. Wilhelm, ibid. 453, 1031 (2008); R. Hanson and D. D. Awschalom, ibid. 453, 1043 (2008). * Leggett (2002) A. J. Leggett, J. Phys.: Condens. Matter 14, R415 (2002). * O’Connell et al. (2010) A. D. O’Connell et al. Nature 464, 697 (2010). * Rafac et al. (2000) R. J. Rafac, B. C. Young, J. A. Beall, W. M. Itano, D. J. Wineland, and J. C. Bergquist, Phys. Rev. Lett. 85, 2462 (2000). * Boyd et al. (2006) M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye, Science 314, 1430 (2006). * Chou et al. (2010a) C. W. Chou, D. B. Hume, D. J. Wineland, and T. Rosenband, Science 329, 5999 (2010a). * Chwalla et al. (2007) M. Chwalla, K. Kim, T. Monz, P. Schindler, M. Riebe, C. F. Roos, and R. Blatt, Appl. Phys. B 89, 483 (2007). * Potzel et al. (1976) W. Potzel, A. Forster, and G. M. Kalvius, J. De Physique C6, 691 (1976). * Dehmelt and Nagourney (1988) H. Dehmelt and W. Nagourney, Proc. Nad. Acad. Sci. 85, 7426 (1988). * Bize et al. (2000) S. Bize, Y. Sortais, P. Lemonde, S. Zhang, P. Laurent, G. Santarelli, C. Salomon, and A. Clairon, Ultrasonics, Ferroelectrics and Frequency Control, IEEE Transactions on 47, 1253 (2000). * Katori et al. (2010) H. Katori, T. Takano, and M. Takamoto, in _Proceedings of the 22nd International Conference on Aomic Physics_ (2010), to be published. * Dick et al. (1987) G. J. Dick, J. D. Prestage, C. A. Greenhall, and L. Maleki, in _Proc. 19th Precise Time and Time interval Mtg._ (1987), p. 133. * Lodewyck et al. (2010) J. Lodewyck, P. G. Westergaard, A. Lecallier, L. Lorini, and P. Lemonde, New J. Phys. 12, 065026 (2010). * Ramsey (1956) N. F. Ramsey, _Molecular Beams_ (Oxford University Press, New York, 1956). * Itano et al. (1993) W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys. Rev. A 47, 3554 (1993). * Chou et al. (2010b) C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010b). * Hume et al. (2007) D. B. Hume, T. Rosenband, and D. J. Wineland, Phys. Rev. Lett. 99, 120502 (2007). * (19) D. B. Hume, Ph.D. Thesis, University of Colorado (2010). * Sivia and Skilling (2006) D. S. Sivia and J. Skilling, _Data Analysis: A Bayesian Tutorial, Second Edition_ (Oxford University Press, Oxford, U.K., 2006). * Vion et al. (2002) D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, and M. H. Devoret, Science 296, 886 (2002). * Rosenband et al. (2007) T. Rosenband et al. Phys. Rev. Lett. 98, 220801 (2007). * Chou et al. (2010c) C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland, and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010c). * Leibfried et al. (2005) D. Leibfried et al. Nature 438, 639 (2005). * (25) T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Han̈sel, M. Hennrich, and R. Blatt, arXiv:1009.6126v1.
arxiv-papers
2011-01-19T20:13:11
2024-09-04T02:49:16.536033
{ "license": "Public Domain", "authors": "C. W. Chou, D. B. Hume, M. J. Thorpe, D. J. Wineland, and T. Rosenband", "submitter": "Chin-wen Chou", "url": "https://arxiv.org/abs/1101.3766" }
1101.3828
# Mean-field Study of Charge, Spin, and Orbital Orderings in Triangular-lattice Compounds $A$NiO2 ($A$=Na, Li, Ag) Hiroshi Uchigaito E-mail address: uchigaito@aion.t.u-tokyo.ac.jp Masafumi Udagawa and Yukitoshi Motome Department of Applied PhysicsDepartment of Applied Physics University of Tokyo University of Tokyo Hongo Hongo Bunkyo-ku Bunkyo-ku Tokyo 113-8656 Tokyo 113-8656 Japan Japan ###### Abstract We present our theoretical results on the ground states in layered triangular- lattice compounds $A$NiO2 ($A$=Na, Li, Ag). To describe the interplay between charge, spin, orbital, and lattice degrees of freedom in these materials, we study a doubly-degenerate Hubbard model with electron-phonon couplings by the Hartree-Fock approximation combined with the adiabatic approximation. In a weakly-correlated region, we find a metallic state accompanied by $\sqrt{3}\times\sqrt{3}$ charge ordering. On the other hand, we obtain an insulating phase with spin-ferro and orbital-ferro ordering in a wide range from intermediate to strong correlation. These phases share many characteristics with the low-temperature states of AgNiO2 and NaNiO2, respectively. The charge-ordered metallic phase is stabilized by a compromise between Coulomb repulsions and effective attractive interactions originating from the breathing-type electron-phonon coupling as well as the Hund’s-rule coupling. The spin-orbital-ordered insulating phase is stabilized by the cooperative effect of electron correlations and the Jahn-Teller coupling, while the Hund’-rule coupling also plays a role in the competition with other orbital-ordered phases. The results suggest a unified way of understanding a variety of low-temperature phases in $A$NiO2. We also discuss a keen competition among different spin-orbital-ordered phases in relation to a puzzling behavior observed in LiNiO2. multi-orbital Hubbard model, electron-phonon coupling, charge order, orbital order, metal-insulator transition, triangular lattice, NaNiO2, LiNiO2, AgNiO2 ## 1 Introduction One of the most distinctive features of strongly-correlated electron systems is diverse cooperative phenomena. [1, 2] Correlated electron systems generally show intricate phase diagrams full of competing or coexisting states, and the phase competition often leads to exotic many-body phenomena. Among many aspects, there are two factors which promote such complexity, i.e., multiple degrees of freedom [2, 3, 4] and geometrical frustration [5]. The multiple degrees of freedom are composed of charge, spin, and orbital of electrons. Strong electron correlations induce interplay among them, resulting in a variety of phases. In addition, the interplay often causes exotic response to external perturbations, such as the colossal magneto-resistance in perovskite manganites [6]. On the other hand, the geometrical frustration promotes the phase competition. In general, the geometrical frustration results in a huge number of low-energy degenerate states, by suppressing conventional long-range orders. The degeneracy yields nontrivial phenomena such as complicated ordering, glassy behavior, and spin-liquid states. It is a long-standing problem in condensed-matter physics to understand a variety of phenomena which emerge from synergetic effects between the multiple degrees of freedom and the geometrical frustration. The family of compounds $A$NiO2 ($A$=Li, Na, Ag) is a typical example of such geometrically-frustrated systems with multiple degrees of freedom. $A$NiO2 takes two different lattice structures depending on the cation $A$, that is, the delafossite structure [7] (AgNiO2) and the ordered rock-salt structure [8] (NaNiO2 and LiNiO2). Both structures are quasi-two-dimensional, composed of stacking of Ni, O, and $A$ layers. The magnetic and transport properties are dominated by Ni cations, which are surrounded by the octahedron of oxygens. The NiO6 octahedra share their edges so that the Ni sites constitute the frustrated triangular layers as shown in Fig. 1. Ni3+ cation has seven $3d$ electrons in the low-spin configuration: Six out of seven fully occupy the lower $t_{2g}$ levels and the remaining one electron enters in the higher $e_{g}$ levels. Hence the doubly-degenerate $e_{g}$ orbital degree of freedom is active in these systems. Figure 1: (Color online). Schematic picture of NiO6 layer in $A$NiO2. NiO6 octahedra share their edges with neighbors to form the triangular lattice of Ni cations. $\gamma$ denotes the Ni-Ni bond directions. Two $e_{g}$ orbitals, $3z^{2}-r^{2}$ and $x^{2}-y^{2}$, are shown. $A$NiO2 shows a variety of behaviors depending on the cation $A$ in spite of the similarity in lattice and electronic structure. NaNiO2 is a Mott insulator with a gap of 0.24eV [9]. This compound shows a first-order structural transition accompanied by cooperative Jahn-Teller distortions at 480K [8, 10] and a second-order magnetic transition at 20K [10, 12, 11]. At the lowest temperature($T$), the system exhibits orbital-ferro and A-type antiferro-spin order (antiferromagnetic stacking of spin-ferro ordered layers)[13, 11]. LiNiO2 is also a Mott insulator with a gap of 0.2eV [14], however, it shows no clear phase transition down to 1.4K in contrast to NaNiO2 [15]. There is strong sample dependence in $T$ dependence of the magnetic susceptibility; the field-cool and zero-field-cool bifurcation appears in different ways depending on samples. This sample dependence strongly suggests the relevance of extrinsic disorder [16]. The ground state as well as the finite-$T$ properties remain controversial [16, 17, 18, 19, 20, 21, 22, 23, 24]. In addition to these insulating materials, recently, a new metallic compound AgNiO2 was synthesized [7, 25]. It shows a structural transition associated with a $\sqrt{3}\times\sqrt{3}$ charge ordering at 365K and antiferromagnetic transition at 20K [26, 27]. The system remains metallic down to the lowest $T$. It was claimed that in the low-$T$ phase the system separates into rather localized spins at 1/3 Ni sites and itinerant electrons at the remaining sites [26, 27]. Magnetic properties at low $T$ were analyzed by considering the competing nearest- and second-neighbor exchange couplings between localized spins [28, 29]. So far, $A$NiO2 has been theoretically studied by the first-principle calculations and the strong-coupling analyses. The orbital and spin ordering observed in NaNiO2 was reproduced by the LSDA+$U$ first-principle calculations [30, 31]. The metallicity as well as the charge ordering in AgNiO2 was also reproduced by the first-principle calculations [25, 26]. On the other hand, effective models in the limit of strong electron correlation, the so-called Kugel-Khomskii models, were studied to understand the orbital and spin ordering in NaNiO2 and the peculiar disordered state in LiNiO2 [20, 32, 33]. For AgNiO2, recently, the magnetic phase diagram was investigated by the classical spin model with ignoring the itinerant electrons [34]. Despite the extensive studies so far, comprehensive understanding of the ground states of $A$NiO2 has not been reached yet. Although the low-$T$ states of NaNiO2 and AgNiO2 are reproduced by the first-principle calculations, the mechanism of stabilizing these states is not fully clarified. In addition, the effective-model approach does not fully succeed in reproducing the ground states of NaNiO2 and LiNiO2 in an unified way. A possible way to explore the comprehensive understanding is to investigate a model in a wide region of interaction parameters systematically beyond the strong-coupling approach. In fact, in both NaNiO2 and LiNiO2, the Mott gap is not large; the gap is comparable to the transfer integrals. Furthermore, the newly-synthesized AgNiO2 shows metallic behavior. These facts suggest an importance of charge fluctuations in weakly- or intermediately-correlated regions. Electron-phonon couplings may be another essential factors, which have not been considered seriously in spite of the experimental facts that the structural transitions are observed in NaNiO2 and AgNiO2. In this study, aiming at a unified picture of these compounds $A$NiO2, we investigate a multi-orbital Hubbard model with electron-phonon couplings on a two-dimensional triangular lattice. Our purpose is to elucidate the microscopic mechanism for the variety of phases in these compounds $A$NiO2. In particular, we focus on electron-phonon couplings and charge degrees of freedom, both of which have not been carefully examined in the previous studies. We clarify that these two elements play an important role in the phase competition in these complicated systems. We obtain a $\sqrt{3}\times\sqrt{3}$ charge-ordered metallic (COM) phase in the weakly- correlated region and an insulating phase with ferro-type spin and orbital ordering in the intermediately- to strongly-correlated region. These two phases reproduce many aspects of the low-$T$ states in AgNiO2 and NaNiO2, respectively. We discuss the peculiar disordered state in LiNiO2 in relation with a keen phase competition in the obtained phase diagram. The organization of this paper is as follows. In Sec. 2, we describe our model and method. We introduce the Hamiltonian term by term, and present the approximations adopted in the calculations. We present our results in Sec. 3. We discuss the parameter region and the mechanism to stabilize the $\sqrt{3}\times\sqrt{3}$ COM phase and the spin-orbital-ordered insulating phase. Finally, Sec. 4 is devoted to summary. ## 2 Model and Method ### 2.1 Model In the present study, to elucidate the phase competition in $A$NiO2, we investigate the ground state of the multi-orbital Hubbard model with electron- phonon couplings. Among the five $3d$ orbitals, we consider only the twofold degenerate $e_{g}$ orbitals, by taking account of the low-spin state of Ni3+ cations. Our Hamiltonian is written as $\mathcal{H}=\mathcal{H}_{\rm{kin}}+\mathcal{H}_{\rm{int}}+\mathcal{H}_{\text{el- ph}}+\mathcal{H}_{\rm{ph}},$ (1) where $\mathcal{H}_{\rm{kin}}$, $\mathcal{H}_{\rm{int}}$, $\mathcal{H}_{\text{el-ph}}$, and $\mathcal{H}_{\rm{ph}}$ represent the kinetic term of electrons, the electron-electron interactions, the electron- phonon couplings, and the elastic term of phonons, respectively. We describe the detailed forms of each term in the following. #### 2.1.1 Kinetic term Due to the spatial anisotropy of the $e_{g}$-orbital wave functions, transfer integrals between Ni sites depend on the bond direction as well as orbital types (see Fig. 1). The kinetic term in eq. (1) is written as $\mathcal{H}_{\rm{kin}}=-\sum_{\langle ij\rangle}\sum_{\alpha,\beta}\sum_{\sigma}t^{\gamma_{ij}}_{\alpha\beta}\bigl{(}c_{i\alpha\sigma}^{\dagger}c_{j\beta\sigma}+\rm{H.c}.\bigr{)}.$ (2) Here, $i$ and $j$ denote the site indices, $\alpha$ and $\beta$ represent the orbital indices with $\alpha=a(b)$ corresponding to the $3z^{2}-r^{2}$ ($x^{2}-y^{2}$) orbital, $\sigma$ is the spin, and $\gamma_{ij}$ denotes the direction of bond between the site $i$ and $j$, as shown in Fig. 1. The sum over $\langle ij\rangle$ is taken for the nearest-neighbor sites on the triangular lattice. The transfer integrals are given by the following matrices; $t^{\gamma=1}=\begin{pmatrix}t&0\\\ 0&t^{\prime}\end{pmatrix},\ t^{\gamma=2}=\begin{pmatrix}\displaystyle t_{2}&t_{3}\\\ t_{3}&t_{4}\end{pmatrix},\ t^{\gamma=3}=\begin{pmatrix}t_{2}&-t_{3}\\\ -t_{3}&t_{4}\end{pmatrix},$ (3) for the two bases of $3z^{2}-r^{2}$ and $x^{2}-y^{2}$ orbitals. From the symmetry of orbitals, we obtain the following relations: $t_{2}=t/4+3t^{\prime}/4$, $t_{3}=\sqrt{3}(t-t^{\prime})/4$, and $t_{4}=3t/4+t^{\prime}/4$, with two independent parameters, $t$ and $t^{\prime}$. We set $t=1$ as an energy scale. The value of $t^{\prime}$ depends on both $d$-$d$ direct transfer integrals and $d$-$p$-$d$ indirect ones in a complicated manner [32, 33]. In the following, we show the results for $t^{\prime}=-1$ by noting that the orbital overlaps between atomic orbitals at the neighboring sites lead to $t^{\prime}/t\sim-1$ when one consider both contributions. An extended study in wider range of $t^{\prime}/t$ for an effective model without phonon is found in Ref. Vernay. The choice of $t$ and $t^{\prime}$ gives the non-interacting bandwidth $8t$. #### 2.1.2 Electron-electron interactions Next we introduce the electron-electron interaction term $\mathcal{H}_{\rm{int}}$ in eq. (1). We consider only the on-site Coulomb interactions. For the doubly-degenerate $e_{g}$ orbital system, $\mathcal{H}_{\rm{int}}$ is written as $\displaystyle\mathcal{H}_{\rm{int}}=\mathcal{H}_{U}+\mathcal{H}_{U^{\prime}}+\mathcal{H}_{J_{\rm{H}}}+\mathcal{H}_{J_{\rm{H}}^{\prime}},$ (4) where $\displaystyle\mathcal{H}_{U}=$ $\displaystyle\ U\sum_{i}\sum_{\alpha}n_{i\alpha\uparrow}n_{i\alpha\downarrow},$ (5) $\displaystyle\mathcal{H}_{U^{\prime}}=$ $\displaystyle\ U^{\prime}\sum_{i}\sum_{\sigma\sigma^{\prime}}n_{ia\sigma}n_{ib\sigma^{{}^{\prime}}},$ (6) $\displaystyle\mathcal{H}_{J_{\rm{H}}}=$ $\displaystyle\ J_{\rm{H}}\sum_{i}\sum_{\sigma\sigma^{\prime}}c^{\dagger}_{ia\sigma}c^{\dagger}_{ib\sigma^{\prime}}c_{ia\sigma^{\prime}}c_{ib\sigma},$ (7) $\displaystyle\mathcal{H}_{J_{\rm{H}}^{\prime}}=$ $\displaystyle\ J_{\rm{H}}^{\prime}\sum_{i}\sum_{\alpha\neq\alpha^{\prime}}c^{\dagger}_{i\alpha\uparrow}c^{\dagger}_{i\alpha\downarrow}c_{i\alpha^{\prime}\downarrow}c_{i\alpha^{\prime}\uparrow}.$ (8) Here $n_{i\alpha\sigma}=c^{\dagger}_{i\alpha\sigma}c_{i\alpha\sigma}$, $U$ and $U^{\prime}$ denote the intra- and inter-orbital Coulomb repulsions, and $J_{\rm{H}}$ and $J_{\rm{H}}^{\prime}$ denote the exchange interaction and the pair hopping, respectively. $\mathcal{H}_{J_{\rm{H}}}$ and $\mathcal{H}_{J_{\rm{H}}^{\prime}}$ are called the Hund’s-rule couplings. Hereafter we assume the relations $\ U^{\prime}=U-2J_{\rm{H}}$ and $J_{\rm{H}}=J_{\rm{H}}^{\prime}$ to retain the rotational symmetry of the Coulomb interaction. #### 2.1.3 Electron-phonon couplings As to the electron-phonon couplings, we consider two relevant distortions of NiO6 octahedra in $A$NiO2, namely, the $A_{1g}$ breathing mode and the $E_{g}$ Jahn-Teller modes. $\mathcal{H}_{\text{el-ph}}$ in eq. (1) is given by the sum of these two contributions as $\mathcal{H}_{\text{el-ph}}=\mathcal{H}_{\text{el- ph}}^{\rm{br}}+\mathcal{H}_{\text{el-ph}}^{\rm{JT}}.$ (9) The $A_{1g}$ mode corresponds to the isotropic expansion (contraction) of NiO6 octahedron [Fig. 2(a)], which lowers (raises) two $e_{g}$ energy levels without lifting their degeneracy. Namely, the $A_{1g}$ mode couples to the local charge on each Ni site, written in the form $\mathcal{H}_{\text{el-ph}}^{\rm{br}}=-\gamma_{\rm{br}}\sum_{i}x_{{\rm br},i}\left(n_{ia}+n_{ib}-1\right),$ (10) where $n_{i\alpha}=\sum_{\sigma}n_{i\alpha\sigma}$, $x_{{\rm br},i}$ is the amplitude of the $A_{1g}$ lattice distortion, and $\gamma_{\rm{br}}>0$ is the corresponding coupling constant. A positive (negative) $x_{{\rm br},i}$ corresponds to an expansion (contraction). The $E_{g}$ mode has two components, $E_{g,u}$ and $E_{g,v}$, as shown in Figs. 2(b) and 2(c), respectively. The $E_{g,u}$ mode corresponds to the $z$-axis elongation of NiO6 octahedron, which splits the energy levels of $x^{2}-y^{2}$ and $3z^{2}-r^{2}$ orbitals, while the $E_{g,v}$ mode causes a mixing of the two orbitals: The coupling to the $E_{g}$ modes is written as $\displaystyle\mathcal{H}_{\text{el-ph}}^{\rm{JT}}=-\gamma_{\rm{JT}}$ $\displaystyle\sum_{i,\sigma}\Bigl{\\{}x_{{\rm JT},i}\left(n_{ia\sigma}-n_{ib\sigma}\right)$ $\displaystyle\quad+\bar{x}_{{\rm JT},i}(c_{ia\sigma}^{\dagger}c_{ib\sigma}+c_{ib\sigma}^{\dagger}c_{ia\sigma})\Bigr{\\}},$ (11) where $x_{{\rm JT},i}$ and $\bar{x}_{{\rm JT},i}$ are the amplitudes of $E_{g,u}$ and $E_{g,v}$ modes, respectively, and $\gamma_{\rm{JT}}$ represents the common coupling constant. Figure 2: Schematic pictures of the displacement of oxygens (open circles) around a Ni cation at the origin in (a) $A_{1g}$ mode, (b) $E_{g,u}$ mode, and (c) $E_{g,v}$ mode. #### 2.1.4 Phonon term The phonon term $\mathcal{H}_{\rm{ph}}$ in eq. (1) consists of the on-site term ($\mathcal{H}_{\rm{ph}}^{\rm{elastic}}$) and the inter-site term ($\mathcal{H}_{\rm{ph}}^{\rm{coop}}$) as, $\displaystyle\mathcal{H}_{\rm{ph}}=$ $\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{elastic}}+\mathcal{H}_{\rm{ph}}^{\rm{coop}}.$ (12) Each term is given by the sum of contributions from the $A_{1g}$ and $E_{g}$ mode phonons. The first term $\mathcal{H}_{\rm{ph}}^{\rm{elastic}}$ is the elastic energy of lattice distortions, which is given by the sum of the following two terms; $\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{elastic,br}}=$ $\displaystyle\frac{1}{2}\sum_{i}x_{{\rm br},i}^{2},$ (13) $\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{elastic,JT}}=$ $\displaystyle\frac{1}{2}\sum_{i}\left(x_{{\rm JT},i}^{2}+\bar{x}_{{\rm JT},i}^{2}\right).$ (14) The elastic constants of $A_{1g}$ mode and $E_{g}$ modes are taken as unity without losing generality, by normalizing the amplitudes of lattice distortions ($x_{{\rm br},i}$, $x_{{\rm JT},i}$, and $\bar{x}_{{\rm JT},i}$) and the coupling constants ($\gamma_{\rm{br}}$ and $\gamma_{\rm{JT}}$). The second term $\mathcal{H}_{\rm{ph}}^{\rm{coop}}$ describes the cooperative couplings of lattice distortions. The precise form of the couplings is, in principle, determined by the phonon dispersion in the materials, but here, for simplicity, we consider the nearest-neighbor couplings only. Then $\mathcal{H}_{\rm{ph}}^{\rm{coop}}$ is defined by the sum of two terms; $\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{coop,br}}=$ $\displaystyle\lambda_{\rm{br}}\sum_{\langle ij\rangle}x_{{\rm br},i}x_{{\rm br},j},$ (15) $\displaystyle\mathcal{H}_{\rm{ph}}^{\rm{coop,JT}}=$ $\displaystyle-\lambda_{\rm{JT}}\sum_{\langle ij\rangle}x_{{\rm JT},i}x_{{\rm JT},j}-\bar{\lambda}_{\rm{JT}}\sum_{\langle ij\rangle}\bar{x}_{{\rm JT},i}\bar{x}_{{\rm JT},j},$ (16) with the coupling constants $\lambda_{\rm{br}}$, $\lambda_{\rm{JT}}$, and $\bar{\lambda}_{\rm{JT}}$. Although the values of $\lambda_{\rm{JT}}$ and $\bar{\lambda}_{\rm{JT}}$ are generally different, we take $\lambda_{\rm{JT}}=\bar{\lambda}_{\rm{JT}}$ for simplicity. It is reasonable to assume $\lambda_{\rm{br}}$ to be positive, i.e., the “antiferro”-type coupling, since the $A_{1g}$-type expansion (contraction) of NiO6 octahedron tends to shrink (expand) the neighboring octahedra due to the edge-sharing network of octahedra. On the other hand, the tendency is opposite for the $E_{g}$ modes; the Jahn-Teller distortion of an octahedra induces the same distortion in the neighboring octahedra. Hence we consider the “ferro”-type coupling $\lambda_{\rm{JT}}>0$ in the following study. In the following calculations, the “antiferro”-type $A_{1g}$ coupling tends to stabilize a charge ordering by differentiating charge density between the neighboring sites, while the “ferro”-type Jahn-Teller coupling favors “ferro”-type orbital ordering. ### 2.2 Method In order to study the ground state of the model (1) in a wide range of parameters, we adopt the Hartree-Fock approximation to decouple the electron- electron interactions, and the adiabatic approximation to treat the electron- phonon couplings. Within the Hartree-Fock approximation, the two-body interaction terms in $\mathcal{H}_{\mathrm{int}}$ are decoupled by introducing mean fields, $\langle c_{i\alpha\sigma}^{\dagger}c_{i\alpha^{\prime}\sigma^{\prime}}\rangle$. The amplitudes of lattice distortions are determined by the adiabatic approximation. Within this approximation, the equilibrium values of $x_{{\rm br},i}$, $x_{{\rm JT},i}$, and $\bar{x}_{{\rm JT},i}$ are determined by using the Hellmann-Feynman theorem as $\Big{\langle}\frac{\partial\mathcal{H}}{\partial x_{{\rm br},i}}\Big{\rangle}=\Big{\langle}\frac{\partial\mathcal{H}}{\partial x_{{\rm JT},i}}\Big{\rangle}=\Big{\langle}\frac{\partial\mathcal{H}}{\partial\bar{x}_{{\rm JT},i}}\Big{\rangle}=0.$ (17) These relations lead to the set of equations in the form $\displaystyle x_{{\rm br},i}+\lambda_{\rm{br}}{\sum_{j}}^{\prime}x_{{\rm br},j}=$ $\displaystyle\gamma_{\rm{br}}\left(\langle n_{ia}\rangle+\langle n_{ib}\rangle-1\right),$ (18) $\displaystyle x_{{\rm JT},i}-\lambda_{\rm{JT}}{\sum_{j}}^{\prime}x_{{\rm JT},j}=$ $\displaystyle\gamma_{\rm{JT}}\sum_{\sigma}\left(\langle n_{ia\sigma}\rangle-\langle n_{ib\sigma}\rangle\right),$ (19) $\displaystyle\bar{x}_{{\rm JT},i}-\lambda_{\rm{JT}}{\sum_{j}}^{\prime}\bar{x}_{{\rm JT},j}=$ $\displaystyle\gamma_{\rm{JT}}\sum_{\sigma}(\langle c_{ia\sigma}^{\dagger}c_{ib\sigma}\rangle+\langle c_{ib\sigma}^{\dagger}c_{ia\sigma}\rangle),$ (20) where the sum $\sum^{\prime}_{j}$ is taken over the nearest neighbors of the site $i$. We determine the mean fields and the lattice distortions in a self-consistent way. For a given set of $\\{\langle c_{i\alpha\sigma}^{\dagger}c_{i\alpha^{\prime}\sigma^{{}^{\prime}}}\rangle,x_{{\rm br},i},x_{{\rm JT},i},\bar{x}_{{\rm JT},i}\\}$, we diagonalize the Hamiltonian under the Hartree-Fock approximation and obtain one-particle eigenenergies and eigenstates, which are used to calculate a new set of $\\{\langle c_{i\alpha\sigma}^{\dagger}c_{i\alpha^{\prime}\sigma^{{}^{\prime}}}\rangle\\}$. These new mean fields are substituted in Eqs. (18)-(20) to determine the new set of $\\{x_{{\rm br},i}$, $x_{{\rm JT},i}$, $\bar{x}_{{\rm JT},i}\\}$. These procedures are repeated until the convergence is reached within the precision less than $10^{-4}$ for all the variables. In the following calculations, we take the unit cell which includes six Ni3+ sites in the triangular lattice as shown in Fig. 3. To incorporate different orbital orderings depending on the way of taking the $\gamma$ direction in Fig. 1, we consider two different ways of embedding the unit cell as shown in Fig. 3. These unit cells accommodate a $\sqrt{3}\times\sqrt{3}$ charge ordering and a two-sublattice ordering such as a stripe-type antiferromagnetic state. Note that the ordering patterns observed in NaNiO2 and AgNiO2 are both included by taking the unit cells. For the initial state in the iteration, we consider more than 30 different states with different symmetry in spin, orbital, and charge sectors, which are relevant in the parameter space we study. The different initial configurations are adopted for each type of the unit cells in Fig. 3. The integration over the wave number in the calculations of the mean fields is replaced by the sum over $24\times 24$ grids in the Brillouin zone for the supercell. Hereafter, we focus on the quarter-filling case, i.e., one electron per site on average, corresponding to one $e_{g}$ electron in the low-spin state of Ni3+. Figure 3: Two different ways of taking the unit cell with six Ni sites on the triangular lattice used in the calculations. The axis $\gamma$ for the orbitals is shown (see also Fig. 1). ## 3 Results and Discussion In this section, we show the results obtained for the ground state of the model (1). In particular, we focus on the roles of $U$, $J_{\rm H}/U$, and $\gamma_{\rm{JT}}$, and discuss how these parameters affect the ground state. We take $\gamma_{\rm{br}}=1.6$ and $\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$, which result in reasonable energy gain in forming CO, as we will see below. As a result, we find that the tendency to charge ordering becomes pronounced by a compromise among $U$, $U^{\prime}$, $J_{\rm H}$, and $\gamma_{\rm br}$. On-site repulsions $U$ and $U^{\prime}$ suppress charge disproportionation, while the Hund’s-rule coupling $J_{\rm H}$ as well as the breathing-type coupling $\gamma_{\rm br}$ works as an inter-orbital effective attractive interaction and promotes charge disproportionation. On the other hand, for larger $U$ and $U^{\prime}$, the system becomes insulating, and the Jahn- Teller coupling $\gamma_{\rm{JT}}$ becomes important and enhances the tendency to orbital ordering, concomitant with magnetic ordering. In order to characterize the orbital-ordered phases, we introduce the pseudospin operators in the orbital sector, defined as $\boldsymbol{\tau}_{i}\equiv\sum_{\sigma}c^{{\dagger}}_{i\alpha\sigma}\boldsymbol{\sigma}_{\alpha\beta}c_{i\beta\sigma},$ (21) where $\boldsymbol{\sigma}_{\alpha\beta}$ denotes the pauli matrix. For example, $\tau_{z}$-OF means a ferro-type order of $\tau_{iz}$, which is the $z$ component of $\boldsymbol{\tau}_{i}$. The main result is summarized as the phase diagrams shown in Fig. 4. In the following, we will focus on the $\sqrt{3}\times\sqrt{3}$-type charger-ordered metallic (COM) phase found in the weak-coupling region [Fig. 4(a)], and the orbital-ferro spin-ferro insulating ($\tau_{z}$-OF SF I) phase obtained in the intermediately- to strongly-correlated region [Figs. 4(b) and 4(c)]. These two phases are candidates for the low-$T$ states of AgNiO2 and NaNiO2, respectively. We will identify the parameter range for these phases, and discuss the origin and the nature of them in the following. Figure 4: (Color online). Ground-state phase diagrams for the model (1) in the plane of $J_{\rm H}/U$ and $\gamma_{\rm JT}$ at (a) $U=4$, (b) $U=8$, and (c) $U=20$. We take $t^{\prime}=-1$, $\gamma_{\rm br}=1.6$, and $\lambda_{\rm br}=\lambda_{\rm JT}=0.05$. The ordering patterns of each phase are shown in Fig. 5. Figure 5: Schematic pictures of charge, spin, and orbital ordering patterns in the six-site unit cell for the phases in Fig. 4. The size of the circles schematically indicates the charge density at each Ni site. The length of the arrows denotes the magnitude of the spin moment at each site. The open circles denote the orbital para state, while the filled or shaded circles show orbitally polarized states. The different patterns denote different orbitally polarized states. See the text for details of (a)-(d). Both $\tau_{z}$-OF SF I and $\tau_{y}$-OF SF are represented by Fig. 5(b) because these phases have the same symmetry (OF SF). Both OF SAF I and $\tau_{x}$-OF SAF M (I) are represented by Fig. 5(e). Typical values of of the charge disproportionation and orbital polarization in (e)-(i) are as follows. (e) OF SAF I: $\langle\tau_{x}\rangle\simeq\langle\tau_{z}\rangle\simeq 0.6$ at all sites. $\tau_{x}$-OF SAF M: $\langle\tau_{x}\rangle\simeq 0.2$ and $\langle\tau_{z}\rangle\simeq-0.06$ at all sites. (f): $\langle n\rangle\simeq 3.0$ and $\langle n\rangle\simeq 1.7$ at the two charge-rich sites and $\langle n\rangle\simeq 0.3$ at the other charge-poor sites. (g): $\langle\tau_{x}\rangle\simeq 0.2$, $\langle\tau_{z}\rangle\simeq 0.7$, and $\langle n\rangle\simeq 1.2$ at the charge rich sites; $\langle\tau_{x}\rangle\simeq 0.1$, $\langle\tau_{z}\rangle\simeq-0.2$, and $\langle n\rangle\simeq 0.8$ at the charge poor sites. (h): $\langle\tau_{x}\rangle\simeq 1.4$, $\langle\tau_{z}\rangle\simeq 0.6$, and $\langle n\rangle\simeq 1.7$ at the charge rich sites and $\langle\tau_{x}\rangle\simeq 0.2$, $\langle\tau_{z}\rangle\simeq 0.0$, and $\langle n\rangle\simeq 0.3$ at the charge poor sites. (i): $\langle n\rangle\simeq 1.02$ at the charge rich sites (1 and 4 in Fig. 3), $\langle n\rangle\simeq 0.98$ at the charge poor sites (3 and 6), and $\langle n\rangle\simeq 1.00$ at the other charge-neutral sites (2 and 5). ### 3.1 Weakly-correlated region #### 3.1.1 Phase diagram A tendency toward charge ordering is widely observed in a weakly-correlated region. Figure 4(a) shows the phase diagram at $U=4$. Among the competing phases, a COM phase with $\sqrt{3}\times\sqrt{3}$-type charge ordering is stabilized in a wide range of $\gamma_{\rm JT}$ at $J_{\rm H}/U\sim 0.25$. The $\sqrt{3}\times\sqrt{3}$ charge ordering is a three-sublattice order, in which one is charge rich (the density is almost two) and the other two charge poor (the density is almost 0.5 per site). This $\sqrt{3}\times\sqrt{3}$ COM state, shown in Fig. 5(a), is remarkable, since it has the same charge ordering pattern as the low-$T$ state of AgNiO2 [26, 27]. The spin state is also interesting; large moments appear at charge-rich sites ($S\simeq 0.6$), while the moments are suppressed at charge-poor sites ($S\simeq 0.05$). A similar differentiation was proposed for the low-$T$ state of AgNiO2 [26, 27], although the calculated spin pattern does not fully agree with the experimental result. We note that importance of interlayer coupling is experimentally suggested for the magnetic ordering [28, 26, 27], which is not taken into account in our model. We will discuss the nature of this COM phase in Sec. 3.1.3 in detail. Around $J_{\rm{H}}/U=0.24$ and $\gamma_{\rm{JT}}=1.1$, we find another COM phase, i.e., the sixfold COM phase. Although this phase is stabilized in a narrow region in the phase diagram, it remains to be metastable in a wide parameter range, with a slightly higher energy than the ground state, as we will discuss in the next section 3.1.2. This phase has the consistent ordering structure with the low-$T$ state of AgNiO2 in terms of both charge and spin, as shown in Fig. 5(c). Strictly speaking, the charge pattern of this phase has lower symmetry compared with AgNiO2, due to the superposition of stripe-type charge modulation onto the $\sqrt{3}\times\sqrt{3}$ charge ordering. However, the magnitude of this modulation is very small: For charge rich sites (sites 1 and 4 in Fig. 3), we have $\langle n_{1}\rangle=1.5040$ and $\langle n_{4}\rangle=1.5027$, while for charge poor sites (sites 2, 3, 5, and 6 in Fig. 3), we have $\langle n_{3}\rangle=\langle n_{5}\rangle=0.7479$ and $\langle n_{2}\rangle=\langle n_{6}\rangle=0.7487$, at $J_{\rm H}/U=0.24$ and $\gamma_{\rm{JT}}=1.1$. The modulation gives very small charge disproportionation of the order of $\sim 0.001$ within charge rich and poor sites. [The small disproportionations are exaggerated in the schematic picture in Fig. 5(c).] In addition to the two COM states, we obtain a variety of ordered phases in the weakly-correlated region. Among them, we focus on two phases which compete with the COM states; the sixfold charge-ordered insulating (sixfold COI) phase in the large $J_{\rm H}/U$ region, and the spin-ferro metallic phase with a weak charge ordering (SF-COM) stabilized for smaller $J_{\rm H}/U$ [Fig. 4(a)]. We argue the stability of the $\sqrt{3}\times\sqrt{3}$ COM as well as the sixfold COM in comparison with the two competing phases in the next section. #### 3.1.2 Stability of the $\sqrt{3}\times\sqrt{3}$ COM phase In order to clarify the competition among the $\sqrt{3}\times\sqrt{3}$ COM, sixfold COM, sixfold COI, and SF-COM phases, we investigate the internal energy in detail by comparing the contributions from different terms in the Hamiltonian; $E_{\rm{kin}}\equiv\langle\mathcal{H}_{\rm{kin}}\rangle$, $E_{U}\equiv\langle\mathcal{H}_{U}\rangle$, $E_{U^{\prime}}\equiv\langle\mathcal{H}_{U^{\prime}}\rangle$, $E_{J_{\rm{H}}}\equiv\langle\mathcal{H}_{J_{\rm{H}}}+\mathcal{H}_{J_{\rm{H}}^{\prime}}\rangle$, $E_{\rm{br}}\equiv\langle\mathcal{H}_{\text{el- ph}}^{\rm{br}}+\mathcal{H}_{\rm{ph}}^{\rm{elestic,br}}+\mathcal{H}_{\rm{ph}}^{\rm{coop,br}}\rangle$, and the total energy $E_{\rm{tot}}\equiv\langle\mathcal{H}\rangle$. We show the comparison as a function of $U$ at $J_{\rm H}/U=0.27$ and $\gamma_{\rm JT}=0.5$ in Fig. 6. For $U\lesssim 3.9$, the sixfold COI state has the lowest energy. As shown in Fig. 5(f), this phase has a polaronic nature, namely, one site is almost fully occupied (the local density is almost 4), and another one site accommodates almost two electrons. At the latter site, spins of two electrons are aligned parallel by the Hund’s-rule coupling. This phase is stabilized in a region where the repulsive Coulomb interactions are compensated by effective attractive interactions originating in the breathing-type electron-phonon coupling as well as the inter-orbital Hund’s-rule coupling. In fact, it is clearly observed in Figs. 6(d) and 6(e) that the energy gain in $E_{J_{\rm{H}}}$ and $E_{\rm{br}}$ contributes to the stabilization of the sixfold COI phase. On the other hand, for $U\gtrsim 4.9$, the SF-COM state is most stabilized. As schematically shown in Fig. 5(d), the charge ordering in this phase is a stripe type, but the charge disproportionation is very small ($\langle n\rangle\sim 1.03-1.09$ at charge rich sites, while $\langle n\rangle\sim 0.91-0.97$ at charge poor sites): the main feature is the spin ferromagnetic ordering. The origin of this phase can be attributed to the Stoner mechanism [35]. As shown in the inset of Fig. 7, the non-interacting Fermi level is located in the vicinity of the steep peak of the density of states (DOS). Consequently, a ferromagnetic instability is caused at a relatively small $U\simeq 3.4$, according to the Stoner’s criterion. The characteristics of Stoner ferromagnet are observed in the energy comparison in Fig. 6(c); $E_{U}$ becomes smallest among the competing phases. The $\sqrt{3}\times\sqrt{3}$ COM state intervenes these two, and has the lowest energy for $4.0\lesssim U\lesssim 4.8$. In the same parameter range, the sixfold COM state appears as a metastable state and stays very close to the ground state, as shown in the inset of Fig. 6(a) (the energy difference is less than 0.02). These two COM states are stabilized by a compromise between the different stabilization mechanisms for the sixfold COI and the SF-COM phases. According to Fig. 6, the COM phases have higher (lower) $E_{J_{\rm{H}}}$ and $E_{\rm{br}}$, while they have lower (higher) $E_{U}$ and $E_{U^{\prime}}$, compared with the sixfold COI phase (the SF-COM phase). Namely, the two COM phases are stabilized in a subtle balance between repulsive Coulomb interactions and effective attractive interactions due to the Hund’s-rule coupling and the breathing-type electron-phonon coupling. Since the COM phases are stabilized in a delicate compromise, it is important to consider their stability against the elements which are ignored in our current analysis, such as fluctuations beyond the mean-field level and the long-range part of electron interactions. First, we note that the Stoner ferromagnetism is fragile when considering the electron correlation effect beyond the mean-field approximation[36]. Therefore, the COM phases are expected to extend to larger $U$ or smaller $J_{\rm H}/U$. Second, the amplitudes of breathing-type distortions are fairly large in the sixfold COI phase compared to those in the other phases. Hence this phase will be suppressed by considering more realistic contributions from phonons, e.g., anharmonic terms of phonons. This may give a chance for the COM phases to become wider also in smaller $U$ or larger $J_{\rm H}/U$ regions. Finally, the long-range part of electron interactions generally works in favor of the charge ordering, in particular, the $\sqrt{3}\times\sqrt{3}$ type and the sixfold COM, as is discussed in several transition metal compounds and organic materials [37]. Therefore, we expect that the COM phases relevant to AgNiO2 become more stable in a wider parameter range when extending the analyses beyond the present model and method. Although the COM phases are robust in this parameter region, the energy difference between the sixfold COM phase and the $\sqrt{3}\times\sqrt{3}$ COM phase is very small, implying that the magnetic ordering pattern might be affected by small perturbations, such as inter-layer coupling. More accurate studies beyond the mean-field approximation are necessary for fully determining the spin pattern of the ground state. Figure 6: (Color online). Energy comparisons among the $\sqrt{3}\times\sqrt{3}$ COM (circle), sixfold COM (triangle), sixfold COI (square), and SF-COM states (diamond): $U$ dependences of (a) the total energy, and the contribution from (b) kinetic term, (c) Coulomb repulsions [closed (open) symbols denote $E_{U}$ ($E_{U^{\prime}}$)], (d) Hund’-rule coupling, and (e) breathing-type electron-phonon coupling. The inset of (a) shows the energy differences between the $\sqrt{3}\times\sqrt{3}$ COM phase and the other competing phases. The parameters are $t^{\prime}=-1$, $J_{\rm{H}}/U=0.27$, $\gamma_{\rm{br}}=1.6$, $\gamma_{\rm{JT}}=0.5$, and $\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$. #### 3.1.3 Nature of the $\sqrt{3}\times\sqrt{3}$ COM phase Reflecting the subtle balance between the attractive and repulsive interactions, the $\sqrt{3}\times\sqrt{3}$ COM phase shows peculiar electronic properties. The density of states (DOS) in the $\sqrt{3}\times\sqrt{3}$ COM phase is shown in Fig. 7. The site- and spin-resolved DOS indicates that the system exhibits a half-metallic nature: Up-spin electrons are localized at charge-rich sites, showing a gap at the Fermi level, on the other hand, down- spin electrons remain conductive, with a finite DOS at the Fermi level. Electron correlations dominantly affect up-spin electrons; down-spin conductive electrons preserve the non-interacting band structure. For comparison, we show DOS for the non-interacting case in the inset of Fig. 7, which is quite similar to DOS of down-spin conductive electrons. DOS in the $\sqrt{3}\times\sqrt{3}$ COM phase reproduces several aspects of the result obtained by the first-principle band calculation [26]: The electrons at charge rich site tend to localize and the electronic structure at the other two charge poor sites resembles each other. This peculiar electronic state can be attributed to the delicate balance between the attractive and repulsive interactions. We plot the effective one- body potential in Fig. 8, which is defined as the sum of the terms in $\mathcal{H}_{\rm{int}}$ and $\mathcal{H}_{\text{el-ph}}$, which couple to the density operator $n_{i}$ at each site under the Hartree-Fock approximation. Figure 8 shows that the charge-rich (-poor) sites bear attractive (repulsive) potentials for up-spin electrons. In contrast, the cancellation between the breathing-type electron-phonon coupling and the repulsive Coulomb interactions leads to an almost flat potential for down-spin electrons. Consequently, these interactions only work as a shift of chemical potential, and the down-spin electrons retain the non-interacting band structure. To conclude the discussions for the weakly-correlated region, the $\sqrt{3}\times\sqrt{3}$ COM phase is stabilized by a compromise between repulsive Coulomb interactions and attractive interactions originating from the breathing-type electron-phonon coupling as well as the Hund’s-rule coupling. This phase shows a half-metallic behavior with large magnetic moments almost localized at charge-rich sites and conduction electrons moving almost freely in the entire lattice. We note that this phase is distinguished from the so-called pinball liquid state, in which the electrons at the charge- rich sites exclude the conduction electrons as hard core potentials and confine them to the honeycomb network of charge-poor sites, as discussed in a spinless tight-binding model with intersite Coulomb repulsion on the triangular lattice [38]. Figure 7: (Color online). DOS per site for the $\sqrt{3}\times\sqrt{3}$ COM phase. The Fermi level is set to be the origin of energy. As to the $\sqrt{3}\times\sqrt{3}$ COM phase, the up-spin (down-spin) DOS is drawn on the upper (lower) side. The parameters are chosen as $U=4$, $J_{\rm{H}}/U=0.27$, $\gamma_{\rm{br}}=1.6$, $\gamma_{\rm{JT}}=0.5$, and $\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$. The inset shows DOS for the non- interacting case. Figure 8: (Color online). (a) Effective one-body potentials at each site for up-spin and down-spin electrons (see the text for details). The horizontal axis denotes the site indices in the unit cell, shown in Fig. 3. The charge-rich (-poor) sites are the site 1 and 4 (2, 3, 5, and 6). The parameters are chosen as $U=4$, $J_{\rm{H}}/U=0.27$, $\gamma_{\rm{br}}=1.6$, $\gamma_{\rm{JT}}=0.5$, and $\lambda_{\rm{br}}=\lambda_{\rm{br}}=0.05$, consistent with Fig. 7. ### 3.2 Intermediately- to strongly-correlated region Next, let us consider the intermediately- to strongly-correlated region. Representative phase diagrams are shown in Figs. 4(b) and 4(c). #### 3.2.1 Phase diagram We show the ground-state phase diagram at $U=8$ in Fig. 4(b). Note that the value of $U$ is comparable with the non-interacting bandwidth., i.e., the system is in the intermediately-correlated region. In this region, we find three dominant orbital-ordered phases; $\tau_{y}$-ordered spin-ferro insulator ($\tau_{y}$-OF SF I), $\tau_{x}$-ordered spin-antiferro insulator ($\tau_{x}$-OF SAF I), and $\tau_{z}$-ordered spin-ferro insulator ($\tau_{z}$-OF SF I). The ordering pattern of $\tau_{y}$\- or $\tau_{z}$-OF SF I ($\tau_{x}$-OF SAF I) is schematically shown in Fig. 5(b) [Fig. 5(e)]. For the small $\gamma_{\rm JT}$ and $J_{\rm H}/U$ region, $\tau_{y}$-OF SF I is stabilized, while it is replaced by $\tau_{x}$-OF SAF I for larger $\gamma_{\rm JT}$ or by a charge-ordered state for larger $J_{\rm H}/U$. Meanwhile, when both $\gamma_{\rm JT}$ and $J_{\rm H}/U$ become large, $\tau_{z}$-OF SF I is stabilized. Among these phases, the $\tau_{z}$-OF SF I phase deserves attention, since the spin and orbital pattern of this phase is consistent with the low-$T$ phase of NaNiO2. Remarkably, the three OF phases remain stable in a wide range of $U$ toward the strongly-correlated regime. Figure 4(c) shows an example of the phase diagram at large $U$. The result indicates that the three phases remain in similar parameter regions of $\gamma_{\rm JT}$ and $J_{\rm H}/U$ compared to the intermediate-$U$ case. This implies a possibility to understand the origin of these phases from the strong-coupling analysis, i.e., by starting from the Mott insulating state at $U=\infty$. In fact, the $\tau_{y}$-OF SF I state was obtained for an effective spin- orbital model in the strong-coupling limit (F3 phase of Fig. 8 in Ref. Vernay). Our result is consistent with the previous study. Meanwhile, the competition between the $\tau_{x}$-OF SAF I phase and the $\tau_{z}$-OF SF I phase is obtained for the first time by explicitly taking account of electron- phonon couplings. In the following, we will consider the mechanism of stabilization of these phases through the detailed study of energetics. #### 3.2.2 Stability of the $\tau_{z}$-OF SF insulating phase In order to understand the stability condition, it is instructive to rewrite $\mathcal{H}^{\rm{JT}}_{\text{el-ph}}$, $\mathcal{H}_{J_{\rm{H}}}$, and $\mathcal{H}_{J^{\prime}_{\rm{H}}}$ by using the pseudospin operators in eq. (21) as $\mathcal{H}_{\text{el-ph}}^{\rm{JT}}=-\gamma_{\rm{JT}}\sum_{i}(x_{{\rm JT},i}\tau_{iz}+\bar{x}_{{\rm JT},i}\tau_{ix}),$ (22) $\mathcal{H}_{J_{\rm{H}}}+\mathcal{H}_{J_{\rm{H}}^{\prime}}=\frac{J_{\rm H}}{2}\sum\limits_{i}\\{\tau_{ix}^{2}-(n_{ia}+n_{ib})\\}.$ (23) These equations clearly show that the Jahn-Teller coupling stabilizes the $\tau_{x}$ and $\tau_{z}$ orbital orderings, while the Hund’s-rule coupling destabilizes the $\tau_{x}$ orbital ordering. In Fig. 9, we compare the energy contributions including these terms, $E_{\rm{JT}}\equiv\langle\mathcal{H}_{\text{el- ph}}^{\rm{JT}}+\mathcal{H}_{\rm{ph}}^{\rm{elastic,JT}}+\mathcal{H}_{\rm{ph}}^{\rm{coop,JT}}\rangle$ and $E_{J_{\rm H}}$, together with other relevant energy contributions, for the three orbital-ordered phases. Figure 9(b) shows that the stability of the $\tau_{y}$-OF SF I phase in the small $\gamma_{\rm{JT}}$ region is attributed to the energy gain in the kinetic energy $E_{\rm{kin}}$. This is consistent with the result of strong- coupling analysis, where the kinetic energy gain through the spin-orbital superexchange interactions is claimed to be the origin of this phase. [33] It is also observed that all energy contributions in this phase are fairly insensitive to $\gamma_{\rm JT}$, as shown in Fig. 9. This is expected from the absence of coupling between $\tau_{y}$ and Jahn-Teller distortions, as is clear from eq. (22). On the other hand, the $\tau_{x}$-OF SAF I and $\tau_{z}$-OF SF I states lower their energy through the coupling to the Jahn-Teller distortions [Fig. 9(e)], as expected from eq. (22); thus they replace the $\tau_{y}$-OF SF I phase and become the ground state for larger $\gamma_{\rm JT}$, as shown in Fig. 9(a). The Hund’s-rule coupling plays an important role in the relative stability between the $\tau_{x}$-OF SAF I and the $\tau_{z}$-OF SF I phases. As is evident from eq. (23), the Hund’s-rule coupling affects only the $\tau_{x}$ ordering, and destabilizes it [Fig. 9(d)]. Furthermore, the $\tau_{z}$-OF SF I phase is stabilized by the kinetic energy gain from the interorbital hoppings, compared with the $\tau_{x}$-OF SAF I phase as shown in Fig. 9(b). In fact, according to the second-order perturbation from the strong coupling limit $U\to\infty$, the $\tau_{z}$-OF SF I phase has lower energy than $\tau_{x}$-OF SAF I phase for $J_{\rm H}/U>\frac{8-\sqrt{10}}{18}\simeq 0.27$. The phase boundary between the two phases in Fig. 4(c) is roughly located around this critical value, which indicates that the phase competition is essentially understood from the strong coupling picture. Figure 9: (Color online). $\gamma_{\rm JT}$ dependences of (a) the total energy, and the contribution from (b) kinetic term, (c) Coulomb repulsions [closed (open) symbols denote $E_{U}$ ($E_{U^{\prime}}$)], (d) Hund’s-rule coupling, and (e) Jahn-Teller coupling. Comparison is made for the $\tau_{z}$-OF SF I (circle), $\tau_{x}$-OF SAF I (square), and $\tau_{y}$-OF SF I states (triangle). The parameters are $t^{\prime}=-1$, $U=20$, $J_{\rm{H}}/U=0.25$, $\gamma_{\rm{br}}=1.6$, and $\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$. #### 3.2.3 Nature of the $\tau_{z}$-OF SF insulating phase Due to the strong electron repulsion, an excitation gap opens at the Fermi level in the $\tau_{z}$-OF SF I phase. Hence, this state is insulating, consistent with the low-$T$ insulating phase of NaNiO2. Figure 10 shows DOS in the $\tau_{z}$-OF SF I phase. DOS is composed of four sectors (two in each spin component), and the total weight of each sector is equal to one electron per site, as indicated in the integrated DOS in the figure. This structure can be qualitatively understood by considering the atomic limit with ignoring the hoppings, $t$ and $t^{\prime}$. Let us assume the perfect $\tau_{z}$-OF SF order, i.e., $\langle n_{i,3z^{2}-r^{2},\uparrow}\rangle=1$ and let other mean fields to be zero. Then the excitation energies in the atomic limit are estimated as $\displaystyle E_{x^{2}-y^{2},\uparrow}$ $\displaystyle=U^{\prime}-J_{\rm{H}}-2E^{*}_{\rm{JT}},$ (24) $\displaystyle E_{x^{2}-y^{2},\downarrow}$ $\displaystyle=U^{\prime}-2E^{*}_{\rm{JT}},$ (25) $\displaystyle E_{3z^{2}-r^{2},\downarrow}$ $\displaystyle=U+2E^{*}_{\rm{JT}},$ (26) where $E_{\alpha\sigma}$ signifies the energy necessary to add one electron with orbital $\alpha$ and spin $\sigma$ to the $(3z^{2}-r^{2},\uparrow)$ ground state. $E^{*}_{\rm{JT}}$ is the energy gain from the Jahn-Teller distortion per site, estimated as $E^{*}_{\rm{JT}}=-\frac{\gamma_{\rm{JT}}^{2}}{2(1-6\lambda_{\rm{JT}})}.$ (27) Substituting the parameters used in Fig. 10 ($U=20$, $U^{\prime}=10$, $J_{{\rm H}}=5$, $\gamma_{\rm JT}=1.0$, and $\lambda_{\rm JT}=0.05$) into these equations, we obtain $E_{x^{2}-y^{2},\uparrow}\simeq 6.4$, $E_{x^{2}-y^{2},\downarrow}\simeq 11$, and $E_{3z^{2}-r^{2},\downarrow}\simeq 19$. These values well correspond to the mean energy of each sector of DOS measured from that for the lowest one in Fig. 10. Eq. (24) gives an estimate of the energy gap $\Delta$ in the atomic limit. This atomic value is reduced for finite $t$ and $t^{\prime}$, since each atomic level is broadened by a renormalized bandwidth $\tilde{W}$: The estimate of energy gap is modified as $\Delta\sim E_{x^{2}-y^{2},\uparrow}-\tilde{W}$. We use this simple estimate with replacing $\tilde{W}$ by the bare bandwidth $W=8t$ for a comparison with the Hartree-Fock solutions. As shown in Fig. 11, our simple estimate from the atomic limit is qualitatively consistent with Hartree-Fock results. This fact also supports that the electronic spectrum of the $\tau_{z}$-OF SF I phase is adiabatically continued from the strong-coupling limit. Figure 10: (Color online). DOS for the $\tau_{z}$-OF SF I state. The parameters are $t^{\prime}=-1.0$, $U=20$, $J_{\rm{H}}/U=0.25$, $\gamma_{\rm{JT}}=1.0$, $\gamma_{\rm{br}}=1.6$, and $\lambda_{\rm{br}}=\lambda_{\rm{JT}}=0.05$. The vertical line denotes the Fermi level. The up-spin (down-spin) DOS is represented on the upper (lower) side. The integrated DOS is also shown. Figure 11: (Color online). Energy gap in the $\tau_{z}$-OF SF I phase as a function of $U$ at $t^{\prime}=-1.0$, $J_{\rm{H}}/U=0.25$, $\gamma_{\rm{JT}}=1.0$, $\gamma_{\rm{br}}=1.6$, and $\lambda_{\rm{br}}=\lambda_{\rm{br}}=0.05$. The line is the simple estimate from the strong-coupling analysis. See the text for details. To conclude this part, three orbital-ordered insulating phases appear dominantly in the intermediately- to strongly-correlated region. Among them, the $\tau_{z}$-OF SF I phase, which is relevant to NaNiO2, becomes stable in the region where both the Jahn-Teller coupling and the Hund’-rule coupling are substantial. This phase is understood by the strong-coupling picture under the Jahn-Teller type electron-phonon couplings. ### 3.3 Comparison to experiments In our calculations, we successfully reproduce the $\sqrt{3}\times\sqrt{3}$ COM phase and the $\tau_{z}$-OF SF I phase, whose ordering patterns are consistent with the low-$T$ phases in AgNiO2 and NaNiO2, respectively. These two phases appear in close parameter regions of $\gamma_{\rm JT}$ and $J_{\rm H}/U$, but for different range of the on-site repulsion $U$. The $\sqrt{3}\times\sqrt{3}$ COM phase is stabilized in the weakly-correlated region, where $U$ is smaller than the bare bandwidth, while the $\tau_{z}$-OF SF I phase is stabilized in the intermediately- to strongly-correlated region. The difference in $U$ may be attributed to the structure of cation bands in these compounds. The magnitude of the effective on-site repulsion for Ni $3d$ electrons is not solely determined by its atomic value, but it is considerably affected by the screening effect brought about by electrons in the $A$ cation and oxygen $p$ bands. According to the first-principle calculations, Ag bands in AgNiO2 reside in the vicinity of the Fermi level [25, 26], while the Na bands in NaNiO2 are located about 4eV above the Fermi level [30, 31]. Consequently, a larger screening effect is expected for AgNiO2, which reduces the magnitude of $U$ considerably, compared to that for NaNiO2. Our results are consistent with this trend. In the first-principle calculations [30, 31, 25, 26], the bandwidth of Ni 3$d$ bands was roughly estimated to be $2\sim 3$eV, leading to a rough estimate of $t=0.25\sim 0.4$eV. In the LSDA+$U$ calculation for NaNiO2 [30, 31], the value of $U$ was taken to be 5eV to reproduce the correct size of excitation gap. On the other hand, a cluster- model analysis of the photoemission spectra gave an estimate of $U$=7.0eV [39]. These studies suggest that $U\simeq 10-30t$ is reasonable, consistent with our results. It is noteworthy that the gap in our calculation at $U=20t$ corresponds to $0.5\sim 0.8$eV, which is in the same order of magnitude as the experimental value $\sim 0.24$eV in NaNiO2 [9]. We also note that the CO stabilization energy, which is estimated from the energy difference between the $\sqrt{3}\times\sqrt{3}$ COM phase and a para phase, is $\sim$ $0.05t$ for the present parameters $\gamma_{\rm br}$ and $\lambda_{\rm br}$: This result leads a rough estimate that the CO stabilization energy is $\sim$ 0.01 - 0.02 eV, which is in the same order of magnitude as the CO transition temperature observed in AgNiO2 (365K [26, 27]). Finally, we make a brief comment on the absence of any explicit ordering in LiNiO2. Since LiNiO2 is also a Mott insulator with a gap of 0.2eV [14], we suppose that the compound is in the strongly-correlated region similar to NaNiO2. In our results, there exists phase competition among three insulating phases with different spin and orbital patterns, the $\tau_{z}$-OF SF, $\tau_{x}$-OF SAF, and $\tau_{y}$-OF SF orderings. The competition brings about a frustration in the spin and orbital sectors in the vicinity of the phase boundaries. To argue the consequence of such frustration, we need to go beyond the present mean-field analysis; however, we can expect severe suppression of the orderings and some liquid-like or glassy behavior in the spin-orbital coupled system. Hence, one possibility is that LiNiO2 is located in such competing regime. It is noteworthy that the competition is brought about by explicitly taking account of the electron-phonon couplings, which have not been considered in the previous effective model approaches [20, 32, 33]. In addition to this intrinsic phase competition, extrinsic defects on Li sites may play an important role in the glassy behavior. Furthermore, a long- range strain effect might also play a role through the frustrating orbital and lattice sectors [24]. It is interesting to take account of these factors explicitly, by extending our model and analysis. We leave this problem for a future study. ## 4 Summary We have investigated the ground state of the multi-orbital Hubbard model with electron-phonon couplings by the Hartree-Fock approximation and the adiabatic approximation, in order to elucidate the origin of various phases observed in $A$NiO2 in a unified way. We found the $\sqrt{3}\times\sqrt{3}$ charge-ordered metallic phase in the weakly-correlated region and the orbital-ferro spin- ferro ordered insulating phase in the strongly-correlated region. The $\sqrt{3}\times\sqrt{3}$ charge-ordered metallic phase is stabilized by a compromise between Coulomb repulsions and effective attractive interactions from the breathing-type electron-phonon coupling as well as the Hund’s-rule coupling. The electronic state is half metallic; up-spin electrons are localized at the charge-rich sites, but down-spin electrons are extended and almost free. On the other hand, the orbital-ferro spin-ferro ordered insulating phase is stabilized by the Jahn-Teller coupling under strong electron correlation, with a help by the Hund’s-rule coupling in the competition with other orbital-ordered phases. These two phases are promising candidates for the low-$T$ phases in AgNiO2 and NaNiO2, respectively. A possible origin of the quite different electron repulsion between AgNiO2 and NaNiO2 might be a screening effect from the cation and oxygen $p$ bands. The puzzling glassy behavior in LiNiO2 might be ascribed to the competition among different spin and orbital ordered states in the strongly-correlated region, which occurs under a substantial Jahn-Teller type electron-phonon coupling. ## Acknowledgements The authors thank M. Imada, S. Watanabe, and Y. Yamaji for fruitful discussions. This work was supported by Grants-in-Aid for Scientific Research (No. 17071003, 17740244, 19014020, and 19052008), Global COE Program “the Physical Sciences Frontier”, the Next Generation Super Computing Project, and Nanoscience Program, from MEXT, Japan. ## References * [1] M. Imada, A. Fujimori, and Y. Tokura: Rev. Mod. Phys. 70 (1998) 1039. * [2] Y. Tokura and N. Nagaosa: Science 288 (2000) 462. * [3] K. I. Kugel and D. I. Khomskii: Sov. Phys. JETP 37 (1973) 725. * [4] K. I. Kugel and D. I. Khomskii: Sov. Phys. Solid State 17 (1975) 285. * [5] “Frustrated Spin Systems”, edited by H. T. 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arxiv-papers
2011-01-20T06:33:14
2024-09-04T02:49:16.542274
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hiroshi Uchigaito, Masafumi Udagawa, and Yukitoshi Motome", "submitter": "Hiroshi Uchigaito", "url": "https://arxiv.org/abs/1101.3828" }
1101.4028
# Who is the best player ever? A complex network analysis of the history of professional tennis Filippo Radicchi Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, US ###### Abstract We consider all matches played by professional tennis players between $1968$ and $2010$, and, on the basis of this data set, construct a directed and weighted network of contacts. The resulting graph shows complex features, typical of many real networked systems studied in literature. We develop a diffusion algorithm and apply it to the tennis contact network in order to rank professional players. Jimmy Connors is identified as the best player of the history of tennis according to our ranking procedure. We perform a complete analysis by determining the best players on specific playing surfaces as well as the best ones in each of the years covered by the data set. The results of our technique are compared to those of two other well established methods. In general, we observe that our ranking method performs better: it has a higher predictive power and does not require the arbitrary introduction of external criteria for the correct assessment of the quality of players. The present work provides a novel evidence of the utility of tools and methods of network theory in real applications. ## I Introduction Social systems generally display complex features castellano09 . Complexity is present at the individual level: the behavior of humans often obeys complex dynamical patterns as for example demonstrated by the rules governing electronic correspondence barabasi05 ; malmgren08 ; radicchi09a ; wu10 . At the same time, complexity is present also at the global level. This can be seen for example when social systems are mathematically represented in terms of graphs or networks, where vertices identify individuals and edges stand for interactions between pairs of social agents. Social networks are in most of the cases scale-free Barabasi1999 , indicating therefore a strong degree of complexity from the topological and global points of view. During last years, the analysis of social systems has become an important topic of interdisciplinary research and as such has started to be not longer of interest to social scientists only. The presence of a huge amount of digital data, describing the activity of humans and the way in which they interact, has made possible the analysis of large-scale systems. This new trend of research does not focus on the behavior of single agents, but mainly on the analysis of the macroscopic and statistical properties of the whole population, with the aim to discover regularities and universal rules. In this sense, professional sports also represent optimal sources of data. Soccer onody04 ; duch10 ; heuer10 , football girvan02 ; naim05 , baseball petersen08 ; sire09 ; saavedra09 ; Petersen2011 and basketball naim07 ; skinner10 are some remarkable cases in which network analysis revealed features not visible with traditional approaches. These are practical examples of the general outcome produced by the intense research activity of last years: network tools and theories do not serve only for descriptive purposes, but have also wide practical applicability. Representing a real system as a network allows in fact to have a global view of the system and simultaneously use the entire information encoded by its complete list of interactions. Particularly relevant results are those regarding: the robustness of networks under intentional attacks albert00 ; the spreading of viruses in graphs satorras01 ; synchronization processes arenas08 , social models castellano09 , and evolutionary and coevolutionary games Szabo2007 ; perc10 taking place on networks. In this context fall also ranking techniques like the PageRank algorithm Brin1 , where vertices are ranked on the basis of their “centrality” in a diffusion process occurring on the graph. Diffusion algorithms, originally proposed for ranking web pages, have been recently applied to citation networks price65 . The evaluation of the popularity of papers chen07 , journals west08 ; west10 and scientists radicchi09 is performed not by looking at local properties of the network (i.e., number of citations) but by measuring their degree of centrality in the flow of information diffusing over the entire graph. The use of the whole network leads to better evaluation criteria without the addition of external ingredients because the complexity of the citation process is encoded by the topology of the graph. Figure 1: Properties of the data set. In panel a, we report the total number of tournaments (top panel) and players (bottom panel) as a function of time. In panel b, we plot the fraction of players having played (black circles), won (red squares) and lost (blue diamonds) a certain number of matches. The black dashed line corresponds to the best power-law fit with exponent consistent with the value $1.2(1)$. In this paper we continue in this direction of research and present a novel example of a real system, taken from the world of professional sports, suitable for network representation. We consider the list of all tennis matches played by professional players during the last $43$ years ($1968$-$2010$). Matches are considered as basic contacts between the actors in the network and weighted connections are drawn on the basis of the number of matches between the same two opponents. We first provide evidence of the complexity of the network of contacts between tennis players. We then develop a ranking algorithm similar to PageRank and quantify the importance of tennis players with the so-called “prestige score”. The results presented here indicate once more that ranking techniques based on networks outperform traditional methods. The prestige score is in fact more accurate and has higher predictive power than well established ranking schemes adopted in professional tennis. More importantly, our ranking method does not require the introduction of external criteria for the assessment of the quality of players and tournaments. Their importance is self-determined by the various competitive processes described by the intricate network of contacts. Our algorithm does nothing more than taking into account this information. ## II Methods ### II.1 Data set Data were collected from the web site of the Association of Tennis Professionals (ATP, www.atpworldtour.com). We automatically downloaded all matches played by professional tennis players from January $1968$ to October $2010$. We restrict our analysis only to matches played in Grand Slams and ATP World Tour tournaments for a total of $3\,640$ tournaments and $133\,261$ matches. For illustrative purposes, in the top plot of the panel a of Figure 1, we report the number of tournaments played in each of the years covered by our data set. With the exception of the period between $1968$ and $1970$, when ATP was still in its infancy, about $75$ tournaments were played each year. Two periods of larger popularity were registered around years $1980$ and $1992$ when more than $90$ tournaments per year were played. The total number of different players present in our data set is $3\,700$, and in the bottom plot of panel a of Figure 1 we show how many players played at least one match in each of the years covered by our analysis. In this case, the function is less regular. On average, $400$ different players played in each of the years between $1968$ and $1996$. Large fluctuations are anyway visible and a very high peak in $1980$, when more than $500$ players participated in ATP tournaments, is also present. Between $1996$ and $2000$, the number of players decreased from $400$ to $300$ in an almost linear fashion. After that, the number of participants in ATP tournaments started to be more constant with small fluctuations around an average of about $300$ players. ### II.2 Network representation Figure 2: Top player network and scheme for a single tournament. In panel a, we draw the subgraph of the contact network restricted only to those players who have been number one in the ATP ranking. Intensities and widths are proportional to the logarithm of the weight carried by each directed edge. In panel b, we report a schematic view of the matches played during a single tournament, while in panel c we draw the network derived from it. We represent the data set as a network of contacts between tennis players. This is a very natural representation of the system since a single match can be viewed as an elementary contact between two opponents. Each time the player $i$ plays and wins against player $j$, we draw a directed connection from $j$ to $i$ [$j\to i$, see Figure 2]. We adopt a weighted representation of the contacts Barrat2004 , by assigning to the generic directed edge $j\to i$ a weight $w_{ji}$ equal to the number of times that player $j$ looses against player $i$. Our data are flexible and allow various levels of representation by including for example only matches played in a certain period of time, on a certain type of surface, etc. An example is reported in panel a of Figure 2 where the network of contacts is restricted only to the $24$ players having been number one in the official ATP ranking. In general, networks obtained from the aggregation of a sufficiently high number of matches have topological complex features consistent with the majority of networked social systems so far studied in literature Reka:RevModPhys2002 ; Newman2003 . Typical measures revealing complex structure are represented by the probability density functions of the in- and out-strengths of vertices Barrat2004 , both following a clear power-law behavior [see Figure 1, panel b]. In our social system, this means that most of the players perform a small number of matches (won or lost) and then quit playing in major tournaments. On the other hand, a small set of top players performs many matches against worse opponents (generally beating them) and also many matches (won or lost) against other top players. This picture is consistent with the so-called “Matthew effect” in career longevity recently observed also in other professional sports petersen08 ; Petersen2011 . ### II.3 Prestige score The network representation can be used for ranking players. In our interpretation, each player in the network carries a unit of “tennis prestige” and we imagine that prestige flows in the graph along its weighted connections. The process can be mathematically solved by determining the solution of the system of equations $P_{i}\;=\;\left(1-q\right)\,\sum_{j}\,P_{j}\,\frac{w_{ji}}{s_{j}^{out}}\;+\;\frac{q}{N}\;+\;\frac{1-q}{N}\,\sum_{j}\,P_{j}\,\delta\left(s_{j}^{out}\right)\;\;,$ (1) valid for all nodes $i=1,\ldots,N$, with the additional constraint that $\sum_{i}P_{i}=1$. $N$ indicates the total number of players (vertices) in the network, while $s_{j}^{out}=\sum_{i}\,w_{ji}$ is the out-strength of the node $j$ (i.e., the sum of the weight of all edges departing from vertex $j$). $P_{i}$ is the “prestige score” assigned to player $i$ and represents the fraction of the overall tennis prestige sitting, in the steady state of the diffusion process, on vertex $i$. In Eqs. (1), $q\in\left[0,1\right]$ is a control parameter which accounts for the importance of the various terms contributing to the score of the nodes. The term $\left(1-q\right)\,\sum_{j}\,P_{j}\,\frac{w_{ji}}{s_{j}^{out}}$ represents the portion of score received by node $i$ in the diffusion process: vertices redistribute their entire credit to neighboring nodes proportionally to the weight of the connections linking to them. $\frac{q}{N}$ stands for a uniform redistribution of tennis prestige among all nodes according to which each player in the graph receives a constant and equal amount of credit. Finally the term $\frac{1-q}{N}\,\sum_{j}\,P_{j}\,\delta\left(s_{j}^{out}\right)$ [with $\delta\left(\cdot\right)$ equal to one only if its argument is equal to zero, and zero otherwise] serves as a correction in the case of existence of dandling nodes (i.e., nodes with null out-strength), which otherwise would behave as sinks in the diffusion process. Our prestige score is analogous to the PageRank score Brin1 , originally formulated for ranking web pages and more recently applied in different contexts. In general topologies, analytical solutions of Eqs. (1) are hard to find. The stationary values of the scores $P_{i}$s can be anyway computed recursively, by setting at the beginning $P_{i}=1/N$ (but the results do not depend on the choice of the initial value) and iterating Eqs. (1) until they converge to values stable within a priori fixed precision. #### II.3.1 Single tournament Figure 3: Prestige score in a single tournament. Prestige score $P_{r}$ as a function of the number of victories $r$ in a tournament with $\ell=7$ rounds (Grand Slam). Black circles are obtained from Eqs. (7) and valid for $q=0$. All other values of $q>0$ have been calculated from Eqs. (6): red squares stand for $q=0.15$, blue diamonds for $q=0.5$, violet up-triangles for $q=0.85$ and green down-triangles for $q=1$. In the simplest case in which the graph is obtained by aggregating matches of a single tournament only, we can analytically determine the solutions of Eqs. (1). In a single tournament, matches are hierarchically organized in a binary rooted tree and the topology of the resulting contact network is very simple [see Figure 2, panels b and c]. Indicate with $\ell$ the number of matches that the winner of the tournament should play (and win). The total number of players present at the beginning of the tournament is $N=2^{\ell}$. The prestige score is simply a function of $r$, the number of matches won by a player, and can be denoted by $P_{r}$. We can rewrite Eqs. (1) as $P_{r}=P_{0}\;+\;\left(1-q\right)\,\sum_{v=1}^{r}P_{v-1}\;\;,$ (2) where $P_{0}=\frac{1-q}{N}\,P_{\ell}\;+\;\frac{q}{N}$ and $0\leq r\leq\ell$. The score $P_{r}$ is given by the sum of two terms: $P_{0}$ stands for the equal contribution shared by all players independently of the number of victories; $\left(1-q\right)\,\sum_{v=1}^{r}P_{v-1}$ represents the score accrued for the number of matches won. The former system of equations has a recursive solution given by $P_{r}\;=\;\left(2-q\right)P_{r-1}\;=\ldots\;=\;\left(2-q\right)^{r}\,P_{0}\;\;,$ (3) which is still dependent on a constant that can be determined by implementing the normalization condition $\sum_{r=0}^{\ell}\,n_{r}\,P_{r}=1\;\;.$ (4) In Eq. (4), $n_{r}$ indicates the number of players who have won $r$ matches. We have $n_{r}=2^{\ell-r-1}$ for $0\leq r<\ell$ and $n_{\ell}=1$ and Eqs. (3) and (4) allow to compute $\begin{array}[]{ccc}\left(P_{0}\right)^{-1}=&\sum_{r=0}^{\ell-1}\,\left(2-q\right)^{r}\,2^{\ell-1-r}&+\;\left(2-q\right)^{\ell}\\\ &2^{\ell-1}\,\sum_{r=0}^{\ell-1}\,\left(\frac{2-q}{2}\right)^{r}&+\;\left(2-q\right)^{\ell}\\\ &2^{\ell-1}\,\frac{1-\left[\left(2-q\right)/2\right]^{\ell}}{\left[\left(2-q\right)/2\right]}&+\;\left(2-q\right)^{\ell}\\\ &\frac{2^{\ell}-\left(2-q\right)^{\ell}}{q}&+\;\left(2-q\right)^{\ell}\end{array}\;\;.$ In the former calculations, we have used the well known identity $\sum_{r=0}^{v}\,x^{r}\,=\,\frac{1-x^{v+1}}{1-x}$, valid for any $\left|x\right|<1$ and $v\geq 0$, which respectively means $0<q\leq 1$ and $\ell>0$ in our case. Finally, we obtain $P_{0}\;=\;\frac{q}{2^{\ell}\,+\,\left(2-q\right)^{\ell}\,\left(q-1\right)}\;\;,$ (5) which together with Eqs. (3) provides the solution $P_{r}\;=\;\frac{q\;\left(2-q\right)^{r}}{2^{\ell}\,+\,\left(2-q\right)^{\ell}\,\left(q-1\right)}\;\;.$ (6) It is worth to notice that for $q=1$, Eqs. (6) correctly give $P_{r}=2^{-\ell}$ for any $r$, meaning that, in absence of diffusion, prestige is homogeneously distributed among all nodes. Conversely, for $q=0$ the solution is $P_{r}\;=\;\frac{2^{r}}{2^{\ell-1}\,\left(\ell+2\right)}\;\;.$ (7) In Figure 3, we plot Eqs. (6) and (7) for various values of $q$. In general, sufficiently low values of $q$ allow to assign to the winner of the tournament a score which is about two order of magnitude larger than the one given to players loosing at the first round. The score of the winner is an exponential function of $\ell$, the length of the tournament. Grand Slams have for instance length $\ell=7$ and their relative importance is therefore two or four times larger than the one of other ATP tournaments, typically having lengths $\ell=6$ or $\ell=5$. ## III Results Rank | Player | Country | Hand | Start | End ---|---|---|---|---|--- $1$ | Jimmy Connors | United States | L | $1970$ | $1996$ $2$ | Ivan Lendl | United States | R | $1978$ | $1994$ $3$ | John McEnroe | United States | L | $1976$ | $1994$ $4$ | Guillermo Vilas | Argentina | L | $1969$ | $1992$ $5$ | Andre Agassi | United States | R | $1986$ | $2006$ $6$ | Stefan Edberg | Sweden | R | $1982$ | $1996$ $7$ | Roger Federer | Switzerland | R | $1998$ | $2010$ $8$ | Pete Sampras | United States | R | $1988$ | $2002$ $9$ | Ilie Năstase | Romania | R | $1968$ | $1985$ $10$ | Björn Borg | Sweden | R | $1971$ | $1993$ $11$ | Boris Becker | Germany | R | $1983$ | $1999$ $12$ | Arthur Ashe | United States | R | $1968$ | $1979$ $13$ | Brian Gottfried | United States | R | $1970$ | $1984$ $14$ | Stan Smith | United States | R | $1968$ | $1985$ $15$ | Manuel Orantes | Spain | L | $1968$ | $1984$ $16$ | Michael Chang | United States | R | $1987$ | $2003$ $17$ | Roscoe Tanner | United States | L | $1969$ | $1985$ $18$ | Eddie Dibbs | United States | R | $1971$ | $1984$ $19$ | Harold Solomon | United States | R | $1971$ | $1991$ $20$ | Tom Okker | Netherlands | R | $1968$ | $1981$ $21$ | Mats Wilander | Sweden | R | $1980$ | $1996$ $22$ | Goran Ivanišević | Croatia | L | $1988$ | $2004$ $23$ | Vitas Gerulaitis | United States | R | $1971$ | $1986$ $24$ | Rafael Nadal | Spain | L | $2002$ | $2010$ $25$ | Raúl Ramirez | Mexico | R | $1970$ | $1983$ $26$ | John Newcombe | Australia | R | $1968$ | $1981$ $27$ | Ken Rosewall | Australia | R | $1968$ | $1980$ $28$ | Yevgeny Kafelnikov | Russian Federation | R | $1992$ | $2003$ $29$ | Andy Roddick | United States | R | $2000$ | $2010$ $30$ | Thomas Muster | Austria | L | $1984$ | $1999$ Table 1: Top $30$ players in the history of tennis. From left to right we indicate for each player: rank position according to prestige score, full name, country of origin, the hand used to play, and the years of the first and last ATP tournament played. Players having been at the top of ATP ranking are reported in boldface. Figure 4: Relation between prestige rank and other ranking techniques. In panel a, we present a scatter plot of the prestige rank versus the rank based on the number of victories (i.e., in-strength). Only players ranked in the top $30$ positions in one of the two lists are reported. Rank positions are calculated on the network corresponding to all matches played between $1968$ and $2010$. In panel b, a similar scatter plot is presented, but now only matches of year $2009$ are considered for the construction of the network. Prestige rank positions are compared with those assigned by ATP. Year | Prestige | ATP year-end | ITF ---|---|---|--- $1968$ | Rod Laver | - | - $1969$ | Rod Laver | - | - $1970$ | Rod Laver | - | - $1971$ | Ken Rosewall | - | - $1972$ | Ilie Năstase | - | - $1973$ | Tom Okker | Ilie Năstase | - $1974$ | Björn Borg | Jimmy Connors | - $1975$ | Arthur Ashe | Jimmy Connors | - $1976$ | Jimmy Connors | Jimmy Connors | - $1977$ | Guillermo Vilas | Jimmy Connors | - $1978$ | Björn Borg | Jimmy Connors | Björn Borg $1979$ | Björn Borg | Björn Borg | Björn Borg $1980$ | John McEnroe | Björn Borg | Björn Borg $1981$ | Ivan Lendl | John McEnroe | John McEnroe $1982$ | Ivan Lendl | John McEnroe | Jimmy Connors $1983$ | Ivan Lendl | John McEnroe | John McEnroe $1984$ | Ivan Lendl | John McEnroe | John McEnroe $1985$ | Ivan Lendl | Ivan Lendl | Ivan Lendl $1986$ | Ivan Lendl | Ivan Lendl | Ivan Lendl $1987$ | Stefan Edberg | Ivan Lendl | Ivan Lendl $1988$ | Mats Wilander | Mats Wilander | Mats Wilander $1989$ | Ivan Lendl | Ivan Lendl | Boris Becker $1990$ | Stefan Edberg | Stefan Edberg | Ivan Lendl $1991$ | Stefan Edberg | Stefan Edberg | Stefan Edberg $1992$ | Pete Sampras | Jim Courier | Jim Courier $1993$ | Pete Sampras | Pete Sampras | Pete Sampras $1994$ | Pete Sampras | Pete Sampras | Pete Sampras $1995$ | Pete Sampras | Pete Sampras | Pete Sampras $1996$ | Goran Ivanišević | Pete Sampras | Pete Sampras $1997$ | Patrick Rafter | Pete Sampras | Pete Sampras $1998$ | Marcelo Ríos | Pete Sampras | Pete Sampras $1999$ | Andre Agassi | Andre Agassi | Andre Agassi $2000$ | Marat Safin | Gustavo Kuerten | Gustavo Kuerten $2001$ | Lleyton Hewitt | Lleyton Hewitt | Lleyton Hewitt $2002$ | Lleyton Hewitt | Lleyton Hewitt | Lleyton Hewitt $2003$ | Roger Federer | Andy Roddick | Andy Roddick $2004$ | Roger Federer | Roger Federer | Roger Federer $2005$ | Roger Federer | Roger Federer | Roger Federer $2006$ | Roger Federer | Roger Federer | Roger Federer $2007$ | Rafael Nadal | Roger Federer | Roger Federer $2008$ | Rafael Nadal | Rafael Nadal | Rafael Nadal $2009$ | Novak Djoković | Roger Federer | Roger Federer $2010$ | Rafael Nadal | Rafael Nadal | Rafael Nadal Table 2: Best players of the year. For each year we report the best player according to our ranking scheme and those of ATP and ITF. Best year-end ATP players are listed for all years from $1973$ on. ITF world champions have started to be nominated since $1978$ only. We set $q=0.15$ and run the ranking procedure on several networks derived from our data set. The choice $q=0.15$ is mainly due to tradition. This is the value originally used in the PageRank algorithm Brin1 and then adopted in the majority of papers about this type of ranking procedures chen07 ; radicchi09 ; west08 ; west10 . It should be stressed that $q=0.15$ is also a reasonable value because it ensures a high relative score for the winner of the tournament as stated in Eqs. (6). In Table 1, we report the results obtained from the analysis of the contact network constructed over the whole data set. The method is very effective in finding the best players of the history of tennis. In our top $10$ list, there are $9$ players having been number one in the ATP ranking. Our ranking technique identifies Jimmy Connors as the best player of the history of tennis. This could be a posteriori justified by the extremely long and successful career of this player. Among all top players in the history of tennis, Jimmy Connors has been undoubtedly the one with the longest and most regular trend, being in the top $10$ of the ATP year-end ranking for $16$ consecutive years ($1973$-$1988$). Prestige score is strongly correlated with the number of victories, but important differences are evident when the two techniques are compared. Panel a of Figure 4 shows a scatter plot, where the rank calculated according to our score is compared to the one based on the number of victories. An important outlier is this plot is represented by the Rafael Nadal, the actual number one of the ATP ranking. Rafael Nadal occupies the rank position number $40$ according to the number of victories obtained in his still young career, but he is placed at position number $24$ according to prestige score, consistently with his high relevance in the recent history of tennis. A similar effect is also visible for Björn Borg, whose career length was shorter than average. He is ranked at position $17$ according to the number of victories. Prestige score differently is able to determine the undoubted importance of this player and, in our ranking, he is placed among the best $10$ players of the whole history of professional tennis. In general, players still in activity are penalized with respect to those who have ended their careers. Prestige score is in fact strongly correlated with the number of victories [see panel a of Figure 4] and still active players did not yet played all matches of their career. This bias, introduced by the incompleteness of the data set, can be suppressed by considering, for example, only matches played in the same year. Table 2 shows the list of the best players of the year according to prestige score. It is interesting to see how our score is effective also here. We identify Rod Laver as the best tennis player between $1968$ and $1971$, period in which no ATP ranking was still established. Similar long periods of dominance are also those of Ivan Lendl ($1981-1986$), Pete Sampras ($1992-1995$) and Roger Federer ($2003-2006$). For comparison, we report the best players of the year according to ATP (year-end rank) and ITF (International Tennis Federation, www.itftennis.com) rankings. In many cases, the best players of the year are the same in all lists. Prestige rank seems however to have a higher predictive power by anticipating the best player of the subsequent year according to the two other rankings. John McEnroe is the top player in our ranking in $1980$ and occupies the same position in the ATP and ITF lists one year later. The same happens also for Ivan Lendl, Pete Sampras, Roger Federer and Rafael Nadal, respectively best players of the years $1984$, $1992$, $2003$ and $2007$ according to prestige score, but only one year later placed at the top position of ATP and ITF rankings. The official ATP rank and the one determined on the basis of the prestige score are strongly correlated, but small differences between them are very interesting. An example is reported in panel b of Figure 4, where the prestige rank calculated over the contact network of $2009$ is compared with the ATP rank of the end of the same year (official ATP year-end rank as of December $28$, $2009$). The top $4$ positions according to prestige score do not corresponds to those of the ATP ranking. The best player of the year, for example, is Novak Djoković instead of Roger Federer. We perform also a different kind of analysis by constructing networks of contacts for decades and for specific types of playing surfaces. According to our score, the best players per decade are [Table 3 lists the top $30$ players in each decade] : Jimmy Connors ($1971-1980$), Ivan Lendl ($1981-1990$), Pete Sampras ($1991-2000$) and Roger Federer ($2001-2010$). Prestige score identifies Guillermo Vilas as the best player ever in clay tournaments, while on grass and hard surfaces the best players ever are Jimmy Connors and Andre Agassi, respectively [see Table 4 for the list of the top $30$ players of a particular playing surface]. ## IV Discussion Tools and techniques of complex networks have wide applicability since many real systems can be naturally described as graphs. For instance, rankings based on diffusion are very effective since the whole information encoded by the network topology can be used in place of simple local properties or pre- determined and arbitrary criteria. Diffusion algorithms, like the one for calculating the PageRank score Brin1 , were first developed for ranking web pages and more recently have been applied to citation networks chen07 ; radicchi09 ; west08 ; west10 . In citation networks, diffusion algorithms generally outperform simple ranking techniques based on local network properties (i.e., number of citations). When the popularity of papers is in fact measured in terms of mere citation counts, there is no distinction between the quality of the citations received. In contrast, when a diffusion algorithm is used for the assessment of the quality of scientific publications, then it is not only important that popular papers receive many citations, but also that they are cited by other popular articles. In the case of citation networks however, possible biases are introduced in the absence of a proper classification of papers in scientific disciplinesradicchi08 . The average number of publications and citations strongly depend on the popularity of a particular topic of research and this fact influences the outcome of a diffusion ranking algorithm. Another important issue in paper citation networks is related to their intrinsic temporal nature: connections go only backward in time, because papers can cite only older articles and not vice versa. The anisotropy of the underlying network automatically biases any method based on diffusion. Possible corrections can be implemented: for example, the weight of citations may be represented by an exponential decaying function of the age difference between citing and cited papers chen07 . Though these corrections can be reasonable, they are ad hoc recipes and as such may be considered arbitrary. Here we have reported another emblematic example of a real social system suitable for network representation: the graph of contacts (i.e., matches) between professional tennis players. This network shows complex topological features and as such the understanding of the whole system cannot be achieved by decomposing the graph and studying each component in isolation. In particular, the correct assessment of players’ performances needs the simultaneously consideration of the whole network of interactions. We have therefore introduced a new score, called “prestige score”, based on a diffusion process occurring on the entire network of contacts between tennis players. According to our ranking technique, the relevance of players is not related to the number of victories only but mostly to the quality of these victories. In this sense, it could be more important to beat a great player than to win many matches against less relevant opponents. The results of the analysis have revealed that our technique is effective in finding the best players of the history of tennis. The biases mentioned in the case of citation networks are not present in the tennis contact graph. Players do not need to be classified since everybody has the opportunity to participate to every tournament. Additionally, there is not temporal dependence because matches are played between opponents still in activity and the flow does not necessarily go from young players towards older ones. In general, players still in activity are penalized with respect to those who already ended their career only for incompleteness of information (i.e., they did not play all matches of their career) and not because of an intrinsic bias of the system. Our ranking technique is furthermore effective because it does not require any external criteria of judgment. As term of comparison, the actual ATP ranking is based on the amount of points collected by players during the season. Each tournament has an a priori fixed value and points are distributed accordingly to the round reached in the tournament. In our approach differently, the importance of a tournament is self-determined: its quality is established by the level of the players who are taking part of it. In conclusion, we would like to stress that the aim of our method is not to replace other ranking techniques, optimized and almost perfected in the course of many years. Prestige rank represents only a novel method with a different spirit and may be used to corroborate the accuracy of other well established ranking techniques. ###### Acknowledgements. We thank the Association of Tennis Professionals for making publicly available the data set of all tennis matches played during last $43$ years. Helpful discussions with Patrick McMullen are gratefully acknowledged as well. | $1971-1980$ | $1981-1990$ | $1991-2000$ | $2001-2010$ ---|---|---|---|--- Rank | Player | Country | Player | Country | Player | Country | Player | Country $1$ | Jimmy Connors | United States | Ivan Lendl | United States | Pete Sampras | United States | Roger Federer | Switzerland $2$ | Björn Borg | Sweden | John McEnroe | United States | Andre Agassi | United States | Rafael Nadal | Spain $3$ | Ilie Năstase | Romania | Mats Wilander | Sweden | Michael Chang | United States | Andy Roddick | United States $4$ | Guillermo Vilas | Argentina | Stefan Edberg | Sweden | Goran Ivanišević | Croatia | Lleyton Hewitt | Australia $5$ | Arthur Ashe | United States | Jimmy Connors | United States | Yevgeny Kafelnikov | Russian Federation | Nikolay Davydenko | Russian Federation $6$ | Brian Gottfried | United States | Boris Becker | Germany | Jim Courier | United States | Ivan Ljubičić | Croatia $7$ | Manuel Orantes | Spain | Andrés Gómez | Ecuador | Richard Krajicek | Netherlands | Juan Carlos Ferrero | Spain $8$ | Eddie Dibbs | United States | Yannick Noah | France | Thomas Muster | Austria | Novak Djoković | Serbia $9$ | Harold Solomon | United States | Brad Gilbert | United States | Wayne Ferreira | South Africa | David Nalbandian | Argentina $10$ | Stan Smith | United States | Tomáš Šmíd | Czech Republic | Thomas Enqvist | Sweden | Tommy Robredo | Spain $11$ | Roscoe Tanner | United States | Henri Leconte | France | Boris Becker | Germany | David Ferrer | Spain $12$ | Raúl Ramírez | Mexico | Tim Mayotte | United States | Stefan Edberg | Sweden | Fernando González | Chile $13$ | Tom Okker | Netherlands | Anders Jarryd | Sweden | Sergi Bruguera | Spain | Andy Murray | Great Britain $14$ | John Alexander | Australia | Miloslav Mečíř Sr. | Slovakia | Marc Rosset | Switzerland | Carlos Moyá | Spain $15$ | Vitas Gerulaitis | United States | Kevin Curren | United States | Petr Korda | Czech Republic | Mikhail Youzhny | Russian Federation $16$ | Ken Rosewall | Australia | Aaron Krickstein | United States | Todd Martin | United States | James Blake | United States $17$ | John Newcombe | Australia | Guillermo Vilas | Argentina | Cédric Pioline | France | Tommy Haas | United States $18$ | Wojtek Fibak | Poland | Joakim Nystrom | Sweden | Michael Stich | Germany | Fernando Verdasco | Spain $19$ | Dick Stockton | United States | Emilio Sánchez | Spain | Àlex Corretja | Spain | Marat Safin | Russian Federation $20$ | John McEnroe | United States | Johan Kriek | United States | Patrick Rafter | Australia | Tomás̆ Berdych | Czech Republic $21$ | Adriano Panatta | Italy | Martin Jaite | Argentina | Magnus Gustafsson | Sweden | Juan Ignacio Chela | Argentina $22$ | Jan Kodeš | Czech Republic | Jakob Hlasek | Switzerland | Andrei Medvedev | Ukraine | Radek Štěpánek | Czech Republic $23$ | Jaime Fillol Sr. | Chile | Jimmy Arias | United States | Francisco Clavet | Spain | Andre Agassi | United States $24$ | Robert Lutz | United States | Pat Cash | Australia | Marcelo Ríos | Chile | Robin Söderling | Sweden $25$ | Marty Riessen | United States | Ramesh Krishnan | India | Greg Rusedski | Great Britain | Rainer Schüttler | Germany $26$ | Rod Laver | Australia | José-Luis Clerc | Argentina | Fabrice Santoro | France | Feliciano López | Spain $27$ | Tom Gorman | United States | Eliot Teltscher | United States | Magnus Larsson | Sweden | Tim Henman | Great Britain $28$ | Vijay Amritraj | India | Thierry Tulasne | France | Tim Henman | Great Britain | Jarkko Nieminen | Finland $29$ | Mark Cox | Great Britain | Scott Davis | United States | Alberto Berasategui | Spain | Mardy Fish | United States $30$ | Onny Parun | New Zealand | Vitas Gerulaitis | United States | Albert Costa | Spain | Gastón Gaudio | Argentina Table 3: Top $30$ players per decade. | Clay | Grass | Hard ---|---|---|--- Rank | Player | Country | Player | Country | Player | Country $1$ | Guillermo Vilas | Argentina | Jimmy Connors | United States | Andre Agassi | United States $2$ | Manuel Orantes | Spain | Boris Becker | Germany | Jimmy Connors | United States $3$ | Thomas Muster | Austria | Roger Federer | Switzerland | Ivan Lendl | United States $4$ | Ivan Lendl | United States | John Newcombe | Australia | Pete Sampras | United States $5$ | Carlos Moyá | Spain | John McEnroe | United States | Roger Federer | Switzerland $6$ | Eddie Dibbs | United States | Pete Sampras | United States | Stefan Edberg | Sweden $7$ | José Higueras | Spain | Tony Roche | Australia | Michael Chang | United States $8$ | Björn Borg | Sweden | Stefan Edberg | Sweden | John McEnroe | United States $9$ | Ilie Năstase | Romania | Roscoe Tanner | United States | Andy Roddick | United States $10$ | Andrés Gómez | Ecuador | Lleyton Hewitt | Australia | Lleyton Hewitt | Australia $11$ | Àlex Corretja | Spain | Ken Rosewall | Australia | Brad Gilbert | United States $12$ | Rafael Nadal | Spain | Arthur Ashe | United States | Jim Courier | United States $13$ | José-Luis Clerc | Argentina | Stan Smith | United States | Brian Gottfried | United States $14$ | Sergi Bruguera | Spain | Phil Dent | Australia | Thomas Enqvist | Sweden $15$ | Mats Wilander | Sweden | Björn Borg | Sweden | Stan Smith | United States $16$ | Albert Costa | Spain | Goran Ivanišević | Croatia | Boris Becker | Germany $17$ | Gastón Gaudio | Argentina | Pat Cash | Australia | Wayne Ferreira | South Africa $18$ | Juan Carlos Ferrero | Spain | Andy Roddick | United States | Ilie Năstase | Romania $19$ | Harold Solomon | United States | Ivan Lendl | United States | Roscoe Tanner | United States $20$ | Emilio Sánchez | Spain | Tim Henman | Great Britain | Tommy Haas | United States $21$ | Adriano Panatta | Italy | Rod Laver | Australia | Rafael Nadal | Spain $22$ | Félix Mantilla | Spain | Mark Edmondson | Australia | Tim Henman | Great Britain $23$ | Francisco Clavet | Spain | John Alexander | Australia | Mats Wilander | Sweden $24$ | Balázs 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arxiv-papers
2011-01-20T21:01:27
2024-09-04T02:49:16.551913
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Filippo Radicchi", "submitter": "Filippo Radicchi", "url": "https://arxiv.org/abs/1101.4028" }
1101.4072
# Early phase of massive star formation: A case study of Infrared dark cloud G084.81$-$01.09 S. B. Zhang11affiliation: Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; shbzhang@pmo.ac.cn , J. Yang11affiliation: Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; shbzhang@pmo.ac.cn , Y. Xu11affiliation: Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China; shbzhang@pmo.ac.cn , J. D. Pandian22affiliation: Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822, USA , K. M. Menten33affiliation: Max-Planck-Institut f$\ddot{u}$r Radioastronomie, Auf dem H$\ddot{u}$gel 69, 53121 Bonn, Germany , C. Henkel33affiliation: Max-Planck-Institut f$\ddot{u}$r Radioastronomie, Auf dem H$\ddot{u}$gel 69, 53121 Bonn, Germany ###### Abstract We mapped the MSX dark cloud G084.81$-$01.09 in the NH3 (1,1) - (4,4) lines and in the $J$ = 1-0 transitions of 12CO, 13CO, C18O and HCO+ in order to study the physical properties of infrared dark clouds, and to better understand the initial conditions for massive star formation. Six ammonia cores are identified with masses ranging from 60 to 250 $M_{\sun}$, a kinetic temperature of 12 K, and a molecular hydrogen number density $n({\rm H_{2}})\sim 10^{5}$ cm-3. In our high mass cores, the ammonia line width of 1 km s-1 is larger than those found in lower mass cores but narrower than the more evolved massive ones. We detected self-reversed profiles in HCO+ across the northern part of our cloud and velocity gradients in different molecules. These indicate an expanding motion in the outer layer and more complex motions of the clumps more inside our cloud. We also discuss the millimeter wave continuum from the dust. These properties indicate that our cloud is a potential site of massive star formation but is still in a very early evolutionary stage. stars: formation – ISM: molecules – ISM: kinematics and dynamics – ISM: individual (G084.81$-$01.09) ## 1 INTRODUCTION Infrared dark clouds (IRDCs) are identified as small areas with high extinction against the bright diffuse mid-infrared background along the Galactic Plane. These “flecks” in the sky were first discovered by the 15 $\mu$m ISOGAL survey (the inner Galactic disk survey of the Infrared Space Observatory, Pérault et al. 1996) and later cataloged and studied using the 8.3 $\mu$m survey data of the Mid-course Space Experiment (MSX, Egan et al. 1998). Egan et al. found about 2000 IRDCs and note that these objects are cold and dense molecular cores. These results were refined by later molecular line observations (Carey et al., 1998; Teyssier et al., 2002) and radio continuum observations (Carey et al., 2000). These authors concluded that these clouds have low kinetic temperatures of 10-20 K and a gas density over $10^{5}$ cm-3. Their small line widths of 1 - 3 km s-1 can be related to a very early phase of star formation. Therefore, the IRDCs are good candidates for pre- protostellar cores and may be the primary sites for future massive star formation, which plays an important role in the evolution of our galaxy. However, IRDCs remain mysterious in at least some aspects. A controversial issue being discussed is whether the cloud evolution is quasi-static or whether it is undergoing a dynamical process caused by lack of equilibrium (Bergin & Tafalla, 2007). Understanding the role of supersonic turbulence in dark clouds is important in settling this argument. Evidence from current observations of IRDCs seems to support both views. Most IRDC cores are supported by turbulent pressure but appear to be virialised (Pillai et al., 2006). Furthermore, the origin of the turbulence in the molecular clouds and the transition from the turbulent cloud to the quiescent core regime is still unclear. Also, more systematic studies need to be done to compare them with other relatively nearby massive star-forming clouds. The answers will provide a critical element in our general understanding of star formation. In this study, we have obtained spectroscopic data in several molecules towards the dark cloud G048.81–01.09, a source in the catalog of Simon et al. (2006). The cloud has an elongated morphology (N-S) with an extent of $\sim 15$ arcmin, and is situated in a region of high extinction ($A_{v}>20$, Camberésy et al. 2002) of the dark cloud LDN935 (Lynds, 1962) which separates the W80 HII region into the North America (NGC 7000) and Pelican (IC 5070) nebulae (Comeron & Pasquali, 2005). Comeron & Pasquali (2005) proposed that the ionizing source of the complex is 2MASS J205551.25+435224.6, an O5V star located behind LDN935. The IRDC is almost starless with no associated IRAS or MSX point sources. Previous molecular line observations indicated the presence of large amounts of gas with different velocity components and signs of ongoing star formation in LDN935 around our dark cloud (Feldt & Wendker, 1993). A single pointed CO observation towards our cloud conducted by Milman (1975) suggests however that the gas is not appreciably heated. There is no direct distance measurement for G084.81$-$01.09, but it is associated with dark cloud LDN935 for which there are a number of distance estimates. A radio continuum study carried out by Wendker et al. (1983) gave a distance estimate of 500 pc. Straižys et al. (1993) obtained a consistent result of 580 pc on the basis of photometric results of 564 stars toward three areas near the North America and Pelican Nebulae complexes. New measurements done by Cersosimo et al. (2007) derived a distance of about 0.7 kpc, which was obtained from radio recombination line observations at a frequency around 1.4 GHz and by applying a flat rotation-curve model. Comeron & Pasquali (2005) obtained a distance of 610 pc to the ionizing star, 2MASS J205551.25+435224.6. Here we adopt a distance of 580 pc in our calculation of physical parameters. ## 2 OBSERVATIONS AND DATA REDUCTION ### 2.1 Effelsberg 100 m Observations We mapped the infrared dark cloud G084.81$-$01.09 in the NH3 (1,1), (2,2), (3,3) and (4,4) transitions using the Effelsberg 100 m telescope of the Max- Planck-Institut für Radioastronomie (MPIfR) during December 12 - 14, 2007 (see Table 1 for adopted line frequencies). The spectrometer backend employed was the 8192 channel AK90 auto-correlator which consists of 8 individual 1024 channel correlators of 20 MHz bandwidth each. Additional data of NH3 (1,1), (2,2) and (3,3) were obtained on February 21 - 24, 2008 using the Fast Fourier Transform Spectrometer (FFTS), which has 16382 channels covering a bandwidth of 500 MHz. The data were taken in using frequency switching with a frequency throw of 7.5 MHz. The map covers the area with notable extinction in the MSX 8.3 $\mu$m image with a grid spacing of 30″. The pointing was checked roughly every hour by observations of nearby continuum sources and was found to be accurate to better than 10″. Continuum scans were used for calibrating the absolute flux density. The raw data were scaled from arbitrary units to main beam brightness temperature by applying gain-elevation corrections and the flux scale was set using NGC 7027 by assuming it to have a flux density of 5.4 Jy at our observing frequency Ott et al. (1994). Our absolute calibration uncertainty is $\pm$10%. ### 2.2 PMODLH 13.7 m Observations The 12CO (1$-$0), 13CO (1$-$0), C18O (1$-$0) and HCO+ (1$-$0) observations of the MSXDC G084.81$-$01.09 (line frequencies shown in Table 1) were carried out at the Purple Mountain Observatory Delingha (PMODLH) 13.7 m telescope from March to June in 2008. The 3 mm SIS receiver was used in double sideband mode. Using three 1024 channel AOS spectrometers with bandwidths of 145.43 MHz, 42.87 MHz and 43.3 MHz, the three CO lines were observed simultaneously. The source was mapped using position switched observations, with the standard chopper wheel method for calibration (Penzias & Burrus, 1973). Standard sources were checked roughly every 2 hours. To compare with the NH3 data, our CO map covered the area mapped with NH3 and extended a few arcmins to the northeast. The grid spacing was 30″ in the center and 60″ in other regions. The HCO+ map covered a region similar to that of the NH3 map with a typical grid spacing of 30″. A summary of all observation parameters is provided in Table 1. ### 2.3 Data Reduction We used the CLASS software package111CLASS is part of the GILDAS software suite available from http://www.iram.fr/IRAMFR/GILDAS for the spectral data reduction. After discarding the bad scans, the spectra of each position were averaged and a polynomial baseline of orders 1-3 was subtracted. Bad channels were excluded from the baseline fitting. The method “NH3(1,1)” in CLASS (Buisson et al., 2005) was used to fit the hyperfine structure of the NH3 (1,1) spectra and to derive optical depths and line widths. The (2,2) line is so weak that only the main line could be dectected towards a few positions. Therefore, we simply fitted the (2,2) line with a single Gaussian to derive its main beam brightness temperature and an optical depth was not calculated in this case. Table 2 summarizes the fit parameters toward the cores detected in the data. No NH3 (3,3) or (4,4) lines were detected in our observations. The excitation temperature, rotational temperature, kinetic temperature and ammonia column density toward the cores are listed in Table 3. These physical parameters were derived using the standard formulae for NH3 spectra (Ho & Townes, 1983). The CO and HCO+ main beam brightness temperatures were derived from the antenna temperature, $T_{A}^{*}$, using a main beam efficiency, $\eta_{\rm mb}$, of 61%. A first order baseline was applied for the CO spectra and 1st-3rd order baselines were used for the HCO+ spectra. A single Gaussian fit to the spectra provided the line velocity with respect to the local standard of rest (LSR) and the line width for later discussion. The fit results are given in Table 4. ## 3 RESULTS ### 3.1 Identification and Morphology of the Cloud Fig. 1 shows the NH3 integrated intensity maps created from the velocity range -2.5 to 4.5 km s-1 (which excludes the satellite lines). Our images of ammonia emission toward G084.81$-$01.09 reveal extended, filamentary molecular emission that closely matches the morphology of the Spitzer mid-infrared extinction. Also, the ammonia maps (both NH3 (1,1) and (2,2)) allow us to identify two main molecular condensations: one in the northeast, containing ammonia cores 1, 2 and 6, and one in the south, consisting of cores 3, 4 and 5. All cores defined in the NH3 (1,1) map follow the standards given by Tachihara et al. (2000) and the name of the cores were designated in descending order of their integrated intensities, as tabulated in Table 2. The NH3 (1,1) core sizes range from 0.25 to 0.42 pc. The NH3 molecular emission also matches the 1.1 mm dust emission of Bolocam Galactic Plane Survey (BGPS, Aguirre et al. 2010) remarkably well as shown in Fig. 2a. Furthermore, with the blue circles marking the clump peaks of continuum emission identified by Rosolowsky et al. (2009), we note that the ammonia cores are coincident with the peaks in 1.1 mm emission. The integrated intensity maps of 13CO (1$-$0) and C18O (1$-$0) are shown in Fig. 2b and 2c. The 13CO and C18O emission lines show an extended feature along north-south direction, with a sharp cutoff toward the northwest. The extent of C18O emission is larger than that of both the ammonia and continuum emission. The 13CO line, which tends to trace a lower gas density than C18O, appears to be more extended. The strongest part of 13CO emission coincides with the northern condensation (peak at the position of ammonia core 1) and shows a tail to the east, while the 13CO emission in the southern part is flat with no peak associated with that in the C18O map. Fig. 2d shows the integrated intensity of HCO+ (1$-$0) overlaid on dust emission. The HCO+ emission shows a different distribution compared to CO. Since the line is optically thick (see §3.3), it is likely to trace the envelope surrounding a dense core, while the C18O line is expected to trace the dense core itself. A candidate massive young stellar object (MYSO), G084.7847$-$01.1709, identified by Urquhart et al. (2009) in their 6 cm VLA survey is marked with cross symbol on Fig. 1. It is spatially coincident with the north condensation of our cloud. However, no compact source can be found near this object in the Spitzer 24 or 70 $\mu$m data. The lack of infrared emission could be due to the source being an extragalactic object rather than a Galactic MYSO. Further observations at centimeter wavelengths are needed to determine the spectral index of emission and the nature of this object. ### 3.2 Line intensities and kinetic temperature The excitation temperature, $T_{\rm ex}$, of the NH3 (1,1) transition (Column $T_{\rm ex}$ in Table 3) was obtained from the optical depth via the relation $T_{\rm mb}={h\nu\over k}[J(T_{\rm ex})-J(T_{\rm bg})](1-e^{-\tau}),J(T)=(e^{h\nu\over kT}-1)^{-1},$ where $T_{\rm mb}$ and $\tau$ represent the temperature and the optical depth from CLASS fitting procedures and $T_{\rm bg}$ equals 2.7 K. The rotational temperature between the NH3 (1,1) and (2,2) inversion doublets were then derived from the excitation temperature using the method given by Ho & Townes (1983). We then estimate the kinetic temperature using the expression of Tafalla et al. (2004). The typical kinetic temperature in the cores is about 12 K as seen in Table 3. No obvious difference in temperature appears among the six cores. The kinetic temperature in the envelope is about 2 K higher than in the cores. Désert et al. (2008) carried out a low resolution ($\sim$12 arcmin) survey at four (sub)millimeter wavelengths and derived a dust temperature of 8.5 K with a fixed emissivity law exponent of 2 toward our cloud, which is slightly lower than our gas temperature, probably because of the different beam size. The physical parameters of CO are calculated under the assumption of Local Thermodynamic Equilibrium (LTE), in which the 13CO and C18O lines are assumed to reach the same excitation temperature as 12CO line. The 12CO (1-0) line is optically thick, so we can derive the kinetic temperature directly from its brightness temperature. The kinetic temperature is found to be 12 – 14 K as listed in Table 5 and agrees with that derived from ammonia. The 13CO and C18O line intensity ratio indicates the different abundances and optical depths in the cloud. As shown in Fig. 3, the 13CO to C18O ratios in regions with strong emission are low (about 1.4-2.5), while the ratios of faint emission areas are high and close to the local interstellar [13CO]/[C18O] ratio of about 7.3 (Wilson & Rood, 1994; Teyssier et al., 2002). The observation of variable line ratios as a function of intensities is consistent with that of Teyssier et al. (2002). ### 3.3 Line widths and profiles The NH3 (1,1) line widths in the cloud vary from 0.4 to 2.8 km s-1, which is consistent with the value reported from massive dense cores (Bensen & Myers, 1989; Pillai et al., 2006). The distribution of ammonia line widths is shown in Fig. 4. The line widths in the peripheral regions ($>$1.3 km s-1) are generally larger than those in the clump centers (0.7 - 1.4 km s-1). The line width of Core 6 is influenced and broadened by the extended part of core 1 and double peaked spectra appear in the central part of core 6 (shown in Fig. 5, P6). For cores 2, 4, 5 and 6, the (2,2) line widths are larger than the (1,1) line widths, which suggests that the (2,2) line may not trace the same gas as the (1,1) line toward these cores (Pillai et al., 2006). As seen in Fig. 5, the CO lines show multiple velocity components. Most of the 13CO spectra in the cores are blended with a component around 5.5 km s-1, which also appears in some C18O spectra. This component is consistent with the extended part of another cloud to the west of core 1, reported by Feldt & Wendker (1993). A few of the 13CO spectra are flat topped, indicating that the line is optically thick at these locations. As with the 13CO line profiles, the C18O spectra are also slightly asymmetric, especially in core 2. The line widths over the core regions range between 2.3 and 4.6 km s-1 for 13CO and between 1.5 and 2.7 km s-1 for C18O. The thermal line width, $[(8\ln 2)kT_{kin}/m]^{1/2}$ ($k$ is the Boltzmann constant and $m$ is the mean molecular mass), at a kinetic temperature, $T_{kin}$, of 12 K is 0.18 km s-1 for NH3 and 0.14 km s-1 for 13CO and C18O. Therefore, the non-thermal line widths are significantly larger than the thermal line widths and near the observed line widths. This suggests that non- thermal broadening mechanisms (rotation, turbulence, etc) play a dominant role in producing the observed line profiles. The HCO+ (1-0) spectra as shown in Fig. 6 display three types of profiles: an asymmetric, double-peaked shape in the northern C18O condensation (e.g., B, C, P1, P2, etc.), spectra with a blue shoulder in the far north (e.g., A, P6, etc.) and a single blue-shifted peak in the south (e.g., D, etc.). The single peaked C18O line appears to bisect the HCO+ profiles in velocity, indicating that the HCO+ lines are likely optically thick and self-absorbed. The blue-shifted dip due to self-absorption may indicates that this occurs in an expanding envelope. By fitting Gaussian line profiles to these spectra and comparing the peak velocity of HCO+ and C18O spectra, one can roughly estimate an expansion velocity (Aguti et al., 2007). We measure a velocity of $\delta V=|v_{\rm peak}({\rm C^{18}O})-v_{\rm peak}({\rm HCO^{+}})|\approx 1.05$ km s-1. This value is less than the C18O line width at corresponding positions but is supersonic. On the contrary, the blue peaked profile in the south can be associated with infall asymmetry and may suggest infall motions of the outer cloud material. However, due to the low signal to noise ratio of the HCO+ spectra nearby and the relatively large beam sizes involved, this motion can not be confirmed at all positions with blue-shifted peak. In order to better understand the spatial distribution of the HCO+ line profile, we have compared the velocity of peak emission in the HCO+ and C18O lines for each location in the northern condensation, and mapped the velocity difference, $\delta V=v_{\rm peak}({\rm C^{18}O})-v_{\rm peak}({\rm HCO^{+}})$ (Fig. 6). It can be seen that the outflow asymmetry (coded blue) dominates the central and northern regions of the cloud, while the infall asymmetry (coded red) appears on the southern edge. ### 3.4 Velocity structure From Table 3, we note that the line width of core 2 is rather small (0.95 km s-1) but remarkably increases to 1.2 km s-1 when the spectra are averaged over the core area, which reflects the existence of velocity gradients in the core. This is also seen in the NH3 (1,1) channel maps shown in Fig. 7 for velocities from $-$1.0 to 3.0 km s-1 where different clumps appear at separate velocities. The clump structure is complex and covers a wide velocity range. In the lower velocity channels ($-$1.0 to 0.0 km s-1), the gas is concentrated in a clump in the east of northern condensation, while the higher velocity channels (0.5 to 2.0 km s-1) indicate the clump in the two peak position and a southern clump. The CO presents multiple peaks along the line of sight (cf. CO spectra in Fig. 5). The channel maps of 13CO (1-0) and C18O (1-0) (Fig. 8) show that the different velocity components present various shapes and orientations. C18O shows a velocity structure that is similar to that of ammonia, shown in Fig. 7. The velocity increases from the position of core 1 at about 1 km s-1 to 2 km s-1 towards the north, which can be seen in position-velocity maps (Fig. 9) and the 13CO, C18O and NH3 channel maps. To estimate the magnitude and direction of the velocity gradient in each core, we adopt the method of Goodman et al. (1993) by fitting a linear velocity gradient to the projected velocity field along the line of sight. Thus the observed velocity $v_{\rm LSR}$ can be expressed as $v_{\rm LSR}=v_{0}+\mathscr{G}\Delta\alpha\sin\Theta_{\mathscr{G}}+\mathscr{G}\Delta\delta\cos\Theta_{\mathscr{G}},$ where $\mathscr{G}$ is the magnitude of the velocity gradient, $\Delta\alpha$ and $\Delta\delta$ represent offsets in right ascension and declination in arcseconds, $v_{0}$ is the systemic velocity of the cloud, and $\Theta_{\mathscr{G}}$ is the direction of increasing velocity, measured east of north. We carry out a least-squares fit to the two dimensional velocity field of 13CO (1-0), C18O (1-0), NH3 (1,1) and NH3 (2,2) weighted by $1/\sigma_{\rm LSR}^{2}$, where $\sigma_{\rm LSR}$ is the uncertainty in fitting $v_{\rm LSR}$. In Table 6, we list our fitting results ($\mathscr{G}$ and $\Theta_{\mathscr{G}}$) with their formal errors ($\sigma_{\mathscr{G}}$ and $\sigma_{\Theta_{\mathscr{G}}}$). We exclude fits that fail the $3\sigma$ criterion ($\mathscr{G}\geq 3\sigma_{\mathscr{G}}$ mentioned by (Goodman et al., 1993)) or those with a random velocity field as seen from the velocity map. Both cores 1 and 6 present significant velocity gradients in CO and NH3 lines. The magnitude of the velocity gradient is greater in NH3, and the direction of the gradient is somewhat different from that seen in CO. In cores 3, 4 and 5, the velocity gradient is detected only in NH3 (1,1). The NH3 gas associated with core 2 also exhibits a clear velocity gradient in Fig. 4. The peak lies on a velocity ridge with a gradient of 3.15 km s-1 pc-1 to the east and 1.93 km s-1 pc-1 to the west. The distinct velocity gradients given by different molecules are probably due to the different densities they trace (Goodman et al., 1993). Velocity gradients seen in the lower density tracers, 13CO or C18O, probably arise in the envelope of the northern condensations. This gradient is dominated by unresolved clump-clump motions within the envelope, and may also be contaminated by the higher velocity component. The higher density tracers (e.g., NH3), on the other hand, reveal the gradients within the small clumps themselves. ### 3.5 Density and abundance The column densities of CO are estimated using the equation given by Scoville et al. (1986): $N={3k^{2}\over 4h\pi^{3}\mu^{2}\nu^{2}}\exp\left({h\nu J\over kT_{\rm ex}}\right)\times{T_{\rm ex}+h\nu/6k(J+1)\over{\rm exp}(-h\nu/kT_{\rm ex})}{\tau\over 1-{\rm exp}(-\tau)}\int{T_{R}^{*}dV},$ where $J$ is the rotational quantum number of the lower state in the observed transition, $\mu$ is the permanent dipole moment and $\tau$ is the optical depth derived from the observed ratios of 12CO and 13CO (or C18O) emission. The derived physical properties are listed in Table 5. We also derive the column densities of NH3 for the cores, which range from $15-34\times 10^{14}$ cm-2 as shown in Table 3. Since 13CO is not optically thin in some cores, we estimate the H2 column densities based on the C18O column densities where a factor $N({\rm H_{2}})/N({\rm C^{18}O})=7\times 10^{6}$ was adopted (Frerking et al., 1982; Kramer et al., 1999). This gives H2 column densities around $\sim 10^{22}$ cm-2 as listed in column (2) of Table 7. With the H2 column densities derived from CO emission, we determine the ammonia abundances and list them in Table 7. The ammonia abundance in the six peak positions is $3-5.5\times 10^{-8}$, which agrees with the parameters reported by Pillai et al. (2006), though their abundances are derived from the dust emission of SCUBA. When averaged over the cores, the abundances are reduced to $2.3\times 10^{-8}$. The ammonia abundance and the correlation of its spatial distribution with the morphology of the dust emission is consistent with a chemical model given by Bergin & Langer (1997), in which NH3 is not depleted. We then compare its abundance with that of 13CO toward cores in Fig. 10. They differ slightly from each other with core 1 showing the highest abundance among the cores. This reveals the complex chemical environment in the cloud. We obtain the gas mass of cores by integrating the total H2 column density over the cores and show the results in Table 8. The mass range from 60 $M_{\sun}$ to 250 $M_{\sun}$. The gas density can be derived by assuming the cores to be spherical and dividing the mass by the volume of the cores. An average density of $\sim 1.1\times 10^{5}$ cm-3 was derived as shown in Table 8. The total H2 column density can also be related to the 1.1 mm flux by assuming the continuum emission to be optically thin. After convolving the data to the resolution of the 100 m Effelsberg (40″) and 13.7 m Delingha (60″) beams, the dust based column density is given by $N({\rm H_{2}})_{\rm 1.1~{}mm}={S_{\rm 1.1~{}mm}\over\Omega_{\rm beam}B_{\nu}(T_{d})\kappa_{\rm 1.1~{}mm}m},$ where $\Omega_{\rm beam}$ is the telescope beam solid angle, $B_{\nu}(T_{d})$ is the Planck function at the dust temperature $T_{d}$ (assumed to be equal to the gas temperature deduced from NH3), $\kappa_{\rm 1.1~{}mm}$ is the dust opacity at 1.1 mm, and $m$ is the mean molecular mass. Here we adopt $\kappa_{\rm 1.1~{}mm}$ to be $0.0114$ cm2 g-1 (Ossenkopf & Henning, 1994; Enoch et al., 2008) and assume a gas to dust mass ratio of 100. The results have the same magnitude as those derived from C18O and are listed in Table 7. We also compare our molecular emission with the BGPS survey, as shown in Fig. 11. The integrated intensity of NH3 (1,1) shows a better correlation with the 1.1 mm dust emission compared to C18O as mentioned previously. The diversity in column densities derived from molecular gas and dust may be due to different structures and components they trace. The CO column density, because of the low density it traces, is highly affected by the structure of the cloud along the line of sight, especially at the positions where clumps with similar velocities blend together. Also the molecular observations are affected by high opacity or chemical depletion at high densities. On the other hand, calculations related to the dust emission are based on the value of dust opacity and assumptions about the dust to gas ratio. The dust opacity changes between different dust models and in environments of different density (Ossenkopf & Henning, 1994). ## 4 DISCUSSION ### 4.1 Evidence of massive star formation In this section, we will compare our results with those from other low-mass star forming regions, and discuss the evidence for massive star formation and the evolutionary state of the cloud. The mean intrinsic line width for NH3 (1,1) in the samples given by Jijina et al. (1999) (most of the objects are low-mass cores) is 0.5 km s-1. A recent work by Foster et al. (2009) also reported a small observed line width of 0.3 km s-1 for low mass NH3 cores in the Perseus Molecular cloud. Crapsi et al. (2005) undertook a survey toward low-mass starless cores and also reported a narrow line width of 0.5 km s-1 for C18O (1-0). A broader line width of 1.2 km s-1 for C18O (1-0) is derived from a JCMT observation by Jørgensen et al. (2002), which is still less than those given in our observed region. Meanwhile, the dominant non-thermal line widths in our cloud indicate more non-thermal support than those in low-mass star forming regions. Although line widths can be broadened by active outflows in low-mass protostellar regions to 2 km s-1 for C18O (1-0) (Swift & Welch, 2008) and 1.2 km s-1 for NH3 (1,1 ) (Rudolph et al., 2001), at this stage, an active outflow has not developed yet. The NH3 line widths here are comparable to those IRDCs of Pillai et al. (2006). Moreover, our derived column densities of NH3, a few times 1015 cm-2, are higher than those of the low-mass cores ($\sim 10^{13}-10^{15}$ cm-2) given by Suzuki et al. (1992), and are also comparable to those of the massive IRDCs given by Pillai et al. (2006). In addition, the supersonic motions of the envelope and the large velocity gradients in our cloud also indicate a more dynamic motion compared to low- mass cores. Therefore, we suggest that MSXDC G084.81$-$01.09 is potentially at an early evolutionary state of massive star formation. ### 4.2 Gravitational stability of the cores The expanding envelope detected from the line profiles described in §3.3 lead us to consider the gravitational stability of MSXDC G084.81$-$01.09. By using the molecular mass, $M_{Molecular}$, derived from C18O (1-0) and the core radius $R$ determined from the NH3 (1,1) maps, we obtain an escape velocity ($\sqrt{2GM/R}$) which ranges from $\sim 1.9$ km s-1 in core 6 to a maximum of $\sim 3.2$ km s-1 in core 2 (Table 8). The three-dimensional velocity dispersion of the cores provided by the NH3 line width lies in the range of 1.2 – 2.4 km s-1 which is less than the escape velocity in each core. However, the C18O velocity dispersions (2.5 - 4 km s-1) exceed the escape velocity in some regions of the cloud, especially for core 6, which has a small escape velocity. This may lead to instability for core 6. Additionally, an expansion speed of 1.0 km s-1 in the envelope of the core (cf. §3.3), though significantly less than the escape velocity, may aggravate the situation. To examine the gravitational stability of the cores, it is useful to calculate the virial mass of the cores as well as the virial parameter, the ratio between the virial mass and core mass. The virial parameter given by Bertoldi & McKee (1992) can be expressed as $\alpha=5\sigma^{2}R/GM$, where R is the radius of the core, $\sigma=\sqrt{3/(8\ln 2)}\times FWHM$, and $M$ is the core mass. The virial mass and virial parameter can be found in Table 8. The average virial parameter is about 1.1 which suggests that most of the cores are virialised. However, the virial parameters of cores 2 and 6 deviate significantly from unity. This can be caused by a number of factors. One explanation is the potential for the presence of spatially unresolved stellar clusters in the cores, which can be identified from the color temperature of the dust continuum map. For $M_{\rm Virial}>M_{\rm Molecular}$, beam filling factors smaller than unity or streaming motions may adversely affect the estimate and may lead to an underestimated column density or an overestimated line width. Alternatively, it is possible that the cores may not actually be in virial equilibrium, and may be transient entities (Ballesteros-Paredes, 2006), which is probably the case in core 6. We derive a Jeans mass greater than 170 $M_{\sun}$ for each core using the equation $M_{Jeans}=17.3~{}{T_{kin}}^{1.5}n^{-0.5}M_{\sun}$, where $T_{kin}$ is the kinetic temperature derived from NH3 data, and $n$ is the average density as listed in Table. 8. We note that the masses of cores 1 and 2 are comparable to their Jeans masses, but the other cores are significantly less massive. However, considering the clumping in cores 1 and 2, it is still possible for them to form several individual protostars. Therefore, we suggest that cores 1 - 5 are gravitationally bound, among which cores 1 and 2 are marginally stable against collapse. However, core 6 is probably transient. ### 4.3 Comparison with more evolved clouds Our dark cloud has a lower kinetic temperature than other evolved clouds. Wu et al. (2006) reported a mean temperature of 19 K in massive water maser sources excluding known HII regions, and in ultra-compact HII (UC HII) regions. Churchwell et al. (1990) give a significantly higher temperature spread from 15 K to $>$60 K for UC HII regions. The temperature of the cloud is a good tracer to determine its evolutionary stage in the context of massive star formation. The low temperatures in our cloud suggests that it is not an active high-mass star forming region yet. The C18O line widths are typically a factor of two to five times smaller than the line widths in more evolved massive star forming regions traced by other molecules with similar densities. The NH3 line widths we find are comparable to the massive cores reported by Pillai et al. (2006) and Wu et al. (2006), but are smaller than those associated with water masers or are in proximity to UC HII regions. More specifically, the line widths of our cloud are similar to those of the most quiescent NH3 cores with no methanol maser or 24 GHz continuum emission described by Longmore et al. (2007) and smaller than those in more evolved cores with maser or continuum emission. Our cloud also presents similar NH3 line width to those bright-rimmed clouds which are undergoing recently initiated star formation and being subjected to intense levels of ionizing radiation (Morgan et al., 2010). After removing the thermal broadening contributions to the line widths, the velocity dispersions associated with turbulence are supersonic and lie between those of the sources triggered by the radiatively driven implosion and the non-triggered ones. For the more evolved clouds in the UC HII phase, Churchwell et al. (1990) reported significantly larger line widths of $\sim 3$ km s-1. The large line widths in more evolved stages are likely due to a combination of warmer temperatures and broadening from dynamics such as outflows. The HCO+ (1$-$0) spectra associated with hyper-compact H II regions have a considerably larger line width (15-60 km s-1) reported by Churchwell et al. (2010). For our cloud, our HCO+ line profiles are similar to the samples of Brand et al. (2001) and Cesaroni et al. (1999), which are all in the pre-UC HII stage of star formation, though some sources in Cesaroni et al. (1999) are proven to be more evolved than those in Brand et al. (2001). We find similar line widths and profiles as theirs, with a self-absorption dip between the blue and red peak. However, our cloud is colder than the pre-UC HII sources and do not have any associated IRAS point source. The optical depths of NH3 (1,1) are comparable to the cores given by Pillai et al. (2006) and Longmore et al. (2007). Our core 4, 5 and 6 have a lower optical depth than the NH3 sources ($\sim 2.7-2.9$) associated with methanol masers or 24-GHz continuum emission in Longmore et al. (2007), while core 1, 2 and 3 near the cloud center gives higher optical depth but still lower than those samples ($\sim 3.1$) with only NH3 association. However, due to the large uncertainty, without further observation, it would be premature to classify these sources into more detailed evolutionary stages based on optical depth. Our cloud density is about an order of magnitude lower than the typical value in more evolved massive star-forming regions: an average density of $10^{6}$ cm-3 is given by Beuther et al. (2002) using LVG calculations in clouds prior to building up an UC HII region, and a similar value is given by Pillai et al. (2007) in clouds harboring a UC HII region. Our NH3 (1,1) core sizes lie between the mean value of 0.28 pc given by Longmore et al. (2007) and 0.57 pc given by Pillai et al. (2006), but are significantly smaller than the mean size of 1.6 pc reported by Wu et al. (2006). The small values of Longmore et al. (2007) probably result from their small beam size (11″), but the large values of Wu et al. (2006) are likely a result of evolution. All these aspects suggest that our cloud is in an early evolutionary stage of massive star formation. These quiescent cores are likely the candidates for pre-protostellar cores. ### 4.4 Rotation in the cores One of the possible generators of the velocity gradients mentioned in §3.4 is rotation in the cores. Goodman et al. (1993) has discussed the relation between rotation and the geometric properties of the cores. They found no causal relation between velocity gradient direction and core elongation, nor any relationship between the magnitude of the gradient and core shape, on the size scale of $10^{17}$ cm. An examination of the gradients in our clouds also reveals no obvious relation of this kind. In view of the early evolutionary state and the dominant role of non-thermal line broadening in the cloud, we believe that the velocity gradients in the cloud are more affected by stochastic processes, such as fragmentation, collisions, and nonuniform magnetic fields rather than ordered motions such as rotation. We use the parameter $\beta$ as defined by Goodman et al. (1993) to compare the rotational kinetic energy to the gravitational energy. Thus $\beta$ can be written as $\beta={(1/2)I\omega^{2}\over qGM^{2}/R}={1\over 2}{p\over q}{\omega^{2}R^{3}\over GM},$ where $I$ is the moment of inertia given by $I=pMR^{2}$, $qGM^{2}/R$ represents the gravitational potential energy of the mass $M$ within a radius $R$, and $\omega=\mathscr{G}/\sin i$, where $i$ is the inclination of the core along the line of sight. We assume $p/q=0.22$ as for a sphere with an $r^{-2}$ density profile and $\sin i=1$. Our $\beta$ values as listed in Table 6 are consistent with the result of Goodman et al. (1993) that most clouds have $\beta\leq 0.05$. The small values of $\beta$ show that the effect of rotation is not significant in maintaining the overall dynamical stability for the cloud. The results also indicate that these cores are unlikely to experience instabilities driven by rotation (e.g., bars, fission, or rings), if the magnetic fields are not taken into account. ## 5 Summary The multiple molecular line observations of the infrared dark cloud MSXDC G084.81$-$01.09 were analyzed. The results are summarized as below. The cloud has a low gas temperature of 12 K, a column density of $10^{22}$ cm-2 and masses of the cores range from 60 to 250 $M_{\sun}$. The spatial distribution of the ammonia emission correlates well with the mid-infrared absorption and millimeter dust emission. All 6 ammonia cores are associated with their corresponding dust peaks. The line width of 1 km s-1 for ammonia indicates the dominant role of non-thermal broadening in the cloud. The abundances of ammonia range from $2-3.5\times 10^{-8}$. These facts show that the cloud is a site of potential massive star formation at a very early evolutionary stage. 13CO and C18O trace a more extended cloud structure. Together with the HCO+ emission, we detect an expanding envelope in part of the cloud with a expansion velocity of 1 km s-1. Five of the cores are gravitationally bound (four are virialised) with Jeans masses comparable or exceeding their molecular mass except a possible unbound transient core. We found velocity gradients in the cores using different molecular tracers. The different directions and magnitudes of the gradients using the different tracers indicate clumping motions on different scales. The observations presented here are part of a larger program designed to search and investigate potential massive star forming regions. Higher resolution observations with interferometers would be needed to study turbulence and fragmentation inside our cores. Additional submillimeter mapping observations could derive the dust temperature and structure associated with the gas. We could also use radio polarization observation to study the relationship between magnetic field and cloud structure at the early phrases of massive star forming. This work is made based on observations with the 100-m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg and Delingha 13.7-m telescope of Purple Mountain Observatory. This work is also based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. The authors appreciate all the staff members of the observatories for their help during the observations. This work was supported by the Chinese NSF through grants NSF 11073054, NSF 10733030, NSF 10703010 and NSF 10621303, and NBRPC (973 Program) under grant 2007CB815403. ## References * Aguirre et al. 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A., & Myers, P. C. 1989, ApJ, 346, 168 Figure 1: Spitzer MIPS 24 $\mu$m image overlaid with an NH3 (1,1) contour map of integrated intensity from the MSX dark cloud G084.81$-$01.09. The spectra were integrated over $-$2.5 to 4.5 km s-1. The peak contour flux is 8 K km s-1 with contour interval 1 K km s-1. The red star indicates the ionizing star of W80 complex, 2MASS J205551.25+435224.6, reported by Comeron & Pasquali (2005). The cross indicates the MYSO candidate, G084.7847$-$01.1709, identified by Urquhart et al. (2009). The green triangles indicate the star clusters detected by Camberésy et al. (2002). The white circle shows the beamsize of the NH3 (1,1) observation. Figure 2: CSO BOLOCAM 1.1 mm continuum image overlaid with an integrated intensity contour map of observed molecular lines. (a) NH3 (1,1) integrated over $-$2.5 to 4.5 km s-1, with a bold line at 50% of maximum intensity of 8.7 K km s-1 in steps of 9%. (b) 13CO(1$-$0) integrated over $-$0.5 to 2.5 km s-1, with a bold line at 70% of maximum intensity of 38.2 K km s-1 in steps of 5%. (c) C18O(1$-$0) integrated over $-$0.5 to 2.5 km s-1, with bold line at 70% of the maximum intensity of 8.3 K km s-1 in steps of 6%. (d) HCO+(1$-$0) integrated over $-$2.5 to 4.5 km s-1, with bold line at 70% of the maximum intensity of 7.7 K km s-1 in steps of 8%. Four peaks in the HCO+ map are designated as A-D. The cross indicates the MYSO candidate, G084.7847$-$01.1709, identified by Urquhart et al. (2009). The small circles indicate the peak positions of clumps identified from the continuum map (Rosolowsky et al., 2009). The white circle at the bottom left shows the beamsize of the NH3 (1,1) observation. Figure 3: $\rm T_{MB}$ of 13CO(1$-$0) vs. that of C18O(1$-$0) for G084.81$-$01.09. The data are extracted within a velocity range of $-$1.5 to 3.7 km s-1 and smoothed to 0.2 km s-1 spectral resolution. Each dot represents the $\rm T_{MB}$ of the same channel at same position for 13CO and C18O. Data points from the northern and southern clumps are shown in black and grey colors respectively. The dashed-line represents the local ISM ratio. Note that the northern and southern clumps have a different trend for the intensity ratio at high intensities. Figure 4: Left panel: Color-coded map of the NH3 (1,1) full width at half maximum (FWHM) line width in the MSX dark cloud G084.81$-$01.09 overlaid on a contour map of the integrated intensity. Right panel: Color-coded map of the local standard of rest velocity (Vlsr) overlaid on a contour map of the integrated intensity. Only spectra with signal greater than 5$\sigma$ (FWHM map) and 3$\sigma$ (Vlsr map) are shown. The circle in the bottom right shows the beamsize. Figure 5: Spectra of G084.81$-$01.09 toward the six peak positions (labeled as P1–P6). The first two rows show NH3 (1,1) and (2,2) spectra. The NH3 (2,2) lines are moved upward in each panel for clarity. The main (1,1) line feature of P6 was extracted and drawn beside the spectra. The next two rows show the 13CO (1$-$0) and C18O (1$-$0) spectra. In all cases, the stronger line belongs to 13CO. The peak positions are referred to in Table 2. Figure 6: Left: HCO+ (solid) and C18O (dashed) spectra at the positions of the 4 NH3 cores and the 4 HCO+ peaks (indicated by red characters in Fig. 2) from north to south. The vertical dashed lines indicate the Vlsr of the NH3 (1,1) line, obtained from a Gaussian fit. Right: Map of $\delta V$, the velocity difference between the peak of C18O and HCO+ overlaid on a C18O contour map of the region. Figure 7: NH3 (1,1) channel maps for G084.81$-$01.09. The central velocity of each channel, in km s-1, is marked on the top-left corner for each map. The reference position is at 20h57m00.9s,+43°44′15.4″. Figure 8: 13CO(1$-$0) channel maps (upper panels) for G084.81$-$01.09 and C18O(1$-$0) channel maps (lower panels) for G084.81$-$01.09. The central velocity of each channel, in km s-1, is marked on the top-left corner for each map, and the plus signs indicate the positions of 6 cores identified from NH3 (1,1) map (see Fig. 1). The reference position is the same as in Fig. 7. Figure 9: The position-velocity map of 13CO (top), C18O (second row), HCO+ (third row) and NH3 (1,1) main component (bottom) along the Declination axis for the MSX dark cloud G084.81$-$01.09, at Right Ascension of 20h56m47.1s (left panel with offset $\Delta\delta$=$-$150″) and Right Ascension of 20h57m03.7s (right panel with offset $\Delta\delta$=30″). The declination reference coordinate is the same as in Fig. 7. Figure 10: Abundances relative to 13CO toward peak positions (Table 2). The relative abundances were shown in different grey scale. Figure 11: Relation between the NH3 (1,1) and C18O (1-0) integrated intensity and the 1.1 mm flux density. Continuum data were smoothed to corresponding resolution. The uncertanties of integrated intensity are the rms noise of the spectra and the error in dust flux is calculated following Rosolowsky et al. (2009). The solid line represents the least-square fit of the data weighted in both coordinates. Table 1: Observation parameters Line | $\nu_{0}$ | Telescope | During | HPBW | $\delta\nu$ | $\delta v$ | $T_{\rm sys}$ ---|---|---|---|---|---|---|--- | (GHz) | | | (″) | (KHz) | (km s-1) | (K) NH3 ($J,K$=1, 1) | 23.694496 | Effelsberg 100m | 2007 Dec. | 40 | 19.5 | 0.24 | 30-150 | | | 2008 Feb. | 40 | 30.5 | 0.38 | 30-45 NH3 ($J,K$=2, 2) | 23.722633 | Effelsberg 100m | 2007 Dec. | 40 | 19.5 | 0.24 | 30-140 | | | 2008 Feb. | 40 | 30.5 | 0.38 | 30-45 NH3 ($J,K$=3, 3) | 23.870130 | Effelsberg 100m | 2007 Dec. | 40 | 19.5 | 0.24 | 30-140 | | | 2008 Feb. | 40 | 30.5 | 0.38 | 30-45 NH3 ($J,K$=4, 4) | 24.139417 | Effelsberg 100m | 2007 Dec. | 40 | 19.5 | 0.24 | 30-160 12CO ($J$=1-0) | 115.271204 | DLH 13.7m | 2008 Mar.-Apr. | 60 | 142.1 | 0.37 | 200-400 13CO ($J$=1-0) | 110.201353 | DLH 13.7m | 2008 Mar.-Apr. | 64 | 41.7 | 0.11 | 200-400 C18O ($J$=1-0) | 109.782183 | DLH 13.7m | 2008 Mar.-Apr. | 64 | 42.1 | 0.11 | 200-400 HCO+ ($J$=1-0) | 89.188530 | DLH 13.7m | 2008 May.-Jun. | $\sim$78 | 42.1 | 0.14 | 270-420 Note. — $\nu_{0}$ is the rest frequency of the line, HPBW is the half power beam width of the telescope, $\delta\nu$ and $\delta v$ represent the frequency and velocity resolutions Table 2: Cores detected in the NH3 observations of G084.81$-$01.09 Peak | $\alpha$ | $\delta$ | Transition | $V_{\rm LSR}$ | $T_{\rm MB}$ | $FWHM$ | $\tau_{\rm main}$ | $\int{T_{A}^{*}dv}$ ---|---|---|---|---|---|---|---|--- | (J2000) | (J2000) | | (km s-1) | (K) | (km s-1) | | (K km s-1) 1 …… | 20 56 47.1 | +43 44 45 | 1-1 | 1.25$\pm$0.01 | 3.84$\pm$0.24 | 1.38$\pm$0.03 | 3.63$\pm$0.20 | 8.73$\pm$0.20 | | | 2-2 | 1.30$\pm$0.04 | 1.14$\pm$0.19 | 0.96$\pm$0.14 | | 1.39$\pm$0.19 2 …… | 20 56 47.1 | +43 43 15 | 1-1 | 1.23$\pm$0.01 | 4.98$\pm$0.37 | 0.75$\pm$0.02 | 3.61$\pm$0.24 | 7.26$\pm$0.19 | | | 2-2 | 1.20$\pm$0.03 | 1.51$\pm$0.17 | 0.88$\pm$0.09 | | 1.61$\pm$0.19 3 …… | 20 57 03.7 | +43 40 45 | 1-1 | 1.22$\pm$0.01 | 3.67$\pm$0.36 | 0.89$\pm$0.04 | 3.25$\pm$0.29 | 6.03$\pm$0.19 | | | 2-2 | 1.20$\pm$0.04 | 1.07$\pm$0.19 | 0.63$\pm$0.08 | | 1.09$\pm$0.19 4 …… | 20 57 00.9 | +43 38 45 | 1-1 | 0.99$\pm$0.01 | 3.71$\pm$0.31 | 1.00$\pm$0.03 | 2.22$\pm$0.20 | 5.91$\pm$0.18 | | | 2-2 | 0.84$\pm$0.10 | 0.58$\pm$0.14 | 1.40$\pm$0.28 | | 0.78$\pm$0.18 5 …… | 20 57 03.7 | +43 37 15 | 1-1 | 1.20$\pm$0.01 | 3.52$\pm$0.25 | 0.97$\pm$0.03 | 2.32$\pm$0.17 | 5.53$\pm$0.10aaPeak 5 was observed in FFT mode with lower noise compared to Peak 1-4 and 6 in frequency switch mode. | | | 2-2 | 1.21$\pm$0.06 | 0.72$\pm$0.09 | 1.32$\pm$0.21 | | 1.02$\pm$0.11 6 …… | 20 56 55.4 | +43 47 15 | 1-1 | 2.01$\pm$0.02 | 2.51$\pm$0.23 | 1.40$\pm$0.04 | 2.08$\pm$0.21 | 5.02$\pm$0.18 | | | 2-2 | 2.07$\pm$0.14 | 0.53$\pm$0.12 | 1.98$\pm$0.34 | | 1.39$\pm$0.19 Note. — Columns are peak number, offset position from map center, NH3 (J,K) transition, the local standard rest velocity, main beam brightness temperature, full width at half maximum, optical depth and integrated intensity of the main line. Values in column (4), (5), (6) and (7) are the HFS or GAUSS fitting results and errors estimated in CLASS. Integrated intensities are calculated from $-$2.5 to 4.5 km s-1 with errors derived from rms. Table 3: Physical properties of the NH3 cores Number | $T_{\rm ex}$ | $T_{\rm rot}$aaNH3 rotational temperature given by Ho & Townes (1983) as $T_{\rm rot}=-41.5\div\ln\left\\{{-0.282\over\tau_{m}(1,1)}\ln\left[1-{T_{R}^{*}(2,2,m)\over T_{R}^{*}(1,1,m)}\times(1-e^{-\tau_{m}(1,1)})\right]\right\\}.$ | $T_{\rm kin}$bbNH3 kinetic temperature $T_{\rm kin}={T_{\rm rot}\over 1-{T_{\rm rot}\over 42}\ln[1+1.1\exp(-16/T_{\rm rot})]}$. | $N(\rm NH_{3})$ccNH3 column density derived from its optical depth via the relation given by Bachiller et al. (1987) as $N({\rm NH_{3}})=2.784\times 10^{13}\tau J(T_{\rm ex})\Delta V\times Q/Q_{1},$ where $Q$ is the partition function and $Q/Q_{1}={1\over 3}e^{23.4/T_{\rm rot}}+1+{5\over 3}e^{-41.5/T_{\rm rot}}+{14\over 3}e^{-101.5/T_{\rm rot}}+...$. | $\Delta v_{1}$ddThe mean line width is the fitting error-weighted average line width over the core regions. | $\Delta v_{2}$eeThe line width of the mean spectrum is the line width of mean spectrum of the spectra inside the core regions. ---|---|---|---|---|---|--- | (K) | (K) | (K) | ($10^{14}\rm cm^{-2}$) | (km s-1) | (km s-1) 1 …… | 6.7$\pm$0.5 | 11.4$\pm$0.6 | 12.2$\pm$0.7 | 33.5$\pm$2.9 | 1.30$\pm$0.01 | 1.56$\pm$0.01 2 …… | 7.8$\pm$0.8 | 11.5$\pm$0.5 | 12.3$\pm$0.6 | 21.1$\pm$2.2 | 0.95$\pm$0.01 | 1.21$\pm$0.01 3 …… | 6.5$\pm$0.8 | 11.7$\pm$0.7 | 12.5$\pm$0.9 | 18.3$\pm$2.5 | 0.87$\pm$0.01 | 0.98$\pm$0.02 4 …… | 6.9$\pm$0.8 | 10.5$\pm$0.7 | 11.1$\pm$0.9 | 17.5$\pm$2.2 | 1.06$\pm$0.01 | 1.19$\pm$0.02 5 …… | 6.6$\pm$0.6 | 11.2$\pm$0.5 | 11.9$\pm$0.6 | 15.3$\pm$1.6 | 1.01$\pm$0.01 | 1.05$\pm$0.02 6 …… | 5.6$\pm$0.8 | 11.5$\pm$0.8 | 12.4$\pm$1.0 | 16.1$\pm$2.2 | 1.21$\pm$0.02 | 1.49$\pm$0.03 Note. — Columns are peak number, excitation temperature, NH3 rotational temperature, kinetic temperature, NH3 column density derived from the optical depth, the mean line width and the line width of the mean spectrum of the regions. Table 4: CO and HCO+ line parameters at the positions of the NH3 cores. Peak | Molecule | $V_{\rm LSR}$ | $T^{*}_{\rm R}$ | $\Delta V$ | $\int{T_{R}^{*}dV}$ | $\sigma$ ---|---|---|---|---|---|--- | | (km s-1) | (K) | (km s-1) | (K km s-1) | (K) 1 …… | 12CO (1$-$0) | 2.31 | 10.76 | 6.38 | 89.21$\pm$0.44 | 0.20 | 13CO (1$-$0) | 0.96 | 8.00 | 3.19 | 31.44$\pm$0.22 | 0.22 | C18O (1$-$0) | 0.83 | 3.04 | 2.17 | 6.41$\pm$0.10 | 0.17 | HCO+ (1$-$0) | 1.44 | 1.00 | 4.76 | 4.99$\pm$0.25 | 0.21 2 …… | 12CO (1$-$0) | 2.67 | 11.02 | 6.47 | 89.02$\pm$0.72 | 0.33 | 13CO (1$-$0) | 1.03 | 7.48 | 3.08 | 30.84$\pm$0.22 | 0.22 | C18O (1$-$0) | 1.04 | 2.75 | 1.83 | 5.21$\pm$0.13 | 0.21 | HCO+ (1$-$0) | 1.41 | 1.61 | 3.11 | 5.83$\pm$0.20 | 0.17 3 …… | 12CO (1$-$0) | 2.21 | 9.86 | 7.24 | 95.16$\pm$0.51 | 0.23 | 13CO (1$-$0) | 0.90 | 5.87 | 4.05 | 28.53$\pm$0.25 | 0.25 | C18O (1$-$0) | 1.08 | 2.51 | 1.72 | 4.49$\pm$0.15 | 0.24 | HCO+ (1$-$0) | $-$0.52 | 1.08 | 2.11 | 2.44$\pm$0.24 | 0.20 4 …… | 12CO (1$-$0) | 2.64 | 9.79 | 6.69 | 85.58$\pm$0.44 | 0.20 | 13CO (1$-$0) | 1.11 | 5.71 | 3.30 | 25.15$\pm$0.21 | 0.20 | C18O (1$-$0) | 1.04 | 2.83 | 1.62 | 4.80$\pm$0.10 | 0.15 5 …… | 12CO (1$-$0) | 2.80 | 9.76 | 6.34 | 77.12$\pm$0.43 | 0.20 | 13CO (1$-$0) | 1.32 | 5.57 | 3.14 | 23.28$\pm$0.21 | 0.21 | C18O (1$-$0) | 1.24 | 2.47 | 1.50 | 3.99$\pm$0.11 | 0.17 6 …… | 12CO (1$-$0) | 2.47 | 10.89 | 5.85 | 77.47$\pm$0.45 | 0.20 | 13CO (1$-$0) | 1.67 | 8.34 | 2.70 | 28.82$\pm$0.23 | 0.23 | C18O (1$-$0) | 1.57 | 2.62 | 2.08 | 5.18$\pm$0.10 | 0.16 | HCO+ (1$-$0) | 2.08 | 1.35 | 3.33 | 4.91$\pm$0.19 | 0.16 Note. — 12CO, 13CO, C18O and HCO+ line parameters are calculated within the velocity range from -3.5 to 6.8 km s-1, -2.7 to 7.7 km s-1, -1.0 to 3.0 km s-1 and 0.0 to 9.0 km s-1, respectively. The HCO+ line was not observed towards cores P4 and P5 due to low signal-to-noise ratio. Table 5: Physical properties derived from CO. | $\rm{}^{12}CO$ | | $\rm{}^{13}CO$ | | $\rm C^{18}O$ ---|---|---|---|---|--- Peak | $T_{\rm ex}$ | | $\tau$ | $N$ | | $\tau$ | $N$ | (K) | | | ($10^{15}$cm-2) | | | ($10^{15}$cm-2) 1 …… | 14.17$\pm$0.20 | | 1.26$\pm$0.09 | 58.36$\pm$2.12 | | 0.35$\pm$0.03 | 8.91$\pm$0.31 2 …… | 14.43$\pm$0.33 | | 1.18$\pm$0.09 | 56.27$\pm$2.32 | | 0.31$\pm$0.03 | 6.35$\pm$0.33 3 …… | 13.26$\pm$0.22 | | 0.90$\pm$0.07 | 45.31$\pm$1.55 | | 0.34$\pm$0.03 | 6.37$\pm$0.32 4 …… | 12.90$\pm$0.19 | | 0.91$\pm$0.06 | 39.87$\pm$1.18 | | 0.39$\pm$0.03 | 6.63$\pm$0.30 5 …… | 12.25$\pm$0.20 | | 1.01$\pm$0.08 | 37.02$\pm$1.29 | | 0.33$\pm$0.03 | 4.70$\pm$0.26 6 …… | 14.30$\pm$0.21 | | 1.35$\pm$0.10 | 54.75$\pm$2.19 | | 0.30$\pm$0.02 | 7.88$\pm$0.29 Note. — First column is core number. Column densities are calculated under the assumption of local thermodynamic equilibrium (LTE), in which 13CO and C18O have the same excitation temperature as that of optically thick 12CO. Table 6: Velocity gradients detected in the cores Core | Molecule | $v_{0}\pm\sigma_{v_{0}}$ | $\mathscr{G}\pm\sigma_{\mathscr{G}}$ | $\mathscr{G}$ at 580 pc | $\Theta_{\mathscr{G}}\pm\sigma_{\Theta_{\mathscr{G}}}$ | $\beta$ ---|---|---|---|---|---|--- | | (km s-1) | (m s-1 arcsec-1) | (km s-1 pc-1) | (deg E of N) | 1 …… | 13CO (1$-$0) | 1.01$\pm$0.003 | 04.25$\pm$0.096 | 1.51 | $-$8.3$\pm$0.8 | | C18O (1$-$0) | 0.91$\pm$0.007 | 03.69$\pm$0.205 | 1.31 | $-$12.1$\pm$2.0 | | NH3 (1,1) | 1.15$\pm$0.014 | 10.76$\pm$0.339 | 3.83 | $-$37.2$\pm$1.2 | 1.3E-2 | NH3 (2,2) | 1.28$\pm$0.024 | 08.76$\pm$0.927 | 3.12 | $-$50.5$\pm$4.4 | 3 …… | NH3 (1,1) | 1.19$\pm$0.012 | 03.69$\pm$0.337 | 1.31 | 165.0$\pm$4.0 | 2.3E-3 4 …… | NH3 (1,1) | 1.01$\pm$0.011 | 04.32$\pm$0.419 | 1.54 | $-$64.5$\pm$4.0 | 3.6E-3 5 …… | NH3 (1,1) | 1.12$\pm$0.013 | 02.31$\pm$0.497 | 0.82 | $-$123.8$\pm$8.5 | 6.2E-4 6 …… | 13CO (1$-$0) | 1.66$\pm$0.004 | 02.87$\pm$0.188 | 1.02 | 16.5$\pm$2.2 | | C18O (1$-$0) | 1.54$\pm$0.010 | 04.19$\pm$0.507 | 1.49 | 12.6$\pm$4.5 | | NH3 (1,1) | 1.81$\pm$0.018 | 09.45$\pm$0.772 | 3.36 | 90.2$\pm$3.9 | 1.5E-2 | NH3 (2,2) | 1.88$\pm$0.048 | 14.92$\pm$2.772 | 5.31 | 45.6$\pm$7.6 | Note. — Columns are core number, molecules fitted, systemic velocity, magnitude of the velocity gradient, $\mathscr{G}$ at cloud distance, direction of increasing velocity (measured east of north), and parameter $\beta$ given in §4.4. Errors quoted are $1~{}\sigma$ uncertainties. Table 7: NH3 abundance in the cores Peak | $N(\rm H_{2})_{C^{18}O}$ | $\chi_{\rm{NH}_{3}}$ | $N(\rm H_{2})_{1.12mm}$ | $\chi^{\prime}_{\rm{NH}_{3}}$ | $\chi~{}{\rm in~{}core}$ ---|---|---|---|---|--- | ($10^{22}\rm cm^{-2}$) | ($10^{-8}$) | ($10^{22}\rm cm^{-2}$) | ($10^{-8}$) | ($10^{-8}$) 1 …… | 6.23 | 5.4 | 5.36 | 6.3 | 2.41 2 …… | 4.45 | 4.7 | 5.54 | 3.8 | 2.74 3 …… | 4.46 | 4.1 | 2.85 | 6.4 | 2.46 4 …… | 4.64 | 3.8 | 2.16 | 8.1 | 2.24 5 …… | 3.29 | 4.7 | 0.96 | 15.9 | 1.88 6 …… | 5.52 | 2.9 | 1.25 | 12.9 | 2.04 Note. — The columns show the core number, H2 column density and NH3 abundance $\chi$ based on C18O observations (column 2 and 3) and 1.1 mm dust flux (column 4 and 5), and the average NH3 abundance over the cores. Table 8: Masses of the NH3 cores Region | R | $n({\rm H_{2}})$ | $M_{\rm Jeans}$ | $M_{\rm Virial}$ | $M_{\rm Molecular}$ | $\alpha$ | $v_{\rm escape}$ ---|---|---|---|---|---|---|--- | (arcsec) | ($10^{4}\rm cm^{-3}$) | ($M_{\sun}$) | ($M_{\sun}$) | ($M_{\sun}$) | | (km s-1) 1 …… | 69.0 | 13.8 | 198 | 206 | 208 | 0.99 | 3.03 2 …… | 74.9 | 13.2 | 205 | 121 | 255 | 0.41 | 3.22 3 …… | 69.0 | 8.7 | 259 | 92 | 130 | 0.71 | 2.40 4 …… | 60.1 | 10.4 | 198 | 119 | 104 | 1.15 | 2.30 5 …… | 46.7 | 16.6 | 174 | 85 | 77 | 1.10 | 2.24 6 …… | 49.7 | 11.0 | 228 | 129 | 62 | 2.08 | 1.95 Note. — Columns are core number, size of the cores, H2 densities derived from C18O, Jeans mass, virial masses from NH3 and gas mass, virial parameter $\alpha$, and escape velocity.
arxiv-papers
2011-01-21T05:31:11
2024-09-04T02:49:16.559837
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. B. Zhang, J. Yang, Y. Xu, J. D. Pandian, K. M. Menten, C. Henkel", "submitter": "Shaobo Zhang", "url": "https://arxiv.org/abs/1101.4072" }
1101.4161
# Parameter rigid actions of simply connected nilpotent Lie groups Hirokazu Maruhashi 111Research Fellow of the Japan Society for the Promotion of Science Department of Mathematics, Kyoto University ###### Abstract We show that for a locally free $C^{\infty}$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [8] concerning the Heisenberg groups. ## 1 Introduction Let $G$ be a connected Lie group with Lie algebra ${\mathfrak{g}}$ and $M$ a $C^{\infty}$-manifold without boundary. Let $\rho:M\times G\to M$ be a $C^{\infty}$ right action. We call $\rho$ locally free if every isotropy subgroup of $\rho$ is discrete in $G$. Assume that $\rho$ is locally free. Then we have the orbit foliation ${\mathcal{F}}$ of $\rho$ whose tangent bundle $T{\mathcal{F}}$ is naturally isomorphic to a trivial bundle $M\times{\mathfrak{g}}$. The action $\rho$ is parameter rigid if any action $\rho^{\prime}$ of $G$ on $M$ with the same orbit foliation ${\mathcal{F}}$ is $C^{\infty}$-conjugate to $\rho$, more precisely, there exist an automorphism $\Phi$ of $G$ and a $C^{\infty}$-diffeomorphism $F$ of $M$ which preserves each leaf of ${\mathcal{F}}$ and homotopic to identity through $C^{\infty}$-maps preserving each leaf of ${\mathcal{F}}$ such that $F(\rho(x,g))=\rho^{\prime}(F(x),\Phi(g))$ for all $x\in M$ and $g\in G$. Parameter rigidity of actions has been studied by several authors, for instance, Katok and Spatzier [3], Matsumoto and Mitsumatsu [4], Mieczkowski [5], dos Santos [8] and Ramírez [7]. Most of known examples of parameter rigid actions are those of abelian groups and nonabelian actions have not been considered so much. Parameter rigidity is closely related to cocycles over actions. Now suppose $G$ is contractible and $M$ is compact. Let $H$ be a Lie group. A $C^{\infty}$-map $c:M\times G\to H$ is called a $H$-valued cocycle over $\rho$ if $c$ satisfies $c(x,gg^{\prime})=c(x,g)c(\rho(x,g),g^{\prime})$ for all $x\in M$ and $g,g^{\prime}\in G$. A cocycle $c$ is constant if $c(x,g)$ is independent of $x$. A constant cocycle is just a homomorphism $G\to H$. $H$-valued cocycles $c,c^{\prime}$ are cohomologous if there exists a $C^{\infty}$-map $P:M\to H$ such that $c(x,g)=P(x)^{-1}c^{\prime}(x,g)P(\rho(x,g))$ for all $x\in M$ and $g\in G$. The action $\rho$ is $H$-valued cocycle rigid if every $H$-valued cocycle over $\rho$ is cohomologous to a constant cocycle. ###### Proposition 1 ([4]). If $\rho$ is $G$-valued cocycle rigid, then it is parameter rigid. ###### Remark. In [4] Matsumoto and Mitsumatsu assume that $\rho$ has at least one trivial isotropy subgroup, but this assumption is not necessary. ###### Proposition 2 ([4]). When $G={\mathbb{R}}^{n}$, the following are equivalent: 1. 1. $\rho$ is ${\mathbb{R}}$-valued cocycle rigid. 2. 2. $\rho$ is ${\mathbb{R}}^{n}$-valued cocycle rigid. 3. 3. $\rho$ is parameter rigid. ###### Remark. The equivalence of the first two conditions is obvious. In this paper we consider actions of simply connected nilpotent Lie groups. In [8], dos Santos proved that for actions of a Heisenberg group $H_{n}$, ${\mathbb{R}}$-valued cocycle rigidity implies $H_{n}$-valued cocycle rigidity and using this, he constructed parameter rigid actions of Heisenberg groups. To the best of my knowledge these are the only known nontrivial parameter rigid actions of nonabelian nilpotent Lie groups. We prove the following. ###### Theorem 1. Let $N$ be a connected and simply connected nilpotent Lie group, $M$ a compact manifold and $\rho$ a locally free $C^{\infty}$-action of $N$ on $M$. Then, the following are equivalent: 1. 1. $\rho$ is ${\mathbb{R}}$-valued cocycle rigid. 2. 2. $\rho$ is $N$-valued cocycle rigid. 3. 3. $\rho$ is parameter rigid and every orbitwise constant real valued $C^{\infty}$-function of $\rho$ on $M$ is constant on $M$. This theorem enables us to construct parameter rigid actions of nilpotent Lie groups. The most interesting one is the following. ###### Theorem 2 ([7]). Let $N$ denote the group of all upper triangular real matrices with $1$ on the diagonal, $\Gamma$ a cocompact lattice of ${\rm SL}(n,{\mathbb{R}})$ and $\rho$ the action of $N$ on $\Gamma\backslash{\rm SL}(n,{\mathbb{R}})$ by right multiplication. If $n\geq 4$, $\rho$ is ${\mathbb{R}}$-valued cocycle rigid. ###### Remark. In [7], Ramírez proved more general theorems. ###### Corollary. The above action $\rho$ is parameter rigid. In Section 4 we construct parameter rigid actions of nilpotent groups using Theorem 1. It is a generalization of dos Santos’ example. Let $N$ be a connected and simply connected nilpotent Lie group and $\Gamma$, $\Lambda$ be lattices in $N$. Consider the action of $\Lambda$ on ${\Gamma\backslash N}$ by right multiplication and let $\tilde{\rho}$ be its suspended action of $N$. ###### Theorem 3. If $\Lambda$ is Diophantine with respect to $\Gamma$, then the action $\tilde{\rho}$ of $N$ is parameter rigid. For the definition of Diophantine lattices, see Section 4. ## 2 Preliminaries Let $G$ be a contractible Lie group with Lie algebra ${\mathfrak{g}}$, $M$ a compact manifold and $\rho$ a locally free action of $G$ on $M$ with orbit foliation ${\mathcal{F}}$. Let $H$ be a Lie group with Lie algebra ${\mathfrak{h}}$. We denote by $\Omega^{p}({\mathcal{F}},{\mathfrak{h}})$ the set of all $C^{\infty}$-sections of $\mathop{\mathrm{Hom}}\nolimits(\bigwedge^{p}T{\mathcal{F}},{\mathfrak{h}})$. The exterior derivative ${d_{\mathcal{F}}}:\Omega^{p}({\mathcal{F}},{\mathfrak{h}})\to\Omega^{p+1}({\mathcal{F}},{\mathfrak{h}})$ is defined since $T{\mathcal{F}}$ is integrable. By differentiating, $H$-valued cocycles over $\rho$ are in one-to-one correspondence with ${\mathfrak{h}}$-valued leafwise one forms $\omega\in\Omega^{1}({\mathcal{F}},{\mathfrak{h}})$ such that ${d_{\mathcal{F}}}\omega+[\omega,\omega]=0.$ ###### Proposition 3. Let $c_{1},c_{2}$ be $H$-valued cocycles over $\rho$ and let $\omega_{1},\omega_{2}$ be corresponding differential forms. For a $C^{\infty}$-map $P:M\to H$, the following are equivalent: 1. 1. $c_{1}(x,g)=P(x)^{-1}c_{2}(x,g)P(\rho(x,g))$ for all $x\in M$ and $g\in G$. 2. 2. $\omega_{1}=\mathop{\mathrm{Ad}}\nolimits(P^{-1})\omega_{2}+P^{*}\theta$ where $\theta\in\Omega^{1}(H,{\mathfrak{h}})$ is the left Maurer-Cartan form on $H$. ###### Corollary ([4]). The following are equivalent: 1. 1. $\rho$ is $G$-valued cocycle rigid. 2. 2. For each $\omega\in\Omega^{1}({\mathcal{F}},{\mathfrak{g}})$ such that ${d_{\mathcal{F}}}\omega+[\omega,\omega]=0$, there exist a endomorphism $\Phi:{\mathfrak{g}}\to{\mathfrak{g}}$ of Lie algebra and a $C^{\infty}$-map $P:M\to G$ such that $\omega=\mathop{\mathrm{Ad}}\nolimits(P^{-1})\Phi+P^{*}\theta.$ Proposition 3 is obtained by examining the proof of Corollary Corollary in [4]. In this paper, we will identify a cocycle with its corresponding differential form. Let us consider real valued cocycles. A real valued cocycle over $\rho$ is given by $\omega\in\Omega^{1}({\mathcal{F}},{\mathbb{R}})$ satisfying ${d_{\mathcal{F}}}\omega=0$. Two real valued cocycles $\omega_{1},\omega_{2}$ are cohomologous if and only if $\omega_{1}=\omega_{2}+{d_{\mathcal{F}}}P$ for some $C^{\infty}$-function $P:M\to{\mathbb{R}}$. Leafwise cohomology $H^{*}({\mathcal{F}})$ of ${\mathcal{F}}$ is the cohomology of the cochain complex $(\Omega^{*}({\mathcal{F}},{\mathbb{R}}),{d_{\mathcal{F}}})$. Thus $H^{1}({\mathcal{F}})$ is the set of all equivalence classes of real valued cocycles. The identification $T{\mathcal{F}}\simeq M\times{\mathfrak{g}}$ induces a map $H^{*}({\mathfrak{g}})\to H^{*}({\mathcal{F}})$ where $H^{*}({\mathfrak{g}})$ is the cohomology of the Lie algebra ${\mathfrak{g}}$. By the compactness of $M$, this map is injective on $H^{1}({\mathfrak{g}})$. Hence we identify $H^{1}({\mathfrak{g}})$ with its image. Note that $H^{1}({\mathfrak{g}})$ is the set of all equivalence classes of constant real valued cocycles. Thus real valued cocycle rigidity is equivalent to $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{g}})$. Notice that $H^{0}({\mathcal{F}})$ is the set of leafwise constant real valued $C^{\infty}$-functions of ${\mathcal{F}}$ on $M$ and $H^{0}({\mathfrak{g}})$ consists of constant functions on $M$. Therefore the equivalence of 1 and 3 in Theorem 1 can be stated as follows: $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$ if and only if $\rho$ is parameter rigid and $H^{0}({\mathcal{F}})=H^{0}({\mathfrak{n}})$. ## 3 Proof of Theorem 1 Let $N$ be a simply connected nilpotent Lie group with Lie algebra ${\mathfrak{n}}$, $M$ a compact manifold and $\rho$ a locally free action of $N$ on $M$ with orbit foliation ${\mathcal{F}}$. We first prove that $N$-valued cocycle rigidity implies real valued cocycle rigidity. There exist closed subgroups $N^{\prime}$ and $A$ of $N$ such that $N^{\prime}\triangleleft N$, $N=N^{\prime}\rtimes A$ and $A\simeq{\mathbb{R}}$. Let $c$ be any real valued cocycle over $\rho$. We regard $c$ as a $N$-valued cocycle over $\rho$ via the inclusion ${\mathbb{R}}\simeq A\hookrightarrow N$. By $N$-valued cocycle rigidity, there exist an endomorphism $\Phi$ of $N$ and a $C^{\infty}$-map $P:M\to N$ such that $c(x,g)=P(x)^{-1}\Phi(g)P(\rho(x,g))$ for all $x\in M$ and $g\in N$. Applying the natural projection $\pi:N\to A\simeq{\mathbb{R}}$, we obtain $c(x,g)=(\pi\circ P)(x)^{-1}(\pi\circ\Phi)(g)(\pi\circ P)(\rho(x,g))$. Thus $c$ is cohomologous to a constant cocycle $\pi\circ\Phi$. Next we assume $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$ and prove $N$-valued cocycle rigidity. We need the following two lemmata. ###### Lemma 1. Let $V$ be a finite dimensional real vector space. Assume that $\omega\in\Omega^{1}({\mathcal{F}},V)$ satisfies the equation ${d_{\mathcal{F}}}\omega=\varphi$, where $\varphi\in\mathop{\mathrm{Hom}}\nolimits(\bigwedge^{2}{\mathfrak{n}},V)$ is a constant leafwise two form. Then there exists a constant leafwise one form $\psi\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{n}},V)$ with $\varphi={d_{\mathcal{F}}}\psi$. ###### Proof. Since $N$ is amenable, there exists a $N$-invariant Borel probability measure $\mu$ on $M$. Define $\psi\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{n}},V)$ by $\psi(X)=\int_{M}\omega(X)d\mu$ where $X\in{\mathfrak{n}}$. Since $\varphi(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y])$ for all $X,Y\in{\mathfrak{n}}$, we obtain $\varphi(X,Y)=-\int_{M}\omega([X,Y])d\mu.$ Thus ${d_{\mathcal{F}}}\psi(X,Y)=-\psi([X,Y])=-\int_{M}\omega([X,Y])d\mu=\varphi(X,Y),$ hence ${d_{\mathcal{F}}}\psi=\varphi$. ∎ Set ${\mathfrak{n}}^{1}={\mathfrak{n}}$, ${\mathfrak{n}}^{i}=[{\mathfrak{n}},{\mathfrak{n}}^{i-1}]$. Then ${\mathfrak{n}}^{s}\neq 0$, ${\mathfrak{n}}^{s+1}=0$ for some $s$. For each $1\leq i\leq s$, choose a subspace $V_{i}$ with ${\mathfrak{n}}^{i}=V_{i}\oplus{\mathfrak{n}}^{i+1}$, so that ${\mathfrak{n}}=\bigoplus_{i=1}^{s}V_{i}$. ###### Lemma 2. Let $\omega\in\Omega^{1}({\mathcal{F}},{\mathfrak{n}})$ be such that ${d_{\mathcal{F}}}\omega+[\omega,\omega]=0$. Decompose $\omega$ as $\omega=\xi+\omega_{k}+\omega_{k+1}$ where $\xi\in\Omega^{1}({\mathcal{F}},\bigoplus_{i=1}^{k-1}V_{i})$, $\omega_{k}\in\Omega^{1}({\mathcal{F}},V_{k})$ and $\omega_{k+1}\in\Omega^{1}({\mathcal{F}},{\mathfrak{n}}^{k+1})$. If $\xi$ is constant, then there exists $\omega^{\prime}\in\Omega^{1}({\mathcal{F}},{\mathfrak{n}})$ with ${d_{\mathcal{F}}}\omega^{\prime}+[\omega^{\prime},\omega^{\prime}]=0$ which is cohomologous to $\omega$ and such that $\omega^{\prime}=\xi^{\prime}+\omega_{k+1}^{\prime}$ where $\xi^{\prime}\in\Omega^{1}({\mathcal{F}},\bigoplus_{i=1}^{k}V_{i})$ is constant and $\omega_{k+1}^{\prime}\in\Omega({\mathcal{F}},{\mathfrak{n}}^{k+1})$. ###### Proof. By cocycle equation, $0={d_{\mathcal{F}}}\xi+{d_{\mathcal{F}}}\omega_{k}+{d_{\mathcal{F}}}\omega_{k+1}+[\xi,\xi]+\mbox{an element of}\ \Omega^{2}({\mathcal{F}},{\mathfrak{n}}^{k+1}).$ Comparing $V_{k}$ component, we see that ${d_{\mathcal{F}}}\omega_{k}$ is constant. Hence by Lemma 1, ${d_{\mathcal{F}}}\omega_{k}={d_{\mathcal{F}}}\psi$ for some $\psi\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{n}},V_{k})$. Since we are assuming that $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$, there exists $\psi^{\prime}\in\mathop{\mathrm{Hom}}\nolimits({\mathfrak{n}},V_{k})$ and $C^{\infty}$-map $h:M\to V_{k}$ such that $\omega_{k}=\psi+\psi^{\prime}+{d_{\mathcal{F}}}h.$ Put $P=e^{h}:M\to N$. Let $x\in M$ and $X\in T_{x}{\mathcal{F}}$. Choose a path $x(t)$ such that $X=\frac{d}{dt}x(t)\big{|}_{t=0}$. Let $\theta\in\Omega^{1}(N,{\mathfrak{n}})$ be the left Maurer-Cartan form on $N$. Then $\displaystyle P^{*}\theta(X)$ $\displaystyle=\frac{d}{dt}P(x)^{-1}P(x(t))\Big{|}_{t=0}=\frac{d}{dt}e^{-h(x)}e^{h(x(t))}\Big{|}_{t=0}$ $\displaystyle=\frac{d}{dt}\exp(-h(x)+h(x(t))+\mbox{an element of }{\mathfrak{n}}^{k+1})\Big{|}_{t=0}$ $\displaystyle={d_{\mathcal{F}}}h(X)+\mbox{an element of }{\mathfrak{n}}^{k+1}.$ Thus $P^{*}\theta={d_{\mathcal{F}}}h+\mbox{an element of }\Omega^{1}({\mathcal{F}},{\mathfrak{n}}^{k+1})$. Note that $\mathop{\mathrm{Ad}}\nolimits(P^{-1})=\exp\mathop{\mathrm{ad}}\nolimits(-h)$ is identity on $\bigoplus_{i=1}^{k}V_{i}$ and preserves ${\mathfrak{n}}^{k+1}$. Hence $\displaystyle\omega-P^{*}\theta$ $\displaystyle=\xi+\psi+\psi^{\prime}+\mbox{an element of }\Omega^{1}({\mathcal{F}},{\mathfrak{n}}^{k+1})$ $\displaystyle=\mathop{\mathrm{Ad}}\nolimits(P^{-1})(\xi+\psi+\psi^{\prime}+\mbox{an element of }\Omega^{1}({\mathcal{F}},{\mathfrak{n}}^{k+1})).$ ∎ Let $\omega$ be any $N$-valued cocycle. Using Lemma 2, we can exchange $\omega$ for a cohomologous cocycle whose $V_{1}$-component is constant. Applying Lemma 2 repeatedly, we eventually get a constant cocycle cohomologous to $\omega$. This proves $N$-valued cocycle rigidity. Next we assume that $\rho$ is parameter rigid and $H^{0}({\mathcal{F}})=H^{0}({\mathfrak{n}})$. Let ${\mathfrak{n}}^{i}$ and $V_{i}$ be as above. Note that ${\mathfrak{n}}^{s}$ is central in ${\mathfrak{n}}$. Fix a nonzero $Z\in{\mathfrak{n}}^{s}$. Let $[\omega]\in H^{1}({\mathcal{F}})$. Let $\omega_{0}$ be the $N$-valued cocycle over $\rho$ corresponding to the constant cocycle $\mathrm{id}:N\to N$. We call $\omega_{0}$ the canonical $1$-form of $\rho$. Fix a $\epsilon>0$ and put $\eta:=\omega_{0}+\epsilon\omega Z$. $\eta$ is an $N$-valued cocycle over $\rho$ since ${d_{\mathcal{F}}}\eta+[\eta,\eta]={d_{\mathcal{F}}}\omega_{0}+\epsilon({d_{\mathcal{F}}}\omega)Z+[\omega_{0},\omega_{0}]=0.$ Since $M$ is compact, we can assume $\eta_{x}:T_{x}{\mathcal{F}}\to{\mathfrak{n}}$ is bijective for all $x\in M$ by choosing $\epsilon>0$ small. There exists a unique action $\rho^{\prime}$ of $N$ on $M$ whose orbit foliation is ${\mathcal{F}}$ and canonical $1$-form is $\eta$. See [1]. By parameter rigidity $\rho^{\prime}$ is conjugate to $\rho$. Thus there exist a $C^{\infty}$-map $P:M\to N$ and an automorphism $\Phi$ of $N$ satisfying $\omega_{0}+\epsilon\omega Z=\mathop{\mathrm{Ad}}\nolimits(P^{-1})\Phi_{*}\omega_{0}+P^{*}\theta.$ (1) Note that $\mathop{\mathrm{log}}\nolimits:N\to{\mathfrak{n}}$ is defined since $N$ is simply connected and nilpotent. Let us decompose $\omega_{0}=\sum_{i=1}^{s}\omega_{0i}$, $\Phi_{*}\omega_{0}=\sum_{i=1}^{s}\omega_{0i}^{\prime}$ and $\mathop{\mathrm{log}}\nolimits P=\sum_{i=1}^{s}P_{i}$ according to the decomposition ${\mathfrak{n}}=\bigoplus_{i=1}^{s}V_{i}$. ###### Lemma 3. Assume that $P_{1}=\dotsm=P_{k-1}=0$ i.e. $\mathop{\mathrm{log}}\nolimits P\in{\mathfrak{n}}^{k}$. 1. 1. If $k<s$, then there exist a $C^{\infty}$-map $Q:M\to N$ and an automorphism $\Psi$ of $N$ such that $\omega_{0}+\epsilon\omega Z=\mathop{\mathrm{Ad}}\nolimits(Q^{-1})\Psi_{*}\omega_{0}+Q^{*}\theta$ and $Q_{1}=\dotsm=Q_{k}=0$ where $\mathop{\mathrm{log}}\nolimits Q=\sum_{i=1}^{s}Q_{i}$. 2. 2. If $k=s$, then $\omega$ is cohomologous to a constant cocycle. ###### Proof. For all $X=\frac{d}{dt}x(t)\big{|}_{t=0}\in T_{x}{\mathcal{F}}$, $\displaystyle P^{*}\theta(X)$ $\displaystyle=\frac{d}{dt}P(x)^{-1}P(x(t))\Big{|}_{t=0}=\frac{d}{dt}\exp\left(-\sum_{i=k}^{s}P_{i}(x)\right)\exp\left(\sum_{i=k}^{s}P_{i}(x(t))\right)\Big{|}_{t=0}$ $\displaystyle=\frac{d}{dt}\exp\left\\{\sum_{i=k}^{s}\left(P_{i}(x(t))-P_{i}(x)\right)+\mbox{an element of }{\mathfrak{n}}^{k+1}\right\\}\Big{|}_{t=0}$ $\displaystyle=\frac{d}{dt}\exp\left(P_{k}(x(t))-P_{k}(x)+\mbox{an element of }{\mathfrak{n}}^{k+1}\right)\Big{|}_{t=0}$ $\displaystyle={d_{\mathcal{F}}}P_{k}(X)+\mbox{an element of }{\mathfrak{n}}^{k+1}.$ We have $\displaystyle\mathop{\mathrm{Ad}}\nolimits(P^{-1})\Phi_{*}\omega_{0}$ $\displaystyle=\exp\left(\mathop{\mathrm{ad}}\nolimits\left(-\sum_{i=k}^{s}P_{i}\right)\right)\sum_{i=1}^{s}\omega_{0i}^{\prime}$ $\displaystyle=\sum_{i=1}^{s}\omega_{0i}^{\prime}+\mbox{an element of }{\mathfrak{n}}^{k+1}.$ Comparing the $V_{k}$-component of (1) we get $\omega_{0k}+\delta_{ks}\epsilon\omega Z=\omega_{0k}^{\prime}+{d_{\mathcal{F}}}P_{k}.$ When $k=s$ the equation $\omega Z=\epsilon^{-1}(\omega_{0s}^{\prime}-\omega_{0s})+{d_{\mathcal{F}}}(\epsilon^{-1}P_{s})$ shows that $\omega$ is cohomologous to a constant cocycle. If $k<s$, then ${d_{\mathcal{F}}}P_{k}=\phi\circ\omega_{0}$ for some linear map $\phi:{\mathfrak{n}}\to V_{k}$. For any $X\in{\mathfrak{n}}$, let $\tilde{X}$ denote the vector field on $M$ determined by $X$ via $\rho$. We have $\tilde{X}P_{k}=\phi(X)$ and by integrating over an integral curve $\gamma$ of $\tilde{X}$ we get $P_{k}(\gamma(T))-P_{k}(\gamma(0))=\phi(X)T$ for all $T>0$. Since $M$ is compact, $\phi(X)=0$. Therefore ${d_{\mathcal{F}}}P_{k}=0$, so that $P_{k}$ is constant on each leaf of ${\mathcal{F}}$. Thus $P_{k}$ is constant on $M$ by our assumption. Put $g:=\exp(-P_{k})$ and $Q:=gP=\exp\left(\sum_{i=k+1}^{s}P_{i}+\mbox{an element of }{\mathfrak{n}}^{k+1}\right)$. Then $\displaystyle\omega_{0}+\epsilon\omega Z$ $\displaystyle=\mathop{\mathrm{Ad}}\nolimits(Q^{-1}g)\Phi_{*}\omega_{0}+(L_{g^{-1}}\circ Q)^{*}\theta$ $\displaystyle=\mathop{\mathrm{Ad}}\nolimits(Q^{-1})\Psi_{*}\omega_{0}+Q^{*}\theta$ where $\Psi_{*}:=\mathop{\mathrm{Ad}}\nolimits(g)\Phi_{*}$. ∎ Applying Lemma 3 repeatedly, we see that $\omega$ is cohomologous to a constant cocycle. Finally we assume $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$ and prove that $\rho$ is parameter rigid and $H^{0}({\mathcal{F}})=H^{0}({\mathfrak{n}})$. Parameter rigidity of $\rho$ follows from Proposition 1. Let $f\in H^{0}({\mathcal{F}})$. Fix a nonzero $\phi\in H^{1}({\mathfrak{n}})$ and denote the corresponding leafwise $1$-form on $M$ by $\tilde{\phi}$. Then $f\tilde{\phi}\in H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}})$. Thus there exist $\psi\in H^{1}({\mathfrak{n}})$ and a $C^{\infty}$-function $g:M\to{\mathbb{R}}$ such that $f\tilde{\phi}-\tilde{\psi}={d_{\mathcal{F}}}g$ where $\tilde{\psi}$ is the leafwise $1$-form corresponding to $\psi$. Choose $X\in{\mathfrak{n}}$ satisfying $\phi(X)\neq 0$. Let $x(t)$ be an integral curve of $\tilde{X}$ where $\tilde{X}$ is the vector field corresponding to $X$. We have $f(x(t))\phi(X)-\psi(X)=\tilde{X}_{x(t)}g=\frac{d}{dt}g(x(t))$. By integrating over $[0,T]$, we get $T(f(x(0))\phi(X)-\psi(X))=g(x(T))-g(x(0))$. Since $g$ is bounded, $f(x(0))\phi(X)-\psi(X)$ must be zero. Then $f(x(0))=\frac{\psi(X)}{\phi(X)}$ and $f$ is constant on $M$. This completes the proof of Theorem 1. ## 4 A construction of parameter rigid actions Let us now construct real valued cocycle rigid actions of nilpotent groups. For the structure theory of nilpotent Lie groups, see [2]. A basis $X_{1},\dots,X_{n}$ of ${\mathfrak{n}}$ is called a strong Malcev basis if $\mathop{\mathrm{span}}\nolimits_{{\mathbb{R}}}\\{X_{1},\dots,X_{i}\\}$ is an ideal of ${\mathfrak{n}}$ for each $i$. If $\Gamma$ is a lattice in $N$, there exists a strong Malcev basis $X_{1},\dots,X_{n}$ of ${\mathfrak{n}}$ such that $\Gamma=e^{{\mathbb{Z}}X_{1}}\cdots e^{{\mathbb{Z}}X_{n}}$. Such a basis is called a strong Malcev basis strongly based on $\Gamma$. Let $\Gamma$ and $\Lambda$ be lattices in $N$. ###### Definition 1. $\Lambda$ is Diophantine with respect to $\Gamma$ if there exists a strong Malcev basis $X_{1},\dots,X_{n}$ of ${\mathfrak{n}}$ strongly based on $\Gamma$ and a strong Malcev basis $Y_{1},\dots,Y_{n}$ of ${\mathfrak{n}}$ strongly based on $\Lambda$ such that $Y_{i}=\sum_{j=1}^{i}a_{ij}X_{j}$ for every $1\leq i\leq n$, where $a_{ii}$ is Diophantine. Let $\rho$ be the action of $\Lambda$ on ${\Gamma\backslash N}$ by right multiplication. First we will prove the following. ###### Theorem 4. If $\Lambda$ is Diophantine with respect to $\Gamma$, then every real valued $C^{\infty}$ cocycle $c:{\Gamma\backslash N}\times\Lambda\to{\mathbb{R}}$ over $\rho$ is cohomologous to a constant cocycle. Note that $X_{1}$ is in the center of ${\mathfrak{n}}$. Let $\pi:N\to\bar{N}:={e^{{\mathbb{R}}X_{1}}}\backslash N$ be the projection. Since $\Gamma\cap{e^{{\mathbb{R}}X_{1}}}=e^{{\mathbb{Z}}X_{1}}$ is a cocompact lattice in ${e^{{\mathbb{R}}X_{1}}}$, $\bar{\Gamma}:=\pi(\Gamma)={e^{{\mathbb{R}}X_{1}}}\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}$ is a cocompact lattice in $\bar{N}$. Let $\bar{{\mathfrak{n}}}={\mathbb{R}}X_{1}\backslash{\mathfrak{n}}$, then $\bar{X_{2}},\dots,\bar{X_{n}}$ is a strong Malcev basis of $\bar{{\mathfrak{n}}}$ strongly based on $\bar{\Gamma}$. We will see that the naturally induced map $\bar{\pi}:{\Gamma\backslash N}\to{\bar{\Gamma}\backslash\bar{N}}$ is a principal $S^{1}$-bundle. Indeed, $\Gamma\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}\hookrightarrow{\Gamma\backslash N}\twoheadrightarrow\Gamma{e^{{\mathbb{R}}X_{1}}}\backslash N$ is a principal $\Gamma\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}$-bundle and we have $\Gamma\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}\simeq\Gamma\cap{e^{{\mathbb{R}}X_{1}}}\backslash{e^{{\mathbb{R}}X_{1}}}=e^{{\mathbb{Z}}X_{1}}\backslash{e^{{\mathbb{R}}X_{1}}}\simeq{\mathbb{Z}}\backslash{\mathbb{R}}$ and $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 25.19348pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\\\&&\crcr}}}\ignorespaces{\hbox{\kern-25.19348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{e^{{\mathbb{R}}X_{1}}}\backslash\Gamma{e^{{\mathbb{R}}X_{1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 25.19348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 49.19348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 49.19348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{e^{{\mathbb{R}}X_{1}}}\backslash N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 110.99802pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 68.09575pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 110.99802pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\Gamma{e^{{\mathbb{R}}X_{1}}}\backslash N}$}}}}}}}{\hbox{\kern-3.0pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 57.59573pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\bar{\Gamma}\backslash\bar{N}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 91.30661pt\raise-15.71594pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.28406pt\hbox{$\scriptstyle{\sim}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 124.09853pt\raise-5.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 130.02528pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$ Since $\Lambda\cap{e^{{\mathbb{R}}X_{1}}}=\Lambda\cap e^{{\mathbb{R}}Y_{1}}=e^{{\mathbb{Z}}Y_{1}}$ is a cocompact lattice in $e^{{\mathbb{R}}X_{1}}$, $\bar{\Lambda}:=\pi(\Lambda)$ is a cocompact lattice in $\bar{N}$. $\bar{Y_{2}},\dots,\bar{Y_{n}}$ is a strong Malcev basis of $\bar{{\mathfrak{n}}}$ strongly based on $\bar{\Lambda}$ and $\bar{Y_{i}}=\sum_{j=2}^{i}a_{ij}\bar{X_{j}}$ where $a_{ii}$ is Diophantine. Therefore $\bar{\Lambda}$ is Diophantine with respect to $\bar{\Gamma}$. Since $\bar{\pi}$ is $\Lambda$-equivariant, the action $\rho$ of $\Lambda$ when restricted to $e^{{\mathbb{Z}}Y_{1}}$, preserves fibers of $\bar{\pi}$. Let $z\in{\bar{\Gamma}\backslash\bar{N}}$. Choose a point $\Gamma x$ in ${\bar{\pi}^{-1}(z)}$. Then we have a trivialization $\iota_{\Gamma x}:{\mathbb{Z}}\backslash{\mathbb{R}}\simeq{\bar{\pi}^{-1}(z)}$ of ${\bar{\pi}^{-1}(z)}$ given by $\iota_{\Gamma x}(s)=\Gamma e^{sX_{1}}x$. Note that if we take another point $\Gamma y\in{\bar{\pi}^{-1}(z)}$, $\iota_{\Gamma y}^{-1}\circ\iota_{\Gamma x}:{\mathbb{Z}}\backslash{\mathbb{R}}\to{\mathbb{Z}}\backslash{\mathbb{R}}$ is a rotation. Let $Y_{1}=aX_{1}$ where $a$ is Diophantine. If we identify ${\bar{\pi}^{-1}(z)}$ with ${\mathbb{Z}}\backslash{\mathbb{R}}$ by $\iota_{\Gamma x}$, then the action of $e^{Y_{1}}$ on ${\mathbb{Z}}\backslash{\mathbb{R}}$ is $s\mapsto s+a$. Let $\mu_{z}$ be the normalized Haar measure naturally defined on $\bar{\pi}^{-1}(z)$, $\mu$ the $N$-invariant probability measure on ${\Gamma\backslash N}$ and $\nu$ the $\bar{N}$-invariant probability measure on ${\bar{\Gamma}\backslash\bar{N}}$. For any $f\in C({\Gamma\backslash N})$, $\int_{{\Gamma\backslash N}}fd\mu=\int_{{\bar{\Gamma}\backslash\bar{N}}}\int_{\bar{\pi}^{-1}(z)}fd\mu_{z}d\nu.$ (2) ###### Lemma 4. $\rho$ is ergodic with respect to $\mu$. ###### Proof. We use induction on $n$. If $n=1$, $\rho$ is an irrational rotation on ${\mathbb{Z}}\backslash{\mathbb{R}}$, hence the result is well known. In general, Let $f:{\Gamma\backslash N}\to{\mathbb{C}}$ be a $\Lambda$-invariant $L^{2}$-function with $\int_{{\Gamma\backslash N}}fd\mu=0$. Since the action of $e^{{\mathbb{Z}}Y_{1}}$ on ${\bar{\pi}^{-1}(z)}$ is ergodic, $f|_{{\bar{\pi}^{-1}(z)}}$ is constant $\mu_{z}$-almost everywhere. We denote this constant by $g(z)$. Then $g:{\bar{\Gamma}\backslash\bar{N}}\to{\mathbb{C}}$ is $\bar{\Lambda}$-invariant measurable function. By induction, $g$ is constant $\nu$-almost everywhere. By (2), this constant must be zero. Therefore $f$ is zero $\mu$-almost everywhere. ∎ Let $c:{\Gamma\backslash N}\times\Lambda\to{\mathbb{R}}$ be a $C^{\infty}$-cocycle over $\rho$. We must show that $c$ is cohomologous to a constant cocycle $c_{0}:\Lambda\to{\mathbb{R}}$ where $c_{0}(\lambda):=\int_{{\Gamma\backslash N}}c(x,\lambda)d\mu(x)$. Therefore we may assume that $\int_{{\Gamma\backslash N}}c(x,\lambda)d\mu(x)=0$ for all $\lambda\in\Lambda$, and we will show that $c$ is a coboundary. We prove this by induction on $n$. When $n=1$, $\rho$ is a Diophantine rotation on ${\mathbb{Z}}\backslash{\mathbb{R}}$, hence the result is well known. ###### Lemma 5. For all $m\in{\mathbb{Z}}$, $\int_{{\bar{\pi}^{-1}(z)}}c(s,e^{mY_{1}})d\mu_{z}(s)=0.$ ###### Proof. Fix $m$ and put $g(z)=\int_{{\bar{\pi}^{-1}(z)}}c(s,e^{mY_{1}})d\mu_{z}(s)$. For any $\lambda\in\Lambda$, cocycle equation gives $c(x,\lambda)+c(x\lambda,e^{mY_{1}})=c(x,e^{mY_{1}})+c(xe^{mY_{1}},\lambda)$. By integrating this equation on ${\bar{\pi}^{-1}(z)}$, we get $g(z\pi(\lambda))=g(z)$. Since the action of $\bar{\Lambda}$ on ${\bar{\Gamma}\backslash\bar{N}}$ is ergodic, $g$ is constant. By (2), $g$ must be zero. ∎ Let $f:{\mathbb{Z}}\backslash{\mathbb{R}}\xrightarrow{\iota_{\Gamma x}}{\bar{\pi}^{-1}(z)}\xrightarrow{c(\cdot,e^{Y_{1}})}{\mathbb{R}}$. We define $h_{z}(\iota_{\Gamma x}(s))=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{\hat{f}(k)}{-1+e^{2\pi ika}}e^{2\pi iks}.$ Then $h_{z}:{\bar{\pi}^{-1}(z)}\to{\mathbb{R}}$ is $C^{\infty}$, since $f$ is $C^{\infty}$ and $a$ is Diophantine. By Lemma 5, we have $c(\iota_{\Gamma x}(s),e^{Y_{1}})=-h_{z}(\iota_{\Gamma x}(s))+h_{z}(\iota_{\Gamma x}e^{Y_{1}}).$ If we choose another point $\Gamma e^{s_{0}X_{1}}x\in{\bar{\pi}^{-1}(z)}$ to define $h_{z}$, $\displaystyle h_{z}(\iota_{\Gamma x}(s))$ $\displaystyle=h_{z}(\Gamma e^{sX_{1}}x)=h_{z}(\iota_{\Gamma e^{s_{0}X_{1}}x}(s-s_{0}))$ $\displaystyle=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi ika}}\int_{0}^{1}c(\Gamma e^{(u+s_{0})X_{1}}x,e^{Y_{1}})e^{-2\pi iku}du\ e^{2\pi ik(s-s_{0})}$ $\displaystyle=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi ika}}\int_{0}^{1}f(u+s_{0})e^{-2\pi iku}du\ e^{2\pi ik(s-s_{0})}$ $\displaystyle=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{\hat{f}(k)}{-1+e^{2\pi ika}}e^{2\pi iks},$ so that $h_{z}$ is determined only by $z$. Define $h:{\Gamma\backslash N}\to{\mathbb{R}}$ by $h|_{{\bar{\pi}^{-1}(z)}}=h_{z}$. Then for all $x\in{\Gamma\backslash N}$ and $m\in{\mathbb{Z}}$, $c(x,e^{mY_{1}})=-h(x)+h(xe^{mY_{1}})$. Let $U\subset{\bar{\Gamma}\backslash\bar{N}}$ be open and $\sigma:U\to\bar{\pi}^{-1}(U)$ a section of $\bar{\pi}$. Then we have a trivialization ${\mathbb{Z}}\backslash{\mathbb{R}}\times U\simeq\bar{\pi}^{-1}(U)$ which sends $(s,z)$ to $\iota_{\sigma(z)}(s)=\Gamma e^{sX_{1}}\sigma(z)$. Hence $h(\iota_{\sigma(z)}(s))=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi ika}}\int_{0}^{1}c(\iota_{\sigma(z)}(u),e^{Y_{1}})e^{-2\pi iku}du\ e^{2\pi iks}$ on $\bar{\pi}^{-1}(U)$. The following lemma shows $h$ is $C^{\infty}$ on ${\Gamma\backslash N}$. ###### Lemma 6. Let $U\subset{\mathbb{R}}^{n}$ be open and let $f:{\mathbb{Z}}\backslash{\mathbb{R}}\times U\to{\mathbb{R}}$ be a $C^{\infty}$-function. Define $h(s,z)=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi ika}}\widehat{f_{z}}(k)e^{2\pi iks}$ where $f_{z}(u)=f(u,z)$. Then $h:{\mathbb{Z}}\backslash{\mathbb{R}}\times U\to{\mathbb{R}}$ is $C^{\infty}$. ###### Proof. Let $V$ be open such that $\bar{V}\subset U$ and $\bar{V}$ is compact. We will show that $h$ is $C^{\infty}$ on ${\mathbb{Z}}\backslash{\mathbb{R}}\times V$. Choose constants $C,\alpha>0$ such that $\lvert-1+e^{2\pi ika}\rvert\geq C\lvert k\rvert^{-\alpha}$ for all $k\in{\mathbb{Z}}\setminus\\{0\\}$. We will first prove that $h$ is continuous. Since for any $m\in{\mathbb{Z}}_{>0}$, $\frac{\partial^{m}f_{z}}{\partial s^{m}}(s)=\sum_{k\in{\mathbb{Z}}}(2\pi ik)^{m}\widehat{f_{z}}(k)e^{2\pi iks}$ in $L^{2}({\mathbb{Z}}\backslash{\mathbb{R}})$, $\displaystyle\Bigl{\lVert}\frac{\partial^{m}f_{z}}{\partial s^{m}}\Bigr{\rVert}_{2}^{2}$ $\displaystyle=\sum_{k\in{\mathbb{Z}}}\lvert(2\pi ik)^{m}\widehat{f_{z}}(k)\rvert^{2}$ $\displaystyle\geq(2\pi)^{2m}\lvert k\rvert^{2m}\lvert\widehat{f_{z}}(k)\rvert^{2}\geq\lvert k\rvert^{2m}\lvert\widehat{f_{z}}(k)\rvert^{2}.$ Since $\Bigl{\lVert}\frac{\partial^{m}f_{z}}{\partial s^{m}}\Bigr{\rVert}_{2}=\left(\int_{0}^{1}\Bigl{\lvert}\frac{\partial^{m}}{\partial s^{m}}f(s,z)\Bigr{\rvert}^{2}ds\right)^{\frac{1}{2}}$ is continuous in $z$, there exists $M>0$ such that $\Bigl{\lVert}\frac{\partial^{m}f_{z}}{\partial s^{m}}\Bigr{\rVert}_{2}<M$ for every $z\in\bar{V}$. Hence for all $k\in{\mathbb{Z}}$ and $z\in\bar{V}$, $\lvert k\rvert^{m}\lvert\widehat{f_{z}}(k)\rvert\leq M$. Therefore, for any $z\in\bar{V}$, $\displaystyle\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\Bigl{\lvert}\frac{1}{-1+e^{2\pi ika}}\widehat{f_{z}}(k)e^{2\pi iks}\Bigr{\rvert}$ $\displaystyle\leq C^{-1}\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{\lvert k\rvert^{2}}\lvert k\rvert^{\alpha+2}\lvert\widehat{f_{z}}(k)\rvert$ $\displaystyle\leq C^{-1}M\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{\lvert k\rvert^{2}}<\infty.$ This implies continuity of $h$ on ${\mathbb{Z}}\backslash{\mathbb{R}}\times\bar{V}$. We have $\frac{\partial h}{\partial s}(s,z)=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{2\pi ik}{-1+e^{2\pi ika}}\widehat{f_{z}}(k)e^{2\pi iks}.$ Thus a similar argument shows that $\frac{\partial h}{\partial s}$ is continuous. Let $z=(z_{1},\dots,z_{n})$. For any $z\in\bar{V}$, $\displaystyle\Bigl{\lvert}\frac{\partial}{\partial z_{j}}\left(\frac{1}{-1+e^{2\pi ika}}\widehat{f_{z}}(k)e^{2\pi iks}\right)\Bigr{\rvert}$ $\displaystyle=\Bigl{\lvert}\frac{1}{-1+e^{2\pi ika}}\widehat{\frac{\partial f}{\partial z_{j}}(\cdot,z)}(k)e^{2\pi iks}\Bigr{\rvert}$ $\displaystyle\leq C^{-1}\frac{1}{\lvert k\rvert^{2}}\lvert k\rvert^{\alpha+2}\Bigl{\lvert}\widehat{\frac{\partial f}{\partial z_{j}}(\cdot,z)}(k)\Bigr{\rvert}$ $\displaystyle\leq C^{-1}M^{\prime}\frac{1}{\lvert k\rvert^{2}}\ \in L^{1}({\mathbb{Z}}\setminus\\{0\\}).$ Thus $\frac{\partial h}{\partial z_{j}}(s,z)=\sum_{k\in{\mathbb{Z}}\setminus\\{0\\}}\frac{1}{-1+e^{2\pi ika}}\widehat{\frac{\partial f}{\partial z_{j}}(\cdot,z)}(k)e^{2\pi iks}.$ Hence $\frac{\partial h}{\partial z_{j}}$ is continuous by an argument similar to those above. For higher derivatives of $h$, continue this procedure. ∎ Set $c_{1}(x,\lambda)=c(x,\lambda)+h(x)-h(x\lambda)$. $c_{1}:{\Gamma\backslash N}\times\Lambda\to{\mathbb{R}}$ is a $C^{\infty}$-cocycle and $c_{1}(x,e^{mY_{1}})=0$. Thus for any $\lambda\in\Lambda$, cocycle equation implies $c_{1}(x,\lambda)=c_{1}(xe^{Y_{1}},\lambda)$. Since the action of $e^{{\mathbb{Z}}Y_{1}}$ on ${\bar{\pi}^{-1}(z)}$ is ergodic, $c_{1}(x,\lambda)$ is constant on ${\bar{\pi}^{-1}(z)}$. Therefore we can define a cocycle $\bar{c}:{\bar{\Gamma}\backslash\bar{N}}\times\bar{\Lambda}\to{\mathbb{R}}$ by $\bar{c}(\bar{\pi}(x),\pi(\lambda))=c_{1}(x,\lambda)$. Indeed, if $\bar{\pi}(x)=\bar{\pi}(y)$ and $\pi(\lambda)=\pi(\lambda^{\prime})$, then there exists a $m\in{\mathbb{Z}}$ with $\lambda=e^{mY_{1}}\lambda^{\prime}$, so that $c_{1}(x,\lambda)=c_{1}(x,e^{mY_{1}}\lambda^{\prime})=c_{1}(xe^{mY_{1}},\lambda^{\prime})=c_{1}(y,\lambda^{\prime}).$ Furthermore, $\displaystyle\int_{{\bar{\Gamma}\backslash\bar{N}}}\bar{c}(x,\pi(\lambda))d\nu(z)$ $\displaystyle=\int_{{\bar{\Gamma}\backslash\bar{N}}}\int_{{\bar{\pi}^{-1}(z)}}c_{1}(s,\lambda)d\mu_{z}(s)d\nu(z)$ $\displaystyle=\int_{{\Gamma\backslash N}}c_{1}(x,\lambda)d\mu(x)=0.$ By induction, there exists a $C^{\infty}$-function $P:{\bar{\Gamma}\backslash\bar{N}}\to{\mathbb{R}}$ such that $\bar{c}(z,\pi(\lambda))=-P(z)+P(z\pi(\lambda))$. Put $Q=P\circ\bar{\pi}$. Then $c_{1}(x,\lambda)=\bar{c}(\bar{\pi}(x),\pi(\lambda))=-Q(x)+Q(x\lambda)$. This proves Theorem 4. Let $\tilde{\rho}:M\times N\to M$ be the suspension of $\rho:{\Gamma\backslash N}\times\Lambda\to{\Gamma\backslash N}$ where $M={\Gamma\backslash N}\times_{\Lambda}N$ is a compact manifold. Then $\tilde{\rho}$ is locally free and let ${\mathcal{F}}$ be its orbit foliation. We have $H^{1}({\mathcal{F}})\simeq H^{1}(\Lambda,C^{\infty}({\Gamma\backslash N}))$ by [6] where the right hand side is the first cohomology of $\Lambda$-module $C^{\infty}({\Gamma\backslash N})$ obtained by $\rho$. It is easy to prove that $\mathop{\mathrm{Hom}}\nolimits(\Lambda,{\mathbb{R}})\to H^{1}(\Lambda,C^{\infty}({\Gamma\backslash N}))$ is injective. By Theorem 4, $H^{1}(\Lambda,C^{\infty}({\Gamma\backslash N}))=\mathop{\mathrm{Hom}}\nolimits(\Lambda,{\mathbb{R}}).$ ###### Lemma 7. $\mathop{\mathrm{dim}}\nolimits\mathop{\mathrm{Hom}}\nolimits(\Lambda,{\mathbb{R}})=\mathop{\mathrm{dim}}\nolimits H^{1}({\mathfrak{n}}).$ ###### Proof. Recall that $[N,N]\backslash\Lambda[N,N]$ is a cocompact lattice in $[N,N]\backslash N$ and that $[\Lambda,\Lambda]\backslash\left(\Lambda\cap[N,N]\right)$ is finite. Since $0\to[\Lambda,\Lambda]\backslash\left(\Lambda\cap[N,N]\right)\to[\Lambda,\Lambda]\backslash\Lambda\to[N,N]\backslash\Lambda[N,N]\to 0$ is exact, we have $\mathop{\mathrm{rank}}\nolimits[\Lambda,\Lambda]\backslash\Lambda=\mathop{\mathrm{rank}}\nolimits[N,N]\backslash\Lambda[N,N]=\mathop{\mathrm{dim}}\nolimits[N,N]\backslash N.$ Thus $\displaystyle\mathop{\mathrm{dim}}\nolimits\mathop{\mathrm{Hom}}\nolimits(\Lambda,{\mathbb{R}})$ $\displaystyle=\mathop{\mathrm{dim}}\nolimits\mathop{\mathrm{Hom}}\nolimits([\Lambda,\Lambda]\backslash\Lambda,{\mathbb{R}})$ $\displaystyle=\mathop{\mathrm{rank}}\nolimits[\Lambda,\Lambda]\backslash\Lambda$ $\displaystyle=\mathop{\mathrm{dim}}\nolimits[N,N]\backslash N$ $\displaystyle=\mathop{\mathrm{dim}}\nolimits\mathop{\mathrm{Hom}}\nolimits_{{\mathbb{R}}}([{\mathfrak{n}},{\mathfrak{n}}]\backslash{\mathfrak{n}},{\mathbb{R}})$ $\displaystyle=\mathop{\mathrm{dim}}\nolimits H^{1}({\mathfrak{n}}).$ ∎ Therefore we obtain $H^{1}({\mathcal{F}})=H^{1}({\mathfrak{n}}).$ This proves Theorem 3. ## 5 Existence of Diophantine lattices Let ${\mathfrak{n}}_{{\mathbb{Q}}}$ be a rational structure of ${\mathfrak{n}}$. We construct Diophantine lattices when ${\mathfrak{n}}_{{\mathbb{Q}}}$ admits a graduation. Namely, we assume that ${\mathfrak{n}}_{{\mathbb{Q}}}$ has a sequence $V_{i}$ of ${\mathbb{Q}}$-subspaces such that ${\mathfrak{n}}_{{\mathbb{Q}}}=\bigoplus_{i=1}^{k}V_{i}$ and $[V_{i},V_{j}]\subset V_{i+j}$. Let $X_{1},\dots,X_{n}$ be a ${\mathbb{Q}}$-basis of ${\mathfrak{n}}_{{\mathbb{Q}}}$ such that $X_{1},\dots,X_{i_{1}}\in V_{k}$, $X_{i_{1}+1},\dots,X_{i_{2}}\in V_{k-1}$$,\dots$, $X_{i_{k-1}+1},\dots,X_{n}\in V_{1}$. Then $X_{1},\dots,X_{n}$ is a strong Malcev basis of ${\mathfrak{n}}$ with rational structure constants. Multiplying $X_{1},\dots,X_{n}$ by a integer if necessary, we may assume that $\Gamma:=e^{{\mathbb{Z}}X_{1}}\cdots e^{{\mathbb{Z}}X_{n}}$ is a cocompact lattice in $N$. Let $\alpha$ be a root of a irreducible polynomial of degree $k+1$ over ${\mathbb{Q}}$. Since $\alpha,\alpha^{2},\dots,\alpha^{k}$ are irrational algebraic numbers, they are Diophantine. If we define a linear map $\varphi:{\mathfrak{n}}\to{\mathfrak{n}}$ by $\varphi(X)=\alpha^{i}X$ for $X\in V_{i}\otimes{\mathbb{R}}$, then $\varphi$ is an automorphism of Lie algebra ${\mathfrak{n}}$. Put $Y_{i}=\varphi(X_{i})$. $Y_{1},\dots,Y_{n}$ is a strong Malcev basis of ${\mathfrak{n}}$ strongly based on $\Lambda:=e^{{\mathbb{Z}}Y_{1}}\cdots e^{{\mathbb{Z}}Y_{n}}$. Thus $\Lambda$ is Diophantine with respect to $\Gamma$. ## Acknowledgement The author would like to thank his advisor, Masayuki Asaoka, for helpful comments. ## References * [1] M. Asaoka. Deformation of lacally free actions and the leafwise cohomology. arXiv:1012.2946. * [2] L. Corwin and F. P. Greenleaf. Representations of nilpotent Lie groups and their applications. Part 1:Basic theory and examples. Cambridge studies in advanced mathematics, vol. 18, Cambridge University Press, Cambridge, 1990. * [3] A. Katok and R. J. Spatzier. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Inst. Hautes Études Sci. Publ. Math. 79(1994), 131–156. * [4] S. Matsumoto and Y. Mitsumatsu. Leafwise cohomology and rigidity of certain Lie group actions. Ergod. Th. & Dynam. Sys. 23(2003), 1839–1866. * [5] D. Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. J. Mod. Dyn. 1(2007), 61–92. * [6] M. S. Pereira and N. M. dos Santos. On the cohomology of foliated bundles. Proyecciones 21(2)(2002), 175–197. * [7] F. A. Ramírez. Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups. J. Mod. Dyn. 3(2009), 335–357. * [8] N. M. dos Santos. Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys. 27(2007), 1719–1735.
arxiv-papers
2011-01-21T15:38:38
2024-09-04T02:49:16.570774
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Hirokazu Maruhashi", "submitter": "Hirokazu Maruhashi", "url": "https://arxiv.org/abs/1101.4161" }
1101.4306
# A Matrix-Analytic Solution for Randomized Load Balancing Models with Phase- Type Service Times Quan-Lin Li1 John C.S. Lui2 Wang Yang3 1 School of Economics and Management Sciences Yanshan University, Qinhuangdao 066004, China 2 Department of Computer Science & Engineering The Chinese University of Hong Kong, Shatin, N.T, Hong Kong 3 Institute of Network Computing & Information Systems Peking University, China ###### Abstract In this paper, we provide a matrix-analytic solution for randomized load balancing models (also known as _supermarket models_) with phase-type (PH) service times. Generalizing the service times to the phase-type distribution makes the analysis of the supermarket models more difficult and challenging than that of the exponential service time case which has been extensively discussed in the literature. We first describe the supermarket model as a system of differential vector equations, and provide a doubly exponential solution to the fixed point of the system of differential vector equations. Then we analyze the exponential convergence of the current location of the supermarket model to its fixed point. Finally, we present numerical examples to illustrate our approach and show its effectiveness in analyzing the randomized load balancing schemes with non-exponential service requirements. ## 1 Introduction In the past few years, a number of companies (e.g., Amazon, Google, ,…etc) are offering the _cloud computing_ service to enterprises. Furthermore, many content publishers and application service providers are increasingly using _Data Centers_ to host their services. This emerging computing paradigm allows service providers and enterprises to concentrate on developing and providing their own services/goods without worrying about computing system maintenance or upgrade, and thereby significantly reduce their operating cost. For companies that offer cloud computing service in their data centers, they can take advantage of the variation of computing workloads from these customers and achieve the computational multiplexing gain. One of the important technical challenges that they have to address is how to utilize these computing resources in the data center efficiently since many of these servers can be virtualized. There is a growing interest to examine simple and robust load balancing strategies to efficiently utilize the computing resource of the server farms. Distributed load balancing strategies, in which individual job (or customer) decisions are based on information on a limited number of other processors, have been studied analytically by Eager, Lazokwska and Zahorjan [4, 5, 6] and through trace-driven simulations by Zhou [26]. Further, randomized load balancing is a simple and effective mechanism to fairly utilize computing resources, and also can deliver surprisingly good performance measures such as reducing collisions, waiting times, backlogs,… etc. In a supermarket model, each arriving job randomly picks a small subset of servers and examines their instantaneous workload, and the job is routed to the least loaded server. When a job is committed to a server, jockeying is not allowed and each server uses the first-come-first-service (FCFS) discipline to process all jobs, e.g., see Mitzenmacher [11, 12]. For the supermarket models, most of recent research applied density dependent jump Markov processes to deal with the simple case with Poisson arrival processes and exponential service times, and illustrated that there exists a fixed point which decreases doubly exponentially. Readers may refer to, such as, a simple supermarket model by [1, 24, 11, 12]; simple variations by [19, 13, 14, 17, 23, 18, 25]; load information by [20, 3, 16, 18]; fast Jackson network by Martin and Suhov [10, 9, 21]; and general service times by Bramson, Lu and Prabhakar [2]. When the arrival processes or the service times are more general, the available results of the supermarket models are few up to now. The purpose of this paper is to provide a novel approach for studying a supermarket model with PH service times, and show that the fixed point decreases doubly exponentially. The remainder of this paper is organized as follows. In the next section, we describe the supermarket model with the PH service times as a system of differential vector equations based on the density dependent jump Markov processes. In Section 3, we set up a system of nonlinear equations satisfied by the fixed point, provide a doubly exponential solution to the system of nonlinear equations, and compute the expected sojourn time of any arriving customer. In Section 4, we study the exponential convergence of the current location of the supermarket model to its fixed point. In Section 5, numerical examples illustrate that our approach is effective in analyzing the supermarket models from non-exponential service time requirements. Some concluding remarks are given in Section 6. ## 2 Supermarket Model In this section, we describe a supermarket model with the PH service times as a system of differential vector equations based on the density dependent jump Markov processes. Let us formally describe the supermarket model, which is abstracted as a multi-server multi-queue stochastic system. Customers arrive at a queueing system of $n>1$ servers as a Poisson process with arrival rate $n\lambda$ for $\lambda>0$. The service times of these customers are of phase type with irreducible representation $\left(\alpha,T\right)$ of order $m$. Each arriving customer chooses $d\geq 1$ servers independently and uniformly at random from these $n$ servers, and waits for service at the server which currently contains the fewest number of customers. If there is a tie, servers with the fewest number of customers will be chosen randomly. All customers in every server will be served in the FCFS manner. Please see Figure 1 for an illustration. Figure 1: The supermarket model: each customer can probe the loading of $d$ servers For the supermarket models, the PH distribution allows us to model more realistic systems and understand their performance implication under the randomized load balancing strategy. As indicated in [7], the process lifetime of many parallel jobs, in particular, jobs to data centers, tends to be non- exponential. For the PH service time distribution, we use the following irreducible representation: $\left(\alpha,T\right)$ of order $m$, the row vector $\alpha$ is a probability vector whose $j$th entry is the probability that a service begins in phase $j$ for $1\leq j\leq m$; $T$ is an $m\times m$ matrix whose $\left(i,j\right)^{th}$ entry is denoted by $t_{i,j}$ with $t_{i,i}<0$ for $1\leq i\leq m$, and $t_{i,j}\geq 0$ for $1\leq i,j\leq m$ and $i\neq j$. Let $T^{0}=-Te\gvertneqq 0$, where $e$ is a column vector of ones with a suitable dimension in the context. The expected service time is given by $1/\mu=-\alpha T^{-1}e$. Unless we state otherwise, we assume that all random variables defined above are independent, and that the system is operating in the stable region $\rho=\lambda/\mu<1$. We define $n_{k}^{\left(i\right)}\left(t\right)$ as the number of queues with at least $k$ customers and the service time in phase $i$ at time $t\geq 0$. Clearly, $0\leq n_{k}^{\left(i\right)}\left(t\right)\leq n$ for $k\geq 0$ and $1\leq i\leq m$. Let $X_{n}^{\left(0\right)}\left(t\right)=\frac{n}{n}=1,$ and $k\geq 1$ $X_{n}^{\left(k,i\right)}\left(t\right)=\frac{n_{k}^{\left(i\right)}\left(t\right)}{n},$ which is the fraction of queues with at least $k$ customers and the service time in phase $i$ at time $t\geq 0$. We write $X_{n}^{\left(k\right)}\left(t\right)=\left(X_{n}^{\left(k,1\right)}\left(t\right),X_{n}^{\left(k,2\right)}\left(t\right),\ldots,X_{n}^{\left(k,m\right)}\left(t\right)\right),\text{ \ }k\geq 1,$ $X_{n}\left(t\right)=\left(X_{n}^{\left(0\right)}\left(t\right),X_{n}^{\left(1\right)}\left(t\right),X_{n}^{\left(2\right)}\left(t\right),\ldots\right).$ The state of the supermarket model may be described by the vector $X_{n}\left(t\right)$ for $t\geq 0$. Since the arrival process to the queueing system is Poisson and the service times of each server are of phase type, the stochastic process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ describing the state of the supermarket model is a Markov process whose state space is given by $\displaystyle\Omega_{n}$ $\displaystyle=$ $\displaystyle\\{\left(g_{n}^{\left(0\right)},g_{n}^{\left(1\right)},,g_{n}^{\left(2\right)}\ldots\right):g_{n}^{\left(0\right)}=1,g_{n}^{\left(k-1\right)}\geq g_{n}^{\left(k\right)}\geq 0,$ $\displaystyle\text{and \ \ }ng_{n}^{\left(k\right)}\text{ \ is a vector of nonnegative integers for }k\geq 1\\}.$ Let $s_{0}\left(n,t\right)=E\left[X_{n}^{\left(0\right)}\left(t\right)\right]$ and $k\geq 1$ $s_{k}^{\left(i\right)}\left(n,t\right)=E\left[X_{n}^{\left(k,i\right)}\left(t\right)\right].$ Clearly, $s_{0}\left(n,t\right)=1$. We write $S_{k}\left(n,t\right)=\left(s_{k}^{\left(1\right)}\left(n,t\right),s_{k}^{\left(2\right)}\left(n,t\right),\ldots,s_{k}^{\left(m\right)}\left(n,t\right)\right),\text{ \ }k\geq 1.$ As shown in Martin and Suhov [10] and Luczak and McDiarmid [8], the Markov process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ is asymptotically deterministic as $n\rightarrow\infty$. Thus the limits $\lim_{n\rightarrow\infty}E\left[X_{n}^{\left(0\right)}\left(t\right)\right]$ and $\lim_{n\rightarrow\infty}E\left[X_{n}^{\left(k,i\right)}\right]$ always exist by means of the law of large numbers. Based on this, we write $S_{0}\left(t\right)=\lim_{n\rightarrow\infty}s_{0}\left(n,t\right)=1,$ for $k\geq 1$ $s_{k}^{\left(i\right)}\left(t\right)=\lim_{n\rightarrow\infty}s_{k}^{\left(i\right)}\left(n,t\right),$ $S_{k}\left(t\right)=\left(s_{k}^{\left(1\right)}\left(t\right),s_{k}^{\left(2\right)}\left(t\right),\ldots,s_{k}^{\left(m\right)}\left(t\right)\right)$ and $S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right).$ Let $X\left(t\right)=\lim_{n\rightarrow\infty}X_{n}\left(t\right)$. Then it is easy to see from the Poisson arrivals and the PH service times that $\left\\{X\left(t\right),t\geq 0\right\\}$ is also a Markov process whose state space is given by $\Omega=\left\\{\left(g^{\left(0\right)},g^{\left(1\right)},g^{\left(2\right)},\ldots\right):g^{\left(0\right)}=1,g^{\left(k-1\right)}\geq g^{\left(k\right)}\geq 0\right\\}.$ If the initial distribution of the Markov process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ approaches the Dirac delta- measure concentrated at a point $g\in$ $\Omega$, then its steady-state distribution is concentrated in the limit on the trajectory $S_{g}=\left\\{S\left(t\right):t\geq 0\right\\}$. This indicates a law of large numbers for the time evolution of the fraction of queues of different lengths. Furthermore, the Markov process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ converges weakly to the fraction vector $S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right)$, or for a sufficiently small $\varepsilon>0$, $\lim_{n\rightarrow\infty}P\left\\{||X_{n}\left(t\right)-S\left(t\right)||\geq\varepsilon\right\\}=0,$ where $||a||$ is the $L_{\infty}$-norm of vector $a$. In what follows we provide a system of differential vector equations in order to determine fraction vector $S\left(t\right)$. To that end, we introduce the _Hadamard Product_ of two matrices $A=\left(a_{i,j}\right)$ and $B=\left(b_{i,j}\right)$ as follows: $A\odot B=\left(a_{i,j}b_{i,j}\right).$ Specifically, for $k\geq 2$, we have $A^{\odot k}=\underset{k\text{ matrix }A}{\underbrace{A\odot A\odot\cdots\odot A}}.$ To determine the fraction vector $S\left(t\right)$, we need to set up a system of differential vector equations satisfied by $S\left(t\right)$ by means of the density dependent jump Markov process. To that end, we provide a concrete example for $k\geq 2$ to indicate how to derive the the system of differential vector equations. Consider the supermarket model with $n$ servers, and determine the expected change in the number of queues with at least $k$ customers over a small time period of length d$t$. The probability vector that during this time period, any arriving customer joins a queue of size $k-1$ is given by $n\left[\lambda S_{k-1}^{\odot d}\left(n,t\right)-\lambda S_{k}^{\odot d}\left(n,t\right)\right]\text{d}t.$ Similarly, the probability vector that a customer leaves a server queued by $k$ customers is given by $n\left[S_{k}\left(n,t\right)T+S_{k+1}\left(n,t\right)T^{0}\alpha\right]\text{d}t.$ Therefore we can obtain $\displaystyle\text{d}E\left[n_{k}\left(n,t\right)\right]=$ $\displaystyle n\left[\lambda S_{k-1}^{\odot d}\left(n,t\right)-\lambda S_{k}^{\odot d}\left(n,t\right)\right]\text{d}t$ $\displaystyle+n\left[S_{k}\left(n,t\right)T+S_{k+1}\left(n,t\right)T^{0}\alpha\right]\text{d}t,$ which leads to $\frac{\text{d}S_{k}\left(n,t\right)}{\text{d}t}=\lambda S_{k-1}^{\odot d}\left(n,t\right)-\lambda S_{k}^{\odot d}\left(n,t\right)+S_{k}\left(n,t\right)T+S_{k+1}\left(n,t\right)T^{0}\alpha.$ Taking $n\rightarrow\infty$ in the both sides of Equation (LABEL:Equ1), we have $\frac{\text{d}S_{k}\left(t\right)}{\text{d}t}=\lambda S_{k-1}^{\odot d}\left(t\right)-\lambda S_{k}^{\odot d}\left(t\right)+S_{k}\left(t\right)T+S_{k+1}\left(t\right)T^{0}\alpha.$ Using a similar analysis to Equation (LABEL:Equ2), we can obtain a system of differential vector equations for the fraction vector $S\left(t\right)=\left(S_{0}\left(t\right),S_{1}\left(t\right),S_{2}\left(t\right),\ldots\right)$ as follows: $S_{0}\left(t\right)=1,$ $\frac{\mathtt{d}}{\text{d}t}S_{0}\left(t\right)=-\lambda S_{0}^{d}\left(t\right)+S_{1}\left(t\right)T^{0},$ (1) $\frac{\mathtt{d}}{\text{d}t}S_{1}\left(t\right)=\lambda\alpha S_{0}^{d}\left(t\right)-\lambda S_{1}^{\odot d}\left(t\right)+S_{1}\left(t\right)T+S_{2}\left(t\right)T^{0}\alpha,$ (2) and for $k\geq 2$, $\frac{\mathtt{d}}{\text{d}t}S_{k}\left(t\right)=\lambda S_{k-1}^{\odot d}\left(t\right)-\lambda S_{k}^{\odot d}\left(t\right)+S_{k}\left(t\right)T+S_{k+1}\left(t\right)T^{0}\alpha.$ (3) ###### Remark 1 Mitzenmacher [11, 12] provided an heuristical and interesting method to establish such systems of differential equations, but they lack a rigorous mathematical meaning for understanding the stochastic process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ in which $X_{n}\left(t\right)=(X_{n}^{\left(0\right)}\left(t\right),X_{n}^{\left(1\right)}\left(t\right),$ $X_{n}^{\left(2\right)}\left(t\right),\ldots)$ and $X_{n}^{\left(k\right)}\left(t\right)=n_{k}\left(t\right)/n$ for $k\geq 0$. This section, following Martin and Suhov [10] and Luczak and McDiarmid [8], gives some necessary mathematical analysis for the stochastic process $\left\\{X_{n}\left(t\right),t\geq 0\right\\}$ and the system of differential vector equations (1), (2) and (3). ## 3 A Matrix-Analytic Solution In this section, we provide a doubly exponential solution to the fixed point of the system of differential vector equations (1), (2) and (3). A row vector $\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ is called a fixed point of the fraction vector $S\left(t\right)$ if $\lim_{t\rightarrow+\infty}S\left(t\right)=\pi$. In this case, it is easy to see that $\lim_{t\rightarrow+\infty}\left[\frac{\mathtt{d}}{\text{d}t}S\left(t\right)\right]=0.$ Therefore, as $t\rightarrow+\infty$ the system of differential vector equations (1), (2) and (3) can be simplified as $-\lambda\pi_{0}^{d}+\pi_{1}T^{0}=0,$ (4) $\lambda\alpha\pi_{0}^{d}-\lambda\pi_{1}^{\odot d}+\pi_{1}T+\pi_{2}T^{0}\alpha=0,$ (5) and for $k\geq 2$, $\lambda\pi_{k-1}^{\odot d}-\lambda\pi_{k}^{\odot d}+\pi_{k}T+\pi_{k+1}T^{0}\alpha=0.$ (6) In general, it is more difficult and challenging to express the fixed point of the supermarket models with more general arrival processes or service times, because the systems of nonlinear equations are more complicated for computation. Fortunately, we can derive a closed-form expression for the fixed point $\pi=(\pi_{0},\pi_{1},\pi_{2},...)$ for the supermarket model with PH service times by means of a novel matrix-analytic approach given as follows. Noting that $S_{0}\left(t\right)=1$ for all $t\geq 0$, it is easy to see that $\pi_{0}=1$. It follows from Equation (4) that $\pi_{1}T^{0}=\lambda.$ (7) To solve Equation (7), we denote by $\omega$ the stationary probability vector of the irreducible Markov chain $T+T^{0}\alpha$. Obviously, we have $\omega T^{0}=\mu,$ $\frac{\lambda}{\mu}\omega T^{0}=\lambda.$ (8) Thus, we obtain $\pi_{1}=\frac{\lambda}{\mu}\omega=\rho\cdot\omega$. Based on the fact that $\pi_{0}=1$ and $\pi_{1}=\rho\cdot\omega$, it follows from Equation (5) that $\lambda\alpha-\lambda\rho^{d}\cdot\omega^{\odot d}+\rho\cdot\omega T+\pi_{2}T^{0}\alpha=0,$ which leads to $\lambda-\lambda\rho^{d}\cdot\omega^{\odot d}e+\rho\cdot\omega Te+\pi_{2}T^{0}=0.$ Note that $\omega Te=-\mu$, we obtain $\pi_{2}T^{0}=\lambda\rho^{d}\omega^{\odot d}e.$ Let $\theta=\omega^{\odot d}e$. Then it is easy to see that $\theta\in\left(0,1\right)$, and $\pi_{2}T^{0}=\lambda\theta\rho^{d}.$ Using a similar analysis to Equation (8), we have $\pi_{2}=\frac{\lambda\theta\rho^{d}}{\mu}\omega=\theta\rho^{d+1}\cdot\omega.$ (9) Based on $\pi_{1}=\rho\cdot\omega$ and $\pi_{2}=\theta\rho^{d+1}\cdot\omega$, it follows from Equation (6) that for $k=2$, $\lambda\rho^{d}\cdot\omega^{\odot d}-\lambda\theta^{d}\rho^{d^{2}+d}\cdot\omega^{\odot d}+\theta\rho^{d+1}\cdot\omega T+\pi_{3}T^{0}\alpha=0,$ which leads to $\lambda\theta\rho^{d}-\lambda\theta^{d+1}\rho^{d^{2}+d}+\theta\rho^{d+1}\cdot\omega Te+\pi_{3}T^{0}=0,$ thus we obtain $\pi_{3}T^{0}=\lambda\theta^{d+1}\rho^{d^{2}+d}.$ Using a similar analysis on Equation (8), we have $\pi_{3}=\frac{\lambda\theta^{d+1}\rho^{d^{2}+d}}{\mu}\omega=\theta^{d+1}\rho^{d^{2}+d+1}\cdot\omega.$ (10) Based on Equations (9) and (10), we may infer that there is a structured expression $\pi_{k}=\theta^{d^{k-2}+d^{k-3}+\cdots+d+1}\rho^{d^{k-1}+d^{k-2}+\cdots+d+1}\cdot\omega$, for $k\geq 1$. To that end, the following theorem states this important result. ###### Theorem 1 The fixed point $\pi=\left(\pi_{0},\pi_{1},\pi_{2},\ldots\right)$ is unique and is given by $\pi_{0}=1,\hskip 14.45377pt\pi_{1}=\rho\cdot\omega$ and for $k\geq 2,$ $\pi_{k}=\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\omega,$ (11) or $\displaystyle\pi_{k}$ $\displaystyle=$ $\displaystyle\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\omega=\rho^{d^{k-1}}\left(\theta\rho\right)^{\frac{d^{k-1}-1}{d-1}}\cdot\omega.$ Proof: By induction, one can easily derive the above result. It is clear that Equation (11) is correct for the cases with $l=2,3$ according to Equations (9) and (10). Now, we assume that Equation (11) is correct for the cases with $l=k$. Then it follows from Equation (6) that for $l=k+1$, we have $\displaystyle\lambda$ $\displaystyle\theta^{d^{k-2}+d^{k-3}+\cdots+d}\rho^{d^{k-1}+d^{k-2}+\cdots+d}\cdot\omega^{\odot d}-\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d}\rho^{d^{k}+d^{k-1}+\cdots+d}\cdot\omega^{\odot d}$ $\displaystyle+\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\omega T+\pi_{k+1}T^{0}\alpha=0,$ which leads to $\displaystyle\lambda$ $\displaystyle\theta^{d^{k-2}+d^{k-3}+\cdots+d+1}\rho^{d^{k-1}+d^{k-2}+\cdots+d}-\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}$ $\displaystyle+\theta^{d^{k-2}+d^{k-3}+\cdots+1}\rho^{d^{k-1}+d^{k-2}+\cdots+1}\cdot\omega Te+\pi_{k+1}T^{0}=0,$ thus we obtain $\pi_{k+1}T^{0}=\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}.$ By a similar analysis to (8), we have $\displaystyle\pi_{k+1}$ $\displaystyle=\frac{\lambda\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d}}{\mu}\omega$ $\displaystyle=\theta^{d^{k-1}+d^{k-2}+\cdots+d+1}\rho^{d^{k}+d^{k-1}+\cdots+d+1}\cdot\omega.$ This completes the proof. Now, we compute the expected sojourn time $T_{d}$ that a tagged arriving customer spends in the supermarket model. For the PH service times, a tagged arriving customer is the $k$th customer in the corresponding queue with probability vector $\pi_{k-1}^{\odot d}-\pi_{k}^{\odot d}$. When $k\geq 1$, the head customer in the queue has been served, and so its service time is residual and is denoted as $X_{R}$. Let $X$ be of phase type with irreducible representation $\left(\alpha,T\right)$. Then $X_{R}$ is of phase type with irreducible representation $\left(\omega,T\right)$. Clearly, we have $E\left[X\right]=\alpha\left(-T\right)^{-1}e,\text{ \ }E\left[X_{R}\right]=\omega\left(-T\right)^{-1}e.$ Thus it is easy to see that the expected sojourn time of the tagged arriving customer is given by $\displaystyle E\left[T_{d}\right]$ $\displaystyle=\left(\pi_{0}^{d}-\pi_{1}^{\odot d}e\right)E\left[X\right]+\sum_{k=1}^{\infty}\left(\pi_{k}^{\odot d}-\pi_{k+1}^{\odot d}\right)e\left\\{E\left[X_{R}\right]+kE\left[X\right]\right\\}$ $\displaystyle=\pi_{1}^{\odot d}e\left\\{E\left[X_{R}\right]-E\left[X\right]\right\\}+E\left[X\right]\left[1+\sum_{k=1}^{\infty}\pi_{k}^{\odot d}e\right]$ $\displaystyle=\rho^{d}\theta\left(\omega-\alpha\right)\left(-T\right)^{-1}e+\alpha\left(-T\right)^{-1}e\left(1+\sum_{k=1}^{\infty}\theta^{\frac{d^{k}-1}{d-1}}\rho^{\frac{d^{k+1}-d}{d-1}}\right).$ When the arrival process and the service time distribution are Poisson and exponential, respectively, it is clear that $\alpha=\omega=\theta=1$ and $\alpha\left(-T\right)^{-1}e=1/\mu$, thus we have $E\left[T_{d}\right]=\frac{1}{\mu}\sum_{k=0}^{\infty}\rho^{\frac{d^{k+1}-d}{d-1}},$ which is the same as Corollary 3.8 in Mitzenmacher [12]. In what follows we consider an interesting problem: how many moments of the service time distribution are needed to obtain a better accuracy for computing the fixed point or the expected sojourn time. It is well-known from the theory of probability distributions that the first three moments is basic for analyzing such an accuracy, and we can construct a PH distribution of order 2 by using the first three moments. Telek and Heindl [22] provided a fitting procedure for matching a PH distribution of order 2 from the first three moments exactly. It is necessary to list the fitting procedure as follows: For a nonnegative random variable $X$, let $m_{n}=E\left[X^{n}\right]$, $n\geq 1$. We take a PH distribution of order 2 with the canonical representation $\left(\alpha,T\right)$, where $\mathbf{\alpha=}\left(\eta,1-\eta\right)$ and $T\mathbf{=}\left(\begin{array}[]{cc}-\xi_{1}&\xi_{1}\\\ 0&-\xi_{2}\end{array}\right),$ $0\leq\eta\leq 1$ and $0<\xi_{1}\leq\xi_{2}$. Note that the three unknown parameters $\eta$, $\xi_{1}$ and $\xi_{2}$ can be obtained from the first three moments $m_{1}$, $m_{2}$ and $m_{3}$ of an arbitrary general distribution. Table 1: Specific Bounds of the First Three Moments Moment | Condition | Bounds ---|---|--- $m_{1}$ | | $0<m_{1}<\infty$ $m_{2}$ | | $1.5m_{1}^{2}\leq m_{2}$ $m_{3}$ | $0.5\leq c_{X}^{2}\leq 1$ | $3m_{1}^{3}\left(3c_{X}^{2}-1+\sqrt{2}\left(1-c_{X}^{2}\right)^{\frac{3}{2}}\right)\leq m_{3}\leq 6m_{1}^{3}c_{X}^{2}$ | $1<c_{X}^{2}$ | $\frac{3}{2}m_{1}^{3}\left(1+c_{X}^{2}\right)^{2}<m_{3}<\infty$ In Table 1, $c_{X}^{2}=m_{2}\diagup m_{1}^{2}-1$ is the squared coefficient of variation. If the moments do not satisfy these conditions in Table 1, then we may analyze the following four cases: (a.1) if $m_{2}<1.5m_{1}^{2}$, then we take $m_{2}=1.5m_{1}^{2}$; (a.2) if $0.5\leq c_{X}^{2}\leq 1$, and $m_{3}<3m_{1}^{3}\left(3c_{X}^{2}-1+\sqrt{2}\left(1-c_{X}^{2}\right)^{\frac{3}{2}}\right)$, then we take $m_{3}=3m_{1}^{3}\left(3c_{X}^{2}-1+\sqrt{2}\left(1-c_{X}^{2}\right)^{\frac{3}{2}}\right)$; (a.3) if $0.5\leq c_{X}^{2}\leq 1$, and $m_{3}>6m_{1}^{3}c_{X}^{2}$, then we take $m_{3}=6m_{1}^{3}c_{X}^{2}$; and (a.4) if $1<c_{X}^{2}$, and $m_{3}\leq\frac{3}{2}m_{1}^{3}\left(1+c_{X}^{2}\right)^{2}$, then we take $m_{3}=\frac{3}{2}m_{1}^{3}\left(1+c_{X}^{2}\right)^{2}$. Let $c=3m_{2}^{2}-2m_{1}m_{3}$, $d=2m_{1}^{2}-m_{2}$, $b=3m_{1}m_{2}-m_{3}$ and $a=b^{2}-6cd$. If the moments respectively satisfy their specific bounds __ shown in Table 1 or the exceptive four cases, then three unknown parameters $\eta$, $\xi_{1}$ and $\xi_{2}$ can be computed in the following three cases. (1) If $c>0$, then $\eta=\frac{-b+6m_{1}d+\sqrt{a}}{b+\sqrt{a}},\text{ }\xi_{1}=\frac{b-\sqrt{a}}{c},\text{ }\xi_{2}=\frac{b+\sqrt{a}}{c}.$ (2) If $c<0$, then $\eta=\frac{b-6m_{1}d+\sqrt{a}}{-b+\sqrt{a}},\text{ }\xi_{1}=\frac{b+\sqrt{a}}{c},\text{ }\xi_{2}=\frac{b-\sqrt{a}}{c}.$ (3) If $c=0$, then $\eta=0,\text{ }\xi_{1}>0,\text{ }\xi_{2}=\frac{1}{m_{1}}.$ From the above discussion, we can always construct a PH distribution of order 2 to approximate an arbitrary general distribution with the same first three moments. In fact, such an approximation achieves a better accuracy in computation. For the PH distribution of order 2, we have $T+T^{0}\alpha=\left(\begin{array}[]{cc}-\xi_{1}&\xi_{1}\\\ 0&-\xi_{2}\end{array}\right)+\left(\begin{array}[]{c}0\\\ \xi_{2}\end{array}\right)\left(\begin{array}[]{cc}\eta&1-\eta\end{array}\right)=\left(\begin{array}[]{cc}-\xi_{1}&\xi_{1}\\\ \xi_{2}\eta&-\xi_{2}\eta\end{array}\right),$ which leads to $\omega=\left(\frac{\xi_{2}\eta}{\xi_{1}+\xi_{2}\eta},\frac{\xi_{1}}{\xi_{1}+\xi_{2}\eta}\right)$ and $\theta=\frac{\xi_{1}^{d}+\xi_{2}^{d}\eta^{d}}{\left(\xi_{1}+\xi_{2}\eta\right)^{d}}.$ Thus, the PH distribution of order 2 can effectively approximates an arbitrary general service time distribution in the supermarket model under the same first three moments, and specifically, all the computations are very simple to implement. ###### Remark 2 Bramson, Lu and Prabhakar [2] provided a modularized program based on ansatz for treating the supermarket model with a general service time distribution. They organized a functional equation $\pi=F\left(G\left(\pi\right)\right)$ for analyzing the stationary probability vector $\pi$ in terms of insensitivity and generalized Fibonacci sequences, although the operators $F$ and $G$ are not easy to be given for this supermarket model. This paper studies the supermarket model with a PH service time distribution, provides the doubly exponential solution to the fixed point, and is specifically related to the phase type environment by means of the crucial factor $\theta=\omega^{\odot d}e$. Note that the PH distributions are dense in the set of all nonnegative random variables, this paper can numerically provide necessary understanding for the role played by the general service time distribution in performance analysis of the supermarket model by means of the PH approximation of order 2. ## 4 Exponential convergence to the fixed point In this section, we study the exponential convergence of the current location $S\left(t\right)$ of the supermarket model to its fixed point $\pi$. For the supermarket model, the initial point $S\left(0\right)$ can affect the current location $S\left(t\right)$ for each $t>0$, since the service process in the supermarket model is under a unified structure. To that end, we provide some notation for comparison of two vectors. Let $a=\left(a_{1},a_{2},a_{3},\ldots\right)$ and $b=\left(b_{1},b_{2},b_{3},\ldots\right)$. We write $a\prec b$ if $a_{k}<b_{k}$ for some $k\geq 1$ and $a_{l}\leq b_{l}$ for $l\neq k,l\geq 1$; and $a\preceq b$ if $a_{k}\leq b_{k}$ for all $k\geq 1$. Now, we can obtain the following useful proposition whose proof is clear from a sample path analysis and thus is omitted here. ###### Proposition 1 If $S\left(0\right)\preceq\widetilde{S}\left(0\right)$, then $S\left(t\right)\preceq\widetilde{S}\left(t\right)$. Based on Proposition 1, the following theorem shows that the fixed point $\pi$ is an upper bound of the current location $S\left(t\right)$ for all $t\geq 0$. ###### Theorem 2 For the supermarket model, if there exists some $k$ such that $S_{k}\left(0\right)=0$, then the sequence $\left\\{S_{k}\left(t\right)\right\\}$ has an upper bound sequence which decreases doubly exponentially for all $t\geq 0$, that is, $S\left(t\right)\preceq\pi$ for all $t\geq 0$. Proof: Let $\widetilde{S}_{k}\left(0\right)=\pi_{k}$ for $k\geq 1$. Then for each $k\geq 1$, $\widetilde{S}_{k}\left(t\right)=\widetilde{S}_{k}\left(0\right)=\pi_{k}$ for all $t\geq 0$, since $\widetilde{S}\left(0\right)=\left(\widetilde{S}_{1}\left(0\right),\widetilde{S}_{2}\left(0\right),\widetilde{S}_{2}\left(0\right),\ldots\right)$ is a fixed point in the supermarket model. If $S_{k}\left(0\right)=0$ for some $k$, then $S_{k}\left(0\right)\prec\widetilde{S}_{k}\left(0\right)$ and $S_{j}\left(0\right)\preceq\widetilde{S}_{j}\left(0\right)$ for $j\neq k,j\geq 1$, thus $S\left(0\right)\preceq\widetilde{S}\left(0\right)$. It is easy to see from Proposition 1 that $S_{k}\left(t\right)\preceq\widetilde{S}_{k}\left(t\right)=\pi_{k}$ for all $k\geq 1$ and $t\geq 0$. Thus we obtain that for all $k\geq 1$ and $t\geq 0$ $S_{k}\left(t\right)\leq\theta^{\frac{d^{k-1}-1}{d-1}}\rho^{\frac{d^{k}-1}{d-1}}\cdot\omega.$ This completes the proof. To show the exponential convergence, we define a Lyapunov function $\Phi\left(t\right)$ as $\Phi\left(t\right)=\sum_{k=1}^{\infty}w_{k}\left[\pi_{k}-S_{k}\left(t\right)\right]e$ in terms of the fact that $S_{k}\left(t\right)\preceq\pi_{k}$ for $k\geq 1$ and $\pi_{0}=S_{0}\left(t\right)=1$, where $\left\\{w_{k}\right\\}$ is a positive scalar sequence with $w_{k+1}\geq w_{k}\geq w_{1}=1$ for $k\geq 2$. The following theorem measures the distance $\Phi\left(t\right)$ of the current location $S\left(t\right)$ for $t\geq 0$ to the fixed point $\pi$, and illustrates that this distance between the fixed point and the current location is very close to zero with exponential convergence. This shows that from a suitable starting point, the supermarket model can be quickly close to the fixed point. ###### Theorem 3 For $t\geq 0$, $\Phi\left(t\right)\leq c_{0}e^{-\delta t}$, where $c_{0}$ and $\delta$ are two positive constants. In this case, the potential function $\Phi\left(t\right)$ is exponentially convergent. Proof: Note that $\Phi\left(t\right)=\sum_{k=1}^{\infty}w_{k}\left[\pi_{k}-S_{k}\left(t\right)\right]e,$ we have $\frac{d}{dt}\Phi\left(t\right)=-\sum_{k=1}^{\infty}w_{k}\frac{d}{dt}S_{k}\left(t\right)e.$ It follows from Equations (1) to (3) that $\displaystyle\frac{d}{dt}\Phi\left(t\right)=$ $\displaystyle-w_{1}[\lambda S_{0}^{d}\left(t\right)\alpha-\lambda S_{1}^{\odot d}\left(t\right)+S_{1}\left(t\right)T+S_{2}\left(t\right)T^{0}\alpha]e$ $\displaystyle-\sum_{k=1}^{\infty}w_{k}[\lambda S_{k-1}^{\odot d}\left(t\right)-\lambda S_{k}^{\odot d}\left(t\right)+S_{k}\left(t\right)T+S_{k+1}\left(t\right)T^{0}\alpha]e.$ By means of $S_{0}\left(t\right)=1$ and $Te=-T^{0}$, we can obtain $\displaystyle\frac{d}{dt}\Phi\left(t\right)=$ $\displaystyle- w_{1}[\lambda-\lambda S_{1}^{\odot d}\left(t\right)e-S_{1}\left(t\right)T^{0}+S_{2}\left(t\right)T^{0}]$ $\displaystyle-\sum_{k=2}^{\infty}w_{k}[\lambda S_{k-1}^{\odot d}\left(t\right)e-\lambda S_{k}^{\odot d}\left(t\right)e-S_{k}\left(t\right)T^{0}+S_{k+1}\left(t\right)T^{0}].$ (12) We take some nonnegative constants $c_{k}\left(t\right)$ and $d_{k}\left(t\right)$ for $k\geq 1$ such that $\lambda=f_{1}\left(t\right)S_{1}\left(t\right)T^{0},$ for $k\geq 1$ $\lambda S_{k}^{\odot d}\left(t\right)e=c_{k}\left(t\right)\left[\pi_{k}-S_{k}\left(t\right)\right]e$ and $S_{k}\left(t\right)T^{0}=d_{k}\left(t\right)\left[\pi_{k}-S_{k}\left(t\right)\right]e.$ Then it follows from (12) that $\displaystyle\frac{d}{dt}\Phi\left(t\right)$ $\displaystyle=-\left\\{\left[\left(w_{2}-w_{1}\right)\right]c_{1}\left(t\right)+w_{1}\left[f_{1}\left(t\right)-1\right]d_{1}\left(t\right)\right\\}\cdot\left[\pi_{1}-S_{1}\left(t\right)\right]e$ $\displaystyle-\sum_{k=2}^{\infty}\left[\left(w_{k+1}-w_{k}\right)c_{k}\left(t\right)+\left(w_{k-1}-w_{k}\right)d_{k}\left(t\right)\right]\cdot\left[\pi_{k}-S_{k}\left(t\right)\right]e.$ For a constant $\delta>0$, we take $w_{1}=1,$ $\left[\left(w_{2}-w_{1}\right)\right]c_{1}\left(t\right)+w_{1}\left[f_{1}\left(t\right)-1\right]d_{1}\left(t\right)\geq\delta w_{1}$ and $\left(w_{k+1}-w_{k}\right)c_{k}\left(t\right)+\left(w_{k-1}-w_{k}\right)d_{k}\left(t\right)\geq\delta w_{k}.$ In this case, it is easy to see that $w_{2}\geq 1+\frac{\delta+1-f_{1}\left(t\right)}{c_{1}\left(t\right)}$ and for $k\geq 2$ $w_{k+1}\geq w_{k}+\frac{\delta w_{k}}{c_{k}\left(t\right)}+\frac{d_{k}\left(t\right)}{c_{k}\left(t\right)}\left(w_{k}-w_{k-1}\right).$ Thus we have $\frac{d}{dt}\Phi\left(t\right)\leq-\delta\sum_{k=0}^{\infty}w_{k}\left[\pi_{k}-S_{k}\left(t\right)\right]e=-\delta\Phi\left(t\right),$ which can leads to $\Phi\left(t\right)\leq c_{0}e^{-\delta t}.$ This completes the proof. ## 5 Numerical examples In this section, we provide some numerical examples to illustrate that our approach is effective and efficient in the study of supermarket models with non-exponential service requirements, including Erlang service time distributions, hyper-exponential service time distributions and PH service time distributions. Example one (Erlang Distribution) We consider an $m$-order Erlang distribution with the irreducible PH representation $(\alpha,T)$, where$\alpha=\left(1,0,\ldots,0,0\right)$ and $T=\left(\begin{array}[]{ccccc}-\eta&\eta&&&\\\ &-\eta&\eta&&\\\ &&\ddots&\ddots&\\\ &&&-\eta&\eta\\\ &&&&-\eta\end{array}\right),\text{ \ \ }T^{0}=\left(\begin{array}[]{c}0\\\ 0\\\ \vdots\\\ 0\\\ \eta\end{array}\right).$ It is clear that $T+T^{0}\alpha=\left(\begin{array}[]{ccccc}-\eta&\eta&&&\\\ &-\eta&\eta&&\\\ &&\ddots&\ddots&\\\ &&&-\eta&\eta\\\ \eta&&&&-\eta\end{array}\right),$ which leads to the stationary probability vector of the Markov chain $T+T^{0}\alpha$ as follows: $\omega=\left(\frac{1}{m},\frac{1}{m},\ldots\frac{1}{m},\frac{1}{m}\right);\hskip 7.22743pt\mu=\omega T^{0}=\frac{\eta}{m};\hskip 7.22743pt\rho=\frac{\lambda}{\mu}=\frac{m\lambda}{\eta};\hskip 7.22743pt\theta=m\left(\frac{1}{m}\right)^{d}=m^{1-d}.$ Thus we obtain $\displaystyle\pi_{k}$ $\displaystyle=m^{1-d^{k-1}}\left(\frac{m\lambda}{\eta}\right)^{\frac{d^{k}-1}{d-1}}\left(\frac{1}{m},\frac{1}{m},\ldots\frac{1}{m},\frac{1}{m}\right)$ $\displaystyle=m^{\frac{d^{k-1}+d-2}{d-1}}\left(\frac{\lambda}{\eta}\right)^{\frac{d^{k}-1}{d-1}}\left(\frac{1}{m},\frac{1}{m},\ldots\frac{1}{m},\frac{1}{m}\right).$ Let $\lambda=1$. If $\rho=\frac{m\lambda}{\eta}<1$, then this supermarket model is stable. In the stable case, $\eta>m$. We may consider the following simple cases: (a) If $m=2$ and $d=2$, then $\pi_{k}=2^{2^{k-1}}\eta^{1-2^{k}}$. (b) If $m=3$ and $d=2$, then $\pi_{k}=3^{2^{k-1}}\eta^{1-2^{k}}$. Based on the two simple examples with $\lambda=1$ and $d=2$, we need to illustrate how the fixed point depends on the stage number $m$ and the exponential service rate $\eta$. To that end, we write $\pi_{k}\left(m,\eta\right)$. It is easy to see that for a given pair $\left(k,\eta\right)$ for $\eta>m$ and $k=1,2,\ldots,$ we have $\pi_{k}\left(1,\eta\right)<\pi_{k}\left(2,\eta\right)<\cdots<\pi_{k}\left(m,\eta\right)<\cdots.$ On the other hand, for a given pair $\left(k,m\right)$ for $m,k=1,2,\ldots,$ we can see that $\pi_{k}\left(m,\eta\right)$ is a decreasing function of $\eta$. Let us consider the average response time of the supermarket model with an $m-$stage Erlang distribution. We first consider a parallel system with $n=100$ servers and the service time distribution is exponential. We normalize the average service time to unity and vary the arrival rate $\lambda$. For the $m-$stage Erlang distribution, the bigger the number $m$ is, the bigger its variance is. Table 2 illustrates the average response time under different probe size $d$. One can observe that there is a dramatic improvement (or reduction) in the average response time when increasing the probe size $d$. Table 2: Average response time for exponential service time number of servers ($n$) | probe size ($d$) | arrival rate ($\lambda$) | response time ($E[\mathcal{T}]$) ---|---|---|--- 100 | 2 | 0.500000 | 1.395977 100 | 2 | 0.700000 | 1.768194 100 | 2 | 0.800000 | 2.072020 100 | 2 | 0.900000 | 2.721852 100 | 3 | 0.500000 | 1.395320 100 | 3 | 0.700000 | 1.604113 100 | 3 | 0.800000 | 1.802933 100 | 3 | 0.900000 | 2.209601 100 | 5 | 0.900000 | 1.916280 We further analyze the cases that the service time is either distributed according to $2$-stage Erlang or $3$-stage Erlang distribution. Similarly, we normalized the total average service time as unity and we vary the arrival rate $\lambda$. Tables 3 and 4 illustrate the average response time under different probe size $d$. One can observe that * • Simple probing size $d$ can significantly improve the performance by lowering the average response time. * • When the service time has lower variance, the average response time is lower. Table 3: Average response time for $2-$stage Erlang service time number of servers ($n$) | probe size ($d$) | arrival rate ($\lambda$) | response time ($E[\mathcal{T}]$) ---|---|---|--- 100 | 2 | 0.500000 | 1.353783 100 | 2 | 0.700000 | 1.599851 100 | 2 | 0.800000 | 1.829199 100 | 2 | 0.900000 | 2.298470 100 | 3 | 0.500000 | 1.325610 100 | 3 | 0.700000 | 1.492651 100 | 3 | 0.800000 | 1.639987 100 | 3 | 0.900000 | 1.941196 100 | 5 | 0.900000 | 1.739867 Table 4: Average response time for $3-$stage Erlang service time number of servers ($n$) | probe size ($d$) | arrival rate ($\lambda$) | response time ($E[\mathcal{T}]$) ---|---|---|--- 100 | 2 | 0.500000 | 1.322544 100 | 2 | 0.700000 | 1.539621 100 | 2 | 0.800000 | 1.739972 100 | 2 | 0.900000 | 2.148191 100 | 3 | 0.500000 | 1.298863 100 | 3 | 0.700000 | 1.452785 100 | 3 | 0.800000 | 1.581663 100 | 3 | 0.900000 | 1.834704 100 | 5 | 0.900000 | 1.678233 Example two (Hyper-Exponential Distribution) We consider an $m$-order hyper- exponential distribution $F\left(x\right)=1-\sum\limits_{k=1}^{m}\alpha_{k}\exp\left\\{-\eta_{k}x\right\\}$, or the probability density function $f\left(x\right)=\sum\limits_{k=1}^{m}\alpha_{k}\eta_{k}\exp\left\\{-\eta_{k}x\right\\}$. It is clear that the hyper-exponential distribution is of phase type with the irreducible representation $(\alpha,T)$, where $\alpha=\left(\alpha_{1},\alpha_{2},\ldots,\alpha_{m}\right)$, and $T=\left(\begin{array}[]{cccc}-\eta_{1}&&&\\\ &-\eta_{2}&&\\\ &&\ddots&\\\ &&&-\eta_{m}\end{array}\right),\text{ \ }T^{0}=\left(\begin{array}[]{c}\eta_{1}\\\ \eta_{2}\\\ \vdots\\\ \eta_{m}\end{array}\right),$ which lead to $T+T^{0}\alpha=\left(\begin{array}[]{cccc}-\eta_{1}\left(1-\alpha_{1}\right)&\eta_{1}\alpha_{2}&\cdots&\eta_{1}\alpha_{m}\\\ \eta_{2}\alpha_{1}&-\eta_{2}\left(1-\alpha_{2}\right)&\cdots&\eta_{2}\alpha_{m}\\\ \vdots&\vdots&&\vdots\\\ \eta_{m}\alpha_{1}&\eta_{m}\alpha_{2}&\cdots&-\eta_{m}\left(1-\alpha_{m}\right)\end{array}\right).$ In general, the system of equations $\omega\left(T+T^{0}\alpha\right)=0$ and $\omega e=1$ does not admit a simple analytic solution. For a convenient description, we only consider a simple one with $m=2$. In this case, we obtain $\omega=\left(\frac{\alpha_{1}\eta_{2}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}},\frac{\alpha_{2}\eta_{1}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right),\hskip 14.45377pt\mu=\frac{\eta_{1}\eta_{2}\left(\alpha_{1}+\alpha_{2}\right)}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}},$ $\rho=\frac{\lambda}{\mu}=\frac{\lambda\left(\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}\right)}{\eta_{1}\eta_{2}\left(\alpha_{1}+\alpha_{2}\right)},\hskip 7.22743pt\theta=\left(\frac{\alpha_{1}\eta_{2}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right)^{d}+\left(\frac{\alpha_{2}\eta_{1}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right)^{d}$ and $\displaystyle\pi_{k}=$ $\displaystyle\left[\left(\frac{\alpha_{1}\eta_{2}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right)^{d}+\left(\frac{\alpha_{2}\eta_{1}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right)^{d}\right]^{\frac{d^{k-1}-1}{d-1}}$ $\displaystyle\cdot\left[\frac{\lambda\left(\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}\right)}{\eta_{1}\eta_{2}\left(\alpha_{1}+\alpha_{2}\right)}\right]^{\frac{d^{k}-1}{d-1}}\left(\frac{\alpha_{1}\eta_{2}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}},\frac{\alpha_{2}\eta_{1}}{\alpha_{1}\eta_{2}+\alpha_{2}\eta_{1}}\right).$ Tables 5 and 6 indicate how the doubly exponential solution ($\pi_{1}$ to $\pi_{5}$) depends on the vectors $\eta=\left(\eta_{1},\eta_{2}\right)$ and $\alpha=\left(\alpha_{1},\alpha_{2}\right)$, respectively. Table 5: The doubly exponential solution depends on $\eta$ | $\eta=(3,3)$ | $\eta=(3,10)$ | $\eta=(3,20)$ ---|---|---|--- $\pi_{1}$ | (0.1667, 0.1667) | (0.1667, 0.0500) | (0.1667, 0.0250) $\pi_{2}$ | (0.0093, 0.0093) | (0.0050, 0.0015) | (0.0047, 0.0007) $\pi_{3}$ | (2.858e-05, 2.858e-05) | (4.626e-06, 1.388e-06) | (3.819e-06, 5.728e-07) $\pi_{4}$ | (2.722e-10, 2.722e-10) | (3.888e-12, 1.166e-12) | (2.485e-12, 3.728e-13) $\pi_{5}$ | (2.470e-20, 2.470e-20) | (2.746e-24, 8.238e-25) | (1.053e-24, 1.579e-25) Table 6: The doubly exponential solution depends on $\alpha$ | $\alpha=(0.5,\;0.5)$ | $\alpha=(0.2,\;0.8)$ | $\alpha=(0.8,\;0.2)$ ---|---|---|--- $\pi_{1}$ | (0.1667, 0.1667) | (0.0667, 0.0267) | (0.2667, 0.0067) $\pi_{2}$ | (0.0047, 0.0005) | (0.0003, 0.0001) | (0.0190, 0.0005) $\pi_{3}$ | (3.680e-06, 3.680e-07) | (9.136e-09, 3.654e-09) | (9.607e-05, 2.402e-06) $\pi_{4}$ | (2.280e-12, 2.280e-13) | (6.454e-18, 2.582e-18) | (2.463e-09, 6.157e-11) $\pi_{5}$ | (8.752e-25, 8.752e-26) | (3.221e-36, 1.289e-36) | (1.618e-18, 4.046e-20) Let us consider the average response time of the supermarket model with an $m$-stage hyper-exponential service time distribution. We consider a parallel system with $n=100$ servers and the probability density function of the service time of a customer is given by $f(x)=0.5\times(2\times e^{-2x})+0.25\times(0.5\times e^{-0.5x})+0.25\times(e^{-x}).$ Note that the total average service time is normalized to unity and we vary the arrival rate $\lambda$. Table 7 illustrates the average response time under different probe size $d$. One can observe that there is a dramatic reduction in the average response time when increasing the probe size. Furthermore, when the service time has a higher variance (we here compare it with the exponential distribution or $m-$stage Erlang distribution), the average service time is much higher. This indicates that we improve the performance of the supermarket model, one has to increase the probe size $d$. Table 7: Average response time for $3-$stage Hyper-exponential service time number of servers ($n$) | probe size ($d$) | arrival rate ($\lambda$) | response time ($E[\mathcal{T}]$) ---|---|---|--- 100 | 2 | 0.500000 | 1.552282 100 | 2 | 0.700000 | 1.969132 100 | 2 | 0.800000 | 2.360255 100 | 2 | 0.900000 | 3.225117 100 | 3 | 0.500000 | 1.462128 100 | 3 | 0.700000 | 1.723764 100 | 3 | 0.800000 | 1.947548 100 | 3 | 0.900000 | 2.476718 100 | 5 | 0.900000 | 2.066462 Example three (PH Distribution) We consider an $m$-order PH distribution with irreducible representation $\left(\alpha,T\right)$. For $m=2,d=2,\alpha=\left(1/2,1/2\right)$ and $T\left(1\right)=\left(\begin{array}[]{cc}-4&3\\\ 2&-7\end{array}\right),T\left(2\right)=\left(\begin{array}[]{cc}-5&3\\\ 2&-7\end{array}\right),T\left(3\right)=\left(\begin{array}[]{cc}-4&4\\\ 2&-7\end{array}\right),$ Table 8 illustrates how the doubly exponential solution depends on the PH matrices $T\left(1\right)$, $T\left(2\right)$ and $T\left(3\right)$, respectively. Table 8: The doubly exponential solution depends on the PH matrices $T(i)$ | $T(1)$ | $T(2)$ | $T(3)$ ---|---|---|--- $\pi_{1}$ | (0.2045, 0.1591) | (0.1410, 0.1026) | (0.3125, 0.2500) $\pi_{2}$ | (0.0137, 0.0107) | (0.0043, 0.0031) | (0.0500, 0.0400) $\pi_{3}$ | (6.193e-05, 4.817e-05) | (3.965e-06, 2.884e-06) | (0.0013 , 0.0010) $\pi_{4}$ | (1.259e-09, 9.793e-10) | (3.390e-12, 2.465e-12) | (8.446e-07, 6.757e-07) $\pi_{5}$ | (5.204e-19, 4.048e-19) | (2.478e-24, 1.802e-24) | (3.656e-13, 2.925e-13) To discuss how different caused by a non-exponential distribution versus an exponentially distributed service time with the same mean, for the above three PH distributions we take three corresponding exponential distributions with service rates $\mu(1)=2.7500,\mu(2)=3.4118$ and $\mu(3)=2.3529$, respectively. Table 9 illustrates how the doubly exponential solution ($\pi_{1}$ to $\pi_{5}$) depends on the three service rates. Since the exponential distribution has a lower variance than the PH distribution, it is seen from Tables 8 and 9 that the service time has lower variance, $\pi_{k}$(Exp)$<\pi_{k}$(PH)$e$. Table 9: The doubly exponential solution depends on exponential service rates $\mu(i)$ | $\mu(1)=2.7500$ | $\mu(2)=3.4118$ | $\mu(3)=2.3529$ ---|---|---|--- $\pi_{1}$ | 0.3636 | 0.2931 | 0.4250 $\pi_{2}$ | 0.0481 | 0.0252 | 0.0768 $\pi_{3}$ | 8.408e-04 | 1.858e-04 | 0.0025 $\pi_{4}$ | 2.571e-07 | 1.012e-08 | 2.667e-06 $\pi_{5}$ | 2.402e-14 | 3.004e-17 | 3.030e-12 For the PH and exponential service times, the following two figures provides a comparison for the expected sojourn time. Clearly, the PH service time makes the lower expected sojourn time. Figure 2: $E\left[T_{d}\right]$s of the PH and exponential distributions for $T(1)$ and $T(2)$, respectively For $m=3,d=5,\alpha\left(1\right)=\left(1/3,1/3,1/3\right)$ and $\alpha\left(2\right)=\left(1/12,7/12,1/3\right)$, $T=\left(\begin{array}[]{ccc}-10&2&4\\\ 3&-7&4\\\ 0&2&-5\end{array}\right).$ Table 10 shows how the doubly exponential solution ($\pi_{1}$ to $\pi_{4}$) depends on the vectors $\alpha\left(1\right)$ and $\alpha\left(2\right)$, respectively. Table 10: The doubly exponential solution depends on the vectors $\alpha$ | $\alpha=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ | $\alpha=(\frac{1}{12},\frac{7}{12},\frac{1}{3})$ ---|---|--- $\pi_{1}$ | (0.0741, 0.1358 , 0.2346) | (0.0602, 0.1728, 0.2531) $\pi_{2}$ | (5.619e-05, 1.030e-05, 1.779e-04 ) | (7.182e-05, 2.063e-04, 3.020e-04) $\pi_{3}$ | (1.411e-20, 2.587e-20, 4.469e-20) | (1.739e-19, 4.993e-19, 7.311e-19) $\pi_{4}$ | (1.410e-98, 2.586e-98, 4.466e-98) | (1.444e-92, 4.148e-92, 6.074e-92) ## 6 Concluding remarks In this paper, we provide a matrix-analytic solution for supermarket models. We describe the supermarket model with PH service times as a system of differential vector equations, and provide a doubly exponential solution to the fixed point of the system of differential vector equations. We also provide some numerical examples to illustrate that our approach is effective and efficient in the study of randomized load balancing schemes with non- exponential service requirements, such as, Erlang service time distributions, hyper-exponential service time distributions and PH service time distributions. We expect that this approach will be applicable to study other randomized load balancing schemes, for example, generalizing the arrival process to non-Poisson such as renewal process or Markovian arrival process, generalizing the service times to general probability distributions, and analyzing retrial and processor-sharing service disciplines. ## References * [1] Y. Azar, A.Z. Broder, A.R. Karlin and E. Upfal (1999). Balanced allocations. SIAM Journal on Computing 29, 180–200. A preliminary version of this paper appeared in Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, 1994. * [2] M. Bramson, Y. Lu and B. Prabhakar (2010). 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arxiv-papers
2011-01-22T18:35:45
2024-09-04T02:49:16.578805
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Quan-Lin Li, John C.S. Lui, Yang Wang", "submitter": "Quan-Lin Li", "url": "https://arxiv.org/abs/1101.4306" }
1101.4435
# Solutions for the MIMO Gaussian Wiretap Channel with a Cooperative Jammer††thanks: The authors are with the Dept. of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697-2625, USA. e-mail:{afakoori, swindle}@uci.edu††thanks: This work was supported by the U.S. Army Research Office under the Multi-University Research Initiative (MURI) grant W911NF-07-1-0318. S. Ali. A. Fakoorian*, Student Member, IEEE and A. Lee Swindlehurst, Fellow, IEEE ###### Abstract We study the Gaussian MIMO wiretap channel with a transmitter, a legitimate receiver, an eavesdropper and an external helper, each equipped with multiple antennas. The transmitter sends confidential messages to its intended receiver, while the helper transmits jamming signals independent of the source message to confuse the eavesdropper. The jamming signal is assumed to be treated as noise at both the intended receiver and the eavesdropper. We obtain a closed-form expression for the structure of the artificial noise covariance matrix that guarantees no decrease in the secrecy capacity of the wiretap channel. We also describe how to find specific realizations of this covariance matrix expression that provide good secrecy rate performance, even when there is no non-trivial null space between the helper and the intended receiver. Unlike prior work, our approach considers the general MIMO case, and is not restricted to SISO or MISO scenarios. ###### Index Terms: Physical-layer security, interference channel, MIMO wiretap channel, cooperative jamming. EDICS: WIN-CONT, WIN-PHYL, WIN-INFO, MSP-CAPC ## I Introduction Recent information-theoretic research on secure communication has focused on enhancing security at the physical layer. The wiretap channel, first introduced and studied by Wyner [1], is the most basic physical layer model that captures the problem of communication security. This work led to the development of the notion of perfect secrecy capacity, which quantifies the maximum rate at which a transmitter can reliably send a secret message to its intended recipient, without it being decoded by an eavesdropper. The Gaussian wiretap channel, in which the outputs of the legitimate receiver and the eavesdropper are corrupted by additive white Gaussian noise, was studied in [2]. The secrecy capacity of a Gaussian wiretap channel, which is in general a difficult non-convex optimization problem, has been addressed and solved for in [3]-[7]. The secrecy capacity under an average power constraint is treated in [4] and [5], where in [4] a beamforming approach, based on the generalized singular value decomposition (GSVD), is proposed that achieves the secrecy capacity in the high SNR regime. In [5], we propose an optimal power allocation that achieves the secrecy capacity of the GSVD-based multiple- input, multiple-output (MIMO) Gaussian wiretap channel for any SNR. In [7], a closed-form expression for the secrecy capacity is derived under a certain power-covariance constraint. It was shown in [8] that, for a wiretap channel without feedback, a non-zero secrecy capacity can only be obtained if the eavesdropper’s channel is of lower quality than that of the intended recipient. Otherwise, it is infeasible to establish a secure link under Wyner’s wiretap channel model. In such situations, one approach is to exploit user cooperation in facilitating the transmission of confidential messages from the source to the destination. In [9]-[13], for example, a four-terminal relay-eavesdropper channel is considered, where a source wishes to send messages to a destination while leveraging the help of a relay/helper node to hide the messages from the eavesdropper. While the relay can assist in the transmission of confidential messages, its computational cost may be prohibitive and there are difficulties associated with the coding and decoding schemes at both the relay and the intended receiver. Alternatively, a cooperating node can be used as a helper that simply transmits jamming signals, independent of the source message, to confuse the eavesdropper and increase the range of channel conditions under which secure communications can take place. The strategy of using a helper to improve the secrecy of the source-destination communication is generally known as cooperative jamming [9, 11] or noise-forwarding [12] in prior work. In [9], the scenario where multiple single-antenna users communicate with a common receiver (i.e., the multiple access channel) in the presence of an eavesdropper is considered, and the optimal transmit power allocation that achieves the maximum secrecy sum-rate ia obtained. The work of [9] shows that any user prevented from transmitting based on the obtained power allocation can help increase the secrecy rate for other users by transmitting artificial noise to the eavesdropper (cooperative jamming). In [11], a source-destination system in the presence of multiple helpers and multiple eavesdroppers is considered, where the helpers can transmit weighted jamming signals to degrade the eavesdropper’s ability to decode the source. While the objective is to select the weights so as to maximize the secrecy rate under a total power constraint, or to minimize the total power under a secrecy rate constraint, the results in [11] yield sub-optimal weights for both single and multiple eavesdroppers, due to the assumption that the jamming signal must be nulled at the destination. The noise forwarding scheme of [12] requires that the interferer’s codewords be decoded by the intended receiver. A generalization of [9, 11] and [12] is proposed in [13], in which the helper’s codewords do not have to be decoded by the receiver. The prior work in [9]-[13] assumes single antenna nodes and models single- input, single-output (SISO) or multiple-input, single-output (MISO) cases. A more general MIMO case with multiple cooperative jammers was studied in [14], in which the jammers aligned their interference to lie within a pre-specified “jamming subspace” at the receiver, but the dimensions of the subspace and the power allocation were not optimized. In this paper, we also address the general MIMO case, where the transmitter, legitimate receiver, eavesdropper and helper are in general all equipped with multiple antennas. The transmitter sends confidential messages to its intended receiver, while the helper node assists the transmitter by sending jamming signals independent of the source message to confuse the eavesdropper. While the previous work on this problem shows the fundamental role of jamming as a means to increase secrecy rates, it also emphasizes the fact that that non-carefully designed jamming strategies can preclude secure communication [15]. In this work, we derive a closed-form expression for the structure of the artificial noise covariance matrix of a cooperating jammer that guarantees no decrease in the secrecy capacity of the wiretap channel, assuming the jamming signal from the helper is treated as noise at both the intended receiver and the eavesdropper. We describe algorithms for finding specific realizations of this covariance expression that provide good secrecy rate performance, and show that even when there is no non-trivial nullspace between the helper and the intended receiver, the helper can still transmit artificial noise that does not impact the mutual information between the transmitter and the intended receiver, while decreasing the mutual information between the transmitter and the eavesdropper. Hence, the secrecy level of the confidential message is increased. The situation we consider is different from the one in [16], where the transmitter itself rather than an external helper broadcasts artificial noise to degrade the eavesdropper’s channel. However, both approaches are able to achieve a positive perfect secrecy rate in scenarios where the secrecy capacity in the absence of jamming is zero. The remainder of the paper is organized as follows. In Section II, we describe the system model for the helper-assisted Gaussian MIMO wiretap channel and formulate the problem to be solved. In Sections III and IV, we derive the artificial noise covariance matrix that guarantees no decrease in the secrecy capacity of the wiretap channel. Numerical results in Section V are presented to illustrate the proposed solution. Finally, Section VI concludes the paper. Notation: Throughout the paper, we use boldface uppercase letters to denote matrices. Vector-valued random variables are written with non-boldface uppercase letters (e.g., $X$), while the corresponding lowercase boldface letter (${\mathbf{x}}$) denotes a specific realization of the random variable. Scalar variables are written with non-boldface (lowercase or uppercase) letters. We use $(.)^{T}$ to represent matrix transposition, $(.)^{H}$ the Hermitian (i.e., conjugate) transpose, Tr(.) the matrix trace, $E$ the expectation operator, I the identity matrix, and 0 a matrix or vector with all zeros. Mutual information between the random variables $A$ and $B$ is denoted by $I(A;B)$, and $\mathcal{CN}(0,1)$ represents the complex circularly symmetric Gaussian distribution with zero mean and unit variance. ## II System Model We consider a MIMO wiretap channel that includes a transmitter, an intended receiver, a helping interferer and an eavesdropper, with $n_{t}$, $n_{r}$, $n_{h}$ and $n_{e}$ antennas, respectively. The transmitter sends a confidential message to the intended receiver with the aid of the helper, in the presence of an eavesdropper. We assume that the helper does not know the confidential message and transmits only a Gaussian jamming signal which is not known at the intended receiver nor the eavesdropper and which is treated as noise at both receivers. The mathematical model for this scenario is given by: $\displaystyle{\mathbf{y}}_{1}$ $\displaystyle=$ $\displaystyle{\mathbf{H}}_{1}{\mathbf{x}}_{1}+{\mathbf{G}}_{2}{\mathbf{x}}_{2}+{\mathbf{z}}_{1}$ (1) $\displaystyle{\mathbf{y}}_{2}$ $\displaystyle=$ $\displaystyle{\mathbf{H}}_{2}{\mathbf{x}}_{2}+{\mathbf{G}}_{1}{\mathbf{x}}_{1}+{\mathbf{z}}_{2}\;,$ (2) where ${\mathbf{x}}_{1}$ is a zero-mean $n_{t}\times 1$ transmitted signal vector, ${\mathbf{x}}_{2}$ is a zero-mean $n_{h}\times 1$ jamming vector transmitted by the helper, and ${\mathbf{z}}_{1}\in\mathbb{C}^{n_{r}\times 1}$, ${\mathbf{z}}_{2}\in\mathbb{C}^{n_{e}\times 1}$ are additive white Gaussian noise (AWGN) vectors at the intended receiver and the eavesdropper, respectively, with i.i.d. entries distributed as $\mathcal{CN}(0,1)$. The matrices ${\mathbf{H}}_{1},{\mathbf{G}}_{1}$ represent the channels from the transmitter to the intended receiver and eavesdropper, respectively, while ${\mathbf{H}}_{2},{\mathbf{G}}_{2}$ are the channels from the helper to the eavesdropper and intended receiver, respectively. The channels are assumed to be independent of each other and full rank with arbitrary dimensions. We also assume that the transmitter has full channel state information and is aware of the effective noise covariance at both receivers, where the effective noise is the background noise plus the received artificial noise. Both the helper and the eavesdropper are also aware of all channel matrices as well. The jamming signal transmitted by the helper satisfies an average power constraint: $\text{Tr}(E\\{X_{2}X_{2}^{H}\\})=\text{Tr}({\mathbf{K}}_{w})\leq P_{h}$ (3) where $X_{2}$ is the random variable associated with the specific realization ${\mathbf{x}}_{2}$ and ${\mathbf{K}}_{w}$ is the corresponding covariance matrix. The channel input is subject to a matrix power constraint [7, 17] $E\\{X_{1}X_{1}^{H}\\}={\mathbf{K}}_{x}\preceq{\mathbf{S}}$ (4) where ${\mathbf{K}}_{x}$ is the input covariance matrix, ${\mathbf{S}}$ is a positive semi-definite matrix, and “$\preceq$” denotes that ${\mathbf{S}}-{\mathbf{K}}_{x}$ is positive semi-definite. Note that (4) is a rather general power constraint that subsumes many other important power constraints, including the average total and per-antenna power constraints as special cases. The approach developed in this paper will assume that $P_{h}$ and ${\mathbf{S}}$ (or $\text{Tr}({\mathbf{S}})\leq P_{t}$) are fixed, and that power is not allocated jointly between the transmitter and helper. The numerical results presented later, however, will illustrate the trade-off associated with the power allocation when $P_{h}+P_{t}$ is fixed. As mentioned before, we assume Gauusian signaling for the helper. Thus the effective noise at both receivers is Gaussian and consequently the above MIMO wiretap channel model is Gaussian. For this case, a Gaussian input signal is the optimal choice [6, 17]. Hence, the general optimization problem is equivalent to finding the matrices ${\mathbf{K}}_{x}\succeq 0$ and ${\mathbf{K}}_{w}\succeq 0$ that allow the secrecy capacity of the network to be obtained. A matrix characterization of this optimization problem is given by: $\displaystyle C_{sec}$ $\displaystyle=$ $\displaystyle\max_{{\mathbf{K}}_{x}\succeq 0,{\mathbf{K}}_{w}\succeq 0}[I(X_{1};Y_{1})-I(X_{1};Y_{2})]$ (5) $\displaystyle=$ $\displaystyle\max_{{\mathbf{K}}_{x}\succeq 0,{\mathbf{K}}_{w}\succeq 0}\log|{\mathbf{K}}_{x}{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+\textbf{I})^{-1}{\mathbf{H}}_{1}+\textbf{I}|$ $\displaystyle\qquad\qquad\quad-\log|{\mathbf{K}}_{x}{\mathbf{G}}_{1}^{H}({\mathbf{H}}_{2}{\mathbf{K}}_{w}{\mathbf{H}}_{2}^{H}+\textbf{I})^{-1}{\mathbf{G}}_{1}+\textbf{I}|\;,$ where the non-convex maximization problem in carried out under the power constraints given in (3) and (4). Lemma 1: For a given ${\mathbf{K}}_{w}$, the maximum of (5) is given by $C_{sec}({\mathbf{S}})=\sum_{i=1}^{\rho}\log\gamma_{i}$ (6) where $\gamma_{i}$, $i=1,\cdots,\rho$, are the generalized eigenvalues of the pencil $({\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+\textbf{I})^{-1}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I},\quad{\mathbf{S}}^{\frac{1}{2}}{\mathbf{G}}_{1}^{H}({\mathbf{H}}_{2}{\mathbf{K}}_{w}{\mathbf{H}}_{2}^{H}+\textbf{I})^{-1}{\mathbf{G}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I})$ (7) that are greater than 1. Proof: When the optimization problem in (5) is performed over ${\mathbf{K}}_{x}$ under the matrix power constraint (4) for a given ${\mathbf{K}}_{w}$, it is equivalent to a simple MIMO Gaussian wiretap channel without a helper, where the noise covariance matrices at the receiver and the eavesdropper are $({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+\textbf{I})$ and $({\mathbf{H}}_{2}{\mathbf{K}}_{w}{\mathbf{H}}_{2}^{H}+\textbf{I})$, respectively. The above lemma is a natural extension of [7] and [17, Theorem 3] for the standard MIMO Gaussian wiretap channel. Note that since both elements of the pencil (7) are strictly positive definite, all of the generalized eigenvalues are real and positive [17, 18]. In (6), a total of $\rho$ of them are assumed to be greater than one. Clearly, if there are no such eigenvalues, then the information signal received at the intended receiver is a degraded version of that of the eavesdropper, and in this case the secrecy capacity is zero. Note also that Lemma 1 only provides the secrecy capacity for the optimal ${\mathbf{K}}_{x}$, but does not give an explicit expression for this ${\mathbf{K}}_{x}$. A general expression for the maximizing ${\mathbf{K}}_{x}$ will be given in the next section. To solve the general optimization problem in (5), we would need to find the ${\mathbf{K}}_{w}$ that maximizes (6). Unfortunately, this appears to be a very difficult problem to solve without resorting to some type of ad hoc search. In the following we obtain a sub-optimal closed-form solution for the artificial noise covariance matrix ${\mathbf{K}}_{w}$ that guarantees no decrease in the mutual information between the transmitter and the intended receiver compared with the case where ${\mathbf{K}}_{w}=\mathbf{0}$, while maintaining the power constraint in (5). Hence, the new non-zero ${\mathbf{K}}_{w}$ will only interfere with the eavesdropper, and the secrecy level of the confidential message will be increased. Once such a ${\mathbf{K}}_{w}$ is found, additional improvement in the secrecy rate can be achieved if the transmitter updates its covariance matrix ${\mathbf{K}}_{x}$ for the obtained ${\mathbf{K}}_{w}$. The final secrecy rate for this method is obtained by simply computing (6) and (7) for the resulting ${\mathbf{K}}_{w}$. Note that we will not propose an iterative algorithm that would further alternate between calculating ${\mathbf{K}}_{x}$ and ${\mathbf{K}}_{w}$. We will see in the next section that there is no clear way to update ${\mathbf{K}}_{w}$ from a known non-zero value. ## III Analytical Method We begin with the case where the helper transmits no signal $({\mathbf{K}}_{w}=0)$. In this case, the communication system is reduced to a simple MIMO Gaussian wiretap channel without helper. Based on Lemma 1, the maximum of (5) when ${\mathbf{K}}_{w}=0$ is obtained by applying the generalized eigenvalue decomposition to the following two Hermitian positive definite matrices [7, 17]: ${\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I},\quad{\mathbf{S}}^{\frac{1}{2}}{\mathbf{G}}_{1}^{H}{\mathbf{G}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\;.$ In particular, there exists an invertible generalized eigenvector matrix ${\mathbf{C}}$ such that [18] ${\mathbf{C}}^{H}\left[{\mathbf{S}}^{\frac{1}{2}}{\mathbf{G}}_{1}^{H}{\mathbf{G}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right]{\mathbf{C}}=\textbf{I}$ (8) ${\mathbf{C}}^{H}\left[{\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right]{\mathbf{C}}=\mathbf{\Lambda}$ (9) where $\mathbf{\Lambda}=\text{diag}\\{\lambda_{1},...,\lambda_{n_{t}}\\}$ is a positive definite diagonal matrix and $\lambda_{1},...,\lambda_{n_{t}}$ represent the generalized eigenvalues. Without loss of generality, we assume the generalized eigenvalues are ordered as $\lambda_{1}\geq...\geq\lambda_{b}>1\geq\lambda_{b+1}\geq...\geq\lambda_{n_{t}}>0$ so that a total of $b$ $(0\leq b\leq n_{t})$ are assumed to be greater than 1. Hence, we can write $\mathbf{\Lambda}$ as $\mathbf{\Lambda}=\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&\mathbf{\Lambda}_{2}\end{array}\right]$ (10) where $\mathbf{\Lambda}_{1}=\text{diag}\\{\lambda_{1},...,\lambda_{b}\\}$ and $\mathbf{\Lambda}_{2}=\text{diag}\\{\lambda_{b+1},...,\lambda_{n_{t}}\\}$. Also, we can write ${\mathbf{C}}$ as ${\mathbf{C}}=[{\mathbf{C}}_{1}\quad{\mathbf{C}}_{2}]$ (11) where ${\mathbf{C}}_{1}$ is the $n_{t}\times b$ submatrix representing the generalized eigenvectors corresponding to $\\{\lambda_{1},...,\lambda_{b}\\}$ and ${\mathbf{C}}_{2}$ is the $n_{t}\times(n_{t}-b)$ submatrix representing the generalized eigenvectors corresponding to $\\{\lambda_{b+1},...,\lambda_{n_{t}}\\}$. For the case of ${\mathbf{K}}_{w}=0$, the secrecy capacity of (5) under the matrix power constraint (4) is given by (Lemma 1 or [17, Theorem 3]): $C_{sec}=\sum_{i=1}^{b}\log\lambda_{i}=\log|\mathbf{\Lambda}_{1}|$ (12) and the input covariance matrix ${\mathbf{K}}_{x}^{*}$ that maximizes (5) is given by ([7, 17]): ${\mathbf{K}}_{x}^{*}={\mathbf{S}}^{\frac{1}{2}}{\mathbf{C}}\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\ 0&0\end{array}\right]{\mathbf{C}}^{H}{\mathbf{S}}^{\frac{1}{2}}\;.$ (13) Note that (13) is a general expression for the ${\mathbf{K}}_{x}$ that optimizes (5) for a given ${\mathbf{K}}_{w}$ even when ${\mathbf{K}}_{w}\neq 0$, although in this case the ${\mathbf{C}}$ will be the generalized eigenvector matrix of the pencil (7). From (9) we note that ${\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}$ can be written as ${\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}={\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&\mathbf{\Lambda}_{2}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}\;.$ (14) The following lemma gives the mutual information $I(X_{1};Y_{1})$ between the transmitter and the intended receiver when ${\mathbf{K}}_{w}=0$ and ${\mathbf{K}}_{x}$ is given by (13). Lemma 2: The following equality holds: $I(X_{1};Y_{1})|_{{\mathbf{K}}_{w}=0,{\mathbf{K}}_{x}={\mathbf{K}}_{x}^{*}}=\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+\textbf{I}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|\;.$ (15) Proof: Following the same steps as the proof of [7, App. D] and using (13) and (14), we have $\displaystyle\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+\textbf{I}\right|$ $\displaystyle=$ $\displaystyle\left|{\mathbf{S}}^{\frac{1}{2}}{\mathbf{C}}\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\ 0&0\end{array}\right]{\mathbf{C}}^{H}\times\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&\mathbf{\Lambda}_{2}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}+\textbf{I}\right|$ (20) $\displaystyle=$ $\displaystyle\left|\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\ 0&0\end{array}\right]\times\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&\mathbf{\Lambda}_{2}\end{array}\right]-\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\ 0&0\end{array}\right]{\mathbf{C}}^{H}{\mathbf{C}}+\textbf{I}\right|$ (27) $\displaystyle=$ $\displaystyle\left|\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}&0\\\ 0&0\end{array}\right]-\left[\begin{array}[]{ccc}\textbf{I}&({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\\\ 0&0\end{array}\right]+\textbf{I}\right|$ (32) $\displaystyle=$ $\displaystyle\left|\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}&-({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\\\ 0&{\mathbf{I}}\end{array}\right]\right|$ (35) $\displaystyle=$ $\displaystyle\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|$ (36) where (27) follows from the fact that $\left|{\mathbf{A}}{\mathbf{B}}+{\mathbf{I}}\right|=\left|{\mathbf{B}}{\mathbf{A}}+{\mathbf{I}}\right|$, and (32) follows since ${\mathbf{C}}^{H}{\mathbf{C}}=\left[{\mathbf{C}}_{1}\quad{\mathbf{C}}_{2}\right]^{H}\left[{\mathbf{C}}_{1}\quad{\mathbf{C}}_{2}\right]=\left[\begin{array}[]{ccc}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}&{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\\\ {\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}&{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}\end{array}\right]\;.$ We now return to the general optimization problem in (5) with non-zero ${\mathbf{K}}_{w}$. As the helper begins to broadcast artificial noise, both the mutual information between the transmitter and the intended receiver $I(X_{1};Y_{1})$ and the mutual information between the transmitter and the eavesdropper $I(X_{1};Y_{2})$ are in general decreased. Both of these functions are non-increasing in ${\mathbf{K}}_{w}$ since $\frac{\left|{\mathbf{A}}+{\mathbf{B}}\right|}{\left|{\mathbf{B}}\right|}\geq\frac{\left|{\mathbf{A}}+{\mathbf{B}}+\mathbf{\bigtriangleup}\right|}{\left|{\mathbf{B}}+\mathbf{\bigtriangleup}\right|}$ when ${\mathbf{A}}$, $\mathbf{\bigtriangleup}\succeq 0$ and ${\mathbf{B}}\succ 0$ [20]. A favorable choice for ${\mathbf{K}}_{w}$ would be one that reduces $I(X_{1};Y_{2})$ more than $I(X_{1};Y_{1})$. Since the optimal solution to (5) is intractable, we propose a suboptimal approach that introduces an additional constraint; namely, we search among those ${\mathbf{K}}_{w}$ matrices that guarantee no decrease in the favorable term $I(X_{1};Y_{1})$ while the power constraint (3) is satisfied. It should be noted that this approach is more general than the cooperative jamming schemes proposed in [10, 11] for the MISO case where the jamming signal is nulled out at the destination. Clearly, such sub-optimal solutions are restricted to the case where there exists a null space between the helper and the intended receiver. In the following, we obtain an expression that represents all ${\mathbf{K}}_{w}\succeq 0$ matrices with the power constraint $\text{Tr}({\mathbf{K}}_{w})=P_{h}$ that do not impact the mutual information between the transmitter and the intended receiver; i.e., $I(X_{1};Y_{1})|_{{\mathbf{K}}_{w}\succeq 0,{\mathbf{K}}_{x}={\mathbf{K}}_{x}^{*}}=I(X_{1};Y_{1})|_{{\mathbf{K}}_{w}=0,{\mathbf{K}}_{x}={\mathbf{K}}_{x}^{*}}\;,$ or from (15) $\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}+{\mathbf{I}}\right|=\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|.$ (37) Note that, without loss of generality, we have used an equality power constraint $\text{Tr}({\mathbf{K}}_{w})=P_{h}$ since for the desired ${\mathbf{K}}_{w}$ the best performance is in general obtained when helper transmits at maximum power. Theorem 1: All ${\mathbf{K}}_{w}\succeq 0$ matrices for which $\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}+{\mathbf{I}}\right|=\log\left|{\mathbf{K}}_{x}^{*}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|$ satisfy the following relation: ${\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}={\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}$ (38) where $\begin{array}[]{c}\mathbf{\Lambda}_{22}\preceq{\mathbf{N}}\preceq\mathbf{\Lambda}_{2}\\\ \mathbf{\Lambda}_{22}={\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}+{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}(\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\end{array}$ (39) and $\mathbf{\Lambda}_{1}$, $\mathbf{\Lambda}_{2}$, ${\mathbf{C}}$, ${\mathbf{C}}_{1}$ and ${\mathbf{C}}_{2}$ are defined in (8)-(11). Proof: In Appendix A, using similar steps as those used to obtain (36), we show that all $\mathbf{\Sigma}\succeq 0$ matrices for which $\log\left|{\mathbf{K}}_{x}^{*}\mathbf{\Sigma}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|$ must have the following form $\mathbf{\Sigma}={\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&{\mathbf{M}}\\\ {\mathbf{M}}^{H}&{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}\;.$ (40) In the following, we obtain matrices ${\mathbf{N}}\succeq 0$ and ${\mathbf{M}}$ and complete the proof by considering the following specific choice for $\mathbf{\Sigma}$: $\mathbf{\Sigma}={\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}\;.$ (41) For the specific $\mathbf{\Sigma}$ in (41), it is evident that $0\preceq\mathbf{\Sigma}\preceq{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}.$ (42) By applying the constraint $\mathbf{\Sigma}\preceq{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}$ on (40) and using (14), it is enough to show that: $\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&{\mathbf{M}}\\\ {\mathbf{M}}^{H}&{\mathbf{N}}\end{array}\right]\preceq\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&\mathbf{\Lambda}_{2}\end{array}\right]$ or equivalently that $\left[\begin{array}[]{ccc}0&-{\mathbf{M}}\\\ -{\mathbf{M}}^{H}&\mathbf{\Lambda}_{2}-{\mathbf{N}}\end{array}\right]\succeq 0\;.$ By applying the Schur Complement Lemma [18], the above relationship is true _iff_ $\mathbf{\Lambda}_{2}-{\mathbf{N}}\succeq 0$ and $-{\mathbf{M}}(\mathbf{\Lambda}_{2}-{\mathbf{N}})^{-1}{\mathbf{M}}^{H}\succeq 0$, which in turn is true only when $\displaystyle{\mathbf{M}}$ $\displaystyle=$ $\displaystyle 0$ (43) $\displaystyle\mathbf{\Lambda}_{2}-{\mathbf{N}}$ $\displaystyle\succeq$ $\displaystyle 0\;.$ (44) Applying the results of (43) and (44) in (40) for the specific choice of $\mathbf{\Sigma}={\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}$, we have: $\mathbf{\Sigma}={\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}\;.$ (45) Based on (42), we also need to show that $\mathbf{\Sigma}\succeq 0$. From (45), it is enough to show that $\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&{\mathbf{N}}\end{array}\right]-{\mathbf{C}}^{H}{\mathbf{C}}=\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}&-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\\\ -{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}&{\mathbf{N}}-{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}\end{array}\right]\succeq 0\;.$ By applying the Schur Complement Lemma, the above relationship is true _iff_ $\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}\succeq 0$ and ${\mathbf{N}}-{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}-{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}(\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\succeq 0$. Using Eqs. (8)-(10), it is evident that $\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}={\mathbf{C}}_{1}^{H}\left[{\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right]{\mathbf{C}}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1}={\mathbf{C}}_{1}^{H}{\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}{\mathbf{C}}_{1}\succeq 0$ and finally the lower bound for ${\mathbf{N}}$ is given by ${\mathbf{N}}\succeq{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{2}+{\mathbf{C}}_{2}^{H}{\mathbf{C}}_{1}(\mathbf{\Lambda}_{1}-{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{C}}_{1}^{H}{\mathbf{C}}_{2}\succ 0\;,$ which completes the proof. It should be noted that as ${\mathbf{N}}\rightarrow\mathbf{\Lambda}_{22}$, we have Tr$({\mathbf{K}}_{w})\rightarrow\infty$. Moreover, Tr$({\mathbf{K}}_{w})=0$ is achieved by ${\mathbf{N}}=\mathbf{\Lambda}_{2}$. Hence, for each scalar $P_{h}$, there always exists an ${\mathbf{N}}$ in the range $\mathbf{\Lambda}_{22}\preceq{\mathbf{N}}\preceq\mathbf{\Lambda}_{2}$ that will lead to a ${\mathbf{K}}_{w}$ that satisfies (38) with Tr$({\mathbf{K}}_{w})=P_{h}$. Thus far, we have not made any assumption on the number of antennas at each node. But it is clear from (38) that, for example when ${\mathbf{G}}_{2}$ has more columns than rows, for a fixed ${\mathbf{N}}$ in the acceptable range (39) there will be an infinite number of ${\mathbf{K}}_{w}$ matrices that satisfy (38) and consequently do not decrease $I(X_{1};Y_{1})$. In fact, in this example, a common policy for the helper is to simply transmit artificial noise in the null space of ${\mathbf{G}}_{2}$. A more interesting case occurs when no such null space exists, i.e., when the number of antennas at the helper is less than or equal to that of the intended receiver ($n_{h}\leq n_{r}$). The above result demonstrates the non-trivial fact that even when $n_{h}\leq n_{r}$, it is possible to find a non-zero jamming signal that does not impact $I(X_{1};Y_{1})$ even when the jamming signal can not be nulled by the channel. In the next section, we find more constructive expressions for the ${\mathbf{K}}_{w}$ matrices that satisfy (38) for various combinations of the number of antennas at different nodes. In particular, we show that when $n_{h}\leq n_{r}$, a closed-form expression for ${\mathbf{K}}_{w}$ can be found. ## IV Results for Different Scenarios In this section, we consider all possible combinations of the number of antennas at the transmitter, helper and intended receiver, and obtain constructive methods for computing specific ${\mathbf{K}}_{w}$ matrices that satisfy (38). Such ${\mathbf{K}}_{w}$ will have no impact on $I(X_{1};Y_{1})$, but will in general decrease $I(X_{1};Y_{2})$, the mutual information between the transmitter and the eavesdropper, compared with the case that there is no helper. Hence, the secrecy level of the confidential message is increased. As mentioned before, additional improvement in the secrecy rate can be achieved if the transmitter updates its covariance matrix ${\mathbf{K}}_{x}$ once ${\mathbf{K}}_{w}$ is computed. Note, however, that such an iterative process will not be pursued beyond updating ${\mathbf{K}}_{x}$; unlike the first step, where ${\mathbf{K}}_{w}$ was updated from its initial value of zero, there is no guarantee that finding a new ${\mathbf{K}}_{w}$ will reduce $I(X_{1};Y_{2})$. Hence, the final secrecy rate for the proposed method is obtained by simply computing (6) and (7) for the resulting ${\mathbf{K}}_{w}$ matrices derived in this section. ### IV-A Case 1: $n_{h}\leq\min\\{n_{r},n_{t}\\}$ We show here that for the case where $n_{h}\leq\min\\{n_{r},n_{t}\\}$ and for a fixed ${\mathbf{N}}$ in the acceptable range (39), there is only one ${\mathbf{K}}_{w}$ matrix that satisfies (38) and consequently does not decrease $I(X_{1};Y_{1})$. Using the matrix inversion lemma, Eq. (38) can be written as: $\displaystyle{\mathbf{H}}_{1}^{H}({\mathbf{G}}_{2}{\mathbf{K}}_{w}{\mathbf{G}}_{2}^{H}+{\mathbf{I}})^{-1}{\mathbf{H}}_{1}$ $\displaystyle=$ $\displaystyle{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}-{\mathbf{H}}_{1}^{H}{\mathbf{G}}_{2}({\mathbf{G}}_{2}^{H}{\mathbf{G}}_{2}+{\mathbf{K}}_{w}^{-1})^{-1}{\mathbf{G}}_{2}^{H}{\mathbf{H}}_{1}$ (48) $\displaystyle=$ $\displaystyle{\mathbf{S}}^{-1/2}\left[{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}&0\\\ 0&{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}-\textbf{I}\right]{\mathbf{S}}^{-1/2}\;.$ Replacing ${\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}$ with (14), we have: ${\mathbf{H}}_{1}^{H}{\mathbf{G}}_{2}({\mathbf{G}}_{2}^{H}{\mathbf{G}}_{2}+{\mathbf{K}}_{w}^{-1})^{-1}{\mathbf{G}}_{2}^{H}{\mathbf{H}}_{1}={\mathbf{S}}^{-1/2}{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}0&0\\\ 0&\mathbf{\Lambda}_{2}-{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}{\mathbf{S}}^{-1/2}\;.$ (49) Since we have assumed that the channels are full rank, in the case of $n_{h}\leq n_{r}\leq n_{t}$ or $n_{h}\leq n_{t}\leq n_{r}$, it is clear that rank$({\mathbf{G}}_{2}^{H}{\mathbf{H}}_{1})=n_{h}.$ Thus, from (49) we have: $({\mathbf{G}}_{2}^{H}{\mathbf{G}}_{2}+{\mathbf{K}}_{w}^{-1})^{-1}={\mathbf{O}}^{H}{\mathbf{S}}^{-1/2}{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}0&0\\\ 0&\mathbf{\Lambda}_{2}-{\mathbf{N}}\end{array}\right]{\mathbf{C}}^{-1}{\mathbf{S}}^{-1/2}{\mathbf{O}}$ (50) where ${\mathbf{O}}$ is the right inverse of ${\mathbf{G}}_{2}^{H}{\mathbf{H}}_{1}$, which, for example when $n_{h}\leq n_{r}\leq n_{t}$, can be written as ${\mathbf{O}}={\mathbf{H}}_{1}^{H}({\mathbf{H}}_{1}{\mathbf{H}}_{1}^{H})^{-1}{\mathbf{G}}_{2}({\mathbf{G}}_{2}^{H}{\mathbf{G}}_{2})^{-1}$. The following lemma is a direct result of Eqs. (49) and (50). Lemma 3: For the case of $n_{h}\leq\min\\{n_{r},n_{t}\\}$ and for a fixed ${\mathbf{N}}$ in the acceptable range (39), the ${\mathbf{K}}_{w}\succeq 0$ matrix for which (38) is satisfied and $I(X_{1};Y_{1})$ is not decreased is given by ${\mathbf{K}}_{w}={\mathbf{Q}}-{\mathbf{Q}}{\mathbf{G}}_{2}^{H}({\mathbf{G}}_{2}{\mathbf{Q}}{\mathbf{G}}_{2}^{H}-{\mathbf{I}})^{-1}{\mathbf{G}}_{2}{\mathbf{Q}}$ (51) where ${\mathbf{Q}}$ is the RHS of (50). Proof: After applying the matrix inversion lemma on the LHS of (50), a straightforward computation yields (51). As is evident from Eqs. (50)-(51), we still have a design parameter, ${\mathbf{N}}$, that should be chosen in its acceptable range $\mathbf{\Lambda}_{22}\preceq{\mathbf{N}}\preceq\mathbf{\Lambda}_{2}$ such that the power constraint $\text{Tr}({\mathbf{K}}_{w})=P_{h}$ is satisfied. Finding the optimal ${\mathbf{N}}$ that minimizes $I(X_{1};Y_{2})$ when ${\mathbf{K}}_{x}$ and ${\mathbf{K}}_{w}$ are given by (13) and (51), respectively, is as intractable as the general optimization problem in (5). Instead, we simply restrict the ${\mathbf{N}}$ we consider to those that can be linearly parameterized within the acceptable range, as follows: ${\mathbf{N}}=\mathbf{\Lambda}_{22}+t\left(\mathbf{\Lambda}_{2}-\mathbf{\Lambda}_{22}\right)\;.$ (52) Consequently the term $\mathbf{\Lambda}_{2}-{\mathbf{N}}$ in Eq. (51) becomes $\mathbf{\Lambda}_{2}-{\mathbf{N}}=(1-t)\left(\mathbf{\Lambda}_{2}-\mathbf{\Lambda}_{22}\right)$ where the scalar $0\leq t\leq 1$ is chosen such that the power constraint $\text{Tr}({\mathbf{K}}_{w})=P_{h}$ is satisfied. Note that as $t\rightarrow 0$ $({\mathbf{N}}\rightarrow\mathbf{\Lambda}_{22})$ then $\text{Tr}({\mathbf{K}}_{w})\rightarrow\infty$, and as $t\rightarrow 1$ $({\mathbf{N}}\rightarrow\mathbf{\Lambda}_{2})$ then $\text{Tr}({\mathbf{K}}_{w})\rightarrow 0$. Thus, we are guaranteed that an acceptable ${\mathbf{N}}$ can be found in this way. ### IV-B Case 2: $n_{h}>\min\\{n_{r},n_{t}\\}$ As mentioned before, for the case of $n_{h}>n_{r}$ and for a fixed ${\mathbf{N}}$ in the acceptable range (39), there are many ${\mathbf{K}}_{w}$ matrices that satisfy (38) and consequently do not decrease $I(X_{1};Y_{1})$. A common policy for the helper in this case is to transmit artificial noise in the null space of ${\mathbf{G}}_{2}$. However, as (38) shows, this policy is sufficient but it is not necessary. In other words, it is possible that the optimal ${\mathbf{K}}_{w}$ satisfying (38) has elements outside the null space of ${\mathbf{G}}_{2}$. Because of the non-linear constraint in (38), finding the optimal ${\mathbf{K}}_{w}$ is intractable. A similar discussion applies for the case of $n_{t}<n_{h}\leq n_{r}$. In this section, we present an approach for computing a suitable ${\mathbf{K}}_{w}$. Consider the following jamming signal covariance matrix: ${\mathbf{K}}_{w}=\mathbf{\Gamma}\,\mathbf{\Pi}\,\mathbf{\Gamma}^{H}\;,$ (53) where $\mathbf{\Pi}$ is a $d\times d$ positive semidefinite matrix, and $\mathbf{\Gamma}$ is an $n_{h}\times d$ matrix. For the case of $n_{t}<n_{h}\leq n_{r}$ or $n_{h}>n_{r}$, we can choose $\mathbf{\Gamma}$ such that ${\mathbf{G}}_{2}\,\mathbf{\Gamma}$ is orthogonal to ${\mathbf{H}}_{1}\,{{\mathbf{K}}_{x}^{*}}^{\frac{1}{2}}$, i.e., ${{\mathbf{K}}_{x}^{*}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{G}}_{2}\,\mathbf{\Gamma}={\boldsymbol{0}}$. For example, $\mathbf{\Gamma}$ can be chosen as the $d$ right singular vectors in the nullspace of ${{\mathbf{K}}_{x}^{*}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{G}}_{2}$. Since ${\mathbf{K}}_{x}$ will often be rank deficient, the value of $d$ will typically be larger than $n_{h}-n_{t}$ for the case of $n_{t}<n_{h}\leq n_{r}$, and larger than $n_{h}-n_{r}$ for the case of $n_{h}>n_{r}$. For this choice of $\mathbf{\Gamma}$, the resulting ${\mathbf{K}}_{w}$ in (53) satisfies (38), and doesn’t decrease $I(X_{1};Y_{1})$ for ${\mathbf{N}}=\mathbf{\Lambda}_{2}$, as is clear from (38). Given $\mathbf{\Gamma}$, the choice of $\mathbf{\Pi}$ can be made to maximize the transfer of the “information” in the helper’s jamming signal to the eavesdropper. In particular, note that at the eavesdropper, the covariance of the helper’s jamming signal will be given by ${\mathbf{H}}_{2}\mathbf{\Gamma\Pi\Gamma}^{H}{\mathbf{H}}_{2}^{H}$. If the eigenvalue decomposition of $\mathbf{\Gamma}^{H}{\mathbf{H}}_{2}^{H}{\mathbf{H}}_{2}\mathbf{\Gamma}$ is written as $\mathbf{\Gamma}^{H}{\mathbf{H}}_{2}^{H}{\mathbf{H}}_{2}\mathbf{\Gamma}={\mathbf{U}}\,{\mathbf{D}}\,{\mathbf{U}}^{H}$ with ${\mathbf{U}}$ unitary and ${\mathbf{D}}$ square and diagonal, then $\mathbf{\Pi}$ can be found via waterfilling; i.e., $\mathbf{\Pi}={\mathbf{U}}\,\mathbf{\Delta}\,{\mathbf{U}}^{H}\;,$ where $\mathbf{\Delta}=\left[\eta{\mathbf{I}}-{\mathbf{D}}^{-1}\right]^{+}$, the operation $[{\mathbf{A}}]^{+}$ zeros out any negative elements, and the water-filling level $\eta$ is chosen such that $\text{Tr}({\mathbf{K}}_{w})=\text{Tr}(\mathbf{\Delta})=P_{h}$. ## V Numerical Results In this section, we present numerical results to illustrate our theoretical findings. In all of the following figures, channels are assumed to be quasi- static flat Rayleigh fading and independent of each other. The channel matrices ${\mathbf{H}}_{1}\in\mathbb{C}^{n_{r}\times n_{t}}$ and ${\mathbf{G}}_{2}\in\mathbb{C}^{n_{r}\times n_{h}}$ have i.i.d. entries distributed as $\mathcal{CN}(0,\sigma_{d}^{2})$, while ${\mathbf{G}}_{1}\in\mathbb{C}^{n_{e}\times n_{t}}$ and ${\mathbf{H}}_{2}\in\mathbb{C}^{n_{e}\times n_{h}}$ have i.i.d. entries distributed as $\mathcal{CN}(0,\sigma_{c}^{2})$. In each figure, values for the number of antennas at each node, as well as $\sigma_{d}^{2}$ and $\sigma_{c}^{2}$, will be depicted. Unless otherwise indicated, results are calculated based on an average of at least 500 independent channel realizations. In the first example, Fig. 1, we randomly generate positive definite matrices ${\mathbf{S}}$ such that $\text{Tr}({\mathbf{S}})\leq P_{t}$. For each ${\mathbf{S}}$, we compute the secrecy capacity of the MIMO Gaussian wiretap channel without helper (${\mathbf{K}}_{w}={\boldsymbol{0}}$) as given by (12). Next, using (51), we obtain a ${\mathbf{K}}_{w}$ with the average power constraint $\text{Tr}({\mathbf{K}}_{w})=P_{h}$ that does not decrease $I(X_{1};Y_{1})$, and then update ${\mathbf{K}}_{x}$ and compute $C_{sec}({\mathbf{S}})$, using (6) and (7), accordingly. Fig. 1 compares the secrecy capacity of the wiretap channel with (solid lines) and without (dotted lines) the helper. Note that the vertical difference between the solid curves (about 0.6 bps/channel use) represents the role of the transmit power $P_{t}$ on the secrecy capacity with helper when $P_{t}$ changes from 100 to 150 and $P_{h}=20$. This relatively small difference indicates that, in this example, $P_{t}$ does not have a big impact on the secrecy capacity. Its role is even more negligible when $P_{h}=0$, where only an increase of $0.3$ bps/channel use is obtained as $P_{t}$ increases from 100 to 150. The role of the helper on the other hand is significantly more important; increasing $P_{h}$ from 0 to 20 while holding $P_{t}$ fixed results in an increase on the order of 3 bps/channel use. Furthermore, the use of the helper with a total power of only 120 ($P_{t}=100,P_{h}=20$) provides significantly better secrecy performance than not using the helper and transmitting with total power equal to 150 ($P_{t}=150,P_{h}=0$). In the next examples, we calculate the secrecy capacity of the proposed algorithms under the assumption of an average power constraint $P_{t}$ at the transmitter, and under the constraint that the helper does not reduce the mutual information between the transmitter and receiver. While Eqs. (6) and (7) provide the performance for a specific ${\mathbf{S}}$, one must solve [17], [20, Lemma 1] $C_{sec}(P_{t})=\max_{{\mathbf{S}}\succeq 0,\text{Tr}({\mathbf{S}})\leq P_{t}}C_{sec}({\mathbf{S}})$ (54) to find the secrecy capacity over all ${\mathbf{S}}$ that satisfy the average power constraint. In the examples that follow, we perform a numerical search to solve (54) and compute the secrecy capacity. Fig. 2 shows the secrecy capacity versus $P_{h}$ for a fixed total average power $P_{t}+P_{h}=110$. In this figure, we consider a situation in which $\sigma_{c}>\sigma_{d}$, or in other words where the channel between the transmitter and the intended receiver is weaker than the channel between the transmitter and the eavesdropper, and the channel between the helper and the intended receiver is weaker than the channel between the helper and the eavesdropper. The arrow in the figure shows the secrecy capacity without the helper $(P_{h}=0)$. The figure shows that a helper with just a single antenna can provide a dramatic improvement in secrecy rate with very little power allocated to the jamming signal; in fact, the optimal rate is obtained when $P_{h}$ is less than 2% of the total available transmit power. If the number of antennas at the helper increases, a much higher secrecy rate can be obtained, but at the expense of allocating more power to the helper and less to the signal for the desired user. In Fig. 3, we consider a situation in which, unlike the above example, we have $\sigma_{d}>\sigma_{c}$. Thus, the intended receiver, in comparison with the eavesdropper, receives a weaker information signal and a stronger jamming signal than the eavesdropper. It might seem that in this situation, the helper cannot be very useful, but the figure shows that even in this case we can have a notable improvement in the secrecy rate (about 4 bps/channel use) by increasing the number of antennas at the helper, and with an appropriate power assignment between the transmitter and the helper, without requiring extra total transmit power for the helper node. In Fig. 4, we consider a specific scenario where the secrecy capacity in the absence of the helper node is zero. While channel matrices ${\mathbf{H}}_{2}$ and ${\mathbf{G}}_{2}$ are generated randomly with i.i.d. entries distributed as $\mathcal{CN}(0,\sigma_{c}^{2})$ and $\mathcal{CN}(0,\sigma_{d}^{2})$, respectively, we assume the following specific choices for ${\mathbf{H}}_{1}$ and ${\mathbf{G}}_{1}$: ${\mathbf{H}}_{1}=\left[\begin{array}[]{ccc}-0.25+0.5i&-0.35&-1.25-0.9i\\\ -0.4+0.1i&-0.2+0.75i&-i\end{array}\right]$ ${\mathbf{G}}_{1}=\left[\begin{array}[]{ccc}2+0.25i&1.5+0.5i&2i\\\ 0.25+0.25i&-0.7+1.5i&0.5+0.33i\\\ -1.5&-0.5-i&-2.9i\end{array}\right].$ Since ${\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}\preceq{\mathbf{G}}_{1}^{H}{\mathbf{G}}_{1}$, all the generalized eigenvalues of the pencil $\left({\mathbf{S}}^{\frac{1}{2}}{\mathbf{H}}_{1}^{H}{\mathbf{H}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right)-\gamma\left({\mathbf{S}}^{\frac{1}{2}}{\mathbf{G}}_{1}^{H}{\mathbf{G}}_{1}{\mathbf{S}}^{\frac{1}{2}}+\textbf{I}\right)$ are zero for all ${\mathbf{S}}\succeq 0$ and consequently, the secrecy capacity without helper will be zero. In this example, we also assume that not only is the total power fixed at $P_{t}+P_{h}=110$, but also the total number of transmit antennas is fixed at $n_{t}+n_{h}=3$. As in the other examples, the secrecy rate of the wiretap channel is considerably improved with the helper. In this case, the best performance is obtained when the helper has only a single antenna. Finally, in Fig. 5, we consider the role of number of antennas at the helper, $n_{h}$, in the secrecy rate for the specific matrix power constraint ${\mathbf{S}}=\frac{P_{t}}{n_{t}}{\mathbf{I}}$. Note that the solution of Section IV-A applies for $n_{h}\leq 3$, while the solution of Section IV-B holds for $n_{h}>3$. In all cases, we see that the secrecy rate increases considerably as $n_{h}$ increases. ## VI Conclusions In this paper, we have studied the Gaussian MIMO Wiretap channel in the presence of an external jammer/helper, where the helper node assists the transmitter by sending artificial noise independent of the source message to confuse the eavesdropper. The jamming signal from the helper is not required to be decoded by the intended receiver and is treated as noise at both the intended receiver and the eavesdropper. We obtained a closed-form relationship for the structure of the helper’s artificial noise covariance matrix that guarantees no decrease in the mutual information between the transmitter and the intended receiver. We showed how to find appropriate solutions within this covariance matrix framework that provide very good secrecy rate performance, even when there is no non-trivial null space between the helper and the intended receiver. The proposed scheme is shown to achieve a notable improvement in secrecy rate even for a fixed average total power and a fixed total number of antennas at the transmitter and the helper, without requiring extra power or antennas to be allocated to the helper node. ## Appendix A We are interested in finding a relationship that represents all matrices $\mathbf{\Sigma}\succ 0$ for which $\log\left|{\mathbf{K}}_{x}^{*}\mathbf{\Sigma}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|\;,$ (55) where ${\mathbf{K}}_{x}^{*}={\mathbf{S}}^{\frac{1}{2}}{\mathbf{C}}\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\ 0&0\end{array}\right]{\mathbf{C}}^{H}{\mathbf{S}}^{\frac{1}{2}}\;.$ (56) Using the fact that $|{\mathbf{A}}{\mathbf{B}}+{\mathbf{I}}|=|{\mathbf{B}}{\mathbf{A}}+{\mathbf{I}}|$, it is clear that $\mathbf{\Sigma}$ will have the form $\mathbf{\Sigma}={\mathbf{S}}^{-\frac{1}{2}}{\mathbf{C}}^{-H}{\mathbf{X}}{\mathbf{C}}^{-1}{\mathbf{S}}^{-\frac{1}{2}}$ for some matrix ${\mathbf{X}}={\mathbf{X}}^{H}$. Substituting this expression for $\mathbf{\Sigma}$ into (55) results in the following equation that must be solved for ${\mathbf{X}}$: $\log\left|\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\ 0&0\end{array}\right]{\mathbf{X}}+{\mathbf{I}}\right|=\log\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}\mathbf{\Lambda}_{1}\right|\;.$ (57) Write ${\mathbf{X}}$ as ${\mathbf{X}}=\left[\begin{array}[]{ccc}{\mathbf{X}}_{1}&{\mathbf{X}}_{2}\\\ {\mathbf{X}}_{2}^{H}&{\mathbf{X}}_{3}\end{array}\right]$ so that we have $\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}&0\\\ 0&0\end{array}\right]{\mathbf{X}}+{\mathbf{I}}=\left[\begin{array}[]{ccc}({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{X}}_{1}+{\mathbf{I}}&({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{X}}_{2}\\\ 0&{\mathbf{I}}\end{array}\right]\;,$ and note that the determinant of the above matrix is given by $\left|({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})^{-1}{\mathbf{X}}_{1}+{\mathbf{I}}\right|$. By comparing this result with (55), we see that ${\mathbf{X}}_{1}=\mathbf{\Lambda}_{1}-({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})$. Consequently, we have: $\mathbf{\Sigma}={\mathbf{S}}^{-\frac{1}{2}}{\mathbf{C}}^{-H}\left[\begin{array}[]{ccc}\mathbf{\Lambda}_{1}-({\mathbf{C}}_{1}^{H}{\mathbf{C}}_{1})&{\mathbf{X}}_{2}\\\ {\mathbf{X}}_{2}^{H}&{\mathbf{X}}_{3}\end{array}\right]{\mathbf{C}}^{-1}{\mathbf{S}}^{-\frac{1}{2}}$ (58) where ${\mathbf{X}}_{2}$ and ${\mathbf{X}}_{3}$ are still unknown and must be found as described in the text. It is clear that (58) and (40) are equivalent. ## References * [1] A. Wyner, “The wire-tap channel,” _Bell. Syst. Tech. J._ , vol. 54, no. 8, pp. 1355-1387, Jan. 1975. * [2] S. K. Leung-Yan-Cheong and M. E. Hellman, “The Gaussian wire-tap channel,” _IEEE Trans. Inf. Theory_ , vol. 24, pp. 451-456, Jul. 1978. * [3] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap channel,” in _Proc. IEEE Int. Symp. Information Theory_ Toronto, ON, Canada, Jul. 2008, pp. 524-528. * [4] A. Khisti and G. Wornell, “Secure transmission with multiple antennas II: The MIMOME wiretap channel,” to appear, _IEEE Trans. Inf. Theory_ , 2010. Available at: http://allegro.mit.edu/pubs/posted/journal/2008-khisti-wornell-it.pdf * [5] S. Ali. A. Fakoorian and A. L. Swindlehurst, “Optimal power allocation for the GSVD based MIMO Gaussian wiretap channel,” submitted to _IEEE Trans. Inf. Theory_ , Available: http://arxiv.org/abs/1006.1890 * [6] T. Liu and S. Shamai (Shitz), “A note on secrecy capacity of the multi-antenna wiretap channel,” _IEEE Trans. Inf. Theory_ , vol. 55, no. 6, pp. 2547-2553, 2009. * [7] R. Bustin, R. Liu, H. V. Poor, and S. Shamai (Shitz), “A MMSE approach to the secrecy capacity of the MIMO Gaussian wiretap channel,” _EURASIP Journal on Wireless Communications and Networking_ , vol. 2009, Article ID 370970, 8 pages, 2009. * [8] I. Csiszar and J. Korner, “Broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , vol. 24, pp. 339-348, May 1978. * [9] E. Tekin and A. Yener, “The general Gaussian multiple access and two-way wire-tap channels: Achievable rates and cooperative jamming,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 6, pp. 2735 2751, Jun. 2008. * [10] L. Dong, Z. Han, A. P. Petropulu, H. V. Poor, “Cooperative jamming for wireless physical layer security”, in Proc. of _IEEE Workshop on Statistical Signal Processing_ , Cardiff, Wales, U.K. 2009 * [11] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Improving wireless physical layer security via cooperating relays,” _IEEE Trans. Signal Proc._ , vol. 58, NO. 3, pp. 1875-1888, Mar. 2010. * [12] L. Lai and H. El Gamal, “The relay-eavesdropper channel: Cooperation for secrecy,” _IEEE Trans. Inf. Theory_ , vol. 54, no. 9, pp. 4005 4019, Sep. 2008. * [13] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor, “The Gaussian wiretap channel with a helping interferer,” in _Proc. IEEE Int. Symp. Inf. Theory_ , Toronto, ON, Canada, Jul. 2008. * [14] J. Wang and A. Swindlehurst, “Cooperative jamming in MIMO ad hoc networks,” in _Proc. Asilomar Conf. on Signals, Systems and Computers_ , pp. 1719-1723, Nov., 2009. * [15] E. MolavianJazi, M. Bloch, and J. N. Laneman, “Arbitrary jamming can preclude secure communication,” in _Proc. Allerton Conf._ Communications, Control, and Computing, Monticello, IL, Sept. 2009. * [16] S. Goel and R. Negi, “Guaranteeing secrecy using artificial noise,” _IEEE Trans. Wireless Commun_., vol. 7, no. 6, pp. 2180-2189, June 2008. * [17] Ruoheng Liu, Tie Liu, H. Vincent Poor, and Shlomo Shamai (Shitz), “Multiple-input multiple-output Gaussian broadcast channels with confidential messages,” _IEEE Trans. Inf. Theory_ , to appear. * [18] R. A. Horn and C. R. Johnson, _Matrix Analysis_ , University Press, Cambridge, UK, 1985. * [19] S. W. Peters and R. W. Heath, Jr., “Interference alignment via alternating minimization,” in Proc. of _IEEE ICASSP_ , April 2009, Taiwan. * [20] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The capacity region of the Gaussian multiple-input multiple-output broadcast channel,” _IEEE Trans. Inf. Theory_ , vol. 52, no. 9, pp. 3936-3964, 2006 Figure 1: Comparison of secrecy capacity for MIMO Gaussian wiretap channel with and without helper for different $P_{t}$ and $P_{h}$. Figure 2: Comparison of the secrecy capacity for the MIMO Gaussian wiretap channel with and without a helper versus $P_{h}$ for different number of antennas at the helper, $P_{t}+P_{h}=110$, assuming the eavesdropper’s channels are stronger than those of the receiver ($\sigma_{d}^{2}=1,\sigma_{c}^{2}=5$). Figure 3: Comparison of the secrecy capacity for the MIMO Gaussian wiretap channel with and without a helper versus $P_{h}$ for different number of antennas at the helper, $P_{t}+P_{h}=110$, assuming the receiver’s channels are stronger than those of the eavesdropper ($\sigma_{d}^{2}=2,\sigma_{c}^{2}=1$). Figure 4: Comparison of the secrecy capacity for the MIMO Gaussian wiretap channel with and without a helper versus $P_{h}$ for different number of antennas at the helper, $P_{t}+P_{h}=110$, and $n_{t}+n_{h}=3$. Figure 5: Secrecy data rate versus $n_{h}$ for a specific matrix power constraint ${\mathbf{S}}=\frac{P_{t}}{n_{t}}{\mathbf{I}}$.
arxiv-papers
2011-01-24T03:09:16
2024-09-04T02:49:16.586353
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. Ali. A. Fakoorian and A. Lee Swindlehurst", "submitter": "Ali Fakoorian", "url": "https://arxiv.org/abs/1101.4435" }
1101.4458
arxiv-papers
2011-01-24T07:49:23
2024-09-04T02:49:16.592269
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Qun Mo and Yi Shen", "submitter": "Yi Shen", "url": "https://arxiv.org/abs/1101.4458" }
1101.4464
# Spatially Modulated Interaction Induced Bound States and Scattering Resonances Ran Qi and Hui Zhai Institute for Advanced Study, Tsinghua University, Beijing, 100084, China ###### Abstract We study the two-body problem with a spatially modulated interaction potential using a two-channel model, in which the inter-channel coupling is provided by an optical standing wave and its strength modulates periodically in space. As the modulation amplitudes increases, there will appear a sequence of bound states. Part of them will cause divergence of the effective scattering length, defined through the phase shift in the asymptotic behavior of scattering states. We also discuss how the local scattering length, defined through short-range behavior of scattering states, modulates spatially in different regimes. These results provide a theoretical guideline for new control technique in cold atom toolbox, in particular, for alkali-earth-(like) atoms where the inelastic loss is small. Feshbach resonances (FR) and optical lattices (OL) are two major techniques in cold atom toolbox. FR can be used to control the interactions by tuning a bound state in the so-called “closed channel” to the scattering threshold via magnetic field, laser field or external confinement RMP ; OFR ; CIR . At resonance, the $s$-wave scattering length diverges and the system becomes a strongly interacting one. OL can strongly modify the single particle spectrum of atoms, which suppress the kinetic energy so that the interaction effects are enhanced. With these two methods, many interesting many-body physics, such as BEC-BCS crossover, superfluid to Mott insulator transition and strongly correlated quantum fluids in low dimensions, have been studied extensively in cold atom systems during the last decade review . In this letter we theoretically study a new control tool for cold atom system. It is analogous to OL because it also makes use of two counter propagating laser fields that lead to a periodic modulation of laser intensity in space; however, its main effect is not acted on single particle, but manifested on the interaction term when two particles collide with each other. It generates a spatial modulation of two-body interaction, i.e. the two-body interaction potential not only depends on the relative coordinate of two particles under collision, but also depends on their center-of-mass coordinate. As far as we know, this is a situation not encountered in interacting systems studied before, ranging from high-energy and nuclear physics to condensed matter systems. One can expect a spatial modulated interaction potential will result in many fascinating phenomena. The explicit model under consideration is schematically shown in Fig. 1. The open and closed channels are different orbital states and are coupled by laser field. Such an optical FR has been studied before for uniform laser intensity OFR , and has been observed experimentally for both alkaline and alkaline- earth-like Yb atoms exp . In this letter we shall consider the situation that the laser field is a standing wave whose intensity, and therefore the coupling strength between open and closed channel, is modulated periodically in space. Such a setup allows one to control spatial modulation of inter-atomic interaction on the scale of sub-micron. Recently, this has been realized in 174Yb condensation Kyoto , although the optical standing wave is a pulsed one. Alkiline-earth(-like) atom like Yb is particular suitable for such an experiment because the narrow ${}^{1}S_{0}$-${}^{3}P_{1}$ inter-combination transition line can avoid large inelastic scattering loss. In this experiment, a spatially modulated mean-field energy has been observed from diffraction pattern in a time-of-flight imagine Kyoto . However, the theoretical study of this system is still very limited, and even the two-body problem has not been studied. In this work we show that surprises indeed arise even in the two-body problem of this model. Figure 1: A schematic of the model we studied. See text for detail description. Coupled Two-channel Model. We consider a two-body Hamiltonian for a FR in which the open and closed channels are modeled by two square well potentials cheng $\mathcal{H}=-\frac{\hbar^{2}}{4m}\nabla^{2}_{\bf{R}}-\frac{\hbar^{2}}{m}\nabla^{2}_{\bf{r}}+v({\bf R},{\bf r})$ (1) where ${\bf R}=({\bf r_{1}}+{\bf r_{2}})/2$ and ${\bf r}={\bf r_{1}}-{\bf r_{2}}$. For $r<r_{0}$, $v({\bf R},{\bf r})=\left(\begin{array}[]{cc}-V_{\text{o}}&\hbar\Omega({\bf R})\\\ \hbar\Omega({\bf R})&-V_{\text{c}}\end{array}\right)$ (2) and for $r>r_{0}$, $v({\bf R},{\bf r})=\left(\begin{array}[]{cc}0&0\\\ 0&+\infty\end{array}\right).$ (3) In this model $V_{\text{o}}$ is given by the background scattering length $a_{\text{bg}}$ as $\tan(k_{\text{o}}r_{0})/(k_{\text{o}}r_{0})=1-a_{\text{bg}}/r_{0}$ where $k_{\text{o}}=\sqrt{mV_{\text{o}}}/\hbar$, and $V_{\text{c}}$ is determined by the binding energy of closed channel molecule $\epsilon_{\text{c}}$ through $V_{\text{c}}=\hbar^{2}\pi^{2}/(mr^{2}_{0})-\epsilon_{\text{c}}$. The size of inter-atomic potential $r_{0}$ is much smaller than all the other length scales. Conventionally, the inter-channel coupling $\Omega({\bf R})$ is a constant independent of ${\bf R}$. Such a model captures all key features of a FR RMP ; cheng . A bound state appears at threshold and causes scattering resonance at $\Omega_{0}=\sqrt{\epsilon_{\text{c}}/|\beta|}$, and the scattering length is given by cheng $a_{\text{s}}=a_{\text{bg}}\left(1-\frac{\beta\Omega^{2}}{\epsilon_{\text{c}}+\beta\Omega^{2}}\right)$ (4) where $\beta=32r_{0}a_{\text{bg}}/(9\pi^{2})$. Figure 2: First four bound states as a function of the amplitude of coupling $\Omega$. The parameters for this plot are $q=0$, $\epsilon_{\text{c}}=0.05E_{\text{R}}$, $Ka_{\text{bg}}=-0.01$ and $Kr_{0}=10^{-3}$. $E_{\text{R}}=\hbar^{2}K^{2}/m$ is taken as energy unit. Now consider the situation $\Omega$ depends on ${\bf R}$. Solving the Schrödinger equation follows two steps: (i) in the regime $r<r_{0}$, because the $-\hbar^{2}\nabla^{2}_{{\bf r}}/m$ term commutes with the Hamiltonian, we consider the wave function of following form $\psi_{\text{o}}=\frac{\sin kr}{r}a({\bf R});\ \ \psi_{\text{c}}=\frac{\sin kr}{r}b({\bf R})$ (5) where $a({\bf R})$ and $b({\bf R})$ satisfy a coupled equation $\displaystyle\left[-\frac{\hbar^{2}}{4m}\nabla^{2}_{\bf R}-V_{\text{o}}\right]a({\bf R})+\Omega({\bf R})b({\bf R})=\epsilon a({\bf R})$ (6) $\displaystyle\left[-\frac{\hbar^{2}}{4m}\nabla^{2}_{\bf R}-V_{\text{c}}\right]b({\bf R})+\Omega({\bf R})a({\bf R})=\epsilon b({\bf R})$ (7) where $\epsilon=E-\hbar^{2}k^{2}/m$. There will be a set of eigen-function $a_{l}({\bf R})$, $b_{l}({\bf R})$ and $k_{l}$ that give rise to the same energy $E$. The eigen wave function in the regime $r<r_{0}$ should be assumed as $\psi({\bf R},{\bf r})=\sum\limits_{l}A_{l}\frac{\sin k_{l}r}{r}\left(\begin{array}[]{c}a_{l}({\bf R})\\\ b_{l}({\bf R})\end{array}\right)$ (8) (ii) The superposition coefficient $A_{l}$, the binding energy $E$ for bound states, as well as the phase shift $\delta(E)$ for scattering states, are determined by matching the wave function in the regime of $r>r_{0}$ at $r=r_{0}$ for any ${\bf R}$. Hereafter we will consider an explicit situation where $\Omega({\bf R})=\Omega\cos(Kx)$ ($x$ denotes the $x$-component of ${\bf R}$). Note that there is still a discrete translation symmetry $x\rightarrow x+2\pi/K$, we can introduce a good quantum number “crystal momentum” $q$. In the regime $r>r_{0}$, $\psi_{\text{c}}=0$, and for the bound states whose energy $E<\hbar^{2}q^{2}/(4m)$, $\psi^{q}_{\text{o}}(x,r)$ can always be expanded as $\psi^{q}_{\text{o}}(x,r)=e^{iqx}\sum\limits_{n}U^{q}_{n}e^{inKx}\frac{e^{-r\sqrt{(q+nK)^{2}-4mE/\hbar^{2}}}}{r}$ (9) Eq. (9) can be viewed as the Bloch wave function for molecules. And for the low energy scattering state whose energy is greater than but close to $\hbar^{2}q^{2}/(4m)$, we have $\displaystyle\psi^{q}_{\text{o}}(x,r)$ $\displaystyle=e^{iqx}\left(U_{0}\frac{\sin(kr-\delta)}{r\sin\delta}\right.$ $\displaystyle\left.+\sum\limits_{n\neq 0}U^{q}_{n}e^{inKx}\frac{e^{-r\sqrt{(q+nK)^{2}-4mE/\hbar^{2}}}}{r}\right)$ (10) where $k=\sqrt{mE/\hbar^{2}-q^{2}/4}$, and $\delta$ is a function of $k$. Figure 3: $rw(x,r)$ (where $w(x,r)$ is the “wannier wave function”) for the first four bound states. $a=2\pi/K$ is the “lattice spacing”. The parameters for this plot are $\epsilon_{\text{c}}=0.05E_{\text{R}}$, $Ka_{\text{bg}}=-0.01$, $Kr_{0}=10^{-3}$ and $Kr=0.1$. Results 1– Bound States: In contrast to the uniform case where there is only one bound state when $\Omega>\Omega_{0}$, in this case we find a sequence of bound states as $\Omega$ increases, as shown in Fig. 2. This is because the periodic structure of coupling $\Omega({\bf R})$ leads to a “ band structure ” for the molecules, and as the coupling strength increases, the molecules with zero crystal momentum but in different bands touch the scattering threshold one after the other. We can introduce the “wannier” wave function as $w(x-x_{0},r)=\int_{-K/2}^{K/2}e^{iqx_{0}}\psi^{q}_{\text{o}}(x,r)dq$ (11) As shown in Fig. 3, the “wannier” function for the bound states that appear at larger $\Omega$ have more oscillation, which means that they come from higher bands. This can also be illustrated from the symmetry of $U_{n}$ in the Bloch function of Eq. (9), as summarized in the Table 1 for the first four bound states. The first two bound state has even parity while the other two have odd parity. Figure 4: The effective scattering length defined as Eq. (12) $a_{\text{eff}}/|a_{\text{bg}}|$ as a function of $\Omega/\Omega_{0}$. (b) and (c) are enlarged plot around $\Omega/\Omega_{0}=2.64$ (b), $9.20$ (c). The arrows indicate the positions at which we plot the local scattering length $a_{\text{loc}}$ in Fig. 5(a-d). | $U_{-2}$ | $U_{-1}$ | $U_{0}$ | $U_{1}$ | $U_{2}$ ---|---|---|---|---|--- 1st | 0 | $+$ | 0 | $+$ | 0 2nd | $+$ | 0 | $+$ | 0 | $+$ 3rd | $+$ | 0 | 0 | 0 | $-$ 4th | 0 | $+$ | 0 | $-$ | 0 Table 1: Symmetry of Bloch wave function for the first four bound states Results 2 – Effective Scattering Length: For the scattering state wave function, at large ${\bf r}$ only the first term in Eq. (10) will not exponentially decay, and the asymptotic behavior of the scattering wave function is still the same as that in the uniform case. Hence we can introduce an effective scattering length as $a_{\text{eff}}=\lim_{k\rightarrow 0}\frac{\tan\delta(k)}{k}.$ (12) Note that though the interaction is spatially dependent, the effective scattering length defined as Eq. (12) is a spatial independent one. Among the first four bound states, $a_{\text{eff}}$ only diverges when the second bound state appears at threshold, as one can see by comparing Fig. 4(a) with Fig. 2. This is because the divergence of $a_{\text{eff}}$ implies the first term in Eq. (10) goes like $1/r$, which should be smoothly connected to a zero-energy bound state with non-zero $U_{0}$. Therefore, for the other three bound states whose $U_{0}=0$, their coupling to the low-energy scattering states vanish and will not cause divergency of $a_{\text{eff}}$. In Fig. 4(c) we show that $a_{\text{eff}}$ diverges when the sixth bound state (whose $U_{0}\neq 0$) appears at scattering threshold, but the width of resonance becomes narrower compared to Fig. 4(b) because this bound state comes from higher band and its coupling to low-energy scattering state ( i.e. the absolute value of $U_{0}$) is smaller. Figure 5: The local scattering length $a_{\text{loc}}$ as a function of position $x/a$ for $\Omega/\Omega_{0}=0.71,2.55,2.64$ and $2.7$ (a-d). The solid blue line is calculated results, the black dashed line is the fitting formula Eq(17) or (18), and the green dash-dotted line in (a) is from simple replacement formula Eq. (15). Results 3 – Local Scattering Length: At short distance the wave function Eq. (10) satisfies the Bethe-Peierls contact condition and display $1/r-1/a_{\text{loc}}(x)$ behavior, hence we can introduce a local scattering length as $a_{\text{loc}}(x)=-\lim\limits_{r\rightarrow r_{0}}\frac{r\psi_{\text{o}}(x,r)}{\partial_{r}(r\psi_{\text{o}}(x,r))}$ (13) Unlike in the uniform case, $a_{\text{eff}}$ and $a_{\text{loc}}$ are different. Similar situation has also been encountered for scattering in confined geometry CIR , lattices Cui and mixed dimension tan . What is unique here is that $a_{\text{loc}}$ is spatially dependent. Naively, one may think that $a_{\text{loc}}(x)$ can be obtained by replacing $\Omega$ in Eq. (4) by local $\Omega(x)$, i.e. $\displaystyle a_{\text{loc}}(x)$ $\displaystyle=a_{\text{bg}}\left(1-\frac{\beta\Omega^{2}\cos^{2}(Kx)}{\epsilon_{\text{c}}+\beta\Omega^{2}\cos^{2}(Kx)}\right)$ (14) $\displaystyle\approx a_{\text{bg}}\left[1-\beta\Omega^{2}\cos^{2}(Kx)/\epsilon_{\text{c}}\right]$ (15) where the second line is valid for small $\Omega$. This formula in fact corresponds to an oversimplified approximation in our model that the kinetic energy term of the center-of-mass motion ($-\hbar^{2}\nabla^{2}_{\bf{R}}/(4m)$) is completely ignored in Eq. (1). In fact, what we really obtained from the wave function Eq. (10) is $\displaystyle a_{\text{loc}}(x)$ $\displaystyle=\frac{1-\sum_{m\neq 0}U_{m}\cos(mKx)/U_{0}}{a^{-1}_{\text{eff}}-\sum_{m\neq 0}U_{m}|m|K\cos(mKx)/(2U_{0})}$ $\displaystyle\approx\frac{1-2U_{2}\cos(2Kx)/U_{0}}{a^{-1}_{\text{eff}}-2U_{2}K\cos(2Kx)/U_{0}}$ (16) The second line is also valid when $\Omega$ is not too large, so the coefficient $U_{m>2}$ is small enough that can be ignored. Away from a resonance, $Ka_{\text{eff}}\ll 1$, Eq. (16) can be well approximated as $a_{\text{loc}}(x)=a_{\text{eff}}\left[1-\frac{2U_{2}}{U_{0}}\cos(2Kx)\right]$ (17) In fact, we show in Fig. 5(a), (b) and (d) that the formula Eq. (17) (dashed black line) is a very good approximation to the actual results (solid blue line). In Fig. 5(a) we show the simple replacement formula Eq. (14) already significantly deviates from the actual results in weak coupling regime. From Fig. 5(b) and (d) one can also see that the mean value of $a_{\text{loc}}(x)$ changes sign as $a_{\text{eff}}$ changes sign. At resonance, $a^{-1}_{\text{eff}}\rightarrow 0$, Eq. (16) can be approximated as $a_{\text{loc}}(x)=\frac{1}{K}\left[1-\frac{U_{0}}{2U_{2}\cos(2Kx)}\right]$ (18) We show in Fig. 5(c) that Eq. (18) is also a very good approximation to actual $a_{\text{loc}}$ at resonance. Hence, we show that $a_{\text{loc}}$ behaves very differently in the regime nearby or away from a scattering resonance. Implications to Many-body Physics: In summary, we have revealed a number of novel features in the two-body problem with a spatially modulated interaction potential, which have strong implications for many-body physics and provide new insights for developing new tools for quantum control in cold atom systems. First, when $a_{\text{eff}}$ diverges, the system enters a strongly interacting regime and is expected to exhibit universal behavior, which can even be manifested in the high temperature regime high-T . For a two-component Fermi gas, it provides a new route toward BEC-BCS crossover physics, and “high-temperature” superfluid may exist in this regime. The periodic structure will add new ingredient to the crossover physics. Secondly, for the low-energy states whose energy $|E|\ll E_{\text{R}}$, the energy dependence of scattering length can be ignored and the many-body system can be effectively described by a pseudo-potential model: $\hat{H}=-\sum\limits_{i}\frac{\hbar^{2}\nabla^{2}_{\bf{r}_{i}}}{2m}+\sum\limits_{ij}\frac{4\pi\hbar^{2}a_{\text{loc}}({\bf R}_{ij})}{m}\delta^{3}(\mathbf{r_{ij}})\frac{\partial}{\partial r_{ij}}r_{ij},$ (19) where ${\bf R}_{ij}=({\bf r}_{i}+{\bf r}_{j})/2$ and $r_{ij}={\bf r}_{i}-{\bf r}_{j}$. It is very important that $a_{\text{loc}}({\bf R})$ in the pseudo- potential of Eq. (19) is given by Eq. (16) from the two-body calculation, so that a two-body problem of the Hamiltonian Eq. (19) can produce correct low- energy eigen-wave function and the effective scattering length as from model potential. For bosons, with a mean-field approximation, Eq. (19) implies that the interaction energy should take the form $E_{\text{mf}}=\frac{4\pi\hbar^{2}}{m}\int a_{\text{loc}}(x)n^{2}(x)dx$ (20) which leads to a modulation of condensate density $n(x)$ and self-trapping nearby the minimum of $a_{\text{loc}}(x)$. It is very likely a strong enough modulation of condensate density will eventually result in the loss of superfluidity and the system enters an insulating phase. If so, it provides a completely different mechanism for superfluid to insulator transition where the transition is not driven by suppression of kinetic energy as in conventional OL. Final Comments: In this work we choose a coupled two square-well model whose advantage is that the physics can be demonstrated in a simple and transparent way. However, some more sophisticated effects in real system, such as the inelastic loss, are ignored. We have also implemented more systematic scattering theory which includes these effects and found that the physics discussed here will remain qualitatively unchanged. These results will be published elsewhere Qiran . Moreover, the formalism used in this work can be easily generalized to other realizations of spatial modulation of interactions. For instance, in a magnetic FR, one can consider the presence of a magnetic field gradient so that the closed channel molecular energy varies spatially. This effect is particularly important for a narrow resonance. One can also optically couple the closed channel molecule to another molecular state via a bound-bound transition, which leads to a periodic variation of molecule energy Rempe . Similar effects as discussed in Results 1-3 also present in these cases Qiran . Acknowledgements. We thank Xiaoling Cui, Zeng-Qiang Yu, Peng Zhang and Zhenhua Yu for helpful discussions. This work is supported by Tsinghua University Initiative Scientific Research Program, NSFC under Grant No. 11004118 and NKBRSFC under Grant No. 2011CB921500. ## References * (1) C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010). * (2) P. O. Fedichev, Yu. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996); J. L. Bohn and P. S. Julienne, Phys. Rev. A 56, 1486 (1997). * (3) M. Olshanii, Phys. Rev. Lett. 81, 938 (1998); T. Bergeman, M. G. Moore, and M. Olshanii, Phys. Rev. Lett. 91, 163201 (2003). * (4) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008). * (5) F. K. Fatemi, K. M. Jones, and P. D. Lett, Phys. Rev. Lett. 85, 4462 4465 (2000); M. Theis, et al. Phys. Rev. Lett. 93, 123001 (2004) and K. Enomoto, K. Kasa, M. Kitagawa, and Y. Takahashi, Phys. Rev. Lett. 101, 203201 (2008). * (6) R Yamazaki, S. Taie, S. Sugawa, and Y. Takahashi, Phys. Rev. Lett. 105, 050405 (2010). * (7) C. Chin, arXiv: 0506313. * (8) X. Cui, Y. Wang, and F. Zhou, Phys. Rev. Lett. 104, 153201 (2010) and H. P. Büchler, Phys. Rev. Lett. 104, 090402 (2010) * (9) Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401 (2008) * (10) T. L. Ho and E. J. Mueller, Phys. Rev. Lett. 92, 160404 (2004). * (11) R. Qi, P. Zhang and H. Zhai in preparation. * (12) D. M. Bauer, et al. Nature Physics 5, 339 (2009). Appendix: In this appendix, we present some details of solving the two-body Schrödinger equation. Using the discrete translation symmetry, we expand $\displaystyle a^{q}(x)=e^{iqx}\sum_{n}e^{inKx}a^{q}_{n}$ (21) $\displaystyle b^{q}(x)=e^{iqx}\sum_{n}e^{inKx}b^{q}_{n},$ (22) $a_{n}$ and $b_{n}$ satisfy coupled matrix equation $\displaystyle\left(\frac{\hbar^{2}(q+nK)^{2}}{4m}-V_{\text{o}}\right)a^{q}_{n}+\frac{\Omega}{2}\left(b^{q}_{n-1}+b^{q}_{n+1}\right)=\epsilon^{q}a^{q}_{n}$ (23) $\displaystyle\left(\frac{\hbar^{2}(q+nK)^{2}}{4m}-V_{\text{c}}\right)b^{q}_{n}+\frac{\Omega}{2}\left(a^{q}_{n-1}+a^{q}_{n+1}\right)=\epsilon^{q}b^{q}_{n}$ (24) This matrix has a set of eigen-values $\epsilon^{q}_{l}$ and their eigen- vectors $\\{a^{q}_{l,n},b^{q}_{l,n}\\}$. Hence there are a set of wave functions sharing the same energy $E$, which ar $\displaystyle\psi^{q}_{\text{o},l}=e^{iqx}\frac{\sin(k^{q}_{l}r)}{r}\sum\limits_{n}e^{inKx}a^{q}_{l,n}$ (25) $\displaystyle\psi^{q}_{\text{c},l}=e^{iqx}\frac{\sin(k^{q}_{l}r)}{r}\sum\limits_{n}e^{inKx}b^{q}_{l,n}$ (26) where $k^{q}_{l}=\sqrt{m(E-\epsilon^{q}_{l})}/\hbar$ is a function of $E$. In general, the eigen-states take the form $\psi_{q}=\sum\limits_{l}A^{q}_{l}\left(\begin{array}[]{c}\psi^{q}_{\text{o},l}\\\ \psi^{q}_{\text{c},l}\end{array}\right)=e^{iqx}\sum\limits_{n}e^{inKx}\left(\begin{array}[]{c}\varphi^{q}_{\text{o},n}\\\ \varphi^{q}_{\text{c},n}\end{array}\right)$ (27) where $\displaystyle\varphi^{q}_{\text{o},n}({\bf r})=\sum\limits_{l}A^{q}_{l}\frac{\sin(k^{q}_{l}r)}{r}a^{q}_{l,n};\ \ \varphi^{q}_{\text{c},n}({\bf r})=\sum\limits_{l}A^{q}_{l}\frac{\sin(k^{q}_{l}r)}{r}b^{q}_{l,n}$ For bound states, to match the boundary condition with Eq. (9) at $r_{0}$ in both open and closed channels, we obtain a matrix equation $M^{q}_{kl}(E)A^{q}_{l}=0$ where $\displaystyle M^{q}_{2n+1,l}=\sin(k^{q}_{l}r_{0})b^{q}_{l,n}$ $\displaystyle M^{q}_{2n,l}=(k^{q}_{l}\cos(k^{q}_{l}r_{0})+\sqrt{(q+nK)^{2}-\frac{4mE}{\hbar^{2}}}\sin(k^{q}_{l}r_{0}))a^{q}_{l,n}$ Therefore for a given $q$ the eigen-energy $E$ is determined by $\text{Det}(M)=0$ and $U^{q}_{n}=e^{r_{0}\sqrt{\frac{\hbar^{2}(q+nK)^{2}}{4m}-E}}\sum\limits_{l}A_{l}a^{l}_{n}\sin(k^{l}r_{0})$ (28) For the scattering states, $M^{q}_{2n+1,l}$ and $M^{q}_{2n,l}$ ($n\neq 0$) are the same as bound state, while $\displaystyle M^{q}_{0,l}=\sin(k^{q}_{l}r_{0})\cos(kr_{0}-\delta)k-\cos(k^{q}_{l}r_{0})\sin(kr_{0}-\delta)a^{l}_{0}$ where $k=\sqrt{mE/\hbar^{2}-q^{2}/4}$. In this case $\text{Det}(M)=0$ gives rise to the relation between phase shift $\delta$ and energy $E$. $U^{q}_{n}$ ($n\neq 0$) is also the same as Eq. (28), while for $n=0$, $U^{q}_{0}=\frac{\sin\delta}{\sin(kr_{0}-\delta)}\sum\limits_{l}A_{l}\sin(k^{q}_{l}r_{0})a^{l}_{0}$ (29)
arxiv-papers
2011-01-24T08:36:31
2024-09-04T02:49:16.595851
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ran Qi and Hui Zhai", "submitter": "Ran Qi", "url": "https://arxiv.org/abs/1101.4464" }
1101.4556
# Distinct Fermi Surface Topology and Nodeless Superconducting Gap in (Tl0.58Rb0.42)Fe1.72Se2 Superconductor Daixiang Mou1, Shanyu Liu1, Xiaowen Jia1, Junfeng He1, Yingying Peng1, Lin Zhao1, Li Yu1, Guodong Liu1, Shaolong He1, Xiaoli Dong1, Jun Zhang1, Hangdong Wang2, Chiheng Dong2, Minghu Fang2, Xiaoyang Wang3, Qinjun Peng3, Zhimin Wang3, Shenjin Zhang3, Feng Yang3, Zuyan Xu3, Chuangtian Chen3 and X. J. Zhou1,∗ 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2Department of Physics, Zhejiang University, Hangzhou 310027, China 3Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China (January 24, 2011) ###### Abstract High resolution angle-resolved photoemission measurements have been carried out to study the electronic structure and superconducting gap of the (Tl0.58Rb0.42)Fe1.72Se2 superconductor with a Tc=32 K. The Fermi surface topology consists of two electron-like Fermi surface sheets around $\Gamma$ point which is distinct from that in all other iron-based compounds reported so far. The Fermi surface around the M point shows a nearly isotropic superconducting gap of $\sim$12 meV. The large Fermi surface near the $\Gamma$ point also shows a nearly isotropic superconducting gap of $\sim$15 meV while no superconducting gap opening is clearly observed for the inner tiny Fermi surface. Our observed new Fermi surface topology and its associated superconducting gap will provide key insights and constraints in understanding superconductivity mechanism in the iron-based superconductors. ###### pacs: 74.70.-b, 74.25.Jb, 79.60.-i, 71.20.-b The discovery of the Fe-based superconductorsKamihara ; ZARenSm ; RotterSC ; MKWu11 ; CQJin111 has attracted much attention because it represents the second class of high temperature superconductors in addition to the copper- oxide (cuprate) superconductorsBednorz . It is important to explore whether the high-Tc superconductivity mechanism in this new Fe-based system is conventional, parallel to that in cuprates, or along a totally new routeNPReview . Different from the cuprates where the low-energy electronic structure is dominated by Cu 3d${}_{x^{2}-y^{2}}$ orbital, the electronic structure of the Fe-based compounds involves all five Fe 3d-orbitals forming multiple Fermi surface sheets: hole-like Fermi surface sheets around $\Gamma$(0,0) and electron-like ones around M($\pi$,$\pi$)DJSingh1111 ; Kuroki . It has been proposed that the interband scattering between the hole-like bands near $\Gamma$ and electron-like bands near M gives rise to electron pairing and superconductivityKuroki ; FeSCMagnetic . An alternative picture is also proposed based on the interaction of local Fe magnetic momentJ1J2Picture . The latest discovery of superconductivity with a Tc above 30 K in a new AxFe2-ySe2 (A=K, Tl, Cs, Rb and etc.) systemJGGuo ; Switzerland ; MHFang ; GFChen is surprising that provides new perspectives in understanding Fe-based compounds. First, it may involve Fe vacancies in the FeSe layerMHFang ; GFChen ; ZWang . This is against the general belief that a perfect Fe-sublattice is essential for superconductivity in the Fe-based compounds, similar to that perfect CuO2 plane is considered essential for cuprate superconductors. Second, the superconductivity of AxFe2-ySe2 is realized in a close proximity to an antiferromagnetic semiconducting (insulating) phaseMHFang ; GFChen ; QMSi . This is in strong contrast to other Fe-based compounds where the parent compounds are spin-ordered metalsDongSDW ; RotterParent ; PCDai ; BFSNeutron3 . Third, the intercalation of A=(K,Tl,Cs,Rb and etc.) in AxFe2-ySe2 is expected to introduce large number of electrons into the system; this usually would lead to suppression or disappearance of superconductivity like in heavily electron-doped Ba(Fe,Co)2As2 systemCo122 . The existence of superconductivity in AxFe2-ySe2 at such a high Tc (over 30 K) with so high electron doping is unexpected. Most interestingly, band structure calculationsLJZhang ; IRShein ; XWYan and electronic structure measurementsYZhang ; TQian all suggest that high electron doping in AxFe2-ySe2 may lead to the disappearance of the hole-like Fermi surface sheets around $\Gamma$. This would render it impossible for the electron scattering between the hole-like bands near $\Gamma$ and electron-like bands near M that is considered to be important for the electron pairing in Fe-based superconductors by some theoriestsKuroki ; FeSCMagnetic . Therefore, investigations of the AxFe2-ySe2 system would be important in searching for common ingredients underlying the physical properties, especially the superconductivity of the Fe-based superconductors. Figure 1: Fermi surface of (Tl0.58Rb0.42)Fe1.72Se2 superconductor (Tc=32 K). (a). Spectral weight distribution integrated within [-20meV,10meV] energy window near the Fermi level as a function of kx and ky measured using h$\nu$=21.2 eV light source. Two Fermi surface sheets are observed around $\Gamma$ point which are marked as $\alpha$ for the inner small sheet and $\beta$ for the outer large one. Near the M($\pi$,$\pi$) point, one Fermi surface sheet is clearly observed which is marked as $\gamma$. (b) Three- dimensional image of Fig. 1a. (c). Fermi surface mapping measured using h$\nu$=40.8 eV light source Although the signal is relatively weak, one can see traces of two Fermi surface sheets around $\Gamma$ and one around M. In this paper, we report observation of a distinct Fermi surface topology and nearly isotropic nodeless superconducting gap in (Tl0.58Rb0.42)Fe1.72Se2 superconductor (Tc=32 K) from high resolution angle-resolved photoemission (ARPES) measurements. We have observed an electron-like Fermi surface sheet near M($\pi$,$\pi$) and two electron-like Fermi surface sheets near $\Gamma$(0,0). This Fermi surface topology is distinct from the hole-like Fermi surface sheets near the $\Gamma$ point found in other Fe-based compoundsDJSingh1111 ; Kuroki or disappearance of hole-like Fermi surface sheets near $\Gamma$ in AxFe2-ySe2 compoundsLJZhang ; IRShein ; XWYan ; YZhang ; TQian . We observe nearly isotropic superconducting gap around the Fermi surface sheets near $\Gamma$ ($\sim$15 meV) and M ($\sim$12 meV); no gap node is observed in both Fermi surface sheets. These rich information on this new Fe-based superconductor will provide key insights on the superconductivity mechanism in the Fe-based superconductors. High resolution angle-resolved photoemission measurements were carried out on our lab system equipped with a Scienta R4000 electron energy analyzerGDLiu . We use Helium discharge lamp as the light source which can provide photon energies of h$\upsilon$= 21.218 eV (Helium I) and 40.8 eV (Helium II). The energy resolution was set at 10 meV for the Fermi surface mapping (Fig. 1a) and band structure measurements (Fig. 2) and at 4 meV for the superconducting gap measurements (Figs. 3 and 4). The angular resolution is $\sim$0.3 degree. The Fermi level is referenced by measuring on a clean polycrystalline gold that is electrically connected to the sample. The (Tl,Rb)Fe2-ySe2 crystals were grown by the Bridgeman methodMHFang . Their composition determined by using an Energy Dispersive X-ray Spectrometer (EDXS) measurement is (Tl0.58Rb0.42)Fe1.72Se2. The crystals have a sharp superconducting transition at Tc(onset)=32 K with a transition width of $\sim$1 K. The crystal was cleaved in situ and measured in vacuum with a base pressure better than 5$\times$10-11 Torr. Figure 2: Band structure and photoemission spectra of (Tl0.58Rb0.42)Fe1.72Se2 measured along two high symmetry cuts. (a). Band structure along the Cut 1 crossing the $\Gamma$ point; the location of the cut is shown on top of Fig. 2a. The $\alpha$ band and two Fermi crossings of the $\beta$ band ($\beta_{L}$ and $\beta_{R}$) are marked. (b). Corresponding EDC second derivative image of Fig. 2a. (c). Band structure along the Cut 2 crossing M point; the location of the cut is shown on top of Fig. 2c. The two Fermi crossings of the $\gamma$ band ($\gamma_{L}$ and $\gamma_{R}$) are marked. (d). Corresponding EDC second derivative image of Fig. 2c. (e). EDCs corresponding to Fig. 2a for the Cut 1. (f). EDCs corresponding to Fig. 2c for the Cut 2. Fig. 1 shows Fermi surface mapping of the (Tl0.58Rb0.42)Fe1.72Se2 superconductor covering multiple Brillouin zones. The band structure along two typical high symmetry cuts are shown in Fig. 2. An electron-like Fermi surface is clearly observed around M($\pi$,$\pi$) (Fig. 1a, Figs. 2c and 2d). This Fermi surface (denoted as $\gamma$ hereafter) is nearly circular with a Fermi momentum (kF) of 0.35 in a unit of $\pi$/a (lattice constant a=3.896 $\AA$). The Fermi surface near the $\Gamma$ point consists of two sheets. The inner tiny pocket (denoted as $\alpha$) is electron-like with a band bottom barely touching the Fermi level ( Figs. 2a and 2b for the Cut 1). The outer larger Fermi surface sheet around $\Gamma$ (denoted as $\beta$) (Fig. 1a) is electron-like (Figs. 2a and 2b) with a Fermi momentum of 0.35 $\pi$/a. The observation of two electron-like Fermi surface sheets, $\alpha$ and $\beta$, around $\Gamma$ in (Tl0.58Rb0.42)Fe1.72Se2 is distinct from that observed in other Fe-based compounds where hole-like pockets are expected around the $\Gamma$ pointDJSingh1111 ; Kuroki . It is also different from the band structure calculationsLJZhang ; IRShein ; XWYan ; YZhang ; TQian and previous ARPES measurementsYZhang ; TQian on AxFe2-ySe2 that suggest disappearance of hole-like Fermi surface sheets near $\Gamma$ because of the lifted chemical potential due to a large amount of electron doping. Figure 3: Temperature dependence of energy bands and superconducting gap near $\Gamma$ and M points. (a). Photoemission images along the Cut B (bottom-right inset). (b). Photoemission spectra at the Fermi crossing of $\gamma$ Fermi surface and their corresponding symmetrized spectra (c) measured at different temperatures. (d). Temperature dependence of the superconducting gap. The dashed line is a BCS gap form. (e). Photoemission images along the Cut A (bottom-right inset) at different temperatures. The original EDCs at the $\Gamma$ point and at the Fermi crossing of $\beta$ Fermi surface measured at different temperatures are shown in (f) and their corresponding symmetrized EDCs are shown in (g). One immediate question is on the origin of the electron-like $\beta$ band around $\Gamma$. The first possibility is whether it could be a surface state. While surface state on some Fe-based compounds like the “1111” system was observed beforeHYLiu , it has not been observed in the “11”-type Fe(Se,Te) systemFeSTARPES . The second possibility is whether the $\beta$ band can be caused by the folding of the electron-like $\gamma$ Fermi surface near M. It is noted that the Fermi surface size, the band dispersion, and the band width of the $\beta$ band at $\Gamma$ is similar to that of the $\gamma$ band near M. A band folding picture would give a reasonable account for such a similarity if there exists a ($\pi$,$\pi$) modulation in the system that can be either structural or magnetic. An obvious issue with this scenario is that, in this case, one should also expect the folding of the $\alpha$ band near $\Gamma$ onto the M point; but such a folding is not observed at the M point (Fig. 1a and Fig. 2c). The third possibility is whether the measured $\beta$ sheet is a Fermi surface at a special kz cut. Although the Fermi surface at $\Gamma$ is absent in TlFe2Se2 from the band structure calculationsLJZhang , there is a 3-dimensional Fermi pocket that is present near the zone center at kz=$\pi$$/$c when x is close to 1 in KxFe2Se2IRShein and CsxFe2Se2XWYan . We note that the electron doping in (Tl0.58Rb0.42)Fe1.72Se2 is lower than that of (K,Cs)Fe2Se2. Also we observed similar $\beta$ Fermi surface at different photon energies (Fig. 1a and Fig. 1c) which corresponds to different kz. The final resolution of this possibility needs further detailed photon energy dependent measurements. The clear identification of various Fermi surface sheets makes it possible to investigate the superconducting gap in this new superconductor. We start first by examining the superconducting gap near the M point. Fig. 3a shows the photoemission images along the Cut B near M (its location shown in the bottom- right inset of Fig. 3) at different temperatures. The corresponding photoemission spectra (energy distribution curves, EDCs) on the Fermi momentum at different temperatures are shown in Fig. 3b. To visually inspect possible gap opening and remove the effect of Fermi distribution function near the Fermi level, we have symmetrized these original EDCs to get spectra in Fig. 3c, following the procedure that is commonly used in high temperature cuprate superconductorsMNorman . For the $\gamma$ pocket near M, there is a clear gap opening at low temperature (15 K), as indicated by an obvious dip at the Fermi energy in the symmetrized EDCs (Fig. 3c). With increasing temperature, the dip at EF is gradually filled up and is almost fully filled above Tc=32 K. The gap size at different temperatures is extracted from the peak position of the symmetrized EDCs or fitted by the phenomenological formulaMNorman (Fig. 3d); it is $\sim$11 meV at 15 K. The temperature dependence of the gap size roughly follows the BCS-type form (Fig. 3d). Similar temperature dependent measurements of the superconducting gap were also carried out along the $\Gamma$-M cut near $\Gamma$ (Figs. 3e-g). The Fermi crossing on the $\beta$ Fermi surface also displays a clear superconducting gap in the superconducting state which is closed above Tc (lower curves in Figs. 3f and 3g). For the peculiar tiny $\alpha$ pocket near $\Gamma$, we do not find signature of clear superconducting gap opening below Tc (upper curves in Figs. 3f and 3g). Now we come to the momentum-dependent measurements of the superconducting gap. For this purpose we took high resolution Fermi surface mapping (energy resolution of 4 meV) of the $\gamma$ pocket at M (Fig. 4a) and the $\beta$ pocket at $\Gamma$ (Fig. 4b). Fig. 4c shows photoemission spectra around the $\gamma$ Fermi surface measured in the superconducting state (T= 15 K); the corresponding symmetrized photoemission spectra are shown in Fig. 4d. The extracted superconducting gap (Fig. 4g) is nearly isotropic with a size of (12$\pm$2) meV. The superconducting gap around the $\beta$ Fermi surface near $\Gamma$ is also nearly isotropic with a size of (15$\pm$2) meV (Figs. 4e, 4f and 4g). Figure 4: Momentum dependent superconducting gap along the $\gamma$ and the $\beta$ Fermi surface sheets measured at T=15 K. Fermi surface mapping near M (a) and near $\Gamma$ (b) and the corresponding Fermi crossings marked by red circles. (c). EDCs along the $\gamma$ Fermi surface and their corresponding symmetrized EDCs (d). (e). EDCs along the $\beta$ Fermi surface and their corresponding symmetrized EDCs (f). (g). Momentum dependence of the superconducting gap along the $\gamma$ Fermi surface sheet (red circles) and along the $\beta$ Fermi surface sheet (blue circles). The observation of a distinct Fermi surface topology in (Tl0.58Rb0.42)Fe1.72Se2 has important implications to the understanding of superconductivity in Fe-based superconductors. The realization of high Tc in this new superconductor with a distinct Fermi surface topology is helpful to sort out key electronic structure ingredient that is responsible for superconductivity. With the electron-like $\beta$ Fermi surface present in (Tl0.58Rb0.42)Fe1.72Se2, the possibility of electron scattering between the $\Gamma$ Fermi surface sheet(s) and the M Fermi surface sheet(s), proposed by some theories to account for superconductivity in the Fe-based superconductorKuroki ; FeSCMagnetic , cannot be ruled out. However, the electron scattering between two electron-like bands may have different effect on the electron pairing from that between an electron-like band and a hole- like band. The nearly isotropic superconducting gap on the $\beta$ and $\gamma$ Fermi surface sheets, together with the absence of gap nodes, appears to favor s-wave superconducting gap symmetry in (Tl0.58Rb0.42)Fe1.72Se2. This is similar to that in (Ba0.6K0.4)Fe2As2122Gap and NdFeAsO0.9F0.1Kaminski1111 . The gap size of 12 (for $\gamma$) and 15 meV (for $\beta$) gives a ratio of 2$\Delta$/kTc=9 and 11, respectively, which is significantly larger than the traditional BCS weak-coupling value of 3.52 for an s-wave gap. This indicates that this new superconductor is at least in the strong coupling regime in terms of the BCS picture. These will put strong constraints on various proposed gap symmetries and the underlying pairing mechanisms for the iron- based superconductors. In summary, we have identified a distinct Fermi surface topology in the new (Tl0.58Rb0.42)Fe1.72Se2 superconductor that is different from all other Fe- based superconductors reported so far. Near the $\Gamma$ point, two electron- like Fermi surface sheets are observed that are different from the band structure calculations and previous ARPES measurement results. We observed nearly isotropic superconducting gap around the Fermi surface sheets near $\Gamma$ and M without gap nodes. These rich information will shed more light on the nature of superconductivity in the Fe-based superconductors. XJZ and MHF thank the funding support from NSFC (Grant No. 10734120 and 10974175) and the MOST of China (973 program No: 2011CB921703 and 2011CBA00103). ∗Corresponding author: XJZhou@aphy.iphy.ac.cn ## References * (1) Y. Kamihara et al., J. Am. Chem. Soc. 130, 3296 (2008). * (2) Z. A. Ren et al., Chin. Phys. Lett. 25, 2215 (2008). * (3) M. Rotter et al., Phys. Rev. Lett. 101, 107006(2008). * (4) F. 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arxiv-papers
2011-01-24T14:56:57
2024-09-04T02:49:16.602615
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Daixiang Mou, Shanyu Liu, Xiaowen Jia, Junfeng He, Yingying Peng, Lin\n Zhao, Li Yu, Guodong Liu, Shaolong He, Xiaoli Dong, Jun Zhang, Hangdong Wang,\n Chiheng Dong, Minghu Fang, Xiaoyang Wang, Qinjun Peng, Zhimin Wang, Shenjin\n Zhang, Feng Yang, Zuyan Xu, Chuangtian Chen and X. J. Zhou", "submitter": "Xingjiang Zhou", "url": "https://arxiv.org/abs/1101.4556" }
1101.4580
# $B$-physics with dynamical domain-wall light quarks and relativistic $b$-quarks Ruth S. Van de Water and Brookhaven National Laboratory, Department of Physics, Upton, NY 11973, USA E-mail ###### Abstract: We report on our progress in calculating the $B$-meson decay constants and $B^{0}$-$\bar{B}^{0}$ mixing parameters using domain-wall light quarks and relativistic $b$-quarks. We present our computational method and show some preliminary results obtained on the coarser ($a\approx 0.11$fm) $24^{3}$ lattices. This work is presented on behalf of the RBC and UKQCD collaborations. ## 1 Introduction The study of $B$-meson physics on the lattice is of special phenomenological interest because it allows one to obtain constraints on the CKM unitarity triangle. In the standard global unitarity triangle fit, the apex of the CKM unitarity triangle is constrained using lattice input for neutral $B$-meson mixing. Experimentally, $B_{q}$–$\bar{B}_{q}$ mixing is well measured in terms of mass differences (oscillation frequencies) $\Delta m_{q}$; in the standard model it is parameterized by [1] $\displaystyle\Delta m_{q}=\frac{G_{F}^{2}m^{2}_{W}}{6\pi^{2}}\eta_{B}S_{0}m_{B_{q}}{f_{B_{q}}^{2}B_{B_{q}}}\lvert V_{tq}^{*}V_{tb}\rvert^{2},$ (1) where the index $q$ denotes either a $d$\- or a $s$-quark, $m_{B_{q}}$ is the mass of the $B_{q}$-meson, and $V_{tq}^{*}$ and $V_{tb}$ are CKM matrix elements. The Inami-Lim function, $S_{0}$ [2], and the QCD coefficient, $\eta_{B}$ [1], can be computed perturbatively, whereas the non-perturbative input is given by $f_{q}^{2}B_{B_{q}}$: the leptonic decay constant $f_{B_{q}}$ and the $B$-meson bag parameter $B_{B_{q}}$. Computing the ratio $\Delta m_{s}$/$\Delta m_{d}$ is particularly advantageous because statistical and systematic uncertainties largely cancel and, moreover, the ratio of CKM matrix elements becomes accessible $\displaystyle\frac{\Delta m_{s}}{\Delta m_{d}}=\frac{m_{B_{s}}}{m_{B_{d}}}\,{\xi^{2}}\,\frac{\lvert V_{ts}\rvert^{2}}{\lvert V_{td}\rvert^{2}}.$ (2) The non-perturbative input is now solely contained in the $SU(3)$-breaking ratio $\displaystyle\xi$ $\displaystyle=\frac{f_{B_{s}}\sqrt{B_{B_{s}}}}{f_{B_{d}}\sqrt{B_{B_{d}}}}.$ (3) An alternative way of constraining the CKM triangle has been proposed by Lunghi and Soni [3] that uses the CKM matrix elements $V_{ub}$ or $V_{cb}$, thereby avoiding the tension between inclusive and exclusive determinations of both $V_{ub}$ (3$\sigma$) and $V_{cb}$ (2$\sigma$). This alternative method, however, requires precise knowledge of the decay constant $f_{B}$ as well as $BR(B\to\tau\nu)$ and $\Delta m_{s}$. Moreover, Lunghi and Soni point out [4, 5] that already the current data may show signs of physics beyond the standard model manifesting themselves as deviations of experimental values for $\sin(2\beta)$ (3.3$\sigma$) and $BR(B\to\tau l\nu)$ (2.8$\sigma$) from the Standard Model. They argue that the most likely sources for new physics are in $B_{q}$ mixing and $\sin(2\beta)$. Therefore, it is timely for the lattice community to determine these parameters precisely and prove by using statistically and systematically independent setups that all sources of errors are well under control. Currently, there are three determinations for $\xi$ using 2+1-flavor gauge field configurations: an exploratory study by the RBC-UKQCD collaboration [6] with non-competitive errors and two precise determinations by Fermilab-MILC [7, 8] and HPQCD [9], which however both rely on the same set of MILC gauge field configurations. The latter two collaborations also use the same configurations to obatin the decay constants $f_{B_{d}}$ and $f_{B_{s}}$ and find good agreement. Here we present the first results of our project to determine $B$-meson parameters using domain-wall light quarks and relativistic $b$-quarks. ## 2 Computational setup Our project is based on using the RBC-UKQCD 2+1-flavor domain wall lattices with the Iwasaki gauge action for the gluons. Our first results presented in these proceedings are obtained on the coarser $a\approx 0.11$fm lattices with a spatial volume of $24^{3}$ in lattice units [10], whereas the full project will also utilize the finer $a\approx 0.08$fm lattices ($32^{3}$) [11] (for details see Tab. 1). | | | | | approx. | # time ---|---|---|---|---|---|--- L | $a$(fm) | $m_{l}$ | $m_{s}$ | $m_{\pi}$(MeV) | # configs. | sources 24 | $\approx$ 0.11 | 0.005 | 0.040 | 331 | 1640 | 1 24 | $\approx$ 0.11 | 0.010 | 0.040 | 419 | 1420 | 1 24 | $\approx$ 0.11 | 0.020 | 0.040 | 558 | 350 | 8 32 | $\approx$ 0.08 | 0.004 | 0.030 | 307 | 600 | 1 32 | $\approx$ 0.08 | 0.006 | 0.030 | 366 | 900 | 1 32 | $\approx$ 0.08 | 0.008 | 0.030 | 418 | 550 | 1 Table 1: Overview of the gauge field ensembles to be used for this project. For the heavy quarks we use a relativistic formulation derived from the “Fermilab action” [12] in which we tune the three relevant parameters of the action non-perturbatively. As demonstraged by Christ, Li and Lin, this requires only one additional experimental input compared to the formulation of Fermilab [13]. The relativistic heavy quark (RHQ) action is given by $\displaystyle S=\sum_{n,n^{\prime}}\bar{\Psi}_{n}\left\\{\\!m_{0}+\gamma_{0}D_{0}-\\!\frac{aD_{0}^{2}}{2}+\zeta\left[\vec{\gamma}\cdot\vec{D}-\frac{a\left(\vec{D}\right)^{2}}{2}\right]\\!-a\sum_{\mu\nu}\frac{ic_{P}}{4}\sigma_{\mu\nu}F_{\mu\nu}\\!\right\\}_{\\!\\!n,n^{\prime}}\\!\\!\\!\\!\Psi_{n^{\prime}},$ (4) where the covariant derivative is denoted by $D$, $F_{\mu\nu}$ is the field strength tensor and we need to tune the three parameters $m_{0}a$, $c_{P}$ and $\zeta$. Exploratory studies on how to tune these parameters have been performed by Li and Peng [14, 15, 16]. The idea is to use experimental values for the spin averaged meson mass ($\overline{m}=(m_{B_{s}}+3m_{B_{s}^{*}})/4$) and the hyperfine splitting ($\Delta_{m}=m_{B^{*}_{s}}-m_{B_{s}}$) as inputs together with the constraint from the dispersion relation that the rest mass $m_{1}$ equal the kinetic mass $m_{2}$. These three quantities are computed for a set of seven trial parameters determined by making an initial guess for $m_{0}a$, $c_{P}$, and $\zeta$ and then varying it by a chosen uncertainty $\pm\sigma_{\\{m_{0}a,c_{P},\zeta\\}}$: $\displaystyle\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\ \zeta\\\ \end{array}\right],\left[\\!\\!\begin{array}[]{c}m_{0}a-\sigma_{m_{0}a}\\\ c_{P}\\\ \zeta\\\ \end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a+\sigma_{m_{0}a}\\\ c_{P}\\\ \zeta\\\ \end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}-\sigma_{c_{P}}\\\ \zeta\\\ \end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}+\sigma_{c_{P}}\\\ \zeta\\\ \end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\ \zeta-\sigma_{\zeta}\\\ \end{array}\\!\\!\right],\;\left[\\!\\!\begin{array}[]{c}m_{0}a\\\ c_{P}\\\ \zeta+\sigma_{\zeta}\\\ \end{array}\\!\\!\right]$ (26) (see Fig. 1). We iterate over the parameters $\\{m_{0}a,\,c_{P},\,\zeta\\}$ until we determine the values that reproduce the known experimental measurements for $\overline{m},\,\Delta_{m},\,m_{1}/m_{2}$: $\displaystyle\left[\begin{array}[]{c}m_{0}a\\\ c_{P}\\\ \zeta\end{array}\right]^{\text{RHQ}}=J^{-1}\times\left(\left[\begin{array}[]{c}\overline{m}\\\ \Delta_{m}\\\ \frac{m_{1}}{m_{2}}\end{array}\right]^{\text{PDG}}-A\right)$ (33) with $\displaystyle J=\left[\frac{Y_{3}-Y_{2}}{2\sigma_{m_{0}a}},\,\frac{Y_{5}-Y_{4}}{2\sigma_{c_{P}}},\,\frac{Y_{7}-Y_{6}}{2\sigma_{\zeta}}\right]\qquad\text{and}\quad A=Y_{1}-J\times\left[m_{0}a,\,c_{P},\,\zeta\right]^{t}.$ (34) In (34) we use the vectors $Y_{i}$ as shorthand notation for $[\overline{m},\,\Delta_{m},\,m_{1}/m_{2}]^{t}_{i}$ obtained for the input parameters as given in Eq. (26), labeled 1-7 from left to right. Eq. (33) assumes a linear dependence of the meson masses on the parameters of the action; therefore we must be in a linear regime to reliably extract $\\{m_{0}a,\,c_{P},\,\zeta\\}$. We stop the iteration once the tuned values lie within our variation range. Figure 1: Initial guess for the RHQ parameters and their uncertainties. In order to finally compute $B$-meson decay constants and mixing parameters as precisely as possible we repeated the original tuning presented at Lattice 2008 [14] with increased statistics and optimized smeared wavefunction parameters used for generating the heavy quark propagators. Moreover, performing the tuning on the same configurations we intend to use for computing weak matrix elements such as $f_{B}$ will allow for an improved error analysis in which the correlations among the three RHQ parameters can be fully taken into account. Our method for computing the $\Delta B=2$ four-quark operators requires the light quarks to be generated with a point source and sink, but for the heavy quarks we are free to explore different smearing choices. In addition to point sources and sinks, we tried Gaussian smeared sources/sinks and also varied the radius of the Gaussian smearing. As a first guess we chose the radius of the Gaussian source to be the rms radius of the $b\bar{b}$\- and $c\bar{c}$ states [17]. Later we extended the radius and found that $r_{\text{rms}}=0.634$fm gives the best signal as can be seen in Fig. 2. Figure 2: $B_{l}$-meson effective masses for different $b$-quark spatial wavefunctions; in each case the light quark has a point source. Using this setup we obtain the RHQ parameters on the three $24^{3}$ ensembles as given in Table 2. In contrast to earlier work [14] the determination uses only quantities from the heavy-light system. We expect these values to be close to our final determination. Moreover, we observe that within statistical uncertainties there is no dependence on the light sea quark mass ($m_{\text{sea}}^{l}$). $m_{\text{sea}}^{l}$ | $m_{0}a$ | $c_{P}$ | $\zeta$ ---|---|---|--- 0.005 | 8.41(9) | 5.7(2) | 3.1(2) 0.010 | 8.4(1) | 5.8(2) | 3.1(1) 0.020 | 8.4(1) | 5.6(2) | 3.1(1) Table 2: Preliminary determination of the RHQ parameters on the three $24^{3}$ ensembles with $a\approx 0.11$fm. ## 3 Computation of decay constants Using these newly determined RHQ parameters we compute as a first non-trivial test the decay constants of the unitary ($B_{l}$) and strange ($B_{s}$) mesons via the relation $\displaystyle f_{B_{q}}=Z_{\Phi}\;\Phi_{B_{q}}\;a^{-3/2}/\sqrt{m_{B_{q}}},$ (35) where $\Phi_{B}$ is the lattice decay amplitude, $Z_{\Phi}$ is the renormalization factor. For the $B_{s}$ meson (domain-wall valence quark has mass of physical $s$-quark) we generated data for the set of seven different RHQ input parameters. Therefore we can use similar equations to (33) and (34) in order to determine the decay amplitude $\Phi_{B_{s}}$ at the tuned RHQ parameters: $\displaystyle\Phi^{\text{RHQ}}=J_{\Phi}^{(1\times 3)}\times\left[\begin{array}[]{c}m_{0}a\\\ c_{P}\\\ \zeta\end{array}\right]^{\text{RHQ}}+A_{\Phi}$ (39) with $\displaystyle\displaystyle J_{\Phi}=\left[\frac{\Phi_{3}-\Phi_{2}}{2\sigma_{m_{0}a}},\,\frac{\Phi_{5}-\Phi_{4}}{2\sigma_{c_{P}}},\,\frac{\Phi_{7}-\Phi_{6}}{2\sigma_{\zeta}}\right]\qquad\text{and}\qquad A_{\Phi}=\Phi_{1}-J_{\Phi}\times\left[m_{0}a,\,c_{P},\,\zeta\right]^{t}.$ (40) In Fig. 3 we show the results for $\Phi_{B_{s}}$ obtained on the ensemble with $m_{\text{sea}}^{l}=0.005$. To test the linearity with respect to the input parameters $\\{m_{0}a,\,c_{P},\,\zeta\\}$ we used three different sets of seven RHQ parameters all centered around the same point. The vertical black line with the gray error band indicates the tuned value of $m_{0}a$, $c_{P}$ or $\zeta$ and allows for a simple estimate of the error in $\Phi_{B_{s}}$ due to the uncertainty in each of the three parameters. These plots show that the effect of the uncertainty in $m_{0}a$ and $c_{P}$ is negligible, whereas $\zeta$ contributes an error of about 1%. Moreover, we see that $\Phi_{B_{s}}$ depends linearly on $m_{0}a$, $c_{P}$ and $\zeta$ in the range of interest. Figure 3: Dependence of $\Phi_{B_{s}}$ on the three RHQ parameters $\\{m_{0}a,\,c_{P},\,\zeta\\}$; results are shown for the lightest sea quark mass $m_{\text{sea}}^{l}=0.005$ on the $24^{3}$ ensembles. Alternatively, one may simply use the tuned RHQ values to compute the needed correlation functions. We followed this procedure to calculate the $B_{l}$ meson (domain-wall valence quark mass equals the light sea quark mass) the decay amplitude $\Phi_{B_{l}}$. Atfer renormalizing $\Phi_{B}$ multiplicatively at 1-loop [18] we obtain the decay constants which are given in Tab. 3 and shown in Fig. LABEL:Fig-fB. As mentioned above, the values for $f_{B_{l}}$ are obtained by simulating directly at the tuned RHQ parameters, whereas the $f_{B_{s}}$ values are extracted using Eq. (39). The statistical errors on $f_{B_{s}}$ are therefore larger because they take into account the statistical uncertainties in the three RHQ parameters. For a better comparison, we increase the errors in $f_{B_{l}}$ in the Fig. LABEL:Fig-fB by the error due to the statistical uncertainty in the RHQ parameters estimated in Fig. 3. Moreover, we emphasize that this computation is performed without $O(a)$ improvement. ## 4 Conclusion We presented our first results computing $B$-meson decay constants using domain-wall light and relativistic $b$-quarks with all parameters of the RHQ action tuned non-perturbatively. Currently, our results still need to be ${\cal O}(a)$ improved. For $f_{B_{s}}$ we expect only a mild chiral extrapolation but of course need to perform an extrapolation to the continuum and estimate other systematic errors. Despite these caveats the small statistical errors and the fact that our central values lie in the same ballpark as results of other collaborations indicates the promise of our method. $m_{sea}^{l}$ | $f_{B_{l}}$(MeV) | $f_{B_{s}}$(Mev) ---|---|--- 0.005 | 188(2) | 215(3) 0.010 | 194(2) | 214(4) 0.020 | — | 221(2) Table 3: Preliminary results for the decay constants using 1-loop multiplicative renormalization, but without $O(a)$ improvement of the axial- current operator. ## Acknowledgments We are thankful to all the members of the RBC and UKQCD collaborations. Numerical computations for this work utilized USQCD resources and were performed on the kaon and jpsi clusters at FNAL, in part funded by the Office of Science of the U.S. Department of Energy. This manuscript has been authored by an employee of Brookhaven Science Associates, LLC under Contract No. DE- AC02-98CH10886 with the U.S. Department of Energy. ## References * [1] A. J. Buras, M. Jamin, and P. H. Weisz, Nucl. Phys. B347, 491 (1990) * [2] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981) * [3] E. Lunghi and A. Soni, Phys.Rev.Lett. 104, 251802 (2010), arXiv:0912.0002 [hep-ph] * [4] E. Lunghi and A. Soni, Phys. Lett. B666, 162 (2008), arXiv:0803.4340 [hep-ph] * [5] E. Lunghi and A. Soni, arXiv:1010.6069 [hep-ph] * [6] C. Albertus, Y. Aoki, P. Boyle, N. Christ, T. Dumitrescu, _et al._ , Phys.Rev. D82, 014505 (2010), arXiv:1001.2023 [hep-lat] * [7] R. T. Evans, E. Gamiz, and A. X. El-Khadra, PoS LAT2008, 052 (2008) * [8] R. T. Evans, E. Gamiz, A. El-Khadra, and A. Kronfeld (Fermilab Lattice and MILC Collaborations), PoS LAT2009, 245 (2009), arXiv:0911.5432 [hep-lat] * [9] E. Gamiz _et al._ (HPQCD), Phys. Rev. D80, 014503 (2009), arXiv:0902.1815 [hep-lat] * [10] C. Allton _et al._ (RBC-UKQCD), Phys. Rev. D78, 114509 (2008), arXiv:0804.0473 [hep-lat] * [11] Y. Aoki _et al._ (RBC and UKQCD Collaborations), arXiv:1011.0892 [hep-lat] * [12] A. X. El-Khadra, A. S. Kronfeld, and P. B. Mackenzie, Phys. Rev. D55, 3933 (1997), arXiv:hep-lat/9604004 * [13] N. H. Christ, M. Li, and H.-W. Lin, Phys.Rev. D76, 074505 (2007), arXiv:hep-lat/0608006 * [14] M. Li (RBC and UKQCD collaborations), PoS LATTICE2008, 120 (2008), arXiv:0810.0040 [hep-lat] * [15] H. Peng (RBC and UKQCD collaborations), PoS LATTICE2009, 094 (2009) * [16] H. Peng (RBC and UKQCD collaborations), PoS LATTICE2010, 107 (2010) * [17] D. P. Menscher, “Charmonium and Charmed Mesons with Improved Lattice QCD,” (2005), PhD thesis * [18] N. Yamada, S. Aoki, and Y. Kuramashi, Nucl. Phys. B713, 407 (2005), arXiv:hep-lat/0407031
arxiv-papers
2011-01-24T16:13:41
2024-09-04T02:49:16.607567
{ "license": "Public Domain", "authors": "Ruth S. Van de Water and Oliver Witzel", "submitter": "Oliver Witzel", "url": "https://arxiv.org/abs/1101.4580" }
1101.4632
# A Secure Web-Based File Exchange Server Software Requirements Specification Document CIISE Security Investigation Initiative Represented by: Serguei A. Mokhov Marc-André Laverdière Ali Benssam Djamel Benredjem {mokhov,ma_laver,d_benred,al_ben}@ciise.concordia.ca Montréal, Québec, Canada (December 14, 2005) ###### Contents 1. 1 Introduction 1. 1.1 Purpose 2. 1.2 Scope 3. 1.3 Definitions and Acronyms 2. 2 Overall Description 1. 2.1 Product Perspective 1. 2.1.1 System interfaces 2. 2.1.2 User Interfaces 3. 2.1.3 Hardware Interfaces 4. 2.1.4 Software Interfaces 2. 2.2 Product Functions 3. 3 Specific Requirements 1. 3.1 Functional Requirements 1. 3.1.1 Domain Model 2. 3.1.2 Use Case Model 2. 3.2 Software System Attributes 1. 3.2.1 Security 2. 3.2.2 Reliability 3. 3.2.3 Availability 4. 3.2.4 Maintainability 5. 3.2.5 Portability 3. 3.3 Logical Database Requirements ###### List of Figures 1. 3.1 SFS system packages 2. 3.2 Normal user use-case 3. 3.3 Administrator use-case ## Chapter 1 Introduction Building Trust is the basis of all communication, especially electronic one, as the identity of the other entity remains concealed. To address problems of trust, authentication and security over the network, electronic communications and transactions need a framework that provides security policies, encryption mechanisms and procedures to generate manage and store keys and certificates. This software requirements specification (SRS) document demonstrates all the concerns and specifications of the secure web-based file exchange server (SFS). SFS is a security architecture that we propose here to provide an increased level of confidence for exchanging information over increasingly insecure networks, such as the Internet. SFS is expected to offer users a secure and trustworthy electronic transaction. ### 1.1 Purpose The intent of implementation and deployment of SFS facilities is to meet its basic purpose of providing Trust. Presently, SFS needs to perform the following security functions: * • _Mutual authentication of entities taking part in the communication:_ Only authenticated principals can access files to which they have privileges. * • _Ensure data integrity:_ By issuing digital certificates which guarantee the integrity of the public key. Only the public key for a certificate that has been authenticated by a certifying authority should work with the private key possessed by an entity. This eliminates impersonation and key modification. * • _Enforce security:_ Communications are more secure by using SSL to exchange information over the network. ### 1.2 Scope SFS is implemented to secure sensitive resources of the organization and avoid security breaches. The SFS allows trustworthy communication between the different principals. These principals must be authenticated and the access to the resources (files) should be secured and regulated. Any principal wants to access to the database needs to perform the following steps: * • _Mutual authentication:_ The Web Server via which the database is contacted authenticates the principal using its digital certificate and username to ensure that it is who it claims to be . The principal authenticates also the server using its certificate information. * • _Principal validation:_ To validate the principal, the server looks up information from an LDAP server which contains the hierarchy of all principals along with certificates and credentials. * • _Enforcing security:_ The security is enforced by using SSL to communicate between the Web Server and the LDAP server, the Web Server and the database and between the principal and Web Server. * • Principal authentication: Upon successful authentication, the Web Server will allow the principal to perform actions on the database according to a pre specified Access Control List. * • Kinds of principals: There are two kinds of principals, administrators and clients: _Clients_ have the ability to upload, download, delete and view files. _Administrators_ have the ability to: Upload, download, delete and view files; Add, delete and modify users; Generate user’s certificate, with all required information; Generate ACL to users; Manage groups, Perform maintenance. Finally, this infrastructure allows additional features such as the ability to assign users to groups in order to provide users with the access to files prepared by other group members. ### 1.3 Definitions and Acronyms * • PKI: Public Key Infrastructure * • OpenLDAP : is a free, open source implementation of the Lightweight Directory Access Protocol (LDAP). * • OpenSSL: an open source SSL library and certificate authority * • Apache Tomcat: A Java based Web Application container that was created to run Servlets and JavaServer Pages (JSP) in Web applications * • PostgreSQL: An open source object-relational database server * • SSL: Secure Socket Layer * • JSP: Java Server Pages * • JCE: Java Cryptography Extension * • API: Application Programming Interface * • JDBC: Java Database Connectivity * • JNDI: Java Naming and Directory Interface * • LDAP: Lightweight Directory Access Protocol * • X.509: A standard for defining a Digital Certificate used by SSL * • SRS: Specification Request Document * • SDD: Specification Design Document * • DER: Distinguished Encoding Rules * • Mutual Authentication: The process of two principals proving their identities to each other * • SFS: Secure File Exchange Server, this product * • COTS: Commercial Off The Shelf, common commercially or freely available software The coming sections of the SRS are a description of all the requirements to be implemented in SFS system. The requirements specifications are organized in two major sections: Overall Description and Specific Requirements. ## Chapter 2 Overall Description In this chapter we provide an overall insight of the general factors that affect the SFS system and its requirements. ### 2.1 Product Perspective The SFS system is intended to operate in a distributed environment: clients machines, application server, database server, and LDAP server. SFS is accessed via secure connections we intend to provide in this work. The system’s user can be either a normal user or an administrator. A normal user has the ability to upload, download, delete and view files. An administrator is able to: Upload, download, delete and view files; add, delete and modify users; generate user certificates along with all required information; generate an ACL to each and every user; manage groups; perform various maintenance actions such as: check log files, delete files, etc. #### 2.1.1 System interfaces The various parts of the SFS application will be installed on client machines and different servers. The client that uses the system has to have the certificate installed on his machine to provide client authentication. The servers are responsible of one of the following functions: provide the database, provide the LDAP server functionalities, and provide the application server for different clients. #### 2.1.2 User Interfaces The user interfaces consist of web-based graphical components that allow the user to interact with the SFS system. The user will use a web browser to send and receive data. If the user is the administrator, s/he will have more options to add, delete, etc. users, generate users certificates, generate ACL for each user, etc. #### 2.1.3 Hardware Interfaces The hardware interfaces will be achieved through the abstraction layer of the Java Virtual Machine (JVM). The keyboard and the mouse are examples of such hardware interfaces that allow users to interact with the SFS system. #### 2.1.4 Software Interfaces Among the most important software interfaces used in this project, we have: * • The SFS system is OS-independent due to the cross platform Java implementation. It will support web browsers such as Internet Explorer, Mozilla Fireworks, etc. * • Access to databases will be provided by JDBC 3 on both Windows and Linux environment. * • JXplorer? will be used to provide a graphical access to LDAP server. * • OpenLDAP? is used to host users’ certificates. * • Java 1.5 JDK from Sun. * • JRE 1.5 from Sun. * • Servlets for client and administrator interfaces. * • Apache Tomcat5 server as the web server used in this project. * • PostegreSQL? is the database used to host users information, files, etc. * • OpenSSL toolkit to generate the certificates for users. Hereafter, we provide the software and documentation’s locations related to these interfaces: * • OpenLDAP? software and documentation found at: _http://www.openldap.org/_ * • PostgreSQL database and documentation found at: _http://www.postgresql.org/_ * • Java development kit 1.5 available at: _http://java.sun.com/j2se/_ * • JSP documentation found at: _http://java.sun.com/products/jsp/_ * • OpenSSL Toolkit found at: _http://www.openssl.org/_ * • Apache Tomcat 5.0 web server found at: _http://tomcat.apache.org/_ ### 2.2 Product Functions The SFS system will implement the following functionalities: * • _Server authentication:_ This use case allow the user to authenticate the web server is connecting to. * • _Client authentication:_ This is use case allow the server to authenticate the user he is trying to connect to. * • _Secure communication:_ Between users over the network. * • _Files handling:_ Such as downloading, uploading, and deletion files. * • Users management: Administrator has the ability to add and delete users, add and delete groups, and assign users to groups. ## Chapter 3 Specific Requirements In this section we describe the software requirements to design the SFS system. The system design should satisfy the following requirements. ### 3.1 Functional Requirements Hereafter, we express the expectations in terms of system functions and constraints. This includes the domain model and the most important use case diagrams of the SFS system. #### 3.1.1 Domain Model The SFS system domain model consists of many packages. * • _Client authentication module:_ used to provide users the ability to authenticate the server. * • _Server authentication module:_ used to provide the server the ability to authenticate the clients. * • _LDAP connection module:_ used to provide connection to the LDAP server in order to check clients’ credentials. * • _Database connection module:_ used to provide users the ability to connect to the database server in a secure mode. Figure 3.1: SFS system packages #### 3.1.2 Use Case Model The SFS system consists of a set of use cases manageable by the users of the application. There are two types of users of the system: normal users and administrator. Normal can view, delete, download, and upload files. For the administrator, s/he can: add, delete, etc. users; generate users’ certificates; generate ACL for each user, etc. The diagram in Figure 3.2 shows the capabilities of a normal user. Figure 3.2: Normal user use-case The diagram in Figure 3.3 shows the capabilities of the administrator user. Figure 3.3: Administrator use-case ### 3.2 Software System Attributes There are a number of software attributes that can serve as requirements. It is important that required attributes by specified so that their achievement can be objectively verified. The following items are some of the most important ones: security, reliability, availability, maintainability, and portability. #### 3.2.1 Security Security is the most important attribute of the SFS design and implementation. The mutual authentication between the server and clients is crucial for the system use. The system should be able to authenticate users and differentiate among them, either are normal users or administrator. In order to achieve the security feature expected from the SFS system, the following tasks have to be realized: * • Utilize cryptographic techniques * • Check users’ credentials before using the system and accessing the database * • Provide secure communications between different parts of the system #### 3.2.2 Reliability The basis for the definition of reliability is the probability that a system will fail during a given period. The reliability of the whole system depends on the reliability of its components and on the reliability of the communication between its components. The SFS system is based mainly on some standard components such as OpenLDAP, OpenSSL, JDBC, PostgreSQL, etc. The reliability of these service components is already proved. This fact improves the reliability of the system and restricts the proof work on assuring only of the reliability of the added components and the communications between the different components. In addition, the system must ensure the security of the communications which is the most important issue of the SFS system. #### 3.2.3 Availability The SFS system must be able to work continuously in order to provide users with an access to different server’s parts of the system. However, since this system depends on distributed information systems and databases, many constraints should be taken into account such as: * • The connection to the web server that provides access to the system * • The interconnection between different parts of the system should always be available; otherwise, the users cannot complete their tasks using the system. * • The database should be available in the database server side * • The LDAP server should be always available in order to check users’ credentials * • The web server should be also available in order to allow users connecting to the system. #### 3.2.4 Maintainability Maintainability is defined as the capacity to undergo repairs and modifications. The main goal in designing SFS system is to keep it easy to be modified and extended. #### 3.2.5 Portability The portability is one of the main specifications of Java. Since SFS is implemented using the Java programming language, the portability is automatically satisfied and the system is able to run on any machine or operating system which supports the execution of a Java virtual machine. ### 3.3 Logical Database Requirements The rationale behind SFS system is to provide secure connections for users accessing databases to view, delete, upload, and download files through a web server and LDAP server. After analyzing the requirements we propose using a relational database model to meet our requirements. This database is required to store information about users, files, groups of users, etc. The database is expected to work on 24 hours and 7 days in order to provide nonstop access to the users. Therefore, backup of the database should be taken periodically (daily or weekly. The relational database itself guarantees the flexibility, simplicity and elimination of redundancy once designed carefully. The entity relationship model will be elaborated in detail in the database design of the design part, in this document. ## Bibliography * [Apa11] Apache Foundation. Apache Jakarta Tomcat. [online], 1999–2011. http://jakarta.apache.org/tomcat/index.html. * [CBH03] Suranjan Choudhury, Katrik Bhatnagar, and Wisam Haque. Public Key Infrastructure, Implementation and Design. NIIT Books, 2003. * [Deb05] Mourad Debbabi. INSE 6120: Cryptographic protocols and network security, lecture notes. Concordia Institute for Information Systems Engineering, Concordia University, Montreal, Canada, 2005. http://users.encs.concordia.ca/~debbabi/inse6120.html. * [E+11] Eclipse contributors et al. Eclipse Platform. eclipse.org, 2000–2011. http://www.eclipse.org, last viewed February 2010. * [GB11] Erich Gamma and Kent Beck. JUnit. [online], Object Mentor, Inc., 2001–2011. http://junit.org/. * [Lar06] Craig Larman. Applying UML and Patterns: An Introduction to Object-Oriented Analysis and Design and Iterative Development. Pearson Education, third edition, April 2006. ISBN: 0131489062. * [Leu05] Leuvens Universitair. A few frequently used SSL commands. [online], Leuvens Universitair Dienstencentrum voor Informatica en Telematica, 2005. http://shib.kuleuven.be/docs/ssl_commands.shtml. * [Ope05a] OpenLDAP Community. OpenLDAP: The open source for LDAP software and information. [online], 2005. www.openldap.org. * [Ope05b] OpenSSL Community. OpenSSL: The open source toolkit for SSL/TLS. [online], 2005. http://www.openssl.org. * [O’R05a] O’Reilly. Home of com.oreilly.servlet. [online], 2005. http://servlets.com/cos. * [O’R05b] O’Reilly. JXplorer – a Java LDAP browser. [online], 2005. http://sourceforge.net/projects/jxplorer/. * [Sun05a] Sun Microsystems, Inc. Java servlet technology. [online], 1994–2005. http://java.sun.com/products/servlets. * [Sun05b] Sun Microsystems, Inc. JavaServer pages technology. [online], 2001–2005. http://java.sun.com/products/jsp/. * [The11] The PostgreSQL Global Development Group. PostgreSQL – the world’s most advanced open-source database. [ditigal], 1996–2011. http://www.postgresql.org/, last viewed January 2010. ## Index * Introduction Chapter 1
arxiv-papers
2011-01-24T19:49:14
2024-09-04T02:49:16.612528
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Serguei A. Mokhov and Marc-Andr\\'e Laverdi\\`ere and Ali Benssam and\n Djamel Benredjem", "submitter": "Serguei Mokhov", "url": "https://arxiv.org/abs/1101.4632" }
1101.4634
# Upper Bound on Fidelity of Classical Sagnac Gyroscope Thomas B. Bahder Aviation and Missile Research, Development, and Engineering Center, US Army RDECOM, Redstone Arsenal, AL 35898, U.S.A. ###### Abstract Numerous quantum mechanical schemes have been proposed that are intended to improve the sensitivity to rotation provided by the classical Sagnac effect in gyroscopes. A general metric is needed that can compare the performance of the new quantum systems with the classical systems. The fidelity (Shannon mutual information between the measurement and the rotation rate) is proposed as a metric that is capable of this comparison. A theoretical upper bound is derived for the fidelity of an ideal classical Sagnac gyroscope. This upper bound for the classical Sagnac gyroscope should be used as a benchmark to compare the performance of proposed enhanced classical and quantum rotation sensors. In fact, the fidelity is general enough to compare the quality of two different apparatuses (two different experiments) that attempt to measure the same quantity. ###### pacs: PACS number 07.60.Ly, 03.75.Dg, 06.20.Dk, 07.07.Df The Sagnac effect Sagnac (1913a, b, 1914); Post (1967) is the basis for all modern rotation sensors Lefevre (1993) and their applications to inertial navigation systems Titterton and Weston (2004). Besides its practical applications, the Sagnac effect is being contemplated for studying general relativistic effects, such as Lense-Thirring frame dragging, detecting gravitational waves, and testing local Lorentz invariance Chow et al. (1985); Stedman (1997). The original experiments of Sagnac consisted of mirrors mounted on a rotating disk, see Fig. 1. The mirrors define two paths, one clockwise (CW) and the other counter-clockwise (CCW) on the disk. A source of light at point $S$ was mounted on the rotating disk, having wavelength $\lambda=2\pi c/\omega$, where $\omega$ is the angular frequency as measured in an inertial frame and $c$ is the speed of light in vacuum. The beam is split at the beam splitter $BS$ and light is propagated along the clockwise and counter-clockwise paths. The two beams are brought together at the beam splitter $BS$ and observed at point $O$. When the interferometer is rotated at angular velocity $\Omega$, with light source and detector mounted on the rotating disk, a fringe shift $\Delta N$ is observed with respect to the fringe position for the stationary interferometer, given by Post (1967); Chow et al. (1985) $\Delta N=\frac{4{\bf A}\cdot{\bf\Omega}}{\lambda\,c}$ (1) where ${\bf A}$ is a vector perpendicular to the area enclosed by the paths, having magnitude $A=|{\bf A}|$, and ${\bf\Omega}$ is the vector in the direction of the angular velocity of rotation, with magnitude $|{\bf\Omega}|=\Omega$. The fringe shift, $\Delta N=\Delta L/\lambda=c\Delta t/\lambda$, can be expressed in terms of the path length difference, $\Delta L$, or time difference, $\Delta t=4{\bf A}\cdot{\bf\Omega}/c^{2}$, for the CW and CCW paths, as measured in an inertial frame. For typical rotation rates in the laboratory, the classical Sagnac effect is small, and the effect has to be enhanced to make a practical rotation sensor. The classical Sagnac effect is exploited for sensing rotation by either measuring a frequency shift or a phase shift. In the active ring laser gyroscope (RLG), where the optical medium is inside the cavity Chow et al. (1985), or in a resonant fiber-optic gyroscope (R-FOG) Lefevre (1993), a measurement is made of the frequency shift, $\Delta\omega$, between the CW and CCW propagating modes Post (1967); Chow et al. (1985) $\Delta\omega=\frac{4A\,\Omega}{L\,c}\omega=S\,\Omega$ (2) where $L$ is the length of the perimeter of the path measured in an inertial frame, and $S$ is commonly called the scale factor. In a passive fiber ring interferometer (I-FOG) Lefevre (1993), or a passive ring laser gyroscope with light source outside the medium Chow et al. (1985), the phase shift $\Delta\phi$, is measured between CW and CCW propagating beams, $\Delta\phi=\frac{4A\Omega}{c^{2}}\omega$ (3) For a fiber-optic gyroscope with phase shift enhanced by $N$ turns, the frequency shift is $\Delta\phi_{N}=N\Delta\phi$. Figure 1: Schematic of a Sagnac interferometer is shown, with light from source $S$ incident on beam splitter $BS$, mirrors at $M_{1}$, $M_{2}$, and $M_{3}$, and the observer at $O$ that detects the frequency shift $\Delta\omega$. Recently, much effort has been expended on experiments with quantum Sagnac interferometers, using single-photons Bertocchi et al. (2006), using cold atoms Gustavson et al. (2000); Gilowski et al. (2009) and using Bose-Einstein condensates(BEC) Gupta et al. (2005); Wang et al. (2005); Tolstikhin et al. (2005), in efforts to make improvements over the sensitivity to rotation of the classical Sagnac effect. Schemes have also been proposed to improve the sensitivity of rotation sensing using multi-photon correlations Kolkiran and Agarwal (2007) and using entangled particles, which are expected to have Heisenberg limited precision that scales as $1/N$, where $N$ is the number of particles Cooper et al. (2010). The utility of these quantum systems as rotation sensors must be compared with the classical Sagnac effect using classical light. The metric used to compare the classical and quantum systems must be sufficiently general to treat both types of systems on an equal basis. Information measures are examples of such metrics because they are general enough to compare quantum and classical systems. The determination of the rotation rate is a specific example of the more general problem of parameter estimation, whose goal is to determine one or more parameters from measurements Cramér (1958); Helstrom (1967, 1976); Holevo (1982); Braunstein and Caves (1994); Braunstein et al. (1996); Barndorff- Nielsen and Gill (2000); Barndorff-Nielsen et al. (2003). The Cramér-Rao theoremCramér (1958); Cover and Thomas (2006) can be applied to get an upper bound on the variance of an unbiased estimator of a parameter of interest (here the rotation rate) in terms of the classical and quantum Fisher information, see Ref. Bahder (2010) and references contained therein. One potential drawback of this approach is that the Fisher information, and therefore the upper bound on the variance of the estimator, can depend on the true value of the parameter to be determined Bahder (2010). Of course, the true value of the parameter is unknown. Specifically, the Fisher information can depend on the true angular rotation rate when the Sagnac interferometer is not unitary, which occurs when scattering or dissipation are present Bahder (2010). Consequently, I propose to characterize a rotation sensor by its fidelity, which is the Shannon mutual information, a quantity that does not depend on the true rotation rate. In this letter, I calculate a fundamental upper bound on the rotation sensitivity of a classical Sagnac gyroscope that follows Eq. (2). This upper bound can be used as a benchmark to compare the performance of rotation sensors based on newly proposed quantum and classical methodologies, see earlier discussion. In addition, the fidelity is a useful measure to compare other proposed gyroscopes based on slow light generated by electromagnetically induced transparency Leonhardt and Piwnicki (2000) and other classical optical enhancements Matsko et al. (2004); Smith et al. (2008, 2009). The fidelity Bahder and Lopata (2006); Bahder (2010) is the Shannon mutual information Shannon (1948); Cover and Thomas (2006) between the measurement (the frequency shift), $\Delta\omega$, and the parameter to be measured (the rotation rate) $\Omega$: $\displaystyle H$ $\displaystyle=$ $\displaystyle\int_{-\infty}^{+\infty}d({\Delta\omega})\,\,\int_{-\infty}^{+\infty}d\Omega\,\,p(\Delta\omega|\Omega)\,p(\Omega)\,\,\times\,$ (4) $\displaystyle\log_{2}\left[\frac{p(\Delta\omega|\Omega)\,}{\int_{-\infty}^{+\infty}\,\,\,p(\Delta\omega|\Omega^{\prime})\,p(\Omega^{\prime})\,\,d\,\Omega^{\prime}}\right].$ where $p(\Delta\omega|\Omega)$ is the conditional probability density of measuring a frequency shift $\Delta\omega$, given a true rotation rate $\Omega$. Our prior information on the rotation rate is given by the probability density $p(\Omega)$. The fidelity, $H$, is the Shannon mutual information in a communication problem between Alice and Bob, wherein Alice sends messages to Bob Shannon (1948); Cover and Thomas (2006). The fidelity $H$ does not depend on the measurements, $\Delta\omega$, or on the parameter, $\Omega$, because it is an average over all possible measurements and parameter values. If we are completely ignorant of the rotation rate, we can take the prior probability as flat distribution, $p(\Omega)=1/(2\pi)$, indicative of no prior information on our part. In this case, the fidelity $H$ characterizes the quality of the Sagnac interferometer itself, in terms of mutual information that the measurement of $\Delta\omega$ carries about the parameter of interest, the rotation rate $\Omega$. In fact, the fidelity $H$ is a specific example of a general way to characterize the quality of all physical measurements. The fidelity in Eq.(4) is a completely general way to describe any classical or quantum measurement experiment. The classical or quantum apparatus is viewed as a channel through which information flows from the phenomena to be measured to the measurements. The fundamental quantity that describes this process is the conditional probability of a measurement and the probability distribution that describes our prior information, above notated as $p(\Delta\omega|\Omega)$ and $p(\Omega^{\prime})$, respectively. In the language of communication, there is mutual information $H$ between the continuous alphabet of the parameter, $\Omega$, and the continuous alphabet of the measurements, $\Delta\omega$. In order to compute the fidelity for the classical Sagnac gyroscope from Equation (4) a model is needed for the conditional probability density $p(\Delta\omega|\Omega)$. In the case of a quantum system, these probabilities are given by a trace: $p(\Delta\omega|\Omega,\rho)=\rm{tr}\left(\hat{\rho}\,\hat{\Pi}\left(\Delta\omega\right)\right)$ (5) where the state is specified by the density matrix, $\hat{\rho}$, and the measurements are defined by the positive-operator valued measure (POVM), $\hat{\Pi}(\Delta\omega)$. For a classical Sagnac system, I obtain an upper bound on the fidelity in Eq.(4). I consider classical light of bandwidth $\Delta\omega$ and center frequency $\bar{\omega}$, input into a Sagnac gyroscope that satisfies Eq.(2). Therefore, I define the classical measurement probabilities, $p(\Delta\omega|\Omega)$, by $p(\Delta\omega|\Omega)=\sum\limits_{n=0}^{\infty}{p\left({\Delta\omega|\Omega,\omega_{n}}\right)\,P_{in}\left({\omega_{n}}\right)}$ (6) where $p\left({\Delta\omega|\Omega,\omega_{n}}\right)$ is the probability density for measuring $\Delta\omega$, given the that the true rotation rate is $\Omega$, and the input was monochromatic at frequency $\omega_{n}$. In Eq. (6), for convenience, I assume that the allowed frequency modes are discrete, $\omega_{n}$, for $n=0,1,\cdots\infty$. The probability $P_{in}\left(\omega\right)$ gives the distribution of input frequencies, which has center frequency $\bar{\omega}$ and bandwidth $\Delta\omega$. As an example, I can take $P_{in}\left(\omega\right)$ to be a Gaussian distribution of input frequencies with mean $\bar{\omega}$ and standard deviation $\sigma_{\omega}$ $P_{in}\left({\omega_{n}}\right)=\left({\frac{{\delta\omega}}{{2\pi\bar{\omega}}}}\right)^{1/2}\,\exp\left[{-\frac{{\left({\omega_{n}-\bar{\omega}}\right)^{2}}}{{2\,\delta\omega\,\bar{\omega}}}}\right]$ (7) where $\delta\omega=\omega_{n+1}-\omega_{n}$ and the variance is given by $\sigma_{\omega}^{2}=\delta\omega\,\bar{\omega}$. The distribution of frequencies, $P_{in}\left({\omega_{n}}\right)$, is normalized $\sum\limits_{n=0}^{\infty}{P_{in}\left({\omega_{n}}\right)=1}$ (8) in the limit $\bar{\omega}/\delta\omega\gg 1$. The size of bandwidth, $\sigma_{\omega}$, is due to fundamental physical processes in the experiment. I want to obtain an upper bound on the fidelity in Eq.(4) for a classical system. Therefore, I assume that classical measurements are have no noise and no bias. The classical measurement probability, $p(\Delta\omega|\Omega)$, in Eq. (6) is obtained from Eq. (2) as $p\left({\Delta\omega|\Omega,\omega}\right)=\delta\left({\Delta\omega-\frac{{4A\omega}}{{Lc}}\Omega}\right)$ (9) where $\delta(x)$ is the Dirac delta function. Using Eq. (9) in Eq.(6) gives the classical probability of measuring $\Delta\omega$ given the true rotation rate is $\Omega$: $p(\Delta\omega|\Omega)=\left|{\frac{{Lc}}{{4A\Omega}}}\right|\,P_{in}\left({\frac{{Lc}}{{4A{\kern 1.0pt}\Omega}}\,\Delta\omega}\right)$ (10) Note that Eq. (10) is valid for an arbitrary input frequency distribution $P_{in}\left({\omega}\right)$. As an example, for a monochromatic frequency $\bar{\omega}$ input, $P_{in}\left(\omega\right)=\delta\left({\omega-\bar{\omega}}\right)$ (11) Eq.(10) gives the probability of classical measurement as expected: $p\left({\Delta\omega|\Omega}\right)=\delta\left({\Delta\omega-\frac{{4A{\kern 1.0pt}\Omega}}{{Lc}}\bar{\omega}}\right)$ (12) For classical light input, with the Gaussian distribution in Eq. (7), Eq.(10) gives the probability of classical measurement as $p\left({\Delta\omega|\Omega}\right)=\left({\frac{1}{{2\pi}}}\right)^{1/2}\frac{{Lc}}{{4A\left|\Omega\right|{\kern 1.0pt}\sigma_{\omega}}}\exp\left[{-\frac{{\left({\frac{{Lc}}{{4A{\kern 1.0pt}\Omega}}\Delta\omega-\bar{\omega}}\right)^{2}}}{{2{\kern 1.0pt}\sigma_{\omega}^{2}}}}\right]$ (13) The conditional probability density in Eq. (13) can be inverted by using Bayes’ rule $p\left({\Omega|\Delta\omega}\right)=\frac{{p\left({\Delta\omega|\Omega}\right)p\left(\Omega\right)}}{{\int\limits_{-\infty}^{+\infty}{p\left({\Delta\omega|\Omega^{\prime}}\right)p\left({\Omega^{\prime}}\right)\,d\Omega^{\prime}}}}$ (14) where $p\left(\Omega\right)$ specifies the prior probability distribution on rate of rotation, based on our prior information on the rotation rate. With the probability distribution in Eq. (13), the conditional probability distribution $p\left({\Omega|\Delta\omega}\right)$ defined by Eq. (14) has a divergence. However, our prior information on the rotation rate, given by the distribution $p\left(\Omega\right)$ provides a natural cutoff on the integral in Eq. (14). We can be reasonably sure that $p(\Omega)\rightarrow 0$ as $|\Omega|\rightarrow\pm\infty$. For example, we can take $p\left(\Omega\right)=\left\\{{\begin{array}[]{*{20}c}{\frac{1}{{2\Omega_{\max}}},}&{-\Omega_{\max}<\Omega<+\Omega_{\max}}\\\ {0,}&{{\rm{otherwise}}}\\\ \end{array}}\right.$ (15) where $\Omega_{\max}$ represents the maximum expected rotation rate on physical grounds. For the monochromatic input frequency in Eq. (11), the probability of rotation is $p\left(\Omega|\Delta\omega\right)=\delta\left(\Omega-\frac{Lc}{4A}\frac{\Delta\omega}{\bar{\omega}}\right)$ (16) For the Gaussian frequency distribution in Eq. (7), Eq. (14) gives the probability of rotation as $p\left({\Omega|\Delta\omega}\right)=\frac{{\bar{\omega}}}{{\sqrt{2\pi}{\kern 1.0pt}\sigma_{\omega}}}\frac{1}{{\left|\Omega\right|}}\exp\left[{-\frac{1}{{2\sigma_{\omega}^{2}}}\left({\frac{{Lc{\kern 1.0pt}\Delta\omega}}{{4A}}}\right)^{2}\left({\frac{1}{\Omega}-\frac{{4A\bar{\omega}}}{{Lc{\kern 1.0pt}\Delta\omega}}}\right)^{2}}\right]$ (17) In the limit $\Omega\rightarrow\infty$, the probability distribution for $\Omega$, defined by Eq.(17) approaches the function $p\left({\Omega|\Delta\omega}\right)=\frac{{\bar{\omega}}}{{\sqrt{2\pi}{\kern 1.0pt}\sigma_{\omega}}}\frac{1}{{\left|\Omega\right|}}\exp\left[{-\frac{1}{2}\left({\frac{{\bar{\omega}}}{{\sigma_{\omega}}}}\right)^{2}}\right]$ (18) and hence it is not a normalizable probability distribution because its integral diverges logarithmically like $\log\Omega$ for $\Omega\rightarrow\infty$. However, this divergence is multiplied by the exponentially small factor $\frac{{\bar{\omega}}}{{\sigma_{\omega}}}\exp\left[{-\frac{1}{2}\left({\frac{{\bar{\omega}}}{{\sigma_{\omega}}}}\right)^{2}}\right]\ll 1$ (19) since $\bar{\omega}/\sigma_{\omega}\gg 1$. Note that the peak in the probability distribution for $\Omega$ in Eq. (17) occurs at a value $\bar{\Omega}=Lc\Delta\omega/(4A\bar{\omega})$, which is consistent with Eq.(2). The probability distribution for the rotation rate in Eq. (17) is not a Gaussian distribution. However it is possible to define a width, $\sigma_{\Omega}$, that depends on $\Omega$: $\sigma_{\Omega}=\frac{{\sigma_{\omega}}}{{\bar{\omega}}}\Omega$ (20) Equation (20) gives the uncertainty in the rotation rate, $\sigma_{\Omega}$, in terms of the true rotation rate, $\Omega$, the bandwidth of the input classical light, $\sigma_{\omega}$, and the center frequency, $\bar{\omega}$, used in the classical Sagnac gyroscope. As expected, the uncertainty in the rotation rate, $\sigma_{\Omega}$, is proportional to the bandwidth of the input light, $\sigma_{\omega}$. The uncertainty also decreases with higher input frequency, ${\bar{\omega}}$. The upper bound on the fidelity (Shannon mutual information), $H_{\max}$, for the classical Sagnac gyroscope is given by Eq. (4) using Eq. (13) and Eq. (15): $H_{\max}=\frac{1}{2}{\kern 1.0pt}\log_{2}\left[{\left({\frac{e}{{2\pi}}}\right)^{1/2}\frac{{\bar{\omega}}}{{\sigma_{\omega}}}}\right]$ (21) Equation (21) represents a fundamental theoretical upper bound on the information (in bits) that an ideal classical Sagnac gyroscope can provide to a user, for each measurement of frequency shift, $\Delta\omega$. The value in Eq. (21) is an upper bound because we have assumed an ideal classical measurement that has no associated noise. Therefore, the upper bound in Eq. (21) for the classical Sagnac gyroscope is a benchmark to which we can compare rotation sensors based on new quantum technologies, see references above. In summary, I have proposed the use of a new metric, the Shannon mutual information (called the fidelity) between the rotation rate and the measurements (frequency shift) to judge the quality of a rotation sensor. The fidelity metric is general enough to allow comparison of classical and quantum rotation sensors. For an ideal classical Sagnac gyroscope, I have computed a theoretical upper bound on the mutual information that the gyroscope can give to a user by assuming a classical measurement model with no noise. Consequently, $H_{\max}$ in Eq. (21) is the Shannon capacity of a classical Sagnac gyroscope. This upper bound is a benchmark to compare the performance of new rotation sensors based on improved classical and quantum technologies. In addition, in Eq. (20) I have derived a relation between the bandwidth of light input into a classical Sagnac gyroscope, $\sigma_{\omega}$, and an estimate of the uncertainty in the rotation rate, $\sigma_{\Omega}$. Finally, the fidelity defined in Eq. (4) is general enough to describe the quality of any physical measurement. Consequently, the fidelity can be used to compare the quality of two different apparatuses (two different experiments) that attempt to measure the same quantity. ## References * Sagnac (1913a) G. Sagnac, Compt. Rend. 157, 708 (1913a). * Sagnac (1913b) G. Sagnac, Compt. Rend. 157, 1410 (1913b). * Sagnac (1914) G. Sagnac, J. Phys. Radium 5th Series 4, 177 (1914). * Post (1967) E. J. Post, Rev. Mod. Phys. 39, 475 (1967). * Lefevre (1993) H. Lefevre, _The fiber-optic gyroscope_ (Artech House, Boston, USA, 1993). * Titterton and Weston (2004) D. H. Titterton and J. Weston, _Strapdown Inertial Navigation Technology_ (The Insitution of Engineering and Technology and The American Institute of Aeronautics, London, U.K. and Reston, Virginia, USA, 2004), second edition ed. * Chow et al. (1985) W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985). * Stedman (1997) G. E. Stedman, Rep. Prog. Phys. 60, 615 (1997). * Bertocchi et al. (2006) G. Bertocchi, O. Alibart, D. B. Ostrowsky, S. Tanzilli, and P. Baldi, J. Phys. B 39, 1011 (2006). * Gustavson et al. (2000) T. L. Gustavson, A. Landragin, and M. A. Kasevich, Class. Quantum Grav. 17, 2385 (2000). * Gilowski et al. (2009) M. Gilowski, C. Schubert, T. Wendrich, P. Berg, G. Tackmann, W. Ertmer, and E. M. Rasel, Frequency Control Symposium, 2009 Joint with the 22nd European Frequency and Time forum. IEEE International pp. 1173 – 1175 (2009). * Gupta et al. (2005) S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Phys. Rev. Lett. 95, 143201 (2005). * Wang et al. (2005) Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, Phys. Rev. Lett. 94, 090405 (2005). * Tolstikhin et al. (2005) O. I. Tolstikhin, T. Morishita, and S. Watanabe, Phys. Rev. A 72, 051603(R) (2005). * Kolkiran and Agarwal (2007) A. Kolkiran and G. S. Agarwal, Optics Express 15, 6798 (2007). * Cooper et al. (2010) J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, Phys. Rev. A 81, 043624 (2010). * Cramér (1958) H. Cramér, _Mathematical Methods of Statistics_ (Princeton University Press, Princeton, 1958), eight printing. * Helstrom (1967) C. W. Helstrom, Phys. Lett. A 25, 101 (1967). * Helstrom (1976) C. W. Helstrom, _Quantum Detection and Estimation Theory_ (Academic Press, New York, 1976). * Holevo (1982) A. S. Holevo, _Probabilistic and Statistical Aspects of Quantum Theory_ (North-Holland, Amsterdam, 1982). * Braunstein and Caves (1994) S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994). * Braunstein et al. (1996) S. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. of Phys. 247, 135 (1996). * Barndorff-Nielsen and Gill (2000) O. E. Barndorff-Nielsen and R. D. Gill, J. Phys. A: Math. Gen. 33, 4481 (2000). * Barndorff-Nielsen et al. (2003) O. E. Barndorff-Nielsen, R. D. Gill, and P. E. Jupp, J. Roy. Stat. Soc. B 65, 775 (2003), URL http://arxiv.org/abs/quant-ph/0307191. * Cover and Thomas (2006) T. M. Cover and J. A. Thomas, _Elements of Information Theory_ (J. Wiley & Sons, Inc., Hoboken, New Jersey, 2006), second edition ed. * Bahder (2010) T. B. Bahder, submitted to Phys. Rev. A xx, xxx (2010), URL http://arxiv.org/abs/1012.5293. * Leonhardt and Piwnicki (2000) U. Leonhardt and P. Piwnicki, Phys. Rev. A 62, 055801 (2000). * Matsko et al. (2004) A. B. Matsko, A. A. Savchenkov, V. S. Ilchenko, and L. Maleki, Opt. Commun. 233, 107 (2004). * Smith et al. (2008) D. D. Smith, H. Chang, L. Arissian, and J. C. Diels, Phys. Rev. A 78, 053824 (2008). * Smith et al. (2009) D. D. Smith, K. Myneni, J. A. Odutola, and J. C. Diels, Phys. Rev. A 80, 011809 (2009). * Bahder and Lopata (2006) T. B. Bahder and P. A. Lopata, Phys. Rev. A 74, 051801R (2006), URL http://arxiv.org/abs/quant-ph/0602123. * Shannon (1948) C. E. Shannon, The Bell System Technical Journal 27, 379 (1948).
arxiv-papers
2011-01-24T19:52:52
2024-09-04T02:49:16.617103
{ "license": "Public Domain", "authors": "Thomas B. Bahder", "submitter": "Thomas B. Bahder", "url": "https://arxiv.org/abs/1101.4634" }
1101.4640
# Design and Implementation of a Secure Web-Based File Exchange Server Specification Design Document CIISE Security Investigation Initiative Represented by: Serguei A. Mokhov Marc-André Laverdière Ali Benssam Djamel Benredjem {mokhov,ma_laver,al_ben,d_benred}@ciise.concordia.ca Montréal, Québec, Canada (December 14, 2005) ###### Contents 1. 1 Introduction 1. 1.1 Purpose 2. 1.2 Scope 3. 1.3 Definitions and Acronyms 2. 2 System Architecture 1. 2.1 Architectural Philosophy 2. 2.2 Components 1. 2.2.1 User Interface 2. 2.2.2 Web Server 3. 2.2.3 Database Server 4. 2.2.4 LDAP Server 5. 2.2.5 Certificate Authority 6. 2.2.6 Logging Engine 3. 2.3 Interactions 4. 2.4 Software Interface Design 1. 2.4.1 Web Server 2. 2.4.2 Database Server 3. 2.4.3 LDAP Server 4. 2.4.4 Certificate Authority 5. 2.4.5 Logging Engine 5. 2.5 Hardware Environment 6. 2.6 Code View 3. 3 Detailed System Design 1. 3.1 Class Diagrams 2. 3.2 User Interface 3. 3.3 Class Diagrams 1. 3.3.1 LDAPConnection 2. 3.3.2 UserCredentials 3. 3.3.3 OptionsFileLoaderSingleton 4. 3.3.4 DatabaseConnection 4. 3.4 Data Storage Format 1. 3.4.1 Entity Relationship Diagram 5. 3.5 Options File 6. 3.6 Directory Configuration 7. 3.7 External System Interfaces 1. 3.7.1 External Systems and Databases 8. 3.8 User Scenarios 1. 3.8.1 Typical Scenario 2. 3.8.2 Variant scenario 3. 3.8.3 Variant scenario 9. 3.9 Administrator Scenarios 1. 3.9.1 Typical Scenario 2. 3.9.2 Variant Scenario 3. 3.9.3 Variant Scenario ## Chapter 1 Introduction Building Trust is the basis of all communication, especially electronic one, as the identity of the other entity remains concealed. To address problems of trust, authentication and security over the network, electronic communications and transactions need a framework that provides security policies, encryption mechanisms and procedures to generate manage and store keys and certificates. The Public Key Infrastructure (PKI) is a security architecture that has been introduced to provide an increased level of confidence for exchanging information over increasingly insecure networks, such as the Internet. A PKI infrastructure is expected to offer its users a secure and trustworthy electronic transaction. ### 1.1 Purpose The intent of implementation and deployment of PKI facilities is to meet its basic purpose of providing Trust. Presently, PKI needs to perform the following security functions: * • _Mutual authentication of entities taking part in the communication:_ Only authenticated principals can access files to which they have privileges. * • _Ensure data integrity:_ By issuing digital certificates which guarantee the integrity of the public key. Only the public key for a certificate that has been authenticated by a certifying authority should work with the private key possessed by an entity. This eliminates impersonation and key modification. * • _Enforce security:_ Communications are more secure by using SSL to transmit information. ### 1.2 Scope PKI is implemented to secure sensitive resources of the organization and avoid security breaches. The PKI environment allows trustworthy communication between the different principals. These principals must be authenticated and the access to the resources (files) should be secured and regulated. Any principal wants to access to the database needs to perform the following steps: * • _Mutual authentication:_ The Web Server via which the database is contacted authenticates the principal using its digital certificate and username to ensure that it is who it claims to be . The principal authenticates also the server using its certificate information. * • _Principal validation:_ To validate the principal, the server looks up information from an LDAP server which contains the hierarchy of all principals along with certificates and credentials. The LDAP service is compliant with the X.500 database structure. * • _Enforcing security:_ The security is enforced by using SSL to communicate between the Web Server and the LDAP server, the Web Server and the database and between the principal and Web Server. * • _Principal authentication:_ Upon successful authentication, the Web Server will allow the principal to perform actions on the database according to a pre specified Access Control List. * • _Kinds of users:_ We distinguish between a normal and an administrator. While a normal user can upload, download, delete and view files; the administrator has the ability to: upload, download, delete and view files; add, delete and modify users; generate user’s certificate, with all required information; generate ACL to users; manage groups, perform maintenance. Finally, this infrastructure allows additional features such as the ability to assign users to groups in order to provide users with the access to files prepared by other group members. ### 1.3 Definitions and Acronyms * • PKI: Public Key Infrastructure * • OpenLDAP : is a free, open source implementation of the Lightweight Directory Access Protocol (LDAP). * • OpenSSL: an open source SSL library and certificate authority * • Apache Tomcat: A Java based Web Application container that was created to run Servlets and JavaServer Pages (JSP) in Web applications * • PostgreSQL: An open source object-relational database server * • SSL: Secure Socket Layer * • JSP: Java Server Pages * • JCE: Java Cryptography Extension * • API: Application Programming Interface * • JDBC: Java Database Connectivity * • JNDI: Java Naming and Directory Interface * • LDAP: Lightweight Directory Access Protocol * • X.509: A standard for defining a Digital Certificate used by SSL * • SRS: Specification request Document * • SDD: Specification Design document * • DER: Distinguished Encoding Rules * • Mutual Authentication: The process of two principals proving their identities to each other * • SFS: Secure File Exchange Server, this product * • COTS: Commercial Off The Shelf, common commercially or freely available software ## Chapter 2 System Architecture This chapter is intended to provide an overview of the whole system as proposed in the previous requirements and specification document. It describes the product’s perspective, interfaces and design constraints as we have assumed. We will first describe the architectural guidelines for this product, followed by software interface design design, and hardware environment. ### 2.1 Architectural Philosophy The SFS technology hereby implemented is running on architecture that provides a high level of secrecy and integrity for exchanging information. The system is externally visible only through a web application for normal users, and is also entirely visible and accessible for administrators in the scope of normal operations. In addition, the system assumes an internal certificate authority which is explicitly trusted by all principals using SFS. For the proposed architecture, it requires mutual authentication between the user and the web server, an LDAP validation of the user by using digital certificates, the use of the SSL protocol to enforce the security over the communication between modules and the preservation of files in a database. This architecture must respect the following properties: * • _Security:_ The confidentiality, integrity and availability of information. This is to be implemented by supporting the data encryption and certificates mechanisms for secured communication, as well as specifying an access control mechanism for the files stored * • _Trustworthiness:_ The use of electronic certificates internally generated, and of specific use for the application, allow an high trust to be given to the user. * • _Scalability:_ SFS can be expanded easily to cope with large loads. Methods such as load balancing and replication can be easily integrated. * • _Openness:_ The proposed architecture can be implemented and deployed using Java Technologies and open source tools that are well-used and rely on standards. The SFS itself is an open source product developed to achieve security objectives. * • _Component-Based Software Engineering:_ The SFS framework may be treated as components (modules) . We have already mentioned the relevant technologies that can better fit for each of these modules. * • _Usability:_ The SFS service must be designed for high usability. All the required information for a single operation should be grouped in a single screen, with a minimum number of screens needed for all the application. This architecture aims at maximizing software reuse by the integration of COTS applications, portability and interoperability by the use of standards, security and scalability by the user of a single access point. ### 2.2 Components The Figure 2.1 describes the overall architecture of the SFS system. We see four major interacting components: the user interface, the web server, the database and the LDAP server. Two components are not displayed on this figure , which are the certificate authority and the logging engine. Figure 2.1: Main system architecture #### 2.2.1 User Interface The user interface constitutes of HTML web pages that the user uses through a web browser. Those web pages are generated by the web server and interact exclusively with the SFS web server. #### 2.2.2 Web Server The web server is the single access point of the system. It handles authentication responsibilities, database access and user interaction. This is to be handled by the Apache Tomcat 5.0 server and custom J2SE 5.0 code. #### 2.2.3 Database Server This database server contains all the information about the files and their access control rights. It contains also a subset of the user information. This is to be handled by PostgreSQL. #### 2.2.4 LDAP Server This specialized server holds the user credentials (notably user name and password). It could be extended to include user certificates. This module will be realized by OpenLDAP. #### 2.2.5 Certificate Authority This responsibility is manually managed by administrators. Using software tools, they are able to generate the user and server certificates. In our case, we use OpenSSL to perform those functions. #### 2.2.6 Logging Engine This component is responsible for collecting the audit trails and debug information from other components and store it locally. We wanted to use log4j, but we finally opted for the logging mechanisms available in the tools we are using, notably by using Tomcat’s logging. ### 2.3 Interactions We will now describe the inter-module interactions by the use of a system scenario. The user, with a web browser, connects to the web server using SSL. The web server, being configured as to require client authentication, both parties exchange their certificates and validate their peer’s identity. The web server then prompts the user for a user name and password, thereby enforcing 2-factor authentication. Upon receiving this information, the web server queries the LDAP directory based on the user name and retrieves the user’s password hash and certificate (if any is defined). The web server then proceeds to hash the plaintext password (using SHA1) received from the user and compare with the one from the LDAP server. If that information (as well as the certificates, if any) matches, the user is logged in the system. The web server will then query the database server for the access rights of the user (administrator or normal user) and the list of files the user has access to. Based on this information, it will display the appropriate user interface functions and the file list. On user requests to upload, delete or download files, the web server will request the database server to perform the needed transactions. ### 2.4 Software Interface Design This section describes the software interfaces (commonly referred to as APIs) to be used to communicate between each component. #### 2.4.1 Web Server The web server is reached by the client through an HTTPS connection to the single point of access for the web application, the login screen. #### 2.4.2 Database Server The database server is reached using the JDBC API with the official PostgreSQL JDBC driver. #### 2.4.3 LDAP Server The LDAP server is reached using the default Sun LDAP JNDI driver. The LDAP protocol is used for the queries and is encapsulated by JNDI. #### 2.4.4 Certificate Authority This component is not integrated with others and, as such, does not have a software interface to document. #### 2.4.5 Logging Engine On the Web server, this component is called automatically by the use of the default output and default error streams, which will write the information to a log instead of a console. ### 2.5 Hardware Environment The SFS system is expected to work in a networked environment, possibly a LAN, but not necessarily. We assume that the principals have a network connection allowing to communicate with each other. Only typical low-end workstation hardware (such as a Pentium III system with 256+ MB of RAM, 2GB hard drive with a 10BaseT Ethernet connection) is required to operate all the components of the system, which may be distributed or centralized as needed. Ordinary P III with 256+ MB of Ram and 2GB HDD are the minimum requirements needed to deploy the system. ### 2.6 Code View We decided to structure our software in a few main packages, as illustrated in Figure 2.2. Figure 2.2: Java Packages Those packages hold as follows: 1. 1. _securefileserver:_ root of our custom code 2. 2. _base64:_ Library for encoding and decoding Base64-encoded data 3. 3. _apps:_ Various applications 4. 4. _webapps:_ The web application code 5. 5. _conn:_ Connection abstraction code, such as SSL connections, LDAP connections and database connections 6. 6. _cert:_ Certificate authority code 7. 7. _util:_ Utility classes, such as the configuration file loader Since we had to deal with a large set of non-code artifacts, we structured our repository as illustrated in Figure 2.3.s Figure 2.3: Repository Structure Those directories hold as follows: 1. 1. root directory: contains the Ant makefile and all the other subdirectories 2. 2. _cert:_ certificates generated manually using OpenSSL 3. 3. _conf:_ configuration files 4. 4. _CVS:_ repository management code, handled automatically by CVS 5. 5. _images:_ documentation-related images 6. 6. _lib:_ libraries in JAR format 7. 7. _sql:_ database initialization script 8. 8. _src:_ Java source code 9. 9. _tex:_ documentation in LaTeX2e format 10. 10. _txt:_ various notes in text format 11. 11. _design:_ Rational Rose model of the system ## Chapter 3 Detailed System Design In this section of the specification document we elaborate the detailed description of the main modules and subprograms of the SFS system. We provide the important class diagrams for the different packages of the design phase as mentioned above. Please consult Figure 3.1 for a high-level view of the class diagram of our application. Please note that the servlets login and User also have a fair amount of business logic integrated in them. This situation is due to the evolutionary nature of the development method used in this project which, combined with tight deadlines, did not allow for a proper refactoring of the classes. We can also take note the presence of test classes in our class diagram, which are JUnit test cases that allowed to perform some unit testing. The smallness of the class diagram is mostly explained by the fact that most of the functionality was implemented in COTS programs that needed only some configuration. ### 3.1 Class Diagrams The following diagrams show some of the important user interfaces of the SFS software system. Figure 3.1: SFS main class diagram Figure 3.2: Conn Package Diagram Figure 3.3: Application Package Diagram Figure 3.4: The util Package Diagram Figure 3.5: The cert Package Diagram ### 3.2 User Interface The following diagrams show some of the important user interfaces of the SFS software system. On figure 3.6, we see the interface allowing clients to log in. Figure 3.6: User interface Log on Once the client mutual authentication is achieved and user allowed to use the system he will get the following screen (3.7). Figure 3.7: User operations displayed When the client chooses the file to download, he will be prompted to open the file or give the path he want to save the file in. Figure 3.8: User operations displayed Figure 3.9 shows the upload operation, so the user will be prompted to select the file he want to upload. Figure 3.9: User operations displayed Figure 3.10 shows the administrator capabilities: adding users, remove users, adding groups, setting rights, etc. Figure 3.10: User operations displayed The snapshot of the SFS provides the login interface via which the services of the SFS system can not be utilized unless the user is already logged in. ### 3.3 Class Diagrams Figure 3.1 includes most of the classes already present. We will describe a few classes in detail here. All the details regarding the classes are available in the Javadoc. #### 3.3.1 LDAPConnection This class provides an abstraction of an LDAP JNDI context, as well as pre- made queries for obtaining user credentials, deleting an user and adding an user in the database. It depends on SSLConnectionFactory for ensuring that our SSL Connections are set with the proper keys. It also depends on UserCredentials, since this is the data type it returns on a query. #### 3.3.2 UserCredentials This class encapsulates a user name, a password, and a certificate. It integrates the hashing of plaintext passwords, as well as a matching comparison between two sets of credentials. It depends on TestSSHA for generating and validating the salted SHA1 hash. #### 3.3.3 OptionsFileLoaderSingleton This class is a singleton, meaning that only up to one instance can exist at any time. It loads and parses a configuration file. It also includes many default keys of the configuration file as constant strings. #### 3.3.4 DatabaseConnection This class provides an abstraction for an SSL-enabled database connection. Contrary to LDAPConnection, it does not provide high-level methods in it, leaving to the calling code the responsibility to formulate proper SQL queries. ### 3.4 Data Storage Format In this section, we provide a description for the database handling the security aspect of the system. It consists of the following entity relation model. #### 3.4.1 Entity Relationship Diagram The Groupe entity contains the list of groups a user may belong to. The User entity contains the list of users having right to use the system. The File entity contains information about different files a user can upload, download, delete and view. A user may be an Administrator or normal user. The other relationships (group_user, group_files) are defined between entities Groupe and User and Groupe and File to host different information related to both of them respectively. Hereafter, we provide in Figure 3.11 the Entity Relationship Model of the Security Database. Figure 3.11: Entity Relationship Diagram ### 3.5 Options File A .config file is expected in the execution root in order to read the configuration options. Lines can be comments (#), blank, or containing a key=value pair. The expected configuration options are: * • _ca_server:_ host name of the CA server. This option is reserved for future use. * • _db_server:_ host name of the database server. * • _keystore_filepath:_ relative or absolute path for the web server’s keys * • _keystore_password:_ password for the previously specified keystore * • _ca_certificate_filepath:_ path to the CA certificate’s keystore * • _ca_certificate_password:_ password for the CA certificate * • _ldap_password:_ administrator password for the LDAP server * • _ldap_principal:_ administrator user name for the LDAP server * • _ldap_server:_ host name of the LDAP server ### 3.6 Directory Configuration The LDAP directory is to be structured in an abritarily manner, as the DN is not used in queries. However, the UID parameter is used for querying based on the user name. The user information is of type inetOrgPerson, with the fields uid, userPassword and userCertificate for, respectively, the user name, its password (hashed with salted SHA) and its certificate. ### 3.7 External System Interfaces The only externally reachable interface to the SFS system is the login page of the web application. This page should be located at /sfs/login.jsp. It is also symlinked to it via index.jsp. #### 3.7.1 External Systems and Databases The SFS system is not designed for interacting with any other systems than those described as part of our architecture. ### 3.8 User Scenarios #### 3.8.1 Typical Scenario 1. 1. User connects to remote server using a web browser on a secure connection 2. 2. User is prompted with a logon screen, and provides a username and password 3. 3. After validation, the user is logged as a normal user and is shown a list of files to which he or she has access to, as well as their rights 4. 4. The user clicks to download a file and, if the system validates its rights, the download begins #### 3.8.2 Variant scenario 4\. The user chooses to upload a file and the upload begins. The file will be modifyable by the user and according to the default new file ACL. #### 3.8.3 Variant scenario 4\. The user chooses to delete a file to which it has delete rights and the systems perform the deletion ### 3.9 Administrator Scenarios #### 3.9.1 Typical Scenario 1. 1. User connects to remote server using a web browser on a secure connection 2. 2. User is prompted with a logon screen, and provides a username and password 3. 3. After validation, the user is logged as an administrator and is shown a menu of options, and chooses to view a list of files in the system 4. 4. The user clicks on the user edit button and changes the access control list of the object. #### 3.9.2 Variant Scenario 1. 1. User connects to the certificate administration service and generate a certificate for a given subject through a secure connection 2. 2. User connects to remote server on a secure connection 3. 3. User is prompted with a logon screen, and provices username and password 4. 4. After validation, the user is logged as an administrator and is shown the menu 5. 5. The user clicks on the certificate edit button and is shown a screen of certificate maintenance 6. 6. The user uploads the certificate and the system binds it with the certificate’s defined principal (or creates the user if none exists already) #### 3.9.3 Variant Scenario 1. 1. User connects to remote server using a web browser on a secure connection 2. 2. User is prompted with a logon screen, and provides a username and password 3. 3. After validation, the user is logged as an administrator and is shown a menu of options, and chooses to edit the certificates 4. 4. The user chooses to remove the certificate from a given principal 5. 5. The user connects to the certificate administration service and issues a certificate revocation. ## Bibliography * [Apa11] Apache Foundation. Apache Jakarta Tomcat. [online], 1999–2011. http://jakarta.apache.org/tomcat/index.html. * [CBH03] Suranjan Choudhury, Katrik Bhatnagar, and Wisam Haque. Public Key Infrastructure, Implementation and Design. NIIT Books, 2003. * [E+11] Eclipse contributors et al. Eclipse Platform. eclipse.org, 2000–2011. http://www.eclipse.org, last viewed February 2010. * [GB11] Erich Gamma and Kent Beck. JUnit. [online], Object Mentor, Inc., 2001–2011. http://junit.org/. * [Lar06] Craig Larman. Applying UML and Patterns: An Introduction to Object-Oriented Analysis and Design and Iterative Development. Pearson Education, third edition, April 2006. ISBN: 0131489062. * [Leu05] Leuvens Universitair. A few frequently used SSL commands. [online], Leuvens Universitair Dienstencentrum voor Informatica en Telematica, 2005. http://shib.kuleuven.be/docs/ssl_commands.shtml. * [Ope05a] OpenLDAP Community. OpenLDAP: The open source for LDAP software and information. [online], 2005. www.openldap.org. * [Ope05b] OpenSSL Community. OpenSSL: The open source toolkit for SSL/TLS. [online], 2005. http://www.openssl.org. * [O’R05] O’Reilly. Home of com.oreilly.servlet. [online], 2005. http://servlets.com/cos. * [Sun05a] Sun Microsystems, Inc. Java servlet technology. [online], 1994–2005. http://java.sun.com/products/servlets. * [Sun05b] Sun Microsystems, Inc. JavaServer pages technology. [online], 2001–2005. http://java.sun.com/products/jsp/. * [The11] The PostgreSQL Global Development Group. PostgreSQL – the world’s most advanced open-source database. [ditigal], 1996–2011. http://www.postgresql.org/, last viewed January 2010. ## Index * Introduction Chapter 1
arxiv-papers
2011-01-24T20:30:17
2024-09-04T02:49:16.622345
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Serguei A. Mokhov, Marc-Andr\\'e Laverdi\\`ere, Ali Benssam, Djamel\n Benredjem", "submitter": "Serguei Mokhov", "url": "https://arxiv.org/abs/1101.4640" }
1101.4891
# The effect of dust cooling on low-metallicity star-forming clouds Gustavo Dopcke, Simon C. O. Glover, Paul C. Clark and Ralf S. Klessen Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany ###### Abstract The theory for the formation of the first population of stars (Pop III) predicts a IMF composed predominantly of high-mass stars, in contrast to the present-day IMF, which tends to yield stars with masses less than 1 M⊙. The leading theory for the transition in the characteristic stellar mass predicts that the cause is the extra cooling provided by increasing metallicity and in particular the cooling provided at high densities by dust. The aim of this work is to test whether dust cooling can lead to fragmentation and be responsible for this transition. To investigate this, we make use of high- resolution hydrodynamic simulations. We follow the thermodynamic evolution of the gas by solving the full thermal energy equation, and also track the evolution of the dust temperature and the chemical evolution of the gas. We model clouds with different metallicities, and determine the properties of the cloud at the point at which it undergoes gravitational fragmentation. We follow the further collapse to scales of an AU when we replace very dense, gravitationally bound, and collapsing regions by a simple and nongaseous object, a sink particle. Our results suggest that for metallicities as small as 10${}^{-5}\rm Z_{\odot}$, dust cooling produces low-mass fragments and hence can potentially enable the formation of low mass stars. We conclude that dust cooling affects the fragmentation of low-metallicity gas clouds and plays an important role in shaping the stellar IMF even at these very low metallicities. ###### Subject headings: early universe — hydrodynamics — methods: numerical — stars: formation — stars: luminosity function, mass function ## 1\. Introduction The first burst of star formation in the Universe is thought to give rise to massive stars, the so-called Population III, with current theory predicting masses in the range 20-150 M⊙ (Abel et al., 2002; Bromm et al., 2002; O’Shea & Norman, 2007; Yoshida et al., 2008). This contrasts with present-day star formation, which tends to yield stars with masses less than 1 M⊙ (Kroupa, 2002; Chabrier, 2003), and so at some point in the evolution of the Universe there must have been a transition from primordial (Pop. III) star formation to the mode of star formation we see today (Pop. II/I). When gas collapses to form stars, gravitational energy is transformed to thermal energy and unless this can be dissipated in some fashion, it will eventually halt the collapse. Thermal energy can be dissipated by processes such as atomic fine structure line emission, molecular rotational or vibrational line emission, or the heating of dust grains. In some cases, these processes are able to cool the gas significantly during the collapse. This temperature drop can promote gravitational fragmentation (Mac Low & Klessen, 2004; Bonnell et al., 2007) by diminishing the Jeans mass, which means that instead of forming very massive clumps, with fragment masses corresponding to the initial Jeans mass in the cloud, it can instead form even more fragments with lower masses. If the gas is cooled only by molecular hydrogen emission, numerical simulations show that the stars should be very massive (Abel et al., 2002; Bromm et al., 2002; O’Shea & Norman, 2007; Yoshida et al., 2008). This happens because the H2 cooling becomes inefficient for temperatures bellow 200K and densities above $10^{4}\rm cm^{-3}$. At this temperature and density, the mean Jeans mass at cloud fragmentation is 1,000 times larger than in present-day molecular clouds, $M_{\rm frag}\approx 700M_{\odot}\left(\frac{T_{\rm frag}}{200\rm K}\right)^{3/2}\left(\frac{n_{\rm frag}}{10^{4}\rm cm^{-3}}\right)^{-1/2},$ (1) for an atomic gas with temperature $T_{\rm frag}$ and number density $n_{\rm frag}$. Metal line cooling and dust cooling are effective at lower temperatures and larger densities, and so the most widely accepted cause for the transition from Pop. III to Pop. II is metal enrichment of the interstellar medium by the previous generations of stars. This suggests that there might be a critical metallicity Zcrit at which the mode of star formation changes. The main coolants that have been studied in the literature are CII and OI fine structure emission (Bromm et al., 2001; Bromm & Loeb, 2003; Santoro & Shull, 2006; Frebel et al., 2007), and dust emission. C and O are identified as the key species because in the temperature and density conditions that characterise the early phases of Pop. III star formation, the OI and CII fine- structure lines dominate over all other metal transitions (Hollenbach & McKee, 1989). By equating the CII or OI fine structure cooling rate to the compressional heating rate due to free-fall collapse, one can define critical abundances $[\rm C/H]=-3.5$ and $[\rm O/H]=-3.0$111$[\rm X/\rm Y]=log_{10}(N_{\rm X}/N_{\rm Y})_{\star}-log_{10}(N_{\rm X}/N_{\rm Y})_{\odot}$, for elements X and Y, where $\star$ denotes the gas in question, and where $\rm N_{X}$ and $\rm N_{Y}$ are the mass fractions of the elements X and Y. for efficient metal line cooling (Bromm & Loeb, 2003). However, previous works (Jappsen et al., 2009a, b) show that this metallicity threshold does not represent a critical metallicity: the fact that metal-line cooling has a larger value than the compressional heating does not necessarily lead to fragmentation. Dust-cooling models predict a much lower critical metallicity $(\rm Z_{\rm crit}\approx 10^{-5}\rm Z_{\odot})$. The conditions for fragmentation in the low-metallicity dust cooling model are predicted to occur in high density gas, where the distances between the fragments can be very small (Omukai, 2000; Omukai et al., 2005; Schneider et al., 2002, 2006; Schneider & Omukai, 2010). In this regime, interactions between fragments will be common, and analytic models of fragmentation are unable to predict the mass distribution of the fragments. A full 3D treatment, following the fragments, is needed. Initial attempts were made by Tsuribe & Omukai (2006, 2008) and Clark et al. (2008). However, these treatments used a tabulated equation of state, based on results from previous one-zone chemical models (Omukai et al., 2005), to determine the thermal energy. This approximation assumes that the gas temperature adjusts instantaneously to a new equilibrium temperature whenever the density changes and hence ignores thermal inertia effects. This may yield too much fragmentation. In this work, we improve upon these previous treatments by solving the full thermal energy equation, and calculating the dust temperature through the energy equilibrium equation. We assume currently that the only significant external heat source is the CMB, and include its effects in the calculation of the dust temperature. ## 2\. Simulations ### 2.1. Numerical method We model the collapse of a low-metallicity gas cloud using a modified version of the Gadget 2 (Springel, 2005) smoothed particle hydrodynamics (SPH) code. To enable us to continue our simulation beyond the formation of the first very high density protostellar core, we use a sink particle approach (Bate et al., 1995), based on the implementation of Jappsen et al. (2005). Sink particles are created once the SPH particles are bound, collapsing, and within an accretion radius, $h_{acc}$, which is taken to be 1.0 AU. The threshold number density for sink particle creation is $5.0\times 10^{13}\rm cm^{-3}$. At the threshold density, the Jeans length at the minimum temperature reached by the gas is approximately one AU, while at higher densities the gas becomes optically thick and begins to heat up. Further fragmentation on scales smaller than the sink particle scale is therefore unlikely to occur. For further discussion see Clark et al. (2011). To treat the chemistry and thermal balance of the gas, we use the same approach as in Clark et al. (2011), with two additions: the inclusion of the effects of dust cooling, as described below, and formation of H2 on the surface of dust grains (see Hollenbach & McKee, 1979). The Clark et al. (2011) chemical network and cooling function were designed for treating primordial gas and do not include the chemistry of metals such as carbon or oxygen, or the effects of cooling from these atoms, or molecules containing them such as CO or H2O. We justify this approximation by noting that previous studies of very low-metallicity gas (e.g. Omukai et al., 2005, 2010) find that gas-phase metals have little influence on the thermal state of the gas. Omukai et al. (2010) showed that H2O and OH are efficient coolants at $10^{8}<n<10^{10}\rm cm^{-3}$ for their one-zone model. In their hydrodynamical calculations, however, the collapse is faster, and the effect of H2O and OH is not perceptible. Therefore we do not expect oxygen-bearing molecules to have a big effect on the thermal evolution of the gas. For the metallicities and dust-to- gas ratios considered in this study, the dominant sources of cooling are the standard primordial coolants (H2 bound-bound emission and collision-induced emission) and energy transfer from the gas to the dust. Figure 1.— Results of our low-resolution simulations, showing the dependence of gas and dust temperatures on gas density for metallicities $10^{-4}$ and $10^{-5}$ times the solar value. In red, we show the gas temperature, and in blue the dust temperature for the turbulent and rotating cloud. The simple core collapse is overploted in dark red and green. The points with thinner features are from the simulations without rotation or turbulence, while those showing more scatter come from the simulations with rotation and turbulence. The dashed lines show constant Jeans mass values. #### 2.1.1 Dust cooling Collisions between gas particles and dust grains can transfer energy from the gas to the dust (if the gas temperature $T$ is greater than the dust temperature $T_{\rm gr}$), or from the dust to the gas (if $T_{\rm gr}>T$). The rate at which energy is transferred from gas to dust is given by (Hollenbach & McKee, 1979) $\Lambda_{\rm gr}=n_{\rm gr}n\bar{\sigma}_{\rm gr}v_{\rm p}f(2kT-2kT_{\rm gr})\>\>{\rm erg}\>{\rm s^{-1}}\>{\rm cm^{-3}},$ (2) where $n_{\rm gr}$ is the number density of dust grains, $n$ is the number density of hydrogen nuclei, $\bar{\sigma}_{\rm gr}$ is the mean dust grain cross-section, $v_{\rm p}$ is the thermal speed of the proton, and $f$ is a factor accounting for the ontribution of species other than protons, as well as for charge and accommodation effects. We assume that $\bar{\sigma}_{\rm gr}$ is the same as for Milky Way dust, and that the number density of dust grains is a factor ${\rm Z}/{\rm Z_{\odot}}$ smaller than the Milky Way value. To compute the rate at which the dust grains radiate away energy, we use the approximation (Stamatellos et al., 2007) $\Lambda_{\rm rad}=4\sigma_{\rm sb}n_{\rm gr}\frac{(T_{\rm gr}^{4}-T_{\rm cmb}^{4})}{\Sigma^{2}\kappa_{R}+\kappa_{P}^{-1}},$ (3) where $T_{\rm cmb}$ is the CMB temperature, $\sigma_{\rm sb}$ is the Stefan- Boltzmann constant, $\kappa_{P}$ and $\kappa_{R}$ are the Planck and Rosseland mean opacities and $\Sigma$ is the column density of gas measured along a radial ray from the particle to the edge of the cloud. As explained by Stamatellos et al. (2007), this expression has the correct behaviour in the optically thin and optically thick limits, and interpolates between these two limits in a smooth fashion. In practice, we approximate further by assuming that the Planck and Rosseland mean opacities are equaland by using the fact that $\Sigma\sim\rho L_{\rm J}$ for a gravitationally collapsing gas, where $\rho$ is the mass density of the gas, and $L_{\rm J}$ is the Jeans length, given by $L_{\rm J}=(\pi c_{s}^{2}/G\rho)^{1/2}$, where $c_{s}$ is the speed of sound in the gas. By approximating $\Sigma$ in this fashion, we avoid the computational difficulties involved with measuring column densities directly in the simulation, while still following the behaviour of the gas reasonably accurately in the optically thick regime. In any case, most of the interesting behaviour that we find in our simulations occurs while dust cooling remains in the optically thin regime. To compute the temperature of the dust grains, we assume that the dust is in thermal equilibrium, and hence solve the equilibrium equation $\Lambda_{\rm gr}-\Lambda_{\rm rad}=0.$ (4) This equation is transcendental, so we solve it numerically. #### 2.1.2 Dust opacity We follow the dust opacity model of Goldsmith (2001), and we calculate the opacity as a function of the dust temperature in the same fashion as in Banerjee et al. (2006). To convert from the frequency-dependent opacity given in Goldsmith (2001) to our desired temperature-dependent mean opacity, we assume that for dust with temperature $T_{\rm gr}$, the dominant contribution to the mean opacity comes from frequencies close to a frequency $\bar{\nu}$ that is given by $h\bar{\nu}=\alpha kT_{\rm gr}$, where $\alpha=2.70$. At a reference temperature $T_{0}=6.75$ K, this procedure yields an opacity $\begin{array}[]{rl}\kappa(T_{0})=&3.3\times 10^{-26}\alpha(\rm n/2\rho_{\rm gas})\\\ =&2.664\times 10^{-2}/(1+4[\rm He])\end{array}$ (5) where [He] is the helium abundance, and $n$ is the number density of hydrogen nuclei. At other temperatures, $\kappa\propto T_{\rm gr}^{2}$, so long as $T_{\rm gr}<200$ K. For grain temperatures larger than 200 K, it is necessary to account for the effects of ice-mantle evaporation, while at much higher grain temperatures, the opacity falls off extremely rapidly due to the melting of the grains. We account for these effects (see Semenov et al., 2003) and so our opacity varies with dust temperature following the relationship $\kappa=\kappa(T_{0})\times\left\\{\begin{array}[]{ll}T^{2}&\hskip 36.135pt\mbox{T $<$ 200K}\\\ T^{0}&\hskip 36.135pt\mbox{200K $<$ T $<$ 1500K}\\\ T^{-12}&\hskip 36.135pt\mbox{T $>$ 1500K}\end{array}\right.$ (6) ### 2.2. Setup and Initial conditions Resolution | Number of | Particle | Turbulence | Angular ---|---|---|---|--- Level | Particles | Mass | | Momentum | | ($10^{-5}\rm M_{\odot}$) | ($E_{\rm turb}/|E_{\rm grav}|$) | ($E_{\rm rot}/|E_{\rm grav}|$) High | $40\times 10^{6}$ | $2.5$ | 0.1 | 0.02 Low | $4\times 10^{6}$ | $25.0$ | 0.1 | 0.02 | | | 0.0 | 0.00 Table 1Simulation properties. We performed three sets of simulations, two at low resolution and one at high resolution. The details are shown in Table 1. Our low resolution simulations were performed to explore the thermal evolution of the gas during the collapse, and had 4 million SPH particles which was insufficient to fully resolve fragmentation. We used these simulations to model the collapse of an initially uniform gas cloud with an initial number density of $10^{5}\>{\rm cm^{-3}}$ and an initial temperature of $300\>{\rm K}$. We modelled two different metallicities (10${}^{-4}{\rm Z}_{\odot}$ and 10${}^{-5}{\rm Z}_{\odot}$). The initial cloud mass was $1000\>{\rm M_{\odot}}$, and the mass resolution was $25\times 10^{-3}\>{\rm M_{\odot}}$. In one set of low- resolution simulations the gas was initially at rest, while in the other, we included small amounts of turbulent and rotational energy, with $E_{\rm turb}/|E_{\rm grav}|=0.1$ and $\beta=E_{\rm rot}/|E_{\rm grav}|=0.02$, where $E_{\rm grav}$ is the gravitational potential energy, $E_{\rm turb}$ is the turbulent kinetic energy and $E_{\rm rot}$ is the rotational energy. For our high resolution simulations, which were designed to investigate whether the gas would fragment, we employed 40 million SPH particles. We adopted initial conditions similar to those in the low-resolution run with turbulence and rotation. Again, we simulated two metallicities, 10${}^{-4}{\rm Z}_{\odot}$ and 10${}^{-5}{\rm Z}_{\odot}$. The mass resolution (taken to be 100 times the SPH particle mass) was $2.5\times 10^{-3}{\rm M}_{\odot}$. ## 3\. Analysis ### 3.1. Thermodynamical evolution of gas and dust Figure 2.— Number density maps for a slice through the high density region. The image shows a sequence of zooms in the density structure in the gas immediately before the formation of the first protostar. In Figure 1, we compare the evolution of the dust and gas temperatures in the low-resolution simulations. The dust temperature, shown in the lower part of the panels, varies from the CMB temperature in the low density region to the gas temperature at much higher densities. At densities higher than $10^{11}$–$10^{12}\>\rm cm^{-3}$, dust cooling starts to be effective and begins to cool the gas. The gas temperature decreases to roughly 600 K in the $10^{-5}\>\rm Z_{\odot}$ simulations, and 300 K in the Z $=10^{-4}\rm Z_{\odot}$ case. This temperature decrease significantly increases the number of Jeans masses present in the collapsing region, making the gas unstable to fragmentation. The dust and the gas temperatures couple for densities higher then $10^{13}\rm cm^{-3}$, when the compressional heating starts to dominate again over the dust cooling. The subsequent evolution of the gas is close to adiabatic. If we compare the results of the runs with and without rotation and turbulence, then the most obvious difference is the much greater scatter in the $n-T$ diagram in the former case. Variations in the infall velocity lead to different fluid elements undergoing different amounts of compressional heating. The overall effect is to reduce both the infall velocity and the average compressional heating rate. This allows dust cooling to dominate at a density that is up to five times smaller than in the case without rotation or turbulence. The gas also reaches a lower temperature, cooling down to $\approx$ 200K (instead of 300K) for the Z $=10^{-4}\rm Z_{\odot}$ case, and to $\approx$ 400K (instead of 600K) for the Z $=10^{-5}\rm Z_{\odot}$ case. This behavior shows that it is essential to use 3D simulations to follow the evolution of the collapsing gas. A similar effect can be seen in Clark et al. (2011). ### 3.2. Fragmentation We follow the thermodynamical evolution of the gas up to very high densities of order $10^{17}\rm cm^{-3}$, where the Jeans mass is $\approx 10^{-2}\rm M_{\odot}$, and so we need a high resolution simulation to study the fragmentation behaviour. The transport of angular momentum to smaller scales during the collapse leads to the formation of a dense disk-like structure, supported by rotation which then fragments into several objects. Figure 2 shows the density structure in the gas immediately before the formation of the first protostar. The top-left panel shows a density slice on a scale comparable to the size of the initial gas distribution. The structure is very filamentary and there are two main overdense clumps in the center. If we zoom in on one of the clumps, we see that its internal structure is also filamentary. We can follow the collapse down to scales of the order of an AU, but at this point we reach the limit of our computational approach: as the gas collapses further, the Courant timestep becomes very small, making it difficult to follow the further evolution of the cloud. In order to avoid this difficulty, we replace very dense, gravitationally bound, and collapsing regions by sink particles. Once the conditions for sink particle creation are met, they start to form in the highest density regions (Figure 3). Due to interactions with other sink particles that result in an increase in velocity, some sink particles can be ejected from the high-density region, but most of the particles still remain within the dense gas. Within 137 years of the formation of the first sink particle, 45 sink particles have formed. At this time, approximately $4.6\rm M_{\odot}$ of gas has been accreted by the sink particles. Figure 3.— Number density map showing a slice in the densest clump, and the sink formation time evolution, for the 40 million particles simulation, and Z = 10-4Z⊙. The box is 100AU x 100AU and the time is measured from the formation of the first sink particle. ### 3.3. Properties of the fragments Figure 4 shows the mass distribution of sink particles when we stop the calculation. We typically find masses below $1\rm M_{\odot}$, with somewhat smaller values in the $10^{-4}\rm Z_{\odot}$ case compared to the $10^{-5}\rm Z_{\odot}$ case. Both histograms have the lowest sink particle mass well above the resolution limit of $0.0025M_{\odot}$. Note that in both cases, we are still looking at the very early stages of star cluster evolution. As a consequence, the sink particle masses in Figure 4 are not the same as the final protostellar masses – there are many mechanisms that will affect the mass function, such as continuing accretion, mergers between the newly formed protostars, feedback from winds, jets and luminosity accretion, etc. Nevertheless, we can speculate that the typical stellar mass is similar to what is observed for Pop II stars in the Milky Way. This suggests that the transition from high-mass primordial stars to Population II stars with mass function similar to that at the present day occurs early in the metal evolution history of the universe, at metallicities $\rm Z_{crit}<10^{-5}Z_{\odot}$. The number of protostars formed by the end of the simulation, for both $10^{-4}Z_{\odot}$ (45) and $10^{-5}Z_{\odot}$ (19) cases, is much larger than the initial number of Jeans masses (3) in the cloud. Figure 4.— Sink particle mass function at the end of the simulations. High and low resolution results and corresponding resolution limits are shown. To resolve the fragmentation, the mass resolution should be smaller than the Jeans mass at the point in the temperature-density diagram where dust and gas couple and the compressional heating starts to dominate over the dust cooling. At the time shown, around 5 M⊙ of gas had been accreted by the sink particles in each simulation. ## 4\. Conclusions In this paper we have addressed the question of whether dust cooling can lead to the fragmentation of low-metallicity star-forming clouds. For this purpose we performed numerical simulations to follow the thermodynamical and chemical evolution of collapsing clouds. The chemical model included a primordial chemical network together with a description of dust evolution, where the dust temperature was calculated by solving self-consistently the thermal energy equilibrium equation. We performed three sets of simulations, two at low resolution and one at high resolution (Table 1). All simulations had an initial cloud mass of 1000 M⊙, number density of 105 cm-3, and temperature of 300K. We tested two different metallicities (10${}^{-4}{\rm Z}_{\odot}$ and 10${}^{-5}{\rm Z}_{\odot}$), and also the inclusion of small amounts of turbulent and rotational energies. We found in all simulations that dust can effectively cool the gas, for number densities higher than $10^{11}\rm cm^{-3}$. An increase in metallicity implies a higher dust-to-gas ratio, and consequently stronger cooling by dust. This is reflected in a lower temperature of the dense gas in the higher metallicity simulation. For the low resolution case, we tested the effect of adding turbulence and rotation. These diminish the infall velocity, leading to different fluid elements undergoing different amounts of compressional heating. This lack of heating allows the gas to reach a lower temperature. We found that the transport of angular momentum to smaller scales lead to the formation of a disk-like structure, which then fragmented into a number of low mass objects. We conclude that the dust is already an efficient coolant even at metallicities as low as 10-5 or 10${}^{-4}Z_{\odot}$, in agreement with previous works (Clark et al., 2008; Omukai et al., 2010; Schneider et al., 2002, 2006; Tsuribe & Omukai, 2006, 2008). Our results support the idea that dust cooling can play an important role in the fragmentation of molecular clouds and the evolution of the stellar IMF. We thank Tom Abel, Volker Brom, Kazuyuki Omukai, Raffaella Schneider, Rowan Smith, and Naoki Yoshida for useful comments. The present work is supported by the _Landesstiftung Baden Württemberg_ via their program International Collaboration II (grant P-LS-SPII/18), the German _Bundesministerium für Bildung und Forschung_ via the ASTRONET project STAR FORMAT (grant 05A09VHA), a Frontier grant of Heidelberg University sponsored by the German Excellence Initiative, the International Max Planck Research School for Astronomy and Cosmic Physics at the University of Heidelberg (IMPRS-HD). All computations described here were performed at the _Leibniz-Rechenzentrum_ , National Supercomputer HLRB-II (_Bayerische Akademie der Wissenschaften_), and on the HPC-GPU Cluster Kolob (University of Heidelberg) . ## References * Abel et al. (2002) Abel, T., Bryan, G. L., & Norman, M. L. 2002, Science, 295, 93 * Banerjee et al. (2006) Banerjee, R., Pudritz, R. E., & Anderson, D. W. 2006, MNRAS, 373, 1091 * Bate et al. (1995) Bate, M. R., Bonnell, I. A., & Price, N. M. 1995, MNRAS, 277, 362 * Bonnell et al. (2007) Bonnell, I. 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S., Larson, R. B., Li, Y., & Mac Low, M. 2005, A&A, 435, 611 * Jappsen et al. (2009b) Jappsen, A., Mac Low, M., Glover, S. C. O., Klessen, R. S., & Kitsionas, S. 2009b, ApJ, 694, 1161 * Kroupa (2002) Kroupa, P. 2002, Science, 295, 82 * Mac Low & Klessen (2004) Mac Low, M., & Klessen, R. S. 2004, Reviews of Modern Physics, 76, 125 * Omukai (2000) Omukai, K. 2000, ApJ, 534, 809 * Omukai et al. (2010) Omukai, K., Hosokawa, T., & Yoshida, N. 2010, ApJ, 722, 1793 * Omukai et al. (2005) Omukai, K., Tsuribe, T., Schneider, R., & Ferrara, A. 2005, ApJ, 626, 627 * O’Shea & Norman (2007) O’Shea, B. W., & Norman, M. L. 2007, ApJ, 654, 66 * Santoro & Shull (2006) Santoro, F., & Shull, J. M. 2006, ApJ, 643, 26 * Schneider et al. (2002) Schneider, R., Ferrara, A., Natarajan, P., & Omukai, K. 2002, ApJ, 571, 30 * Schneider & Omukai (2010) Schneider, R., & Omukai, K. 2010, MNRAS, 402, 429 * Schneider et al. (2006) Schneider, R., Omukai, K., Inoue, A. K., & Ferrara, A. 2006, MNRAS, 369, 1437 * Semenov et al. (2003) Semenov, D., Henning, T., Helling, C., Ilgner, M., & Sedlmayr, E. 2003, A&A, 410, 611 * Springel (2005) Springel, V. 2005, MNRAS, 364, 1105 * Stamatellos et al. (2007) Stamatellos, D., Whitworth, A. P., Bisbas, T., & Goodwin, S. 2007, A&A, 475, 37 * Tsuribe & Omukai (2006) Tsuribe, T., & Omukai, K. 2006, ApJ, 642, L61 * Tsuribe & Omukai (2008) —. 2008, ApJ, 676, L45 * Yoshida et al. (2008) Yoshida, N., Omukai, K., & Hernquist, L. 2008, Science, 321, 669
arxiv-papers
2011-01-25T18:32:29
2024-09-04T02:49:16.632254
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Gustavo Dopcke, Simon C. O. Glover, Paul C. Clark and Ralf S. Klessen", "submitter": "Gustavo Dopcke", "url": "https://arxiv.org/abs/1101.4891" }
1101.4915
National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A. ∗Corresponding author: gnave@nist.gov # Wavelengths of the $\rm\bf 3d^{6}(^{5}D)4s\,a^{6}D-3d^{6}\,(^{5}D)4p\,y^{6}P$ Multiplet of Fe II (UV 8) Gillian Nave∗, Craig J. Sansonetti ###### Abstract We investigate the wavenumber scale of Fe I and Fe II lines using new spectra recorded with Fourier transform spectroscopy and using a re-analysis of archival spectra. We find that standards in Ar II, Mg I, Mg II and Ge I give a consistent wavenumber calibration. We use the recalibrated spectra to derive accurate wavelengths for the a6D-y6P multiplet of Fe II (UV 8) using both directly measured lines and Ritz wavelengths. Lines from this multiplet are important for astronomical tests of the invariance of the fine structure constant on a cosmological time scale. We recommend a wavelength of 1608.45081 Å with a one standard deviation uncertainty of 0.00007 Å for the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ transition. 300.621, 300.6300, 300.6540 ## 1 Introduction The universality and constancy of the laws of nature rely on the invariance of the fundamental constants. However, some recent measurements of quasar (Quasi- stellar objects - QSO) absorption line spectra suggest that the fine-structure constant, $\alpha$, [1] may have had a different value during the early universe [2]. Other measurements (e.g. [3]) do not show any change. The attempt to resolve these discrepancies can probe deviations from the standard model of particle physics and thus provide tests of modern theories of fundamental interactions that are hard to attain in other ways. QSO absorption lines are used in these investigations to measure the wavelength separations of atomic lines in spectra of different elements - the many-multiplet method [4] \- and compare their values at large redshifts with their values today. Any difference in the separations would suggest a change in $\alpha$. Since this method uses many different species in the analysis that have differing sensitivities to changes in $\alpha$, it can be much more sensitive than previous methods, such as the alkali-doublet method [5], that use just one species. However, it requires very accurate laboratory wavelengths to be used successfully, since the observed changes in $\alpha$ are only a few parts in 105, requiring laboratory wavelengths to 1:107 or better. This has led to several recent measurements of ultraviolet wavelengths using both Fourier transform (FT) spectroscopy [6, 7, 8] and frequency comb metrology [9, 10, 11]. One spectral line of particular interest is the $\rm 3d^{6}(^{5}D)4s\,a^{6}D_{9/2}-3d^{5}\,(^{6}S)4s4p(^{3}P)\,y^{6}P_{7/2}$ line of Fe II at 1608.45 Å. This line is prominent in many QSO spectra and its variation with $\alpha$ has the opposite sign from that of other nearby lines [12]. However, measurement of its wavelength using frequency comb metrology, which is at present the most accurate method, is extremely difficult due to its short wavelength. Although this line is strong in many of the FT spectra of iron-neon hollow cathode lamps recorded at the National Institute of Standards and Technology (NIST) and Imperial College, London, UK (IC), these spectra display inconsistencies in the wavelength of the 1608 Å line of around 1.5 parts in 107 \- too great for use in the many-multiplet method to detect changes in $\alpha$. We discussed some of these discrepancies in our previous paper [13] presenting reference wavelengths in the spectra of iron, germanium and platinum around 1935 Å. Here we present a re-analysis of spectra taken at NIST and IC in order to resolve these discrepancies and provide a better value for the wavelength of the 1608.45 Å line of Fe II. The papers involved in this re-analysis are listed in table 1, together with the proposed corrections to the wavenumber scale. The proposed corrections are up to three times the previous total uncertainty, depending on the wavenumber. In section 2 we discuss previous measurements of the a6D - y6P multiplet. Section 3 describes the archival data we use to obtain improved wavelengths for this multiplet Additional spectra taken at NIST in order to re-evaluate the calibration of these archival data are described in section 4. The accuracy of this calibration in the visible and ultraviolet wavelength regions is also discussed in section 4. Section 5 describes three different methods for obtaining the wavelengths of the a6D - y6P multiplet. The first method uses intermediate levels determined using strong Fe II lines in the visible and ultraviolet in order to obtain the values of the y6P levels and Ritz wavenumbers for the a6D - y6P multiplet. The second method uses energy levels optimized by using a large number of spectral lines to derive Ritz wavenumbers for this multiplet. Although better accuracy is achieved using this method than the first method because of the increased redundancy, the way in which the y6P levels are determined is less transparent. The third method uses experimental wavelengths determined in spectra that are recalibrated from spectra in which we have re-evaluated the wavenumber calibration. In section 6 we re-examine the Fe II wavenumbers in our previous paper [13]. All uncertainties in this paper are reported at the one standard deviation level. ## 2 Previous measurements of the a6D - y6P multiplet The region of the a6D - y6P multiplet is shown in figure 1 as observed in a FT spectrum taken at IC. Nave, Johansson & Thorne [14] report Ritz and experimental wavelengths for six of the nine lines of the a6D - y6P multiplet. The Ritz wavelengths are based on energy levels optimized to spectral lines covering wavelengths from 1500 Å to 5.5 $\rm\mu$m measured with FT spectroscopy. The estimated uncertainties are about 2x10-4 Å or about 1.2 parts in 107. The published lines do not include the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ line. Johansson [15] contains Ritz wavelengths for all nine lines, based on unpublished interferometric measurements of Norlén, with a value of 1608.451 Å for this line. The estimated uncertainty of the $\rm y^{6}P_{7/2}$ level with respect to the ground state $\rm a^{6}D_{9/2}$ of Fe II is 0.02 cm-1. The uncertainty of the 1608 Å line can be derived directly from the uncertainty of the $\rm y^{6}P_{7/2}$ level, and corresponds to a wavelength uncertainty at 1608 Å of 0.0005 Å. Wavelengths for all nine lines measured using FT spectroscopy are also given in Pickering et al. [16] in a paper devoted to oscillator strength measurements. No details of the calibration of these lines or their uncertainties are given. The wavelength value recommended by Murphy et al. [12] is 1608.45080$\pm$0.00008 Å, with a reference to Pickering et al. However, this is not the value given in Pickering et al. and the small uncertainty is improbable without additional confirmation. The source of this wavelength is unclear. In addition to these published values, lines from this multiplet are present in some unpublished archival spectra from IC and NIST. The most important spectra for the current work are summarized in Table 2. The spectra on which Nave, Johansson & Thorne [14] is based are part of a much larger set of Fe II spectra covering all wavelengths from 900 Å to 5.5 $\rm\mu$m. Two of these spectra cover the region around 1600 Å and contain all nine lines of the multiplet. The wavelength standards for these spectra are traceable to a set of Ar II lines between 3729 Å and 5146 Å (see section 4 for details). The weighted average wavelength for the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ line in these unpublished archival spectra is 1608.45075$\pm$0.00018 Å. The spectra in Nave & Sansonetti [13] were calibrated with respect to Ge standards of Kaufman & Andrew [17]. In addition to the spectra used in that paper, we recorded a spectrum using FT spectroscopy with a pure iron cathode that covers the wavelength region of the $\rm a^{6}D-y^{6}P$ multiplet (fe1115 in Table 2). It was calibrated with iron lines measured in one of the spectra used for ref. [13] (lp0301 in Table 2). The resulting value for the wavelength of the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ line was 1608.45050$\pm$0.00004 Å, 1.5x10-7 times smaller than the wavelength obtained from the archival spectra and outside their joint uncertainty. This inconsistency is also larger than the uncertainty required for measurements of possible changes in $\alpha$. ## 3 Summary of current experimental data The spectra we re-analyzed are the same as those used in previous studies of Fe I and Fe II [18, 19, 14, 13]. Three different spectrometers were used: the f/60 IR-visible-UV FT spectrometer at the National Solar Observatory, Kitt Peak, AZ (NSO), the f/25 vacuum ultraviolet (VUV) FT spectrometer at IC [20], and the f/25 VUV spectrometer at NIST [21]. The light sources for all of the spectra were high-current hollow cathode lamps containing a cathode of pure iron run in either neon or argon. Gas pressures of 100 Pa to 500 Pa (0.8 Torr to 4 Torr) were used with currents from 0.32 A to 1 A. The total number of FT spectra was 31, covering wavelengths from about 1500 Å to 5 $\rm\mu$m (2000 cm-1 to 66000 cm-1). The wavenumber, intensity and width for all the lines were obtained with Brault’s decomp program [22] or its modification xgremlin [23]. Further details of the experiments can be found in [18, 19, 14, 13]. Additional spectra were taken using the NIST 2-m FT spectrometer and are described in section 4.1. ## 4 Calibration of FT spectra All of the spectra were calibrated assuming a linear FT wavenumber scale, so that in principle only one reference line is required to put the measurements on an absolute scale. In practice, many lines are used. To obtain the absolute wavenumbers, a multiplicative correction factor, k${}_{\mbox{eff}}$, is derived from the reference lines and applied to each observed wavenumber $\sigma_{\mbox{obs}}$ so that $\sigma_{\mbox{corr}}=(1+k_{\mbox{eff}})\sigma_{\mbox{obs}}$ (1) where $\sigma_{\mbox{corr}}$ is the corrected wavenumber. All the spectra in Learner & Thorne (3830 Å to 5760 Å) [18] and Nave et al. (1830 Å to 3850 Å) [19] trace their calibration to 28 Ar II lines in the visible. The original calibration in Refs. [18] and [19] used the wavenumbers of Norlén [24] for these lines. Norlén calibrated these Ar II lines with respect to 86Kr I lines emitted from an electrodeless microwave discharge lamp that had in turn been calibrated with respect to an Engelhard lamp, which was the prescribed source for the primary wavelength standard at the time of his measurements. The estimated standard uncertainty of Norlén’s Ar II wavenumbers varies from 0.0007 cm-1 at 19429 cm-1 to 0.001 cm-1 at 22992 cm-1. The Ar II lines were used to calibrate a ‘master spectrum’ (spectrum k19 in Table 2). Additional spectra of both Fe-Ne and Fe-Ar hollow cathode lamps covering wavelengths from 2778 Å to 7387 Å were calibrated from this master spectrum. The ultraviolet spectra reported in [19] were calibrated with respect to the results of Learner & Thorne [18] by using a bridging spectrum. This bridging spectrum used two different detectors, one on each output of the FT spectrometer. The first overlapped with the visible wavenumbers in Ref. [18] in order to obtain a wavenumber calibration and the second covered the UV wavenumbers being measured. Since the two outputs of the FT spectrometer are not exactly in antiphase, the resulting phase correction has a discontinuity in the region around 35 000 cm-1 where the two detectors overlap, as shown in Fig. 1 of [19]. The full procedure is described in detail in [19]. The 28 Ar II lines used as wavenumber standards in Refs. [18, 19] were subsequently re-measured by Whaling et al. [25] using FT spectroscopy with molecular CO lines as standards. The uncertainty of these measurements is 0.0002 cm-1. The molecular CO standards used in Ref. [25] were measured using heterodyne frequency spectroscopy with an uncertainty of around 1:109 and are ultimately traceable to the cesium primary standard [26]. The wavenumbers of Whaling et al. [25] are systematically higher than those of Norlén [24] by 6.7$\pm$0.8 parts in 108, corresponding to a wavenumber discrepancy of about 0.0014 cm-1 at 21000 cm-1. Since the results of Whaling et al. [25] are more accurate and precise than those of Norlén [24], all the wavenumbers in [18], [19], and Table 3 of [14] have been increased by 6.7 parts in 108 wherever they are used in the current work. The spectra in Nave & Sansonetti [13] were calibrated with respect to 29 Ge I Ritz wavenumbers derived from the energy levels of Kaufman & Andrew [17]. However, the Fe II wavenumbers derived using this calibration were found to be greater than those in Nave et al. [19] by about 7 parts in 108, even after the wavenumbers in the latter were adjusted to the wavenumber scale of Whaling et al. [25]. In order to present accurate wavenumbers for Fe II lines around 1600 Å, it is necessary first to confirm the accuracy of the iron lines in the visible that were calibrated with respect to selected lines of Ar II lines [18], to investigate the accuracy with which this calibration is transferred to the VUV, and to resolve the discrepancy between iron and germanium standard wavelengths identified in Ref. [13]. ### 4.1 Calibration of the visible-region spectra In order to confirm the calibration of the master spectrum, k19, used in [18] and [19], we took additional spectra using the NIST 2 m FT spectrometer [23]. The source was a water-cooled high-current hollow cathode lamp with a current of 1.5 A and argon at pressures of 130 Pa to 330 Pa (1 Torr to 2.5 Torr). The spectra covered the region 8500 cm-1 to 37 000 cm-1 with resolutions of either 0.02 cm-1 or 0.03 cm-1. A 1 mm aperture was used in order to minimize possible illumination effects. The detector was a silicon photodiode detector with a 2 mm $\times~{}$2 mm active area. The spectrometer was aligned optimally using a diffused, expanded beam from a helium neon laser, ensuring that the modulation of the laser fringes was maximized throughout the 2 m scan. Before recording some of the spectra, the spectrometer was deliberately misaligned and re-aligned in order to test whether small misalignments that could not be detected using our alignment procedure affected the wavenumber scale. The spectra were calibrated using the values of Whaling et al. [25] for Ar II lines recommended in Ref. [18] that had good signal-to-noise ratio. Wavenumbers of strong iron lines were then measured and compared with iron lines taken from Ref. [18] and [19]. Figure 2 shows the calibration of one of our spectra using Ar II and iron lines from Refs. [25, 18, 19] as standards. The calibration constant k${}_{\mbox{eff}}$ does not depend on wavenumber and is the same for all three sets of standards to within 1:108 when the iron lines from the Refs [18, 19] are adjusted to the wavenumber scale of [25]. The possibility of shifts due to non-uniform illumination of the aperture were investigated by taking a spectrum with the 5 mm diameter image of the hollow cathode lamp offset from the 1 mm aperture by about 2 mm. This spectrum also shows good agreement between the Ar II and iron calibrations. Many of the early interferograms from the NSO FT spectrometer were asymmetrically sampled, with a much larger number of points on one side of zero optical path difference than the other. A Fourier transform of an asymmetrically-sampled interferogram gives a spectrum with a large, antisymmetric imaginary part [27]. A small error in the phase correction causes a small part of this antisymmetric imaginary part to be rotated into the real part of the spectrum, distorting the line profiles and causing a wavenumber shift. The zero optical path difference in spectrum k19 is roughly 1/5 of the way through the interferogram. For a Gaussian profile with a full width at half maximum of W, this produces a wavenumber shift of roughly 0.3W per radian of phase error as shown in Fig. 3 of [27]. We decided to re-examine the phase curve for the master spectrum, k19, against which all the other iron spectra used in [18, 19, 14] were calibrated. The original interferogram for this spectrum was obtained from the NSO Digital Archives [28] and re-transformed using Xgremlin. The phase is plotted in Fig. 3. The residual phase error after fitting an 11th order polynomial is less than 10 mrad for almost all wavenumbers below 35 000 cm-1. This corresponds to an error of 3.6x10-4 cm-1 for a linewidth of 0.12 cm-1. Above 35 000 cm-1 the polynomial no longer fits the points adequately and consequently these points were not used in the comparison. Wavenumbers were measured in the re- transformed spectrum and calibrated with the 28 Ar II lines recommended in Ref. [18] using the values of Ref. [25]. Iron lines were then compared with those from papers [18, 19]. The result is shown in Fig. 4. The two measurements agree to within 1:108. This confirms that the original phase correction of k19 was accurate and the wavenumbers in Ref. [18] and Table 3 of Ref. [19] (2929 Å to 3841 Å) are not affected by phase errors. We conclude that the wavenumbers measured in the master spectrum, k19, are accurate. Although results from this spectrum were used in Learner & Thorne [18] and Table 3 of Nave et al. [19], it did not dominate the weighted average values reported in these papers. ### 4.2 Calibration of the ultraviolet spectra Tables 4 and 5 in Nave et al. [19] cover wavenumbers from 33 695 cm-1 to 54 637 cm-1 in Fe I and Fe II respectively. The wavenumbers in these tables were measured using the vacuum ultraviolet FT spectrometer at IC. The calibration of these spectra was transferred from the master spectrum (k19 in Table 2) using a bridging spectrum (i56 in Table 2), as described in section 4. The principal spectrum covering wavenumbers below 35 000 cm-1 in Table 4 of [19] is i6 in Table 2. It overlaps with the master spectrum between 33 000 cm-1 and 34 000 cm-1. Figure 5 shows a comparison of wavenumbers in i6 with the master spectrum k19. The wavenumbers in spectrum i6 are systematically smaller than in k19 by 3.9$\pm$0.5 parts in 108. Although the region of overlap of i6 with k19 is small and is thus insensitive to non-linearities in the wavenumber scale, this result supports our earlier speculation in Nave & Sansonetti [13] that the calibration of the UV data using the bridging spectrum may be incorrect. Based on the comparison of Fig 5, we conclude the wavenumbers in Tables 4 and 5 of Ref. [19] should be increased by 10.6 parts in 108, consisting of 3.9 parts in 108 to correct the transfer of the calibration to the ultraviolet and an additional 6.7 parts in 108 to put all the spectra on the wavenumber scale of Whaling et al. [25]. We compared our corrected values for iron lines in the UV to results of Aldenius et al. [7, 8], who present wavenumbers of iron lines measured in a high-current hollow cathode lamp using a UV FT spectrometer similar to the one used in [19]. Instead of recording a pure iron spectrum, they included small pieces of Mg, Ti, Cr, Mn and Zn in their Fe cathode. This ensured that spectral lines due to all of these species were placed on the same wavenumber scale, which was calibrated using the Ar II lines of Whaling et al. [25]. Table 3 compares the wavenumbers of Ref. [8] with the corrected values of [19]. Although the wavenumbers of Ref. [8] agree with our revised values within their joint uncertainties, they are systematically smaller by 3.7 parts in 108. Although this might suggest that it is incorrect to increase the wavenumbers of Ref. [19], it might also indicate that the wavenumbers of Ref. [8] need to be increased. Fortunately, there are data that allow us to test these alternatives. In addition to iron lines, the spectra in Ref. [8] contained four lines due to Mg I and Mg II that have since been measured using frequency comb spectroscopy [9, 10, 11] with much higher accuracy than achievable using FT spectroscopy. Table 4 compares the wavenumbers of these four magnesium lines from [8] with those derived from frequency comb measurements of isotopically pure values. For this comparison, the results of [8] have been increased by 3.7 parts in 108, as suggested by the comparison of Fe II lines in Table 3. With this adjustment, the results of Aldenius agree with the frequency comb values within their joint uncertainties, having a mean deviation of -0.7$\pm$3 parts in 108. Without the adjustment the mean deviation would be (-4$\pm$3)$\times$10-8. We conclude that the wavenumbers in Tables 4 and 5 of Nave et al. [19] should be increased by 10.6 parts in 108: 3.9 parts in 108 to correct for the incorrect transfer of the calibration from the master spectrum to the ultraviolet and 6.7 parts in 108 to put all of the spectra on the scale of Whaling et al. [25]. We have performed this correction in the following sections of the current paper. The wavenumbers of Aldenius et al. [8] should be increased by 3.7 parts in 108 to put them on the same scale. This adjustment of scale brings the measurements of lines of Mg I and Mg II in [8] into agreement with the more accurate frequency comb values [9, 10, 11]. ## 5 Wavenumbers of a6D - y6P transitions The wavenumbers of the a6D - y6P transitions can be obtained either from direct measurements or from energy levels derived from a larger set of experimental data (Ritz wavenumbers). Direct measurements will have larger uncertainties due to the cumulative addition of the uncertainties in the transfer of the calibration from the visible to the UV. Ritz wavenumbers are more accurate due to the increased redundancy, but use of a large set of experimental data to derive the energy levels makes it less clear exactly how the Ritz wavenumbers are derived. We illustrate this process by using a small subset of the strongest transitions that determine the y6P levels that are present in the visible and ultraviolet regions of the spectrum where we have corrected the wavenumber calibration. The y6P levels can be determined from three sets of lines in the UV and visible as shown in Fig. 6. The first set of nine lines near 2350 Å determines the three $\rm 3d^{6}(^{6}D)4p\,z^{6}P$ levels. All nine lines are present in archival spectra from IC which we have recalibrated using the results of section 4.2. Two of the nine lines are blended with other lines and a third, between $\rm a^{6}D_{9/2}$ and $\rm z^{6}P_{7/2}$, is self-absorbed in the IC spectra. These lines are unsuitable for determining the z6P levels. The recalibrated values of the remaining six lines are shown in column 4 of Table 5. Each line is observed with a signal-to-noise ratio of over 100 in at least eight spectra, all of which agree within 0.006 cm-1. The wavenumbers in Table 5 are weighted mean values of the individual measurements and the standard deviation in the last decimal place is given in parenthesis following the wavenumber. The lower levels in column 3 are determined from between 10 and 20 different transitions to upper levels and have been optimized to the archival spectra with the program lopt [29] (described below). The total standard uncertainty in the upper levels includes the calibration uncertainty of 2.3x10-8 times the level value. The second set of three transitions around 5000 Å determines the 3d54s2 a6S5/2 level from the three z6P levels. These lines are present in k19 and other archival spectra taken at NSO that we have recalibrated to correspond to the wavenumber scale of Whaling et al. [25]. Each line is present in five spectra, all of which agree within 0.0035 cm-1. Wavenumbers for these transitions are shown in Table 6 and give a mean value of (23317.6344$\pm$0.0010) cm-1 for the 3d54s2 a6S5/2 level. Finally, the y6P levels can be determined from the a6S5/2 level from three lines around 2580 Å, present in the IC spectra. The recalibrated wavenumbers are shown in Table 7 with the resulting y6P level values. These values were used to calculate Ritz wavenumbers for the a6D - y6P transitions, as shown in the third column of Table 8. Alternate values for the Ritz wavenumbers of the a6D-y6P transitions can be obtained from energy levels optimized using wavenumbers from the archival Fe II spectra from NSO and IC corrected according to sections 4.1 and 4.2. The program lopt [29] was used to derive optimized values for 939 energy levels from 9567 transitions. Weights were assigned proportional to the inverse of the estimated variance of the wavenumber. Lines with more than one possible classification, lines that were blended, or for which the identification was questionable were assigned a low weight. Two iterations were made. In the first, lines connecting the lowest a6D term to higher $\rm 3d^{6}\,(^{5}D)4p$ levels were assigned a weight proportional to the inverse of the statistical variance of the wavenumber, omitting the calibration uncertainty. This was done to obtain accurate values and uncertainties for the a6D intervals. These intervals are determined from differences between lines close to one another in the same spectrum sharing the same calibration factor. Hence the calibration uncertainty does not contribute to the uncertainty in the relative values of these energy levels. The values of the a6D levels obtained in this iteration are given in column 3 of table 5. In the second iteration, the a6D levels were fixed to the values and uncertainties determined from the first iteration. The weights of the $\rm a^{6}D-3d^{6}\,(^{5}D)4p$ transitions were assigned by combining in quadrature the statistical uncertainty in the measurement of the line position and the calibration uncertainty in order to obtain accurate uncertainties for the $\rm 3d^{6}\,(^{5}D)4p$ and higher levels. The values of the y6P levels are given in column 4 of table 7. Ritz wavenumbers for the $\rm a^{6}D-3d^{6}\,(^{5}D)4p$ transitions based on these globally optimized level values are presented in column 5 of table 8. The corrected experimental wavenumbers from the archival spectra are given in column 4 of table 8. The main contribution to the uncertainty in the experimental wavenumbers is from the calibration and consists of two components – the uncertainty in the standards and the uncertainty in calibrating the spectrum. The calibration uncertainty is common to all lines in the calibrated spectrum and hence must be added to the uncertainties of wavenumbers measured using transfer standards, rather than added in quadrature as would be the case for random errors. Hence the uncertainty in the wavenumbers increases with each calibration step, resulting in larger uncertainties at the shortest wavenumbers which are furthest from the calibration standards. The experimental standard uncertainties in Table 8 are determined by combining in quadrature the statistical uncertainty in determining the line position and the calibration uncertainty of 4$\times$10-8 times the wavenumber. The experimental wavenumber and both Ritz wavenumbers agree within their joint uncertainties. The Ritz wavenumbers determined from optimized energy levels have the smallest uncertainties. Wavelengths corresponding to these wavenumbers are given in column 7. ## 6 A re-examination of Fe II wavenumbers from Nave & Sansonetti[13] The Fe I lines in Nave & Sansonetti [13] were calibrated with respect to lines of Ge I. Figure 3 of that paper showed that the calibration factor k${}_{\mbox{eff}}$ derived from Fe I and Fe II lines is smaller than that derived from Ge I by 6.5 parts in 108. We attributed this to a possible problem in the transfer of the wavenumber calibration of the Fe I and Fe II lines from the region of the Ar II wavenumber standards to the vacuum ultraviolet, thus suggesting that the wavenumber standards in [19] are too small. In section 4.2 we have confirmed that the wavenumbers in Tables 4 and 5 of Nave et al. [19] should be increased by 3.9 parts in 108 due to the transfer of the calibration. This reduces but does not fully explain the calibration discrepancy in [13]. The Ge I lines used to calibrate the spectra in Ref. [13] were measured by Kaufman and Andrew [17]. The wavenumber standard they used was the 5462 Å line of 198Hg emitted by an electrodeless discharge lamp maintained at a temperature of $19\,^{\circ}\rm{C}$, containing Ar at a pressure of 400 Pa (3 Torr). The vacuum wavelength of this line was assumed to be 5462.27075 Å. This value was based on a vacuum wavelength of 5462.27063 Å measured in the same lamp at $7\,^{\circ}\rm{C}$ [30], with an adjustment for the different temperature using the measurements of Emara [31]. The 5462 Å line was remeasured by Salit et al. [32] using a temperature of $8\,^{\circ}\rm{C}$. A value of (5462.27085$\pm$0.00007) Å was obtained. More recent work by Sansonetti & Veza [33] gives the wavelength of this line as 5462.270825(11) Å, in agreement with [32] but more precise. Adoption of this value for the wavelength of the 198Hg line implies that all of the Ge I wavenumbers in [17] should be decreased by 1.4 parts in 108. Figure 7 shows how Fig. 3 in [13] (Spectrum lp0301 in Table 2) changes with the adjustment of both the iron and germanium wavenumbers. The calibrations based on Ge and Fe lines now differ by only 1.5 parts in 108, which is within the joint uncertainties. We thus conclude that the calibration derived from Fe I and Fe II lines is in agreement with that derived from Ge I when both sets of standards are adjusted to correspond with the most recent measurements. Spectrum lp0301 in Table 2 can be used to calibrate spectrum fe1115 in Table 2 referred to in the last paragraph of section 2. A value of (62171.634$\pm$0.006) cm-1 is obtained for the wavenumber of the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ line, corresponding to a wavelength of (1608.45057$\pm$0.00016) Å, This disagrees with the Ritz value by 1.7 times the joint uncertainty and marginally disagrees with the experimental values of Table 8. The mean difference in the experimental values for all nine $\rm a^{6}D-y^{6}P$ lines is (0.008$\pm$0.004) cm-1. We believe this difference is due to a small slope in the calibration of lp0301 but have been unable to confirm this with our data. The principal contributors to the uncertainty are the uncertainty in the iron and germanium standards, the uncertainty in calibrating the spectrum in ref. [13] from these standards, and the uncertainty in calibrating spectrum fe1115 from spectrum lp0301. ## 7 Conclusions We investigated the wavenumber scale of published Fe I and Fe II lines using new spectra recorded with the NIST 2-m FT spectrometer and a re-analysis of archival spectra. Our new spectra confirm the wavenumber scale of visible- region iron lines calibrated using the Ar II wavenumber standards of Whaling et al. [25]. Having confirmed the wavenumber scale of iron lines in the visible and ultraviolet regions, we have used lines from these spectra to derive Ritz values for the wavenumbers and wavelengths of lines in the $\rm a^{6}D-y^{6}P$ multiplet of Fe II (UV 8). Ritz wavenumbers derived using two different methods agree with one another and with directly measured wavenumbers within the joint uncertainties. We recommend a value of 1608.45081$\pm$0.00007 Å for the wavelength of the $\rm a^{6}D_{9/2}-y^{6}P_{7/2}$ line of Fe II, which is an important line for detection of changes in the fine-structure constant during the history of the Universe using quasar absorption-line spectra. We find that the wavenumbers in Learner & Thorne [18] and Table 3 of Nave et al. [19] should be increased by 6.7 parts in 108 to put them on the scale of the Ar II lines of Whaling et al. [25]. The wavenumbers in Tables 4 and 5 of Ref. [19] should be increased by 10.6 parts in 108 to put them on the Ar II scale of Ref. [25] and to correct for an error in the transfer of this wavenumber scale to the ultraviolet. The Ge I wavenumbers of Kaufman & Andrew [17] and all the wavenumbers in Nave & Sansonetti [13] should be decreased by 1.4 parts in 108 to put them on the scale of recent measurements of the 198Hg line at 5462 Å. ## 8 Acknowledgments We thank Michael T. Murphy for alerting us to the importance of the Fe II line at 1608 Å. We also thank Anne P. Thorne and Juliet C. Pickering for helpful discussions on the calibration of FT spectrometers and the linearity of the FT wavenumber scale. This work is partially supported by NASA inter-agency agreement NNH10AN38I. ## References * [1] P. J. Mohr, B. N. Taylor, and D. B. Newell (2007), “The 2006 CODATA Recommended Values of the Fundamental Physical Constants” (Web Version 5.2). Available: http://physics.nist.gov/constants [2010, May 11]. National Institute of Standards and Technology, Gaithersburg, MD 20899. * [2] M. T. Murphy, J. K. Webb, V. V. Flambaum, “Further evidence for a variable fine-structure constant from Keck/HIRES QSO absorption spectra,” Mon. Not. R. Astron. Soc. 345, 609–638 (2003). * [3] H. Chand, R. Srianand, P. Petitjean, B. Aracil, R. Quast, D. Reimers, “Variation of the fine-structure constant: very high resolution spectrum of QSO HE 0515-4414,” Astron. Astrophys., 451, 45–56 (2006). * [4] V. A. Dzuba, V. V. Flambaum, J. K. Webb, “Space-Time Variation of Physical Constants and Relativistic Corrections in Atoms,” Phys. Rev. Lett. 82, 888–891 (1999). * [5] J. N. Bahcall, W. L. W. Sargent, M. Schmidt, “An Analysis of the Absorption Spectrum of 3c 191,” Astrophys. J. 149, L11–L15 (1967) * [6] J. C. Pickering, A. P. Thorne, J. K. Webb, “Precise laboratory wavelengths of the Mg I and Mg II resonance transitions at 2853, 2803 and 2796 Angstroms,” Mon. Not. R. Astron. Soc., 300, 131–134 (1998). * [7] M. Aldenius, S. Johansson, M. T. Murphy, “Accurate laboratory ultraviolet wavelengths for quasar absorption-line constraints on varying fundamental constants,” Mon. Not. R. Astron. Soc., 370, 444–452 (2006). * [8] M. Aldenius, “Laboratory wavelengths for cosmological constraints on varying fundamental constants,” Phys. Scr. T134, 014008 (2009). * [9] E. J. Salumbides, S. Hannemann, K. S. E. Eikema, W. Ubachs, “Isotopically resolved calibration of the 285-nm MgI resonance line for comparison with quasar absorptions,” Mon. Not. R. Astron. Soc., 373, L41–L44 (2006). * [10] S. Hannemann, E. J. Salumbides, S. Witte, R. Th. Zinkstok, E.-J. van Duijn, K. S. E. Eikema, W. Ubachs, “Frequency metrology on the Mg 3s$\rm{}^{2\,1}S\rightarrow$3s4p 1P line for comparison with quasar data,” Phys. Rev. A, 74 012505 (2006). * [11] V. Batteiger, S. Knünz, M. Herrmann, G. Saathoff, H. A. Schüssler, B. Bernhardt, T. Wilken, R. Holzwarth, T. W. Hänsch, Th. Udem, “Precision spectroscopy of the 3s-3p fine-structure doublet in Mg+,” Phys. Rev. A 80, 022503 (2009). * [12] M. T. Murphy, J. K. Webb, V. V. Flambaum, “Further evidence for a variable fine-structure constant from Keck/HIRES QSO absorption spectra,” Mon. Not. R. Astron. Soc. 345, 609–638 (2003). * [13] G. Nave, C. J. Sansonetti, “Reference wavelengths in the spectra of Fe, Ge, and Pt in the region near 1935 Å,” 21, 442–453 (2004). * [14] G. Nave, S. Johansson, A. P. Thorne, “Precision vacuum-ultraviolet wavelengths of Fe II measured by Fourier-transform and grating spectrometry,” J. Opt. Soc. Am. B 14, 1035–1042 (1997). * [15] S. Johansson, “The spectrum and term system of Fe II,” Phys. Scr. 18, 217-265 (1978). * [16] J. C. Pickering, M. P. Donnelly, H. Nilsson, A. Hibbert, S. Johansson, “The FERRUM Project: Experimental oscillator strengths of the UV 8 multiplet and other UV transitions from the $\mathsf{y^{6}}$P levels of Fe II,” Astron. Astrophys. 396, 715–722 (2002). * [17] V. Kaufman, K. L. Andrew, “Germanium vacuum ultraviolet Ritz standards,” J. Opt. Soc. Am. 52, 1223–1237 (1962). * [18] R. C. M. Learner, A. P. Thorne, “Wavelength calibration of Fourier-transform emission spectra with applications to Fe I,” J. Opt. Soc. Am. B 5, 2045–2059 (1988). * [19] G. Nave, R. C. M. Learner, A. P. Thorne, C. J. Harris, “Precision Fe I and Fe II wavelengths in the ultraviolet spectrum of the iron-neon hollow-cathode lamp,” J. Opt. Soc. Am. B 8, 2028–2041 (1991). * [20] A. P. Thorne, C. J. Harris, I. Wynne-Jones, R. C. M. Learner, G. Cox, “A Fourier transform spectrometer for the vacuum ultraviolet: design and performance,” J. Phys. E 20, 54–60 (1987). * [21] U. Griesmann, R. Kling, J. H. Burnett, L. Bratasz, “NIST FT700 vacuum ultraviolet Fourier transform spectrometer: applications in ultraviolet spectrometry and radiometry,” in Ultraviolet Atmospheric and Space Remote Sensing: Methods and Instrumentation II, G. R. Carruthers & K. F. Dymond, eds., Proc. SPIE 3818, 180–188 (1999). * [22] J. W. Brault, M. C. Abrams, “DECOMP: a Fourier transform spectra decomposition program,” Volume 6 of 1989 OSA Technical Digest Series, 110–112. * [23] G. Nave, C. J. Sansonetti, U. Griesmann, “Progress on the NIST IR-vis-UV Fourier transform spectrometer,” Volume 3 of 1997 OSA Technical Digest Series, 38–40. * [24] G. Norlén, “Wavelengths and energy levels of Ar I and Ar II based on new interferometric measurements in the region 3400-9800 Å,” Phys. Scr. 8, 249–268 (1973). * [25] W. Whaling, W. H. C. Anderson, M. T. Carle, J. W. Brault, H. A. Zarem, “Argon ion linelist and level energies in the hollow-cathode discharge,” J. Quant. Spectrosc. Radiat. Transfer 53, 1–22 (1995). * [26] A. G. Maki, J. S. Wells, “New wavenumber calibration tables from heterodyne frequency measurements,” J. Res. Natl. Inst. Stand. Tech. 97, 409–470 (1992). * [27] R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, M. C. Abrams, “Phase correction of emission line Fourier transform spectra,” 12, 2165–2171 (1995). * [28] NSO Digital Library available online at http://diglib.nso.edu/nso$\\_$user.html * [29] A. E. Kramida, “The program lopt for least-squares optimization of energy levels,” Comp. Phys. Comm. 182, 419-434 (2010). * [30] V. Kaufman, “Wavelengths, Energy Levels, and Pressure Shifts in Mercury 198,” J. Opt. Soc. Am., 52, 866–870 (1962). * [31] S. H .Emara, “Wavelength shifts in 198Hg as a function of temperature,” J. Res. Natl. Bur. Standards 65A, 473–474 (1961). * [32] M. L. Salit, C. J. Sansonetti, D. Veza, and J. C. Travis, “Investigation of Single-Factor Calibration of the Wave-Number Scale in Fourier-Transform Spectroscopy,” J. Opt. Soc. Am B, 21, 1543–1550 (2004). * [33] C. J. Sansonetti, D. Veza, “Doppler-free measurement of the 546-nm line of mercury,” J. Phys. B 43, 205003 (2010). Figure 1: The region of the Fe II $\rm a^{6}D-y^{6}P$ transitions. The labeled lines show the J-values of the lower and upper energy levels respectively. Figure 2: (Color online) Calibration of wavenumbers in spectrum fe0409.002 in Table 2 using Ar II standards from [25], and iron standards taken from Learner & Thorne [18] and Nave et al. [19]. The error bars represent the statistical uncertainty in the measurement of the wavenumber. Figure 3: Phase in the master spectrum, k19, used in Learner & Thorne [18]. The insert shows the residual phase after fitting the points to an 11${}^{\mbox{th}}$ order polynomial. Figure 4: (Color online) Comparison of wavenumbers in the master spectrum, k19, calibrated from Ar II standards from [25] with iron standards taken from [18] and [19] adjusted to the scale of [25]. The error bars represent the statistical uncertainty in the measurement of the wavenumber. Figure 5: Comparison of wavenumbers in the master spectrum, k19, with those in i6, the main spectrum contributing to Table 4 of [19] in this wavelength region. Figure 6: Partial term diagram of Fe II showing the determination of the y6P levels using transitions in the UV and visible regions Figure 7: Figure 3 from Nave & Sansonetti [13], with all of the Ge I wavenumbers reduced by 1.4 parts in 108 and the Fe I and Fe II wavenumbers increased by 3.9 parts in 108. The mean value of k${}_{\mbox{eff}}$ for the Ge I wavenumbers is $(1.221\pm 0.020)\times$10-6, in agreement within the joint uncertainties with the value of $(1.206\pm 0.020)\times$10-6 from the Fe I and Fe II lines Table 1: Proposed corrections to previous papers. Reference | Wavenumber | Previous | Previous | New | Correction to ---|---|---|---|---|--- | range | standard | uncertainty | standard | wavenumber scale | (cm-1) | | (cm-1) | | [18] | 17350 - 26140 | Ar II [24] | 0.001 | Ar II [25] | (+6.7$\pm$1.8)x10-8 Table 3 of [19] | 26027 - 34131 | Ar II [24], i56 | 0.002 | Ar II [25] | (+6.7$\pm$1.8)x10-8 Tables 4 & 5 of [19] | 33695 - 54637 | Ar II [24], i56 | 0.002 | Ar II [25], Fig.5 | (+10.6$\pm$2.3)x10-8 [14] | 50128 - 107887 | Ar II [24], i56 | 0.005 | Ar II [25], Fig.5 | (+10.6$\pm$2.3)x10-8 [13] | 51613 - 51692 | Ge I [17] | 0.002 | 198Hg [33] | (-1.4$\pm$2)x10-8 [8] | 38458 - 44233 | Ar II [25] | 0.002 | Table 3,4 | (+3.7$\pm$3)x10-8 [17] (Ge I & Ge II) | 8283 - 100090 | 198Hg,[30, 31] | $<$0.006 | 198Hg [33] | (-1.4$\pm$1.8)x10-8 [24] (Ar II)a | 4348 - 5145 | 86Kr Engelhard lamp | $<$0.001 | Ar II [25] | (+6.7$\pm$0.8)x10-8 11footnotetext: The proposed correction has only been confirmed for the 28 Ar II lines recommended in [18]. Table 2: Summary of spectra . Spectrum Instrument Date Wavelength Calibration Comments y/m/d Range (Å) Spectrum k19 [19] NSO 81/07/22 2800 to 5600 Ar II[25] 810622R0.009 (NSO[28]) A1 in [18] i56 [19] IC 2270 to 4170 k19 i6 [19] IC 89/11/07 2220 to 3030 i56 lp0301 NIST VUV 02/03/01 1830 to 3194 Ge I,II[17] Figs. 3 &4 in [13] fe1115 NIST VUV 02/11/15 1558 to 2689 lp0301 fe0409.002 NIST 2-m 09/04/09 2748 to 5765 Ar II [25], Fe I,II[18, 19] Table 3: Comparison of wavenumbers of Fe lines in Aldenius et al. [8] and adjusted wavenumbers of Nave et al. [19]. The wavenumbers of [19] have been increased by 10.6 parts in 108. The standard uncertainty in the last digits of the wavenumbers and of the levels is given in parenthesis and is dominated by the calibration uncertainty. . Species Nave et al. (cm-1) Aldenius et al. (cm-1) (column 3 / column 2) - 1 [19] [8] Fe II 38458.9912(20) 38458.9908(20) -1.0x10-8 Fe II 38660.0535(20) 38660.0523(20) -3.1x10-8 Fe II 41968.0687(20) 41968.0654(20) -7.7x10-8 Fe II 42114.8374(20) 42114.8365(20) -2.1x10-8 Fe II 42658.2449(20) 42658.2430(20) -4.5x10-8 Mean (-3.7$\pm$2.6)x10-8 Table 4: Comparison of adjusted wavenumbers of Mg lines in Aldenius et al. [8] with frequency comb measurements (col. 3) taken from paper listed in the reference column. The wavenumbers from [8] have been increased by 3.7 parts in 108. The standard uncertainties in the last digits of the wavenumbers are given in parentheses. Species | Aldenius (cm-1) | Frequency comb (cm-1) | (column 2/column 3) -1 | Reference ---|---|---|---|--- | [8] | | | Mg I | 35051.2817(20) | 35051.2808(2) | 2.6x10-8 | [9] Mg II | 35669.3039(20) | 35669.30440(6) | -1.3x10-8 | [11] Mg II | 35760.8523(20) | 35760.85414(6) | -5.1x10-8 | [11] Mg I | 49346.7730(30) | 49346.77252(7) | 1.0x10-8 | [10] | | Mean | (-0.7$\pm$3)x10-8 | Table 5: Determination of the z6P levels of Fe II from transitions to the ground term around 2350 Å. The statistical uncertainty in the last decimal place of the wavenumbers is given in parenthesis. The total standard uncertainty includes the uncertainty in the calibration. Lower | Upper | lower level value | Wavenumber | Upper level value ---|---|---|---|--- level | level | cm-1 | cm-1 | cm-1 a${}^{6}D_{5/2}$ | z${}^{6}P_{7/2}$ | 667.6829(5) | 41990.5610(3) | 42658.2439(6) a${}^{6}D_{7/2}$ | z${}^{6}P_{7/2}$ | 384.7872(4) | 42273.4573(4) | 42658.2445(6) | | | Mean | 42658.2442(5) | | | Total uncertainty | 0.0011 a${}^{6}D_{5/2}$ | z${}^{6}P_{5/2}$ | 667.6829(5) | 42570.9226(4) | 43238.6055(6) a${}^{6}D_{7/2}$ | z${}^{6}P_{5/2}$ | 384.7872(4) | 42853.8188(4) | 43238.6060(6) | | | Mean | 43238.6058(5) | | | Total uncertainty | 0.0011 a${}^{6}D_{1/2}$ | z${}^{6}P_{3/2}$ | 977.0498(6) | 42643.9332(4) | 43620.9830(7) a${}^{6}D_{5/2}$ | z${}^{6}P_{3/2}$ | 667.6829(5) | 42953.2994(5) | 43620.9823(7) | | | Mean | 43620.9827(6) | | | Total uncertainty | 0.0012 Table 6: Determination of the a6S5/2 level of Fe II using transitions from the z6P levels. The statistical uncertainties in the last digits of the wavenumber and levels are given in parentheses. The total standard uncertainty of the a${}^{6}S$ level includes a contribution of 4$\times 10^{-8}$ times the level uncertainty due to the calibration. Upper level | Upper level value | Wavenumber | a6S level ---|---|---|--- | cm-1 | cm-1 | cm-1 z$\rm{}^{6}P_{7/2}$ | 42658.2442(5) | 19340.6092(2) | 23317.6350(5) z$\rm{}^{6}P_{5/2}$ | 43238.6058(5) | 19920.9733(2) | 23317.6325(5) z$\rm{}^{6}P_{3/2}$ | 43620.9827(6) | 20303.3477(3) | 23317.6350(7) | | Weighted mean | 23317.6340(3) | | Total uncertainty | 0.0010 Table 7: Determination of the y6P levels of Fe II from transitions to the a6S level. The statistical uncertainty in the last digits of the wavenumbers and levels is given in parentheses. The total standard uncertainty is common to all levels and includes a contribution of 4$\times 10^{-8}\sigma$ due to the calibration. The last column contains the level value and standard uncertainty in parenthesis with respect to the ground level obtained from the lopt program as described in section 5 . Upper level | Wavenumber | Upper level value | Level value from lopt ---|---|---|--- | cm-1 | cm-1 | cm-1 y$\rm{}^{6}P_{3/2}$ | 38657.2997(14) | 61974.9347(14) | 61974.9325(24) y$\rm{}^{6}P_{5/2}$ | 38731.4041(7) | 62049.0381(8) | 62049.0408(27) y$\rm{}^{6}P_{7/2}$ | 38853.9885(4) | 62171.6225(5) | 62171.6245(27) | Total uncertainty | 0.003 | Table 8: Experimental and Ritz wavenumbers for the a6D-y6P multiplet. The standard uncertainties in the last digits of the wavenumbers and wavelengths are given in parentheses. Lower | Upper | $\sigma_{a6S}^{a}$ | $\sigma_{exp}^{b}$ | $\sigma_{Ritz}^{c}$ | $\lambda_{Ritz}^{d}$ ---|---|---|---|---|--- level | level | cm-1 | cm-1 | cm-1 | Å a6D1/2 | y6P3/2 | 60997.884(3) | 60997.882(3) | 60997.8827(25) | 1639.40117(7) a6D3/2 | y6P3/2 | 61112.322(3) | 61112.321(3) | 61112.3207(25) | 1636.33125(7) a6D5/2 | y6P3/2 | 61307.251(3) | 61307.247(3) | 61307.2496(25) | 1631.12847(7) a6D3/2 | y6P5/2 | 61186.426(3) | 61186.432(4) | 61186.4290(28) | 1634.34934(7) a6D5/2 | y6P5/2 | 61381.355(3) | 61381.358(3) | 61381.3579(28) | 1629.15914(7) a6D7/2 | y6P5/2 | 61664.251(3) | 61664.255(3) | 61664.2536(27) | 1621.68508(7) a6D5/2 | y6P7/2 | 61503.940(3) | 61503.945(8) | 61503.9416(27) | 1625.91205(7) a6D7/2 | y6P7/2 | 61786.835(3) | 61786.837(3) | 61786.8373(27) | 1618.46769(7) a6D9/2 | y6P7/2 | 62171.623(3) | 62171.626(4) | 62171.6245(27) | 1608.45081(7) 11footnotetext: Wavenumber calculated using a6S as an intermediate level22footnotetext: Experimental wavenumber from archival spectra corrected according to section 4.2.33footnotetext: Ritz wavenumber calculated from all optimized energy levels44footnotetext: Wavelength calculated from the Ritz wavenumber in column 5.
arxiv-papers
2011-01-25T20:15:51
2024-09-04T02:49:16.640159
{ "license": "Public Domain", "authors": "Gillian Nave and Craig J. Sansonetti", "submitter": "Gillian Nave", "url": "https://arxiv.org/abs/1101.4915" }
1101.5225
# Interfacial thermal transport in atomic junctions Lifa Zhang Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore Pawel Keblinski Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, New York, 12180, USA. Jian-Sheng Wang Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore Baowen Li Electronic address: phylibw@nus.edu.sg NUS Graduate School for Integrative Sciences and Engineering, Singapore 117456, Republic of Singapore Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore (30 Oct 2010, Revised 10 Jan 2011 ) ###### Abstract We study ballistic interfacial thermal transport across atomic junctions. Exact expressions for phonon transmission coefficients are derived for thermal transport in one-junction and two-junction chains, and verified by numerical calculation based on a nonequilibrium Green’s function method. For a single- junction case, we find that the phonon transmission coefficient typically decreases monotonically with increasing freqency. However, in the range between equal frequency spectrum and equal acoustic impedance, it increases first then decreases, which explains why the Kapitza resistance calculated from the acoustic mismatch model is far larger than the experimental values at low temperatures. The junction thermal conductance reaches a maximum when the interfacial coupling equals the harmonic average of the spring constants of the two semi-infinite chains. For three-dimensional junctions, in the weak coupling limit, we find that the conductance is proportional to the square of the interfacial coupling, while for intermediate coupling strength the conductance is approximately proportional to the interfacial coupling strength. For two-junction chains, the transmission coefficient oscillates with the frequency due to interference effects. The oscillations between the two envelop lines can be understood analytically, thus providing guidelines in designing phonon frequency filters. ###### pacs: 66.70.-f, 05.60.-k, 44.10.+i, ## I Introduction In the past decade there has been a significant research focus on thermal transport in micro scaleDhar . Several conceptual thermal devices, such as thermal rectifiers/diodes, thermal transistors, thermal logical gates, and thermal memory rectifiers ; transistor ; logicgate ; memory , have been proposed, which, in principle, make it possible to control heat due to phonons and process information with phonons. The issue of quantum thermal transport in nanostructures was also addressed wangjs2008 . In this context, the critical information is in phonon transmission coefficients that in quasi-one- dimensional atomic models can be calculated by transfer matrix method tong1999 ; macia2000 ; cao2005 ; antonyuk2005 . However, the evaluation of the transfer matrix may be numerically unstable, particularly when the system size becomes large. Alternatively, nonequilibrium Green’s function (NEGF) method is an efficient way to calculate the transmission coefficientnegfref . Unfortunately, both of these two methods are numerical in nature and do not give analytical expressions. For thermal transport and control, the interfacial thermal scattering process is becoming increasingly important, especially in practical devices. Two theories, acoustic mismatch model little1959 and the diffuse mismatch model swartz1989 , have been proposed to study the mechanism of the thermal interfacial resistance. However, both models offer limited accuracy in nanoscale interfacial resistance predictions stevens2005 because they neglect atomic details of actual interfaces. A scattering boundary method within the lattice dynamic approach was first proposed by Lumpkin and Saslow to study the Kapitza conductance in a one-dimensional (1D) lattice lumpkin1978 , and was then applied to calculate the Kapitza resistance in two- and three-dimensional (3D) lattices paranjape1987 ; young1989 . This method can predict thermal interfacial conductance between heterogeneous materials with full consideration of the atomic structures in the interface. Recently, this method was applied to study the ballistic thermal transport in nanotube junctionswang2006 , spin chainszhang2008 , and honeycomb lattice ribbons cuansing2009 . In this paper we give an explicit analytical expression of transmission coefficient obtained through the scattering boundary method, and use it to study the interfacial thermal transport across atomic junctions. First, in Sec. II, we introduce a model in which two semi-infinite 1D atomic chains are coupled either via a point junction or an extended junction region. By using the boundary scattering method we derive the exact expressions for phonon transmission coefficients for thermal transport in one-junction and two- junction chains in Sec. III. The role of various parameters on the junction conductance is analyzed and discussed in Sec. IV. In section IV we also estimate the interfacial conductance between two 3D solids. In Sec. V, we introduce briefly the NEGF method, and use it to verify the results from analytical formulae for the thermal transport in our model. A short summary is presented in Sec. VI. Figure 1: (color online) A schematic representation of the 1D atomic chain model. The size of the center part is $N_{C}=8$. The left and right regions are two semi-infinite harmonic atomic chains at different temperatures $T_{L}$ and $T_{R}$. The three parts are coupled by harmonic springs with constant strength $k_{12}$ and $k_{23}$; all of which are harmonic chains with mass and spring constant as $m_{1},k_{1}$, $m_{2},k_{2}$ and $m_{3},k_{3}$, respectively. ## II Model The one-dimensional atomic chain consists of three parts: two semi-infinite leads and an center region (see Fig. 1). The two leads are in equilibrium at different temperatures $T_{L}$ and $T_{R}$. The three parts are coupled by harmonic springs with constant strength $k_{12}$ and $k_{23}$; all of which are harmonic chains with mass and spring constants $m_{1},k_{1}$, $m_{2},k_{2}$ and $m_{3},k_{3}$, respectively. So the total Hamiltonian can be written as $H=\sum\limits_{\alpha=1,2,3}{H_{\alpha}}+\frac{1}{2}k_{12}(x_{1,1}-x_{2,1})^{2}+\frac{1}{2}k_{23}(x_{2,N_{c}}-x_{3,1})^{2};$ (1) here, $H_{\alpha}=\sum\limits_{i=1}^{N_{\alpha}}{\frac{1}{2}m_{\alpha}\dot{x}_{\alpha,i}^{2}+\sum\limits_{i=1}^{N_{\alpha}-1}\frac{1}{2}k_{\alpha}(x_{\alpha,i}-x_{\alpha,i+1})^{2}}.$ (2) Where $x_{\alpha,i}$ is the relative displacement of i-th atom in $\alpha$-th part. If there is no center part, that is, the two semi-infinite leads connected directly by $k_{12}$, then by setting $\alpha=1,2$ and $k_{23}=0$ in Eq. (1), we can obtain the corresponding Hamiltonian. For the semi-infinite leads, $N_{\alpha}=\infty$. ## III Analytical Solution from the Scattering Boundary Method Heat current flowing from left to right through a junction connecting two leads kept at different equilibrium heat-bath temperatures $T_{L}$ and $T_{R}$ is given by the Landauer formula wangjs2008 $I=\frac{1}{{2\pi}}\int_{0}^{\infty}{\hbar\omega\;\bigl{[}f_{L}(\omega)-f_{R}(\omega)\bigr{]}T[\omega]}d\omega,$ (3) which allows us to develop the junction conductance formula $\sigma=\frac{1}{{2\pi}}\int_{0}^{\infty}{d\omega\;\hbar\omega\,T[\omega]\frac{\partial f(\omega)}{\partial T}},$ (4) here, $f_{L,R}=\\{\exp[\hbar\omega/(k_{B}T_{L,R})]-1\\}^{-1}$ is the Bose- Einstein distribution for phonons, and $T[\omega]$ is the frequency dependent transmission coefficient. Therefore, the key step for the thermal transport characterization is to calculate the transmission coefficients. We first consider a point-junction case, that is, two semi-infinite harmonic chains connected by a spring with constant strength $k_{12}$. We assume a wave solution transmitting from the left lead to the right lead. We label the atoms as $-\infty,\cdots,-1,0,1,2,\cdots,+\infty$. Atoms $0$ and $1$ are connected by $k_{12}$ spring. An incident wave from left is assumed as $x_{I}=\lambda_{1}^{j}e^{-i\omega t}$. When it arrives at the interface, it will be partially reflected and partially transmitted. The reflected wave amplitude is $x_{R}=r_{12}\lambda_{1}^{-j}e^{-i\omega t}$ and the transmission wave can be written as $x_{T}=t_{12}\lambda_{2}^{j-1}e^{-i\omega t}$. So at each atom we have $\cdots,\;x_{-1}=(\lambda_{1}^{-1}+r_{12}\lambda_{1})e^{-i\omega t},\;x_{0}=(1+r_{12})e^{-i\omega t}$, $x_{1}=t_{12}e^{-i\omega t},\;x_{2}=t_{12}\lambda_{2}e^{-i\omega t},\;\cdots$. Here, $\lambda_{j}=e^{iq_{j}a_{j}}$, $q_{j}$ is the wave vector, $a_{j}$ is the interatomic spacing. For the atom in the $j-th$ part, we can have the equation of motion as $m_{j}\frac{d^{2}x_{j,n}}{dt^{2}}=k_{j}(x_{j,n+1}-x_{j,n})+k_{j}(x_{j,n}-x_{j,n-1}),$ (5) each wave transport separately and satisfies such equation. Thus $\lambda_{j}$ satisfies the dispersion relation of the corresponding lead as $\omega^{2}m_{j}=-k_{j}\lambda_{j}^{-1}+2k_{j}-k_{j}\lambda_{j}.$ (6) The quadratic equation has two roots. Which one should we choose? Replacing $\omega$ with $\omega+i\eta$, $\eta=0^{+}$, none of the eigenvalues $\lambda$ will have modulus exactly 1. We find for the traveling waves velev2004 $|\lambda|=1-\eta\frac{a}{v},$ (7) thus the forward moving waves with group velocity $v>0$ have $|\lambda|<1$. Therefore we should take the one with $|\lambda|<1$ of the two roots which are given as $\lambda_{j}=\frac{{-h_{j}\pm\sqrt{h_{j}^{2}-4}}}{2},\;\;\;h_{j}=\frac{{m_{j}}}{{k_{j}}}(\omega+i\eta)^{2}-2.$ (8) From the scattering boundary method, the coefficients $r_{12}$, $t_{12}$ can be obtained from the continuity condition at the interface as: $\displaystyle\omega^{2}m_{1}x_{0}=-k_{1}x_{-1}+(k_{1}+k_{12})x_{0}-k_{12}x_{1};$ (9) $\displaystyle\omega^{2}m_{2}x_{1}=-k_{12}x_{0}+(k_{12}+k_{2})x_{1}-k_{12}x_{2}.$ (10) Finally we can get the transmission coefficient as $T[\omega]=1-|r_{12}|^{2}=1-|r_{21}|^{2},$ (11) here, $r_{ij}=\frac{{k_{i}(\lambda_{i}-1/\lambda_{i})(k_{j}-k_{ij}-k_{j}/\lambda_{j})}}{{(k_{i}-k_{ij}-k_{i}/\lambda_{i})(k_{j}-k_{ij}-k_{j}/\lambda_{j})-k_{ij}^{2}}}-1.$ (12) Of course, we can also use $t_{12}$ to express $T[\omega]$ as $\frac{{m_{2}v_{2}/a_{2}}}{{m_{1}v_{1}/a_{1}}}|t_{12}|^{2}$, here the group velocity $v_{i}=\frac{{d\omega}}{{dq_{i}}}=\frac{{a_{i}}}{2}\sqrt{\frac{{4k_{i}}}{{m_{i}}}-\omega^{2}}$, which is derived from the dispersion relation given by Eq. (6). Thus, the transmission coefficient can also be expressed as $T[\omega]=\frac{{\sqrt{4k_{2}m_{2}-\omega^{2}m_{2}^{2}}}}{{\sqrt{4k_{1}m_{1}-\omega^{2}m_{1}^{2}}}}|t_{12}|^{2},$ (13) here $t_{ij}=\frac{{-k_{ij}k_{i}(\lambda_{i}-1/\lambda_{i})}}{{(k_{i}-k_{ij}-k_{i}/\lambda_{i})(k_{j}-k_{ij}-k_{j}/\lambda_{j})-k_{ij}^{2}}}.$ (14) For the long-wave limit, that is, $\omega=0^{+}$, we get $r_{ij}=\frac{{\sqrt{k_{i}m_{i}}-\sqrt{k_{j}m_{j}}}}{{\sqrt{k_{i}m_{i}}+\sqrt{k_{j}m_{j}}}}$; and the transmission is $T[0^{+}]=\frac{{4\sqrt{k_{1}m_{1}k_{2}m_{2}}}}{{(\sqrt{k_{1}m_{1}}+\sqrt{k_{2}m_{2}})^{2}}}.$ (15) This result is consistent with the one obtained for the acoustic mismatch model, i.e., $T=\frac{4Z_{1}Z_{2}}{(Z_{1}+Z_{2})^{2}}.$ little1959 Where the acoustic impedance is $Z_{i}=\rho_{i}v_{i}=(m_{i}/a_{i})v_{i}$, and $Z_{i}(\omega=0^{+})=\sqrt{k_{i}m_{i}}$. We note that in acoustic mismatch model the transmission coefficient is frequency independent, and in reality it only applies in the limit of low frequency/long wavelengths. In this case the phonon sees the interface only as a discontinuity between two semi-infinite media and the transmission does not depend on the coupling spring strength $k_{ij}$. If the two leads have the same acoustic impedance for long wave limit, then $T[0^{+}]=1$; otherwise $T[0^{+}]<1$. For a two-junction case, which is shown in Fig. 1, the transmission wave will be reflected and transmitted by the second boundary, leading to multiple reflections. Finally the total transmitted wave function is obtained as a superposition of multiple reflections and transmissions, resulting in the transmission coefficient through the center part $T[\omega]=\frac{{(1-|r_{12}|^{2})(1-|r_{23}|^{2})}}{{|1-r_{23}r_{21}\lambda_{2}^{2(N_{C}-1)}|^{2}}},$ (16) here $r_{ij}$ and $\lambda_{i}$ are determined by Eq. (12) and Eq. (8); $N_{C}$ is the number of atoms in the center atomic chain. From this expression, we can find that the transmission coefficient oscillates with frequency, and is between the envelope lines of maximum and minimum transmission, which are $T_{{\rm max}}[\omega]=(1-|r_{12}|^{2})(1-|r_{23}|^{2})/(1-|r_{23}r_{21}|)^{2}$ for constructive interference and $T_{{\rm min}}[\omega]=/(1-|r_{12}|^{2})(1-|r_{23}|^{2})/(1+|r_{23}r_{21}|)^{2}$ for destructive interference. Figure 2: (color online) The transmission coefficient vs frequency $\omega$ for different interface coupling $k_{12}$ in one-junction chains. (a) shows the transmission in one junction connected by the same semi-infinite atomic chains with $k_{1}=k_{2}=1.0,\;m_{1}=m_{2}=1.0$; the solid, dashed, dotted and dash-dotted lines correspond to $k_{12}=0.1$, 0.5, 1.0 and 2.0, respectively. (b) shows the transmission in one junction connected by two different semi- infinite atomic chains with $k_{1}=1.0,\;m_{1}=1.0,\;k_{2}=3.0$ and $m_{2}=4.0$; the solid, dashed, dotted, dash-dotted and shot-dashed lines correspond to $k_{12}=0.5$, 1.0, 1.5, 3.0 and 8.0, respectively. Figure 3: (color online) The thermal conductance vs interface coupling $k_{12}$ in point-junction model. Here, $k_{1}=1.0,\;m_{1}=1.0$. ## IV Results and Discussions ### IV.1 Thermal transport in 1D one-junction chains In Sec. III, we have derived the analytical expressions for the phonon transmission coefficient for point-junction and extended-junction (two point junction) cases Eq. (11), Eq. (12) and Eq. (16) by using the scattering boundary method. Using these analytical expressions, we analyze the role of various parameters on the thermal transport in one- and two- point junctions. Figure 2 shows the transmission coefficient as a function of frequency for a different interface spring constant $k_{12}$ for the point-junction model. The maximum frequency at which the transmission coefficient is above zero is equal to the minimum of $2\sqrt{k_{1}/m_{1}}$ and $2\sqrt{k_{2}/m_{2}}$. In Fig. 2(a), the two semi-infinite atomic chains have the same mass and spring constant. When the interface coupling $k_{12}$ equals to that of the chains, the transmission is equal to one in the whole frequency domain, because of the homogeneity of the chain structure. If $k_{12}$ increases or decreases, the transmission coefficient decreases. If we set $k_{1}/m_{1}=k_{2}/m_{2}$, the transmission coefficient exhibits similar behavior, the only difference is that the transmission coefficient changes to the value obtained by Eq. (15). In Fig. 2(b), the two semi-infinite atomic chains have different masses and spring constants. The transmission decreases with increased frequency for all the coupling values $k_{12}$. Also, it appears that for a given frequency the transmission is maximized for a $k_{12}$ value residing between $k_{1}$ and $k_{2}$. From Eq. (11) and Eq. (12), $T[\omega]=0$, if $k_{12}=0$; and $T[\omega]$ has definite value $1-|\frac{{k_{1}(\lambda_{1}-1)-k_{2}(1-\lambda_{2}^{-1})}}{{k_{1}(1-\lambda_{1}^{-1})+k_{2}(1-\lambda_{2}^{-1})}}|^{2}$, if $k_{12}=\infty$. Figure 4: (color online) The thermal conductance vs the ratio of $k_{12}/k_{12m}$ in one-junction atomic chain. Here $k_{12m}$ is the harmonic average of the spring constants of the two semi-infinite leads. (a) $k_{1}=1.0$, $m_{1}=m_{2}=1.0$; the solid, dashed, and dotted lines correspond to $k_{2}=0.1$, 1.0, and 40.0, respectively. (b) $k_{1}=1.0$, $m_{1}=1.0$, $k_{2}=10.0$; the solid, dashed, and dotted lines correspond to $m_{2}=0.01$, 1.0, and 100.0, respectively. The maximum transmission concept results in the maximum junction conductance as shown in Fig. 3. With the increasing of $k_{12}$, we find that the conductance will first increase, then arrive at maximum value, and then slightly decrease and at last it will tend to a constant. We find that the maximum transmission or conductance occurs at $k_{12}$ given by $k_{12}=k_{12m}=\frac{2k_{1}k_{2}}{k_{1}+k_{2}},$ (17) i.e., when the coupling spring stiffness is equal to the harmonic average of spring connecting atoms in the two semi-infinite chains. In Fig. 4, we show the thermal conductance vs the ratio of $k_{12}$ and $k_{12m}$. For the two semi-infinite chains with the same mass $m_{1}=m_{2}$, the maximum conductance occurs exactly at $k_{12m}$. If the two leads have different masses $m_{1}\neq m_{2}$, the maximum conductance is almost exactly at the $k_{12m}$ point, for mass ratios ranging from 0.01 to 100. Figure 5: (color online) The transmission coefficient vs frequency for different mass ratios $m_{2}/m_{1}$ at the interface coupling $k_{12m}$. Here, $k_{1}=1.0$, $k_{2}=3.0$, $k_{12}=k_{12m}=1.5$ and $m_{1}=1.0$. Figure 6: (color online) The transmission coefficient vs frequency for different interface coupling $k_{12m}$. Here, $k_{1}=1.0,m_{1}=1.0$, $k_{2}=0.7,m_{2}=0.3$. In Fig. 5, we show the curves of the transmission as a function of frequency for interface coupling equal to $k_{12m}$. If $k_{1}/m_{1}=k_{2}/m_{2}$, that is, when both chains have the same frequency spectrum of $[0,2\sqrt{k_{1}/m_{1}}]$, the transmission equals to a constant $T[\omega]=T[0^{+}]$, which can be seen from the solid line in Fig. 5, and which is consistent with Fig. 2(a). Thus for chains with matched spectra the transmission is frequency independent. Let us now fix $k_{1},k_{2}$ and $k_{2}$, and decrease $m_{2}$. In the range between the point of equal- spectrum ($\omega_{m}=k_{1}/m_{1}=k_{2}/m_{2}$) and the one of equal-impedance ($Z(\omega=0^{+})=k_{1}m_{1}=k_{2}m_{2}$), the transmission will first increase with frequency and then decrease. Otherwise, there is a monotonic decrease. The former behavior is quite interesting, as one expects that the transmission should be the largest in the long wavelength limit. For highly dissimilar materials, the transmission coefficient in the whole frequency range is much larger than that in the long wave limit $T[\omega=0^{+}]=\frac{4Z_{1}Z_{2}}{(Z_{1}+Z_{2})^{2}}$, thus the real conductance is far larger than that calculated from the acoustic mismatch model. This result explain why the interfacial resistance calculated from the acoustic mismatch model is far lager than the experimental value measured at low temperatures, where the phonon transport can be regarded as ballistic transport. Figure 7: (color online)(a) The cutoff frequency vs interface coupling for 1D one-junction atomic chains. The parameters are: $k_{1}=1.0$, $m_{1}=1.0$. (b) The transmission as function of interface coupling for 1D one-junction atomic chains. The parameters are: $k_{1}=1.0$, $m_{1}=1.0$, $k_{2}=0.7$, $m_{2}=0.3$ In many real interfaces, interface coupling is very weak, that is, the $k_{12}$ is less than $k_{12m}$. So it is desirable to study the thermal transport in atomic chains in the weak coupling limit. Figure 6 shows the transmission coefficient as function of interface coupling. In the weak coupling limit, with the frequency increasing, the transmission decreases rapidly to zero, so the frequency region where phonons are effectively transmitted is very narrow. With interface strength increasing, more and more modes contribute to the transmission and the phonon transmission window widens. If the interface coupling increases further, that is $k_{12}/k_{12m}>0.1$, out of the weak interface coupling limit, all the phonons contribute to the transmission. The only further change with increasing $k_{12}$ is the actual values of the transmission coefficients increase. In Fig. 7(a), we show the transmission cutoff frequency as function of the interface coupling. Here, we define the cutoff frequency $\omega_{\rm cutoff}$ at which the transmission $T(\omega_{\rm cutoff})=0.1T(0^{+})$. We find that the cutoff frequency shows linear dependance on interface coupling in the weak coupling limit $k_{12}<0.1k_{12m}$. If the interface strength increase further, the cutoff frequency is saturated. In Fig. 7(b), we show the transmission as function of interface coupling for several different phonons. We find that in the weak interface coupling region, the transmission is proportional to the square of the interface coupling, which is consistent with the formulas Eq. (13) and Eq. (14). Figure 8: (color online) The thermal conductance vs interface coupling for 1D point-junction atomic chains. The parameters are: $k_{1}=1.0$, $m_{1}=1.0$. In the weak interface coupling region, for the 1D atomic one-junction chains, it is shown that the thermal conductance is linear with the interface coupling (see Fig. 8). If we strengthen the interface coupling between the two chains, the conductance will be linearly enhanced. For different mismatched chains, the absolute values of the conductance are different, but dependence on the coupling strength is the same. Figure 9: (color online) The thermal conductance vs interface coupling for 3D one-junction atomic chains. The parameters are the same with Fig. 8. (a) Interface coupling is far less than the coupling $k_{12m}$: $k_{12m}/k_{12}=0.001-0.1$; (b) Interface coupling is in the region of $0.1k_{12m}\sim 0.9k_{12m}$. ### IV.2 Thermal transport in 3D single-interface structures The thermal conductance Eq. (4) can also be written as hopkins2009 : $\sigma=\int_{0}^{\infty}{d\omega\;\hbar\omega\,T[\omega]\frac{\partial f(\omega)}{\partial T}v(\omega)D(\omega)},$ (18) because of $v(\omega)=\partial\omega/\partial k$ and phonon density of states in 1D structure, $D(\omega)=1/(2\pi v)$, we can obtain Eq. (4). In order to estimate the behavior of the interfacial thermal transport across interfaces in 3D structures, we only need to change the phonon density of states in the above equation. Because the density of states for 3D structure within the Debye approximation is $D(\omega)\sim\omega^{2}$, therefore we can replace $\omega$ with $\omega^{3}$ in Eq. (4); the thermal conductance as a function of the coupling strength is shown in Fig. 9. From Fig. 9(a), we find that in the weak interface limit, conductance is proportional to the square of interface coupling, which is consistent with the results from other models scho1980 ; lavr1998 ; prasher2009 , while it is linear dependent on the interface coupling in 1D junctions. This is due to the fact that in 3D low frequency region contributes relatively little to the conductance as the density of states is low there. If the interface coupling increases further, that is $k_{12}/k_{12m}>0.1$, out of the weak interface coupling limit, all the modes contribute to the transmittance, the conductance is no longer proportional to the square of the interface coupling, and the slope continuously decreases. In some intermediate ranges the conductance is approximately proportional to the interfacial coupling (see Fig. 9(b)), which is consistent with the results from molecular simulation approach hu2009 . For stronger coupling the conductances for the 1D case and 3D one have similar behaviors, the slope of both cases will decrease continuously to be zero at point $k_{12m}$, where the conductance will be maximized and then decrease slightly to a limiting value. ### IV.3 Thermal transport in extended junctions Figure 10: (color online) The transmission coefficient of the two-junction atomic chains. Parameters: $k_{1}=1.0,\,m_{1}=1.0,\,k_{2}=0.9,\,m_{2}=1.6,\,k_{3}=4.5,\,m_{3}=2.0$, The solid, dotted, dashed and shot dashed lines correspond to maximum transmission, minimum transmission, $N_{c}=4$ and $N_{c}=9$, respectively. The interface couplings are different: (a) $k_{12}=0.3,k_{23}=0.7$; (b) $k_{12}=1.0,k_{23}=4.5$. Figure 11: (color online) The transmission coefficient of the two-junction atomic chains. Here, $k_{1}=1.0,\,m_{1}=1.0$. The solid, dotted, dashed and shot dashed lines correspond to maximum transmission, minimum transmission, $N_{c}=4$ and $N_{c}=9$, respectively. (a) $k_{2}=3.0,\,m_{2}=5.0,\,k_{3}=1.0,\,m_{3}=1.0,\,k_{12}=k_{23}=1.0$; (b) $k_{2}=3.0,\,m_{2}=1.0,\,k_{3}=5.0,\,m_{3}=1.0,\,k_{12}=k_{12m}=1.5,\,k_{23}=k_{23m}=3.75$; (c) $k_{2}=3.0,\,m_{2}=3.0,\,k_{3}=5.0,\,m_{3}=5.0,\,k_{12}=k_{12m}=1.5,\,k_{23}=k_{23m}=3.75$. Now we focus on a case where the junction is extended and involves a center part. The overall behavior of the transmission is the combination of the transmission behavior in single point-junction case and the oscillatory behavior due to phonon interferences arising form multiple scattering. We show the transmission coefficient as a function of frequency of an arbitrary case in Fig. 10(a). Here, the three chain parts have different masses and spring constants, and the interface coupling is not special. From the analytical expression of Eq. (16), we plot curves of the maximum transmission and minimum transmission, $N_{c}=4$ and $N_{c}=9$. The transmission oscillates between the envelop lines of maximum and minimum transmission. The maximum transmission line will increase first, and the minimum transmission line will monotonically decrease with frequency. However for interface coupling that is the same with the leads, the two envelop lines will monotonically decrease, which can be seen in Fig. 10(b). For some special cases, the transmission coefficient in the frequency domain has interesting phenomena, which are shown in Fig. 11. In Fig. 11(a), the transmission for the case of two identical leads is shown. In this case, the maximum transmission is equal to one, the infinite-long wavelength phonon and the resonance mode can transmit fully through the center part. The minimum transmission is very low, indicating efficient destructive interference. Figure 11(b) shows the transmission when all three parts are different and connected by interface couplings $k_{12m}$ and $k_{23m}$. We find that overall trend for the maximum and minimum transmission lines is increasing first, then decreasing. If, in addition, the ratios of $k_{i}/m_{i}$ are the same for three parts, then the maximum and minimum transmission are constants in the whole frequency range, and the transmission coefficient through finite-size center part oscillate between the two constants, which can be clearly seen in Fig. 11(c). Therefore, we can use the above properties of transmission to design the frequency filters. Figure 12 shows the maximum and minimum transmission coefficient for the filter. If the spring constant of the center part is very different from the ones of the the two leads, the oscillatory peak is sharp, and transmission for most of the frequency will tend to zero, only few resonant frequency can be transmitted. This finding provides guidelines for the design of selective frequency filters. Figure 12: (color online) The maximum and minimum transmission coefficient of the two-junction atomic chains. Here, $k_{1}=1.0,\,m_{1}=1.0$. The solid, dashed lines correspond to maximum transmission and minimum transmission $k_{3}=1.0,\,m_{3}=1.0$, respectively; the dotted and dash-dotted lines correspond to maximum transmission and minimum transmission $k_{3}=5.0,\,m_{3}=5.0$, respectively. The inset shows the transmission coefficient with frequency for different $k_{2}$. $k_{1}=k_{3}=1.0,\,m_{1}=m_{3}=1.0$. The dotted, dashed, and solid lines correspond to $k_{2}=0.5$, 0.1, and 0.02 respectively. For all the curves, $m_{2}=k_{2}$ and $k_{12}=k_{12m},k_{23}=k_{23m}$. ## V Verification by Nonequilibrium Green’s Function Method The NEGF method is an exact approach to study the ballistic thermal transport through junctions. Following the discussion in Sec. II, if we use a transformation for the coordinates, $u_{j}=\sqrt{m_{j}}x_{j}$, which is called the mass-normalized displacement, then the Hamiltonian can be written as $H=\sum\limits_{\alpha=1,2,3}H_{\alpha}+\sum\limits_{\beta=1,3}{U_{\beta}^{T}V_{\beta,2}U_{2}},$ (19) where $H_{\alpha}=\frac{1}{2}\left(P_{\alpha}^{T}P_{\alpha}+U_{\alpha}^{T}K_{\alpha}U_{\alpha}\right)$. $K_{\alpha}$ is the mass-normalized spring constant matrix, and $V_{12}=(V_{21})^{T}$ is the coupling matrix of the left lead to the central region and similarly for $V_{23}$ is the coupling matrix of the right lead to the central region. As stated in Ref. wangjs2006, the element of the coupling matrix $V_{\alpha,\beta}^{ij}$ is equal to $-k_{ij}/sqrt{m_{i}m_{j}}$ which corresponding to the coupling between the $i_{\rm th}$ atom in region $\alpha$ and the $j_{\rm th}$ atom in region $\beta$. Figure 13: (color online) The comparison of the results from scattering boundary method and nonequilibrium Green’s function method for the transmission coefficient in two-junction atomic chains. The square curve and solid line correspond the parameters: $N_{c}=6,\,k_{1}=1.0,\,m_{1}=1.0,\,k_{2}=1.5,\,m_{2}=1.3,\,k_{3}=2.0,\,m_{3}=1.7,\,k_{12}=1.3,\,k_{23}=0.8$; the circle curve and dashed line correspond the parameters: $N_{c}=13,\,k_{1}=1.0,\,m_{1}=1.0,\,k_{2}=1.5,\,m_{2}=1.3,\,k_{3}=4.0,\,m_{3}=2.7,\,k_{12}=1.3,\,k_{23}=0.8$. The square and circle curves are the results from nonequilibrium Green’s function method; The solid and dash lines are the results from scattering boundary method. We can use the nonequilibrium Green’s function method wangjs2008 to study the thermal transport in the atomic chain. We define the contour-ordered Green’s function as $G^{\alpha\beta}(\tau,\tau^{\prime})\equiv-\frac{i}{\hbar}\left\langle{\mathcal{T}\,U_{\alpha}(\tau)U_{\beta}(\tau^{\prime})^{T}}\right\rangle,$ (20) where $\alpha$ and $\beta$ refer to the region that the coordinates belong to and $\mathcal{T}$ is the contour-ordering operator. Then the equations of motion of the Green’s function can be derived. In particular, the retarded Green’s function for the central region in frequency domain is $G^{r}[\omega]=\Bigl{[}(\omega+i\eta)^{2}-K_{2}-\Sigma^{r}[\omega]\Bigr{]}^{-1}.$ (21) Here, $\Sigma^{r}=\sum\limits_{\alpha=1,3}{\Sigma_{\alpha}^{r}}$, and $\Sigma_{\alpha}=V_{2,\alpha}g_{\alpha}V_{\alpha,2}$ is the self-energy due to interaction with the heat bath, $g_{\alpha}^{r}=[(\omega+i\eta)^{2}-K_{\alpha}]^{-1}$. And in the advanced Green’s function $G^{a}=(G^{r})^{\dagger}$, the transmission coefficient can be calculated by the so-called Caroli formula as $T_{\beta\alpha}[\omega]={\rm{Tr}}(G^{r}\Gamma_{\beta}G^{a}\Gamma_{\alpha}),$ (22) where $\Gamma_{\alpha}=i\bigl{(}\Sigma_{\alpha}^{r}[\omega]-\Sigma_{\alpha}^{a}[\omega]\bigr{)}.$ For single-junction atomic chains, if we regard the two atoms in the interface (atom 0 and atom 1) as the center part, then we can still use the formulae above to study the phonon transmission leading to the exact formula yielding the same result with the one obtained from the scattering boundary method. In Appendix A, We give the analytical proof of this fact. For two-junction atomic chains, according to the NEGF formulas, we do the numerical calculation and plot the curves of the transmission coefficient as a function of frequency and compare them to the results obtained the scattering boundary method (see Fig. 13). We find that for any arbitrary case, the results from the NEGF method and the scattering boundary method are exactly the same. If there is no many-body interaction, that is, for the ballistic thermal transport the scattering matrix approach and the Green’s function method give the same results. These two methods are equivalent, which has been proved from other points of view in Refs. khomyakov2005 ; harbola2008 . ## VI Conclusion In this paper, we study the ballistic interfacial thermal transport in atomic junctions, we give the analytical simple formulae Eq. (11), Eq. (12) and Eq. (16) for the transmission of one-junction and two-junction cases, which are consistent with the results from the NEGF method. For one-junction case, we find the transmission and conductance are maximized when the interface spring constant equals to the harmonic average of the two spring constants of the leads. At the point near $k_{12}=k_{12m}$, the transmission $T[\omega]$ is a constant if $k_{2}/m_{2}=k_{1}/m_{1}$; if not equal, in the range between $k_{1}/m_{1}=k_{2}/m_{2}$ and $k_{1}m_{1}=k_{2}m_{2}$, the transmission coefficient increases first then decreases with the increasing of frequency, otherwise the transmission monotonically decreases as the frequency increasing. For weak interface coupling, the cutoff frequency and the interface conductance for 1D chain is linear dependent with the interface coupling strength. Because of different density of states, we change the formula of conductance to mimic the thermal transport in 3D junctions. In weak interface coupling limit, we find that the conductance is proportional to the square of the interface coupling, which is consistent with the results from other models. The slope of the conductance as function of interfacial coupling strength decreases continuously from two to zero, in certain range of which, the conductance is linear proportional to the interface coupling, which are consistent with the results of other molecular simulations. For two-junction case, the transmission will oscillate with frequency in the envelop lines of maximum and minimum transmission which are determined by the one-junction picture. The transmission sometimes oscillates between two decreasing envelop lines, sometimes between two increasing envelop curves, or between two constants, etc. ## Acknowledgements P. K. is supported by the U.S. Air Force Office of Scientific Research Grant No. MURI FA9550-08-1-0407. J.-S. W. acknowledge support from a NUS research grant R-144-000-257-112. ## Appendix A Analytical proof of the equality of the two methods for one junction In this appendix we give the analytical proof for the equality of the scattering boundary method and the non-equilibrium Green’s function approach for the one-junction atomic chains. From the scattering boundary method, we obtain the transmission Eq. (13) and Eq. (14), that is $T[\omega]=\frac{{\sqrt{4k_{2}m_{2}-\omega^{2}m_{2}^{2}}}}{{\sqrt{4k_{1}m_{1}-\omega^{2}m_{1}^{2}}}}\big{|}\frac{{-k_{12}k_{1}(\lambda_{1}-1/\lambda_{1})}}{{(k_{1}-k_{12}-k_{1}/\lambda_{1})(k_{2}-k_{12}-k_{2}/\lambda_{2})-k_{12}^{2}}}\big{|}^{2},$ (23) From the dispersion relation Eq. (6), we can obtain $k_{j}-k_{j}/\lambda_{j}=\omega^{2}m_{j}-k_{j}(1-\lambda_{j})$ (24) ; and $k_{j}^{2}|\lambda_{j}-1/\lambda_{j}|^{2}=\omega^{2}(4k_{j}m_{j}-\omega^{2}m_{j}^{2})$ (25) , So we can get $T[\omega]=\frac{k_{12}^{2}\omega^{2}\sqrt{4k_{1}m_{1}-\omega^{2}m_{1}^{2}}\sqrt{4k_{2}m_{2}-\omega^{2}m_{2}^{2}}}{\big{|}[\omega^{2}m_{1}-k_{1}(1-\lambda_{1})-k_{12}][\omega^{2}m_{2}-k_{2}(1-\lambda_{2})-k_{12}]-k_{12}^{2}\big{|}^{2}}.$ (26) Using the NEGF formulae, we regard the two atoms in the interface (atom 0 and atom 1) as the center part 0, then the dynamic matrix of the center as $K_{0}=\left({\begin{array}[]{*{20}c}\frac{k_{1}+k_{12}}{m_{1}}&\frac{-k_{12}}{\sqrt{m_{1}m_{2}}}\\\ \frac{-k_{12}}{\sqrt{m_{1}m_{2}}}&\frac{k_{12}+k_{2}}{m_{1}}\\\ \end{array}}\right).$ (27) And the coupling matrices between the leads (parts 1 and 2) and the center (part 0) are $V_{01}=(k_{1}/m_{1}\,,\,0)^{T}$ and $V_{02}=(0\,,\,k_{2}/m_{2})^{T}$, and according to Ref. wang2007 , we can obtain the surface Green’s function as $g_{i}^{r}=-\frac{m_{i}\lambda_{i}}{k_{i}},$ (28) here, $i=1,2$ corresponds to the left and right lead. Then we can get the self energy ($\Sigma^{r}=V_{01}g_{1}^{r}V_{10}+V_{02}g_{2}^{r}V_{20}$) as $\Sigma^{r}=\left({\begin{array}[]{*{20}c}-\frac{k_{1}\lambda_{1}}{m_{1}}&0\\\ 0&-\frac{k_{2}\lambda_{2}}{m_{2}}\\\ \end{array}}\right).$ (29) Thus we can calculate the retarded Green’s function of the center $G^{r}=(\omega^{2}I-K_{0}-\Sigma^{r})^{-1}$, which reads as $G^{r}=\left({\begin{array}[]{cc}A_{1}&B\\\ B&A_{2}\\\ \end{array}}\right)^{-1}=\frac{1}{\Delta}\left({\begin{array}[]{cc}A_{2}&-B\\\ -B&A_{1}\\\ \end{array}}\right),$ (30) here, $I$ is two-dimensional identity matrix and $\displaystyle A_{i}=\omega^{2}-\frac{k_{i}}{m_{i}}(1-\lambda_{i})-\frac{k_{12}}{m_{i}};$ (31) $\displaystyle B=\frac{k_{12}}{\sqrt{m_{1}m_{2}}};\;\Delta=A_{1}A_{2}-B^{2}.$ (32) The advanced Green’s function $G^{a}$ equals to $(G^{r})^{\dagger}$. And from the self energy we can get $\Gamma_{1}=\left({\begin{array}[]{*{20}c}C_{1}&0\\\ 0&0\\\ \end{array}}\right);\;\Gamma_{2}=\left({\begin{array}[]{*{20}c}0&0\\\ 0&C_{2}\\\ \end{array}}\right),$ (33) here, $C_{i}=\frac{\omega}{m_{i}}\sqrt{4k_{i}m_{i}-\omega^{2}m_{i}^{2}}$. Therefore, we can calculate the transmission coefficient from the Caroli formula Eq. (22), at last we obtain $T[\omega]=Tr(G^{r}\Gamma_{1}G^{a}\Gamma_{2})=\frac{B^{2}C_{1}C_{2}}{\Delta\Delta^{*}}=\frac{B^{2}C_{1}C_{2}}{|A_{1}A_{2}-B^{2}|^{2}}$ (34) Inserting the values of $A_{i},B$ and $C_{i}$, we get exactly the same result with Eq. (26). Therefore, the results from the scattering boundary method and non-equilibrium Green’s function approach are equivalent. ## References * (1) A Dhar, Adv. Phys. 57, 457 (2008). * (2) M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett.88, 094302 (2002); B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93, 184301 (2004); D. Segal and A. Nitzan, Phys. Rev. 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Phys. 37, 334 (1959). * (13) E. Swartz and R. Pohl, Rev. Mod. Phys.61, 605 (1989). * (14) R. Stevens, A. Smith, and P. Norris,, J. Heat Transfer 127, 315 (2005). * (15) M. E. Lumpkin and W. M. Saslow, Phys. Rev. B. 17,4295 (1997). * (16) B. V. Paranjape, N. Arimitsu, and E. S. Krebes, J. Appl. Phys. 61, 888 (1987). * (17) D. A. Young and H. J. Maris, Phys. Rev. B. 40,3685 (1989). * (18) J. Wang and J.-S. Wang, Phys. Rev. B. 74,054303 (2006). * (19) L. Zhang, J. -S. Wang and B. Li, Phys. Rev. B 78, 144416 (2008). * (20) E. Cuansing and J.-S. Wang, Eur. Phys. J. B 69, 505 (2009). * (21) J. Velev and W. Butler, J. Phys.: Condens. Matter 16, R637 (2004). * (22) W. A. Little, Can. J. Phys. 37, 334 (1959). * (23) P. E. Hopkins, P. M. Norris, M. S. Tsegaye, and A. W. Glosh, J. Appl. Phys. 106, 063503 (2009). * (24) M. Schoenberg, J. Acoust. Soc. Am. 68, 1516 (1980). * (25) A. I. Lavrentyev and S. I. Rokhlin, J. Acoust. Soc. Am. 103, 657 (1998). * (26) R. Prasher, Appl. Phys. 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arxiv-papers
2011-01-27T08:59:36
2024-09-04T02:49:16.650344
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lifa Zhang, Pawel Keblinski, Jian-Sheng Wang, and Baowen Li", "submitter": "Lifa Zhang", "url": "https://arxiv.org/abs/1101.5225" }
1101.5229
# The phonon Hall effect: theory and application Lifa Zhang,1 Jie Ren,2,1 Jian-Sheng Wang,1 and Baowen Li 1,2 1 Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore 2 NUS Graduate School for Integrative Sciences and Engineering, Singapore 117456, Republic of Singapore (6 June 2011) ###### Abstract We present a systematic theory of the phonon Hall effect in a ballistic crystal lattice system, and apply it on the kagome lattice which is ubiquitous in various real materials. By proposing a proper second quantization for the non-Hermite Hamiltonian in the polarization-vector space, we obtain a new heat current density operator with two separate contributions: the normal velocity responsible for the longitudinal phonon transport, and the anomalous velocity manifesting itself as the Hall effect of transverse phonon transport. As exemplified in kagome lattices, our theory predicts that the direction of Hall conductivity at low magnetic field can be reversed by tuning temperatures, which we hope can be verified by experiments in the future. Three phonon-Hall- conductivity singularities induced by phonon-band-topology change are discovered as well, which correspond to the degeneracies at three different symmetric center points, ${\bf\Gamma}$, ${\bf K}$, ${\bf X}$, in the wave- vector space of the kagome lattice. ###### pacs: 63.22.-m 66.70.-f, 72.20.Pa ## 1 Introduction In recent years, phononics, the discipline of science and technology in processing information by phonons and controlling heat flow, becomes more and more exciting [1, 2]. Various functional thermal devices such as thermal diode [3], thermal transistor [4], thermal logic gates [5] and thermal memory [6], etc., have been proposed to manipulate and control phonons, the carrier of heat energy and information. And very recently, similar to the Hall effect of electrons, Strohm _et al._ observed the phonon Hall effect (PHE) – the appearance of a temperature difference in the direction perpendicular to both the applied magnetic field and the heat current flowing through an ionic paramagnetic dielectric sample [7], which was confirmed later in Ref. [8]. Such observation of the PHE is really surprising because phonons as charge- free quasiparticles, different from electrons, cannot directly couple to the magnetic field through the Lorentz force. Since then, several theoretical explanations have been proposed [9, 10, 11, 12] to understand this novel phenomenon. From the work of the PHE in four-terminal nano-junctions and the phonon Hall conductivity in the two-dimensional periodic crystal lattice, we know that the PHE can exist even in the ballistic system. Geometric phase effects [13, 14] are fundamentally important in understanding electrical transport property in quantum Hall effect [15, 16], anomalous Hall effect[17, 18], and anomalous thermoelectric transport [19]. It is successful in characterizing the underlying mechanism of quantum spin Hall effect [20, 21]. Such an elegant connection between mathematics and physics provides a broad and deep understanding of basic material properties. Although there is a quite difference between phonons and electrons, we still can use the topological description to study the underlying properties of the phonon transport, such as topological phonon modes in dynamic instability of microtubules [22] and in filamentary structures [23], Berry-phase-induced heat pumping [24], and the Berry-phase contribution of molecular vibrational instability [25]. The topological nature of the PHE is recently studied in Ref. [26], where a general expression for phonon Hall conductivity is obtained in terms of the Berry curvatures of band structures. In Ref. [26], the authors find a phase transition in the PHE of the honeycomb lattice, explained from topological nature and dispersion relations. From the Green-Kubo formula and considering the contributions from all the phonon bands, the authors obtain the general formula for the phonon Hall conductivity. Then by looking at the phases of the polarization vectors of both the displacements and conjugate momenta as a function of the wave vector, a Berry curvature can be defined uniquely for each band. Combining the above two steps, at last the phonon Hall conductivity can be written in terms of Berry curvatures. Such derivation gives us a clear picture of the contribution to the phonon Hall current from all phonon branches, as well as the relation between the phonon Hall conductivity and the geometrical phase of the polarization vectors, which thus helps us to understand the topological picture of the PHE. However, the process of going from the Berry phase to the heat flux and the phonon Hall conductivity looks not very clear and natural. We know that for the Hall effect of the electrons, in addition to the normal velocity from usual band dispersion contribution, the Berry curvature induces an anomalous velocity always transverse to the electric field, which gives rise to a Hall current, thus the Hall effect occurs [14]. For the magnon Hall effect [27] recently observed, the authors also found the anomalous velocity due to the Berry connection which is responsible for the thermal Hall conductivity. Therefore in this article we will derive the theory of the PHE in a more natural way where the Berry phase effect inducing the anomalous velocity contributes to the extra term of the heat current. Thus the Berry phase effect is straightforward to take the responsibility of the PHE. A kagome lattice, composed of interlaced triangles whose lattice points each have four neighboring points [29], becomes popular in the magnetic community because the unusual magnetic properties of many real magnetic materials are associated with those characteristic of the kagome lattice [30]. The schematic figure of kagome lattice is shown in Fig. 1. In this paper we also apply the PHE theory to the kagome lattice to investigate whether the mechanism of the phase transition found in Ref. [26] is general and how the phonon Hall conductivity, Chern numbers and the dispersion relation behave and relate to each other. In this paper we organize as follows. In Sec. 2, we give a new systematic derivation of the theory of the PHE in terms of Berry curvatures. In this section, we first introduce the Hamiltonian and the modified second quantization, then derive the heat current density operator which includes both the normal velocity and the anomalous velocity from the Berry-phase effect. Using the Green-Kubo formula, the general formula of the phonon Hall conductivity is obtained. Then we give an application example on the kagome lattice in Sec. 3. In this section the computation details about the dynamic matrix, the Chern numbers and the phonon Hall conductivity are given, and the behaviors and relations between the phonon Hall conductivity, Chern numbers, and the band structures are discussed. In the end a short conclusion is presented in Sec. 4. ## 2 The PHE theory In this section, we give the detailed derivation for the theory of the PHE. We use the Hamiltonian in Refs. [26] and [31], which is a positive definite Hamiltonian to describe the ionic crystal lattice with in an applied magnetic field. ### 2.1 The Hamiltonian and the second quantization The Hamiltonian for an ionic crystal lattice in a uniform external magnetic field [26, 28, 31] can be written in a compact form as $\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{1}{2}(p-{\tilde{A}}u)^{T}(p-{\tilde{A}}u)+\frac{1}{2}u^{T}Ku\qquad$ (1) $\displaystyle=$ $\displaystyle\frac{1}{2}p^{T}p+\frac{1}{2}u^{T}(K-{\tilde{A}}^{2})u+u^{T}\\!{\tilde{A}}\,p.$ Here, $u$ is a column vector of displacements from lattice equilibrium positions for all the degrees of freedom, multiplied by the square root of mass; $p$ is the conjugate momentum vector, and $K$ is the force constant matrix. The superscript $T$ stands for the matrix transpose. ${\tilde{A}}$ is an antisymmetric real matrix, which is block diagonal with elements $\Lambda=\left(\begin{array}[]{rr}0&h\\\ -h&0\\\ \end{array}\right)$ (in two dimensions), where $h$ is proportional to the magnetic field, and has the dimension of frequency. For simplicity, we will call $h$ magnetic field later. According to [9], $h$ is estimated to be $0.1\,{\rm cm}^{-1}\approx 3\times 10^{9}\,{\rm rad\,Hz}$ at a magnetic field $\textbf{\emph{B}}=1\,{\rm T}$ and a temperature $T=5.45\,{\rm K}$, which is within the possible range of the coupling strength in ionic insulators [34, 35]. The on-site term, $u^{T}{\tilde{A}}p$, can be interpreted as the Raman (or spin-phonon) interaction. Based on quantum theory and symmetry consideration, the phenomenological description of the spin-phonon interaction was proposed many years ago [34, 35, 32, 33, 36, 37, 38, 39]. From the first row of Eq. (1), we find both of the two terms are positive definite, thus the Hamiltonian (1) is positive definite. The origin of the Hamiltonian for the PHE is discussed in detail in the supplementary information of Ref. [26]. The Hamiltonian Eq. (1) is quadratic in $u$ and $p$. We can write the linear equation of motion as $\displaystyle\dot{p}$ $\displaystyle=$ $\displaystyle-(K-\tilde{A}^{2})u-\tilde{A}p,$ (2) $\displaystyle\dot{u}$ $\displaystyle=$ $\displaystyle p-\tilde{A}u.$ (3) The equation of motion for the coordinate is, $\ddot{u}+2\tilde{A}\dot{u}+Ku=0.$ (4) Since the lattice is periodic, we can apply the Bloch’s theorem $u_{l}=\epsilon e^{i({\bf R}_{l}\cdot{\bf k}-\omega t)}$. The polarization vector $\epsilon$ satisfies $\bigl{[}(-i\omega+A)^{2}+D\bigr{]}\epsilon=0,$ (5) where $D({\bf k})=-A^{2}+\sum_{l^{\prime}}K_{ll^{\prime}}e^{i({\bf R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf k}}$ denotes the dynamic matrix and $A$ is block diagonal with elements $\Lambda$. $D,K_{l,l^{\prime}},$ and $A$ are all $nd\times nd$ matrices, where $n$ is the number of particles in one unit cell and $d$ is the dimension of the vibration. From Eq. (5), we can require the following relations: $\epsilon_{-k}^{*}=\epsilon_{k};\;\omega_{-k}=-\omega_{k}.$ (6) Here, we use the short-hand notation $k=({\bf k},\sigma)$ to specify both the wavevector and the phonon branch, and $-k$ means $(-{\bf k},-\sigma)$. In normal lattice dynamic treatment, we usually take $\sigma,\omega\geq 0$ as a convention, and require $\omega_{\sigma,{\bf k}}=\omega_{\sigma,-{\bf k}}$. For the current problem, this is not true [26, 31]. It is more convenient to have the frequency taking both positive and negative values and require the above equation (6). And from Eq. (3), the momentum and displacement polarization vectors are related through $\mu_{k}=-i\omega_{k}\epsilon_{k}+A\epsilon_{k}.$ (7) Equation (5) is not a standard eigenvalue problem. However, we can describe the system by the polarization vector $x=(\mu,\epsilon)^{T}$, where $\mu$ and $\epsilon$ are associated with the momenta and coordinates, respectively. Using Bloch’s theorem, Eqs. (2) and (3) can be recasted as: $i\frac{\partial}{{\partial t}}x=H_{\rm eff}x,\;\;\;\;\;\;\;\;\;\;H_{\rm eff}=i\left(\begin{array}[]{cc}-A&-D\\\ I_{nd}&-A\end{array}\right).$ (8) Here the $I_{nd}$ is the $nd\times nd$ identity matrix. Therefore, the eigenvalue problem of the equation of motion (8) reads: $H_{\rm eff}\,x_{k}=\omega_{k}\,x_{k},\;\;\;\;\tilde{x}_{k}^{T}\,H_{\rm eff}=\omega_{k}\,\tilde{x}_{k}^{T}.$ (9) where the right eigenvector $x_{k}=(\mu_{k},\epsilon_{k})^{T}$, the left eigenvector ${\tilde{x}}_{k}^{T}=(\epsilon^{\dagger}_{k},-\mu^{\dagger}_{k})/(-2i\omega_{k})$, in such choice the second quantization of the Hamiltonian Eq. (1) holds, which will be proved later. Because the effective Hamiltonian $H_{\rm eff}$ is not hermitian, the orthonormal condition then holds between the left and right eigenvectors, as ${\tilde{x}_{\sigma,{\bf k}}}^{T}\;x_{\sigma^{\prime},{\bf k}}=\delta_{\sigma\sigma^{\prime}}.$ (10) We also have the completeness relation as $\sum_{\sigma}x_{\sigma,{\bf k}}\otimes{\tilde{x}_{\sigma,{\bf k}}}^{T}=I_{2nd}.$ (11) The normalization of the eigenmodes is equivalent to [11] $\epsilon_{k}^{\dagger}\,\epsilon_{k}+\frac{i}{\omega_{k}}\epsilon_{k}^{\dagger}\,A\,\epsilon_{k}=1.$ (12) From the eigenvalue problem Eq. (9), we know that the completed set contains the branch of the negative frequency. And from the topological nature of the PHE [26], the formula of the phonon Hall conductivity can be written in the form comprises the contribution of all the branches including both positive and negative frequency branches. In order to simplify the notation, for all the branches, we define $a_{-k}=a_{k}^{\dagger}.$ (13) The time dependence of the operators is given by: $\displaystyle a_{k}(t)$ $\displaystyle=$ $\displaystyle a_{k}e^{-i\omega_{k}t},$ (14) $\displaystyle a_{k}^{\dagger}(t)$ $\displaystyle=$ $\displaystyle a_{k}^{\dagger}e^{i\omega_{k}t}.$ (15) The commutation relation is $[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{k,k^{\prime}}{\rm sign}(\sigma).$ (16) And we can get $\displaystyle\langle a_{k}^{\dagger}a_{k}\rangle$ $\displaystyle=$ $\displaystyle f(\omega_{k}){\rm sign}(\sigma);$ (17) $\displaystyle\langle a_{k}a_{k}^{\dagger}\rangle$ $\displaystyle=$ $\displaystyle\bigl{[}1+f(\omega_{k})\bigr{]}{\rm sign}(\sigma).$ (18) Here $f(\omega_{k})=(e^{\hbar\omega_{k}/(k_{B}T)}-1)^{-1}$ is the Bose distribution function. The displacement and momentum operators can be written in the following second quantization forms $\displaystyle u_{l}$ $\displaystyle=$ $\displaystyle\sum_{k}\epsilon_{k}e^{i{\bf R}_{l}\cdot{\bf k}}\sqrt{\frac{\hbar}{2N|\omega_{k}|}}\,a_{k};$ (19) $\displaystyle p_{l}$ $\displaystyle=$ $\displaystyle\sum_{k}\mu_{k}e^{i{\bf R}_{l}\cdot{\bf k}}\sqrt{\frac{\hbar}{2N|\omega_{k}|}}\,a_{k}.$ (20) Here, $|\omega_{k}|=\omega_{k}{\rm sign}(\sigma)$. We can verify that the canonical commutation relations are satisfied: $[u_{l},p_{l^{\prime}}^{T}]=i\hbar\delta_{ll^{\prime}}I_{nd}$ by using the completeness Eq. (11) and the commutation relation Eq. (16). The Hamiltonian Eq. (1) then can be written as [31] $H=\frac{1}{2}\sum_{l,l^{\prime}}\tilde{\chi}^{T}_{l}\left(\begin{array}[]{cc}A\delta_{l,l^{\prime}}&K_{l,l^{\prime}}-A^{2}\delta_{l,l^{\prime}}\\\ -I_{nd}\delta_{l,l^{\prime}}&A\delta_{l,l^{\prime}}\end{array}\right)\chi_{l^{\prime}}$ (21) where $\displaystyle\chi_{l}=\left(\begin{array}[]{rr}p_{l}\\\ u_{l}\end{array}\right)$ $\displaystyle=$ $\displaystyle\sqrt{\frac{\hbar}{N}}\sum_{k}x_{k}e^{i{\bf R}_{l}\cdot{\bf k}}c_{k}\;a_{k};$ (24) $\displaystyle\tilde{\chi}_{l}=\left(\begin{array}[]{rr}u_{l}\\\ -p_{l}\end{array}\right)$ $\displaystyle=$ $\displaystyle\sqrt{\frac{\hbar}{N}}\sum_{k}\tilde{x}_{k}e^{-i{\bf R}_{l}\cdot{\bf k}}\tilde{c}_{k}\;a_{k}^{\dagger}.$ (27) Here $c_{k}=\sqrt{\frac{1}{2|\omega_{k}|}}$ and $\tilde{c}_{k}=(-2i\omega_{k})\sqrt{\frac{1}{2|\omega_{k}|}}$. It is easy to verify that $[{\chi}_{l},\tilde{\chi}^{T}_{l^{\prime}}]=-i\hbar\delta_{ll^{\prime}}I_{2nd}$. Because of $e^{i({\bf R}_{l^{\prime}}\cdot{\bf k^{\prime}}-{\bf R}_{l}\cdot{\bf k})}=e^{i({\bf R}_{l}\cdot({\bf k^{\prime}}-{\bf k})+({\bf R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf k^{\prime}})}$ and the definition of the dynamic matrix $D$, then the Hamiltonian can be written as $\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{\hbar}{2N}\sum_{k,k^{\prime},l}e^{i{\bf R}_{l}\cdot({\bf k^{\prime}}-{\bf k})}\tilde{c}_{k}\,c_{k^{\prime}}\tilde{x}_{k}^{T}\left(\begin{array}[]{cc}A&D({\bf k^{\prime}})\\\ -I_{nd}&A\end{array}\right)x_{k^{\prime}}a_{k}^{\dagger}a_{k^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{\hbar}{2N}\sum_{k,k^{\prime},l}e^{i{\bf R}_{l}\cdot({\bf k^{\prime}}-{\bf k})}\tilde{c}_{k}\,c_{k^{\prime}}\tilde{x}_{k}^{T}iH_{\rm eff}x_{k^{\prime}}a_{k}^{\dagger}a_{k^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{k}\hbar|\omega_{k}|a_{k}^{\dagger}a_{k},$ (31) which contains both the positive and negative branches. Here we use the identity $\sum_{l}e^{i{\bf R}_{l}\cdot({\bf k^{\prime}}-{\bf k})}=N\delta_{\bf k^{\prime}k}$ and the eigenvalue problem Eq. (9). Using the relations Eqs. (13) and (16), it is easy to prove that Eq. (2.1) is equivalent to the form $H=\sum_{\sigma>0,{\bf k}}\hbar\omega_{k}(a_{k}^{\dagger}a_{k}+1/2)$ which only includes the nonnegative branches. ### 2.2 The heat current operator The heat current density can be computed as [40]: ${\bf J}=\frac{1}{2V}\sum_{l,l^{\prime}}({\bf R}_{l}\\!-\\!{\bf R}_{l^{\prime}})u^{T}_{l}K_{ll^{\prime}}\dot{u}_{l^{\prime}},$ (32) where $V$ is the total volume of $N$ unit cells. Because of the equation of motion Eq. (3), we can rewrite the heat current as ${\bf J}=\frac{1}{4V}\sum_{l,l^{\prime}}\tilde{\chi}^{T}_{l}{\bf M}_{l\,l^{\prime}}\chi_{l^{\prime}},$ (33) where ${\bf M}_{l\,l^{\prime}}=\left(\begin{array}[]{cc}({\bf R}_{l}-{\bf R}_{l^{\prime}})K_{ll^{\prime}}&-({\bf R}_{l}-{\bf R}_{l^{\prime}})(K_{ll^{\prime}}A+AK_{ll^{\prime}})\\\ 0&({\bf R}_{l}-{\bf R}_{l^{\prime}})K_{ll^{\prime}}\end{array}\right)$ (34) Inserting the Eqs. (24,27), we obtain ${\bf J}=\frac{\hbar}{4VN}\sum_{k,k^{\prime},l,l^{\prime}}\tilde{c}_{k}c_{k^{\prime}}e^{i({\bf R}_{l^{\prime}}\cdot{\bf k^{\prime}}-{\bf R}_{l}\cdot{\bf k})}\tilde{x}^{T}_{k}{\bf M}_{l\,l^{\prime}}x_{k^{\prime}}a_{k}^{\dagger}a_{k^{\prime}},$ (35) Because of $\sum_{l}e^{i{\bf R}_{l}\cdot({\bf k^{\prime}}-{\bf k})}\sum_{l^{\prime}}e^{i({\bf R}_{l^{\prime}}-{\bf R}_{l})\cdot{\bf k^{\prime}}}({\bf R}_{l}-{\bf R}_{l^{\prime}})K_{ll^{\prime}}=iN\delta_{\bf k^{\prime}k}\frac{\partial D}{\partial{\bf k^{\prime}}},$ (36) the heat current can be written as ${\bf J}=\frac{i\hbar}{4V}\sum_{\sigma,\sigma^{\prime},{\bf k}}\tilde{c}_{\sigma,{\bf k}}c_{\sigma^{\prime},{\bf k}}\tilde{x}^{T}_{\sigma,{\bf k}}\frac{\partial H_{\rm eff}^{2}}{\partial{\bf k}}x_{\sigma^{\prime},{\bf k}}a_{\sigma,{\bf k}}^{\dagger}a_{\sigma^{\prime},{\bf k}},$ (37) here we use $\frac{\partial H_{\rm eff}^{2}}{\partial{\bf k}}=\left(\begin{array}[]{cc}\frac{\partial D}{\partial{\bf k}}&-(A\frac{\partial D}{\partial{\bf k}}+\frac{\partial D}{\partial{\bf k}}A)\\\ 0&\frac{\partial D}{\partial{\bf k}}\end{array}\right)$ (38) by making the first derivative of the square of the effective Hamiltonian Eq. (8) with respect to the wave vector ${\bf k}$. From the eigenvalue problem Eq. (9), we have $H_{\rm eff}X=X\Omega;\;\;\tilde{X}^{T}H_{\rm eff}=\Omega\tilde{X}^{T}.$ (39) Where the ${2nd}\times{2nd}$ matrices $X=(x_{1},x_{2},...,x_{2nd})=\\{x_{\sigma}\\}$ (the system has ${2nd}$ phonon branches), $\tilde{X}=\\{\tilde{x}_{\sigma}\\}$, and $\Omega={\rm diag}(\omega_{1},\omega_{2},...,\omega_{2nd})=\\{\omega_{\sigma}\\}$. Because of the completeness relation Eq. (11), $X\tilde{X}^{T}=I_{2nd}$, we get $H_{\rm eff}^{2}=X\Omega^{2}\tilde{X}^{T}.$ (40) By calculating the derivative of the above equation, and using the definition of Berry connection, ${\bf\mathcal{A}}=\tilde{X}^{T}\frac{\partial X}{\partial{\bf k}}.$ (41) Taking the first derivative of Eq. (40) with respect to ${\bf k}$, we obtain $\frac{\partial H_{\rm eff}^{2}}{\partial{\bf k}}=X\left(\frac{\partial\Omega^{2}}{\partial{\bf k}}+[{\bf\mathcal{A}},\Omega^{2}]\right)\tilde{X}^{T}.$ (42) Because of the orthogonality relation between left and right eigenvector Eq. (10), at last we obtain the heat current as ${\bf J}=\frac{i\hbar}{4V}\sum_{\sigma,\sigma^{\prime},{\bf k}}\tilde{c}_{\sigma,{\bf k}}c_{\sigma^{\prime},{\bf k}}a_{\sigma,{\bf k}}^{\dagger}\left(\frac{\partial\Omega^{2}}{\partial{\bf k}}+[{\bf\mathcal{A}},\Omega^{2}]\right)_{\sigma,\sigma^{\prime}}a_{\sigma^{\prime},{\bf k}}.$ (43) The first term $\frac{\partial\Omega^{2}}{\partial{\bf k}}$ in the bracket is a diagonal one corresponding to $\omega_{\sigma}\frac{\partial\omega_{\sigma}}{\partial{\bf k}}$ relating the group velocity. The second term in the bracket $[{\bf\mathcal{A}},\Omega^{2}]$ gives the off-diagonal elements of the heat current density, which can be regarded as the contribution from anomalous velocities similar to the one in the intrinsic anomalous Hall effect. The Berry connection ${\bf\mathcal{A}}$, or we can call it Berry vector potential matrix (the Berry vector potential defined in Ref. [26], ${\bf A}^{\sigma}({\bf k})$, is equal to $i{\bf\mathcal{A}}^{\sigma\sigma}=i\tilde{x}_{\sigma}^{T}\frac{\partial x_{\sigma}}{\partial{\bf k}}$), induces the anomalous velocities to the heat current, which will take the responsibility of the PHE. Therefore, the Berry vector potential comes naturally into the heat current and the PHE. Such a picture is clearer than that in Ref. [26]. ### 2.3 The phonon Hall conductivity Inserting the coefficients $\tilde{c}$ and $c$ to Eq. (43), we get ${\bf J}=\frac{\hbar}{4V}\sum_{\sigma,\sigma^{\prime},{\bf k}}\frac{\omega_{\sigma,{\bf k}}}{\sqrt{|\omega_{\sigma,{\bf k}}\omega_{\sigma^{\prime},{\bf k}}|}}a_{\sigma,{\bf k}}^{\dagger}\left(\frac{\partial\Omega^{2}}{\partial{\bf k}}+[{\bf\mathcal{A}},\Omega^{2}]\right)_{\sigma,\sigma^{\prime}}a_{\sigma^{\prime},{\bf k}}.$ (44) This expression is equivalent to that given in Refs. [26] and [31]. Based on such expression of heat current, the phonon Hall conductivity can be obtained through the Green-Kubo formula [41]: $\kappa_{xy}=\frac{V}{\hbar T}\int_{0}^{\hbar/(k_{B}T)}\\!\\!\\!\\!d\lambda\int_{0}^{\infty}\\!dt\,\bigl{\langle}J^{x}(-i\lambda)J^{y}(t)\bigr{\rangle}_{\rm eq},$ (45) where the average is taken over the equilibrium ensemble with Hamiltonian $H$. The time dependence of the creation and annihilation operators are given as Eqs. (14) and (15), which are also true if $t$ is imaginary. From the Wick theorem, we have $\displaystyle\langle a_{\sigma,{\bf k}}^{\dagger}a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle$ $\displaystyle=$ $\displaystyle\langle a_{\sigma,{\bf k}}^{\dagger}a_{\sigma^{\prime},{\bf k}}\rangle\langle a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle$ (46) $\displaystyle+$ $\displaystyle\langle a_{\sigma,{\bf k}}^{\dagger}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}\rangle\langle a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle$ $\displaystyle+$ $\displaystyle\langle a_{\sigma,{\bf k}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle\langle a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}\rangle.$ Using the properties of the operators $a^{\dagger}$ and $a$ as Eq. (17), we have $\begin{array}[]{ll}\langle a_{\sigma,{\bf k}}^{\dagger}a_{\sigma^{\prime},{\bf k}}\rangle\langle a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle&\\\ =f(\omega_{\sigma,{\bf k}})f(\omega_{\bar{\sigma},{\bf\bar{k}}})\delta_{\sigma\sigma^{\prime}}\delta_{\bar{\sigma}\bar{\sigma^{\prime}}}{\rm sign}(\sigma){\rm sign}(\bar{\sigma}),&\\\ \langle a_{\sigma,{\bf k}}^{\dagger}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}\rangle\langle a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle&\\\ =f(\omega_{\sigma,{\bf k}})(f(\omega_{\sigma^{\prime},{\bf k}})+1)\delta_{\bf\bar{k},-k}\delta_{\sigma,-\bar{\sigma}}\delta_{\sigma^{\prime},-\bar{\sigma^{\prime}}}{\rm sign}(\sigma){\rm sign}(\sigma^{\prime})&\\\ \langle a_{\sigma,{\bf k}}^{\dagger}a_{\bar{\sigma^{\prime}},{\bf\bar{k}}}\rangle\langle a_{\sigma^{\prime},{\bf k}}a_{\bar{\sigma},{\bf\bar{k}}}^{\dagger}\rangle&\\\ =f(\omega_{\sigma,{\bf k}})(f(\omega_{\sigma^{\prime},{\bf k}})+1)\delta_{\bf\bar{k},k}\delta_{\sigma,\bar{\sigma^{\prime}}}\delta_{\sigma^{\prime},\bar{\sigma}}{\rm sign}(\sigma){\rm sign}(\sigma^{\prime}).&\end{array}$ (47) Similar as that in Ref. [26], the diagonal term $\frac{\partial\Omega^{2}}{\partial{\bf k}}$ in the bracket corresponding to $\omega_{\sigma}\frac{\partial\omega_{\sigma}}{\partial{\bf k}}$ has no contribution to the phonon Hall conductivity because which is an odd function of ${\bf k}$. Because of the off-diagonal term $[\mathcal{A}_{k_{\alpha}},\Omega^{2}]_{\sigma,\sigma^{\prime}}=(\omega_{\sigma^{\prime}}^{2}-\omega_{\sigma}^{2})\mathcal{A}_{k_{\alpha}}^{\sigma\sigma^{\prime}}$ (48) and $\mathcal{A}_{k_{\alpha}}^{\sigma\sigma^{\prime}}=\tilde{x}_{\sigma}^{T}\frac{\partial x_{\sigma^{\prime}}}{\partial{k_{\alpha}}}$ from the definition, the phonon Hall conductivity can be written as $\displaystyle\kappa_{xy}$ $\displaystyle=$ $\displaystyle\frac{\hbar}{{8VT}}\sum_{{\bf k},\sigma,\sigma^{\prime}\neq\sigma}[f(\omega_{\sigma})-f(\omega_{\sigma^{\prime}})](\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$ (49) $\displaystyle\times\frac{i}{4\omega_{\sigma}\omega_{\sigma^{\prime}}}\frac{{\epsilon_{\sigma}^{\dagger}\frac{{\partial D}}{{\partial k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial D}}{{\partial k_{y}}}\epsilon_{\sigma}}}{{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}}.$ Here we simplify the notation of the subscripts of $\omega,\epsilon$ which have the same wave vector ${\bf k}$. We can prove $\kappa_{xy}=-\kappa_{yx}$, such that $\kappa_{xy}=\frac{\hbar}{{16VT}}\sum_{{\bf k},\sigma,\sigma^{\prime}\neq\sigma}{[f(\omega_{\sigma})-f(\omega_{\sigma^{\prime}})](\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}B_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}},$ (50) here $\displaystyle B_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}$ $\displaystyle=$ $\displaystyle\frac{i}{{4\omega_{\sigma}\omega_{\sigma^{\prime}}}}\frac{\epsilon_{\sigma}^{\dagger}\frac{{\partial D}}{{\partial k_{x}}}\epsilon_{\sigma^{\prime}}\epsilon_{\sigma^{\prime}}^{\dagger}\frac{{\partial D}}{{\partial k_{y}}}\epsilon_{\sigma}-(k_{x}\leftrightarrow k_{y})}{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}.$ (51) $\displaystyle=$ $\displaystyle i\frac{\tilde{x}_{\sigma}^{T}\frac{{\partial H_{\rm eff}}}{{\partial k_{x}}}x_{\sigma^{\prime}}\tilde{x}_{\sigma^{\prime}}^{T}\frac{{\partial H_{\rm eff}}}{{\partial k_{y}}}x_{\sigma}-(k_{x}\leftrightarrow k_{y})}{(\omega_{\sigma}-\omega_{\sigma^{\prime}})^{2}}.$ $\displaystyle=$ $\displaystyle-i\left(\mathcal{A}_{k_{x}}^{\sigma\sigma^{\prime}}\mathcal{A}_{k_{y}}^{\sigma^{\prime}\sigma}-(k_{x}\leftrightarrow k_{y})\right),$ in the last step we use the relation $\tilde{x}_{\sigma}^{T}\frac{{\partial H_{\rm eff}}}{{\partial k_{x}}}x_{\sigma^{\prime}}=(\omega_{\sigma^{\prime}}-\omega_{\sigma})\tilde{x}_{\sigma}^{T}\frac{{\partial}}{{\partial k_{x}}}x_{\sigma^{\prime}}$ and the definition of ${\bf\mathcal{A}}$ in Eq. (41). And the Berry curvature is $\displaystyle B_{k_{x}k_{y}}^{\sigma}$ $\displaystyle=$ $\displaystyle\sum_{\sigma^{\prime}\neq\sigma}B_{k_{x}k_{y}}^{\sigma\sigma^{\prime}}$ (52) $\displaystyle=$ $\displaystyle-i\sum_{\sigma^{\prime}}\left(\mathcal{A}_{k_{x}}^{\sigma\sigma^{\prime}}\mathcal{A}_{k_{y}}^{\sigma^{\prime}\sigma}-(k_{x}\leftrightarrow k_{y})\right)$ $\displaystyle=$ $\displaystyle i\left(\frac{\partial}{\partial k_{x}}\mathcal{A}_{k_{y}}^{\sigma\sigma}-(k_{x}\leftrightarrow k_{y})\right)$ The definition of Berry curvature here is the same as that of Ref. [26], that is, $B_{k_{x}k_{y}}^{\sigma}=\frac{\partial}{{\partial k_{x}}}{\bf A}_{k_{y}}^{\sigma}-\frac{\partial}{{\partial k_{y}}}{\bf A}_{k_{x}}^{\sigma}$. From the above derivation, we find that a Berry curvature can be defined uniquely for each band by looking at the phases of the polarized vectors of both the displacements and conjugate momenta as functions of the wave vector. If we only look at the polarized vector $\epsilon$ of the displacement, a Berry curvature cannot properly be defined. We need both $\epsilon$ and $\mu$. The nontrivial Berry vector potential takes the responsibility of the PHE. The associated topological Chern number is obtained through integrating the Berry curvature over the first Brillouin zone as $C^{\sigma}=\frac{1}{{2\pi}}\int_{{\rm BZ}}{dk_{x}dk_{y}B_{k_{x}k_{y}}^{\sigma}}=\frac{{2\pi}}{{L^{2}}}\sum\limits_{\bf k}{B_{k_{x}k_{y}}^{\sigma}},$ (53) where, $L$ is the length of the sample. ## 3 Application on the kagome lattice Figure 1: (color online) The schematic picture of kagome lattice. Each unit cell has three atoms such as the number shown 1,2,3. The coupling between the atoms are $K_{01},K_{02},K_{03}$. Each unit cell has six nearest neighbors; the coupling between the unit cell and the neighbors are $K_{1},K_{2},...,K_{6}$. In Ref. [26], we provide a topological understanding of the PHE in dielectrics with Raman spin-phonon coupling for the honeycomb lattice structure. Because of the nature of phonons, the phonon Hall conductivity, which is not directly proportional to the Chern number, is not quantized. We observed a phase transition in the PHE, which corresponds to the sudden change of band topology, characterized by the altering of integer Chern numbers. Such PHE can be explained by touching and splitting of phonon bands. To check whether the mechanism of the PHE is universal, in the following we apply the theory to the kagome lattice, which has been used to model many real materials [30]. ### 3.1 Calculation of the dynamic matrix $D$ In order to calculate the phonon Hall conductivity, we first need to calculate the dynamic matrix $D({\bf k})$, for the two-dimensional kagome lattice. As shown in Fig. 1, each unit cell has three atoms, thus $n=3$. We only consider the nearest neighbor interaction. The spring constant matrix along $x$ direction is assumed as $K_{x}=\left(\begin{array}[]{cc}K_{L}&0\\\ 0&K_{T}\\\ \end{array}\right).$ (54) $K_{L}=0.144\,$eV/(uÅ2) is the longitudinal spring constant and the transverse one $K_{T}$ is 4 times smaller. The unit cell lattice vectors are $(a,0)$ and $(a/2,a\sqrt{3}/2)$ with $a=1\,$Å. To obtain the explicit formula for the dynamic matrix, we first define a rotation operator in two dimensions as: $U(\theta)=\left({\begin{array}[]{*{20}c}{\cos\theta}&{-\sin\theta}\\\ {\sin\theta}&{\cos\theta}\\\ \end{array}}\right).$ The three kinds of spring-constant matrices between two atoms are $K_{01}=K_{x}$ (between atoms 1 and 2 in Fig. 1), $K_{02}=U(\pi/3)K_{x}U(-\pi/3)$ (between atoms 2 and 3), $K_{03}=U(-\pi/3)K_{x}U(\pi/3)$ (between atoms 3 and 1), which are $2\times 2$ matrices. Then we can obtain the on-site spring-constant matrix and the six spring-constant matrices between the unit cell and its nearest neighbors as: $K_{0}=\left({\begin{array}[]{ccc}2(K_{01}+K_{02})&-K_{01}&-K_{02}\\\ -K_{01}&2(K_{01}+K_{03})&-K_{03}\\\ -K_{02}&-K_{03}&2(K_{02}+K_{03})\end{array}}\right),$ $\displaystyle K_{1}$ $\displaystyle=$ $\displaystyle\left({\begin{array}[]{ccc}0&0&0\\\ -K_{01}&0&0\\\ 0&0&0\end{array}}\right),\;K_{2}=\left({\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ -K_{02}&0&0\end{array}}\right),$ (61) $\displaystyle K_{3}$ $\displaystyle=$ $\displaystyle\left({\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&-K_{03}&0\end{array}}\right),\;K_{4}=\left({\begin{array}[]{ccc}0&-K_{01}&0\\\ 0&0&0\\\ 0&0&0\end{array}}\right),$ (68) $\displaystyle K_{5}$ $\displaystyle=$ $\displaystyle\left({\begin{array}[]{ccc}0&0&-K_{02}\\\ 0&0&0\\\ 0&0&0\end{array}}\right),\;K_{6}=\left({\begin{array}[]{ccc}0&0&0\\\ 0&0&-K_{03}\\\ 0&0&0\end{array}}\right),$ (75) which are $6\times 6$ matrices. Finally we can obtain the $6\times 6$ dynamic matrix $D({\bf k})$ as $\displaystyle D({\bf k})$ $\displaystyle=$ $\displaystyle-A^{2}+K_{0}+K_{1}e^{ik_{x}}+K_{2}e^{i(\frac{k_{x}}{2}+\frac{\sqrt{3}k_{y}}{2})}$ (76) $\displaystyle+K_{3}e^{i(-\frac{k_{x}}{2}+\frac{\sqrt{3}k_{y}}{2})}+K_{4}e^{-ik_{x}}$ $\displaystyle+K_{5}e^{i(-\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})}+K_{6}e^{i(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})},$ where, $A^{2}=-h^{2}\cdot I_{6}$, here $I_{6}$ is the $6\times 6$ identity matrix. Figure 2: (color online) The contour map of dispersion relations for the positive frequency bands. For all the insets, the horizontal and vertical axes correspond to wave vector $k_{x}$ and $k_{y}$, respectively. The upper six insets are the dispersion relations for bands 1 to 6 (from left to right) at $h=0$, respectively. And $h=10$ rad/ps for the lower ones. ### 3.2 The PHE and the associated phase transition After we get the expression for the dynamic matrix, we can calculate the eigenvalues and eigenvectors of the effective Hamiltonian. Inserting the eigenvalues, eigenvectors and the $D$ matrix to the formula Eq. (50), we are able to compute the phonon Hall conductivity. As is well known, in quantum Hall effect for electrons, the Hall conductivity is just the Chern number in units of $e^{2}/h$ ($h$ is the Planck constant); thus with the varying of magnetic field, the abrupt change of Chern numbers directly induces the obvious discontinuity of the Hall conductivity. However, for the PHE, there is an extra weight of $(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$ in Eq. (50), which can not be moved out from the summation. As a consequence, the change of phonon Hall conductivity is smoothened at the critical magnetic field. However, in the study on the PHE in the honeycomb lattice system [26], from the first derivative of phonon Hall conductivity with respect to the magnetic field $h$, at the critical point $h_{c}$, we still can observe the divergence (singularity) of $d\kappa_{xy}/dh$, where the phase transition occurs corresponding to the sudden change of the Chern numbers. Can such mechanism be applied for the kagome lattice system? In the following, we give a detailed discussion on it. Inserting the dynamic matrix Eq. (76) to the effective Hamiltonian Eq. (9), we calculate eigenvalues and eigenvectors of the system, and also get the dispersion relation of the system. Because each unit cell has three atoms, and we only consider the two-dimensional motion, we get six phonon branches with positive frequencies. The branches with negative frequencies have similar behavior because of $\omega_{-k}=-\omega_{k}$. We show the contour map of the dispersion relation in Fig. 2. We can see that the dispersion relations have a 6-fold symmetry. For different bands, they are different. With a changing magnetic field, the dispersion relations vary. The point ${\bf\Gamma}$ (${\bf k}=(0,0)$) is the 6-fold symmetric center; the point ${\bf K}$ (${\bf k}=(\frac{4\pi}{3},0)$) is 3-fold symmetric center; and the middle point of the line between two 6-fold symmetric centers, ${\bf X}$ (${\bf k}=(\pi,\frac{\sqrt{3}\pi}{3})$) is a 2-fold symmetric center. In the following discussion, we will see the possible bands touching at these symmetric centers. Figure 3: (color online) The phonon Hall conductivity vs magnetic field at different temperatures. The inset is the zoom-in curve of the phonon Hall conductivity at weak magnetic field. Here the sample size $N_{L}$=400. Using the formula Eq. (50), we calculate the phonon Hall conductivity of the kagome lattice systems, the results are shown in Fig. 3. Similar as shown in Ref. [26], we find the nontrivial behavior of the phonon Hall conductivity as a function of the magnetic field. When $h$ is small, $\kappa_{xy}$ is proportional to $h$, which is shown in the inset of Fig.3; while the dependence becomes nonlinear when $h$ is large. As $h$ is further increased, the magnitude of $\kappa_{xy}$ increases before it reaches a maximum magnitude at certain value of $h$. Then the magnitude of $\kappa_{xy}$ decreases and goes to zero at very large $h$. The on-site term $\tilde{A}^{2}$ in the Hamiltonian (1) increases with $h$ quadratically so as to blockade the phonon transport, which competes with the spin-phonon interaction. Because of the coefficient of $f(\omega_{\sigma})$ in the summation of the formula Eq. (50), the sign of the Hall conductivity will change with temperatures, which is clearly shown in the inset of Fig.3. While the phonon hall conductivity at weak magnetic field is always positive for the honeycomb lattice, the sign reverse of the phonon Hall conductivity with temperature for the kagome lattices is novel and interesting, which could be verified by future experimental measurements. Figure 4: (color online) The Chern numbers and the phonon Hall conductivity vs magnetic field. The dashed line and the dotted line correspond to the Chern numbers of phonon bands 2 and 3 (left scale). The solid line correspond to the phonon Hall conductivity (right scale) at $T=50$ K. We plot the curves of the Chern numbers of bands 2 and 3 as a function of the magnetic field in Fig. 4. The phonon Hall conductivity at $T=50K$ is also shown for comparison. To calculate the integer Chern numbers, large number of ${\bf k}$-sampling points $N$ is needed. However there is always a zero eigenvalue at the ${\bf\Gamma}$ point of the dispersion relation, which corresponds to a singularity of the Berry curvature. Therefore, we cannot sum up the Berry curvature very near this point to obtain Chern number of this band, unless we add a negligible on-site potential $\frac{1}{2}u^{T}V_{\mathrm{onsite}}u$ to the original Hamiltonian [26], which will not change the topology of the space of the eigenvectors. In Fig. 4, we set $V_{\mathrm{onsite}}=10^{-3}K_{L}$. The Chern numbers of bands 2 and 3 have three jumps with the increasing of the magnetic field, although the phonon Hall conductivity is continuous. For other bands, the Chern numbers keep constant: $C^{1}=C^{4}=-1$, $C^{5}=0$, and $C^{6}=1$. For the electronic Hall effect, we know it is quantized because the Hall conductivity is directly proportional to the quantized Chern numbers. Here we also find the quantized effect of the Chern numbers from Fig. 3, while there is no quantized effect for the phonon Hall conductivity. Such difference of the PHE from the electronic Hall effect comes from the different nature of the phonons respective to the electrons. In Eq. (50), in the summation, an extra term $(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$ relating to the phonon energy which is an analog of the electrical charge term $e^{2}$ in the electron Hall effect, can not be moved out from the summation. Combining the Bose distribution, the term $f(\omega_{\sigma})(\omega_{\sigma}+\omega_{\sigma^{\prime}})^{2}$ make the phonon Hall conductivity smooth, no discontinuity comes out although the Chern numbers have some sudden jumps. From the discussion in Ref. [26], the discontinuity of the Chern numbers corresponds to the phase transitions and would relate to the divergency of derivative of the phonon Hall conductivity. Figure 5: (color online) The first derivative of the phonon Hall conductivity $dk_{xy}/dh$ at $T=50K$ and the Chern numbers of bands 2 and 3 in the vicinity of the magnetic fields. The solid line correspond to the $dk_{xy}/dh$ at $T=50$K (left scale); the dashed and dotted lines correspond to the Chern numbers of bands 2 and 3, respectively (right scale). The inset shows the second derivative with respective to the magnetic field $dk_{xy}^{2}/dh^{2}$ (vertical axis) vs magnetic field $h$ (horizontal axis) at $T=50$ K. Figure 6: (color online) The dispersion relations around the critical magnetic fields. (a), (b), and (c) show the dispersion relations along the direction from ${\bf\Gamma}$ (${\bf k}$=(0,0)) to ${\bf K}$ (${\bf k}=(\frac{4\pi}{3},0)$) and to ${\bf X}$ (${\bf k}=(\pi,\frac{\sqrt{3}\pi}{3})$) at the critical magnetic fields $h_{c1}=5.07$rad/ps, $h_{c2}=6.75$rad/ps, and $h_{c3}=20.39$rad/ps, respectively. (d)-(f) show the contour maps of the dispersion relation of band 2 at the three critical magnetic fields. (g)-(i) show the contour maps of the dispersion relation of band 3 at the three critical magnetic fields. The squares with number 1, 2, and 3 are marked for the touching points. In (d), (g) and (e), (h), we only mark one of the six symmetric points by squares of number 1 and 2 for simplicity. Figure 5 shows the curves of the derivative of the phonon Hall conductivity and the Chern numbers at the critical magnetic fields. The first derivative of phonon Hall conductivity has a minimum or maximum at the magnetic fields $h_{c1}=5.07,h_{c2}=6.75,{\rm and}h_{c3}=20.39$ rad/ps for the finite-size sample (the sample has $N=N_{L}^{2}$ unit cells). The first derivative $d\kappa_{xy}/dh$ at the points $h_{c1},h_{c2},h_{c3}$ diverges when the system size increases to infinity [26]. At the three critical points the second derivative $d^{2}\kappa_{xy}/dh^{2}$ is discontinuous, which is shown in the inset of Fig. 5, across which phase transitions occur. For different temperatures, the phase transitions occur at exactly the same critical values. Thus the temperature-independent phase transition does not come from the thermodynamic effect, but is induced by the topology of the phonon band structure, which corresponds to the sudden change of the Chern numbers. While there is one discontinuity of the Chern numbers for the honeycomb lattice system, for the kagome lattice system, there are three ones corresponding to the divergency of the derivative of the phonon conductivity, which can be seen in Fig. 5. The touching and splitting of the phonon bands near the critical magnetic field induces the abrupt change of Chern numbers of the phonon band [26]. In Ref. [26], for the PHE in the honeycomb lattices, we know that band $2$ and $3$ are going to touch with each other at the ${\bf\Gamma}$ point if the magnetic field increases to $h_{c}$; at the critical magnetic field, the degeneracy occurs and the two bands possess the cone shape; above the critical point $h_{c}$, the two bands split up. Therefore, the difference between the two bands decreases below and increases above the critical magnetic field, and is zero at the critical point. The eigenfrequecy difference is in the denominator of the Berry curvature, thus the variation of the difference around the critical magnetic field, dramatically affects the Berry curvature of the corresponding bands. In the kagome lattice systems, we find that the touching and splitting of the phonon bands not only occurs at the ${\bf\Gamma}$ point, but also occurs at other points, which is shown in Fig. 6. At the first critical points $h_{c1}$, the bands 2 and 3 touch at the point ${\bf K}$ (marked by a square with number 1); at $h_{c2}$ the two bands touch at ${\bf X}$ ( marked by a square with number 2); while only for the third critical one $h_{c3}$, band 2 and 3 degenerate at the point ${\bf\Gamma}$ (marked by a square with number 3). From the contour maps of bands 2 and 3, we clearly see that the critical magnetic fields $h_{c1}$, $h_{c2}$, and $h_{c3}$, there are local maximum for the band 2 and the local minimum for the band 3. Thus for all the critical magnetic fields where the Chern numbers have abrupt changes, in the wave-vector space we can always find the phonon bands touching and splitting at some symmetric center points. Therefore, through the study of the PHE in both honeycomb lattices [26] and kagome lattices, we find discontinuous jumps in Chern numbers, which manifest themselves as singularities of the first derivative of the phonon Hall conductivity with respect to the magnetic field. Such associated phase transition is connected with the crossing of band 2 and band 3, which corresponds to the touching between a acoustic band and a optical band. However, we can not observe the similar associated phase transition in triangular lattices because of no optical bands, where the Chern number of each band keeps zero while the phonon Hall conductivity is nonzero because of the nonzero Berry curvatures. ## 4 Conclusion We present a new systematic theory of the PHE in the ballistic crystal lattice system, and give an example application of the PHE in the kagome lattice which is a model structure of the many real magnetic materials. By the proper second quantization for the Hamiltonian, we obtain the formula for the heat current density, which considers all the phonon bands including both positive and negative frequencies. The heat current density can be divided into two parts, one is the diagonal, another is off-diagonal. The diagonal part corresponds to the normal velocity; and the off-diagonal part corresponds to the anomalous velocity which is induced by the Berry vector potential. Such anomalous velocity induces the PHE in the crystal lattice. Based on such heat current density we derive the formula of the phonon Hall conductivity which is in terms of the Berry curvatures. From the application on the kagome lattices, we find that at weak magnetic field, the phonon Hall conductivity changes sign with varying temperatures. It is also found that the mechanism on the PHE about the relation between the phonon Hall conductivity, Chern numbers and the phonon band structure can be generally appllied for kagome lattices. While there is only one discontinuity in PHE of the honeycomb lattices, in the kagome lattices there are three singularities induced by the abrupt change of the phonon band topology, which correspond to the touching and splitting at three different symmetric center points in the wave-vector space. ## Acknowledgements L.Z. thanks Bijay Kumar Agarwalla for fruitful discussions. J.R. acknowledges the helpful communication with Hosho Katsura. This project is supported in part by Grants No. R-144-000-257-112 and No. R-144-000-222-646 of NUS. ## References ## References * [1] Wang L and Li B, Physics World 21, No.3, 27 (2008). * [2] Wang J-S, Wang J, and Lü J T, Eur. Phys. J. B 62, 381 (2008). * [3] Li B, Wang L, and Casati G, Phys. Rev. Lett. 93 184301 (2004); Chang C W, Okawa D, Majumdar A, and Zettl A, Science 314, 1121 (2006). * [4] Li B, Wang L and Casati G, Appl. Phys. Lett. 88, 143501 (2006). * [5] Wang L and Li B, Phys. Rev. Lett. 99, 177208 (2007). * [6] Wang L and Li B, Phys. Rev. Lett. 101, 267203 (2008). * [7] Strohm C, Rikken G L J A, and Wyder P, Phys. Rev. Lett. 95, 155901 (2005). * [8] Inyushkin A V and Taldenkov A N, JETP Lett. 86, 379 (2007). * [9] Sheng L, Sheng D N, and Ting C S, Phys. Rev. Lett. 96, 155901 (2006). * [10] Kagan Y and Maksimov L A, Phys. Rev. Lett. 100, 145902 (2008). * [11] Wang J-S and Zhang L, Phys. Rev. B 80, 012301 (2009). * [12] Zhang L, Wang J-S, and Li B, New J. 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arxiv-papers
2011-01-27T09:09:29
2024-09-04T02:49:16.657344
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lifa Zhang, Jie Ren, Jian-Sheng Wang, and Baowen Li", "submitter": "Lifa Zhang", "url": "https://arxiv.org/abs/1101.5229" }
1101.5280
# Extended Detailed Balance for Systems with Irreversible Reactions A. N. Gorban ag153@le.ac.uk Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK G. S. Yablonsky Parks College, Department of Chemistry, Saint Louis University, Saint Louis, MO 63103, USA ###### Abstract The principle of detailed balance states that in equilibrium each elementary process is equilibrated by its reverse process. For many real physico-chemical complex systems (e.g. homogeneous combustion, heterogeneous catalytic oxidation, most enzyme reactions etc), detailed mechanisms include both reversible and irreversible reactions. In this case, the principle of detailed balance cannot be applied directly. We represent irreversible reactions as limits of reversible steps and obtain the principle of detailed balance for complex mechanisms with some irreversible elementary processes. We prove two consequences of the detailed balance for these mechanisms: the structural condition and the algebraic condition that form together the extended form of detailed balance. The algebraic condition is the principle of detailed balance for the reversible part. The structural condition is: the convex hull of the stoichiometric vectors of the irreversible reactions has empty intersection with the linear span of the stoichiometric vectors of the reversible reactions. Physically, this means that the irreversible reactions cannot be included in oriented cyclic pathways. The systems with the extended form of detailed balance are also the limits of the reversible systems with detailed balance when some of the equilibrium concentrations (or activities) tend to zero. Surprisingly, the structure of the limit reaction mechanism crucially depends on the relative speeds of this tendency to zero. ###### keywords: reaction network , detailed balance , microreversibility , pathway , irreversibility , kinetics ###### PACS: 82.40.Qt , 82.20.-w , 82.60.Hc , 87.15.R- ††journal: Chemical Engineering Science ## 1 Introduction ### 1.1 Detailed Balance for Systems with Irreversible Reactions: the Grin of the Vanishing Cat The principle of detailed balance was explicitly introduced and effectively used for collisions by Boltzmann (1964). In 1872, he proved his $H$-theorem using this principle. In its general form, this principle is formulated for kinetic systems which are decomposed into elementary processes (collisions, or steps, or elementary reactions). At equilibrium, each elementary process should be equilibrated by its reverse process. The arguments in favor of this property are founded upon microscopic reversibility. The microscopic “reversing of time” turns at the kinetic level into the “reversing of arrows”: the elementary processes transform into their reverse processes. For example, the reaction $\sum_{i}\alpha_{i}A_{i}\to\sum_{j}\beta_{j}B_{j}$ transforms into $\sum_{j}\beta_{j}B_{j}\to\sum_{i}\alpha_{i}A_{i}$ and conversely. The equilibrium ensemble should be invariant with respect to this transformation because of microreversibility and the uniqueness of thermodynamic equilibrium. This leads us immediately to the concept of detailed balance: each process is equilibrated by its reverse process. For a given equilibrium, the principle of detailed balance results in a system of linear conditions on kinetic constants (or collision kernels). On the contrary, if we postulate just the existence of an a priori unknown equilibrium state with the detailed balance property then a system of nonlinear conditions on kinetic constants appear. These conditions were introduced in by Wegscheider (1901) and used later by Onsager (1931). They are known now as the Wegscheider conditions. For linear kinetics, the Wegscheider conditions have a very simple and transparent form: for each oriented cycle of elementary processes the product of kinetic constants is equal to the product of kinetic constants of the reverse processes. However, many mechanisms of complex chemical and biochemical reactions, in particular mechanisms of combustion and enzyme reaction, include some irreversible (unidirectional) reactions. In many cases, complex mechanisms consist of some reversible and some irreversible reactions, equilibrium concentrations and rates of reactions become zeroes, and the standard forms of the detailed balance do not have a sense. In physical chemistry, the feasibility of a reaction depends on the energies and entropies of system states, initial, final, and transition ones. Nevertheless, some combinations of irreversible reactions are impossible irrespective of the values of thermodynamic functions. Since Wegscheider’s time it is known that the cyclic sequence of irreversible reactions (the completely irreversible cycle) is impossible. It is forbidden by the principle of detailed balance. In a similar way, the reaction mechanism $A\rightleftharpoons B$, $A\to C$, $C\to B$ is forbidden as well as $A\rightleftharpoons B$, $A\rightleftharpoons C$, $C\to B$. Two fundamental problems can be posed: (1) Which mechanisms with irreversible steps are allowed, and which such mechanisms are forbidden by the principle of detailed balance? In accordance with our knowledge, this question was not answered rigorously and the general problem was not solved. Beside that, the procedure of determining the forbidden mechanisms was not described. (2) Let a mechanism with some irreversible steps be not forbidden. Do we still have some relationships between kinetic constants of this mechanism? In our paper, both problems are analyzed based on the same procedure. Substituting the zero kinetic constants by small, however not zero values we return to the fully ’reversible case’, in which all steps of the reaction mechanism are reversible. Then, we analyze a limit case, in which small kinetic parameters tend to reach 0. Such an idea was applied previously to several examples. In particular, Chu (1971) used this idea for a three-step mechanism, demonstrating that the mechanism $A\rightleftharpoons B$, $A\to C$, $B\to C$ can appear as a limit of reversible mechanisms which obey the principle of detailed balance, whereas the system $A\rightleftharpoons B$, $A\to C$, $C\to B$ cannot appear in such a limit. However, this approach was not applied to the general analysis of multi-step mechanisms, only to a few systems of low dimensions. Since Lewis Carroll’s “Alice’s Adventures in Wonderland”, the Cheshire Cat is well known, in particular its inscrutable grin. Finally this cat disappears gradually until nothing is left but its grin. Alice makes a remark she has often seen a cat without a grin but never a grin without a cat. The detailed balance for systems with irreversible reactions can be compared with this grin of the Cheshire cat: the whole cat (the reversible system with detailed balance) vanishes but the grin persists. ### 1.2 Detailed Balance: the Classical Relations First, let us consider linear systems and write the general first order kinetic equations: $\dot{p}_{i}=\sum_{j}(k_{ij}p_{j}-k_{ji}p_{i})\,.$ (1) Here, $p_{i}$ is the probability of a state $A_{i}$ ($i=1,\ldots,n$) (or, for monomolecular reactions, the concentration of a reagent $A_{i}$). The kinetic constant $k_{ij}\geq 0$ ($i\neq j$) is the intensity of the transitions $A_{j}\to A_{i}$ (i.e., $k_{ij}$ is $k_{i\leftarrow j}$). The rate of the elementary process $A_{i}\to A_{j}$ is $k_{ji}p_{i}$. The class of equations (1) includes the Kolmogorov equation for finite Markov chains, the Master equation in physical kinetics and the chemical kinetics equations for monomolecular reactions. Let $p_{i}^{\rm eq}>0$ be a positive equilibrium distribution. According to the principle of detailed balance, the rate of the elementary process $A_{i}\to A_{j}$ at equilibrium coincides with the rate of the reverse process $A_{i}\leftarrow A_{j}$: $k_{ij}p_{j}^{\rm eq}=k_{ji}p_{i}^{\rm eq}\,.$ (2) For a given equilibrium, $p_{i}^{\rm eq}$, the principle of detailed balance is equivalent to this system (2) of linear equalities. To find the conditions of the existence of such a positive equilibrium that (2) holds, it is sufficient to write equations (2) in the logarithmic form, $\ln p_{i}^{\rm eq}-\ln p_{j}^{\rm eq}=\ln k_{ij}-\ln k_{ji}$, to consider this system as a system of linear equations with respect to the unknown $\ln p_{i}^{\rm eq}$, and to formulate the standard solvability condition. After some elementary transformation this condition gives: a positive equilibrium with detailed balance (2) exists if and only if 1. 1. If $k_{ij}>0$ then $k_{ji}>0$ (reversibility); 2. 2. For each oriented cycle of elementary processes, $A_{i_{1}}\to A_{i_{2}}\to\ldots A_{i_{q}}\to A_{i_{1}}$, the product of the kinetic constants is equal to the product of the kinetic constants of the reverse processes: $\prod_{j=1}^{q}k_{i_{j+1}i_{j}}=\prod_{j=1}^{q}k_{i_{j}i_{j+1}}$ (3) where the cyclic numeration is used, $i_{q+1}=i_{1}$. Of course, it is sufficient to use in (3) a basis of independent cycles (see, for example the review of Schnakenberg (1976)). Let us introduce the more general Wegscheider conditions for nonlinear kinetics and the generalized mass action law. (For a more detailed exposition we refer to the textbook of Yablonskii et al (1991).) The elementary reactions are given by the stoichiometric equations $\sum_{i}\alpha_{ri}A_{i}\to\sum_{j}\beta_{rj}A_{j}\;\;(r=1,\ldots,m)\,,$ (4) where $A_{i}$ are the components and $\alpha_{ri}\geq 0$, $\beta_{rj}\geq 0$ are the stoichiometric coefficients. The reverse reactions with positive constants are included in the list (4) separately. We need this separation of direct and reverse reactions to apply later the general formalism to the systems with some irreversible reactions. The stoichiometric matrix is $\boldsymbol{\Gamma}=(\gamma_{ri})$, $\gamma_{ri}=\beta_{ri}-\alpha_{ri}$ (gain minus loss). The stoichiometric vector $\gamma_{r}$ is the $r$th row of $\boldsymbol{\Gamma}$ with coordinates $\gamma_{ri}=\beta_{ri}-\alpha_{ri}$. According to the generalized mass action law, the reaction rate for an elementary reaction (4) is $w_{r}=k_{r}\prod_{i=1}^{n}a_{i}^{\alpha_{ri}}\,,$ (5) where $a_{i}\geq 0$ is the activity of $A_{i}$. The list (4) includes reactions with the reaction rate constants $k_{r}>0$. For each $r$ we define $k_{r}^{+}=k_{r}$, $w_{r}^{+}=w_{r}$, $k_{r}^{-}$ is the reaction rate constant for the reverse reaction if it is on the list (4) and 0 if it is not, $w_{r}^{-}$ is the reaction rate for the reverse reaction if it is on the list (4) and 0 if it is not. For a reversible reaction, $K_{r}=k_{r}^{+}/k_{r}^{-}$ The principle of detailed balance for the generalized mass action law is: For given values $k_{r}$ there exists a positive equilibrium $a_{i}^{\rm eq}>0$ with detailed balance, $w_{r}^{+}=w_{r}^{-}$. This means that the system of linear equations $\sum_{i}\gamma_{ri}x_{i}=\ln k_{r}^{+}-\ln k_{r}^{-}=\ln K_{r}$ (6) is solvable ($x_{i}=\ln a_{i}^{\rm eq}$). The following classical result gives the necessary and sufficient conditions for the existence of the positive equilibrium $a_{i}^{\rm eq}>0$ with detailed balance (see, for example, the textbook of Yablonskii et al (1991)). ###### Proposition 1. Two conditions are sufficient and necessary for solvability of (6): 1. 1. If $k_{r}^{+}>0$ then $k_{r}^{-}>0$ (reversibility); 2. 2. For any solution $\boldsymbol{\lambda}=(\lambda_{r})$ of the system $\boldsymbol{\lambda\Gamma}=0\;\;\left(\mbox{i.e.}\;\;\sum_{r}\lambda_{r}\gamma_{ri}=0\;\;\mbox{for all}\;\;i\right)$ (7) the Wegscheider identity holds: $\prod_{r=1}^{m}(k_{r}^{+})^{\lambda_{r}}=\prod_{r=1}^{m}(k_{r}^{-})^{\lambda_{r}}\,.$ (8) ###### Remark 1. It is sufficient to use in (8) a basis of solutions of the system (7): $\boldsymbol{\lambda}\in\\{\boldsymbol{\lambda}^{1},\cdots,\boldsymbol{\lambda}^{g}\\}$. ###### Remark 2. The Wegscheider condition for the linear systems (3) is a particular case of the general Wegscheider identity (8). Therefore, the solutions $\boldsymbol{\lambda}$ of equation (7) are generalizations of the (non- oriented) cycles in the reaction networks. The basis of solutions corresponds to the basic cycles. This basis is, obviously, not unique. ###### Remark 3. In equation (6) unknown $x_{i}=\ln a_{i}$ are independent variables and vector $\boldsymbol{x}$ can take any value in $R^{n}$. In practice, this is not always true. For example, for heterogeneous systems with solid components some activities may vary in a narrow interval or may be even constant (see the more detailed discussion below in Section 3.5). We do not study multiphase equilibiria in our paper. ###### Remark 4. All the closed chemical systems have linear conservation laws: conservation of mass, various sorts of atoms, electric charge and other conserved quantities. They are linear functions of the amounts $N_{i}$ of chemical components $A_{i}$. There is a problem of uniqueness and existence of a positive equilibrium with detailed balance or without it for every set of values of the independent conservation laws. To solve this problem we need some properties of the connection between activities and concentrations, $(a_{i})\leftrightarrow(c_{i})$. We do not assume any hypothesis about this connection and study just existence of a positive equilibrium with detailed balance in the space of activities. The Wegscheider identity (8) gives a necessary and sufficient condition for this existence. In practice, very often $k_{r}^{-}=0$ for some $r$, whereas $k_{r}^{+}>0$. In these cases, the standard forms of the detailed balance have no sense. Indeed, let us consider a linear reversible cycle with an irreversible buffer: $A_{1}\rightleftharpoons A_{2}\rightleftharpoons\ldots A_{n}\rightleftharpoons A_{1}\to A_{0}\,.$ This system converges to the state where only $p_{0}>$ and $p_{i}=0$ for $i>0$. In this state, trivially, $w_{r}^{+}=w_{r}^{-}=0$ and it seems that the standard principle of detailed balance does not imply any restriction on the kinetic constants. Of course, this impression is wrong. Let us consider this system as a limit of the system with a reversible buffer, $A_{1}\rightleftharpoons A_{0}$ (both reaction rate constants are positive), when the constant of the reverse reaction is positive but tends to zero: $k_{1\leftarrow 0}\to 0$, $k_{1\leftarrow 0}>0$. For each positive value $k_{1\leftarrow 0}>0$ the condition of detailed balance $w_{r}^{+}=w_{r}^{-}$ gives the Wegscheider identity (3) for the cycle $A_{1}\rightleftharpoons A_{2}\rightleftharpoons\ldots A_{n}\rightleftharpoons A_{1}$: The product of direct reaction rate constants is equal to the product of the reverse reaction rate constants. This condition holds also in the limit $k_{1\leftarrow 0}\to 0$. So, any practically negligible but positive value of the reverse kinetic constant implies the nontrivial Wegscheider condition for the other constants. If we assume that the negligible values of the constants should not affect the kinetic systems then this Wegscheider condition should hold for the system with fully irreversible steps as well. Therefore, the following way for the formalization of the principle of detailed balance for irreversible reactions is proposed. We return to reversible reactions in which the principle of detailed balance is assumed by the introduction of small $k_{r}^{-}>0$. Then we go to the limit $k_{r}^{-}\to 0$ ($k_{r}^{-}>0$) for these reactions. It is worth mentioning that the free energy has no limit when some of the reaction equilibrium constants tend to zero. For example, for the ideal gases $F=\sum_{i}N_{i}(RT\ln c_{i}+\mu_{i}^{0}-RT)$, where $c_{i}$ is the concentration, $N_{i}$ is the amount and $\mu_{i}^{0}$ is the standard chemical potential of the component $A_{i}$. In the irreversible limit some $\mu_{i}^{0}\to\infty$. On the contrary, the activities $a_{i}=\exp\left(\frac{\mu_{i}-\mu_{i}^{0}}{RT}\right)\,$ (9) remain finite (for the ideal gases, for example, $a_{i}=c_{i}$) and the approach based on the generalized mass action law and the equations $w_{r}^{+}=w_{r}^{-}$ can be applied in the irreversible limit. Below, we study systems with irreversible reactions as the limits of the systems with reversible reactions and detailed balance, when some reaction rate constants go to zero. We formulate the restrictions on the constants in this limit and find the finite number of conditions that is necessary and sufficient to check. First of all we have to discuss the necessary notion of cycles for general reaction networks. ### 1.3 Main Results We develop three approaches to the detailed balance conditions for the systems with some irreversible reactions. The first and the most physical idea is to consider an irreversible reaction as a limit of a reversible reaction when the reaction rate constant for a reverse reaction tends to zero. The limits of systems of reversible reactions with detailed balance conditions cannot be arbitrary systems with some irreversible reactions and we study the structural and algebraic restrictions for these systems. The second approach is based on the technical idea to study the limits of the Wegscheider identities (8). Here, it is very useful to apply the concept of the general (nonlinear) irreversible cycles or pathways developed recently far enough for our purposes by Schuster et al (2000); Gagneur & Klamt (2004) and other. Let us write all reactions separately (including direct and reverse reactions) (4). The general oriented cycle or pathway is a relation between vectors $\gamma_{r}$ with non-negative coefficients : $\sum_{r}\lambda_{r}\gamma_{r}=0$, $\lambda_{r}\geq 0$ and $\sum_{r}\lambda_{r}>0$. For each system with all reversible reactions and detailed balance the Wegscheider identity (8) holds for any oriented cycle. Therefore, if an oriented cycle persists in the limit with some irreversible reactions, then, for $\lambda_{r}>0$, the $r$th reaction should remain reversible and for this cycle the Wegscheider condition persists. This property motivates the definition of the extended (or weakened, Yablonsky et al (2010)) form of detailed balance in Section 3.1 through the general oriented cycles and the Wegscheider conditions. Theorem 1 states that a system satisfies the extended form of detailed balance if and only if it is a limit of systems with all reversible reactions and detailed balance. One part of this theorem (“if”) is proved immediately in Section 3.1, the proof of the second part (“only if”) exploits the third approach and is postponed till Section 4. The third idea is to study the limits when some equilibrium concentrations (or, more general, activities) tend to zero. For systems with all reversible reactions, we can explicitly express the constants of the reverse reactions through the constants of the direct reactions and the equilibrium activities: just use the detailed balance conditions, $w^{+}(a^{\rm eq})=w^{-}(a^{\rm eq})$. Here, instead of 2$m$ parameters, $k^{\pm}_{r}$ ($m$ is the number of reactions) we use $m+n$ parameters: $m$ reaction rate constants $k^{+}_{r}$ and $n$ equilibrium activities $a_{i}^{\rm eq}$. In this description of the reversible reactions, the principle of detailed balance is trivially satisfied. Some reactions become irreversible in the limits when some of the equilibrium activities tends to zero. Surprisingly, the structure of the limit reaction mechanism crucially depends on the relative speeds of this tendency to zero. In Section 4, we assume that $a_{i}^{\rm eq}={\rm const_{i}}\times\varepsilon^{\delta_{i}}$ and study the limits $\varepsilon\to 0$. The $n$-dimensional space of exponents $\delta=(\delta_{i})$ is split by $m$ hyperplanes $(\gamma_{r},\delta)=0$ on convex cones. Each of these cones is given by a set of inequalities $(\gamma_{r},\delta)\lesseqqgtr 0$ ($r=1,\ldots,m$). In every such a cone, the limit reaction mechanism for $\varepsilon\to 0$ is constant. Using this approach, we prove the second part of Theorem 1 and even more: if a system satisfies the extended form of detailed balance then it may be obtained in the limit $\varepsilon\to 0$ from a system with all reversible reactions with given $k_{r}^{+}$ and $a_{i}^{\rm eq}={\rm const_{i}}\times\varepsilon^{\delta_{i}}$ for some exponents ${\delta_{i}}$ (Theorem 4). So, all the three approaches to the consequences of the principle of detailed balance for the systems with some irreversible reactions are equivalent. The computational problem associated with the extended form of detailed balance is nontrivial. For example, the oriented cycles (pathways) form a convex polyhedral cone and we have to formulate the structural condition of the extended form of detailed balance for all extreme rays (extreme pathways) of this cone (Theorem 2): if $\lambda_{r}>0$ for a vector $\boldsymbol{\lambda}$ from an extreme ray then the $r$th reaction should remain reversible. Calculation of all these extreme rays is a well known and computational expensive problem (Fukuda & Prodon, 1996; Papin et al, 2003; Gagneur & Klamt, 2004). In Theorem 3, we significantly reduce the dimension of the problem. Instead of the set of all stoichiometric vectors $\gamma_{r}$ ($r=1,\ldots,m$) in the whole space of composition $\mathbb{R}^{n}$ ($n$ is the number of components, $m$ is the number of reactions) it is sufficient to consider the set of the stoichiometric vectors of the irreversible reactions in the quotient space $\mathbb{R}^{n}/S$, where $S$ is spanned by the stoichiometric vectors of all reversible reactions. The simple exclusion of the linear conservation laws provides additional dimensionality reduction. The application of reduction methods is demonstrated in the case study in Section 3.5. In Section 3.3, we formulate the main results for the simple case of linear (monomolecular) systems. Sections 3.4 and 3.5 are devoted to examples of nonlinear systems. In Section 3.4, the simple examples with obvious lists of the extreme pathways are collected. In Section 3.5, we analyze the possible irreversible limits for a complex reaction of methane reforming with CO2. ## 2 Cycles and Pathways in General Reaction Networks There exist several graph representations of general reaction networks (Yablonskii et al, 1991; Temkin et al, 1996) and each of them implies the correspondent notion of a cycle. For example, each input and output formal sum in the reaction mechanism (4) can be considered as a vertex (a complex) and then a reaction with the positive rate constant is an oriented edge. This graph of the transformation of complexes is convenient for the analysis of the complex balance condition (Feinberg, 1972). The bipartite graphs of reactions (Volpert & Khudyaev, 1985) gives us another example: it includes two types of vertices: components (correspond to $A_{i}$) and reactions (correspond to elementary reactions from (4)). There is an edge from the $i$th component to the $s$th reaction if $\alpha_{ri}>0$ and from the $s$th reaction to the $i$th component if $\beta_{ri}>0$. The correspondent stoichiometric coefficients are the weights of the edges. This graph is convenient for the analysis of the system stability, for calculation of Jacobians for the right hand sides of the kinetic equations and for analysis of their signs (Ivanova, 1979; Mincheva & Roussel, 2007). For nonlinear systems, these graphs do not give a simple representation of the detailed balance conditions. We need a special notion of a cycle which corresponds to the algebraic relations between reactions. Let us recall that we include direct and inverse reactions in the reaction mechanism (4) separately. Each solution of (7) may be represented as follows: $\begin{split}+&\left|\underline{\begin{array}[]{l}\lambda_{1}\times\left(\sum_{i}\alpha_{1i}A_{i}\to\sum_{j}\beta_{1j}A_{j}\right)\\\ \lambda_{2}\times\left(\sum_{i}\alpha_{2i}A_{i}\to\sum_{j}\beta_{2j}A_{j}\right)\\\ \cdots\\\ \lambda_{m}\times\left(\sum_{i}\alpha_{mi}A_{i}\to\sum_{j}\beta_{mj}A_{j}\right)\end{array}}\right.\\\ &\;\;\;\;\;\;\;\;\;\;\;=\;\sum_{i}a_{i}A_{i}\to\sum_{j}a_{j}A_{j}\;.\end{split}$ (10) Here, $a_{i}=\sum_{s}\lambda_{s}\alpha_{si}\equiv\sum_{s}\lambda_{s}\beta_{si}$. Therefore, we need the following definition of a cycle. ###### Definition 1. An oriented cycle is a vector of coefficients $\boldsymbol{\lambda}\neq 0$ with all $\lambda_{i}\geq 0$ that satisfies (10). ###### Remark 5. Cycles in catalysis and, especially, in biochemistry are called pathways (Schuster et al, 2000; Papin et al, 2003). An oriented pathway is an oriented cycle from Definition 1. An extreme (oriented) pathway is a direction vector of an extreme ray of the cone $\Lambda_{+}$. A solution of equation (7) (a non-oriented cycle) is a non-oriented pathway. Qualitatively these concepts have been introduced in the early 1940s by Horiuti who applied them to heterogeneous catalytic reactions (Horiuti, 1973). Horiuti used them to eliminate intermediates of the complex catalytic reaction by adding the steps of the detailed mechanism first multiplied by special coefficients. As result of such procedure, the chemical equation with no intermediates is obtained. All oriented cycles form the cone $\Lambda_{+}$ (without the origin). Extreme ray of a convex cone is a face that is, at the same time, a ray. Each ray may be defined by a directional vector $\boldsymbol{\lambda}$ that is an arbitrary nonzero vector from this ray. The cone $\Lambda_{+}$ is defined by a finite system of linear equations (7) and inequalities $\lambda_{r}\geq 0$. Therefore, it has a finite set of extreme rays. For integer stoichiometric coefficients, $\alpha_{si}$, $\beta_{si}$, any extreme ray is defined by an uniform linear systems of equations with integer coefficients supplemented by the conditions $\lambda_{i}\geq 0$ and $\boldsymbol{\lambda}\neq 0$. Therefore, we can always select a direction vector with the integer coefficients. For each extreme ray, there exists a unique direction vector with minimal integer coefficients. For monomolecular reaction networks, these cycles coincide with the oriented cycles in the graph of reactions (where vertices are reagents and edges are reactions). There exists an oriented cycle of the length two for each pair of mutually reverse reactions. For these cycles the Wegscheider identities (8) hold trivially, for any positive values of $k^{\pm}$. ###### Remark 6. The systems without oriented cycles ($\Lambda_{+}=\\{0\\}$) have a simple dynamic behavior. First of all, for such a system the convex hull of the stoichiometric vectors does not include zero: $0\notin{\rm conv}\\{\gamma_{1},\ldots,\gamma_{m}\\}$. Therefore, there exists a linear functional $l$ that separates 0 from $\\{\gamma_{1},\ldots,\gamma_{m}\\}$: $l(\gamma_{s})>0$ for all $s=1,\ldots,m$. This linear function $l(c)$ increases monotonically due to any kinetic equation $\frac{{\mathrm{d}}c}{{\mathrm{d}}t}=\sum_{s}w_{s}\gamma_{s}$ with reaction rates $w_{s}\geq 0$: ${\mathrm{d}}l(c)/{\mathrm{d}}t>0$ if at least one reaction rate $w_{s}>0$. ## 3 Extended Form of Detailed Balance ### 3.1 Definition A practically important reaction mechanism may include reversible and irreversible steps. However, some mechanisms with irreversible steps may be wrong because they cannot appear as the limits of reversible mechanisms with detailed balance. Therefore, the first question is about the mechanism structure: what is allowed? The second question is about the constants: let the mechanism not be forbidden. If it is the limit of a system with detailed balance then the reaction rate constants may be connected by additional algebraic conditions like the Wegscheider conditions (3). We should describe all the necessary conditions. In this Section we answer both questions and formulate both conditions, structural and algebraic. We have to study study the identities (8) in the limit when some $k_{r}^{-}\to 0$. First of all, let us consider reversible reactions: if $k_{r}^{+}>0$ then $k_{r}^{-}>0$. It is sufficient to use in (8) only $\boldsymbol{\lambda}$ with nonnegative coordinates, $\lambda_{r}\geq 0$. Indeed, the direct and reverse reactions are included in the list (4) under different numbers. Assume that $\lambda_{r}<0$ in an identity (8) for some $r$. Let the reverse reaction for this $r$ have number $r^{\prime}$. Let us substitute $(k^{+}_{r})^{\lambda_{r}}$ in the left hand side of (8) by $(k^{+}_{r^{\prime}})^{-\lambda_{r}}$ and $(k^{-}_{r})^{\lambda_{r}}$ in the right hand side by $(k^{-}_{r^{\prime}})^{-\lambda_{r}}$. The new condition is equivalent to the previous one. Let us iterate this operation for various $r$. In the finite number of steps all the powers $\lambda_{r}\geq 0$. Let us use notation $\Lambda$ for the linear space of solutions of (7) and $\Lambda_{+}$ for the cone of positive solutions $\boldsymbol{\lambda}$ ($\lambda_{r}\geq 0$) of (7). For reversible reactions, we proved the following proposition. Let the reactions are reversible and the direct and reverse reactions are included in the list (4) separately. ###### Proposition 2. The Wegscheider identity (8) holds for all $\boldsymbol{\lambda}\in\Lambda$ if and only if it holds for all positive $\boldsymbol{\lambda}\in\Lambda_{+}$. Elementary linear algebra gives the following corollary for reversible reactions. ###### Corollary 1. The solution of the system of linear equations for logarithms of equilibrium activities (6) exists if and only if for any positive solution $\boldsymbol{\lambda}$ ($\lambda_{r}\geq 0$) of the system $\boldsymbol{\lambda}\boldsymbol{\Gamma}=0$ (7) the condition (8) holds. Let us study identity (8) for a positive $\boldsymbol{\lambda}$ when some of $k_{r}\to 0$. In this limit, for every $\boldsymbol{\lambda}\in\Lambda_{+}$ Corollary 1 gives two conditions: ###### Corollary 2. Let $k_{s}>0$, $k_{s}\to k_{s}^{\rm lim}$ and the Wegscheider identity (8) holds for $k_{s}$. Then 1. 1. If $\lambda_{s}>0$ and $k_{s}^{+}\to 0$ for some $s$ then for some $q$ $\lambda_{q}>0$ and $k_{q}^{-}\to 0$; 2. 2. If for all positive components $\lambda_{s}>0$ the limit constants are positive, $k_{s}^{\rm lim\,\pm}>0$, then the condition (8) holds for $k_{s}^{\rm lim\,\pm}$. We can interpret the positive solutions of (7) as oriented cycles (linear or nonlinear). The first condition means that if a cycle is cut by the limit $k_{s}^{+}\to 0$ in one direction then it should be also cut by a limit $k_{q}^{-}\to 0$ in the opposite direction: the irreversible cycle is forbidden. This remark leads to the definition of the structural condition of the extended form of detailed balance. ###### Definition 2. A system of reactions (4) satisfies the structural condition of the extended form of detailed balance if for every $\boldsymbol{\lambda}\in\Lambda_{+}$ the reaction which satisfy $\lambda_{s}>0$ are reversible: if $\lambda_{s}>0$ then $k_{s}^{\pm}>0$. This condition means that all cycles should be reversible. The second condition means that for all cycles $\boldsymbol{\lambda}\in\Lambda_{+}$ which persist in the system with irreversible reactions the correspondent Wegscheider condition (8) holds. This is the algebraic condition of the extended form of detailed balance. Now, we are ready to formulate the definition of the extended form of detailed balance. ###### Definition 3. The subsystem satisfies the extended form of detailed balance if both the structural and the algebraic condition hold for all $\boldsymbol{\lambda}\in\Lambda_{+}$. The following theorem gives the motivation to this definition. ###### Theorem 1. A system with irreversible reactions is a limit of systems with reversible reactions and detailed balance if and only if it satisfies the extended form of detailed balance. ###### Proof. Let us prove the direct statement: if a system is a limit of systems with reversible reactions and detailed balance then it satisfies the extended form of detailed balance. Indeed, let a system of reactions be a limit of systems with reversible reactions and detailed balance. This means that for each $j=1,2,\ldots$ a set of reaction rate constants $k_{s,j}^{\pm}>0$ is given, $k_{s,j}^{\pm}>0$ satisfy the principle of detailed balance for all $j$ and $k_{s}^{\pm}=\lim_{j\to\infty}k_{s,j}^{\pm}\,.$ Assume that the structural condition is violated: there exists such a $\boldsymbol{\lambda}\in\Lambda_{+}$ that $\lambda_{s}>0$ for an irreversible reaction ($k_{s}^{+}>0$, $k_{s}^{-}=0$). For all $j=1,2,\ldots$ the principle of detailed balance gives: $\prod_{r,\,\lambda_{r}>0}(k_{r,j}^{+})^{\lambda_{r}}=\prod_{r,\,\lambda_{r}>0}(k_{r,j}^{-})^{\lambda_{r}}\,.$ (11) If $\lambda_{r}>0$ then $k_{r}^{+}>0$. Therefore, for these $r$, sufficiently large $j$ and some $\varepsilon,\delta>0$ $\delta>k_{r,j}^{\pm}>\varepsilon>0$. The left hand side of (11) is separated from zero. The right hand side of (11) tends to zero because all factors are bounded and at least one of them tends to zero, $k_{r,j}^{-}\to 0$. This contradiction proves the structural condition. To prove the algebraic condition, it is sufficient to notice that the Wegscheider identity for $k_{s,j}^{\pm}>0$ holds for all $j$, hence, it holds in the limit $j\to\infty$. We will prove the converse statement (if a system satisfies the extended form of detailed balance then it is a limit of systems with reversible reactions and detailed balance) in Section 4, in the proof of Theorem 4. ∎ ### 3.2 Criteria All $\boldsymbol{\lambda}\in\Lambda_{+}$ participate in the definition of the extended form of detailed balance. Nevertheless, it is sufficient to use a finite subset of this cone. We can check directly that if for a set $\\{\boldsymbol{\lambda}^{s}\\}$ the structural and the algebraic conditions of the extended form of detailed balance hold then they hold for any conic combination of $\\{\boldsymbol{\lambda}^{s}\\}$, $\boldsymbol{\lambda}=\sum_{s}a_{s}\boldsymbol{\lambda}^{s}$, $a_{s}\geq 0$. Therefore, it is sufficient to check the conditions for the directional vectors of the extreme rays of $\Lambda_{+}$. Let a reaction mechanism satisfy the extended principle of detailed balance. If we delete from this mechanisms any irreversible elementary reaction or any couple of mutually reverse elementary reactions, the resulting mechanism satisfies the extended principle of detailed balance as well. A cone is pointed if the origin is its extreme point or, which is the same, this cone does not include a whole straight line. The cone $\Lambda_{+}$ is pointed because it belongs to the positive orthant $\\{\boldsymbol{\lambda}\ |\ \boldsymbol{\lambda}\geq 0\\}$. It is a standard task of linear programming and computational convex geometry to find all the extreme rays of the polyhedral pointed cone $\Lambda_{+}$ (Bertsimas & Tsitsiklis, 1997; Motzkin et al, 1953; Fukuda & Prodon, 1996). Let the directional vectors of these extreme rays be $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$. ###### Theorem 2. The system satisfies the extended form of detailed balance if and only if the structural and algebraic conditions hold for the directional vectors $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$ of the extreme rays of the cone $\Lambda_{+}$. Theorem 2 follows just from the definition of extreme rays and the Minkowski theorem which states that every pointed cone given by linear inequalities admits a unique representation as a conic hull of a finite set of extreme rays. This criterion can be simplified as well: it is necessary and sufficient to check the structural conditions for the extreme rays of $\Lambda_{+}$ and then the algebraic condition for a maximal linear independent subset of $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$. ###### Corollary 3. The system satisfies the extended form of detailed balance if and only if the structural conditions hold for all directional vectors $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$ of the extreme rays of the cone $\Lambda_{+}$ and the algebraic conditions hold for a maximal linear independent subset of $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$. If, for a given reaction mechanism, the set $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$ of directional vectors of the extreme rays of $\Lambda_{+}$ is known, then it is easy to check, whether this mechanism satisfies the structural conditions of the extended form of detailed balance. It is sufficient to examine for each ${\lambda}^{s}_{r}>0$, whether $k_{r}^{-}>0$. After these conditions are examined, it is a simple task to extract the independent set of the Wegscheider identities that should be valid: just select a maximal linear independent subset from the set of $\boldsymbol{\lambda}^{s}$ and write the correspondent Wegscheider identities. It is convenient to use all the extreme pathways especially if we would like to study all the subsystems of the given system, which satisfy the extended form of detailed balance. On the other hand, it is computationally expensive to find the set $\\{\boldsymbol{\lambda}^{s}\ |\ s=1,\cdots,q\\}$ (see, for example, the paper by Gagneur & Klamt (2004)). The amount of computation could be significantly reduced because it is not necessary to use all the extreme pathways. Let us consider a reaction mechanism, which includes both reversible and irreversible reactions. For the reversible reactions, let us join the direct and reverse reactions. Let $\gamma_{1},\ldots,\gamma_{r}$ be the stoichiometric vectors of the reversible reactions and $\nu_{1},\ldots,\nu_{s}$ be the stoichiometric vectors of the irreversible reactions. We use $\boldsymbol{\Gamma}_{r}$ for the stoichiometric matrix of the reversible reactions and $\Lambda_{r}$ for the solutions of the equations $\boldsymbol{\lambda}\boldsymbol{\Gamma}_{r}=0$. The linear subspace $S={\rm span}\\{\gamma_{1},\ldots,\gamma_{r}\\}\subset\mathbb{R}^{n}$ consists of all linear combinations of the stoichiometric vectors of the reversible reactions. Let us consider the quotient space $\mathbb{R}^{n}/S$. We use notation $\overline{\nu}_{j}$ for the images of $\nu_{j}$ in $\mathbb{R}^{n}/S$. The following theorem gives the criteria of the extended form of detailed balance, which are more efficient for computations. ###### Theorem 3. The system satisfies the extended form of detailed balance if and only if 1. 1. The convex hull of the stoichiometric vectors of irreversible reactions does not intersect $S$, i.e. $0\notin{\rm conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}\,;$ (12) 2. 2. The reversible reactions satisfy the principle of detailed balance. ###### Proof. Let the condition 1 be violated, i.e. $0\in{\rm conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}$. In this case, there exist such a nonnegative $\theta_{i}\geq 0$ that $\sum_{j=1}^{s}\theta_{j}=1$ and $\sum_{j=1}^{s}\theta_{j}\nu_{j}\in S$. This means that $\sum_{j=1}^{s}\theta_{j}\nu_{j}+\sum_{i=1}^{r}\lambda_{i}\gamma_{i}=0$. We can transform the sum $\sum_{i=1}^{r}\lambda_{i}\gamma_{i}$ in a combination with positive coefficients if for any negative $\lambda_{i}$ we substitute $\gamma_{i}$ by the stoichiometric vector of the reverse reaction, that is, $-\gamma_{i}$. As a result, we get the element of $\Lambda_{+}$, a combination of the stoichiometric vectors with nonnegative coefficients, which is equal to zero and includes some stoichiometric vectors of the irreversible reactions with nonzero coefficients. Therefore, the structural condition of the extended form of detailed balance is violated. Let the structural condition be violated. Then there exist a combination $\sum_{j=1}^{s}\theta_{j}\nu_{j}+\sum_{i=1}^{r}\lambda_{i}\gamma_{i}=0$ with $\theta_{j}\geq 0$ and $\sum_{j=1}^{s}\theta_{j}>0$. Let us notice that $\sum_{j=1}^{s}\frac{\theta_{j}}{\sum_{j=1}^{s}\theta_{j}}\nu_{j}=-\sum_{i=1}^{r}\frac{\lambda_{i}}{\sum_{j=1}^{s}\theta_{j}}\gamma_{i}\,,$ and, therefore, $0\in{\rm conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}$. The condition 1 is violated. We proved that the condition 1 is equivalent to the structural condition of the extended form of detailed balance. If the condition 1 holds then the condition 2 is, exactly, the algebraic condition of the extended form of detailed balance. ∎ ###### Remark 7. The first condition of Theorem, $0\notin{\rm conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}$, is equivalent to the existence of such a linear functional $l$ on $\mathbb{R}^{n}$ that $l(\nu_{j})>0$ for all $j=1,\ldots,s$ and $l(\gamma_{j})=0$ for all $j=1,\ldots,r$. ### 3.3 Linear Systems The results of previous Sections for a linear system (1) have a geometrically clear form (see also the paper by Yablonsky et al (2010)). ###### Proposition 3. The necessary and sufficient condition for the extended form of detailed balance is: In any cycle $A_{i_{1}}\to A_{i_{2}}\to\ldots\to A_{i_{q}}\to A_{i_{1}}$ with the strictly positive constants $k_{i_{j+1}i_{j}}>0$ (here $i_{q+1}=i_{1}$) all the reactions are reversible ($k_{i_{j}i_{j+1}}>0$) and the identity (3) holds. The states (reagents) $A_{q}$ and $A_{r}$ ($q\neq r$) are strongly connected if there exist oriented paths both from $A_{q}$ to $A_{r}$ and from $A_{r}$ to $A_{q}$ (each oriented edge corresponds to a reaction with the nonzero reaction rate constant). From Proposition 3 we get the following statement. ###### Corollary 4. Let a linear system satisfy the extended form of detailed balance. Then all reactions in any directed path between strongly connected states are reversible. In brief, a linear system with the extended form of detailed balance can be described as follows: (i) the oriented cycles are reversible and satisfy the classical condition (3), (ii) the system consists of the reversible parts and the irreversible transitions between these parts and (iii) the system of these irreversible transitions is acyclic. For example, let us analyze subsystems of the simple cycle, $A_{1}\rightleftharpoons A_{2}\rightleftharpoons A_{3}\rightleftharpoons A_{1}$. $\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rrrrrr}-1&0&1&1&0&-1\\\ 1&-1&0&-1&1&0\\\ 0&1&-1&0&-1&1\end{array}\right]$ (13) The cone of nonnegative solutions $\Lambda_{+}$ to the equation $\boldsymbol{\lambda\Gamma}=0$ has extreme rays with the following direction vectors: $\boldsymbol{\lambda}^{1}=(1,1,1,0,0,0)$, $\boldsymbol{\lambda}^{2}=(0,0,0,1,1,1)$, $\boldsymbol{\lambda}^{3}=(1,0,0,1,0,0)$, $\boldsymbol{\lambda}^{4}=(0,1,0,0,1,0)$, and $\boldsymbol{\lambda}^{5}=(0,0,1,0,0,1)$. Vectors $\boldsymbol{\lambda}^{3-5}$ give trivial identities (8) $k_{i}^{+}k_{i}^{-}=k_{i}^{-}k_{i}^{+}$ ($i=1,2,3$) and vectors $\boldsymbol{\lambda}^{1,2}$ give the same identity: $k_{1}^{+}k_{2}^{+}k_{3}^{+}=k_{1}^{-}k_{2}^{-}k_{3}^{-}$. If we delete one elementary reaction from the simple cycle (i.e. one column from $\boldsymbol{\Gamma}^{\mathrm{T}}$ (13)) then one of the nonnegative solutions $\boldsymbol{\lambda}^{1,2}$ persists and, due to the extended detailed balance principle, all the reactions should be reversible. This means that the structural condition of extended detailed balance is not satisfied for the simple reversible cycle without one direct or reverse reaction. If two reactions are reversible then the third should be reversible or completely vanish. If we delete one direct reaction (with number 1, 2 or 3) and one reverse reaction (with number 4, 5 or 6) then there remain no non-trivial solutions in $\Lambda_{+}$ and, therefore, no non-trivial relations between the constants persist after deletion of these two reactions. For the linear systems, the oriented cycles in the graph of reactions (where vertices are the components and edges are the reactions) give the positive solutions to the equation (7): for a linear oriented cycle $C$ the sum of the stoichiometric vectors of its reactions is zero. Moreover, any positive solution of (7) is a convex combination of such cyclic solutions and, therefore, the directed vectors of the extreme rays of $\Lambda_{+}$ can be selected in this form. ### 3.4 Simple Examples of Nonlinear Systems In this section, we present several elementary examples. For these examples, the sets of the extreme pathways are obvious. Let us examine a reaction mechanism with irreversible reactions $A\xrightarrow{k_{1}}B$ and $2B\xrightarrow{k_{2}}2A$. $\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rr}-1&2\\\ 1&-2\end{array}\right]\,.$ (14) The cone $\Lambda_{+}$ is a ray with the directional vector $\boldsymbol{\lambda}=(2,1)$. Both $\lambda_{1,2}>0$, hence, both reactions should be reversible and the condition holds: $(k_{1}^{+})^{2}k_{2}^{+}=(k_{1}^{-})^{2}k_{2}^{-}$. Let us slightly modify this example: $2{\rm H}\to{\rm H}_{2}$, ${\rm H}+{\rm H_{2}}\to 3{\rm H}$. $\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rr}-2&2\\\ 1&-1\end{array}\right]\,.$ (15) The cone $\Lambda_{+}$ is a ray with the directional vector $\boldsymbol{\lambda}=(1,1)$. Both $\lambda_{1,2}>0$, hence, both reactions should be reversible and the condition holds: $k_{1}^{+}k_{2}^{+}=k_{1}^{-}k_{2}^{-}$. If we change the direction of one reaction in the previous example then the new irreversible systems satisfies the extended form of detailed balance: $2{\rm H}\to{\rm H}_{2}$, $3{\rm H}\to{\rm H}+{\rm H_{2}}$. $\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rr}-2&-2\\\ 1&1\end{array}\right]\,.$ (16) The cone $\Lambda_{+}$ is trivial (it includes no rays, just the origin), hence, the structural condition holds. The algebraic condition trivially holds, because there is no reversible reaction. Let us add the forth reversible and nonlinear elementary reaction $A_{1}+A_{2}\rightleftharpoons 2A_{3}$ (with the constants $k_{4}^{\pm}$) to a linear reversible cycle. We should add to $\boldsymbol{\Gamma}^{\mathrm{T}}$ (13) two new columns: $\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rrrrrrrr}-1&0&1&-1&1&0&-1&1\\\ 1&-1&0&-1&-1&1&0&1\\\ 0&1&-1&2&0&-1&1&-2\end{array}\right]$ (17) The extreme rays of $\Lambda_{+}$ include four rays that correspond to pairs of mutually reverse reactions ($\boldsymbol{\lambda}^{1-4}$), two rays that correspond to the linear cycle ($\boldsymbol{\lambda}^{5,6}$) and six rays for three nonlinear cycles ($\boldsymbol{\lambda}^{7-12}$): (i) $A_{1}+A_{2}\to 2A_{3}$, $A_{3}\to A_{2}$, $A_{3}\to A_{1}$; (ii) $A_{1}+A_{2}\to 2A_{3}$, $A_{3}\to A_{1}$, $A_{1}\to A_{2}$ and (iii) $A_{1}+A_{2}\to 2A_{3}$, $A_{3}\to A_{2}$, $A_{2}\to A_{1}$: $\displaystyle\boldsymbol{\lambda}^{5}=(1,1,1,0,0,0,0,0),\;\;$ $\displaystyle\boldsymbol{\lambda}^{6}=(0,0,0,0,1,1,1,0),$ $\displaystyle\boldsymbol{\lambda}^{7}=(0,0,1,1,0,1,0,0),\;\;$ $\displaystyle\boldsymbol{\lambda}^{8}=(0,1,0,0,0,0,1,1),$ $\displaystyle\boldsymbol{\lambda}^{9}=(1,0,2,1,0,0,0,0),\;\;$ $\displaystyle\boldsymbol{\lambda}^{10}=(0,0,0,0,1,0,2,1),$ $\displaystyle\boldsymbol{\lambda}^{11}=(0,0,0,1,1,2,0,0),\;\;$ $\displaystyle\boldsymbol{\lambda}^{12}=(1,2,0,0,0,0,0,1)\,.$ We omit $\boldsymbol{\lambda}^{1-4}$ which do not produce nontrivial conditions. For the reversible reaction mechanism (when $k^{\pm}_{1-4}>0$), there are two independent Wegscheider identities (8) that formalize the classical principle of detailed balance: $k_{1}^{+}k_{2}^{+}k_{3}^{+}=k_{1}^{-}k_{2}^{-}k_{3}^{-}$ and $k_{3}^{+}k_{4}^{+}k_{2}^{-}=k_{3}^{-}k_{4}^{-}k_{2}^{+}$. If some of the elementary reactions are irreversible then the direction vectors $\boldsymbol{\lambda}^{5-12}$ produce 8 conditions. For $\boldsymbol{\lambda}^{5,7,9,11}$ these conditions are below. * 1. ($\boldsymbol{\lambda}^{5}$) If $k_{1,2,3}^{+}>0$ then $k_{1,2,3}^{-}>0$ and $k_{1}^{+}k_{2}^{+}k_{3}^{+}=k_{1}^{-}k_{2}^{-}k_{3}^{-}$; * 2. ($\boldsymbol{\lambda}^{7}$) If $k_{3,4}^{+},k_{2}^{-}>0$ then $k_{3,4}^{-},k_{2}^{+}>0$ and $k_{3}^{+}k_{4}^{+}k_{2}^{-}=k_{3}^{-}k_{4}^{-}k_{2}^{+}$; * 3. ($\boldsymbol{\lambda}^{9}$) If $k_{1,3,4}^{+}>0$ then $k_{1,3,4}^{-}>0$ and $k_{1}^{+}(k_{3}^{+})^{2}k_{4}^{+}=k_{1}^{-}(k_{3}^{-})^{2}k_{4}^{-}$; * 4. ($\boldsymbol{\lambda}^{11}$) If $k_{4}^{+},k_{1,2}^{-}>0$ then $k_{4}^{-},k_{1,2}^{+}>0$ and $k_{4}^{+}k_{1}^{-}(k_{2}^{-})^{2}=k_{4}^{-}k_{1}^{+}(k_{2}^{+})^{2}$. To obtain the conditions for $\boldsymbol{\lambda}^{6,8,10,12}$ it is sufficient to change the superscripts + to - and inverse. These 8 conditions represent the extended form of detailed balance for a given mechanism. To check, whether a subsystem of this mechanism satisfies the extended form of detailed balance, it is necessary and sufficient to check these conditions. ### 3.5 Methane Reforming Processes: a Case Study #### 3.5.1 The System Methane reforming with CO2 is a complex reaction network (Benson, 1981). The main reactions in the methane reforming are: 1. 1. ${\rm CO}_{2}+{\rm H_{2}}\rightleftharpoons{\rm CO}+{\rm H}_{2}{\rm O}$ (RWGS, Reverse water-gas shift); 2. 2. ${\rm CH}_{4}+{\rm CO}_{2}\rightleftharpoons{\rm 2CO}+{\rm 2H_{2}}$ (Dry reforming); 3. 3. ${\rm CO}_{2}+4{\rm H}_{2}\rightleftharpoons{\rm CH}_{4}+2{\rm H}_{2}{\rm O}$ (Methanation); 4. 4. ${\rm CH}_{4}+{\rm H}_{2}{\rm O}\rightleftharpoons{\rm CO}+3{\rm H}_{2}$ (Steam reforming); 5. 5. ${\rm CH}_{4}\rightleftharpoons{\rm 2H}_{2}+{\rm C}$ (Methane decomposition); 6. 6. $2{\rm CO}\rightleftharpoons{\rm CO}_{2}+{\rm C}$ (Boudouard reaction); 7. 7. ${\rm C}+{\rm H}_{2}{\rm O}\rightleftharpoons{\rm CO}+{\rm H}_{2}$ (Coal gasification). For the reagents, we use the notations $A_{1}={\rm CH}_{4}$, $A_{2}={\rm CO}_{2}$, $A_{3}={\rm CO}$, $A_{4}={\rm H}_{2}$, $A_{5}={\rm H}_{2}{\rm O}$, $A_{6}={\rm C}$. Amount of $A_{i}$ is $N_{i}$. There exist three independent linear conservation laws: $b_{\rm C}=N_{1}+N_{2}+N_{3}+N_{6}$; $b_{\rm H}=4N_{1}+2N_{4}+2N_{5}$; $b_{\rm O}=2N_{2}+N_{3}+N_{5}$. The number of degrees of freedom in the closed system is three (six components minus three independent conservation laws). This example enriches our discussion because it deviates from the nice abstract scheme discussed above. First of all, the reactions 1–7 are not elementary steps. We consider them as overall reactions which have their own intrinsic and complicated reaction mechanism. This does not cause a serious problem because the generalized mass action law describes the equilibria of the complex overall reactions as well as the equilibria of the elementary ones. Therefore, we can apply the concept of the extended form of detailed balance and our theorems 1–3 to the process network 1–7 build from the complex reactions. Rigorously speaking, we deal not with the elementary reaction steps but with the main equilibria and may discuss, for example, not the “Boudouard reaction” but the “Boudouard equilibrium”. The second problem is the heterogeneity of the system: $A_{1},\ldots,A_{5}$ are gases and $A_{6}={\rm C}$ is solid. Some of the reactions go on the surface of the solid. If a multiphase system is ideal and the solid components are stoichiometric ones (i.e. they have a fixed composition) then the free energy has the form $F=\sum_{A_{i}\;-\;{\rm gas}}N_{i}(RT\ln c_{i}+\mu_{i}^{0}-RT)+\sum_{A_{i}\;-\;{\rm solid}}N_{i}\mu_{i}^{0}\,.$ (18) Here, the free energy of solid components differs from the free energy of gases by the absence of the term $RTN\ln c$. This term corresponds to the ideal gas pressure $PV=NRT$. In our case, $F=\sum_{i=1}^{5}N_{i}(RT\ln c_{i}+\mu_{i}^{0}-RT)+N_{6}\mu_{6}^{0}\,.$ (19) To define the activities, we follow (9). For the ideal gases $a_{i}=c_{i}$ and for the stoichiometric solids $a_{i}\equiv 1$. In section 1.2, we studied homogeneous systems and considered $x_{i}=\ln a_{i}$ as independent unknowns in the detailed balance equations (6): $\sum_{i}\gamma_{ri}x_{i}=\ln K_{r}\;\;\;(x_{i}=\ln a_{i}^{\rm eq})\,.$ (20) Therefore, for any solution of this system, the activities $a_{i}=\exp x_{i}$ represented a positive equilibrium. In a heterogeneous system with the free energy (18) the activities for the solid components are constant, the correspondent $x_{i}\equiv 0$. Let $\boldsymbol{x}=(x_{i})$ be a solution to equations (20), $\boldsymbol{\Gamma x}=\boldsymbol{K}$, where $\boldsymbol{K}$ is the vector of the equilibrium constants. The vector $\boldsymbol{a}=(a_{i})$, $a_{i}=\ln x_{i}$ is a vector of equilibrium activities if and only if $x_{i}=0$ for all the solid components $A_{i}$. Instead of analyzing the solvability of the detailed balance equations (20) we have to study its solvability under additional condition: $x_{i}=0$ for all the solid components $A_{i}$. Let us postpone the discussion of the extended principle of detailed balance in multiphase systems and consider the system “gaseous mixture + one stoichiometric solid”. Let $A_{n}$ be solid. If there is the only solid component then the solvability conditions for the system (20) and for this system with additional condition $x_{n}=0$ coincide. Indeed, there exist a positive stoichiometric linear conservation law: $\sum_{i=1}^{n}\gamma_{ri}b_{i}=0\mbox{ for all }r\mbox{ and }b_{i}>0\mbox{ for all }i\,.$ For example, this may be conservation of mass or of the amount of atoms. Let $\boldsymbol{b}=(b_{i})$. For any solution of the detailed balance conditions (20) $\boldsymbol{x}=(x_{i})$, the vector $\boldsymbol{x}^{\prime}=\boldsymbol{x}-\frac{x_{n}}{b_{n}}\boldsymbol{b}$ is also a solution to (20) with the condition $x^{\prime}_{n}=0$. So, for our example with seven equilibria 1–7 the conditions of the extended principle of detailed balance for the heterogeneous system with solid $A_{6}=$C are described by the theorems 1–3 and we can use the results of the preceding sections. #### 3.5.2 The Classical Wegscheider Conditions To formulate the classical Wegscheider identities, we have to join the direct and inverse reactions and to find the basic solutions of the system of linear equations $\boldsymbol{\lambda}\boldsymbol{\Gamma}=0$. The stoichiometric matrix for this example is: $\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rrrrrrr}0&-1&1&-1&-1&0&0\\\ -1&-1&-1&0&0&1&0\\\ 1&2&0&1&0&-2&1\\\ -1&2&-4&3&2&0&1\\\ 1&0&2&-1&0&0&-1\\\ 0&0&0&0&1&1&-1\end{array}\right]$ (21) The system of seven equations $\boldsymbol{\lambda}\boldsymbol{\Gamma}=0$ is redundant. There are only three independent equations (one equation for every degree of freedom). It is sufficient to take the components of stoichiometric vectors that correspond to the components $A_{2}$, $A_{4}$ , $A_{6}$. Other components satisfy the same linear relations as the selected ones. The reduced matrix $\boldsymbol{\Gamma}_{\rm r}^{\mathrm{T}}$ is $\boldsymbol{\Gamma}_{\rm r}^{\mathrm{T}}=\left[\begin{array}[]{rrrrrrr}-1&-1&-1&0&0&1&0\\\ -1&2&-4&3&2&0&1\\\ 0&0&0&0&1&1&-1\end{array}\right]$ (22) There are four independent solutions of the equations $\boldsymbol{\lambda}\boldsymbol{\Gamma}=0$ (seven variables minus three independent equations). For example, we can take the following basis of solutions: $(-1,1,0,-1,0,0,0)$, $(0,0,0,-1,1,0,1)$, $(1,0,0,0,0,1,1)$, $(1,0,-1,-1,0,0,0)$. The correspondent Wegscheider identities are: $K_{2}=K_{1}K_{4}$, $K_{5}K_{7}=K_{4}$, $K_{1}K_{6}K_{7}=1$, $K_{1}=K_{3}K_{4}$. #### 3.5.3 Allowed and Forbidden Mechanisms In general, all the seven reactions can be considered as reversible but under various conditions some of them are almost irreversible. Let us study which combinations of irreversible reactions are possible in accordance with the extended form of detailed balance. For example, existence of the positive solution $(0,1,0,0,0,1,1)\in\Lambda$ guarantees that the irreversible system ${\rm CO}_{2}+{\rm H_{2}}\to{\rm CO}+{\rm H}_{2}{\rm O}$, ${\rm CH}_{4}+{\rm CO}_{2}\to{\rm 2CO}+{\rm 2H_{2}}$, ${\rm CO}_{2}+4{\rm H}_{2}\to{\rm CH}_{4}+2{\rm H}_{2}{\rm O}$, ${\rm CH}_{4}+{\rm H}_{2}{\rm O}\to{\rm CO}+3{\rm H}_{2}$, ${\rm CH}_{4}\to{\rm 2H}_{2}+{\rm C}$, $2{\rm CO}\to{\rm CO}_{2}+{\rm C}$, ${\rm C}+{\rm H}_{2}{\rm O}\to{\rm CO}+{\rm H}_{2}$ is forbidden by the extended form of detailed balance. This conclusion is also obvious from the correspondent Wegscheider condition $K_{2}K_{6}K_{7}=1$. Indeed, if all the $k_{i}^{-}\to 0$ for bounded from below $k_{i}^{+}>\varepsilon>0$ then all $K_{i}\to\infty$ and $K_{2}K_{6}K_{7}\to\infty$. This contradicts the Wegscheider condition. The first reaction (RWGS, Reverse water-gas shift) is reversible in the wide interval of conditions (Moe, 1962). Let us first study all the reaction mechanisms with the reversible first reaction and the irreversible reactions 2-7. We find the combinations of the directions of the irreversible reactions that satisfy the extended form of detailed balance. As a criterion of the extended form of detailed balance we use Theorem 3. After that, we consider other reactions as the reversible ones (in addition to RWGS) and study the corresponding reaction mechanisms. The space $S$ is a straight line with the directional vector $\gamma_{1}$ with coordinates $(-1,-1,0)$ in the coordinate system $(N_{2},N_{4},N_{6})$ that corresponds to the components $A_{2}$, $A_{4}$, $A_{6}$. Let us represent the quotient space $\mathbb{R}^{3}/S$ in the coordinate system $(N_{2},N_{6})$ that corresponds to the components $A_{2}$, $A_{6}$. For this purpose, we have to eliminate the coordinate $N_{4}$ using vector $\gamma_{1}$. As a result, we get the following vectors: $\begin{split}\overline{\gamma}_{2}=\left(\begin{array}[]{r}-3\\\ 0\end{array}\right)\,,\overline{\gamma}_{3}=\left(\begin{array}[]{r}3\\\ 0\end{array}\right)\,,\overline{\gamma}_{4}=\left(\begin{array}[]{r}-3\\\ 0\end{array}\right)\,,\\\ \overline{\gamma}_{5}=\left(\begin{array}[]{r}-2\\\ 1\end{array}\right)\,,\overline{\gamma}_{6}=\left(\begin{array}[]{r}1\\\ 1\end{array}\right)\,,\overline{\gamma}_{7}=\left(\begin{array}[]{r}-1\\\ -1\end{array}\right)\,.\end{split}$ (23) For example, to find $\overline{\gamma}_{2}$, we take $\gamma_{2}$ (the second column in (22)) and exclude the coordinate $N_{4}$ by adding $2\gamma_{1}$. The result is a vector $\gamma_{2}+2\gamma_{1}$. In coordinates $(N_{2},N_{6})$, this vector gives us $\overline{\gamma}_{2}$. Figure 1: Images of the stoichiometric vectors of irreversible reactions $\overline{\nu}_{j}=\pm\overline{\gamma}_{j}$ in $\mathbb{R}^{3}/S$ for various combinations of directions of reactions (24) in coordinates $N_{2}$ (abscissa), $N_{6}$. The configurations with $0\in{\rm conv}\\{\overline{\nu}_{2},\ldots,\overline{\nu}_{7}\\}$ are outlined. Vectors $\overline{\nu}_{2}$, $\overline{\nu}_{3}$ and $\overline{\nu}_{4}$ coincide as well as vectors $\overline{\nu}_{6}$ and $\overline{\nu}_{7}$. The stoichiometric vectors of irreversible reactions are $+\gamma_{j}$ or $-\gamma_{j}$ ($j=2,\ldots,7$). Their images in the quotient space $\mathbb{R}^{3}/S$ are $+\overline{\gamma}_{j}$ or $-\overline{\gamma}_{j}$. The extended form of detailed balance requires that the convex envelope of these vectors should not include zero. We have to arrange signs in $\pm\gamma_{j}$ to provide this property. First of all, we see immediately from (23) that the second and the forth reaction should have the same directions and the third reaction should have the opposite direction. The directions of the sixth and the seventh reactions should be opposite. Therefore, we have to analyze eight possible reaction mechanisms. Let us represent them by the directions of reactions: $\left[\begin{array}[]{c}({\rm a})\\\ \rightleftharpoons\\\ \to\\\ \leftarrow\\\ \to\\\ \to\\\ \to\\\ \leftarrow\end{array}\right];\left[\begin{array}[]{c}({\rm b})\\\ \rightleftharpoons\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \to\\\ \to\\\ \leftarrow\end{array}\right];\left[\begin{array}[]{c}({\rm c})\\\ \rightleftharpoons\\\ \to\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \to\\\ \leftarrow\end{array}\right];\left[\begin{array}[]{c}({\rm d})\\\ \rightleftharpoons\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \leftarrow\\\ \to\\\ \leftarrow\end{array}\right];\left[\begin{array}[]{c}({\rm e})\\\ \rightleftharpoons\\\ \to\\\ \leftarrow\\\ \to\\\ \to\\\ \leftarrow\\\ \to\end{array}\right];\left[\begin{array}[]{c}({\rm f})\\\ \rightleftharpoons\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \to\end{array}\right];\left[\begin{array}[]{c}({\rm g})\\\ \rightleftharpoons\\\ \to\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \leftarrow\\\ \to\end{array}\right];\left[\begin{array}[]{c}({\rm h})\\\ \rightleftharpoons\\\ \leftarrow\\\ \to\\\ \leftarrow\\\ \leftarrow\\\ \leftarrow\\\ \to\end{array}\right].$ (24) Arrows here correspond to the directions of reactions. For example, the case (a) corresponds to the reaction mechanism 1. 1. ${\rm CO}_{2}+{\rm H_{2}}\rightleftharpoons{\rm CO}+{\rm H}_{2}{\rm O}$; 2. 2. ${\rm CH}_{4}+{\rm CO}_{2}\to{\rm 2CO}+{\rm 2H_{2}}$; 3. 3. ${\rm CO}_{2}+4{\rm H}_{2}\leftarrow{\rm CH}_{4}+2{\rm H}_{2}{\rm O}$; 4. 4. ${\rm CH}_{4}+{\rm H}_{2}{\rm O}\to{\rm CO}+3{\rm H}_{2}$; 5. 5. ${\rm CH}_{4}\to{\rm 2H}_{2}+{\rm C}$; 6. 6. $2{\rm CO}\to{\rm CO}_{2}+{\rm C}$; 7. 7. ${\rm C}+{\rm H}_{2}{\rm O}\leftarrow{\rm CO}+{\rm H}_{2}$. Combinations (c) and (f) contradict the condition 1 from Theorem 3: the origin belongs to the convex envelope of the vectors $\overline{\nu}_{j}$ of irreversible reactions (see Fig. 1). Hence, only six combinations of directions of irreversible reactions satisfy the extended form of detailed balance (from $2^{6}=64$ possible combinations of directions): (a), (b), (d), (e), (g) and (h). Let us extend the list of reversible reactions. If we assume that the second reaction (dry reforming), is reversible together with the first one (RWGS) then the third and the forth reactions should be also reversible because $\gamma_{3}=2\gamma_{1}-\gamma_{2}$ and $\gamma_{4}=\gamma_{2}-\gamma_{1}$, hence, $\gamma_{3,4}\in{\rm span}\\{\gamma_{1},\gamma_{2}\\}$. According to the condition 1 from Theorem 3, this contradicts to the extended form of detailed balance if the first and the second reactions are reversible and the third and the forth are not. Analogously, in addition to the reversible reaction RWGS, the Boudouard equilibrium 6 and coal gasification 7 can be reversible only together because $\gamma_{7}=-\gamma_{1}-\gamma_{6}$. We have to consider three possible sets of reversible reactions: 1. 1. 1, 2, 3 and 4; 2. 2. 1 and 5; 3. 3. 1, 6 and 7. For all three cases, $\dim S=2$ and $\dim(\mathbb{R}^{3}/S)=1$. We will use for the quotient space the coordinate $N_{6}$ which corresponds to $A_{6}={\rm C}$. In the first case, let us exclude the coordinate $N_{4}$ from $\overline{\gamma}_{5,6,7}$ (23) using vector $\overline{\gamma}_{2}$. We get one-dimensional vectors $\overline{\gamma}_{5}=1,\;\overline{\gamma}_{6}=1,\;\overline{\gamma}_{7}=-1\,.$ To satisfy the extended form of detailed balance the directions of the fifth and the sixth reaction should coincide and the direction of the seventh reaction should be opposite: there are two possible combinations of arrows in irreversible reactions 5, 6 and 7 if reactions 1, 2, 3 and 4 are reversible: $5\to,\,6\to,\,7\leftarrow$ and $5\leftarrow,\,6\leftarrow,\,7\to$. In the second case, let us exclude the coordinate $N_{4}$ from $\overline{\gamma}_{2,3,4,6,7}$ (23) using vector $\overline{\gamma}_{5}$. We get one-dimensional vectors $\overline{\gamma}_{2}=-2/3,\;\overline{\gamma}_{3}=2/3,\;\overline{\gamma}_{4}=-2/3,\;\overline{\gamma}_{6}=1/2,\;\overline{\gamma}_{7}=-1/2\,.$ Again, according to the extended form of detailed balance, here are two possibilities of directions of irreversible reactions 2, 3, 4, 6 and 7 if reactions 1 and 5 are reversible: $2\to,\,3\leftarrow,\,4\to,\,6\leftarrow,\,7\to$ and $2\leftarrow,\,3\to,\,4\leftarrow,\,6\to,\,7\leftarrow$. In the third case, let us exclude the coordinate $N_{4}$ from $\overline{\gamma}_{2,3,4,5}$ (23) using vector $\overline{\gamma}_{6}$. We get one-dimensional vectors $\overline{\gamma}_{2}=3,\;\overline{\gamma}_{3}=-3,\;\overline{\gamma}_{4}=3,\;\overline{\gamma}_{5}=3\,.$ According to the extended form of detailed balance, here are two possibilities of directions of irreversible reactions 2, 3, 4, and 5 if reactions 1, 6 and 7 are reversible: $2\to,\,3\leftarrow,\,4\to,\,5\to$ and $2\leftarrow,\,3\to,\,4\leftarrow,\,5\leftarrow$. In the first and the third cases, there are nontrivial Wegscheider identities for the reaction equilibrium constants of reversible reactions. If reactions 1, 2, 3 and 4 are reversible (case 1) then $\dim\Lambda=2$ and the basis of $\Lambda$ is, for example, $\boldsymbol{\lambda}^{1}=(2,-1,-1,0)$ ($2\gamma_{1}-\gamma_{2}-\gamma_{3}=0$) and $\boldsymbol{\lambda}^{2}=(1,-1,0,1)$ ($\gamma_{1}-\gamma_{2}+\gamma_{4}=0$). The two correspondent Wegscheider identities are: $K_{1}^{2}=K_{2}K_{3}$ and $K_{1}K_{4}=K_{2}$ (where $K_{i}=k_{i}^{+}/k_{i}^{-}$). If the reactions 1, 6 and 7 are reversible then $\dim\Lambda=1$ and the basis of $\Lambda$ consists of one vector $\boldsymbol{\lambda}=(1,1,1)$ ($\gamma_{1}+\gamma_{6}+\gamma_{7}=0$). The correspondent Wegscheider identity is: $K_{1}K_{6}K_{7}=1$. If we add one more reversible reaction in cases 1-3 then all the reactions 1-7 should be reversible in according to the extended form of detailed balance. In this case study, we demonstrated also how it is possible to organize computations and reduce the dimension of the computational problems. ## 4 Multiscale Degenerated Equilibria Let in a system of reversible reactions with detailed balance some $k_{s}^{-}\to 0$, when the correspondent $k_{s}^{+}$ remains constant and separated from zero. In this case, some equilibrium activities also tend to zero. Indeed, at equilibrium $w_{s}^{+}=w_{s}^{-}$, $w_{s}^{-}\to 0$ because $k_{s}^{-}\to 0$, hence, $w_{s}^{+}\to 0$ and some of $a_{i}^{\rm eq}$ with $\alpha_{si}>0$ also tend to zero due to the generalized mass action law (5). Therefore, the irreversible limits of the reactions with detailed balance are closely related to the limits when some equilibrium activities tend to zero. (For the usual mass action law is sufficient to replace the “activity $a_{i}$” by the “concentration $c_{i}$”.) In this section we study asymptotics $a_{i}^{\rm eq}={\rm const}\times\varepsilon^{\delta_{i}}$, $\varepsilon\to 0$ for various values of non-negative exponents $\delta_{i}\geq 0$ ($i=1,\ldots,n$). There exists a well known way to satisfy the principle of detailed balance: just write $k^{-}_{r}=k^{+}_{r}/K_{r}$ where $K_{r}$ is the equilibrium constant: $K_{r}=\frac{\prod_{i=1}^{n}(a_{i}^{\rm eq})^{\beta_{ri}}}{\prod_{i=1}^{n}(a_{i}^{\rm eq})^{\alpha_{ri}}}\,.$ We can define the equilibrium constant through the equilibrium thermodynamics as well (see, for example, the classical book by Prigogine & Defay (1962)). In this case, the principle of detailed balance is also satisfied for the mass action law. In this approach, we have to group the direct and reverse reactions together. Therefore, $m$ is here the number of pairs of reactions, direct + inverse ones. We deal with $m+n$ constants ($m$ rate constants $k_{r}^{+}$ for direct reactions and $n$ equilibrium data for individual reagents: equilibrium concentrations or activities) instead of $2m$ constants $k_{r}^{\pm}$. For these $m+n$ constants, the principle of detailed balance produces no restrictions (Gorban et al, 1989; Yang, et al, 2006). It holds “by the construction” for any positive values of these constants if $k^{-}_{r}=k^{+}_{r}/K_{r}$ and the equilibrium constants are calculated in accordance with the equilibrium data. To transform the conditions of $a_{i}^{\rm eq}\to 0$ into irreversibility of some reactions, it is not sufficient to know which $a_{i}^{\rm eq}\to 0$. We have to take into account the rates of these convergence to zero for different $i$. In the simple example, $A_{1}\rightleftharpoons A_{2}\rightleftharpoons A_{3}\rightleftharpoons A_{1}$, if $a_{1,2}^{\rm eq}\to 0$, $a_{1}/a_{2}\to 0$ then in the limit we get the system $A_{1}\to A_{2}$ (because the $A_{1}/A_{2}$ equilibrium is shifted to $A_{2}$), $A_{1}\to A_{3}$, $A_{2}\to A_{3}$. For the inverse relations between $a_{1}$ and $a_{2}$, $a_{2}/a_{1}\to 0$, the limit system is $A_{2}\to A_{1}$ (the $A_{1}/A_{2}$ equilibrium is shifted to $A_{1}$), $A_{1}\to A_{3}$, $A_{2}\to A_{3}$. For the both limit systems, the equilibrium activities of $A_{1}$, $A_{2}$ are zero but the directions of reaction are different. The limit structure of the reaction mechanism when some of $a_{i}^{\rm eq}\to 0$ depends on the behavior of the ratios $a_{i}^{\rm eq}/a_{i}^{\rm eq}$. To formalize this dependence, let us introduce a parameter $\varepsilon>0$ and take $a_{i}^{\rm eq}={\rm const}\times\varepsilon^{\delta_{i}}$. At equilibrium, each monomial in the generalized mass action law is proportional to a power of $\varepsilon$: $w_{r}^{{\rm eq}+}=k_{r}^{+}{\rm const}\times\varepsilon^{\sum_{i}\alpha_{ri}\delta_{i}}\,,\;\;w_{r}^{{\rm eq}-}=k_{r}^{-}{\rm const}\times\varepsilon^{\sum_{i}\beta_{ri}\delta_{i}}\,.$ The principle of detailed balance gives: $w_{r}^{{\rm eq}+}=w_{r}^{{\rm eq}-}$. Therefore, $\frac{k_{r}^{+}}{k_{r}^{-}}={\rm const}\times\varepsilon^{(\gamma_{r},\delta)}\,,$ (25) where $\delta$ is the vector with coordinates $\delta_{i}$. There are three possibilities for the reversibility of an elementary reaction in asymptotic $\varepsilon\to 0$: 1. 1. If $(\gamma_{r},\delta)=0$ then the reaction remains reversible in asymptotic $\varepsilon\to 0$. This means that $0<\lim(k_{s}^{+}/k_{s}^{-})<\infty$. Therefore, if one of the reactions persists in the limit then the reverse reaction also persists. 2. 2. If $(\gamma_{r},\delta)<0$ then in asymptotic $\varepsilon\to 0$ can remain only direct reaction. This means that $\lim(k_{s}^{-}/k_{s}^{+})=0$. 3. 3. If $(\gamma_{r},\delta)>0$ then in asymptotic $\varepsilon\to 0$ can remain only reverse reaction. This means that $\lim(k_{s}^{+}/k_{s}^{-})=0$. It is possible that $(\gamma_{r},\delta)=0$ but both $k_{r}^{{\rm lim}\pm}=0$ just because $k^{+}_{r}=0$ and $k^{-}_{r}=0$ and not because of the equilibrium degeneration. If we delete some irreversible reactions or several pairs of mutually reverse reaction then the extended form of detailed balance persists. Therefore, we do not consider these cases separately and always discuss the limit reaction mechanisms with $\max\\{k_{r}^{{\rm lim}+},k_{r}^{{\rm lim}-}\\}>0$. For each stoichiometric vector $\gamma_{r}$ the $n$-dimensional space of vectors $\delta$ is split in three sets: hyperplane $(\gamma_{r},\delta)=0$ (reaction remains reversible), hemispace $(\gamma_{r},\delta)<0$ (only direct reaction remains) and hemispace $(\gamma_{r},\delta)>0$ (only reverse reaction remains). For the reaction mechanism, intersections of these sets for all $\gamma_{r}$ ($r=1,\ldots,m$) form a tiling of the n-dimensional space of vectors $\delta$. The intersection of all hyperplanes $(\gamma_{r},\delta)=0$ corresponds to the initial reversible reaction mechanism. Other sets from this tiling correspond to the reaction mechanisms that are limits of the initial reaction mechanism when some of the reaction rate constants tend to zero but the principle of detailed balance is valid. In our study, the exponents $\delta_{j}$ should be non-negative, hence, we have to study the tiling of the positive orthant $\delta_{j}\geq 0$ in $\mathbb{R}^{n}$ Description of the tiling produced by a system of hyperplanes $(\gamma_{r},\delta)=0$ is a classical problem of combinatorial geometry. In the usual linear triangle $A_{1}\rightleftharpoons A_{2}\rightleftharpoons A_{3}\rightleftharpoons A_{1}$ we have to consider three hyperplanes in the space of exponents $\delta=(\delta_{1},\delta_{2},\delta_{3})$: $\delta_{1}=\delta_{2}$ ($(\gamma_{1},\delta)=0$), $\delta_{2}=\delta_{3}$ ($(\gamma_{2},\delta)=0$) and $\delta_{3}=\delta_{1}$ ($(\gamma_{3},\delta)=0$). At least one of the exponents should take zero value to keep the overall concentration in equilibrium neither zero nor infinite. Let us take $\delta_{1}=0$. The hyperplanes turn in the straight lines on the plane $(\delta_{2},\delta_{3})$: $0=\delta_{2}$ ($(\gamma_{1},\delta)=0$), $\delta_{2}=\delta_{3}$ ($(\gamma_{2},\delta)=0$) and $\delta_{3}=0$ ($(\gamma_{3},\delta)=0$). The positive octant on the plane $(\delta_{2},\delta_{3})$ is split in five sets (A)–(E), that correspond to the limits with some irreversible reactions, and the origin: * 1. (A) $\delta_{2}=0$, $\delta_{3}>0$, $A_{1}\rightleftharpoons A_{2}$, $A_{3}\to A_{1}$, $A_{3}\to A_{2}$; * 2. (B) $\delta_{3}>\delta_{2}>0$, $A_{3}\to A_{2}\to A_{1}$, $A_{3}\to A_{1}$; * 3. (C) $\delta_{3}=\delta_{2}>0$, $A_{3}\rightleftharpoons A_{2}$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$; * 4. (D) $\delta_{2}>\delta_{3}>0$, $A_{2}\to A_{3}\to A_{1}$, $A_{2}\to A_{1}$ (this case differs from (B) by the transposition $2\leftrightarrow 3$); * 5. (E) $\delta_{2}>0$, $\delta_{3}=0$ $A_{1}\rightleftharpoons A_{3}$ , $A_{2}\to A_{1}$, $A_{2}\to A_{3}$ (this case differs from (A) by the transposition $2\leftrightarrow 3$). * 6. The origin corresponds to the fully reversible mechanism. For a less trivial example, let us analyze the reaction mechanism from Section 3.4: $A_{1}\rightleftharpoons A_{2}\rightleftharpoons A_{3}\rightleftharpoons A_{1}$, $A_{1}+A_{2}\rightleftharpoons 2A_{3}$. This is a reversible cycle supplemented by a nonlinear step. We join the direct and reverse elementary reactions and, therefore, $\boldsymbol{\Gamma}^{\mathrm{T}}=\left[\begin{array}[]{rrrr}-1&0&1&-1\\\ 1&-1&0&-1\\\ 0&1&-1&2\end{array}\right]$ (26) The columns of this matrix are the stoichiometric vectors $\gamma_{r}$. Let us study the tiling of the positive orthant in $\mathbb{R}^{3}$ by the planes $(\gamma_{r},\delta)=0$ ($r=1,\ldots,4$). First of all, it is necessary and sufficient to study this tiling of the positive octants in three planes: $\delta_{1}=0$, or $\delta_{2}=0$, or $\delta_{3}=0$ because at least one equilibrium concentration should not tend to zero and, therefore, has zero exponent. The symmetry between $A_{1}$ and $A_{2}$ allows us to study two planes: $\delta_{1}=0$ or $\delta_{3}=0$. Figure 2: Tiling of the positive octant of the plane $(\delta_{2},\delta_{3})$ ($\delta_{1}=1$) that corresponds to seven irreversible limits of the reaction mechanism. On the plane $\delta_{1}=0$ with coordinates $\delta_{2}$, $\delta_{3}$ we have four straight lines: $(\gamma_{1},\delta)=0$ ($\delta_{2}=0$), $(\gamma_{2},\delta)=0$, ($\delta_{2}=\delta_{3}$), $(\gamma_{3},\delta)=0$ ($\delta_{3}=0$) and $(\gamma_{4},\delta)=0$ ($\delta_{2}=2\delta_{3}$). These lines divide the positive octant ($\delta_{2,3}\geq 0$) into seven parts (Fig. 2) and the origin: 1. 1. (A) $\delta_{2}=0$, $\delta_{3}>0$, $A_{1}\rightleftharpoons A_{2}$, $A_{3}\to A_{1}$, $A_{3}\to A_{2}$, $2A_{3}\to A_{1}+A_{2}$; 2. 2. (B) $\delta_{2}>0$, $\delta_{3}>\delta_{2}$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$, $A_{3}\to A_{2}$, $2A_{3}\to A_{1}+A_{2}$; 3. 3. (C) $\delta_{2}=\delta_{3}>0$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$, $A_{3}\rightleftharpoons A_{2}$, $2A_{3}\to A_{1}+A_{2}$; 4. 4. (D) $0<\delta_{3}<\delta_{2}<2\delta_{3}$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$, $A_{2}\to A_{3}$, $2A_{3}\to A_{1}+A_{2}$; 5. 5. (E) $0<\delta_{2}=2\delta_{3}$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$, $A_{2}\to A_{3}$, $2A_{3}\rightleftharpoons A_{1}+A_{2}$; 6. 6. (F) $\delta_{2}>2\delta_{3}>0$, $A_{2}\to A_{1}$, $A_{3}\to A_{1}$, $A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$; 7. 7. (G) $\delta_{3}=0$, $\delta_{2}>0$, $A_{2}\to A_{1}$, $A_{1}\rightleftharpoons A_{3}$, $A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$; 8. 8. The origin corresponds to the fully reversible mechanism. The same picture gives us the plane $\delta_{2}=0$ with coordinates $\delta_{1}$, $\delta_{3}$: we need just to transpose the indexes, $1\leftrightarrow 2$. On the plane $\delta_{3}=0$ with coordinates $\delta_{1}$, $\delta_{2}$ the positive octant is divided into five parts and the origin: 1. 1. $\delta_{1}=0$, $\delta_{2}>0$, $A_{2}\to A_{1}$, $A_{1}\rightleftharpoons A_{3}$, $A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$ (this is exactly the case (G) from Fig. 2); 2. 2. $0<\delta_{1}<\delta_{2}$, $A_{2}\to A_{1}$, $A_{1}\to A_{3}$, $A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$; 3. 3. $0<\delta_{1}=\delta_{2}$, $A_{1}\rightleftharpoons A_{2}$, $A_{1}\to A_{3}$, $A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$; 4. 4. $\delta_{1}>\delta_{2}>0$, $A_{1}\to A_{2}$, $A_{1}\to A_{3}$, $A_{2}\to A_{3}$, $A_{1}+A_{2}\to 2A_{3}$; 5. 5. $\delta_{2}=0$, $\delta_{1}>0$, $A_{1}\to A_{2}$, $A_{1}\to A_{3}$, $A_{2}\rightleftharpoons A_{3}$, $A_{1}+A_{2}\to 2A_{3}$; 6. 6. The origin corresponds to the fully reversible mechanism. This approach is equivalent to the previous definition of the extended form of detailed balance based on the pathway analysis. Indeed, if the reaction mechanism with some irreversible reactions is a limit of the reversible mechanism with detailed balance then it satisfies the conditions of the extended form of detailed balance. (This is the direct statement of Theorem 1 proved in Section 3.2.) To prove the converse statement, we have to take a system that satisfies the extended form of detailed balance and to find such a set of exponents $\delta_{i}\geq 0$ ($i=1,\ldots,n$) that the system appears in the limit of a reversible system with detailed balance when $\varepsilon\to 0$ and $a_{i}^{\rm eq}={\rm const}\times\varepsilon^{\delta_{i}}$. Let a system with some irreversible reactions satisfy the extended form of detailed balance. We follow the notations of Theorem 3: $\gamma_{j}$ ($j=1,\ldots,r$) are the stoichiometric vectors of the reversible reactions and $\nu_{1},\ldots,\nu_{s}$ are the stoichiometric vectors of the irreversible reactions. The linear subspace $S={\rm span}\\{\gamma_{1},\ldots,\gamma_{r}\\}\subset\mathbb{R}^{n}$ consists of all linear combinations of the stoichiometric vectors of the reversible reactions. We use notation $\overline{\nu}_{j}$ for the images of $\nu_{j}$ in $\mathbb{R}^{n}/S$. Let $k_{j}^{\pm}>0$ ($j=1,\ldots,r$) be the reaction rate constants for the reversible reactions and $q_{j}=q_{j}^{+}>0$ ($j=1,\ldots,s$) be the reaction rate constants for the irreversible reactions. We extend the system by adding the reverse reactions with the constants $q_{j}^{-}>0$. If the extended system satisfies the principle of detailed balance then $\frac{k_{j}^{+}}{k_{j}^{-}}=\prod_{i=1}^{n}(a_{i}^{\rm eq})^{\gamma_{ri}}\;\;\mbox{ and }\;\;\frac{q_{j}^{+}}{q_{j}^{-}}=\prod_{i=1}^{n}(a_{i}^{\rm eq})^{\nu_{ri}}\,,$ (27) where $a_{i}^{\rm eq}$ is a point of detailed balance. ###### Theorem 4. Let the system satisfy the extended form of detailed balance. Then there exists a vector of nonnegative exponents $\delta=(\delta_{i})$ ($i=1,\ldots,n$) and the family of extended systems with equilibria $a_{i}^{\rm eq}=a_{i}^{*}\varepsilon^{\delta_{i}}$ such that condition (27) hold, $k_{j}^{\pm}$ ($j=1,\ldots,r$) and $q_{j}=q_{j}^{+}$ ($j=1,\ldots,s$) do not depend on $\varepsilon$, and $q_{j}^{-}\to 0$ when $\varepsilon\to 0$. ###### Proof. If the system satisfies the extended form of detailed balance then the reversible part satisfies the principle of detailed balance and, hence, there exists a positive point of detailed balance for the reversible part of the system (Theorem 3): $a_{i}^{*}>0$ and $k_{j}^{+}\prod_{i=1}^{n}(a_{i}^{*})^{\alpha_{ri}}=k_{j}^{+}\prod_{i=1}^{n}(a_{i}^{*})^{\beta_{ri}}\,.$ Let us take $a_{i}^{\rm eq}=a_{i}^{*}\varepsilon^{\delta_{i}}$. Due to (27), $k_{j}^{+}/k_{j}^{-}={\rm const}\times\varepsilon^{(\gamma_{j},\delta)}$. To keep the $k_{i}^{\pm}$ independent of $\varepsilon$, we have to provide $(\gamma_{j},\delta)=0$. Analogously, $q_{j}^{+}/q_{j}^{-}={\rm const}\times\varepsilon^{(\nu_{j},\delta)}$. The rate constant $q_{j}^{+}$ should not depend on $\varepsilon$ and $q_{j}^{-}\to 0$ when $\varepsilon\to 0$. Therefore, $(\nu_{j},\delta)<0$. We came to the system of linear equations and inequalities with respect to exponents $\delta_{i}$: $(\gamma_{j},\delta)=0\;(j=1,\ldots,r),\;\;(\nu_{j},\delta)<0\;(j=1,\ldots,s)\,.$ (28) The solvability of this system is equivalent to the condition 1 of Theorem 3 (see Remark 7). To prove the existence of nonnegative exponents $\delta_{i}\geq 0$, we have to use existence of positive conservation law: $b_{i}>0$, $(\gamma_{j},b)=0$, $(\nu_{j},b)=0$. For every solution $\delta$ of (28) and any number $d$, the vector $\delta+db$ is also a solution of (28). Therefore, the nonnegative solution exists. We proved the theorem and the converse statement of Theorem 1. ∎ ###### Proposition 4. Let a system with the stoichiometric vectors $\gamma_{s}$ and the extended detailed balance be obtained from the reversible systems with detailed balance in the limit $a_{i}^{\rm eq}={\rm const}\times\varepsilon^{\delta_{i}}$, $\varepsilon\to 0$. For this system, the linear function $(\delta,c)$ of the concentrations $c$ monotonically decreases in time due to the kinetic equations $\frac{{\mathrm{d}}c}{{\mathrm{d}}t}=\sum_{s}w_{s}\gamma_{s}$. ###### Proof. Indeed, $\frac{{\mathrm{d}}(\delta,c)}{{\mathrm{d}}t}=\sum_{s}w_{s}(\gamma_{s},\delta)$ (compare to Remark 6). For the reversible reactions, the sign of $w_{s}$ is indefinite but $(\gamma_{s},\delta)=0$. For the irreversible reactions, we always can take $w_{s}=w_{s}^{+}\geq 0$ just by the selection of notations. In this case, only $k^{+}_{s}$ survived in the limit $\varepsilon\to 0$, this means that $(\gamma_{s},\delta)<0$. Therefore, $\frac{{\mathrm{d}}(\delta,c)}{{\mathrm{d}}t}\leq 0$ and it is zero if and only if all the reaction rates of the irreversible reactions vanish. ∎ So, the vector of exponents $\delta$ defines the (partially) irreversible limit of the reaction mechanism and, at the same time, gives the explicit construction of the special Lyapunov function for the kinetic equations of the limit system. In this Section, we developed the approach to the systems with some irreversible reactions based on multiscale degeneration of equilibria, when some $a_{i}\to 0$ as $\varepsilon^{\delta_{i}}$. We proved in Theorem 4 that this approach is equivalent to the extended form of detailed balance based on the pathways analysis or on the limits of the systems with detailed balance when some of the reaction rate constants tend to zero. ## 5 Conclusion The classical principle of detailed balance operates with mechanisms, which consist of fully reversible elementary processes (reactions). If such mechanisms have cycles of reactions, each cycle is characterized by one Wegscheider relationship (8) between its rate constants. The number of functionally independent relationships is equal to the number of linearly independent cycles, linear or nonlinear. In difference from this classical case, we analyzed mechanisms, which may include irreversible reactions as well. For such mechanisms we proved an extended form of detailed balance considering the irreversible reactions as limits of reversible steps, when the rate constants of the corresponding reverse reactions approach zero. The novelty of this form is that the extended detailed balance now is presented as a necessary combination of two constituents: * 1. Structural conditions in accordance to which the irreversible reactions cannot be included in oriented cyclic pathways. * 2. Algebraic conditions which are written for the “reversible part” of the complex mechanism taken separately, without irreversible reactions, using the classical Wegscheider relationships. The computational tools combine linear algebra (some standard tools for chemical kinetics) with methods of linear programming. The most expensive computational problem appears when we check the structural condition of the extended form of detailed balance. Let $n$ be the number of components, and let $\mathbb{R}^{n}$ be the composition space. We consider a system with $r$ reversible and $s$ irreversible reactions. Let us use $\gamma_{1},\ldots,\gamma_{r}$ for the stoichiometric vectors of the reversible reactions, $\nu_{1},\ldots,\nu_{s}$ for the stoichiometric vectors of the irreversible reactions and $\overline{\nu}_{j}$ for the images of $\nu_{j}$ in the quotient space $\mathbb{R}^{n}/S$, where $S$ is spanned by the stoichiometric vectors of all reversible reaction, $S={\rm span}\\{\gamma_{1},\ldots,\gamma_{r}\\}\subset\mathbb{R}^{n}$. The reaction mechanism satisfies the structural condition of the extended form of detailed balance if and only if $0\notin{\rm conv}\\{\overline{\nu}_{1},\ldots,\overline{\nu}_{s}\\}\,.$ We have to check whether the origin belongs to the convex hull of the vectors $\overline{\nu}_{1},\ldots,\overline{\nu}_{s}$. In practice, we can always assume that these vectors have exactly known rational (or even integer) coordinates. We combined three approaches to study the restrictions implied by the principle of detailed balance in the systems with some irreversible reactions: 1. 1. Analysis of limits of the systems with all reversible reactions and detailed balance when some of the reaction rate constants tend to zero. 2. 2. Analysis of the Wegscheider identities for elementary pathways when some of the reaction rate constants turn into zero. 3. 3. Analysis of limits of the systems when some equilibrium concentrations (or, more general, activities) tend to zero. We proved that these three approaches are equivalent if we take into account not only which equilibrium concentrations tend to zero, but the speed of this tendency as well. The various partially or fully irreversible limits of the reaction mechanisms are, in this sense, multiscale asymptotics of the reaction networks when some equilibrium concentration tend to zero with different speed. ## References * Benson (1981) Benson, H. E. (1981), Processing of Gasification Products, In: Chemistry of Coal Utilization, Elliot, M., ed., New York, USA: John Wiley and Sons, Ch. 25, 1753–1800. * Bertsimas & Tsitsiklis (1997) Bertsimas, D., Tsitsiklis, J.N. (1997), Introduction to Linear Optimization, Cambridge, MA, USA: Athena Scientific. * Boltzmann (1964) Boltzmann, L. (1964), Lectures on gas theory, Berkeley, CA, USA: U. of California Press. * Chu (1971) Chu, Ch. (1971), Gas absorption accompanied by a system of first-order reactions, Chem. Eng. Sci. 26(3), 305–312. * Feinberg (1972) Feinberg, M. (1972) Complex balancing in general kinetic systems. Arch. Rat. Mechan. Anal. 49 (3), 187–194. * Fukuda & Prodon (1996) Fukuda, K., Prodon A. (1996), Double description method revisited, In: Combinatorics and Computer Science, Lecture Notes in Computer Science, Volume 1120/1996, 91–111. * Gagneur & Klamt (2004) Gagneur, J., Klamt, S. (2004), Computation of elementary modes: a unifying framework and the new binary approach, BMC Bioinformatics 5:175. * Gorban et al (1989) Gorban, A.N., Mirkes, E.M., Bocharov, A.N., Bykov, V.I. (1989), Thermodynamic consistency of kinetic data, Combustion, Explosion, and Shock Waves, 25 (5) , 593–600, * Horiuti (1973) Horiuti, J. (1973), Theory of reaction rates as based on the stoichiometric number concept, Ann. New York Academy Sci. 213, 5-30 * Ivanova (1979) Ivanova, A.N. (1979), Conditions for uniqueness of stationary state of kinetic systems related to structural scheme of reactions, Kinet. Katal., 20(4), 1019–1023. * Mincheva & Roussel (2007) Mincheva, M., Roussel, M.R. (2007), Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models, J. Math. Biol., 55, 61–68. * Moe (1962) Moe, J.M. (1962), Design of water-gas shift reactors, Chem. Eng. Progress 58 (3), 33–36. * Motzkin et al (1953) Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M. (1953), The double description method. In: H.W. Kuhn and A.W.Tucker, eds, Contributions to theory of games, Vol. 2., Princeton, NJ, USA: Princeton University Press, 51–73. * Onsager (1931) Onsager, L. (1931), Reciprocal relations in irreversible processes. I, Phys. Rev. 37, 405–426; II 38, 2265–2279. * Papin et al (2003) Papin, J.A., Price, N.D., Wiback, S.J., Fell, D.A., Palsson, B.O. (2003), Metabolic pathways in the post-genome era, Trends in Biochemical Sciences 28 (5), 250–258. * Prigogine & Defay (1962) Prigogine, I., Defay, R. (1962), Chemical Thermodynamics, New York, USA: Longmans and Green. * Schnakenberg (1976) Schnakenberg, J. (1976), Network theory of microscopic and macroscopic behavior of master equation systems, Rev. Mod. Phys. 48, 571–585. * Schuster et al (2000) Schuster, S., Fell, D.A., Dandekar, T. (2000), A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic networks, Nat. Biotechnol. 18, 326–332. * Temkin et al (1996) Temkin, O.N., Zeigarnik, A.V., Bonchev, D.G. (1996), Chemical reaction networks: a graph-theoretical approach; Boca Raton, FL, USA: CRC Press. * Volpert & Khudyaev (1985) Volpert, A.I., Khudyaev, S.I. (1985), Analysis in classes of discontinuous functions and equations of mathematical physics. Dordrecht, The Netherlands: Nijoff. * Wegscheider (1901) Wegscheider, R. (1911) Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme, Monatshefte für Chemie / Chemical Monthly 32(8), 849–906. * Yablonskii et al (1991) Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I. (1991), Kinetic Models of Catalytic Reactions, Amsterdam, The Netherlands: Elsevier. * Yablonsky et al (2010) Yablonsky, G.S., Gorban, A.N., Constales, D., Galvita, V.V., Marin, G.B. (2011), Reciprocal relations between kinetic curves, EPL 93, 20004. * Yang, et al (2006) Yang, J., Bruno, W.J., Hlavacek, W.S., Pearson, J. (2006), On imposing detailed balance in complex reaction mechanisms. Biophys. J. 91, 1136–1141.
arxiv-papers
2011-01-27T13:24:23
2024-09-04T02:49:16.666979
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "A.N. Gorban, G.S. Yablonsky", "submitter": "Alexander Gorban", "url": "https://arxiv.org/abs/1101.5280" }
1101.5373
# Beam-beam simulation code BBSIM for particle accelerators H.J. Kim hjkim@fnal.gov (H.J. Kim) T. Sen Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA ###### Abstract A highly efficient, fully parallelized, six-dimensional tracking model for simulating interactions of colliding hadron beams in high energy ring colliders and simulating schemes for mitigating their effects is described. The model uses the weak-strong approximation for calculating the head-on interactions when the test beam has lower intensity than the other beam, a look-up table for the efficient calculation of long-range beam-beam forces, and a self-consistent Poisson solver when both beams have comparable intensities. A performance test of the model in a parallel environment is presented. The code is used to calculate beam emittance and beam loss in the Tevatron at Fermilab and compared with measurements. We also present results from the studies of two schemes proposed to compensate the beam-beam interactions: a) the compensation of long-range interactions in the Relativistic Heavy Ion Collider (RHIC) at Brookhaven and the Large Hadron Collider (LHC) at CERN with a current-carrying wire, b) the use of a low- energy electron beam to compensate the head-on interactions in RHIC. ###### keywords: accelerator physics , parallel computing , beam dynamics ###### PACS: 29.27.Bd , 29.27.Fh ††journal: Nucl. Instrum. Methods Phys. Res., Sect. A ## 1 Introduction In high energy storage-ring colliders, the beam-beam interactions are known to cause emittance growth and a reduction of beam life time, and to limit the collider luminosity [1, 2, 3, 4, 5, 6, 7]. It has been a key issue in a high energy collider to simulate the beam-beam interaction accurately and to mitigate the interaction effects. A beam-beam simulation code BBSIM has been developed at Fermilab over the past few years to study the effects of the machine nonlinearities and the beam-beam interactions [8, 9, 10, 11]. The code is under continuous development with the emphasis being on including the important details of an accelerator and the ability to reproduce observations in diagnostic devices. At present, the code can be used to calculate tune footprints, dynamic apertures, beam transfer functions, frequency diffusion maps, action diffusion coefficients, emittance growth, and beam lifetime. Calculation of the last two quantities over the long time scales of interest is time consuming even with modern computer technology. In order to run efficiently on a multiprocessor system, the resulting model was implemented by using parallel libraries which are MPI (inter-processor Message Passing Interface standard) [12], state-of-the-art parallel solver libraries (Portable, Extensible Toolkit for Scientific Calculation, PETSc) [13], and HDF5 (Hierarchical Data Format) [14]. The organization of the paper is as follows: The physical model used in the simulation code is described in Section 2. The parallelization algorithm and performance are described in Section 3. Some applications are presented for the Tevatron, the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) in Section 4. Section 5 summarizes our results. ## 2 Physical model In a collider simulation, the two beams moving in opposite direction are represented by macroparticles. The macroparticles are generated with the same charge to mass ratio as the particles in the accelerator. The number of macroparticles chosen is much less than the bunch intensity of the beam because it becomes prohibitive to follow approximately 1011 particles for even a few revolutions around the accelerator using modern supercomputers. These macroparticles are generated and loaded with an initial distribution chosen for the specific simulation purpose. As an example, a six-dimensional Gaussian distribution is used for long-term beam evolution. The transverse and longitudinal motion of particles is calculated by a sequence of linear and nonlinear transfer maps. During the beam transport, a particle is removed from the distribution if it reaches a predefined boundary in transverse or longitudinal direction. In our simulation model, the following effects are included: head-on and long-range beam-beam interactions, fields of a current- carrying wire and an electron lens, multipole errors in quadrupole magnets in interaction regions, sextupoles for chromaticity correction, ac dipole, resistive wall wake, tune modulation, noise in lattice elements, single and multiple harmonic rf cavities, and crab cavities. The finite bunch length effect of the beam-beam interactions is considered by slicing the beam into several chunks in the longitudinal direction and then applying a synchro-beam map [15]. Each slice in a beam interacts with slices in the other beam in turn at a collision point. In the following, linear and nonlinear tracking models are described in detail. ### 2.1 Transport through an arc The six-dimensional coordinates of a test particle in the accelerator’s coordinate frame are: $\mathbf{x}=\left(x,x^{{}^{\prime}},y,y^{{}^{\prime}},z,\delta\right)^{T}$, where $x$ and $y$ are horizontal and vertical coordinates, $x^{\prime}$ and $y^{\prime}$ the trajectory slopes of the coordinates, $z=-c\Delta t$ the longitudinal distance from the synchronous particle, and $\delta=\Delta p_{z}/p_{0}$ the relative momentum deviation from the synchronous energy [16]. The transverse linear transformation between two elements denoted by $i$ and $j$ can be written as $\mathbf{x}_{j}=\left(\begin{array}[]{cc}\mathcal{M}&\hat{\mathcal{D}}\\\ \hat{\mathcal{A}}&\mathcal{L}\end{array}\right)\mathbf{x}_{i}.$ (1) Here, $\mathcal{M}$ is a coupled transverse map of _off-momentum_ motion defined by $\mathcal{M}=\mathcal{R}_{j}\tilde{\mathcal{M}}_{i\rightarrow j}\mathcal{R}_{i}^{-1}$, where $\tilde{\mathcal{M}}_{i\rightarrow j}$ is the uncoupled linear map described by Twiss functions at $i$ and $j$ elements, and the transverse coupling matrix $\mathcal{R}$ is defined as [17] $\mathcal{R}=\frac{1}{\sqrt{1+\left|C\right|}}\left(\begin{array}[]{cc}I&C^{\dagger}\\\ -C&I\end{array}\right)$ (2) where $C^{\dagger}$ is the $2\times 2$ matrix and the symplectic conjugate of the coupling matrix $C$. The $4\times 2$ dispersion matrix is defined by $\hat{\mathcal{D}}=\left(0,\mathbf{D}\right)$, and the dispersion vector $\mathbf{D}=\left(D_{x},D_{x}^{{}^{\prime}},D_{y},D_{y}^{{}^{\prime}}\right)^{T}$ is characterized by the transverse dispersion functions and the map $\mathcal{M}$, i.e., $\mathbf{D}=\mathbf{D}_{j}-\mathcal{M}\mathbf{D}_{i}$ where $\mathbf{D}_{i},\mathbf{D}_{j}$ are the dispersion vectors at $i,j$. Since the transport matrix has to be symplectic, the matrix $\hat{\mathcal{A}}$ in Eq. (1) is given by $\hat{A}=-\hat{\mathcal{D}}^{T}S^{T}\mathcal{M},$ where $S$ is a rearranging matrix (see subsection 2.7). The longitudinal map $\mathcal{L}$ is given by $\mathcal{L}=\left(\begin{array}[]{cc}1&-\left(\eta/\beta\right)\Delta s\\\ 0&1\end{array}\right)$, where $\eta$ is the slip factor, $\beta=v/c$, and $\Delta s$ the longitudinal distance between the two elements, i.e., $\Delta s=s_{j}-s_{i}$. It is noted that $s$ is the axis along the beam direction. The nonlinearity of synchrotron oscillations is applied by adding the longitudinal momentum change at a rf cavity: $\Delta\delta=\frac{eV_{rf}}{\beta^{2}E}\left(\sin k_{rf}z-\sin\phi_{s}\right)$ (3) where $V_{rf}$ is the voltage of rf cavity, $\phi_{s}$ the phase angle for a synchronous particle with respect to the rf wave, and $k_{rf}$ the wave number of the rf cavity. If there are higher harmonic cavities, their effects are added to the momentum change. ### 2.2 Beam-beam interactions In order to achieve high luminosity in a collider one can increase the number of bunches which reduces the bunch spacing. More bunches can increase the number of parasitic encounters in the interaction regions. Since the calculation of beam-beam forces requires large amounts of computational resources, it has to be executed rapidly and accurately. BBSIM has three different models for this purpose: a weak-strong model for head-on interactions, a look-up table model for long-range interactions, and a Poisson solver model for the head-on interactions when both beams have comparable intensities (“strong-strong” model). #### 2.2.1 Weak-strong model In the weak-strong model we assume that the “weak” beam is affected by the head-on and long-range interactions while the opposing beam or “strong” beam is unaffected. The charge distribution of the strong beam is assumed to be Gaussian: $\rho\left(x,y,z\right)=\frac{Nq}{\left(2\pi\right)^{3/2}\sigma_{x}\sigma_{y}\sigma_{z}}\exp\left(-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{y^{2}}{2\sigma_{y}^{2}}-\frac{z^{2}}{2\sigma_{z}^{2}}\right)$ (4) Here, $N$ is the number of particles per bunch and $q$ is the charge per particle. Note that the coordinates $\left(x,y,z\right)$ are measured in the rest frame of the strong beam. The beam-beam force between two beams with transverse Gaussian distribution $\rho\left(x,y\right)=\int dz\rho\left(x,y,z\right)$ is well-known [18], and the expression for the slope change is given by, for elliptical beam with $\sigma_{x}>\sigma_{y}$: $\left(\begin{array}[]{c}\Delta x^{\prime}\\\ \Delta y^{\prime}\end{array}\right)=\frac{2Nr_{0}}{\gamma}\frac{\sqrt{\pi}}{\sqrt{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}}\left(\begin{array}[]{c}\text{Im}\left[F\left(x,y\right)\right]\\\ \text{Re}\left[F\left(x,y\right)\right]\end{array}\right)$ (5) where $F\left(x,y\right)=w\left(\frac{x+iy}{\sqrt{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}}\right)-e^{-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{y^{2}}{2\sigma_{y}^{2}}}w\left(\frac{\frac{x\sigma_{y}}{\sigma_{x}}+i\frac{y\sigma_{x}}{\sigma_{y}}}{\sqrt{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}}\right).$ (6) Here, $w\left(z\right)$ is the complex error function defined by $w\left(z\right)=e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}dt\,e^{t^{2}}\right)$, and $\gamma$ the Lorentz factor. The constant $r_{0}$ is defined as $r_{0}\equiv qq_{*}/4\pi\epsilon_{0}m_{0}c^{2}$, where $q_{*}$ is the electric charge of the test particle, and $m_{0}$ the rest mass of the particle. #### 2.2.2 Look-up table model The charge distribution of the strong beam in the weak-strong model is not varied during the simulations. It is redundant to re-calculate the beam-beam force at every parasitic location and every turn. A look-up table is one way to avoid it. The look-up table is used to replace a run time computation with an array indexing operation. The beam-beam force of a Gaussian beam distribution is described by the complex error function, as shown in Eq. (6). The calculation of the complex error function can substantially slow the beam- beam simulation. However, the look-up table is pre-calculated and stored in a memory, usually in an array. When the value of the error function is required, it can be retrieved from the table by an interpolation scheme, instead of using Eq. (6). The look-up table method can significantly reduce a computational cost. The property of the complex error functions yields the symmetry relations of function $F\left(z\right)$ as $F\left(-z\right)=-F\left(z\right),\;F\left(\bar{z}\right)=-\overline{F\left(z\right)},\;F\left(-\bar{z}\right)=\overline{F\left(z\right)}$ (7) where $z=x+iy$ is a complex variable. The symmetry conditions of the function $F\left(z\right)$ can reduce memory space to store the function values. It is sufficient to build the table for the values of function $F\left(z\right)$ in the first quadrant of the complex plane, i.e., $\left|x\right|\geq 0$ and $\left|y\right|\geq 0$. Interpolation techniques are required to predict a value of a function at a point inside its domain based upon the known tabulated values. For a given set of data points $\left(z_{i},f_{i}\right)$, $i=0,\dots,N$, where no two $z_{i}$’s are the same, the interpolated value $g\left(z\right)$ at a value $z\neq z_{i}$ is found from $g\left(z\right)=\sum_{i=0}^{N}f_{i}L_{i}\left(z\right)$ (8) where the $L_{i}$ is Lagrange’s $N$-th order polynomials $L_{i}\left(z\right)=\prod_{j=0,j\neq i}^{N}\frac{z-z_{j}}{z_{i}-z_{j}}.$ (9) In order to save the interpolation time further, one can divide $z$-space and apply a different degree of the Lagrange polynomial. For an example, we apply a sixth order polynomial for small amplitudes $\left|z\right|\leq 4\sigma$ while a third order polynomial is applied for $\left|z\right|>4\sigma$, because the function $F\left(z\right)$ varies more rapidly at small $\left|z\right|$ and slowly at large $\left|z\right|$ . #### 2.2.3 Poisson solver model The weak-strong model is a good approximation when one beam has much smaller intensity than the other, but it is not valid when the intensities of the two beams are comparable because each beam’s parameters are changed by the other beam. One has to solve for the field of each beam self-consistently. The fields are the solutions of the Poisson equation given by $\nabla^{2}\phi\left(\mathbf{r}\right)=-4\pi\rho\left(\mathbf{r}\right)$ (10) where $\phi$ is the electrostatic potential and $\rho$ the density function of the beam. The solution can be obtained by $\phi\left(\mathbf{r}\right)=\int G\left(\mathbf{r},\mathbf{r}_{1}\right)\rho\left(\mathbf{r}_{1}\right)d\mathbf{r}_{1}$ (11) where $G$ is the Green’s function of the Poisson equation and in two space dimension, is given by $G\left(x,y:x_{1},y_{1}\right)=-\frac{1}{4\pi}\ln\left[\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}\right].$ (12) Equation (11) can be efficiently calculated using a convolution theorem and inverse Fourier transform: $\phi\left(\mathbf{r}\right)=\mathcal{F}^{-1}\left(\hat{G}\left(\bm{\omega}\right)\hat{\rho}\left(\bm{\omega}\right)\right)$ (13) where $\hat{G}\left(\bm{\omega}\right)=\left(\frac{1}{\sqrt{2\pi}}\right)^{2}\int_{\mathbb{R}^{2}}G\left(\mathbf{r}\right)e^{-i\bm{\omega}\cdot\mathbf{r}}d\mathbf{r}$ and $\hat{\rho}\left(\bm{\omega}\right)=\left(\frac{1}{\sqrt{2\pi}}\right)^{2}\int_{\mathbb{R}^{2}}\rho\left(\mathbf{r}\right)e^{-i\bm{\omega}\cdot\mathbf{r}}d\mathbf{r}$. It is assumed in Eq. (13) that the density function $\rho\left(\mathbf{r}\right)$ is periodic in both $x$ and $y$ directions. However, since the beam has a finite charge distribution surrounded by a conducting wall in an accelerator system, the transverse beam density does not meet the periodicity requirement of FFT techniques. In order to apply the above formalism, the density function should be rewritten by, in the doubled computational domain [19]: $\rho_{new}\left(x,y\right)=\begin{cases}\rho\left(x,y\right)&,\;0<x\leq L_{x},\;0<y\leq L_{y}\\\ 0&,\;L_{x}<x\leq 2L_{x},\;or\;L_{y}<y\leq 2L_{y}.\end{cases}$ (14) Green’s function is defined in the doubled domain, as follows: $G_{new}\left(x,y\right)=\begin{cases}G\left(x,y\right)&,\,0<x\leq L_{x},\;0<y\leq L_{y}\\\ G\left(2L_{x}-x,y\right)&,\,L_{x}<x\leq 2L_{x},\;0<y\leq L_{y}\\\ G\left(x,2L_{y}-y\right)&,\,0<x\leq L_{x},\;L_{y}<y\leq 2L_{y}\\\ G\left(2L_{x}-x,2L_{y}-y\right)&,\,L_{x}<x\leq 2L_{x},\;L_{y}<y\leq 2L_{y}.\end{cases}$ (15) Both $\rho_{new}$ and $G_{new}$ are doubly periodic functions with periods $2L_{x}$ and $2L_{y}$. It is noted that only the potential within a domain $\left(0,L_{x}\right]\times\left(0,L_{y}\right]$ is valid. The potential outside the domain is incorrect, but it doesn’t matter because the physical domain of interest is $\left(0,L_{x}\right]\times\left(0,L_{y}\right]$. When one beam is separated far from the other, one can apply a shifted Green’s function approach [20]. #### 2.2.4 Crossing angle When there exists a finite crossing angle between two colliding beams at an interaction point, the beam-beam force experienced by a test particle will have transverse and longitudinal components because the electric field generated by the opposing beam is not perpendicular to the particle velocity anymore. The existence of a longitudinal force makes it difficult to apply the result of previous sections. A transformation can be used to remedy the difficulty. It transforms a crossing angle collision in the laboratory frame to a head-on collision in the rotated and boosted frame which is called the head-on frame [21, 22]. The transformation can be described by a transformation from the accelerator coordinates to Cartesian coordinates, a Lorentz boost, and again a backward transformation to the accelerator coordinates: $\displaystyle x^{*}$ $\displaystyle=z\cos\alpha\tan\phi+x\left[1+h_{x}^{*}\cos\alpha\sin\phi\right]+yh_{x}^{*}\sin\alpha\sin\phi$ (16) $\displaystyle y^{*}$ $\displaystyle=z\sin\alpha\tan\phi+y\left[1+h_{y}^{*}\sin\alpha\sin\phi\right]+xh_{y}^{*}\cos\alpha\sin\phi$ $\displaystyle z^{*}$ $\displaystyle=\frac{z}{\cos\phi}+h_{z}^{*}\left[x\cos\alpha\sin\phi+y\sin\alpha\sin\phi\right]$ $\displaystyle p_{x}^{*}$ $\displaystyle=\frac{p_{x}}{\cos\phi}-h\cos\alpha\frac{\tan\phi}{\cos\phi}$ $\displaystyle p_{y}^{*}$ $\displaystyle=\frac{p_{y}}{\cos\phi}-h\sin\alpha\frac{\tan\phi}{\cos\phi}$ $\displaystyle p_{z}^{*}$ $\displaystyle=p_{z}-p_{x}\cos\alpha\tan\phi- p_{y}\sin\alpha\tan\phi+h\tan^{2}\phi$ where a star (*) stands for a dynamical variable in the head-on frame, the Hamiltonian $h\left(p_{x},p_{y},p_{z}\right)=p_{z}+1-\sqrt{\left(p_{z}+1\right)^{2}-p_{x}^{2}-p_{y}^{2}}$, $h_{x}^{*}=\partial h^{*}/\partial p_{x}^{*}$, $h^{*}\left(p_{x}^{*},p_{y}^{*},p_{z}^{*}\right)=h\left(p_{x}^{*},p_{y}^{*},p_{z}^{*}\right)$, $\alpha$ the crossing plane angle in the $x-y$ plane, and $\phi$ the half crossing angle in the $\tilde{x}-s$ plane as shown in Fig. 1. Figure 1: Definition of crossing angles $\alpha$ and $\phi$: $\alpha$ is the crossing plane angle in the $x-y$ plane and $\phi$ is the half crossing angle in the $\tilde{x}-s$ plane. $s$ is the axis along the beam direction when there is no crossing angle. The $\tilde{x}-s$ plane is the crossing plane defined by the angle $\alpha$. The beam trajectories, shown by lines with arrows, lie in the crossing plane. ### 2.3 Finite bunch length The effects due to the finite (as opposed to infinitesimal) bunch length need to be considered when the transverse beta functions at the interaction point are small and comparable to $\sigma_{z}$. The finite longitudinal length is considered by dividing the beam into longitudinal slices and by a so called synchro-beam map [15]. We make slices of both beams moving in opposite directions. Each slice of the strong bunch is integrated over its length, and has only a transverse charge distribution at its center. We take into account the collision between a pair of slices: the $i^{th}$ slice of a bunch and the $j^{th}$ slice of a bunch in the other beam. The collision takes place at collision point $S\left(z^{i},z_{*}^{j}\right)=\frac{1}{2}\left(z^{i}-z_{*}^{j}\right)$ which is usually different from the interaction point. For example, the $i^{th}$ slice of a bunch has successive collisions with slices of a bunch in the other beam. In addition, the electric field varies along the bunch due to the inhomogeneity of the charge density in the longitudinal direction, and couples transverse and longitudinal motions. The coupling can be modeled by the synchro-beam map which includes beam-beam interactions due to the longitudinal component of the electric field as well as the transverse components. The transformation is given by [15] $\displaystyle x^{new}$ $\displaystyle=x+S\left(z,z_{*}\right)\left.\frac{\partial U}{\partial x}\right|_{S},\enskip p_{x}^{new}=p_{x}-\left.\frac{\partial U}{\partial x}\right|_{S}$ (17) $\displaystyle y^{new}$ $\displaystyle=y+S\left(z,z_{*}\right)\left.\frac{\partial U}{\partial y}\right|_{S},\;p_{y}^{new}=p_{y}-\left.\frac{\partial U}{\partial y}\right|_{S}$ $\displaystyle z^{new}$ $\displaystyle=z,\>\delta^{new}=\delta-\frac{1}{2}\left.\frac{\partial U}{\partial x}\right|_{S}\left[p_{x}-\frac{1}{2}\left.\frac{\partial U}{\partial x}\right|_{S}\right]-\frac{1}{2}\left.\frac{\partial U}{\partial y}\right|_{S}\left[p_{y}-\frac{1}{2}\left.\frac{\partial U}{\partial y}\right|_{S}\right]-\frac{1}{2}\left.\frac{\partial U}{\partial z}\right|_{S}.$ Here, $\left.\right|_{S}$ represents the evaluation at the collision point $S\left(z,z_{*}\right)$. $U$ is the normalized potential energy $U=q\Phi/E_{0}$ and is given by $U\left(x,y;\sigma_{x}\left(s\right),\sigma_{y}\left(s\right)\right)=\frac{N_{*}r_{0}}{\gamma}\int_{0}^{\infty}d\zeta\frac{-1+\exp\left(-\frac{x^{2}}{2\sigma_{x}^{2}+\zeta}-\frac{y^{2}}{2\sigma_{y}^{2}+\zeta}\right)}{\sqrt{\left(2\sigma_{x}^{2}+\zeta\right)\left(2\sigma_{y}^{2}+\zeta\right)}}.$ (18) The dependence on the bunch length is contained in $\sigma_{x}(s),\sigma_{y}(s)$. The transverse derivatives of the potential energy are $\left.\frac{\partial U}{\partial x}\right|_{S}=-\Delta x^{\prime}\left(X,Y;S\left(z,z_{*}\right)\right),\;\left.\frac{\partial U}{\partial y}\right|_{S}=-\Delta y^{\prime}\left(X,Y;S\left(z,z_{*}\right)\right)$ (19) where $\left(X,Y\right)$ are the transverse coordinates at $S\left(z,z_{*}\right)$, and$\Delta x^{\prime}$ and $\Delta y^{\prime}$ are given by Eq. (5). The longitudinal derivative of the potential energy which is related to the longitudinal beam-beam kicks is expressed by $\displaystyle\left.\frac{\partial U}{\partial z}\right|_{S}$ $\displaystyle=\frac{1}{2}\left.\frac{d\sigma_{x}^{2}}{ds}\frac{\partial U}{\partial\sigma_{x}^{2}}\right|_{s=S\left(z,z_{*}\right)}+\frac{1}{2}\left.\frac{d\sigma_{y}^{2}}{ds}\frac{\partial U}{\partial\sigma_{y}^{2}}\right|_{s=S\left(z,z_{*}\right)}$ (20) $\displaystyle\frac{\partial U}{\partial\sigma_{x}^{2}}$ $\displaystyle=\frac{1}{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}\left[x\Delta x^{\prime}+y\Delta y^{\prime}+\frac{2N_{*}r_{0}}{\gamma}\left(\frac{\sigma_{y}}{\sigma_{x}}e^{-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{y^{2}}{2\sigma_{y}^{2}}}-1\right)\right]$ $\displaystyle\frac{\partial U}{\partial\sigma_{y}^{2}}$ $\displaystyle=\frac{-1}{2\left(\sigma_{x}^{2}-\sigma_{y}^{2}\right)}\left[x\Delta x^{\prime}+y\Delta y^{\prime}+\frac{2N_{*}r_{0}}{\gamma}\left(\frac{\sigma_{x}}{\sigma_{y}}e^{-\frac{x^{2}}{2\sigma_{x}^{2}}-\frac{y^{2}}{2\sigma_{y}^{2}}}-1\right)\right].$ Note that $\frac{d\sigma_{x}^{2}}{ds}$ and $\frac{d\sigma_{y}^{2}}{ds}$ have zero amplitude and change their sign at the interaction point if $\alpha_{x}=\alpha_{y}=0$. Test particles experience longitudinal acceleration and deceleration passing through the bunch moving in the opposite direction. ### 2.4 Compensation schemes In storage-ring colliders, a beam experiences periodic perturbations when it meets the counter-rotating beam in a common beam pipe. The head-on beam-beam interactions occur when the beams collide in the detectors while the long- range interactions occur when the beams are simultaneously present at the same location but are separated transversely. The nonlinear forces due to these beam-beam interactions result in a tune spread and can cause emittance growth, a reduction of beam life time, and therefore reduce the collider luminosity. The combination of beam-beam and machine nonlinearities excites betatron resonances which can cause particles to diffuse into the tails of the beam distribution and even to the physical aperture. Different compensation methods have been proposed: a current-carrying wire for the effects of the long-range interactions [23] and an electron lens for the head-on interactions in proton machines [24, 25, 26]. Beam collisions with a crossing angle at the interaction point are often necessary in colliders to reduce the effects of the long-range interactions. The crossing angle reduces the geometrical overlap of the beams and hence the luminosity. A deflecting mode cavity, also known as a crab cavity, offers a promising way to compensate the crossing angle and to realize effective head-on collisions [27, 28]. We now describe the modeling of these compensation schemes in the program. #### 2.4.1 Current-carrying wire When the separations at long-range interactions are large compared to the rms beam size the strength of these interactions is inversely proportional to the distance. Its effect on a beam can be compensated by a current-carrying wire which creates a magnetic field with the same $\frac{1}{r}$ dependence. This approach is simple and it is possible to deal with all multipole orders at once. For a finite length $l_{w}$ embedded in the middle of a drift length $L$, the transfer map of a wire can be obtained by $\mathcal{M}_{w}^{\left(L\right)}=D_{L/2}\circ\mathcal{M}_{k}^{\left(L\right)}\circ D_{L/2}$ (21) where $D_{L/2}$ is the drift map with a length $\frac{L}{2}$, and $\mathcal{M}_{k}^{\left(L\right)}$ is the wire kick integrated over a drift length. This kick map $\mathcal{M}_{k}^{\left(L\right)}$ is reproduced by the following changes in slope [29] $\left(\begin{array}[]{c}\Delta x^{\prime}\\\ \Delta y^{\prime}\end{array}\right)=\frac{\mu_{0}}{4\pi}\frac{I_{w}l_{w}}{\left(B\rho\right)}\frac{u-v}{x^{2}+y^{2}}\left(\begin{array}[]{c}x\\\ y\end{array}\right)$ (22) where $I_{w}$ is the current of the wire , $u=\sqrt{\left(\frac{L}{2}+l_{w}\right)^{2}+x^{2}+y^{2}}$ and $v=\sqrt{\left(\frac{L}{2}-l_{w}\right)^{2}+x^{2}+y^{2}}$. We also take into account the wire misalignment including pitch and yaw angles $\left(\theta_{x},\theta_{y}\right)$ respectively as well as lateral shifts $\left(\Delta x,\Delta y\right)$. The transfer map of a wire can be written as $\mathcal{M}_{w}=S_{\Delta x,\Delta y}\circ T_{\theta_{x},\theta_{y}}^{-1}\circ D_{L/2}\circ\mathcal{M}_{k}^{\left(L\right)}\circ D_{L/2}\circ T_{\theta_{x},\theta_{y}}$ (23) where $T_{\theta_{x},\theta_{y}}$ represents the tilt of the coordinate system by horizontal and vertical angles $\theta_{x},\theta_{y}$ to orient the coordinate system parallel to the wire, and $S_{\Delta x,\Delta y}$ represents a shift of the coordinate axes to make the coordinate systems after and before the wire agree. When the wire is parallel to the beam, Eq. (23) becomes Eq. (21). For canceling the long-range beam-beam interactions of the round beam with the wire, one can get the desired wire current and length by equating Eq. (22) and Eq. (5); the integrated strength of the wire compensator is related to the integrated current of the beam bunch as $I_{w}l_{w}=cqN$. #### 2.4.2 Electron lens For the head-on proton-proton beam collisions, particles of one proton bunch are focused by a space charge of the counter-rotating proton bunch. The beam- beam effect on the particles of the proton bunch can be compensated by a counter-rotating beam of negatively charged particles, for example, a low- energy electron beam. In order to cancel out the transverse kick by the counter-rotating proton bunch, the electron beam should have the same transverse charge profile and current as the proton bunch. The proton bunch typically exhibits an approximately Gaussian transverse profile. If we choose a Gaussian distribution of the electron beam, the transverse kick on particles of the proton bunch from the electron beam is given by $\left(\begin{array}[]{c}\Delta x^{\prime}\\\ \Delta y^{\prime}\end{array}\right)=-\frac{2N_{e}r_{0}}{\gamma r^{2}}\zeta\left(x,y:\sigma_{e}\right)\left(\begin{array}[]{c}x\\\ y\end{array}\right)$ (24) where $N_{e}$ is the number of electrons of the electron beam adjusted by the electron beam speed, $r_{0}$ the classic proton radius, $\gamma$ the Lorentz factor, $r^{2}=x^{2}+y^{2}$, and $\sigma_{e}$ the transverse beam size of the electron beam. The function $\zeta$ is given by $\zeta\left(x,y:\sigma_{e}\right)=\left[1-\exp\left(-\frac{x^{2}+y^{2}}{2\sigma_{e}}\right)\right].$ (25) For a non-Gaussian electron charge distribution we implement a flat top profile with smooth edges that generates a linear beam-beam force near the beam center. This flat top beam profile $\rho_{e}\left(r\right)=\rho_{0}/\left(1+\left(r/\sigma_{e}\right)^{8}\right)$ delivers the transverse kicks given by Eq. (24), but the function $\zeta$ is as follows: $\zeta=\frac{\sqrt{2}\tilde{\rho}_{0}}{8}\left[\frac{1}{2}\log\left(\frac{\theta_{+}^{2}+1}{\theta_{-}^{2}+1}\right)+\tan^{-1}\theta_{+}+\tan^{-1}\theta_{-}\right]$ (26) where $\tilde{\rho}$ is a constant, and $\theta_{\pm}=\sqrt{2}\left(\frac{r}{\sigma_{e}}\right)^{2}\pm 1$. #### 2.4.3 Crab cavity When a particle passes through a crab cavity structure, it experiences a transverse deflection and a small change in its longitudinal energy. Crab cavities can compensate for the horizontal or vertical crossing angle at the interaction point by delivering oppositely directed transverse kicks to the head and the tail of the bunches. In the case of a horizontal crossing, the kicks from the crab cavity are given by $\Delta x^{\prime}=-\frac{qV}{E_{0}}\sin\left(\phi_{s}+\frac{\omega z}{c}\right),\enskip\Delta\delta=-\frac{qV}{E_{0}}\cos\left(\phi_{s}+\frac{\omega z}{c}\right)\cdot\frac{\omega}{c}x$ (27) where $q$ denotes the particle charge, $V$ the voltage of crab cavity, $E_{0}$ the particle energy, $\phi_{s}$ the phase of the synchronous particle with respect to the crab-cavity rf wave, $\omega$ the angular frequency of the crab cavity, $c$ the speed of light, $z$ the longitudinal coordinate of the particle with respect to the bunch center, and $x$ the horizontal coordinate. In general this is a nonlinear map which introduces synchro-betatron coupling, but for small $z$, this reduces to a linear map in the horizontal-longitudinal plane. The crab cavity causes a closed orbit distortion dependent on the longitudinal position of particles, and the beam envelope is tilted all around the ring. For a bunch shorter than the rf wavelength of the crab cavity deflecting mode, the tilt angle of the beam envelope at a location with a beam position monitor (BPM) is given by $\tan\theta_{crab}=\frac{qV\omega\sqrt{\beta\beta_{crab}}}{c^{2}p_{0}}\left|\frac{\cos\left(\Delta\varphi-\pi Q\right)}{2\sin\pi Q}\right|$ (28) where $\beta$ is the beta function at the BPM position, $\beta_{crab}$ the beta function at the crab cavity, $\Delta\varphi$ the phase advance between the crab cavity location and the BPM, and $Q$ the betatron tune. The simulations of a crab cavity in the SPS accelerator at CERN using BBSIM will be described in another paper. ### 2.5 Particle distribution At the beginning of a simulation, the simulation particles are distributed over the phase space $\mathbf{x}=\left(x,x^{\prime},y,y^{\prime},z,\delta\right)^{T}$, called the initial loading. In any simulation the number of particles $N$ is limited by the computational power. In order to make the best use of a small number of simulation particles compared to the real number of particles in the accelerator, the loading should be optimized. Indeed the initial loading is very important because this choice can reduce the statistical noise in the physical quantities. _Gaussian distribution_ : For long-term particle tracking where we calculate emittance growth, we consider an exponential distribution in action (Gaussian distribution in coordinates) of the form: $\rho\left(\mathbf{x}\right)=\rho_{0}\exp\left(-\frac{J_{x}}{2\sigma_{J_{x}}}-\frac{J_{y}}{2\sigma_{J_{y}}}-\frac{J_{z}}{2\sigma_{J_{z}}}\right)$ (29) where $J_{x}$, $J_{y}$, and $J_{z}$ are the transverse and longitudinal action variables defined by $\displaystyle J_{x}$ $\displaystyle=\frac{1}{2\beta_{x}}\left[x^{2}+\left(\beta_{x}x^{{}^{\prime}}+\alpha_{x}x\right)^{2}\right],\;J_{y}=\frac{1}{2\beta_{y}}\left[y^{2}+\left(\beta_{y}y^{{}^{\prime}}+\alpha_{y}y\right)^{2}\right]$ (30) $\displaystyle J_{z}$ $\displaystyle=\frac{8}{\pi}\frac{R\nu_{s}}{h^{2}\left|\eta\right|}\left[E\left(k\right)-\left(1-k^{2}\right)K\left(k\right)\right]$ where $R$ is the radius of the accelerator, $h$ the harmonic number, $\nu_{s}$ the longitudinal tune, $E$ and $K$ the complete elliptical integrals, and $k^{2}=\frac{1}{4}\frac{h^{2}\eta^{2}}{\nu_{s}^{2}}\left(\frac{\Delta p}{p}\right)^{2}+\sin^{2}\frac{\phi}{2}.$ (31) $\sigma_{J_{x}}$, $\sigma_{J_{y}}$, and $\sigma_{J_{z}}$ are the rms sizes of action variables. The simulation particles are generated by two steps: 1. 1. The action variables $\left(J_{x},J_{y},J_{z}\right)$ of particles can be directly generated from the distribution function by the inverse transform method and the bit-reversed sequence [30]. 2. 2. For example, $x$ and $x^{\prime}$ are correlated and their distribution is $\hat{\rho}\left(x,x^{\prime}\right)=\hat{\rho}_{0}\exp\left(-\frac{x^{2}+\left(\beta_{x}x^{\prime}+\alpha_{x}x\right)^{2}}{2\sigma_{x}^{2}}\right)$. Since the horizontal action $J_{x}$ is determined at the first step, the horizontal coordinates $\left(x,x^{\prime}\right)$ can be obtained from the random variates: $x=\sqrt{J_{x}}\cos\theta_{x},\quad x^{\prime}=\sqrt{J_{x}}\left(\sin\theta_{x}-\alpha_{x}\cos\theta_{x}\right)/\beta_{x}$ where the value of $\theta_{x}$ is randomly distributed within the interval $0\leq\theta_{x}\leq 2\pi$. _Hollow Gaussian distribution_ : In most cases of particle tracking, lost particles are observed only above a certain large transverse action while the beam core is stable. An example is shown in Section 4.1. A hollow beam is a beam with zero central intensity along the longitudinal beam axis. For the generation of a hollow beam, a bunched beam distribution in longitudinal phase space is a Gaussian, but a distribution in transverse phase space is a hollow Gaussian. The procedure of generating the hollow distribution is the same as that for the Gaussian distribution except that the amplitude of transverse action of a particle should be larger than a minimum value, i.e., $J_{x}+J_{y}\geq\sigma_{J}$. Since most of the stable particles are not included in the tracking simulation, the hollow beam model simulates a large transverse amplitude Gaussian distribution using a small number of macro- particles. This distribution is useful when calculating beam lifetimes. ### 2.6 Particle diffusion Diffusion coefficients can characterize the effects of the nonlinearities present in an accelerator, and can be used to find numerical solutions of a diffusion equation for the density [31, 32]. The solutions yield the time evolution of the beam density distribution function for a given set of machine and beam parameters. This technique enables us to follow the beam intensity and emittance growth for the duration of a luminosity store, something that is not feasible with direct particle tracking. The transverse diffusion coefficients can be calculated numerically from $\displaystyle D_{ij}\left(a_{i},a_{j}\right)$ $\displaystyle=\frac{1}{N}\left\langle\left(J_{i}(a_{i},N)-J_{i}(a_{i},0)\right)\left(J_{j}(a_{j},N)-J_{j}(a_{j},0)\right)\right\rangle$ (32) where $J_{i}\left(a_{i},0\right)$ is the initial action at an amplitude $a_{i}$, $J_{i}\left(a_{i},N\right)$ the action with initial amplitude $a_{i}$ after $N$ turns, $\left\langle\right\rangle$ the average over simulation particles, and $(i,j)$ are the horizontal $x$ or the vertical $y$ coordinates. Equation (29) is averaged over a certain number of turns to eliminate the fluctuation in action due to the phase space structure, e.g. resonance islands. These diffusion coefficients can be directly used to compare amplitude growth under different circumstances, e.g with different tunes. Emittance growth and beam lifetimes can be calculated when these coefficients are used in a diffusion equation, as mentioned above. ### 2.7 Symplecticity In the absence of dissipative effects, particle motion in an accelerator can be described by Hamilton’s equations of motion. Hamiltonian systems obey the symplectic condition which guarantees the conservation of phase space volume as the system evolves, this is also known as Liouville’s theorem. For transfer maps described in previous subsections the symplectic condition requires $M^{T}SM=S,\quad S=\left(\begin{array}[]{ccc}s&0&0\\\ 0&s&0\\\ 0&0&s\end{array}\right)$ (33) where $s=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right)$ is an antisymmetric $2\times 2$ matrix, and $M$ is a transfer matrix for a linear system or the Jacobian matrix of a nonlinear map around any particle trajectory. For a nonlinear map $\mathcal{M}:\mathbf{x}\longrightarrow\bar{\mathbf{x}}$, the Jacobian matrix is obtained from first-order partial derivatives of the new coordinates with respect to the old ones. The elements are defined as $M_{ij}=\partial\bar{x}_{i}/\partial x_{j}$. During implementation of the maps for beam dynamics, one should check to ensure that the map is as symplectic as possible. As a measure of the symplecticity, a matrix norm of $\left\|M^{T}SM-S\right\|$ is used in BBSIM. The accuracy of the look-up table model mentioned in subsection 2.2.2, for example, depends on the number of sample points in a given complex space needed for interpolating the function. Poor interpolation accuracy may violate the symplecticity, and lead to emittance blow-up or shrinkage. We use the symplectic norm obtained with the direct calculation of the complex error function as the benchmark. We find for example, that in order to maintain the symplectic norm with 70 long-range beam-beam interactions in the Tevatron, the number of sample points should be more than 4 points per rms beam size. ### 2.8 Diagnostics Numerical simulation enables the generation of very large amounts of data. The BBSIM code monitors physical quantities, for example, particle amplitudes and saves them into an external file during the simulation. According to a problem of interest, the quantities to be saved can be chosen in order to extract valuable information from post-processing. In addition, some diagnostic functions are calculated in the code as follows: _Betatron tune distribution_ : The betatron tune in an accelerator is one of the most important beam parameters. The tune of each particle in the beam distribution is calculated with a Hanning filter applied to an fast-Fourier transform of particle coordinates found from tracking [33]. _Beam transfer function_ : The beam transfer function (BTF) is defined as the beam response to a small external longitudinal or transverse excitation at a given frequency. BTF diagnostics are widely employed in accelerators due to its non-destructive nature. A stripline kicker or rf cavity excites betatron or synchrotron oscillations respectively over the appropriate tune spectrum. The beam response is observed in a downstream pickup. The fundamental applications of BTF are to measure the transverse tune and tune distribution by exciting betatron oscillation, to analyze the beam stability limits, and to determine the impedance characteristics of the chamber wall, and feedback system [34]. In the code, we apply a sinusoidal driving force to a beam in a transverse plane. The driving frequency is swept in equidistant steps over a continuous frequency range which includes the betatron tune. At each new frequency there is initially a transient response which must be allowed to relax before the frequency is extracted from the data. We avoid the issue of the transients in the simulations by reloading the initial particle distribution at each new frequency. _Frequency diffusion_ : We have calculated frequency diffusion maps as another way to investigate the effects of nonlinear forces. The map represents the variation of the betatron tunes over two successive sets of the tunes [35]: The variation can be quantified by $d=\log\sqrt{\Delta\nu_{x}^{2}+\Delta\nu_{y}^{2}}$, where ($\Delta\nu_{x}=\nu_{x}^{\left(2\right)}-\nu_{x}^{\left(1\right)},\Delta\nu_{y}=\nu_{y}^{\left(2\right)}-\nu_{y}^{\left(1\right)}$) are the tune variations between the first set and next set of 1024 turns. If the tunes $\left(\nu_{x}^{\left(1\right)},\nu_{y}^{\left(1\right)}\right)$ are different from $\left(\nu_{x}^{\left(2\right)},\nu_{y}^{\left(2\right)}\right)$, the particle is moving to different amplitudes. A large tune variation is generally an indicator of fast diffusion and reduced stability. _Dynamic aperture_ : The dynamic aperture of an accelerator is defined as the smallest radial amplitude of particles that survive up to a certain time interval, for example, $10^{6}$ turns. As the number of turns increases, the dynamic aperture approaches an asymptotic value. Initial particles are distributed uniformly over the transverse phase space with amplitudes typically varying between 0-20 $\sigma$, where $\sigma$ is the rms transverse beam size. The longitudinal amplitude is chosen as largest value within a bunch. _Emittance_ : The emittance is defined as the area (or volume) of phase space enclosed by the ellipse containing all the particles in its interior. Statistically, the rms beam emittance can be calculated by a determinant of $\Sigma$-matrix of a beam distribution: $\epsilon=\left[\det\left(\Sigma\right)\right]^{1/d}$ (34) where $d$ is the dimension of phase space, the element of $\Sigma$-matrix is $\Sigma_{ij}=\left\langle\left(\zeta_{i}-\left\langle\zeta_{i}\right\rangle\right)\left(\zeta_{j}-\left\langle\zeta_{j}\right\rangle\right)\right\rangle$, and $\zeta=\left\\{x,x^{\prime},y,y^{\prime},z,\delta\right\\}$. For example, horizontal emittance is obtained by $\epsilon_{x}=\left[\det\left(\begin{array}[]{cc}\Sigma_{xx}&\Sigma_{xx^{\prime}}\\\ \Sigma_{x^{\prime}x}&\Sigma_{x^{\prime}x^{\prime}}\end{array}\right)\right]^{1/2}$. In addition to the emittance of each degree of freedom, four- and six- dimensional emittances are calculated to see the correlation and coupling between the phase space coordinates. _Beam loss_ : The beam loss is one of the fundamental observables and it can be directly compared with simulation. During a beam simulation, each particle is monitored if it reaches a physical aperture transversely or the rf bucket longitudinally. The particle passing beyond the aperture is considered as a lost particle. Unlike a real machine, several _virtual_ apertures are placed inside a beam pipe. The multiple apertures are used to find beam losses at different apertures. ## 3 Parallelization Realistic simulations of beam dynamics demand large computational resources. Calculations on these large number of particles can be distributed over several processors of a parallel computer to improve performance. Two basic approaches exist to allocate the calculations to the processors, particle based and domain (space) based partitions. In the former approach, the particles are uniformly allocated to the processors. They are not limited to a certain spatial domain. The completion time of a parallel solution depends on the processor with the maximum computational workload. The particle decomposition can distribute the computational load evenly among all processors while the interaction between particles, for example, intra-beam scattering needs a very large number of communications between processors since the interacting particles can be located in a distant processor. Conversely, in the domain decomposition approach, the spatial domain is partitioned into elementary regions, and each processor is responsible for one of these regions. The particles in the accelerator simulation are transported by the lattice map. The map causes significant particle movement which may cause the load to become quickly unbalanced. The simulation of colliding beams has two aspects, i.e., pure particle transport and electromagnetic field evaluation. The domain deposition approach is an efficient way of parallelizing the field solver. To achieve the workload balanced, our approach is to use both decomposition schemes. We have implemented a parallel calculation in the BBSIM code to perform a tracking simulation of large numbers of particles. When the weak-strong beam- beam model is used, only the particle decomposition scheme can be applied for parallel computation. Its implementation can be made trivially because the macroparticles are never moved from one processor to another. No inter- processor communication is necessary while the particle trajectories are being developed. Most calculations on each node are executed sequentially. In this model the communication between the parallel processes is only required for reading input data, generating an initial beam distribution, calculating diagnostics such as beam emittance, and writing out the diagnostic information. For the Poisson solver model, however, we have used a particle- in-cell (PIC) model to update the electromagnetic field. The PIC model represents the beam as a large number of computational particles moving according to classical mechanics. The PIC algorithm can be characterized as follows: (a) integrate over particles to obtain a charge distribution on the grid point, (b) solve a Poisson equation for the potential, and (c) interpolate the potential or field onto particles for a small interval of time to advance the position and velocity of particles. Part (a) requires $\mathcal{O}\left(N_{g}^{d}\right)$ numeric operations for a FFT Poisson solver, where $N_{g}$ is the number of grid points per dimension and $d$ is the number of degrees of freedom. Part (a) and (c) obviously require $\mathcal{O}\left(N_{p}\right)$ operations, where $N_{p}$ is the number of computation particles. In general, $N_{p}$ is much larger than $N_{g}$ in that the number of particles should increase according to the degree of freedom to maintain the statistical noise to be constant in a higher spatial dimension. The particle calculations thus dominate the overall computational process, which suggests a prior parallelization of particle calculation. Master/slave configuration of computational nodes shown in Fig. 2 is considered due to the difference of numeric operations between particles and field updates. Each processor on the master and slave nodes possesses the same number of particles. All processors are responsible for advancing their particles. On the contrary, the master node may be a single or many processor(s), depending on the number of grid points required. The charge density of a beam is deposited on the computational grids of each processor using standard area weighting (or higher order) methods [36]. The master node gathers the charge density from all processors, and solves the Poisson equations in parallel. The master node broadcasts the solution of the electric field to all processors such that each processor exerts the electromagnetic force on the particles owned by the processor. Figure 2: Master/slave communication diagram. The performance of the master/slave parallelization approach has been investigated using a real lattice of the Tevatron which has two head-on beam- beam collisions and 70 long-range beam-beam interactions. Speedup test has been performed on the Cray XT5 of the National Energy Research Scientific Computing Center at Lawrence Berkeley National Laboratory. The system is built up of 664 nodes with two quad-core AMD 2.4 GHz processors per node. The speedup of a parallel program is a measure of the utilization of parallel resources and is simply defined as the ratio between sequential execution time and parallel execution time [37]: $S_{p}=\frac{T_{1}}{T_{p}}$ (35) where $p$ is the number of processors, $T_{1}$ is the execution time of the sequential algorithm, and $T_{p}$ is the execution time of the parallel algorithm with $p$ processors. For a fixed number of processors $p$, typically the speedup is $0<S_{p}\leq p$. Ideally all parallel programs should exhibit a linear speedup, i.e., $S_{p}=p$, but it is not common because communication between processors is considerably slower than computation in each processor. Figure 3 (a) illustrates the resulting speedup as a function of the number of processors. (a) (b) Figure 3: Plots of (b) parallel speedup versus the number of nodes, and (b) CPU time versus the number of simulation particles. cerf and table represent the weak-strong model, and look-up table model respectively. The parallelization speedup based on the total simulation time is compared for simulations with the weak-strong model and the look-up table model. The speedup curves are very close to the ideal one below a certain number of processors, while they are less than optimal when the number of processors increases above a critical value, for example, $2^{6}$ processors. On large numbers of processors a relative fraction of the communication time in the total computing time becomes large. A parallel efficiency, defined as the speedup factor divided by the number of processors, can be obtained as high as 87% up to the critical number of processors. Though the efficiency falls well below 38% when the number of processors is beyond $2^{10}$, it runs 367 times faster than on a single processor. In order to see the scalability of our parallel code for larger problem sizes, Fig. 3 (b) shows the execution time as a function of the number of macro-particles. Here the number of processors is fixed at $2^{6}$ for all cases. It is seen that with increasing the number of simulation particles, the execution time also increases linearly. ## 4 Applications In high energy storage-ring colliders, the beam-beam interactions cause emittance growth, may reduce beam lifetime, and hence limit the collider luminosity. We have used BBSIM to study beam-beam interactions and their compensations in the Tevatron, in RHIC and in the LHC. ### 4.1 Tevatron The luminosity of a collider is found from $\mathcal{L}=\frac{N_{1}N_{2}fN_{B}}{4\pi\sigma_{x}\sigma_{y}}R$ (36) where $N_{1}$ and $N_{2}$ are the bunch populations of the colliding beams, $f$ the revolution frequency, $N_{B}$ the number of bunches in one beam, $\sigma_{x}$ and $\sigma_{y}$ the horizontal and vertical rms beam sizes at the collision points respectively, and $R$ the luminosity reduction factor due to the “hour-glass” effect and due to non-zero crossing angle at the interaction point. The beam-beam tune shift of beam 1 is proportional to the factor $N_{2}/\sigma_{x}\sigma_{y}$ and experience from colliders worldwide has shown that the achievable tune shift (and hence luminosity) is limited by the dynamics of the beam-beam interaction. In the Tevatron, proton and anti- proton bunches collide at two detectors called CDF and D0. They share the same beam pipe. Since the two beams circulate on helical orbits, the optics and dynamics of the beam-beam interactions are complex. The beam-beam interactions occur all around the ring and at varying betatron phases. In run II, each beam has three trains of 12 bunches [38]. Each bunch experiences 72 interactions: two interactions are the head-on collisions in the detectors. However, the other 70 interactions are long-range, and are placed at different locations for each bunch. Consequently the beam separation distances between proton and anti-proton beams at the long-range locations are different from bunch to bunch. Figure 4 shows the radial beam separation of three anti-proton bunches from the proton bunches in units of the rms beam size of the proton beam at the locations of the beam-beam interactions. Figure 4: Separation distance between proton and anti-proton beams for anti- proton bunches #1, #6 and #12. The separation is normalized by proton beam’s rms size. The long-range interactions of special importance are those on either side of the head-on interaction points. These occur at small separations and the beta functions there are large. It was observed that the emittance growth at the end bunches of each train is smaller than those in the middle of the train. Here we choose two end bunches (#1 and #12) and one middle bunch (#6) of the first train. Beam emittance growth and loss rate are routinely measured during the Tevatron operation. They can be directly compared with numerical simulations but only for relatively short times. Figure 5 (a) shows the time evolution of the four- dimensional emittance of bunches #1, #6, and #12 for 15 hours of high energy physics (HEP) run of store # 7650. (a) (b) (c) (d) Figure 5: (a) Variation of anti-proton emittance of three bunches, #1, #6, and #12, of store #7650, (b) non-luminous loss rates of anti-proton during the first 1 hour of stores #7601-#7650, (c) simulation of anti-proton emittance growth, and (d) simulation of anti-proton beam loss. Here the emittance is plotted as $\epsilon_{4d}=\sqrt{\epsilon_{x}\epsilon_{y}}$. In the simulation, initial anti-proton emittance $\left(\epsilon_{x},\epsilon_{y}\right)$ is (9.0,7.8) mm-mrad, bunch length 1.5 nsec, and bunch intensity $0.86\times 10^{11}$. Proton’s initial emittance is (18,23) mm-mrad, bunch length 1.7 nsec, bunch intensity $2.64\times 10^{11}$. Nominal tune is (20.571, 20.569). Revolution frequency is 47.7 kHz. The emittance is calculated and plotted by $\epsilon_{4d}=\sqrt{\epsilon_{x}\epsilon_{y}}$. It is observed that during the HEP run, the emittance growth is nearly linear. The growth rate is 6.7%/hr. Figure 5 (b) shows the measured beam loss rates of anti-proton bunches during the first 1 hour of store #7601-#7650 at collision energy 960 GeV. In order to see the effects of beam-beam interactions on the beam loss, the loss rate is obtained by subtracting the particle losses due to luminosity at the main interaction points from the total beam loss rate. Averaged loss rates of bunch #1 and #12 are 1.4 %/hr and 1.2 %/hr respectively, while the loss rate of bunch #6 is 2.3 %/hr. We performed the simulations of emittance growth and particle loss of anti-proton beam, as shown in Fig. 5 (c)-(d). The particle tracking is carried out over $10^{7}$ turns corresponding to approximately 3.5 minutes storage time of the Tevatron. In the simulation, nominal tune is (20.571, 20.569). Initial transverse (95% normalized) emittance of anti-protons $\left(\epsilon_{x},\epsilon_{y}\right)$ is set to be (9.0,7.8) mm-mrad from averaging the measured emittances while proton’s initial emittance is (18,23) mm-mrad. Bunch intensities of anti-proton and proton are $0.86\times 10^{11}$ and $2.64\times 10^{11}$ respectively. Figure 5 (c) shows the emittance growth of three bunches during the simulation. The growth rate is approximately 9 %/hr, which is close to the measured growth rate 7 %/hr in Fig. 5 (a). The emittance does not vary from bunch to bunch. However, the beam losses vary considerably from bunch to bunch. As shown in Fig. 5 (d), bunch #6 loses more particles than bunches #1 and #12, which agrees well with the observation. For the simulation of beam loss, we used the hollow Gaussian distribution in transverse action coordinates. Most of the lost particles have large transverse actions as shown in Fig. 6 (a), while the lost particles are distributed over the entire range of longitudinal action, as shown in Fig. 6 (b). (a) (b) Figure 6: (a) Scatter plot of lost particles in action space $\left(\sqrt{J_{x}},\sqrt{J_{y}}\right)$ and (b) plot of lost particles versus $\sqrt{J_{x}+J_{y}}$ for different longitudinal action. The axis variables are normalized by rms size of transverse action. The compensation of long-range effects in the Tevatron with a current-carrying wire was investigated using an earlier version of the code [8]. It was found that a single wire was unable to compensate for all the 70 interactions, since they were all at different betatron phases from the wire. ### 4.2 Relativistic Heavy Ion Collider We have studied the effects of a current-carrying wire on the beam dynamics in RHIC [32]. Two current-carrying wires, one for each beam, have been installed between the magnets $Q3$ and $Q4$ of IP6 in the RHIC tunnel. In the physics run 9, an attempt was made to compensate the long range beam-beam interaction which shows the reduction of beam loss [39]. During the physics run 7 and 8, the impact of current-carrying wires on a beam was measured without an attempt to compensate the beam-beam interactions. However, the experimental results help to understand the beam-beam effects because the wire force is similar to the long-range beam-beam force at large separations. As an example, Fig. 7 plots the beam loss rate due to the wire as a function of beam-wire separation distance. (a) (b) Figure 7: Comparison of the simulated beam loss rates with the measured as a function of separations. (a) gold beam at collision energy, (b) deuteron beam at collision energy [32]. The onset of beam losses is observed at 8 $\sigma$ and 9 $\sigma$ for gold and deuteron beams, respectively. The threshold separation for the onset of sharp losses observed in the measurements and simulations agree to better than 1 $\sigma$. It is also significant that the simulated loss rates at 7 and 8 $\sigma$ separation for the gold beam and 8 and 9 $\sigma$ for the deuteron beam are very close to the measured loss rates. At fixed separation, the wire causes a much higher beam loss with the deuteron beam than with the gold beam. The loss rate for the gold beam at a 8 $\sigma$ separation is about 10 %/hr while for the deuteron beam the loss rate is about an order of magnitude higher both in measurements and simulation. Simulations of the beam loss rate when the wire is present are in good agreement with the experimental observations. In the proton-proton runs of RHIC, the maximum beam-beam parameter reached so far is about $\xi=0.008$. This tune shift is large enough that the combination of beam-beam and machine nonlinearities excite betatron resonances which cause emittance growth and diffuse particles into the tail of beam distribution and beyond. Consequently RHIC is actively developing an electron lens for compensating the head-on interactions [40]. In order to seek the electron lens parameters at which the beam life time is improved, we choose three different electron beam distribution functions: (a) $1\sigma$ Gaussian distribution with the same rms beam size as that of the proton beam $\sigma$, (b) $2\sigma$ Gaussian distribution with rms size twice that of the proton beam, and (c) Smooth-edge-flat-top (SEFT) distribution with an edge around at 4 $\sigma$. When the electron beam profile matches the proton beam, the full compression of the tune spread requires the electron beam intensity $N_{e}=4\times 10^{11}$ which is defined as the electron beam intensity required for full compensation. Table 1 shows the results of particle loss for different intensities with the three electron beam profiles. Profile | Intensity $\left(4\times 10^{11}\right)$ | Particle loss†(%) ---|---|--- $1\sigma$ Gaussian | 1 | 635 | 1/2 | 115 | 1/4 | 63 | 1/8 | 30 $2\sigma$ Gaussian | 4 | 93 | 2 | 10 | 1 | 8 | 1/2 | 6 SEFT | 8 | 330 | 4 | 21 | 2 | 22 | 1 | 6 | 1/2 | 6 †relative to that without beam-beam compensation Table 1: Comparison of particle loss for different electron beam profiles and intensities. At an intensity $N_{e}=4\times 10^{11}$, the particle loss is nearly six times the loss without beam-beam compensation. The beam lifetime at $N_{e}=2\times 10^{11}$ however is comparable with that of no beam-beam compensation. As the electron beam intensity is decreased, the particle loss decreases significantly, and is reduced to 30% of that without beam-beam compensation at $N_{e}=0.5\times 10^{11}$. For the $2\sigma$ Gaussian and SEFT electron beam profiles, we calculated particle loss for different electron beam intensities. The upper limits of the electron beam intensity for these two distributions are chosen so that peak of the electron profile matches that of the full compensation at $1\sigma$ Gaussian. For the intensities $2\times 10^{11}$ and $4\times 10^{11}$ of $2\sigma$ Gaussian profile, there is a significant reduction in beam loss, for example, below 10% of the particle loss without beam-beam compensation when the electron beam intensity is $2\times 10^{11}$. A significant improvement of beam lifetime with the SEFT profile is also observed below $8\times 10^{11}$. There is a threshold electron beam intensity below which beam life time is increased: $2\times 10^{11}$ for the $1\sigma$ Gaussian, $8\times 10^{11}$ for the $2\sigma$ Gaussian, and $16\times 10^{11}$for the SEFT profile. Particle loss is relatively insensitive to electron lens current variations below the threshold current with the $2\sigma$ Gaussian and SEFT profiles. This looser tolerance on the allowed variations in electron intensity will allow greater intensity fluctuations and is likely to be beneficial during experiments. ### 4.3 Large Hadron Collider As mentioned above, long-range beam-beam interactions cause emittance growth or beam loss in the Tevatron and are expected to deteriorate beam quality in the LHC. Increasing the crossing angle to reduce their effects has several undesirable effects, the most important of which is a lower luminosity due to the smaller geometric overlap. For the LHC, a wire compensation scheme has been proposed to compensate the long-range interactions [23]. However, several issues need to be resolved for efficient compensation. With the design bunch spacing, there are about 30 long-range interactions on both sides of an interaction point (IP). The beam-beam separation distance varies from 6.3 $\sigma$ to 12.6 $\sigma$. The resulting beam-beam force is not identical to that generated by a single or multiple wire(s) but can be closely approximated by the wires. Unlike the Tevatron, the long-range forces in the LHC are all at nearly the same betatron phase and this makes the compensation scheme feasible. The wire-beam separation distance is one of the parameters which determine the performance of a wire compensator. Figure 8 (a) shows the beam- beam separation distance normalized by the transverse rms bunch size. Two counter-rotating beams collide at a vertical crossing angle near IP1 while they collide at a horizontal crossing angle near IP5. The separations are asymmetric with respect to the interaction points. The reference wire-beam separation (9 $\sigma$) is chosen as the average of beam-beam separations. (a) (b) Figure 8: Plot of (a) beam-beam separation at IP 1 and 5 and (b) particle loss according to wire separation distance with wire strength 82.8 Am. Figure 8 (b) shows the results of particle loss for different wire-beam separations. The particle loss saturates at large separation while there is a sharp increase of particle loss at small separation. We directly see the minimum particle loss between 0.9 and 1.0 of the reference separation. It reveals that the average of beam-beam separations is close to an optimal separation between the wire and the high energy bunch. ## 5 Summary In this paper, an efficient parallel beam simulation model for circular colliders is presented in order to study the effects of beam-beam interactions and machine nonlinearities, and the effectiveness of beam-beam compensation schemes. We have included the major nonlinearities present in accelerators in our program as well as models for several methods to compensate the effects of beam-beam interactions. A particle-domain decomposition scheme is implemented with the master/slave configuration to achieve a balanced workload in a parallel environment. A performance test of beam-beam interactions indicates that the parallelization scheme scales linearly in both the number of processors and the number of particles in the beam. We have used the program to study the emittance growth and beam loss of different bunches due to the beam-beam interactions in the Tevatron, the compensation of head-on beam-beam interactions with a low energy electron beam in RHIC, and the long-range beam- beam compensation using a current-carrying wire in the Tevatron, RHIC and the LHC. The pattern of beam losses observed in the Tevatron is reproduced in the simulations. In RHIC, simulations of the beam loss rate when the wire is present are in good agreement with the experimental observations. We have several predictions from the results of head-on compensation in RHIC. For example we find that proton beam life time is increased if the electron beam intensity is kept below a threshold intensity. An electron beam wider than the proton beam at the electron lens location is found to increase beam life time. The results of LHC simulation with the current carrying wire show that the particle loss is minimized when the beam-wire separation is close to the average of beam-beam separations. ## 6 Acknowledgments We thank V. Boocha, B. Erdelyi and V. Ranjbar for their contributions to the development of BBSIM. This research used resources of the Accelerator Physics Center at Fermi National Accelerator Laboratory as well as resources of the National Energy Research Scientific Computing Center at Lawrence Berkeley National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy. This work is partially supported by the US Department of Energy through the US LHC Accelerator Research Program (LARP). Fermi National Accelerator Laboratory (Fermilab) is operated by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States Department of Energy. ## References * [1] J.-Y. Hemery, A. Hofmann, J.-P. Koutchouk, S. Myers, L. Vos, Investigation of the coherent beam-beam effects in the ISR, IEEE T. Nucl. Sci. NS-28 (1981) 2497\. * [2] L. Evans, J. Gareyte, M. Meddahi, R. Schmidt, Beam-beam effects in the strong-strong regime at the CERN-SPS, in: Proceedings of the 1889 Particle Accelerator Conference, 1989. * [3] W. Fischer, A. Drees, J. Brennan, R. Connolly, R. Fliller, S. Tepikian, J. van Zeijts, Beam lifetime and emittance growth measurements of gold beams in RHIC at storage, in: Proceedings of the 2001 Particle Accelerator Conference, 2001, p. 21. * [4] T. Sen, B. Erdelyi, M. Xiao, V. Boocha, Beam-beam effects at the Fermilab Tevatron: Theory, Phys. Rev. Spec. Top. Acceler. Beams 7 (2004) 041001. * [5] F. 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arxiv-papers
2011-01-27T20:02:23
2024-09-04T02:49:16.679495
{ "license": "Public Domain", "authors": "Hyung J. Kim and Tanaji Sen", "submitter": "Hyung Jin Kim", "url": "https://arxiv.org/abs/1101.5373" }
1101.5415
###### Abstract Let $M_{R}$ be a module and $\sigma$ an endomorphism of $R$. Let $m\in M$ and $a\in R$, we say that $M_{R}$ satisfies the condition $\mathcal{C}_{1}$ (respectively, $\mathcal{C}_{2}$), if $ma=0$ implies $m\sigma(a)=0$ (respectively, $m\sigma(a)=0$ implies $ma=0$). We show that if $M_{R}$ is p.q.-Baer then so is $M[x;\sigma]_{R[x;\sigma]}$ whenever $M_{R}$ satisfies the condition $\mathcal{C}_{2}$, and the converse holds when $M_{R}$ satisfies the condition $\mathcal{C}_{1}$. Also, if $M_{R}$ satisfies $\mathcal{C}_{2}$ and $\sigma$-skew Armendariz, then $M_{R}$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ ($\sigma\in Aut(R)$) is a p.p.-module. Many generalizations are obtained and more results are found when $M_{R}$ is a semicommutative module. On Skew Polynomials over p.q.-Baer and p.p.-Modules Mohamed Louzari Department of mathematics Abdelmalek Essaadi University B.P. 2121 Tetouan, Morocco mlouzari@yahoo.com This work is dedicated to my Professor El Amin Kaidi Lhachmi from University of Almería on the occasion of his 62nd birthday. Mathematics Subject Classification: 16S36, 16D80, 16W80 Keywords: Semicommutative modules, p.q.-Baer modules, p.p.-modules. 000Published in Inter. Math. Forum, Vol. 6, 2011, no. 35, 1739 - 1747 ## 1 Introduction In this paper, $R$ denotes an associative ring with unity and modules are unitary. We write $M_{R}$ to mean that $M$ is a right module. Throughout, $\sigma$ is an endomorphism of $R$ (unless specified otherwise), that is, $\sigma\colon R\rightarrow R$ is a ring homomorphism with $\sigma(1)=1$. The set of all endomorphisms (respectively, automorphisms) of $R$ is denoted by $End(R)$ (respectively, Aut(R)). In [10], Kaplansky introduced Baer rings as rings in which the right (left) annihilator of every nonempty subset is generated by an idempotent. According to Clark [9], a ring $R$ is said to be quasi-Baer if the right annihilator of each right ideal of $R$ is generated (as a right ideal) by an idempotent. These definitions are left-right symmetric. Recently, Birkenmeier et al. [7] called a ring $R$ a right $($respectively, left$)$ principally quasi-Baer (or simply right $($respectively, left$)$ p.q.-Baer) if the right (respectively, left) annihilator of a principally right (respectively, left) ideal of $R$ is generated by an idempotent. $R$ is called a p.q.-Baer ring if it is both right and left p.q.-Baer. A ring $R$ is a right (respectively, left) p.p.-ring if the right (respectively, left) annihilator of an element of $R$ is generated by an idempotent. $R$ is called a p.p.-ring if it is both right and left p.p.-ring. Lee-Zhou [12] introduced Baer, quasi-Baer and p.p.-modules as follows: $(1)$ $M_{R}$ is called Baer if, for any subset $X$ of $M$, $r_{R}(X)=eR$ where $e^{2}=e\in R$. $(2)$ $M_{R}$ is called quasi-Baer if, for any submodule $N$ of $M$, $r_{R}(N)=eR$ where $e^{2}=e\in R$. $(3)$ $M_{R}$ is called p.p. if, for any $m\in M$, $r_{R}(m)=eR$ where $e^{2}=e\in R$. In [3], a module $M_{R}$ is called principally quasi Baer (p.q.-Baer for short) if, for any $m\in M$, $r_{R}(mR)=eR$ where $e^{2}=e\in R$. It is clear that $R$ is a right p.q.-Baer ring if and only if $R_{R}$ is a p.q.-Baer module. If $R$ is a p.q.-Baer ring, then for any right ideal $I$ of $R$, $I_{R}$ is a p.q.-Baer module. Every submodule of a p.q.-Baer module is p.q.-Baer module. Moreover, every quasi-Baer module is p.q.-Baer, and every Baer module is quasi-Baer module. A ring $R$ is called semicommutative if for every $a\in R$, $r_{R}(a)$ is an ideal of $R$ (equivalently, for any $a,b\in R$, $ab=0$ implies $aRb=0$). In [8], a module $M_{R}$ is semicommutative, if for any $m\in M$ and $a\in R$, $ma=0$ implies $mRa=0$. Let $\sigma$ an endomorphism of $R$, $M_{R}$ is called $\sigma$-semicommutative module [13] if, for any $m\in M$ and $a\in R$, $ma=0$ implies $mR\sigma(a)=0$. According to Annin [1], a module $M_{R}$ is $\sigma$-compatible, if for any $m\in M$ and $a\in R$, $ma=0$ if and only if $m\sigma(a)=0$. In [12], Lee-Zhou introduced the following notations. For a module $M_{R}$, we consider $M[x;\sigma]:=\left\\{\sum_{i=0}^{s}m_{i}x^{i}:s\geq 0,m_{i}\in M\right\\},$ $M[[x;\sigma]]:=\left\\{\sum_{i=0}^{\infty}m_{i}x^{i}:m_{i}\in M\right\\},$ $M[x,x^{-1};\sigma]:=\left\\{\sum_{i=-s}^{t}m_{i}x^{i}:\;t\geq 0,s\geq 0,m_{i}\in M\right\\},$ $M[[x,x^{-1};\sigma]]:=\left\\{\sum_{i=-s}^{\infty}m_{i}x^{i}:s\geq 0,m_{i}\in M\right\\}.$ Each of these is an Abelian group under an obvious addition operation. Moreover $M[x;\sigma]$ becomes a module over $R[x;\sigma]$ under the following scalar product operation: For $m(x)=\sum_{i=0}^{n}m_{i}x^{i}\in M[x;\sigma]$ and $f(x)=\sum_{j=0}^{m}a_{j}x^{j}\in R[x;\sigma]$ $m(x)f(x)=\sum_{k=0}^{n+m}\left(\sum_{k=i+j}m_{i}\sigma^{i}(a_{j})\right)x^{k}$ $None$ Similarly, $M[[x;\sigma]]$ is a module over $R[[x;\sigma]]$. The modules $M[x;\sigma]$ and $M[[x;\sigma]]$ are called the skew polynomial extension and the skew power series extension of $M$, respectively. If $\sigma\in Aut(R)$, then with a scalar product similar to $(*)$ , $M[x,x^{-1};\sigma]$ (respectively, $M[[x,x^{-1};\sigma]]$) becomes a module over $R[x,x^{-1};\sigma]$ (respectively, $R[[x,x^{-1};\sigma]]$). The modules $M[x,x^{-1};\sigma]$ and $M[[x,x^{-1};\sigma]]$ are called the skew Laurent polynomial extension and the skew Laurent power series extension of $M$, respectively. In [13], a module $M_{R}$ is called $\sigma$-skew Armendariz, if $m(x)f(x)=0$ where $m(x)=\sum_{i=0}^{n}m_{i}x^{i}\in M[x;\sigma]$ and $f(x)=\sum_{j=0}^{m}a_{j}x^{j}\in R[x;\sigma]$ implies $m_{i}\sigma^{i}(a_{j})=0$ for all $i$ and $j$. According to Lee-Zhou [12], $M_{R}$ is called $\sigma$-Armendariz, if it is $\sigma$-compatible and $\sigma$-skew Armendariz. In this paper, we show that if $M_{R}$ is p.q.-Baer then so is $M[x;\sigma]_{R[x;\sigma]}$ whenever $M_{R}$ satisfies the condition $\mathcal{C}_{2}$, and the converse holds when $M_{R}$ satisfies the condition $\mathcal{C}_{1}$ (Proposition 2.3). Also, if $M_{R}$ satisfies $\mathcal{C}_{2}$ and $\sigma$-skew Armendariz, then $M_{R}$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ ($\sigma\in Aut(R)$) is a p.p.-module (Proposition 3.1). As a consequence, if $M_{R}$ is semicommutative and $\sigma$-compatible then: $M_{R}$ is a p.p.-module $\Leftrightarrow$ $M_{R}$ is a p.q.-Baer module $\Leftrightarrow$ $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module $\Leftrightarrow$ $M[x;\sigma]_{R[x;\sigma]}$ is a p.q.-Baer module (Theorem 3.6). Moreover, we obtain a generalization of some results in [3, 4, 6, 12]. ## 2 Skew polynomials over p.q.-Baer modules We start with the next definition. ###### Definition 2.1. Let $m\in M$ and $a\in R$. We say that $M_{R}$ satisfies the condition $\mathcal{C}_{1}$ $($respectively, $\mathcal{C}_{2}$$)$, if $ma=0$ implies $m\sigma(a)=0$ $($respectively, $m\sigma(a)=0$ implies $ma=0$$)$. Note that $M_{R}$ is $\sigma$-compatible if and only if it satisfies $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. Let $M_{R}$ be a module and $\sigma\in End(R)$. ###### Lemma 2.2. If $M_{R}$ satisfies $\mathcal{C}_{1}$ or $\mathcal{C}_{2}$, then $me=m\sigma(e)$ for any $m\in M$ and any $e^{2}=e\in R$. ###### Proof. Suppose $\mathcal{C}_{2}$, from $m\sigma(e)(1-\sigma(e))=0$, we have $0=m\sigma(e)(1-e)=m\sigma(e)-m\sigma(e)e$, so $m\sigma(e)e=m\sigma(e)$. From $m(1-\sigma(e))\sigma(e)=0$, we have $0=m(1-\sigma(e))e=me-m\sigma(e)e$, so $m\sigma(e)=m\sigma(e)e=me$. The same for $\mathcal{C}_{1}$. ∎ ###### Proposition 2.3. Let $M_{R}$ be a module and $\sigma\in End(R)$. $(1)$ If $M_{R}$ is a p.q.-Baer module then so is $M[x;\sigma]_{R[x;\sigma]}$, whenever $M_{R}$ satisfies the condition $\mathcal{C}_{2}$. $(2)$ If $M[x;\sigma]_{R[x;\sigma]}$ or $M[[x;\sigma]]_{R[[x;\sigma]]}$ is a p.q.-Baer module then so is $M_{R}$, whenever $M_{R}$ satisfies the condition $\mathcal{C}_{1}$. ###### Proof. $(1)$ Let $m(x)=m_{0}+m_{1}x+\cdots+m_{n}x^{n}\in M[x;\sigma]$. Then $r_{R}(m_{i}R)=e_{i}R$, for some idempotents $e_{i}\in R\;(0\leq i\leq n)$. Let $e=e_{0}e_{1}\cdots e_{n}$, then $eR=\cap_{i=0}^{n}r_{R}(m_{i}R)$. We show that $r_{R[x;\sigma]}(m(x)R[x;\sigma])=eR[x;\sigma]$. Let $\phi(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{p}x^{p}\in r_{R[x;\sigma]}(m(x)R[x;\sigma])$. Since $m(x)R\phi(x)=0$, we have $m(x)b\phi(x)=0$ for all $b\in R$. Then $m(x)b\phi(x)=\sum_{\ell=0}^{n+p}\left(\sum_{\ell=i+j}m_{i}\sigma^{i}(ba_{j})\right)x^{\ell}=0.$ * • $\ell=0$ implies $m_{0}ba_{0}=0$ then $a_{0}\in r_{R}(m_{0}R)=e_{0}R$. * • $\ell=1$ implies $m_{0}ba_{1}+m_{1}\sigma(ba_{0})=0$ $None$ Let $s\in R$ and take $b=se_{0}$, so $m_{0}se_{0}a_{1}+m_{1}\sigma(se_{0}a_{0})=0$, since $m_{0}se_{0}=0$ we have $m_{1}\sigma(se_{0}a_{0})=m_{1}\sigma(sa_{0})=0$, so $m_{1}sa_{0}=0$, thus $a_{0}\in r_{R}(m_{1}R)=e_{1}R$. In equation $(1)$, $m_{1}\sigma(ba_{0})=m_{1}\sigma(be_{1}a_{0})=m_{1}\sigma(b)e_{1}\sigma(a_{0})=0$, by Lemma 2.2. Then equation (1) gives $m_{0}ba_{1}=0$, so $a_{1}\in e_{0}R$. * • $\ell=2$ implies $m_{0}ba_{2}+m_{1}\sigma(ba_{1})+m_{2}\sigma^{2}(ba_{0})=0$ $None$ Let $s\in R$ and take $b=se_{0}e_{1}$, so $m_{0}se_{0}e_{1}a_{2}+m_{1}\sigma(s)e_{0}e_{1}\sigma(a_{1})+m_{2}\sigma^{2}(se_{0}e_{1}a_{0})=0$, but $m_{0}se_{0}e_{1}a_{2}=m_{1}\sigma(s)e_{0}e_{1}\sigma(a_{1})=0$ we have $m_{2}\sigma^{2}(se_{0}e_{1}a_{0})=0$, since $e_{0}e_{1}a_{0}=a_{0}$ we have $m_{2}\sigma^{2}(sa_{0})=0$ and so $m_{2}sa_{0}=0$ for all $s\in R$. Hence $a_{0}\in e_{2}R$ (thus, $a_{0}\in e_{0}e_{1}e_{2}R$). Equation $(2)$, becomes $m_{0}ba_{2}+m_{1}\sigma(ba_{1})+m_{2}\sigma^{2}(b)e_{0}e_{1}e_{2}\sigma^{2}(a_{0})=0$, which gives $m_{0}ba_{2}+m_{1}\sigma(ba_{1})=0$ $None$ Take $b=se_{0}$ in equation $(2^{\prime})$, we have $m_{0}se_{0}a_{2}+m_{1}\sigma(se_{0}a_{1})=0$, but $m_{0}se_{0}a_{2}=0$ so $m_{1}\sigma(se_{0}a_{1})=m_{1}\sigma(sa_{1})=0$ and thus $m_{1}sa_{1}=0$, hence $a_{1}\in e_{1}R$ (so, $a_{1}\in e_{0}e_{1}R$). Equation $(2^{\prime})$ gives $m_{0}ba_{2}=0$, so $a_{2}\in e_{0}R$. At this point, we have $a_{0}\in e_{0}e_{1}e_{2}R,\;a_{1}\in e_{1}e_{2}R$ and $a_{2}\in e_{0}R$. Continuing this procedure yields $a_{i}\in eR$ ($0\leq i\leq n$). Hence $\phi(x)\in eR[x;\sigma]$. Consequently, $r_{R[x;\sigma]}(m(x)R[x;\sigma])\subseteq eR[x;\sigma]$. Conversely, let $\varphi(x)=b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{p}x^{p}\in R[x;\sigma]$. Then $m(x)\varphi(x)e=\sum_{\ell=0}^{n+p}\left(\sum_{\ell=i+j}m_{i}\sigma^{i}(b_{j})\sigma^{\ell}(e)\right)x^{\ell}=\sum_{\ell=0}^{n+p}\left(\sum_{\ell=i+j}m_{i}\sigma^{i}(b_{j})e\right)x^{\ell}.$ Since $e\in\bigcap_{i=0}^{n}r_{R}(m_{i}R)$, then $m_{i}Re=0$ ($0\leq i\leq n$). Thus $m(x)\varphi(x)e=0$, hence $eR[x;\sigma]\subseteq r_{R[x;\sigma]}(m(x)R[x;\sigma])$. Thus $r_{R[x;\sigma]}(m(x)R[x;\sigma])=eR[x;\sigma]$, therefore $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer. $(2)$ Let $0\neq m\in M$. We have $r_{R[x;\sigma]}(mR[x;\sigma])=e{R[x;\sigma]}$ for some idempotent $e=\sum_{i=0}^{n}e_{i}x^{i}\in R[x;\sigma]$. We have $r_{R[x;\sigma]}(mR[x;\sigma])\cap R=e_{0}R$. On other hand, we show that $r_{R[x;\sigma]}(mR[x;\sigma])\cap R=r_{R}(mR)$. Let $a\in r_{R}(mR)$ then $mRa=0$, so $mR\sigma^{i}(a)=0$ for all $i\geq 1$. So $mR[x;\sigma]a=0$. Therefore $a\in r_{R[x;\sigma]}(mR[x;\sigma])\cap R$. Conversely, let $a\in r_{R[x;\sigma]}(mR[x;\sigma])\cap R$, then $mR[x;\sigma]a=0$, in particular $mRa=0$, so $a\in r_{R}(mR)$. Thus $a\in r_{R}(mR)=e_{0}R$, with $e_{0}^{2}=e_{0}\in R$. So $M_{R}$ is p.q.-Baer. The same method for $M[[x;\sigma]]$. ∎ ###### Corollary 2.4 ([3, Theorem 11]). $M_{R}$ is p.q.-Baer if and only if $M[x]_{R[x]}$ is p.q.-Baer. ###### Corollary 2.5 ([6, Theorem 3.1]). $R$ is right p.q.-Baer if and only if $R[x]$ is right p.q.-Baer. $M_{R}$ is called $\sigma$-reduced module by Lee-Zhou [12], if for any $m\in M$ and $a\in R$: $(1)$ $ma=0$ implies $mR\cap Ma=0$, $(2)$ $ma=0$ if and only if $m\sigma(a)=0$. ###### Corollary 2.6 ([3, Theorem 7(1)]). Let $M_{R}$ a $\sigma$-compatible module. Then the following hold: $(1)$ If $M[x;\sigma]_{R[x;\sigma]}$ is a p.q.-Baer module then so is $M_{R}$. The converse holds if in addition $M_{R}$ is $\sigma$-reduced. $(2)$ If $M[[x;\sigma]]_{R[[x;\sigma]]}$ is a p.q.-Baer module then so is $M_{R}$. ###### Corollary 2.7 ([4, Corollary 2.6]). Let $M_{R}$ be a $\sigma$-compatible module. Then $M_{R}$ is p.q.-Baer if and only if $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer. ## 3 Skew polynomials over p.p.-modules Let $M_{R}$ be an $\sigma$-Armendariz module, if $me=0$ where $e^{2}=e\in R$ and $m\in M$, then $mfe=0$ for any $f^{2}=f\in R$ (by [12, Lemma 2.10]). This result still true if we replace the condition “$M_{R}$ is $\sigma$-Armendariz” by “$M_{R}$ is $\sigma$-skew Armendariz satisfying $\mathcal{C}_{2}$”. ###### Proposition 3.1. Let $M_{R}$ be a $\sigma$-skew Armendariz module which satisfies the condition $\mathcal{C}_{2}$. The following statements hold: $(1)$ $M_{R}$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module, $(2)$ Let $\sigma\in Aut(R)$, then $M_{R}$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ is a p.p.-module. ###### Proof. $(1)$$(\Leftarrow)$ Is clear by [12, Theorem 2.11]. $(\Rightarrow)$ Let $m(x)=m_{0}+m_{1}x+\cdots+m_{n}x^{n}\in M[x;\sigma]$, then $r_{R}(m_{i})=e_{i}R$, for some idempotents $e_{i}\in R\;(0\leq i\leq n)$. Let $e=e_{0}e_{1}\cdots e_{n}$, then $m_{i}e=0$ for all $0\leq i\leq n$ ([12, Lemma 2.10]) and by Lemma 2.2, we have $m_{i}\sigma^{j}(e)=0$ for all $0\leq i\leq n$ and $j\geq 0$. Therefore $e\in r_{R[x;\sigma]}(m(x))$, so $eR[x;\sigma]\subseteq r_{R[x;\sigma]}(m(x))$. Conversely, let $\phi(x)=a_{0}+a_{1}x+\cdots+a_{p}x^{p}\in r_{R[x;\sigma]}(m(x))$, then $m(x)\phi(x)=0$. Since $M_{R}$ is $\sigma$-skew Armendariz, we have $m_{i}\sigma^{i}(a_{j})=0$ for all $i,j$ and with the condition $\mathcal{C}_{2}$ we have $m_{i}a_{j}=0$ for all $i,j$. So $a_{j}\in r_{R}(m_{i})=e_{i}R$ for all $i,j$. Thus $a_{j}\in\cap_{i=0}^{n}r_{R}(m_{i})=eR$ for each $j$. Then $\phi(x)\in e{R[x;\sigma]}$, therefore $r_{R[x;\sigma]}(m(x))=eR[x;\sigma]$. With the same method, we can prove $(2)$. ∎ ###### Corollary 3.2 ([12, Theorem 11(1a,2a)]). If $M_{R}$ is $\sigma$-Armendariz. Then: $(1)$ $M_{R}$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module, $(2)$ Let $\sigma\in Aut(R)$, then $M_{R}$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ is a p.p.-module. If $M_{R}$ is a semicommutative module such that, $m\sigma(a)a=0$ implies $m\sigma(a)=0$ for any $m\in M$ and $a\in R$. Then $M_{R}$ is $\sigma$-semicommutative and hence it satisfies the condition $\mathcal{C}_{1}$. To see this, suppose that $ma=0$ then $mRa=0$, in particular $mr\sigma(a)a=0$ for all $r\in R$. By the above condition, $mr\sigma(a)=0$ for all $r\in R$. Thus $M_{R}$ is $\sigma$-semicommutative. ###### Lemma 3.3. If $M_{R}$ is a semicommutative module such that $m\sigma(a)a=0$ implies $m\sigma(a)=0$ for any $m\in M$ and $a\in R$. Then $M_{R}$ is $\sigma$-skew Armendariz. ###### Proof. Let $m(x)=m_{0}+m_{1}x+\cdots+m_{n}x^{n}\in M[x;\sigma]$ and $f(x)=a_{0}+a_{1}x+\cdots+a_{p}x^{p}\in R[x;\sigma]$. From $m(x)f(x)=0$, we have $\sum_{i+j=k}m_{i}\sigma^{i}(a_{j})=0$, for $0\leq k\leq n+p$. So, $m_{0}a_{0}=0$. Assume that $s\geq 0$ and $m_{i}\sigma^{i}(a_{j})=0$ for all $i,j$ with $i+j\leq s$. Note that for $s+1$, we have $m_{0}a_{s+1}+m_{1}\sigma(a_{s})+\cdots+m_{s}\sigma^{s}(a_{1})+a_{s+1}\sigma^{s+1}(a_{0})=0$ $None$ Multiplying $(1)$ by $\sigma^{s}(a_{0})$ from the right hand, we obtain $m_{0}a_{s+1}\sigma^{s}(a_{0})+m_{1}\sigma(a_{s})\sigma^{s}(a_{0})+\cdots+m_{s}\sigma^{s}(a_{1})\sigma^{s}(a_{0})+a_{s+1}\sigma^{s+1}(a_{0})\sigma^{s}(a_{0})=0,$ we have $m_{0}a_{0}=0$, then $m_{0}\sigma^{s}(a_{0})=0$ because $M_{R}$ is $\sigma$-semicommutative, and so $m_{0}a_{s+1}\sigma^{s}(a_{0})=0$. Also, $m_{1}\sigma(a_{0})=0$ then $m_{1}\sigma^{s}(a_{0})=0$, thus $m_{1}\sigma(a_{s})\sigma^{s}(a_{0})=0$. Continuing this process until the step $s$, $m_{s}\sigma^{s}(a_{0})=0$ then $m_{s}\sigma^{s}(a_{1})$ $\sigma^{s}(a_{0})=0$. Therefore $m_{s+1}\sigma^{s+1}(a_{0})\sigma^{s}(a_{0})=0$. But $m_{s+1}\sigma^{s+1}(a_{0})\sigma^{s}(a_{0})=m_{s+1}\sigma[\sigma^{s}(a_{0})]\sigma^{s}(a_{0})=0.$ So $m_{s+1}\sigma^{s+1}(a_{0})=0$ . Therefore, equation $(1)$, becomes $m_{0}a_{s+1}+m_{1}\sigma(a_{s})+\cdots+m_{s}\sigma^{s}(a_{1})=0$ $None$ Multiplying $(2)$, by $\sigma^{s-1}(a_{1})$ from the right hand to obtain $m_{s}\sigma^{s}(a_{1})=0$. Continuing this procedure yields $m_{0}a_{s+1}=m_{1}\sigma(a_{s})=\cdots=m_{s}\sigma^{s}(a_{1})=a_{s+1}\sigma^{s+1}(a_{0})=0.$ A simple induction shows that $m_{i}\sigma^{i}(a_{j})=0$, for all $i,j$. ∎ ###### Proposition 3.4. Let $M_{R}$ be a module such that $m\sigma(a)a=0$ implies $m\sigma(a)=0$ for any $m\in M$ and $a\in R$. If $M_{R}$ is semicommutative then $M[x;\sigma]_{R[x;\sigma]}$ and $M[[x;\sigma]]_{R[[x;\sigma]]}$ are semicommutative. ###### Proof. Let $m(x)=\sum_{i=0}^{n}m_{i}x^{i}\in M[x;\sigma]$, $f(x)=\sum_{j=0}^{q}a_{j}x^{j}\in R[x;\sigma]$ and $\phi(x)=\sum_{k=0}^{p}b_{k}x^{k}\in R[x;\sigma]$. Suppose that $m(x)f(x)=0$. The coefficients of $m(x)\phi(x)f(x)$ are of the form $\sum_{u+v=w}m_{u}\sigma^{u}\left(\sum_{i+j=v}b_{i}\sigma^{i}(a_{j})\right)=\sum_{u+v=w}\left(\sum_{i+j=v}m_{u}\sigma^{u}(b_{i})\sigma^{u+i}(a_{j})\right).$ By Lemma 3.3, $m_{u}\sigma^{u}(a_{j})=0$, for all $u,j$ and by $\mathcal{C}_{1}$, $m_{u}\sigma^{u+i}(a_{j})=0$, for all $i,j,u$. Since $M_{R}$ is semicommutative then $m_{u}\sigma^{u}(b_{i})\sigma^{u+i}(a_{j})=0$, therefore $\sum_{u+v=w}m_{u}\sigma^{u}\left(\sum_{i+j=v}b_{i}\sigma^{i}(a_{j})\right)=0.$ So $m(x)\phi(x)f(x)=0$, then $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative. The same for $M[[x;\sigma]]_{R[[x;\sigma]]}$. ∎ According to Baser and Harmanci [3], a module $M_{R}$ is reduced if for any $m\in M$ and $a\in R$, $ma^{2}=0$ implies $mR\cap Ma=0$. By [2, Lemma 2.11], if $M_{R}$ is semicommutative p.p. or semicommutative p.q.-Baer then it’s reduced. ###### Corollary 3.5. Let $M_{R}$ be a semicommutative module satisfying the condition $\mathcal{C}_{1}$, if $M_{R}$ is p.q.-Baer or p.p. then $M[x;\sigma]_{R[x;\sigma]}$ and $M[[x;\sigma]]_{R[[x;\sigma]]}$ are semicommutative. ###### Proof. Let $a\in R$ and $m\in M$ such that $m\sigma(a)a=0$, then $m(\sigma(a))^{2}=0$ (by $\mathcal{C}_{1}$), since $M_{R}$ is reduced we have $m\sigma(a)=0$. By Proposition 3.4, $M[x;\sigma]_{R[x;\sigma]}$ and $M[[x;\sigma]]_{R[[x;\sigma]]}$ are semicommutative. ∎ ###### Theorem 3.6. If $M_{R}$ is semicommutative and $\sigma$-compatible. Then the following are equivalent: $(1)$ $M_{R}$ is p.p. $(2)$ $M_{R}$ is p.q.-Baer, $(3)$ $M[x;\sigma]_{R[x;\sigma]}$ is p.p., $(4)$ $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer, ###### Proof. $(1)\Leftrightarrow(2)$ By [2, Proposition 2.7]. $(2)\Leftrightarrow(4)$ By Proposition 2.3. $(3)\Rightarrow(4)$ Since $M_{R}$ is a p.p.-module, then Corollary 3.5 implies that $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative. Therefore $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer by [2, Proposition 2.7]. $(4)\Rightarrow(3)$ By Proposition 2.3, $M_{R}$ is p.q.-Baer, since $M_{R}$ is semicommutative then $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative, and so $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module. ∎ ###### Corollary 3.7. Let $M_{R}$ be a semicommutative module. Then the following are equivalent: $(1)$ $M_{R}$ is p.p. $(2)$ $M_{R}$ is p.q.-Baer, $(3)$ $M[x]_{R[x]}$ is p.p., $(4)$ $M[x]_{R[x]}$ is p.q.-Baer, ###### Corollary 3.8 ([5, Theorem 2.8]). If $M_{R}$ is a reduced module. Then the following are equivalent: $(1)$ $M_{R}$ is p.p. $(2)$ $M_{R}$ is p.q.-Baer, $(3)$ $M[x]_{R[x]}$ is p.p., $(4)$ $M[x]_{R[x]}$ is p.q.-Baer, ###### Proof. Every reduced module is semicommutative by [12, Lemma 1.2]. ∎ ## References * [1] S. Annin, Associated primes over skew polynomials rings, Comm. Algebra, 30 (2002), 2511-2528 * [2] M. Baser and N. Agayev, On reduced and semicommutative modules, Turk. J. Math., 30 (2006), 285-291. * [3] M. Baser and A. Harmanci, reduced and p.q.-Baer modules, Taiwanese J. Math., 2 (1) (2007), 267-275. * [4] M. Baser and A. Harmanci, On quasi-Baer and p.q.-Baer modules, Kyungpook Math. Journal, 49 (2009), 255-263. * [5] M. Baser and M.T. Kosan, On quasi-Armendariz modules, Taiwanese J. Math., 12 (3) (2008), 573-582. * [6] G.F. Birkenmeier, J.Y. Kim and J.K. Park, On polynomial extensions of principally quasi-Baer rings, Kyungpook Math. journal, 40 (2) (2000), 247-253. * [7] G.F. Birkenmeier, J.Y. Kim and J.K. Park, Principally quasi-Baer rings, Comm. Algebra, 29 (2) (2001), 639-660. * [8] A.M. Buhphang and M.B. Rege, semicommutative modules and Armendariz modules, Arab J. Math. Sciences, 8 (2002), 53-65. * [9] W.E. Clark, Twisted matrix units semigroup algebras, Duke Math. Soc., 35 (1967), 417-424. * [10] I. Kaplansky, Rings of operators, Math. Lecture Notes series, Benjamin, New York, 1965. * [11] T.K. Kwak, Extensions of extended symmetric rings, Bull. Korean Math. Soc., 44 (2007), 777-788. * [12] T. K. Lee and Y. Lee, Reduced Modules, Rings, modules, algebras and abelian groups, 365-377, Lecture Notes in Pure and App. Math., 236, Dekker, New york, (2004). * [13] C.P Zhang and J.L. Chen, $\sigma$-skew Armendariz modules and $\sigma$-semicommutative modules, Taiwanese J. Math., 12 (2) (2008), 473-486.
arxiv-papers
2011-01-27T23:22:31
2024-09-04T02:49:16.688833
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Mohamed Louzari", "submitter": "Louzari Mohamed", "url": "https://arxiv.org/abs/1101.5415" }
1101.5463
# Walking on a Graph with a Magnifying Glass Stratified Sampling via Weighted Random Walks Maciej Kurant Minas Gjoka Carter T. Butts Athina Markopoulou University of California, Irvine {mkurant, mgjoka, buttsc, athina}@uci.edu (2011) ###### Abstract Our objective is to sample the node set of a large unknown graph via crawling, to accurately estimate a given metric of interest. We design a random walk on an appropriately defined weighted graph that achieves high efficiency by preferentially crawling those nodes and edges that convey greater information regarding the target metric. Our approach begins by employing the theory of stratification to find optimal node weights, for a given estimation problem, under an independence sampler. While optimal under independence sampling, these weights may be impractical under graph crawling due to constraints arising from the structure of the graph. Therefore, the edge weights for our random walk should be chosen so as to lead to an equilibrium distribution that strikes a balance between approximating the optimal weights under an independence sampler and achieving fast convergence. We propose a heuristic approach (stratified weighted random walk, or S-WRW) that achieves this goal, while using only limited information about the graph structure and the node properties. We evaluate our technique in simulation, and experimentally, by collecting a sample of Facebook college users. We show that S-WRW requires 13-15 times fewer samples than the simple re-weighted random walk (RW) to achieve the same estimation accuracy for a range of metrics. ††conference: SIGMETRICS’11, June 7–11, 2011, San Jose, California, USA.00footnotetext: ​​​* This is an extended version of a paper with the same title presented at _SIGMETRICS’11_. This work was supported by SNF grant PBELP2-130871, Switzerland, and by the NSF CDI Award 1028394, USA. ## 1 Introduction Figure 1: Illustrative example. Our goal is to compare the blue and black subpopulations (e.g., with respect to their median income) in population (a). Optimal independence sampler, WIS (b), over-samples the black nodes, under- samples the blue nodes, and completely skips the white nodes. A naive crawling approach, RW (c), samples many irrelevant white nodes. WRW that enforces WIS- optimal probabilities may result in poor or no convergence (d). S-WRW (e) strikes a balance between the optimality of WIS and fast convergence. Many types of online networks, such as online social networks (OSNs), Peer-to- Peer (P2P) networks, or the World Wide Web (WWW), are measured and studied today via sampling techniques. This is due to several reasons. First, such graphs are typically too large to measure in their entirety, and it is desirable to be able to study them based on a small but representative sample. Second, the information pertaining to these networks is often hard to obtain. For example, OSN service providers have access to all information in their user base, but rarely make this information publicly available. There are many ways a graph can be sampled, e.g., by sampling nodes, edges, paths, or other substructures [27, 23]. Depending on our measurement goal, the elements with different properties may have different importance and should be sampled with a different probability. For example, Fig. 1(a) depicts the world’s population, with residents of China (1.3B people) represented by blue nodes, of the Vatican (800 people) by black nodes, and all other nationalities represented by white nodes. Assume that we want to compare the median income in China and Vatican. Taking a uniform sample of size 100 from the entire world’s population is ineffective, because most of the samples will come from countries other than China and Vatican. Even restricting our sample to the union of China and Vatican will not help much, as our sample is unlikely to include any Vatican resident. In contrast, uniformly sampling 50 Chinese and 50 Vaticanese residents would be much more accurate with the same sampling budget. This type of problem has been widely studied in the statistical and survey sampling literature. A commonly used approach is _stratified sampling_ [34, 12, 28], where nodes (e.g., people) are partitioned into a set of non- overlapping _categories_ (or strata). The objective is then to decide how many independent draws to take from each category, so as to minimize the uncertainty of the resulting measurement. This effect can be achieved in expectation by a weighted independence sampler (WIS) with appropriately chosen sampling probabilities $\pi^{\scriptscriptstyle\textrm{WIS}}$. In our example, WIS samples Vatican residents with much higher probabilities than Chinese ones, and avoids completely the rest of the world, as illustrated in Fig. 1(b). However, WIS, as every independence sampler, requires a sampling frame, i.e., a list of all elements we can sample from (e.g., a list of all Facebook users). This information is typically not available in today’s online networks. A feasible alternative is _crawling_ (also known as exploration or link-trace sampling). It is a graph sampling technique in which we can see the neighbors of already sampled users and make a decision on which users to visit next. In this paper, we study how to perform stratified sampling through graph crawling. We illustrate the key idea and some of the challenges in Fig. 1. Fig. 1(c) depicts a social network that connects the world’s population. A simple random walk (RW) visits every node with frequency proportional to its degree, which is reflected by the node size. In this particular example, for a simplicity of illustration, all nodes have the same degree equal to 3. As a result, RW is equivalent to the uniform sample of the world’s population, and faces exactly the same problems of wasting resources, by sampling all nodes with the same probability. We address these problems by appropriately setting the edge weights and then performing a random walk on the weighted graph, which we refer to as _weighted random walk_ (WRW). One goal in setting the weights is to mimic the WIS- optimal sampling probabilities $\pi^{\scriptscriptstyle\textrm{WIS}}$ shown in Fig. 1(b). However, such a WRW might perform poorly due to potentially slow mixing. In our example, it will not even converge because the underlying weighted graph is disconnected, as shown in Fig. 1(d). Therefore, the edge weights under WRW (which determine the equilibrium distribution $\pi^{\scriptscriptstyle\textrm{WRW}}$) should be chosen in a way that strikes a balance between the optimality of $\pi^{\scriptscriptstyle\textrm{WIS}}$ and fast convergence. We propose Stratified Weighted Random Walk (S-WRW), a practical heuristic that effectively strikes such a balance. We refer to our approach as “walking on the graph with a magnifying glass”, because S-WRW over-samples more relevant parts of the graph and under-samples less relevant ones. In our example, S-WRW results in the graph presented in Fig. 1(e). The only information required by S-WRW are the categories of neighbors of every visited node, which is typically available in crawlable online networks, such as Facebook. S-WRW uses two natural and easy-to-interpret parameters, namely: (i) $\tilde{f}_{\ominus}$, which controls the fraction of samples from irrelevant categories and (ii) $\gamma$, which is the maximal resolution of our magnifying glass, with respect to the largest relevant category. The main contributions of this paper are the following. * • We propose to improve the efficiency of crawling-based graph sampling methods, by performing a stratified weighted random walk that takes into account not only the graph structure but also the node properties that are relevant to the measurement goal. * • We design and evaluate S-WRW, a practical heuristic that sets the edge weights and operates with limited information. * • As a case study, we apply S-WRW to sample Facebook and estimate the sizes of colleges. We show that S-WRW requires 13-15 times fewer samples than a simple random walk for the same estimation accuracy. The outline of the rest of the paper is as follows. Section 2 summarizes the most popular graph sampling techniques, including sampling by exploration. Section 3 presents classical stratified sampling. Section 4 combines stratified sampling with graph exploration, presenting a unified WRW approach that takes into account both network structure and node properties; various trade-offs and practical issues are discussed and an efficient heuristic (S-WRW) is proposed based on the insights. Section 5 presents simulation results. Section 6 presents an implementation of S-WRW for the problem of estimating the college friendship graph on Facebook. Section 7 presents related work. Section 8 concludes the paper. ## 2 Sampling techniques ### 2.1 Notation We consider an undirected, static,111Sampling dynamic graphs is currently an active research area [40, 35, 42], but out of the scope of this paper. graph $G=(V,E)$, with $N\\!=\\!|V|$ nodes and $|E|$ edges. For a node $v\in V$, denote by $\deg(v)$ its degree, and by $\mathcal{N}(v)\subset V$ the list of neighbors of $v$. A graph $G$ can be weighted. We denote by $\textrm{w}(u,v)$ the weight of edge $\\{u,v\\}\in E$, and by $\textrm{w}(u)=\sum_{v\in\mathcal{N}(u)}\textrm{w}(u,v)$ (1) the weight of node $u\in V$. For any set of nodes $A\subseteq V$, we define its volume $\textrm{vol}(A)$ and weight $\textrm{w}(A)$, respectively, as $\textrm{vol}(A)=\sum_{v\in A}\deg(v)\quad\textrm{ and }\quad\textrm{w}(A)=\sum_{v\in A}\textrm{w}(v).$ (2) We will often use $f_{A}=\frac{|A|}{|V|}\quad\textrm{ and }\quad f^{\scriptscriptstyle\textrm{vol}}_{A}=\frac{\textrm{vol}(A)}{\textrm{vol}(V)}$ (3) to denote the relative size of $A$ in terms of the number of nodes and the volumes, respectively. Sampling. We collect a sample $S\subseteq V$ of $n\\!=\\!|S|$ nodes. $S$ may contain multiple copies of the same node, i.e., the sampling is with replacement. In this section, we briefly review the techniques for sampling nodes from graph $G$. We also present the weighted random walk (WRW) which is the basic building block for our approach. ### 2.2 Independence Sampling Uniform Independence Sampling (UIS) samples the nodes directly from the set $V$, with replacements, uniformly and independently at random, i.e., with probability $\pi^{\scriptscriptstyle\textrm{UIS}}(v)\ =\ \frac{1}{N}\qquad\textrm{ for every }v\in V.$ (4) Weighted Independence Sampling (WIS) is a weighted version of UIS. WIS samples the nodes directly from the set $V$, with replacements, independently at random, but with probabilities proportional to node weights $\textrm{w}(v)$: $\pi^{\scriptscriptstyle\textrm{WIS}}(v)\ =\ \frac{\textrm{w}(v)}{\sum_{u\in V}\textrm{w}(u)}.$ (5) In general, UIS and WIS are not possible in online networks because of the lack of sampling frame. For example, the list of all user IDs may not be publicly available, or the user ID space may be too sparsely allocated. Nevertheless, we present them as baseline for comparison with the random walks. ### 2.3 Sampling via Crawling In contrast to independence sampling, the crawling techniques are possible in many online networks, and are therefore the main focus of this paper. Simple Random Walk (RW) [29] selects the next-hop node $v$ uniformly at random among the neighbors of the current node $u$. In a connected and aperiodic graph, the probability of being at the particular node $v$ converges to the stationary distribution $\pi^{\scriptscriptstyle\textrm{RW}}(v)\ =\ \frac{\deg(v)}{2\cdot|E|}.$ (6) Metropolis-Hastings Random Walk (MHRW) is an application of the Metropolis- Hastings algorithm [30] that modifies the transition probabilities to converge to a desired stationary distribution. For example, we can achieve the uniform stationary distribution $\pi^{\scriptscriptstyle\textrm{MHRW}}(v)\ =\ \frac{1}{N}$ (7) by randomly selecting a neighbor $v$ of the current node $u$ and moving there with probability $\min(1,\frac{\deg(u)}{\deg(v)})$. However, it was shown in [35, 17] that RW (after re-weighting, as in Section 2.4) outperforms MHRW for most applications. We therefore restrict our attention to comparing against RW. Weighted Random Walk (WRW) is RW on a weighted graph [4]. At node $u$, WRW chooses the edge $\\{u,v\\}$ to follow with probability $P_{u,v}$ proportional to the weight $\textrm{w}(u,v)\geq 0$ of this edge, i.e., $P_{u,v}=\frac{\textrm{w}(u,v)}{\sum_{v^{\prime}\in\mathcal{N}(u)}\textrm{w}(u,v^{\prime})}.$ (8) The stationary distribution of WRW is: $\pi^{\scriptscriptstyle\textrm{WRW}}(v)\ =\ \frac{\textrm{w}(v)}{\sum_{u\in V}\textrm{w}(u)}.$ (9) WRW is the basic building block of our design. In the next sections, we show how to choose weights for a specific estimation problem. Graph Traversals (BFS, DFS, RDS, …) is a family of crawling techniques where no node is sampled more than once. Because traversals introduce a generally unknown bias (see Sec. 7), we do not consider them in this paper. ### 2.4 Correcting the bias RW, WRW, and WIS all produce biased (nonuniform) node samples. But their bias is known and therefore can be corrected by an appropriate re-weighting of the measured values. This can be done using the Hansen-Hurwitz estimator [19] as first shown in [39, 41] for random walks and also used in [35]. Let every node $v\in V$ carry a value $x(v)$. We can estimate the population total $x_{\scriptscriptstyle\textrm{tot}}=\sum_{v}x(v)$ by $\hat{x}_{\scriptscriptstyle\textrm{tot}}=\frac{1}{n}\sum_{v\in S}\frac{x(v)}{\pi(v)},$ (10) where $\pi(v)$ is the sampling probability of node $v$ in the stationary distribution. In practice, we usually know $\pi(v)$, and thus $\hat{x}_{\scriptscriptstyle\textrm{tot}}$, only up to a constant, i.e., we know the (non-normalized) weights $\textrm{w}(v)$. This problem disappears when we estimate the population mean $x_{\scriptscriptstyle\textrm{av}}=\sum_{v}x(v)/N$ as $\hat{x}_{\scriptscriptstyle\textrm{av}}\ =\ \frac{\sum_{v\in S}\frac{x(v)}{\pi(v)}}{\sum_{v\in S}\frac{1}{\pi(v)}}\ =\ \frac{\sum_{v\in S}\frac{x(v)}{\textrm{w}(v)}}{\sum_{v\in S}\frac{1}{\textrm{w}(v)}}.$ (11) For example, for $x(v)\\!=\\!1$ if $\deg(v)\\!=\\!k$ (and $x(v)\\!=\\!0$ otherwise), $\hat{x}_{\scriptscriptstyle\textrm{av}}(k)$ estimates the node degree distribution in $G$. All the results in this paper are presented _after this re-weighting_ step, whenever necessary. ## 3 Stratified Sampling In Sec. 1, we argued that in order to compare the median income of residents of China and Vatican we should take 50 random samples from each of these two countries, rather than taking 100 UIS samples from China and Vatican together (or, even worse, from the world’s population). This problem naturally arises in the field of survey sampling. The most common solution is _stratified sampling_ [34, 12, 28], where nodes $V$ are partitioned into a set $\mathcal{C}$ of non-overlapping node categories (or “strata”), with $\bigcup_{C\in\mathcal{C}}C=V$. Next, we select uniformly at random $n_{i}$ nodes from category $C_{i}$. We are free to choose the allocation $(n_{1},n_{2},\ldots,n_{|\mathcal{C}|})$, as long as we respect the total budget of samples $n\\!=\\!\sum_{i}n_{i}$. Under _proportional allocation_ [28] (or “prop’) we use $n_{i}\propto|C_{i}|$, i.e., $n_{i}^{\scriptscriptstyle\textrm{prop}}\ =\ |C_{i}|\cdot n/N.$ (12) Another possibility is to do an optimal allocation (or “opt”) that minimizes the variance $\mathbb{V}$ of our estimator for the specific problem of interest. For example, assume that every node $v\in V$ carries a value $x(v)$, and we may want to estimate the mean of $x$ in various scenarios, as discussed below. ### 3.1 Examples of Stratified Sampling Problems #### 3.1.1 Estimating the mean across the entire $V$ A classic application of stratification is to better estimate the population mean $\mu$, given several groups (strata) of different properties (e.g., variances). Given $n_{i}$ samples from category $C_{i}$, we can estimate the mean $\mu_{i}=\frac{1}{|C_{i}|}\sum_{v\in C_{i}}x(v)$ over category $C_{i}$ by $\hat{\mu}_{i}=\frac{1}{n_{i}}\sum_{v\in S\cap C_{i}}x(v)\qquad\textrm{ with }\qquad\mathbb{V}(\hat{\mu}_{i})=\frac{\sigma_{i}^{2}}{n_{i}},$ (13) where $\mathbb{V}(\hat{\mu}_{i})$ is the variance of this estimator and $\sigma_{i}^{2}$ is the variance of population $C_{i}$. We can estimate population mean $\mu$ by a weighted average over all $\hat{\mu}_{i}$s [28], i.e., $\hat{\mu}=\sum_{i}\frac{|C_{i}|}{N}\cdot\hat{\mu}_{i}\qquad\textrm{ with }\qquad\mathbb{V}(\hat{\mu})=\sum_{i}\frac{(|C_{i}|)^{2}\cdot\sigma_{i}^{2}}{N^{2}\cdot n_{i}}.$ Under proportional allocation (Eq.(12)), this boils down to $\mathbb{V}(\hat{\mu}^{\scriptscriptstyle\textrm{prop}})\ =\ \frac{1}{N\cdot n}\ \sum_{i}|C_{i}|\cdot\sigma_{i}^{2}$. However, we can apply Lagrange multipliers to find that $\mathbb{V}(\hat{\mu})$ is minimized when $n_{i}^{\scriptscriptstyle\textrm{opt}}=\frac{|C_{i}|\cdot\sigma_{i}}{\sum_{j}|C_{j}|\cdot\sigma_{j}}\cdot n.$ (14) This solution is sometimes called ‘Neyman allocation’ [34]. This gives us the variance under optimal allocation $\mathbb{V}(\hat{\mu}^{\scriptscriptstyle\textrm{opt}})\ =\ \frac{1}{N^{2}\cdot n}\ \left(\sum_{i}|C_{i}|\cdot\sigma_{i}\right)^{2}$. The variances $\mathbb{V}(\hat{\mu}^{\scriptscriptstyle\textrm{prop}})$ and $\mathbb{V}(\hat{\mu}^{\scriptscriptstyle\textrm{opt}})$ are measures of the performance of proportional and optimal allocation, respectively. In order to make their practical interpretation easier, we also show how these variances translate into sample lengths. We define as _gain_ $\alpha$ of ‘opt’ over ‘prop’ the number of times ‘prop’ must be longer than ‘opt’ in order to achieve the same variance $\textrm{gain }\ \alpha\ =\ \frac{n^{\scriptscriptstyle\textrm{prop}}}{n^{\scriptscriptstyle\textrm{opt}}},\ \textrm{ subject to }\ \mathbb{V}^{\scriptscriptstyle\textrm{prop}}\\!=\\!\mathbb{V}^{\scriptscriptstyle\textrm{opt}}.$ In that case, the gain is $\alpha\ \ =\ \ N\cdot\frac{\sum_{i}|C_{i}|\cdot\sigma_{i}^{2}}{\left(\sum_{i}|C_{i}|\cdot\sigma_{i}\right)^{2}}\qquad(\geq 1).$ (15) Notice that this gain does not depend on the sample budget $n$. The gain is one of the main metrics we will use in the evaluation sections to assess the efficiency of our technique compared to the random walk. #### 3.1.2 Highest precision for all categories If we are equally interested in each category, we might want the same (highest possible) precision of estimating $\mu_{i}$ for all categories $C_{i}$. In this case, the metric to minimize is $\mathbb{V}_{\max}\ =\ \max_{i}\left\\{\mathbb{V}(\hat{\mu}_{i})\right\\}\ =\max_{i}\left\\{\frac{\sigma_{i}^{2}}{n_{i}}\right\\}.$ Under proportional allocation, this translates to $\mathbb{V}_{\max}^{\scriptscriptstyle\textrm{prop}}\ =\frac{N}{n}\max_{i}\frac{\sigma_{i}^{2}}{|C_{i}|}$. But the optimal $n_{i}$, which makes $\mathbb{V}(\hat{\mu}_{i})$ equal for all $i$, is $n_{i}^{\scriptscriptstyle\textrm{opt}}=\frac{\sigma^{2}_{i}}{\sum_{j}\sigma^{2}_{j}}\cdot n.$ (16) Consequently, $\mathbb{V}_{\max}^{\scriptscriptstyle\textrm{opt}}\ =\ \frac{\sum_{i}\sigma_{i}^{2}}{n},$ which leads to gain $\alpha=\frac{\max_{i}\left\\{\frac{N}{|C_{i}|}\sigma_{i}^{2}\right\\}}{\sum_{i}\sigma_{i}^{2}}\quad(\geq 1).$ (17) #### 3.1.3 Smallest sum of variances across categories Even if we are interested in all categories, an alternative objective is to maximize the _average_ precision of category pair comparisons (see Sec. 5A.13 in [12]), which is equivalent to minimizing the sum $\mathbb{V}_{\Sigma}=\sum_{i}\mathbb{V}(\hat{\mu}_{i})=\sum_{i}\frac{\sigma_{i}^{2}}{n_{i}}.$ In this case, proportional allocation achieves $\mathbb{V}_{\Sigma}^{\scriptscriptstyle\textrm{prop}}=\frac{N}{n}\sum_{i}\frac{\sigma_{i}^{2}}{|C_{i}|}$. while, using Lagrange multipliers we get $n_{i}^{\scriptscriptstyle\textrm{opt}}=\frac{\sigma_{i}}{\sum_{j}\sigma_{j}}\cdot n\qquad\textrm{ and }\qquad\mathbb{V}_{\Sigma}^{\scriptscriptstyle\textrm{opt}}=\frac{\left(\sum_{i}\sigma_{i}\right)^{2}}{n},$ (18) which leads to gain $\alpha\ =\ \frac{\sum_{i}\frac{N}{|C_{i}|}\sigma_{i}^{2}}{\left(\sum_{i}\sigma_{i}\right)^{2}}\quad(\geq 1).$ (19) #### 3.1.4 Relative sizes of node categories Stratified sampling assumes that we know the sizes $|C_{i}|$ of node categories. In some applications, however, these sizes are unknown and among the values we need to estimate as well (e.g., by using UIS or WIS). We show in Appendix C (for $|\mathcal{C}|\\!=\\!2$) that the optimal sample allocation and the corresponding gain $\alpha$ of WIS over UIS are respectively $n_{i}^{\scriptscriptstyle\textrm{WIS}}=\frac{1}{|\mathcal{C}|}\cdot n\quad\textrm{ and }\quad\alpha\ =\ \frac{N^{2}}{4|C_{1}|\cdot|C_{2}|}.$ (20) #### 3.1.5 Irrelevant category $C_{\ominus}$ (aggregated) In many practical cases, we may want to measure some (but not all) node categories. E.g., in Fig. 1, we are interested in blue and black nodes, but not in white ones. Similarly, in our Facebook study in Section 6 we are only interested in self-declared college students, which accounts for only 3.5% of all users. We group all categories not covered by our measurement objective as a single _irrelevant category_ $C_{\ominus}\in\mathcal{C}$, and we set $n^{\scriptscriptstyle\textrm{opt}}_{\ominus}=0$. In contrast, $n_{\ominus}^{\scriptscriptstyle\textrm{prop}}\ =\ |C_{\ominus}|\cdot n/N$. As a result, under ‘opt’ we have $N/(N\\!-\\!|C_{\ominus}|)$ times more useful samples than under ‘prop’. Now, if we allocate optimally all these useful samples between the relevant categories $\mathcal{C}\setminus\\{C_{\ominus}\\}$, the gain $\alpha$ becomes $\alpha\ \ =\ \ \frac{N}{N-|C_{\ominus}|}\ \cdot\ \alpha(\mathcal{C}\setminus\\{C_{\ominus}\\}),$ (21) where $\alpha(\mathcal{C}\setminus\\{C_{\ominus}\\})$ is the gain (15), (17), (19) or (20), depending on the metric, calculated only within categories $\mathcal{C}\setminus\\{C_{\ominus}\\}$. In other words, gain $\alpha$ is now composed of two factors: (i) gain in avoiding irrelevant categories, and (ii) gain in optimal allocation of samples among the relevant categories. #### 3.1.6 Practical Guideline Let us look at the optimal weights in the above scenarios, when all $\sigma_{i}=\sigma$ are the same. This is a reasonable working assumption in many practical settings, since we typically do not have prior estimates of $\sigma_{i}$. With this simplification, Eq.(14) becomes $n_{i}^{\scriptscriptstyle\textrm{opt}}\ =\ \frac{|C_{i}|}{N}\cdot n\ =\ n_{i}^{\scriptscriptstyle\textrm{prop}}.$ In contrast, Eq.(16), Eq.(18) and Eq.(20) get simplified to $n_{i}^{\scriptscriptstyle\textrm{opt}}\ =\ \frac{1}{|\mathcal{C}|}\cdot n.$ In conclusion, if we are interested in comparing the node categories with respect to some properties (e.g., average node degree, category size), rather than estimating a property across the entire population, we should take an _equal number of samples from every relevant category_. ## 4 Edge weight setting under WRW In the previous section, we studied the optimal sample allocation under (independence) stratified sampling. However, independence node sampling is typically impossible in large online graphs, while crawling the graph is a natural, available exploration primitive. In this section, we show how to perform a weighted random walk (WRW) which approximates the stratified sampling of the previous section. We can formulate the general problem as follows: _Given a measurement objective, error metric and sampling budget $|S|\\!=\\!n$, set the edge weights in graph $G$ such that the WRW measurement error is minimized._ Although we are able to solve this problem analytically for some specific and fully known topologies, it is not obvious how to address it in general, especially under a limited knowledge of $G$. Instead, in this paper, we propose S-WRW, a heuristic to set the edge weights. S-WRW starts from a solution optimal under WIS, and takes into account practical issues that arise in graph exploration. Once the weights are set, we simply perform WRW as described in Section 2.3 and collect samples. ### 4.1 Preliminaries #### 4.1.1 Category-level granularity One can think of the problem in two levels of granularity: the original graph $G\\!=\\!(V,E)$ and the category graph $G^{C}\\!=\\!(\mathcal{C},E^{C})$. In $G^{C}$, nodes represent categories, and every undirected edge $\\{C_{1},C_{2}\\}\in E^{C}$ represents the corresponding non-empty set of edges $E_{C_{1},C_{2}}\subset E$ in the original graph $G$, i.e., $E_{C_{1},C_{2}}\ =\\{\\{u,v\\}\in E:\ u\in C_{1}\textrm{ and }v\in C_{2}\\}\neq\emptyset.$ In our approach, we move from the finer granularity of $G$ to the coarser granularity of $G^{C}$. This means that we are interested in collecting, say, $n_{i}$ samples from category $C_{i}$, but we do not control how these $n_{i}$ nodes are collected (i.e., with what individual sampling probabilities). The rationale for that simplification is twofold. From a theoretical point of view, categories are exactly the properties of interest in the estimation problems we consider. From a practical point of view, it is relatively easy to obtain or infer information about categories, as we show e.g., in Sec. 4.2.1. #### 4.1.2 Stratification in expectation Ideally, we would like to enforce strictly stratified sampling. However, when we use crawling instead of independence sampling, sampling exactly $n_{i}$ nodes from category $C_{i}$ (and no other nodes) is possible only by discarding observations. It is thus more natural to frame the problem in terms of the probability mass placed on each category in equilibrium. This can be achieved by making the weight $\textrm{w}(C_{i})$ of each category proportional to the desired number $n_{i}$ of samples, i.e., $\textrm{w}(C_{i})\ \propto\ n_{i}.$ (22) As a result, we draw $n_{i}$ samples from $C_{i}$ _in expectation_. #### 4.1.3 Main guideline As the main guideline, S-WRW tries to realize the category weights $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ that are optimal under WIS. There are many edge weight settings in $G$ that achieve $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. In our implementation, we observe that $\textrm{vol}(C_{i})$ counts the number of edges incident on nodes of $C_{i}$. Consequently, if for every category $C_{i}$ we set in $G$ the weights of all edges incident on nodes in $C_{i}$ to $\textrm{w}_{e}(C_{i})\ =\ \frac{\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})}{\textrm{vol}(C_{i})}.$ (23) then weight $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ are achieved.222There exist many other edge weight assignments that lead to $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. Eq.(23) has the advantage of distributing the weights evenly across all $\textrm{vol}(C_{i})$ edges. This simple observation is central to the S-WRW heuristic. In order to apply Eq.(23), we first have to calculate or estimate its terms $\textrm{vol}(C_{i})$ and $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$.333In fact, we need to know $\textrm{w}_{e}(C_{i})$ in Eq.(23) only _up to a constant factor_ , because these factors cancel out in the calculation of transition probabilities of WRW in Eq.(8). Consequently, the same applies to $\textrm{vol}(C_{i})$ and $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. Below, we show how to do it in Step 1 and 2, respectively. Next, in Steps 3-5, we show how to modify these terms to account for practical problems arising mainly from the underlying graph structure. Main guideline (to be modified) Set the edge weights in category $C_{i}$ to $\textrm{w}^{\textrm{WIS}}(C_{i})\,/\,\textrm{vol}(C_{i}).$ Step 1: Estimation of Category Volumes Estimate $\textrm{vol}(C_{i})$ with a pilot RW estimator $\hbox to0.0pt{\raisebox{-1.50694pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{i})$ as in Eq.(35). Step 2: Category Weights Optimal Under WIS For given measurement objective, calculate $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ as in Sec. 3. Step 3: Include Irrelevant Categories Modify $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. $\tilde{f}_{\ominus}$ \- desired fraction of irrelevant nodes. Step 4: Tiny and Unknown Categories Modify $\hbox to0.0pt{\raisebox{-1.50694pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{i})$. $\gamma$ \- maximal resolution. Step 5: Edge Conflict Resolution Set the weights of inter-category edges to Eq.(28). WRW sample Use transition probabilities proportional to edge weights (Sec. 2.3). Correct for the bias Apply formulas from Sec. 2.4. Final result Figure 2: Overview of our approach. ### 4.2 Our practical solution: S-WRW #### 4.2.1 Step 1: Estimation of Category Volumes In general, we have no prior information about $G$ or $G^{C}$. Fortunately, it is easy and inexpensive estimate the relative category volumes $f^{\scriptscriptstyle\textrm{vol}}_{i}$ which is the first piece of information we need in Eq.(23) (see footnote 3). Indeed, it is enough to run a relatively short pilot RW, and plug the collected sample $S$ in Eq.(35) derived in Appendix B, as follows $\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{i}\ =\ \ \frac{1}{n}\sum_{u\in S}\left(\frac{1}{\deg(u)}\sum_{v\in\mathcal{N}(u)}\\!\\!1_{\\{v\in C_{i}\\}}\right).$ #### 4.2.2 Step 2: Category Weights Optimal Under WIS In order to find the optimal WIS category weights $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ in Eq.(23), we first calculate $n_{i}^{\scriptscriptstyle\textrm{opt}}$ as shown, under various scenarios, in Sec. 3. Next, we plug the resulting $n_{i}^{\scriptscriptstyle\textrm{opt}}$ in Eq.(22), e.g., by setting $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})=n^{\scriptscriptstyle\textrm{opt}}_{i}$. #### 4.2.3 Step 3: Irrelevant Categories Problem: Potentially poor or no convergence. Consider the toy example in Fig. 3(a). We are interested in finding the relative sizes of red (dark) and green (light) categories. The white node in the middle is irrelevant for our measurement objective. Due to symmetry, we distinguish between two types of edges with weights $w_{1}$ and $w_{2}$. Under WIS, Eq.(20) gives us the optimal weights $w_{1}>0$ and $w_{2}=0$, i.e., WIS samples every non-white node with the same probability and never samples the white one. However, under WRW with these weights, relevant nodes get disconnected into two components and WRW does not converge. We observed a similar problem in Fig. 1. Guideline: Occasionally visit irrelevant nodes. We show in Appendix D that the optimal WRW weights in Fig. 3(a) are $w_{1}=0$ and $w_{2}>0$. In that case, half of the samples are due to visits in the white (irrelevant) node. In other words, WRW may benefit from allocating small weight $\textrm{w}(C_{\ominus})\\!>\\!0$ to category $C_{\ominus}$ that groups all (if any) categories irrelevant to our estimation. The intuition is that irrelevant nodes may not contribute to estimation but may be needed for connectivity or fast mixing. Implementation in S-WRW. In S-WRW, we achieve this goal by replacing $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ with $\tilde{\textrm{w}}^{\scriptscriptstyle\textrm{WIS}}(C_{i})=\left\\{\begin{array}[]{ll}\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})&\textrm{if }C_{i}\neq C_{\ominus}\\\ \tilde{f}_{\ominus}\cdot\sum_{C\neq C_{\ominus}}\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C)&\textrm{if }C_{i}=C_{\ominus}.\end{array}\right.$ (24) The parameter $0\leq\tilde{f}_{\ominus}\ll 1$ controls the desired fraction of visits in $C_{\ominus}$. Figure 3: Optimal edge weights: WIS vs WRW. The objective is to compare the sizes of red (dark) and green (light) categories. #### 4.2.4 Step 4: Tiny and Unknown Categories Problem: “black holes”. Every optical system has a fundamental magnification limit due to diffraction and our “graph magnifying glass” is no exception. Consider the toy graph in Fig. 3(b): it consists of a big clique $C_{\scriptscriptstyle\textrm{big}}$ of 20 red nodes with edge weights $w_{2}$, and a green category $C_{\scriptscriptstyle\textrm{tiny}}$ with two nodes only and edge weights $w_{1}$. In Sec. 3.1.4, we saw that WIS optimally estimates the relative sizes of red and green categories for $\textrm{w}(C_{\scriptscriptstyle\textrm{big}})\\!=\\!\textrm{w}(C_{\scriptscriptstyle\textrm{tiny}})$, i.e., for $w_{1}\\!=\\!190\,w_{2}$. However, for such large values of $w_{1}$, the two green nodes behave as a sink (or a “black hole”) for a WRW of finite length, thus increasing the variance of the category size estimation. Guideline: limit edge weights. In other words, although WIS suggests to over- sample small categories, WRW should “under-over-sample” very small categories to avoid black holes. For example, in Fig. 3(b) $w_{1}\simeq 60\,w_{2}\ (\ll 190w_{2})$ is optimal for WRW of length $n\\!=\\!50$ (simulation results). Implementation in S-WRW. In S-WRW, we achieve this goal by replacing $\textrm{vol}(C_{i})$ in Eq.(23) with $\displaystyle\hbox to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C)$ $\displaystyle=$ $\displaystyle\max\Big{\\{}\hbox to0.0pt{\raisebox{-2.15277pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C),\ \textrm{vol}_{min}\Big{\\}},\quad\textrm{ where }$ (25) $\displaystyle\textrm{vol}_{min}$ $\displaystyle=$ $\displaystyle\frac{1}{\gamma}\cdot\max_{C\neq C_{\ominus}}\\{\hbox to0.0pt{\raisebox{-2.15277pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C)\\}.$ (26) Moreover, this formulation takes care of every category $C$ that was not discovered by the pilot RW in Sec. 4.2.1, by setting $\hbox to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C)\\!=\\!\textrm{vol}_{min}$. #### 4.2.5 Step 5: Edge Conflict Resolution Problem: Conflicting desired edge weights. With the above modifications, our target edge weights defined in Eq.(23) can be rewritten as $\tilde{\textrm{w}}_{e}(C_{i})\ =\ \frac{\tilde{\textrm{w}}^{\scriptscriptstyle\textrm{WIS}}(C_{i})}{\hbox to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{i})}.$ (27) We can directly set the weight $\textrm{w}(u,v)\\!=\\!\tilde{\textrm{w}}_{e}(C(u))\\!=\\!\tilde{\textrm{w}}_{e}(C(v))$ for every intra-category edge $\\{u,v\\}$. However, for every inter-category edge, we usually have “conflicting” weights $\tilde{\textrm{w}}_{e}(C(u))\neq\tilde{\textrm{w}}_{e}(C(v))$ desired at the two ends of the edge. Guideline: prefer inter-category edges. There are several possible edge weight assignments that achieve the desired category node weights. High weights on intra-category edges and small weights on inter-category edges result in WRW staying in small categories $C_{\scriptscriptstyle\textrm{tiny}}$ for a long time. In order to improve the mixing time, we should do exactly the opposite, i.e., assign relatively high weights to inter-category edges (connecting relevant categories). As a result, WRW will enter $C_{\scriptscriptstyle\textrm{tiny}}$ more often, but will stay there for a short time. This intuition is motivated by Monte Carlo variance reduction techniques such as the use of _antithetic variates_ [15], which seek to induce negative correlation between consecutive draws so as to reduce the variance of the resulting estimator. Implementation in S-WRW. We choose to assign an edge weight $\tilde{\textrm{w}}_{e}$ that is in between these two values $\tilde{\textrm{w}}_{e}(C(u))$ and $\tilde{\textrm{w}}_{e}(C(v))$. We considered several candidate such assignments. We may take the arithmetic or geometric mean of the conflicting weights, which we denote by $\textrm{w}^{\scriptscriptstyle\textrm{ar}}(u,v)$ and $\textrm{w}^{\scriptscriptstyle\textrm{ge}}(u,v)$, respectively. We may also use the maximum of the two values, $\textrm{w}^{\scriptscriptstyle\textrm{max}}(u,v)$, which should improve mixing according to the discussion above. However, $\textrm{w}^{\scriptscriptstyle\textrm{max}}(u,v)$ alone would also add high weight to irrelevant nodes $C_{\ominus}$ (possibly far beyond $\tilde{f}_{\ominus}$). To avoid this undesired effect, we distinguish between the two cases by defining a hybrid solution: $\textrm{w}^{\scriptscriptstyle\textrm{hy}}(u,v)=\left\\{\begin{array}[]{ll}\textrm{w}^{\scriptscriptstyle\textrm{ge}}(u,v)&\textrm{if }C_{\ominus}\in\\{C(u),C(v)\\}\\\ \textrm{w}^{\scriptscriptstyle\textrm{max}}(u,v)&\textrm{otherwise.}\end{array}\right.$ (28) This hybrid edge assignment was the one we found to work best in practice - see Section 6. ### 4.3 Discussion #### 4.3.1 Information needed about the neighbors In the pilot RW (Sec. 4.2.1) as well as in the main WRW, we assume that by sampling a node $v$ we also learn the category (but not degree) of each of its neighbors $u\in\mathcal{N}(v)$. Fortunately, such information is often available in most online graphs at no additional cost, especially when scraping html pages (as we do). For example, when sampling colleges in Facebook (Sec. 6), we use the college membership information of all $v$’s neighbors, which, in Facebook, is available at $v$ together with the friends list. #### 4.3.2 Cost of pilot RW The pilot RW volume estimator described in Sec. 4.2.1 considers the categories not only of the sampled nodes, but also of their neighbors. As a result, it achieves high efficiency, as we show in simulations (Sec. 5.3.1) and Facebook measurements (Sec. 6.1). Given that, and high robustness of S-WRW to estimation errors (see Sec. 5.3.5), pilot RW should be only a small fraction of the later WRW (e.g., 6.5% in our Facebook measurements in Sec. 6). #### 4.3.3 Setting the parameters S-WRW sets the edge weights trying to achieve roughly $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ as the main goal. We slightly shape $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ to avoid black holes and improve mixing, which is controlled by two natural and easy- to-interpret parameters, $\tilde{f}_{\ominus}$ and $\gamma$. Irrelevant nodes visits $\tilde{f}_{\ominus}$. The parameter $0\leq\tilde{f}_{\ominus}\ll 1$ controls the desired fraction of visits in $C_{\ominus}$. When setting $\tilde{f}_{\ominus}$, we should exploit the information provided by the pilot crawl. If the relevant categories appear poorly interconnected and often separated by irrelevant nodes, we should set $\tilde{f}_{\ominus}$ relatively high. We have seen an extreme case in Fig. 3(a), with disconnected relevant categories and optimal $\tilde{f}_{\ominus}\\!=\\!0.5$. In contrast, when the relevant categories are strongly interconnected, we should use much smaller $\tilde{f}_{\ominus}$. However, because we can never be sure that the graph induced on relevant nodes is connected, we recommend always using $\tilde{f}_{\ominus}>0$. For example, when measuring Facebook in Sec. 6, we set $\tilde{f}_{\ominus}=1\%$. Maximal resolution $\gamma$. The parameter $\gamma\geq 1$ can be interpreted as the maximal resolution of our “graph magnifying glass”, with respect to the largest relevant category $C_{\scriptscriptstyle\textrm{big}}$. S-WRW will typically sample well all categories that are less than $\gamma$ times smaller than $C_{\scriptscriptstyle\textrm{big}}$; all categories smaller than that are relatively undersampled (see Sec. 6.2.4). In the extreme case, for $\gamma\rightarrow\infty$, S-WRW tries to cover every category, no matter how small, which may cause the “black hole” problem discussed in Sec. 4.2.4. In the other extreme, for $\gamma\\!=\\!1$ (and identical $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$ for all categories, including $C_{\ominus}$), S-WRW reduces to RW. We recommend always setting $1<\gamma<\infty$. Ideally, we know $|C_{\scriptscriptstyle\textrm{smallest}}|$ \- the smallest category size that is still relevant to us. In that case we should set $\gamma=|C_{\scriptscriptstyle\textrm{big}}|/|C_{\scriptscriptstyle\textrm{smallest}}|$.444Strictly speaking, $\gamma$ is related to volumes $\textrm{vol}(C_{i})$ rather than sizes $|C_{i}|$. They are equivalent when category volume is proportional to its size, which is often the case, and is the central assumption in the “scale-up method” [9]. For example, in Sec. 6 the categories are US colleges; we set $\gamma\\!=\\!1000$, because colleges with size smaller than 1/1000th of the largest one (i.e., with a few tens of students) seem irrelevant to our measurement. As another rule of thumb, we should try to set smaller $\gamma$ for relatively small sample sizes and in graphs with tight community structure (see Sec. 5.3.5). #### 4.3.4 Conservative approach Note that a reasonable setting of these parameters (i.e., $\tilde{f}_{\ominus}>0$ and $1<\gamma<\infty$, and any conflict resolution discussed in the paper), increases the weights of large categories (including $C_{\ominus}$) and decreases the weight of small categories, compared to $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{i})$. This makes S-WRW allocate category weights between the two extremes: RW and WIS. Consequently, S-WRW can be considered _conservative_ (with respect to WIS). #### 4.3.5 S-WRW is unbiased It is also important to note that because the collected WRW sample is eventually corrected with the actual sampling weights as described in Sec. 2.4, S-WRW estimation process is _unbiased_ , regardless of the choice of weights (so long as convergence is attained). In contrast, suboptimal weights (e.g., due to estimation error of $\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}$) can increase WRW mixing time, and/or the _variance_ of the resulting estimator. However, our simulations and empirical experiments on Facebook (see Sec. 5 and 6) show that S-WRW is very robust to suboptimal choice of weights. Figure 4: RW and S-WRW under two scenarios: Random (a-g) and Clustered (h-n). In (b,i), we show error of two volume estimators: naive Eq.(32) (dotted) and neighbor-based Eq.(35) (plain). Next, we show error of size estimator as a function of $n$ (c,j) and $w$ (d,g,k,n); in the latter, UIS and RW correspond to WIS and S-WRW for $w\\!=\\!1$. In (e,l), we show the empirical probability that S-WRW visits $C_{\scriptscriptstyle\textrm{tiny}}$ at least once. Finally, (f,m) is gain $\alpha$ of S-WRW over RW under the optimal choice of $w$ (plain), and for fixed $\gamma\\!=\\!w\\!=\\!5$ (dashed). ## 5 Simulation results The gain of our approach compared to RW comes from two main factors. First, S-WRW avoids, to a large extent or completely, the nodes in $C_{\ominus}$ that are irrelevant to our measurement. This fact alone can bring an arbitrarily large improvement ($\frac{N}{N-|C_{\ominus}|}$ under WIS), especially when $C_{\ominus}$ is large compared to $N$. We demonstrate this in the Facebook measurements in Section 6. Second, we can better allocate samples among the relevant categories. This factor is observable in our Facebook measurements as well, but it is more difficult to evaluate due to the lack of ground-truth therein. In this section, we evaluate the optimal allocation gain in a controlled simulation and we demonstrate some key insights. ### 5.1 Setup We consider a graph $G$ with 101K nodes and 505.5K edges organized in two densely (and randomly) connected communities555The term “community” refers to cluster and is defined purely based on topology. The term “category” is a property of a node and is independent of topology. as shown in Fig. 4(h). The nodes in $G$ are partitioned into two node categories: $C_{\scriptscriptstyle\textrm{tiny}}$ with 1K nodes (dark red), and $C_{\scriptscriptstyle\textrm{big}}$ with 100K nodes (light yellow). We consider two extreme scenarios of such a partition. The ‘random’ scenario is purely random, as shown in Fig. 4(a). In contrast, under ‘clustered’, categories $C_{\scriptscriptstyle\textrm{tiny}}$ and $C_{\scriptscriptstyle\textrm{big}}$ coincide with the existing communities in $G$, as shown in Fig. 4(h). It is arguably the worst case scenario for graph sampling by exploration. We fix the edge weights of all internal edges in $C_{\scriptscriptstyle\textrm{big}}$ to 1. All the remaining edges, i.e., all edges incident on nodes in category $C_{\scriptscriptstyle\textrm{tiny}}$, have weight $w$ each, where $w\geq 1$ is a parameter. Note that this is equivalent to setting $\tilde{\textrm{w}}_{e}(C_{\scriptscriptstyle\textrm{big}})\\!=\\!1$, $\tilde{\textrm{w}}_{e}(C_{\scriptscriptstyle\textrm{tiny}})\\!=\\!w$, and ‘max’ or ‘hybrid’ conflict resolution. ### 5.2 Measurement objective and error metric We are mainly interested in measuring the relative sizes $f_{\scriptscriptstyle\textrm{tiny}}$ and $f_{\scriptscriptstyle\textrm{big}}$ of categories $C_{\scriptscriptstyle\textrm{tiny}}$ and $C_{\scriptscriptstyle\textrm{big}}$, respectively. We use Normalized Root Mean Square Error (NRMSE) to assess the estimation error, defined as [37]: ${\small\textrm{NRMSE}}(\widehat{x})=\frac{\sqrt{\mathbb{E}\big{[}(\widehat{x}-x)^{2}\big{]}}}{x},$ (29) where $x$ is the real value and $\widehat{x}$ is the estimated one. ### 5.3 Results #### 5.3.1 Estimating volumes is usually cheap The first step in S-WRW is obtaining category volume estimates $\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{i}$. We achieve it by running a short pilot RW and applying the estimator Eq.(35). We show ${\small\textrm{NRMSE}}(\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{\scriptscriptstyle\textrm{tiny}})$ as plain curves in Fig. 4(b). This estimator takes advantage of the knowledge of the categories of the neighboring nodes, which makes it much more efficient than the naive estimator Eq.(32) shown by dashed curves. Moreover, the advantage of Eq.(35) over Eq.(32) grows with the graph density and the skewness of its degree distribution (not shown here). Note that under ‘random’, RW and WIS (with the sampling probabilities of RW) are almost equally efficient. However, on the other extreme, i.e., under the ‘clustered’ scenario, the performance of RW becomes much worse and the advantage of Eq.(35) over Eq.(32) diminishes. This is because essentially all friends of a node from category $C_{i}$ are in $C_{i}$ too, which reduces formula Eq.(35) to Eq.(32). Nevertheless, we show later in Sec. 5.3.5 that even severalfold volume estimation errors are likely not to affect significantly the results. #### 5.3.2 Visiting the tiny category Fig. 4(e,l) presents the empirical probability $\mathbb{P}[C_{\scriptscriptstyle\textrm{tiny}}\textrm{ visited}]$ that our walk visits at least one node from $C_{\scriptscriptstyle\textrm{tiny}}$. Of course, this probability grows with the sample length. However, the choice of weight $w$ also helps in it. Indeed, WRW with $w>1$ is more likely to visit $C_{\scriptscriptstyle\textrm{tiny}}$ than RW ($w=1$, bottom line). This demonstrates the first advantage of introducing edge weights and WRW. #### 5.3.3 Optimal $w$ and $\gamma$ Let us now focus on the estimation error as a function of $w$, shown in Fig. 4(d,k). Interestingly, this error does not drop monotonically with $w$ but follows a ’U’ shaped function with a clear optimal value $w^{\scriptscriptstyle\textrm{opt}}$. Under WIS, we have $w^{\scriptscriptstyle\textrm{opt}}\simeq 100$, which confirms our findings in Sec. 3.1.4. Indeed, according to Eq.(20), we need the same number of samples from the two categories, and thus $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{tiny}})=\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{big}})$ (by Eq.(22)). By plugging this and $\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})=100\cdot\textrm{vol}(C_{\scriptscriptstyle\textrm{tiny}})$ to Eq.(23), we finally obtain the WIS-optimal edge weights in $C_{\scriptscriptstyle\textrm{tiny}}$, i.e., $w^{\scriptscriptstyle\textrm{opt}}=\textrm{w}_{e}(C_{\scriptscriptstyle\textrm{tiny}})=100\cdot\textrm{w}_{e}(C_{\scriptscriptstyle\textrm{big}})=100$.666For simplicity, we ignored in this calculation the conflicts on the 500 edges between $C_{\scriptscriptstyle\textrm{big}}$ and $C_{\scriptscriptstyle\textrm{tiny}}$. In contrast, WRW is optimized for $w<100$. For the sample length $n\\!=\\!500$ as in Fig. 4(d,k), the error is minimized already for $w^{\scriptscriptstyle\textrm{opt}}\\!\simeq\\!20$ and increases for higher weights. This demonstrates the “black hole” effect discussed in Sec. 4.2.4. It is much more pronounced in the ‘clustered’ scenario, confirming our intuition that black-holes become a problem only in the presence of relatively isolated, tight communities. Of course, the black hole effect diminishes with the sample length $n$ (and completely vanishes for $n\\!\rightarrow\\!\infty$), which can be observed in Fig. 4(g,n), especially in (n). In other words, the optimal assignment of edge weights (in relevant categories) under WRW lies somewhere between RW (all weights equal) and WIS. In S-WRW, we control it by parameter $\gamma$. In this example, we have $\gamma\equiv w$ for $\gamma\leq 100$. Indeed, by combining Eq.(23), Eq.(25), Eq.(26), $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{tiny}})\\!=\\!\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{big}})$, we obtain $\displaystyle w$ $\displaystyle=$ $\displaystyle\frac{w}{1}\ =\ \frac{w_{e}(C_{\scriptscriptstyle\textrm{tiny}})}{w_{e}(C_{\scriptscriptstyle\textrm{big}})}\ =\ \frac{\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{tiny}})/\hbox to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{\scriptscriptstyle\textrm{tiny}})}{\textrm{w}^{\scriptscriptstyle\textrm{WIS}}(C_{\scriptscriptstyle\textrm{big}})/\hbox to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})}$ $\displaystyle=$ $\displaystyle\frac{\hbox to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})}{\hbox to0.0pt{\raisebox{-2.15277pt}{$\tilde{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{\scriptscriptstyle\textrm{tiny}})}\ =\ \frac{\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})}{\frac{1}{\gamma}\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})}\ =\ \gamma.$ Consequently, the optimal setting of $\gamma$ is the same as $w^{\scriptscriptstyle\textrm{opt}}$ discussed above. #### 5.3.4 Gain $\alpha$ The gain $\alpha$ of WIS over UIS is given by Eq.(20). In this case, we have $\alpha=(101K)^{2}\cdot(4\cdot 1K\cdot 100K)^{-1}\simeq 25$. Indeed, WIS with $n\\!=\\!500$ samples shown in Fig. 4(d) achieves ${\small\textrm{NRMSE}}\\!\simeq\\!0.1$, which is the same as UIS of about $\alpha\\!=\\!25$ times more samples (see Fig. 4(c)). This gain due to stratification is smaller for sampling by exploration: a 500-hop-long WRW with $w\\!\simeq\\!20$ yields the same error ${\small\textrm{NRMSE}}\\!\simeq\\!0.3$ as a 2000-hop-long RW. This means that WRW reduces the sampling cost by a factor of $\alpha\simeq 4$. Fig. 4(f) shows that this gain does not vary much with the sampling length. Under ‘clustered’, both RW and WRW perform much worse. Nevertheless, Fig. 4(m) shows that also in this scenario WRW may significantly reduce the sampling cost, especially for longer samples. It is worth noting that WRW can sometimes significantly outperform UIS. This is the case in Fig. 4(d), where UIS is equivalent to WIS with $w\\!=\\!1$. Because no walk can mix faster than UIS (that is independent and thus has perfect mixing), improving the mixing time alone [10, 37, 38, 5] cannot achieve the potential gains of stratification, in general. So far we focused on the smaller set $C_{\scriptscriptstyle\textrm{tiny}}$ only. When estimating the size of $C_{\scriptscriptstyle\textrm{big}}$, all errors are much smaller, but we observe similar gain $\alpha$. #### 5.3.5 Robustness to $\gamma$ and volume estimation The gain $\alpha$ shown above is calculated for the optimal choice of $w$, or, equivalently, $\gamma$. Of course, in practice it might be impossible to obtain this value. Fortunately, S-WRW is relatively robust to the choice of parameters. The dashed lines in Fig. 4(f,m) are calculated for $\gamma$ fixed to $\gamma\\!=\\!5$, rather than optimized. Note that this value is often drastically smaller than the optimal one (e.g., $w^{\scriptscriptstyle\textrm{opt}}\\!\simeq\\!50$ for $n\\!=\\!5000$). Nevertheless, although the performance somewhat drops, S-WRW still reduces the sampling cost about three-fold. This observation also addresses potential concerns one might have regarding the category volume estimation error (see Sec. 4.2.1). Indeed, setting $\gamma\\!=\\!5$ means that every category $C_{i}$ with volume estimated at $\hbox to0.0pt{\raisebox{-2.15277pt}{$\hat{\phantom{\textrm{vol}}}$}\hss}\textrm{vol}(C_{i})\leq\frac{1}{5}\textrm{vol}(C_{\scriptscriptstyle\textrm{big}})$ is treated the same. In Fig. 4(f), the volume of $C_{\scriptscriptstyle\textrm{tiny}}$ would have to be overestimated by more than 20 times in order to affect the edge weight setting and thus the results. We have seen in Sec. 5.3.1 that this is very unlikely, even under smallest sample lengths and most adversarial scenarios. ### 5.4 Summary WRW brings two types of benefits (i) avoiding irrelevant nodes $C_{\ominus}$ and (ii) carefully allocating samples between relevant categories of different sizes. Even when $C_{\ominus}\\!=\\!\emptyset$, WRW can still reduce the sampling cost by 75%. This second benefit is more difficult to achieve when the categories form strong and tight communities, which leads to the “black hole”’ effect. We should then choose smaller, more conservative values of $\gamma$ in S-WRW, which translate into smaller $w$ in our example. In contrast, under a looser community structure this problem disappears and WRW is closer to WIS. ## 6 Implementation in Facebook Figure 5: 5331 colleges discovered and ranked by RW. (a) Estimated relative college sizes $\widehat{f}_{i}$. (b) Absolute number of user samples per college. (c-e) 25 estimates of size $\widehat{f}_{i}$ for three different colleges and sample lengths $n$. (f) Average NRMSE of college size estimation. Results in (a,b,f) are binned. As a concrete application, we apply S-WRW to measure the Facebook social graph, which is our motivating and canonical example. We also note that it is an undirected and can also be considered a static graph, for all practical purposes in this study.777The Facebook characteristics do change but in time scales much longer than the 3-day duration of our crawls. Websites such as Facebook statistics, Alexa etc show that the number of Facebook users is growing with rate 0.1-0.2% per day. In Facebook, every user may declare herself a member of a college888There also exist categories other than colleges, namely “work” and “high school”. Facebook requires a valid category- specific email for verification. he/she attends. This membership information is publicly available by default and allows us to answer some interesting questions. For example, how do the college networks (or “colleges” for short) compare with respect to their sizes? What is the college-to-college friendship graph? In order to answer these questions, we have to collect many college user samples, preferably evenly distributed between colleges. This is the main goal of this section. ### 6.1 Measurement Setup By default, every Facebook user can see the basic information on any other user, including the name, photo, and a list of friends together with their college memberships (if any). We developed a high performance multi-threaded crawler to explore Facebook’s social graph by scraping this web interface. To make informed decision for the parameters of S-WRW, we first ran a short pilot RW (see Sec. 4.2.1) with a total of $65K$ samples (which is only 6.5% of the length of the main S-WRW sample). Although our pilot walk visited only 2000 colleges, it estimated the relative volumes $f^{\textrm{vol}}_{i}$ for about 9500 colleges discovered among friends of sampled users, as discussed in Sec. 4.3.2. In Fig. 6(a), we show that the neighbor-based estimator Eq.(35) greatly outperforms the naive estimator Eq.(32). These volumes cover several decades. Because colleges with only a few tens of users are not of our interest, we set the maximal resolution to $\gamma\\!=\\!1000$ (see the discussion in Sec. 4.3.3). Finally, because the college students looked very well interconnected in our pilot RW, we set the desired fraction of irrelevant nodes to a small number $\tilde{f}_{\ominus}\\!=\\!1\%$. In the main measurement phase, we collected three S-WRW crawls, each with different edge weight conflict resolution (hybrid, geometric, and arithmetic), and one simple RW crawl as a baseline comparison (Table LABEL:tab:fb_datasets). For each crawl type we collected 1 million _unique_ users. Some of them are sampled multiple times (at no additional bandwidth cost), which results in higher total number of samples in the second row of Table LABEL:tab:fb_datasets. Our crawls were performed on Oct. 16-19 2010, and are available at [1]. ### 6.2 Results: RW vs. S-WRW #### 6.2.1 Avoiding irrelevant categories Only 9% of the RW’s samples come from colleges, which means that the vast majority of sampling effort is wasted. In contrast, the S-WRW crawls achieved 6-10 better efficiency, collecting 86% (hybrid), 79% (geometric) and 58% (arithmetic) samples from colleges. Note that these values are significantly lower than the target 99% suggested by our choice of $\tilde{f}_{\ominus}\\!=\\!1\%$, and that S-WRW hybrid reaches the highest number. This is in agreement with our discussion in Sec. 4.2.5. Finally, we also note that S-WRW crawls discovered $1.6-1.9$ times more unique colleges than RW. It might seem surprising that RW samples colleges in 9% of cases while only 3.5% of Facebook users belong to colleges. This can be explained by looking at the last rows of Table LABEL:tab:fb_datasets. Indeed, the college users have on average three times more Facebook friends than average users, and therefore they attract RW approximately three times more often. #### 6.2.2 Stratification The advantage of S-WRW over RW does not lie exclusively in avoiding the nodes in the irrelevant category $C_{\ominus}$. S-WRW can also over-sample small categories (here colleges) at the cost of under-sampling large ones (which are very well sampled anyway). This feature becomes important especially when the category sizes differ significantly, which is the case in Facebook. Indeed, Fig. 5(a) shows that college sizes exhibit great heterogeneity. For a fair comparison, we only include the 5,331 colleges discovered by RW. (In fact, this filtering actually gives preference to RW. S-WRW crawls discovered many more colleges that we do not show in this figure.) They span more than two orders of magnitude and follow a heavily skewed distribution (not shown here). Fig. 5(b) confirms that S-WRW successfully oversamples the small colleges. Indeed, the number of S-WRW samples per college is almost constant (roughly around 100). In contrast, the number of RW samples follows closely the college size, which results in dramatic 100-fold differences between RW and S-WRW for smaller colleges. #### 6.2.3 College size estimation With more samples per college, we naturally expect a better estimation accuracy under S-WRW. We demonstrate it for three colleges of different sizes (in terms of the number of Facebook users): MIT (large), Caltech (medium), and Eindhoven University of Technology (small). Each boxplot in Fig. 5(c-e) is generated based on 25 independent college size estimates $\widehat{f}_{i}$ that come from walks of length $n\\!=\\!4$K (left), 20K (middle), and 40K (right) samples each. For the three studied colleges, RW fails to produce reliable estimates in all cases except for MIT (largest college) under the two longest crawls. Similar results hold for the overwhelming majority of middle- sized and small colleges. The underlying reason is the very small number of samples collected by RW in these colleges, averaging at below 1 sample per walk. In contrast, the three S-WRW crawls contain typically 5-50 times more samples than RW (in agreement with Fig. 5(b)), and produce much more reliable estimates. Finally, we aggregate the results over all colleges and compute the gain $\alpha$ of S-WRW over RW. We calculate the error ${\small\textrm{NRMSE}}(\widehat{f}_{i})$ by taking as our “ground truth” $f_{i}$ the grand average of $\widehat{f}_{i}$ values over all samples collected via all full-length walks and crawl types. Fig. 5(f) presents ${\small\textrm{NRMSE}}(\widehat{f}_{i})$ averaged over all 5,331 colleges discovered by RW, as a function of walk length $n$. As expected, for all crawl types the error decreases with $n$. However, there is a consistent large gap between RW and all three versions of S-WRW. RW needs 13-15 times more samples than S-WRW in order to achieve the same error. Figure 6: Facebook: Pilot RW and other walks of the same length $n\\!=\\!65K$. (a) The performance of the neighbor-based volume estimator Eq.(35) (plain line) and the naive one Eq.(32) (dashed line). As ‘ground-truth’ we used $f^{\textrm{vol}}_{i}$ calculated for all 4$\times$1M collected samples. (b) The effect of the choice of $\gamma$. #### 6.2.4 The effect of the choice of $\gamma$ Recall that in all the S-WRW results described above, we used the resolution $\gamma\\!=\\!1000$. In order to check how sensitive the results are to the choice of this parameter, we also tried a (shorter) S-WRW run with $\gamma\\!=\\!100$, i.e., ten times smaller. In Fig. 6(b), we see that the number of samples collected in the smallest colleges is smaller under $\gamma\\!=\\!100$ than under $\gamma\\!=\\!1000$. In fact, the two curves diverge for colleges about 100 times smaller than the biggest college, i.e., exactly at the maximal resolution $\gamma\\!=\\!100$. In any case, both settings of $\gamma$ perform orders of magnitude better than RW of the same length. ### 6.3 Summary Only about 3.5% of 500M Facebook users are college members. There are more than 10K colleges and they greatly vary in size, ranging from 50 (or fewer) to 50K members (we aggregate students, alumni and staff). In this setting, state- of-the-art sampling methods such as RW are bound to perform poorly. Indeed, UIS, i.e., an idealized version of RW, with as many as 1M samples will collect only one sample from size-500 college, on average. Even if we could magically sample directly only from colleges, we would typically collect fewer than 30 samples per size-500 college. S-WRW solves these problems. We showed that S-WRW of the same length collects typically about 100 samples per size-500 college. As a result, S-WRW outperforms RW by $\alpha=13-15$ times or $\alpha=12-14$ times if we also consider the 6.5% overhead from the initial pilot RW. This huge gain can be decomposed into two factors, say $\alpha=\alpha_{1}\cdot\alpha_{2}$, as we proposed in Eq.(21). Factor $\alpha_{1}\simeq 8$ can be attributed to a about 8 times higher fraction of college samples in S-WRW compared to RW. Factor $\alpha_{2}\simeq 1.5$ is due to over-sampling smaller networks, i.e., by applying stratified sampling. Another important observation is that S-WRW is robust to the way we resolve target edge weight conflicts in Sec. 4.2.5. The differences between the three S-WRW implementations are minor - it is the application of Eq.(27) that brings most of the benefit. ## 7 Related work Graph Sampling by Exploration. Early crawling of P2P, OSN and WWW typically used graph traversals, mainly BFS [3, 32, 31, 43, 33] and its variants. However, incomplete BFS introduces bias towards high-degree nodes that is unknown and thus impossible to correct in general graphs [2, 26, 8, 17, 25]. Later studies followed a more principled approach based on random walks (RW) [29, 4]. The Metropolis-Hasting RW (MHRW) [30, 16] removes the bias during the walk; it has been used to sample P2P networks [40, 35] and OSNs [17]. Alternatively, we can use RW, whose bias is known and can be corrected for [20, 39], thus leading to a re-weighted RW [35, 17]. RW was also used to sample Web [21], P2P networks [40, 35, 18], OSNs [24, 36, 17, 33], and other large graphs [27]. It was empirically shown in [35, 17] that RW outperforms MHRW in measurement accuracy. Therefore, RW can be considered as the state-of- the-art. Random walks have also been used to sample _dynamic graphs_ [40, 35, 42], which are outside the scope of this paper. Fast Mixing Markov Chains. The mixing time of a random walk determines the efficiency of the sampling. On the practical side, the mixing time of RW in many OSNs was found larger than commonly believed [33]. Multiple dependent random walks [37] have been used to sample disconnected and loosely connected graphs. Random walks with jumps have been used to sample large graphs in [38, 5] and in [27]. All the above methods treat all nodes with equal importance, which is orthogonal to our technique. On the theoretical side, in [10], the authors propose a method to set edge weights that achieve the fastest mixing WRW for a given target stationary distribution. This technique, although related, is not applicable in our context. First, [10] requires the knowledge of the graph, which makes it inapplicable to $G$, yet possibly feasible in $G^{C}$ (after estimating some limited information about $G^{C}$ as in Sec. 4.2.1). In the latter case, however, even given a perfect knowledge of $G^{C}$, [10] often assigns weight 0 to some self-loops, which likely makes the underlying graph $G$ disconnected. Finally, and most importantly, [10] takes a target stationary distribution as input. By taking $\textrm{w}^{\scriptscriptstyle\textrm{WIS}}$, we will face exactly the same problems of potentially poor convergence (Sec. 4.2.3) and “black holes” (Sec. 4.2.4) as we addressed by S-WRW. Stratified Sampling. Our approach builds on _stratified sampling_ [34], a widely used technique in statistics; see [12, 28] for a good introduction. A related work in a different networking problem is [14], where threshold sampling is used to vary sampling probabilities of network traffic flows and estimate their volume. Weighted Random Walks for Sampling. Random walks on graphs with weighted edges, or equivalently reversible Markov chains [29, 4], are well studied and heavily used in Monte Carlo Markov Chain simulations [16] to sample a state space with a specified probability distribution. However, to the best of our knowledge, WRWs have not been designed explicitly for measurements of real online systems. In the context of sampling OSNs, the closest works are [38, 5]. Technically speaking, they use WRW. But they set as their only objective the minimization of the mixing time, which makes them orthogonal and complementary to our approach, as we discussed above. Very recent applications of weighted random walks in online social networks include [7, 6]. [7] uses WRW in the context of link prediction. The authors employ supervised learning techniques to set the edge weights, with the goal of increasing the probability of visiting nodes that are more likely to receive new links. [6] introduces WRW-based methods to generate samples of nodes that are internally well-connected but also approximately uniform over the population. In both these papers, WRW is used to predict/extract something from a known graph. In contrast, we use WRW to estimate features of an unknown graph. In the context of World Wide Web crawling, _focused crawling_ techniques [11, 13] have been introduced to follow web pages of specified interest and to avoid the irrelevant pages. This is achieved by performing a BFS type of sample, except that instead of fifo queue they use a priority queue weighted by the page relevancy. In our context, such an approach suffers from the same problems as regular BFS: (i) collected samples strongly depend on the starting point, and (ii) we are not able to unbias the sample. ## 8 Conclusion We introduced Stratified Weighted Random Walk (S-WRW) - an efficient way to sample large, static, undirected graphs via crawling and using minimal information. S-WRW performs a weighted random walk on the graph with weights determined by the estimation problem. We apply our approach to measure the Facebook social graph, and we show that S-WRW greatly outperforms the state- of-art sampling technique, namely the simple re-weighted random walk. 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For example, for a graph that is a path consisting of two nodes ($v_{1}-v_{2}$), it is impossible to achieve $\textrm{w}(v_{1})\neq\textrm{w}(v_{2})$. However, it is always possible to do so, if there are self loops in each node. ###### Observation 1 For any undirected graph $G=(V,E)$ with a self-loop $\\{v,v\\}$ at every node $v\in V$, we can achieve an arbitrary distribution of node weights $\textrm{w}(v)>0,\ v\in V$, by appropriate choice of edge weights $\textrm{w}(u,v)\\!>\\!0,\ \\{u,v\\}\\!\in\\!E$. ###### Proof 8.1. Denote by $\textrm{w}_{\min}$ the smallest of all target node weights $\textrm{w}(v)$. Set $\textrm{w}(u,v)=\textrm{w}_{\min}/N$ for all non self- loop edges (i.e., where $u\neq v$). Now, for every self-loop $\\{v,v\\}\in E$ set $\textrm{w}(v,v)\ \ =\ \ \frac{1}{2}\left(\textrm{w}(v)-\frac{\textrm{w}_{\min}}{N}\cdot(\deg(v)\\!-\\!2)\right).$ It is easy to check that, because there are exactly $\deg(v)\\!-\\!2$ non self-loop edges incident on $v$, every node $v\in V$ will achieve the target weight $\textrm{w}(v)$. Moreover, the definition of $\textrm{w}_{\min}$ guarantees that $\textrm{w}(v,v)>0$ for every $v\in V$. ## Appendix B: Estimating Category Volumes In this section, we derive efficient estimators of the volume ratio $\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}=\frac{\textrm{vol}(C)}{\textrm{vol}(V)}$. Recall that $S\subset V$ denotes an independent sample of nodes in $G$, with replacement. Node sampling If $S$ is a uniform sample UIS, then we can write $\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}\ =\ \frac{\sum_{v\in S}\deg(v)\cdot 1_{\\{v\in C\\}}}{\sum_{v\in S}\deg(v)},$ (30) which is a straightforward application of the classic ratio estimator [28]. In the more general case, when $S$ is selected using WIS, then we have to correct for the linear bias towards nodes of higher weights $\textrm{w}()$, as follows: $\displaystyle\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}$ $\displaystyle=$ $\displaystyle\frac{\sum_{v\in S}\deg(v)\cdot 1_{\\{v\in C\\}}/\textrm{w}(v)}{\sum_{v\in S}\deg(v)/\textrm{w}(v)}.$ (31) In particular, if $\textrm{w}(v)\sim\deg(v)$, then $\displaystyle\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}$ $\displaystyle=$ $\displaystyle\frac{1}{n}\cdot\sum_{v\in S}1_{\\{v\in C\\}}.$ (32) Star sampling Another approach is to focus on the set of all neighbors $\mathcal{N}(S)$ of sampled nodes (with repetitions) rather than on $S$ itself, i.e., to use ‘star sampling’ [23]. The probability that a node $v$ is a neighbor of a node sampled from $V$ by UIS is $\sum_{u\in V}\frac{1}{N}\cdot 1_{\\{v\in\mathcal{N}(u)\\}}\ \ =\ \ \frac{\deg(v)}{N}.$ Consequently, the nodes in $\mathcal{N}(S)$ are asymptotically equivalent to nodes drawn with probabilities linearly proportional to node degrees. By applying Eq.(32) to $\mathcal{N}(S)$, we obtain999As a side note, observe that formula Eq.(33) generalizes the “scale-up method” [9] used in social sciences to estimate the size (here $|C|$) of hidden populations (e.g., of drug addicts). Indeed, if we assume that the average node degree in $V$ is the same as in $C$, then $\textrm{vol}(C)/\textrm{vol}(V)=|C|/N$, which reduces Eq.(32) to the core formula of the scale-up method. $\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}\ \ =\ \ \frac{1}{\textrm{vol}(S)}\sum_{u\in S}\sum_{v\in\mathcal{N}(u)}\\!\\!1_{\\{v\in C\\}},$ (33) where we used $|\mathcal{N}(S)|=\sum_{u\in S}\deg(u)=\textrm{vol}(S)$. In the more general case, when $S$ is selected using WIS, then we correct for the linear bias towards nodes of higher weights $\textrm{w}()$, as follows: $\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}\ \ =\ \ \frac{1}{\displaystyle\sum_{u\in S}\frac{\deg(u)}{\textrm{w}(u)}}\sum_{u\in S}\left(\frac{1}{\textrm{w}(u)}\sum_{v\in\mathcal{N}(u)}\\!\\!1_{\\{v\in C\\}}\right).$ (34) In particular, if $\textrm{w}(v)\sim\deg(v)$, then $\widehat{f}^{\scriptscriptstyle\textrm{vol}}_{C}\ \ =\ \ \frac{1}{n}\sum_{u\in S}\left(\frac{1}{\deg(u)}\sum_{v\in\mathcal{N}(u)}\\!\\!1_{\\{v\in C\\}}\right).$ (35) Note that for every sampled node $v\in S$, the formulas Eq.(33-35) exploit all the $\deg(v)$ neighbors of $v$, whereas Eq.(30-32) rely on one node per sample only. Not surprisingly, Eq.(33-35) performed much better in all our simulations and implementations. ## Appendix C: Relative sizes of node categories Consider a scenario with only two node categories, i.e., $\mathcal{C}=\\{C_{1},C_{2}\\}$. Denote $f_{1}=|C_{1}|/N$ and $f_{2}=|C_{2}|/N$. The goal is to estimate $f_{1}$ and $f_{2}$ based on the collected sample $S$. ##### UIS - Uniform independence sampling Under UIS, the number $X_{1}$ of times we select a node from $C_{1}$ among $n$ attempts follows the Binomial distribution $X_{1}=Binom(f_{1},n)$. Therefore, we can estimate $f_{1}$ as $\hat{f}^{\scriptscriptstyle\textrm{UIS}}_{1}\ =\ \frac{X_{1}}{n}\qquad\textrm{ with }\qquad\mathbb{V}(\hat{f}^{\scriptscriptstyle\textrm{UIS}}_{1})\ =\ \frac{f_{1}f_{2}}{n}.$ (36) ##### WIS - Weighted independence sampling In contrast, under WIS, at every iteration the probability $\pi(v)$ of selecting a node $v$ is: $\pi(v)=\left\\{\begin{array}[]{rl}\pi_{1}=\frac{1}{N}\cdot\frac{w_{1}}{w_{1}f_{1}+w_{2}f_{2}}&\textrm{ if }v\in C_{1},\textrm{ and }\\\ \pi_{2}=\frac{1}{N}\cdot\frac{w_{2}}{w_{1}f_{1}+w_{2}f_{2}}&\textrm{ if }v\in C_{2},\end{array}\right.$ where $w_{1}$ and $w_{2}$ are the weights $\textrm{w}(v)$ of nodes in $C_{1}$ and $C_{2}$, respectively. By applying the Hansen-Hurwitz estimator (separately for nominator and denominator), we obtain $\displaystyle\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}$ $\displaystyle=$ $\displaystyle\frac{|\hat{C}_{1}|}{\hat{N}}\ =\ \frac{\sum_{v\in S}1_{v\in C_{1}}\,/\,\pi(v)}{\sum_{v\in S}1\,/\,\pi(v)}$ (37) $\displaystyle=$ $\displaystyle\frac{X_{1}\,/\,\pi_{1}}{X_{1}\,/\,\pi_{1}\ +\ (n-X_{1})\,/\,\pi_{2}}\ $ $\displaystyle=$ $\displaystyle\frac{X_{1}\cdot\pi_{2}}{X_{1}(\pi_{2}-\pi_{1})+n\cdot\pi_{1}}\ $ $\displaystyle=$ $\displaystyle\frac{X_{1}\cdot\textrm{w}_{2}}{X_{1}(\textrm{w}_{2}-\textrm{w}_{1})+n\cdot\textrm{w}_{1}},$ where $X_{1}$ is the number of samples taken from $C_{1}$. Note, that to calculate $\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}$ we only need values $w_{1}$ and $w_{2}$, which are set by us and thus known. Computing the variance of $\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}$ is a bit more challenging. We use the second-order Taylor expansions (the ’Delta method’) to approximate it as follows: $\displaystyle\frac{\partial\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}}{\partial X_{1}}$ $\displaystyle=$ $\displaystyle\frac{nw_{1}w_{2}}{((w_{2}-w_{1})X_{1}+nw_{1})^{2}},\quad\textrm{ and}$ $\displaystyle\mathbb{V}(\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1})$ $\displaystyle\cong$ $\displaystyle\left(\frac{\partial\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1}}{\partial X_{1}}\big{(}\mathbb{E}(X_{1})\big{)}\right)^{2}\mathbb{V}(X_{1})$ (38) $\displaystyle=$ $\displaystyle\\!\\!\\!\big{(}\ldots\big{)}=\frac{f_{1}f_{2}}{nw_{1}w_{2}}\cdot(f_{1}w_{1}+f_{2}w_{2})^{2}.$ In the above derivation, we used the fact that $\mathbb{E}(X_{1})=nNf_{1}\pi_{1}$ and $\mathbb{V}(X_{1})=nN^{2}f_{1}\pi_{1}f_{2}\pi_{2}$. This comes from the fact that $X_{1}$ actually follows the binomial distribution $X_{1}=Binom(Nf_{1}\pi_{1},n).$ For $w_{1}=w_{2}$, we are back in the UIS case. But this is not necessarily the optimal choice of weights. Indeed, a quick application of Lagrange multipliers reveals that $\mathbb{V}(\hat{f}^{\scriptscriptstyle\textrm{WIS}}_{1})$ is minimized when $w_{1}\,f_{1}=f_{2}\,w_{2}.$ (39) Moreover, analogous analysis shows that Eq.(39) minimizes $\mathbb{V}(\hat{f}_{2}^{\scriptscriptstyle\textrm{WIS}})$ as well. In other words, the estimators of both $f_{1}$ and $f_{2}$ have the lowest variance if the total weighted mass of $C_{1}$ is equal to that of $C_{2}$. This implies, in expectation, equal allocation of samples between $C_{1}$ and $C_{2}$, i.e., $n^{\scriptscriptstyle\textrm{WIS}}_{i}\ =\ \frac{n}{|\mathcal{C}|}.$ Finally, we can use Eq.(36), Eq.(38) and Eq.(39) to calculate the gain $\alpha$ of WIS over UIS $\alpha\ =\ \frac{1}{4f_{1}f_{2}}\quad(\geq 1).$ (40) Note that we always have $\alpha\geq 1$, and $\alpha$ grows quickly with growing difference between $f_{1}$ and $f_{2}$. ## Appendix D: Optimal WRW weights in Fig. 3(a) Every time WRW visits the white node/category in Fig. 3(a), the next node is chosen uniformly from red and green categories. We stay in this selected category for $k$ rounds, where $k$ is a geometric random variable with parameter $p=w_{2}/(w_{1}\\!+\\!w_{2})\in[0,1]$. Next, we come back to the white category, and reiterate the process. So the number $n_{\scriptscriptstyle\textrm{red}}$ of times the red category is sampled is $n_{\scriptscriptstyle\textrm{red}}=\sum_{1}^{Binom(0.5,n_{\scriptscriptstyle\textrm{wh}})}Geom(p),$ where $n_{\scriptscriptstyle\textrm{wh}}$ is the number of visits to the white category. Because the random variables generated by $Binom(0.5,n_{\scriptscriptstyle\textrm{wh}})$ and $Geom(p)$ are independent, we can write $\displaystyle\mathbb{E}[n_{\scriptscriptstyle\textrm{red}}]$ $\displaystyle=$ $\displaystyle\mathbb{E}[Binom(0.5,n_{\scriptscriptstyle\textrm{wh}})]\cdot\mathbb{E}[Geom(p)]\ =\ 0.5n_{\scriptscriptstyle\textrm{wh}}/p$ $\displaystyle\mathbb{V}[n_{\scriptscriptstyle\textrm{red}}]$ $\displaystyle=$ $\displaystyle\mathbb{E}[Binom()]\mathbb{V}[Geom()]+\mathbb{E}^{2}[Geom()]\mathbb{V}[Binom()]$ $\displaystyle=$ $\displaystyle\frac{n_{\scriptscriptstyle\textrm{wh}}}{4p^{2}}(3-2p).$ A possible unbiased estimator of the relative size $f_{\scriptscriptstyle\textrm{red}}$ of red category (among relevant categories) is $\widehat{f}_{\scriptscriptstyle\textrm{red}}=\frac{n_{\scriptscriptstyle\textrm{red}}}{n_{\scriptscriptstyle\textrm{wh}}/p},$ for which we get $\displaystyle\mathbb{E}[\widehat{f}_{\scriptscriptstyle\textrm{red}}]$ $\displaystyle=$ $\displaystyle\frac{\mathbb{E}[n_{\scriptscriptstyle\textrm{red}}]}{n_{\scriptscriptstyle\textrm{wh}}/p}\ =\ \frac{1}{2}\quad\textrm{(unbiased)}$ $\displaystyle\mathbb{V}[\widehat{f}_{\scriptscriptstyle\textrm{red}}]$ $\displaystyle=$ $\displaystyle\frac{\mathbb{V}[n_{\scriptscriptstyle\textrm{red}}]}{(n_{\scriptscriptstyle\textrm{wh}}/p)^{2}}\ =\ \frac{3-2p}{4n_{\scriptscriptstyle\textrm{wh}}}.$ This variance is expressed as a function of $n_{\scriptscriptstyle\textrm{wh}}$, and not of the total sample length $n$. However, note that $n_{\scriptscriptstyle\textrm{wh}}$ drops with decreasing $p$. Consequently, the variance $\mathbb{V}[\widehat{f}_{\scriptscriptstyle\textrm{red}}]$ (expressed as a function of $n_{\scriptscriptstyle\textrm{wh}}$ or of $n$) is minimized for $p=1$, i.e., for $w_{1}=0$ and $w_{2}>0$ (and $n_{\scriptscriptstyle\textrm{wh}}\\!=\\!n/2$).
arxiv-papers
2011-01-28T06:27:45
2024-09-04T02:49:16.694733
{ "license": "Public Domain", "authors": "M. Kurant, M. Gjoka, C. T. Butts, A. Markopoulou", "submitter": "Maciej Kurant", "url": "https://arxiv.org/abs/1101.5463" }
1101.5664
arxiv-papers
2011-01-29T03:51:24
2024-09-04T02:49:16.708286
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zhancheng Li, Ping Wu, Chenxi Wang, Xiaodong Fan, Wenhua Zhang,\n Xiaofang Zhai, Changgan Zeng, Zhenyu Li, Jinlong Yang, and J. G. Hou", "submitter": "Changgan Zeng", "url": "https://arxiv.org/abs/1101.5664" }
1101.5700
# Infraparticles with superselected direction of motion in two-dimensional conformal field theory Wojciech Dybalski111Supported by the DFG grant SP181/25. Zentrum Mathematik, Technische Universität München, D-85747 Garching, Germany E-mail: dybalski@ma.tum.de Yoh Tanimoto222Supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”. Dipartimento di Matematica, Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 - I–00133 Roma, Italy. E-mail: tanimoto@mat.uniroma2.it ###### Abstract Particle aspects of two-dimensional conformal field theories are investigated, using methods from algebraic quantum field theory. The results include asymptotic completeness in terms of (counterparts of) Wigner particles in any vacuum representation and the existence of (counterparts of) infraparticles in any charged irreducible product representation of a given chiral conformal field theory. Moreover, an interesting interplay between the infraparticle’s direction of motion and the superselection structure is demonstrated in a large class of examples. This phenomenon resembles the electron’s momentum superselection expected in quantum electrodynamics. ## 1 Introduction Particle aspects and superselection structure of quantum electrodynamics are plagued by the infrared problem, which has been a subject of study in mathematical physics for more than four decades [46, 28, 29, 39, 21, 40, 18, 33, 19, 30, 6, 7, 8, 10, 16, 41, 42, 22, 23, 24, 48, 34, 44]. The origin of this difficulty, inherited from classical electrodynamics, is the emission of photons which accompanies any change of the electron’s momentum. It has two important consequences which are closely related: Firstly, the electron is not a particle in the sense of Wigner [50], but rather an _infraparticle_ [46] i.e. it does not have a precise mass. Secondly, the electron’s plane wave configurations of different momenta cannot be superposed into normalizable wavepackets. In fact, such configurations have different spacelike asymptotic flux of the electric field, which imposes a superselection rule [7]. The evidence for this phenomenon of the electron’s _momentum superselection_ comes from two sources: On the one hand, it appears in models of non-relativistic QED in the representation structure of the asymptotic electromagnetic field algebra [18]. On the other hand, it is suggested by structural results in the general framework of algebraic quantum field theory [7, 8, 16, 41, 42]. However, no examples of local, relativistic theories, describing infraparticles with superselected momentum, have been given to date. Thus the logical consistency of this property with the basic postulates of quantum field theory remains to be settled. As a step in this direction, we demonstrate in the present paper that a simple variant of this phenomenon - superselection of direction of motion - occurs in a large class of two- dimensional conformal field theories. Conformal field theory has been a subject of intensive research over the last two decades, both from physical and mathematical viewpoints, motivated, in particular, by the search for non-trivial quantum field theories. (See e.g. [3] and references therein). It exhibits particularly interesting properties in two dimensions, where the symmetry group is infinite dimensional. Since the seminal work of Buchholz, Mack and Todorov [15] the superselection structure of these theories has been investigated [31] and deep classification results have been obtained [36, 37]. It has remained unnoticed, however, that two- dimensional conformal field theories have also a rich and interesting particle structure: The concepts of Wigner particles and infraparticles have their natural counterparts in this setting and both types of excitations appear in abundance: Any chiral conformal field theory in a vacuum representation has a complete particle interpretation in terms of Wigner particles. Although such theories are non-interacting, their (Grosse-Lechner) deformations [14] exhibit non-trivial scattering and inherit the property of asymptotic completeness as we show in a companion paper [25]. It is verified in the present work that any charged irreducible product representation of a chiral conformal field theory admits infraparticles. In a large class of examples these infraparticles have superselected direction of motion i.e. their plane wave configurations with opposite directions of momentum cannot be superposed. Thus subtle particle phenomena, which are not under control in physical spacetime, can be investigated in these two-dimensional models. To keep our analysis general, we rely on the setting of algebraic QFT [32]. We base our discussion on the concept of a local net of $C^{*}$-algebras on $\mathbb{R}^{2}$, defined precisely in Subsection 2.1: To any open, bounded region $\mathcal{O}\subset\mathbb{R}^{2}$ we attach a $C^{*}$-algebra ${\mathfrak{A}}(\mathcal{O})$, acting on a Hilbert space $\mathcal{H}$ of physical states. This algebra is generated by observables which can be measured with an experimental device localized in $\mathcal{O}$. It is contained in the quasilocal algebra ${\mathfrak{A}}$, which is the inductive limit of the net $\mathcal{O}\to{\mathfrak{A}}(\mathcal{O})$. Moreover, there acts a unitary representation of translations $\mathbb{R}^{2}\ni x\to U(x)$ on $\mathcal{H}$, whose adjoint action $\alpha_{x}(\,\cdot\,)=U(x)\,\cdot\,U(x)^{*}$ shifts the observables in spacetime. The infinitesimal generators of $U$ are interpreted as the Hamiltonian $H$ and the momentum operator $\boldsymbol{P}$. Their joint spectrum is contained in the closed forward lightcone $V_{+}$, to ensure the positivity of energy. If there exists a cyclic, unit vector $\Omega\in\mathcal{H}$ which is the unique (up to a phase) joint eigenvector of $H$ and $\boldsymbol{P}$ with eigenvalue zero, then we say that the theory is in a vacuum representation. If each of the subspaces $\mathcal{H}_{\pm}=\ker(H\mp\boldsymbol{P})$ includes some vectors orthogonal to $\Omega$, then we say that the theory contains Wigner particles. Since we do not assume that these particles are described by vectors in some irreducible representation space of the Poincaré group, the present definition is less restrictive than the conventional one. However, it is better suited for a description of the dispersionless kinematics of two-dimensional massless excitations. In particular, it allows us to apply the natural scattering theory, developed by Buchholz in [5], which we outline in Subsection 2.2 below. We recall that in [5] these excitations are called ‘waves’, to stress their composite character. Due to this compound structure of Wigner particles (or ‘waves’), asymptotic completeness in a vacuum representation is not in conflict with the existence of charged representations with a non-trivial particle content. However, in charged representations of massless two-dimensional theories Wigner particles may be absent, as noticed in [12]. In this case scattering theory from [5] does not apply and an appropriate framework for the analysis of particle aspects is the theory of _particle weights_ [10, 47, 16, 35, 41, 42, 23, 24], developed by Buchholz, Porrmann and Stein, which we revisit in Subsections 2.3 and 2.4. This theory is based on the concept of the asymptotic functional, given by $\sigma_{\Psi}^{\mathrm{out}}(C)=\lim_{t\to\infty}\int d\boldsymbol{x}\,(\Psi|\alpha_{(t,\boldsymbol{x})}(C)\Psi),$ (1.1) for any vector $\Psi\in\mathcal{H}$ of bounded energy, and suitable observables $C\in{\mathfrak{A}}$. (In general, some time averaging and restriction to a subnet may be needed before taking the limit). We remark for future reference that this functional induces a sesquilinear form $\psi_{\Psi}^{\mathrm{out}}$ on a certain left ideal of ${\mathfrak{A}}$. We show in Theorem 2.11 below, that asymptotic functionals are non-zero in theories of Wigner particles. If non-trivial asymptotic functionals arise in the absence of Wigner particles, then we say that the theory describes infraparticles333The conventional definition of infraparticles requires that both $\mathcal{H}_{+}$ and $\mathcal{H}_{-}$ contain at most multiples of the vacuum vector. Our (less restrictive) definition imposes this requirement on one of these subspaces only. Thus theories containing ‘waves’ running to the right but no ‘waves’ running to the left (or vice versa) describe infraparticles according to our terminology. Such nomenclature turns out to be more convenient in the context of two-dimensional, massless theories.. Using standard decomposition theory, the GNS representation $\pi$ induced by the sesquilinear form $\psi_{\Psi}^{\mathrm{out}}$ can be decomposed into a direct integral of irreducible representations $\pi\simeq\int_{X}d\mu(\xi)\,\pi_{\xi},$ (1.2) where $(X,d\mu)$ is a Borel space and $\simeq$ denotes unitary equivalence [49, 42]. Results from [2, 47] suggest that the measurable field of irreducible representations $\\{\pi_{\xi}\,\\}_{\xi\in X}$ carries information about all the (infra-)particle types appearing in the theory. In particular, there exists a field of vectors $\\{\,q_{\xi}\,\\}_{\xi\in X}$ which can be interpreted as the energy and momentum of plane wave configurations $\\{\psi_{\xi}\\}_{\xi\in X}$ of the respective (infra-)particles [42]. The sesquilinear forms $\\{\psi_{\xi}\\}_{\xi\in X}$, called _pure particle weights_ , induce the representations $\\{\pi_{\xi}\\}_{\xi\in X}$ and satisfy $\psi_{\Psi}^{\mathrm{out}}=\int_{X}d\mu(\xi)\,\psi_{\xi}.$ (1.3) The existence of such a decomposition was shown, under certain technical restrictions, in [41, 42]. The theory of particle weights is sufficiently general to accommodate the phenomenon of the infraparticle’s momentum superselection, discussed above: In this case $q_{\xi}\neq q_{\xi^{\prime}}$ should imply that $\pi_{\xi}$ is not unitarily equivalent to $\pi_{\xi^{\prime}}$ for almost all labels $\xi$, $\xi^{\prime}$ corresponding to the infraparticle in question. Superselection of direction of motion is a milder property: It only requires that plane waves $\psi_{\xi},\psi_{\xi^{\prime}}$, travelling in opposite directions, give rise to representations $\pi_{\xi},\pi_{\xi^{\prime}}$ which are not unitarily equivalent. This latter interplay between the infraparticle’s kinematics and the superselection structure occurs in some two-dimensional conformal field theories, as we explain below. We state this property precisely in Definitions 2.7 and 2.12, where we restrict attention to representations $\pi$ of (Murray- von Neumann) type I with atomic center. This is sufficient for our purposes and allows us to separate our central concept from ambiguities involved in the general decompositions (1.2), (1.3). Our discussion of conformal field theory relies on the notion of a local net of von Neumann algebras on $\mathbb{R}$ which we introduce in Subsection 3.1. (Such nets arise e.g. by restricting the familiar Möbius covariant nets on the circle to the real line). With any open bounded region ${\cal I}\subset\mathbb{R}$ we associate a von Neumann algebra ${\cal A}({\cal I})$, acting on a Hilbert space $\mathcal{K}$, and denote the quasilocal algebra of this net by ${\cal A}$. Moreover, the Hilbert space $\mathcal{K}$ carries a unitary representation of translations $\mathbb{R}\ni s\to V(s)$, whose spectrum coincides with $\mathbb{R}_{+}$. If there exists a cyclic, unit vector $\Omega_{0}\in\mathcal{K}$, which is the unique (up to a phase) non- zero vector invariant under the action of $V$, then we say that the theory is in a vacuum representation. Given such a net, covariant under the action of some internal symmetry group, one can proceed to the fixed-point subnet which has a non-trivial superselection structure. In the simple case, considered in Subsection 3.4, the action of $\mathbb{Z}_{2}$ is implemented by a unitary $W\neq I$ on $\mathcal{K}$ s.t. $W^{2}=I$. The fixed-point subnet ${\cal A}_{\mathrm{ev}}$ consists of all the elements of ${\cal A}$, which commute with $W$. The subspace $\mathcal{K}_{\mathrm{ev}}=\ker(W-I)$ (resp. $\mathcal{K}_{\mathrm{odd}}=\ker(W+I)$) is invariant under the action of ${\cal A}_{\mathrm{ev}}$ and gives rise to a vacuum representation (resp. a charged representation) of the fixed-point theory. Given two nets of von Neumann algebras on the real line, ${\cal A}_{1}$ and ${\cal A}_{2}$, acting on Hilbert spaces $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$, one obtains the two-dimensional chiral net ${\mathfrak{A}}$, acting on $\mathcal{H}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}$, by the standard construction, recalled in Subsection 3.1: The two real lines are identified with the lightlines in $\mathbb{R}^{2}$ and for any double cone $\mathcal{O}={\cal I}\times\mathfrak{J}$ one sets444In the main part of the paper ${\mathfrak{A}}({\cal I}\times\mathfrak{J})$ denotes a suitable weakly dense ‘regular subalgebra’ of ${\cal A}_{1}({\cal I})\otimes{\cal A}_{2}(\mathfrak{J})$. This distinction is not essential for the present introductory discussion. ${\mathfrak{A}}({\cal I}\times\mathfrak{J})={\cal A}_{1}({\cal I})\otimes{\cal A}_{2}(\mathfrak{J})$. If the nets ${\cal A}_{1}$, ${\cal A}_{2}$ are in vacuum representations, with the vacuum vectors $\Omega_{1}\in\mathcal{K}_{1}$, $\Omega_{2}\in\mathcal{K}_{2}$, then ${\mathfrak{A}}$ is also in a vacuum representation, with the vacuum vector $\Omega=\Omega_{1}\otimes\Omega_{2}$. In spite of their simple tensor product structure, chiral nets play a prominent role in conformal field theory. In fact, with any local conformal net on $\mathbb{R}^{2}$ one can associate a chiral subnet by restricting the theory to the lightlines. In the important case of central charge $c<1$ these subnets were instrumental for the classification results, mentioned above, which clarified the superselection structure of a large class of models [43, 36]. As we show in the present work, chiral nets also offer a promising starting point for the analysis of particle aspects of conformal field theories: Any chiral net in a vacuum representation is an asymptotically complete theory of Wigner particles. Moreover, any charged irreducible product representation of such a net contains infraparticles. With this information at hand, we exhibit examples of infraparticles with superselected direction of motion. This construction is summarized briefly in the remaining part of this Introduction. Let us consider two fixed-point nets ${\cal A}_{1,\mathrm{ev}}$, ${\cal A}_{2,\mathrm{ev}}$, obtained from ${\cal A}_{1}$ and ${\cal A}_{2}$ with the help of the unitaries $W_{1}$ and $W_{2}$, implementing the respective actions of $\mathbb{Z}_{2}$. The resulting chiral net ${\mathfrak{A}}_{\mathrm{ev}}$ acts on the Hilbert space $\mathcal{H}=\mathcal{K}_{1}\otimes\mathcal{K}_{2}$, which decomposes into four invariant subspaces with different particle structure: $\mathcal{H}=(\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{ev}})\oplus(\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}})\oplus(\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}})\oplus(\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}).$ (1.4) ${\mathfrak{A}}_{\mathrm{ev}}$ restricted to $\mathcal{H}_{0}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{ev}}$ is a chiral theory in a vacuum representation. Thus it is an asymptotically complete theory of Wigner particles, by the result mentioned above. $\mathcal{H}_{{\rm{R}}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$ contains ‘waves’ travelling to the right, but no ‘waves’ travelling to the left. In $\mathcal{H}_{{\rm{L}}}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}}$ the opposite situation occurs. Thus ${\mathfrak{A}}_{\mathrm{ev}}$ restricted to $\mathcal{H}_{{\rm{R}}}$ or $\mathcal{H}_{{\rm{L}}}$ describes infraparticles, according to our terminology. Finally, ${\mathfrak{A}}_{\mathrm{ev}}$ restricted to $\hat{\mathcal{H}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}$ is a theory of infraparticles which does not contain ‘waves’. In Theorem 3.10 below, which is our main result, we establish superselection of direction of motion for infraparticles described by the net $\hat{\mathfrak{A}}={\mathfrak{A}}_{\mathrm{ev}}|_{\hat{\mathcal{H}}}$. The argument proceeds as follows: ${\mathfrak{A}}_{\mathrm{ev}}$ is contained in ${\mathfrak{A}}$, which is an asymptotically complete theory of Wigner particles. Thus we can use the scattering theory from [5] to compute the asymptotic functionals (1.1) and obtain the decompositions (1.2) of their GNS representations. Interpreted as a state on ${\mathfrak{A}}$, any vector $\Psi_{1}\otimes\Psi_{2}\in\hat{\mathcal{H}}$ consists of two ‘waves’ at asymptotic times: $\Psi_{1}\otimes\Omega_{2}$ travelling to the right and $\Omega_{1}\otimes\Psi_{2}$ travelling to the left. (Cf. Theorem 3.3 below). However, these two vectors belong to different invariant subspaces of ${\mathfrak{A}}_{\mathrm{ev}}$, namely to $\mathcal{H}_{{\rm{R}}}$ and $\mathcal{H}_{{\rm{L}}}$. The corresponding representations of $\hat{\mathfrak{A}}$ are not unitarily equivalent, since they have different structure of the energy-momentum spectrum. Our paper is organized as follows: Section 2, which does not rely on conformal symmetry, concerns two-dimensional, massless quantum field theories and their particle aspects: Preliminary Subsection 2.1 introduces the main concepts. In Subsection 2.2 we recall the scattering theory of two-dimensional, massless Wigner particles developed in [5]. Subsection 2.3 gives a brief exposition of the theory of particle weights and introduces our main concept: superselection of direction of motion. Subsection 2.4 presents our main technical result, stated in Theorem 2.11, which clarifies the structure of asymptotic functionals in theories of Wigner particles. Its proof is given in Appendix A. In Section 3 we apply the concepts and tools presented in Section 2 to chiral conformal field theories. Our setting, which is slightly more general than the usual framework of conformal field theory, is presented in Subsection 3.1. In Subsection 3.2 we show that any chiral theory in a vacuum representation has a complete particle interpretation in terms of Wigner particles. In Subsection 3.3 we demonstrate that charged irreducible product representations of any chiral theory describe infraparticles. Subsection 3.4 presents our main result, that is superselection of the infraparticle’s direction of motion in chiral theories arising from fixed-point nets of $\mathbb{Z}_{2}$ actions. Proofs of some auxiliary lemmas are postponed to Appendix B. In Section 4 we summarize our work and discuss future directions. ## 2 Particle aspects of two-dimensional massless theories ### 2.1 Preliminaries In this section, which does not rely on conformal symmetry, we present some general results on particle aspects of massless quantum field theories in two- dimensional spacetime. We rely on the following variant of the Haag-Kastler axioms [32]: ###### Definition 2.1. A local net of $C^{*}$-algebras on $\mathbb{R}^{2}$ is a pair $({\mathfrak{A}},U)$ consisting of a map $\mathcal{O}\to{\mathfrak{A}}(\mathcal{O})$ from the family of open, bounded regions of $\mathbb{R}^{2}$ to the family of $C^{*}$-algebras on a Hilbert space $\mathcal{H}$, and a strongly continuous unitary representation of translations $\mathbb{R}^{2}\ni x\to U(x)$ acting on $\mathcal{H}$, which are subject to the following conditions: 1. 1. (isotony) If $\mathcal{O}_{1}\subset\mathcal{O}_{2}$, then ${\mathfrak{A}}(\mathcal{O}_{1})\subset{\mathfrak{A}}(\mathcal{O}_{2})$. 2. 2. (locality) If $\mathcal{O}_{1}\perp\mathcal{O}_{2}$, then $[{\mathfrak{A}}(\mathcal{O}_{1}),{\mathfrak{A}}(\mathcal{O}_{2})]=0$, where $\perp$ denotes spacelike separation. 3. 3. (covariance) $U(x){\mathfrak{A}}(\mathcal{O})U(x)^{*}={\mathfrak{A}}(\mathcal{O}+x)$ for any $x\in\mathbb{R}^{2}$. 4. 4. (positivity of energy) The spectrum of $U$ is contained in the closed forward lightcone $V_{+}:=\\{\,(\omega,\boldsymbol{p})\in\mathbb{R}^{2}\,|\,\omega\geq|\boldsymbol{p}|\,\\}$. 5. 5. (regularity) The group of translation automorphisms $\alpha_{x}(\,\cdot\,)=U(x)\,\cdot\,U(x)^{*}$ satisfies $\lim_{x\to 0}\|\alpha_{x}(A)-A\|=0$ for any $A\in{\mathfrak{A}}$. We also introduce the quasilocal $C^{*}$-algebra of this net ${\mathfrak{A}}=\overline{\bigcup_{\mathcal{O}\subset\mathbb{R}^{2}}{\mathfrak{A}}(\mathcal{O})}$. For any given net $({\mathfrak{A}},U)$ there exists exactly one unitary representation of translations $U^{\mathrm{can}}$ s.t. $U^{\mathrm{can}}$ implements $\alpha$, all the operators $U^{\mathrm{can}}(x)$, $x\in\mathbb{R}^{2}$ are contained in ${\mathfrak{A}}^{\prime\prime}$, the spectrum of $U^{\mathrm{can}}$ is contained in $V_{+}$ and has Lorentz invariant lower boundary upon restriction to any subspace of $\mathcal{H}$ invariant under the action of ${\mathfrak{A}}^{\prime\prime}$ [4]. We assume that this _canonical_ representation of translations has been selected above, i.e. $U=U^{\mathrm{can}}$. We denote by $(H,\boldsymbol{P})$ the corresponding energy-momentum operators i.e. $U(x)=e^{iHt-i\boldsymbol{P}\boldsymbol{x}}$, $x=(t,\boldsymbol{x})$. As we are interested in scattering of massless particles, we introduce the single-particle subspaces ${\cal H}_{\pm}:=\ker(H\mp\boldsymbol{P})$ and denote the corresponding projections by $P_{\pm}$. The intersection $\mathcal{H}_{+}\cap\mathcal{H}_{-}$ contains only translationally invariant vectors. If $\mathcal{H}_{+}\neq\mathcal{H}_{+}\cap\mathcal{H}_{-}$ and $\mathcal{H}_{-}\neq\mathcal{H}_{+}\cap\mathcal{H}_{-}$ then we say that the theory describes Wigner particles. If $U$ has a unique (up to a phase) invariant unit vector $\Omega\in\mathcal{H}$ and $\Omega$ is cyclic under the action of ${\mathfrak{A}}$ then we say that the net $({\mathfrak{A}},U)$ is in a vacuum representation. In this case ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$. (Cf. Theorem 4.6 of [1]). Scattering theory for Wigner particles in a vacuum representation, developed in [5], will be recalled in Subsection 2.2. In the absence of Wigner particles we will apply the theory of particle weights [10, 16, 41, 42], outlined in Subsection 2.3, to extract the (infra-)particle content of a given theory. In this context it is necessary to consider various representations of the net $({\mathfrak{A}},U)$. A representation of the net $({\mathfrak{A}},U)$ is, by definition, a family of representations $\\{\pi_{\cal O}\\}$ of local algebras which are consistent in the sense that if ${\cal O}_{1}\subset{\cal O}_{2}$ then it holds that $\pi_{{\cal O}_{2}}|_{{\mathfrak{A}}({\cal O}_{1})}=\pi_{{\cal O}_{1}}$. Since the family of open bounded regions in ${\mathbb{R}}^{2}$ is directed, this representation uniquely extends to a representation $\pi$ of the quasilocal $C^{*}$-algebra ${\mathfrak{A}}$. Conversely, a representation of ${\mathfrak{A}}$ induces a consistent family of representations of local algebras. In the following $\pi$ may refer to a representation of ${\mathfrak{A}}$ or a family of representations. We say that a representation $\pi:{\mathfrak{A}}\to B(\mathcal{H}_{\pi})$ is covariant, if there exists a strongly continuous group of unitaries $U_{\pi}$ on $\mathcal{H}_{\pi}$, s.t. $\pi(\alpha_{x}(A))=U_{\pi}(x)\pi(A)U_{\pi}(x)^{*},\quad A\in{\mathfrak{A}},\,\,x\in\mathbb{R}^{2}.$ (2.1) Moreover, we say that this representation has positive energy, if the joint spectrum of the generators of $U_{\pi}$ is contained in $V_{+}+q$ for some $q\in\mathbb{R}^{2}$. We denote the corresponding canonical representation of translations by $U_{\pi}^{\mathrm{can}}$ and note that $(\pi({\mathfrak{A}}),U^{\mathrm{can}}_{\pi})$ is again a local net of $C^{*}$-algebras in the sense of Definition 2.1. We say that the net $(\pi({\mathfrak{A}}),U^{\mathrm{can}}_{\pi})$ is in a charged irreducible representation, if $\pi({\mathfrak{A}})$ acts irreducibly on a non-trivial Hilbert space $\mathcal{H}_{\pi}$ which does not contain non-zero invariant vectors of $U^{\mathrm{can}}_{\pi}$. We call two representations $(\pi_{1},\mathcal{H}_{\pi_{1}})$ and $(\pi_{2},\mathcal{H}_{\pi_{2}})$ of $({\mathfrak{A}},U)$ unitarily equivalent, (in short $(\pi_{1},\mathcal{H}_{\pi_{1}})\simeq(\pi_{2},\mathcal{H}_{\pi_{2}})$), if there exists a unitary $W:\mathcal{H}_{\pi_{1}}\to\mathcal{H}_{\pi_{2}}$ s.t. $\displaystyle W\pi_{1}(A)=\pi_{2}(A)W,\quad\phantom{444}A\in{\mathfrak{A}}.$ (2.2) If $\pi_{1}$ is a covariant, positive energy representation then so is $\pi_{2}$ and it is easy to see that $WU_{\pi_{1}}^{\mathrm{can}}(x)=U_{\pi_{2}}^{\mathrm{can}}(x)W,\quad x\in\mathbb{R}^{2}.$ (2.3) ###### Remark 2.2. We note that our (non-standard) Definition 2.1 of the local net neither imposes the Poincaré covariance nor the existence of the vacuum vector. Thus it applies both to vacuum representations and charged representations, which facilitates our discussion. Apart from the physically motivated assumptions, we adopt the regularity property 5, which can always be assured at the cost of proceeding to a weakly dense subnet. This property seems indispensable in the general theory of particle weights [41] e.g. in the proof of Proposition 2.10 stated below. For consistency of the presentation, we proceed to regular subnets also in our discussion of conformal field theories in Section 3. We stress, however, that this property is not needed there at the technical level. ### 2.2 Scattering states Scattering theory for Wigner particles in a vacuum representation of a two- dimensional massless theory $({\mathfrak{A}},U)$ was developed in [5]. For the reader’s convenience we recall here the main steps of this construction. Following [5], for any $F\in{\mathfrak{A}}$ and $T\geq 1$ we introduce the approximants: $\displaystyle F_{\pm}(h_{T})=\int h_{T}(t)F(t,\pm t)dt,$ (2.4) where $F(x):=\alpha_{x}(F)$, $h_{T}(t)=|T|^{-\varepsilon}h(|T|^{-\varepsilon}(t-T))$, $0<\varepsilon<1$ and $h\in C_{0}^{\infty}(\mathbb{R})$ is a non-negative function s.t. $\int dt\,h(t)=1$. By applying the mean ergodic theorem, one obtains $\lim_{T\to\infty}F_{\pm}(h_{T})\Omega=P_{\pm}F\Omega.$ (2.5) Moreover, for $F\in{\mathfrak{A}}(\mathcal{O})$ and sufficiently large $T$ the operator $F_{+}(h_{T})$ (resp. $F_{-}(h_{T})$) commutes with any observable localized in the left (resp. right) component of the spacelike complement of $\mathcal{O}$. Exploiting these two facts, the following result was established in [5]: ###### Proposition 2.3 ([5]). Let $F,G\in{\mathfrak{A}}$. Then the limits $\displaystyle\Phi_{\pm}^{\mathrm{out}}(F):=\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;F_{\pm}(h_{T})\quad$ (2.6) exist and are called the (outgoing) asymptotic fields. They depend only on the respective vectors $\Phi_{\pm}^{\mathrm{out}}(F)\Omega=P_{\pm}F\Omega$ and satisfy: 1. (a) $\Phi_{+}^{\mathrm{out}}(F)\mathcal{H}_{+}\subset\mathcal{H}_{+},\quad\Phi_{-}^{\mathrm{out}}(G)\mathcal{H}_{-}\subset\mathcal{H}_{-}$. 2. (b) $\alpha_{x}(\Phi_{+}^{\mathrm{out}}(F))=\Phi_{+}^{\mathrm{out}}(\alpha_{x}(F)),\quad\alpha_{x}(\Phi_{-}^{\mathrm{out}}(G))=\Phi_{-}^{\mathrm{out}}(\alpha_{x}(G))$ for $x\in{\mathbb{R}}^{2}$. 3. (c) $[\Phi_{+}^{\mathrm{out}}(F),\Phi_{-}^{\mathrm{out}}(G)]=0$. The incoming asymptotic fields $\Phi_{\pm}^{\mathrm{in}}(F)$ are constructed analogously, by taking the limit $T\to-\infty$. With the help of the asymptotic fields one defines the scattering states as follows: Since ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$, for any $\Psi_{\pm}\in\mathcal{H}_{\pm}$ we can find $F_{\pm}\in{\mathfrak{A}}$ s.t. $\Psi_{\pm}=F_{\pm}\Omega$ [45]. The vectors $\displaystyle\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}=\Phi_{+}^{\mathrm{out}}(F_{+})\Phi_{-}^{\mathrm{out}}(F_{-})\Omega$ (2.7) are called the (outgoing) scattering states. By Proposition 2.3 they do not depend on the choice of $F_{\pm}$ within the above restrictions. The incoming scattering states $\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}$ are defined analogously. The physical interpretation of these vectors, as two independent excitations travelling in opposite directions at asymptotic times, relies on the following proposition from [5]: ###### Proposition 2.4 ([5]). Let $\Psi_{\pm},\Psi_{\pm}^{\prime}\in\mathcal{H}_{\pm}$. Then: 1. (a) $(\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}|\Psi^{\prime}_{+}\overset{\mathrm{out}}{\times}\Psi^{\prime}_{-})=(\Psi_{+}|\Psi^{\prime}_{+})(\Psi_{-}|\Psi^{\prime}_{-})$, 2. (b) $U(x)(\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-})=(U(x)\Psi_{+})\overset{\mathrm{out}}{\times}(U(x)\Psi_{-})$, for $x\in\mathbb{R}^{2}$. Analogous relations hold for the incoming scattering states. Following [5], we define the subspaces spanned by the respective scattering states: $\mathcal{H}^{\mathrm{in}}=\mathcal{H}_{+}\overset{\mathrm{in}}{\times}\mathcal{H}_{-}\,\,\textrm{ and }\,\,\mathcal{H}^{\mathrm{out}}=\mathcal{H}_{+}\overset{\mathrm{out}}{\times}\mathcal{H}_{-}.$ (2.8) Next, we introduce the wave operators $\Omega^{\mathrm{out}}:\mathcal{H}_{+}\otimes\mathcal{H}_{-}\to\mathcal{H}^{\mathrm{out}}$ and $\Omega^{\mathrm{in}}:\mathcal{H}_{+}\otimes\mathcal{H}_{-}\to\mathcal{H}^{\mathrm{in}}$, extending by linearity the relations $\Omega^{\mathrm{out}}(\Psi_{+}\otimes\Psi_{-})=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}\,\,\textrm{ and }\,\,\Omega^{\mathrm{in}}(\Psi_{+}\otimes\Psi_{-})=\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}.$ (2.9) These operators are isometric in view of Proposition 2.4 (a). The scattering operator $S:\mathcal{H}^{\mathrm{out}}\to\mathcal{H}^{\mathrm{in}}$, given by $S=\Omega^{\mathrm{in}}(\Omega^{\mathrm{out}})^{*},$ (2.10) is also an isometry. Now we are ready to introduce two important concepts: ###### Definition 2.5. (a) If $S=I$ on $\mathcal{H}^{\mathrm{out}}$, then we say that the theory is non-interacting. (b) If $\mathcal{H}^{\mathrm{in}}=\mathcal{H}^{\mathrm{out}}=\mathcal{H}$ then we say that the theory is asymptotically complete (in terms of ‘waves’). We show in Theorem 3.3 below that any chiral conformal field theory in a vacuum representation is both non-interacting and asymptotically complete. (We demonstrated these facts already in [25] in a different context). To conclude this subsection, we introduce some other useful concepts which are needed in Theorem 2.11 below: Let us choose some closed subspaces ${\cal K}_{\pm}\subset\mathcal{H}_{\pm}$, invariant under the action of $U$, and denote by ${\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-}$ the linear span of the respective scattering states. For any $\Psi\in{\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-}\subset\mathcal{H}^{\mathrm{out}}$ we introduce the positive functionals $\rho_{\pm,\Psi}$, given by the relations $\displaystyle\rho_{+,\Psi}(A)$ $\displaystyle=$ $\displaystyle((\Omega^{\mathrm{out}})^{-1}\Psi|(A\otimes I)(\Omega^{\mathrm{out}})^{-1}\Psi),$ (2.11) $\displaystyle\rho_{-,\Psi}(A)$ $\displaystyle=$ $\displaystyle((\Omega^{\mathrm{out}})^{-1}\Psi|(I\otimes A)(\Omega^{\mathrm{out}})^{-1}\Psi),$ (2.12) where $A\in B(\mathcal{H})$ and the embedding ${\cal K}_{+}\otimes{\cal K}_{-}\subset\mathcal{H}\otimes\mathcal{H}$ is understood. These functionals can be expressed as follows $\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n\in\mathbb{N}}(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n}),$ (2.13) where $\Psi_{\pm,n}\in{\cal K}_{\pm}$ and $\sum_{n\in\mathbb{N}}\|\Psi_{\pm,n}\|^{2}=\|\Psi\|^{2}$. It follows easily from Lemma A.2, that for $\Psi\in P_{E}({\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$, where $P_{E}$ is the spectral projection on vectors of energy not larger than $E$, one can choose $\Psi_{\pm,n}\in P_{E}{\cal K}_{\pm}$. We note that for $\|\Psi\|=1$ the functionals $\rho_{\pm,\Psi}$ are just the familiar reduced density matrices. ### 2.3 Particle weights Similarly as in the previous subsection we consider a local net of $C^{*}$-algebras $({\mathfrak{A}},U)$ acting on a Hilbert space $\mathcal{H}$. However, we do not assume that $\mathcal{H}$ contains the vacuum vector or non-trivial single-particle subspaces $\mathcal{H}_{\pm}$. To study particle aspects in this general situation we use the theory of particle weights [10, 16, 41, 42] which we recall in this and the next subsection. With the help of this theory we formulate in Definitions 2.7 and 2.12 below the central notion of this paper: superselection of direction of motion. First, we recall two useful concepts: almost locality and the energy decreasing property. An observable $B\in{\mathfrak{A}}$ is called almost local, if there exists a net of operators $\\{\,B_{r}\in{\mathfrak{A}}(\mathcal{O}_{r})\,|\,r>0\,\\}$, s.t. for any $k\in\mathbb{N}_{0}$ $\lim_{r\to\infty}r^{k}\|B-B_{r}\|=0,$ (2.14) where $\mathcal{O}_{r}=\\{(t,\boldsymbol{x})\in\mathbb{R}^{2}\,|\,|t|+|\boldsymbol{x}|<r\,\\}$. We say that an operator $B\in{\mathfrak{A}}$ is energy decreasing, if its energy-momentum transfer is a compact set which does not intersect with the closed forward lightcone $V_{+}$. We recall that the energy-momentum transfer (or the Arveson spectrum w.r.t. $\alpha$) of an observable $B\in{\mathfrak{A}}$ is the closure of the union of supports of the distributions $(\Psi_{1}|\tilde{B}(p)\Psi_{2})=(2\pi)^{-1}\int d^{2}x\,e^{-ipx}(\Psi_{1}|B(x)\Psi_{2})$ (2.15) over all $\Psi_{1},\Psi_{2}\in\mathcal{H}$, where $p=(\omega,\boldsymbol{p})$, $x=(t,\boldsymbol{x})$ and $px=\omega t-\boldsymbol{p}\boldsymbol{x}$. Following [16, 41], we introduce the subspace $\mathcal{L}_{0}\subset{\mathfrak{A}}$, spanned by operators which are both almost local and energy decreasing, and the corresponding left ideal in ${\mathfrak{A}}$: $\mathcal{L}:=\\{\,AB\,|\,A\in{\mathfrak{A}},B\in\mathcal{L}_{0}\,\\}.$ (2.16) Particle weights form a specific class of sesquilinear forms on $\mathcal{L}$: ###### Definition 2.6. A particle weight is a non-zero, positive sesquilinear form $\psi$ on the left ideal $\mathcal{L}$, satisfying the following conditions: 1. 1. For any $L_{1},L_{2}\in\mathcal{L}$ and $A\in{\mathfrak{A}}$ the relation $\psi(AL_{1},L_{2})=\psi(L_{1},A^{*}L_{2})$ holds. 2. 2. For any $L_{1},L_{2}\in\mathcal{L}$ and $x\in\mathbb{R}^{2}$ the relation $\psi(\alpha_{x}(L_{1}),\alpha_{x}(L_{2}))=\psi(L_{1},L_{2})$ holds. 3. 3. For any $L_{1},L_{2}\in\mathcal{L}$ the map $\mathbb{R}^{2}\ni x\to\psi(L_{1},\alpha_{x}(L_{2}))$ is continuous. Its Fourier transform is supported in a shifted lightcone $V_{+}-q$, where $q\in V_{+}$ does not depend on $L_{1},L_{2}$. Let us now summarize the pertinent properties of particle weights established in [41] (in a slightly different framework). As a consequence of Theorem 2.9, stated below, particle weights satisfy the following clustering property [41] $\int d\boldsymbol{x}\,|\psi(L_{1},\alpha_{\boldsymbol{x}}(L_{2}))|<\infty,$ (2.17) valid for $L_{1}=B_{1}^{*}A_{1}B_{1}^{\prime}$, $L_{2}=B_{2}^{*}A_{2}B_{2}^{\prime}$, where $B_{1},B_{1}^{\prime},B_{2},B_{2}^{\prime}\in\mathcal{L}_{0}$ and $A_{1},A_{2}\in{\mathfrak{A}}$ are almost local. In view of this bound, the GNS representation $(\pi_{\psi},\mathcal{H}_{\pi_{\psi}})$ induced by a particle weight $\psi$ is well suited for a description of physical systems which are localized in space (e.g. configurations of particles). The Hilbert space $\mathcal{H}_{\pi_{\psi}}$ is given by $\mathcal{H}_{\pi_{\psi}}=(\,\mathcal{L}/\\{\,L\in\mathcal{L}\,|\,\psi(L,L)=0\\})^{\mathrm{cpl}}$ (2.18) and the respective equivalence class of an element $L\in\mathcal{L}$ is denoted by $|L\rangle\in\mathcal{H}_{\pi_{\psi}}$. The completion is taken w.r.t. the scalar product $\langle L_{1}|L_{2}\rangle:=\psi(L_{1},L_{2})$. The representation $\pi_{\psi}$ acts on $\mathcal{H}_{\pi_{\psi}}$ as follows $\pi_{\psi}(A)|L\rangle=|AL\rangle,\quad A\in{\mathfrak{A}}.$ (2.19) This representation is covariant and the translation automorphisms are implemented by the strongly continuous group of unitaries $U_{\pi_{\psi}}$, given by $U_{\pi_{\psi}}(x)|L\rangle=|\alpha_{x}(L)\rangle,\quad x\in\mathbb{R}^{2},\,L\in\mathcal{L}$ (2.20) which is called the standard representation of translations in the representation $\pi_{\psi}$. By property 3 in Definition 2.6 above, its spectrum is contained in a shifted closed forward lightcone. The corresponding canonical representation will be denoted by $U_{\pi_{\psi}}^{\mathrm{can}}$. (Cf. the discussion below Definition 2.1). We also introduce operators $(Q^{0},\boldsymbol{Q})$ of _characteristic energy-momentum_ of $\psi$ which are the generators of the following group of unitaries on $\mathcal{H}_{\pi_{\psi}}$ $U^{\mathrm{char}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{\psi}}(x)U_{\pi_{\psi}}(x)^{-1}\in\pi_{\psi}({\mathfrak{A}})^{\prime},$ (2.21) i.e. $U^{\mathrm{char}}_{\pi_{\psi}}(x)=e^{iQ^{0}t-i\boldsymbol{Q}\boldsymbol{x}}$. We call a particle weight _pure_ , if its GNS representation is irreducible. It follows from definition (2.21) that the operator of characteristic energy- momentum of such a weight is a vector $q=(q^{0},\boldsymbol{q})\in\mathbb{R}^{2}$. It can be interpreted as the energy and momentum of the plane wave configuration of the particle described by this weight [2, 41]. To extract properties of elementary subsystems (particles) of a physical system described by a given (possibly non-pure) particle weight, it is natural to study irreducible subrepresentations of its GNS representation. To ensure that there are sufficiently many such subrepresentations, we restrict attention to particle weights $\psi$ whose GNS representations $\pi_{\psi}$ are of type I with atomic center555i.e. whose center is a direct sum of one- dimensional von Neumann algebras.. (In particular, $\pi_{\psi}$ appearing in our examples in Subsection 3.4 below belong to this family). Then, by Theorem 1.31 from Chapter V of [49], there exists a unique family of Hilbert spaces $(\mathfrak{H}_{\alpha},\mathfrak{K}_{\alpha})_{\alpha\in\mathbb{I}}$ and a unitary $W:\mathcal{H}_{\pi_{\psi}}\to\bigoplus_{\alpha\in\mathbb{I}}\\{\mathfrak{H}_{\alpha}\otimes\mathfrak{K}_{\alpha}\\}$ s.t. $\displaystyle W\pi_{\psi}({\mathfrak{A}})^{\prime\prime}W^{-1}$ $\displaystyle=$ $\displaystyle\bigoplus_{\alpha\in\mathbb{I}}\\{B(\mathfrak{H}_{\alpha})\otimes{\mathbb{C}}I\\},$ (2.22) $\displaystyle W\pi_{\psi}({\mathfrak{A}})^{\prime}W^{-1}$ $\displaystyle=$ $\displaystyle\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes B(\mathfrak{K}_{\alpha})\\}.$ (2.23) We note that a subspace ${\cal K}_{\alpha,e}\subset\mathcal{H}_{\pi_{\psi}}$ carries an irreducible subrepresentation $\pi_{\alpha,e}$ of $\pi_{\psi}$, if and only if $W{\cal K}_{\alpha,e}=\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e$ for some $\alpha\in\mathbb{I}$ and $e\in\mathfrak{K}_{\alpha}$. Clearly, $\pi_{\alpha,e}$ and $\pi_{\alpha,e^{\prime}}$ are unitarily equivalent for any fixed $\alpha$ and arbitrary vectors $e,e^{\prime}\in\mathfrak{K}_{\alpha}$. Choosing in any $\mathfrak{K}_{\alpha}$ an orthonormal basis $B_{\alpha}$, we obtain $\pi_{\psi}=\bigoplus_{\begin{subarray}{c}\alpha\in\mathbb{I}\\\ e\in B_{\alpha}\end{subarray}}\pi_{\alpha,e}.$ (2.24) It is clear from the above discussion that any irreducible subrepresentation of $\pi_{\psi}$ is unitarily equivalent to some $\pi_{\alpha,e}$ in the decomposition above. If all the representations in the decomposition (2.24) are unitarily equivalent to some fixed vacuum representation, then we call the particle weight $\psi$ neutral. Otherwise we call $\psi$ charged. In the case of charged particle weights there may occur an interplay between the translational and internal degrees of freedom of the system which we call _superselection of direction of motion_. To introduce this concept, we need some terminology: Let $\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ (resp. $\mathcal{H}_{\pi_{\psi},\mathrm{L}}$) be the spectral subspace of the characteristic momentum operator $\boldsymbol{Q}$ of $\psi$, corresponding to the interval $[0,\infty)$ (resp. $(-\infty,0)$). Let $\pi$ be an irreducible subrepresentation of $\pi_{\psi}$, acting on a subspace ${\cal K}\subset\mathcal{H}_{\pi_{\psi}}$. Then we say that $\pi$ is _right-moving_ (resp. _left-moving_), if ${\cal K}\neq\\{0\\}$ and ${\cal K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ (resp. ${\cal K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{L}}$). By a suitable choice of the bases $B_{\alpha}$ one can ensure that each representation $\pi_{\alpha,e}$, appearing in decomposition (2.24), has one of these properties. (In fact, exploiting relations (2.21), (2.23), one can choose such basis vectors $e\in\mathfrak{K}_{\alpha}$ that $W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e)$ belong to $\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ or $\mathcal{H}_{\pi_{\psi},\mathrm{L}}$). After this preparation we define the central concept of the present paper: ###### Definition 2.7. Let $\mathcal{W}$ be a family of particle weights and assume that their GNS representations $\\{\,(\pi_{\psi},\mathcal{H}_{\pi_{\psi}})\,|\,\psi\in\mathcal{W}\,\\}$ are of type I with atomic centers. Suppose that for any $\psi,\psi^{\prime}\in\mathcal{W}$ the following properties hold: 1. 1. $\pi_{\psi}$ has both left-moving and right-moving irreducible subrepresentations. 2. 2. No right-moving, irreducible subrepresentation of $\pi_{\psi}$ is unitarily equivalent to a left-moving irreducible subrepresentation of $\pi_{\psi^{\prime}}$. Then we say that this family of particle weights has superselected direction of motion. Let us now relate superselection of direction of motion in the above sense to our discussion of this concept in the Introduction. For this purpose we consider a particle weight $\psi$, whose GNS representation is of type I with atomic center and acts on a _separable_ Hilbert space $\mathcal{H}_{\pi_{\psi}}$. Making use of formula (2.24) and identifying unitarily each $\pi_{\alpha,e}$, acting on $W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e)$, with $\pi_{\alpha}:=\pi_{\alpha,e_{0}}$ acting on ${\cal K}_{\alpha}:=W^{-1}(\mathfrak{H}_{\alpha}\otimes{\mathbb{C}}e_{0})$ for some chosen $e_{0}\in B_{\alpha}$, we obtain $\pi_{\psi}({\mathfrak{A}})\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{\pi_{\alpha}({\mathfrak{A}})\otimes{\mathbb{C}}I\\},$ (2.25) where the r.h.s. acts on $\bigoplus_{\alpha\in\mathbb{I}}\\{{\cal K}_{\alpha}\otimes\mathfrak{K}_{\alpha}\\}$. In the sense of the same identification $\pi_{\psi}({\mathfrak{A}})^{\prime}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes B(\mathfrak{K}_{\alpha})\\}.$ (2.26) Now, following [42], we choose a maximal abelian von Neumann algebra ${\cal M}$ in $\pi_{\psi}({\mathfrak{A}})^{\prime}$, containing $\\{\,U^{\mathrm{char}}_{\pi_{\psi}}(x)\,|\,x\in\mathbb{R}^{2}\,\\}$. As a consequence of formula (2.26) ${\cal M}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{{\mathbb{C}}I\otimes{\cal M}_{\alpha}\\},$ (2.27) where ${\cal M}_{\alpha}\subset B(\mathfrak{K}_{\alpha})$ are maximal abelian von Neumann subalgebras. For any such ${\cal M}_{\alpha}$ there exists a Borel space $(Z_{\alpha},d\mu_{\alpha})$ s.t. $({\cal M}_{\alpha},\mathfrak{K}_{\alpha})\simeq(L^{\infty}(Z_{\alpha},d\mu_{\alpha}),L^{2}(Z_{\alpha},d\mu_{\alpha}))$. (This fact uses separability of the Hilbert space. See Theorem II.2.2 of [20]). Adopting this identification in (2.25) and (2.26), we obtain $U^{\mathrm{char}}_{\pi_{\psi}}\simeq\bigoplus_{\alpha\in\mathbb{I}}\\{I\otimes U^{\mathrm{char}}_{\alpha}\\},$ (2.28) where $U^{\mathrm{char}}_{\alpha}(x)\in L^{\infty}(Z_{\alpha},d\mu_{\alpha})$ is the operator of multiplication by (the equivalence class of) the function $Z_{\alpha}\ni z\to e^{iq_{\alpha,z}x}$, where $q_{\alpha,z}=(q_{\alpha,z}^{0},\boldsymbol{q}_{\alpha,z})\in\mathbb{R}^{2}$. Introducing the field of representations $(\pi_{\alpha,z},\mathfrak{H}_{\alpha,z})_{z\in Z_{\alpha}}$ s.t. $\pi_{\alpha,z}=\pi_{\alpha}$ and $\mathfrak{H}_{\alpha,z}={\cal K}_{\alpha}$ for all $z\in Z_{\alpha}$, we obtain from relation (2.25) the existence of a unitary $\tilde{W}:\mathcal{H}_{\pi_{\psi}}\to\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}d\mu_{\alpha}(z)\,\mathfrak{H}_{\alpha,z}$ s.t. $\tilde{W}\pi_{\psi}(\,\cdot\,)\tilde{W}^{-1}=\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}_{Z_{\alpha}}d\mu_{\alpha}(z)\,\pi_{\alpha,z}(\,\cdot\,).$ (2.29) This is an example of decomposition (1.2), stated in the Introduction. Moreover, as a consequence of (2.28), $\tilde{W}U^{\mathrm{char}}_{\pi_{\psi}}(x)\tilde{W}^{-1}=\bigoplus_{\alpha\in\mathbb{I}}\int^{\oplus}_{Z_{\alpha}}d\mu_{\alpha}(z)\,e^{iq_{\alpha,z}x},$ (2.30) where $\\{q_{\alpha,z}\\}_{z\in Z_{\alpha}}$ is the field of characteristic energy-momentum vectors666This terminology is consistent with the discussion after formula (2.21). In fact, under some technical restrictions each $\pi_{\alpha,z}$ is induced by some pure particle weight $\psi_{\alpha,z}$, whose characteristic energy-momentum vector is $q_{\alpha,z}$ [41, 42]. Cf. also formula (1.3). of the representations $(\pi_{\alpha,z},\mathfrak{H}_{\alpha,z})_{z\in Z_{\alpha}}$. As we required in the Introduction, for any particle weight with superselected direction of motion, the relation $\boldsymbol{q}_{\alpha,z}\cdot\boldsymbol{q}_{\alpha^{\prime},z^{\prime}}\leq 0$ should imply that $\pi_{\alpha,z}$ is not unitarily equivalent to $\pi_{\alpha^{\prime},z^{\prime}}$ for almost all $z,z^{\prime}$. This is in fact the case in view of the following proposition. ###### Proposition 2.8. Suppose that $\psi$ belongs to a family of particle weights which has superselected direction of motion in the sense of Definition 2.7 and s.t. its GNS representation acts on a separable Hilbert space. Then $\\{\pi_{\alpha,z}\\}_{z\in Z_{\alpha}}$, appearing in the decomposition (2.29) of $\pi_{\psi}$, is a field of right-moving (resp. left-moving) representations, if and only if $\boldsymbol{q}_{\alpha,z}\geq 0$ (resp. $\boldsymbol{q}_{\alpha,z}<0$) for almost all $z\in Z_{\alpha}$. Proof. Suppose that $\pi_{\alpha}$ is a right-moving subrepresentation of $\pi_{\psi}$ i.e. ${\cal K}_{\alpha}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$. We recall that $\pi_{\alpha}$ coincides with $\pi_{\alpha,e_{0}}$ acting on ${\cal K}_{\alpha,e_{0}}={\cal K}_{\alpha}$. Since every $\pi_{\alpha,e}$, $e\in\mathfrak{K}$, is unitarily equivalent to $\pi_{\alpha,e_{0}}$, the property of superselection of direction of motion implies that ${\cal K}_{\alpha,e}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ for all $e\in\mathfrak{K}$. Consequently, $\mathcal{H}_{\alpha}=W^{-1}(\mathfrak{H}_{\alpha}\otimes\mathfrak{K}_{\alpha})\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$. Since the projection $P_{\alpha}$ on $\mathcal{H}_{\alpha}$ is central, this subspace is invariant under the action of $U^{\mathrm{char}}_{\pi_{\psi}}$. Formula (2.28) gives $U^{\mathrm{char}}_{\pi_{\psi}}(x)P_{\alpha}\simeq I\otimes U^{\mathrm{char}}_{\alpha}(x),$ (2.31) thus the spectra of the generators of $\mathbb{R}^{2}\ni x\to U^{\mathrm{char}}_{\alpha}(x)$ and $\mathbb{R}^{2}\ni x\to U^{\mathrm{char}}_{\pi_{\psi}}(x)P_{\alpha}$ coincide. In particular the spectrum of the generator of space translations of $U^{\mathrm{char}}_{\alpha}$ is contained in $[0,\infty)$. The opposite implication follows immediately from relation (2.31). $\Box$ ### 2.4 Asymptotic functionals In this subsection we consider a concrete class of particle weights, introduced in [9, 10, 41], which have applications in scattering theory. Their construction relies on the following result due to Buchholz (which remains valid in higher dimensions). ###### Theorem 2.9 ([10]). Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on $\mathbb{R}^{2}$. Then, for any $E\geq 0$, $L\in\mathcal{L}$, $\left\|P_{E}\int_{K}d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})P_{E}\right\|\leq c,$ (2.32) where $P_{E}$ is the spectral projection on vectors of energy not larger than $E$, $K\subset\mathbb{R}$ is a compact interval, and $c$ is a constant independent of $K$. Following [41], we introduce the algebra of detectors $\mathcal{C}=\mathrm{span}\\{L_{1}^{*}L_{2}:L_{1},L_{2}\in\mathcal{L}\\}$ and equip it with a locally convex topology, given by the family of seminorms $p_{E}(C)=\sup\left\\{\int d\boldsymbol{x}\,|(\Psi|C(\boldsymbol{x})\Psi)|\,\,|\,\,\Psi\in P_{E}\mathcal{H},\,\|\Psi\|\leq 1\,\right\\},\,\quad C\in\mathcal{C},$ (2.33) labelled by $E\geq 0$, which are finite by Theorem 2.9. Next, for any $\Psi\in\mathcal{H}$ of bounded energy, (i.e. belonging to $P_{E}\mathcal{H}$ for some $E\geq 0$), we define a sequence of functionals $\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}}$ from the topological dual of $\mathcal{C}$: $\sigma_{\Psi}^{(T)}(C):=\int dt\,h_{T}(t)\int d\boldsymbol{x}\,(\Psi|C(t,\boldsymbol{x})\Psi),\quad C\in\mathcal{C}.$ (2.34) As this sequence is uniformly bounded in $T$ w.r.t. any seminorm $p_{E}$, the Alaoglu-Bourbaki theorem gives limit points $\sigma_{\Psi}^{\mathrm{out}}\in\mathcal{C}^{*}$ as $T\to\infty$, which are called the asymptotic functionals. The following fact was shown in [41]: ###### Proposition 2.10 ([41]). If $\sigma_{\Psi}^{\mathrm{out}}\neq 0$, then the sesquilinear forms on $\mathcal{L}$, given by $\psi_{\Psi}^{\mathrm{out}}(L_{1},L_{2}):=\sigma_{\Psi}^{\mathrm{out}}(L_{1}^{*}L_{2}),$ (2.35) are particle weights, in the sense of Definition 2.6. Fundamental results from [2] suggest a physical interpretation of the particle weights $\psi_{\Psi}^{\mathrm{out}}$ as mixtures of plane wave configurations of all the particle types described by the theory. (Cf. formulas (1.2), (1.3)). Accordingly, we say that a given theory has a non-trivial _particle content_ , if it admits some non-zero asymptotic functionals $\sigma_{\Psi}^{\mathrm{out}}$. This is the case in any massless two- dimensional theory of Wigner particles (in a vacuum representation) as a consequence of the following theorem. A proof of this statement, which is our main technical result, is given in Appendix A. ###### Theorem 2.11. Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on $\mathbb{R}^{2}$ in a vacuum representation, acting on a Hilbert space $\mathcal{H}$. Then, for any $\Psi\in P_{E}\mathcal{H}^{\mathrm{out}}$, $E\geq 0$, $\displaystyle\psi_{\Psi}^{\mathrm{out}}(L_{1},L_{2})$ $\displaystyle=$ $\displaystyle\lim_{T\to\infty}\int dt\,h_{T}(t)\int d\boldsymbol{x}\,(\Psi|(L_{1}^{*}L_{2})(t,\boldsymbol{x})\Psi)$ (2.36) $\displaystyle=$ $\displaystyle\int d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L_{1}^{*}L_{2})(\boldsymbol{x})\big{)},$ where the functionals $\rho_{\pm,\Psi}$ are defined by (2.11), (2.12). In particular, $\psi_{\Psi}^{\mathrm{out}}=0$, if and only if $\Psi\in{\mathbb{C}}\Omega$. In a theory of Wigner particles ${\mathbb{C}}\Omega\neq\mathcal{H}_{\pm}\subset\mathcal{H}^{\mathrm{out}}$, thus the particle content is non-trivial by the above result. However, non- zero asymptotic functionals may also appear in the absence of Wigner particles i.e. when one or both of the subspaces $\mathcal{H}_{\pm}$ equal $\mathcal{H}_{+}\cap\mathcal{H}_{-}$. If this is the case, then we say that the net $({\mathfrak{A}},U)$ describes _infraparticles_. Theorem 3.6 below provides a large class of such theories. In Theorem 3.10 we show that some of these models describe excitations whose direction of motion is superselected in the following sense: ###### Definition 2.12. Let $({\mathfrak{A}},U)$ be a net describing infraparticles. We say that the infraparticles of the net $({\mathfrak{A}},U)$ have superselected direction of motion, if $\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in P_{E}\mathcal{H},\,E\geq 0\,\\}$ is a family of particle weights with superselected direction of motion in the sense of Definition 2.7. ## 3 Particle aspects of conformal field theories ### 3.1 Preliminaries In this section we are interested in particle aspects of chiral conformal field theories. To emphasize the relevant properties of these models, we base our investigation on the concept of a local net of von Neumann algebras on ${\mathbb{R}}$, defined below. There are many examples of such nets. In particular, they arise from Möbius covariant nets on $S^{1}$ by means of the Cayley transform and the subsequent restriction to the real line. The simplest example is the so-called $U(1)$-current net [15], whose subnets and extensions are well-studied. For certain classes of nets on $S^{1}$ even classification results have been obtained [36, 37]. ###### Definition 3.1. A local net of von Neumann algebras on $\mathbb{R}$ is a pair $({\cal A},V)$ consisting of a map ${\cal I}\to{\cal A}({\cal I})$ from the family of open, bounded subsets of $\mathbb{R}$ to the family of von Neumann algebras on a Hilbert space $\mathcal{K}$ and a strongly continuous unitary representation of translations $\mathbb{R}\ni s\to V(s)$, acting on $\mathcal{K}$, which are subject to the following conditions: 1. 1. (isotony) If ${\cal I}\subset\mathfrak{J}$, then ${\cal A}({\cal I})\subset{\cal A}(\mathfrak{J})$. 2. 2. (locality) If ${\cal I}\cap\mathfrak{J}=\varnothing$, then $[{\cal A}({\cal I}),{\cal A}(\mathfrak{J})]=0$. 3. 3. (covariance) $V(s){\cal A}({\cal I})V(s)^{*}={\cal A}({\cal I}+s)$ for any $s\in\mathbb{R}$. 4. 4. (positivity of energy) The spectrum of $V$ coincides with $\mathbb{R}_{+}$. We also denote by ${\cal A}$ the quasilocal $C^{*}$-algebra of this net i.e. ${\cal A}=\overline{\bigcup_{{\cal I}\subset\mathbb{R}}{\cal A}({\cal I})}$. Since we assumed that ${\cal A}({\cal I})$ are von Neumann algebras, we cannot demand norm continuity of the functions $s\to\beta_{s}(A)$, $A\in{\cal A}$, where $\beta_{s}(\,\cdot\,)=V(s)\,\cdot\,V(s)^{*}$. This regularity property holds, however, on the following weakly dense subnet of $C^{*}$-algebras ${\cal I}\to\bar{\cal A}({\cal I}):=\\{\,A\in{\cal A}({\cal I})\,|\,\lim_{s\to 0}\|\beta_{s}(A)-A\|=0\,\\}.$ (3.1) The corresponding quasilocal algebra is denoted by $\bar{\cal A}$. If $V$ has a unique (up to a phase) invariant, unit vector $\Omega_{0}\in\mathcal{K}$ and $\Omega_{0}$ is cyclic under the action of any ${\cal A}({\cal I})$ (the Reeh-Schlieder property) then we say that the net $({\cal A},V)$ is in a vacuum representation. In this case ${\cal A}$ acts irreducibly on $\mathcal{K}$. In the course of our analysis we will also consider other representations of $({\cal A},V)$. We say that a representation $\pi:{\cal A}\to B(\mathcal{K}_{\pi})$ is covariant, if there exists a strongly continuous group of unitaries $V_{\pi}$ on $\mathcal{K}_{\pi}$, s.t. $\pi(\alpha_{s}(A))=V_{\pi}(s)\pi(A)V_{\pi}(s)^{*},\quad A\in{\cal A},\,\,s\in\mathbb{R}.$ (3.2) Moreover, we say that this representation has positive energy, if the spectrum of $V_{\pi}$ coincides with $\mathbb{R}_{+}$. If $\pi$ is locally normal (i.e. its restriction to any local algebra ${\cal A}({\cal I})$ is normal) then $(\pi({\cal A}),V_{\pi})$ is again a net of von Neumann algebras in the sense of Definition 3.1. Let $({\cal A}_{1},V_{1})$ and $({\cal A}_{2},V_{2})$ be two nets of von Neumann algebras on ${\mathbb{R}}$, acting on Hilbert spaces $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$. To construct a local net $({\mathfrak{A}},U)$ on ${\mathbb{R}}^{2}$, acting on the tensor product space ${\cal H}={\cal K}_{1}\otimes{\cal K}_{2}$, we identify the two real lines with the lightlines $I_{\pm}=\\{\,(t,\boldsymbol{x})\in\mathbb{R}^{2}\,|\,\boldsymbol{x}\mp t=0\,\\}$ in ${\mathbb{R}}^{2}$. We first specify the unitary representation of translations $U(t,\boldsymbol{x}):=V_{1}\left(\frac{1}{\sqrt{2}}(t-\boldsymbol{x})\right)\otimes V_{2}\left(\frac{1}{\sqrt{2}}(t+\boldsymbol{x})\right),$ (3.3) whose spectrum is easily seen to coincide with $V_{+}$ as a consequence of property 4 from Definition 3.1. We mention for future reference that if $\alpha_{(t,\boldsymbol{x})}(\,\cdot\,):=U(t,\boldsymbol{x})\,\,\cdot\,\,U(t,\boldsymbol{x})^{*}$ is the corresponding group of translation automorphisms and $\beta^{(1/2)}_{s}(\,\cdot\,):=V_{1/2}(s)\,\,\cdot\,\,V_{1/2}(s)^{*}$, then $\alpha_{(t,\boldsymbol{x})}(A_{1}\otimes A_{2})=\beta^{(1)}_{\frac{1}{\sqrt{2}}(t-\boldsymbol{x})}(A_{1})\otimes\beta^{(2)}_{\frac{1}{\sqrt{2}}(t+\boldsymbol{x})}(A_{2}),\quad A_{1}\in{\cal A}_{1},\,A_{2}\in{\cal A}_{2}.$ (3.4) Any double cone $D\subset{\mathbb{R}}^{2}$ can be expressed as a product of intervals on lightlines $D={\cal I}_{1}\times{\cal I}_{2}$. We define the corresponding local von Neumann algebra by ${\mathfrak{A}}^{\textrm{vN}}(D):={\cal A}_{1}({\cal I}_{1})\otimes{\cal A}_{2}({\cal I}_{2})$, and for a general open region $\mathcal{O}$ we put ${\mathfrak{A}}^{\textrm{vN}}(\mathcal{O})=\bigvee_{D\subset\mathcal{O}}{\mathfrak{A}}^{\textrm{vN}}(D)$. The net of von Neumann algebras $({\mathfrak{A}}^{\textrm{vN}},U)$, which we call the chiral net, satisfies all the properties from Definition 2.1 except for the regularity property 5. Therefore, we introduce the following weakly dense subnet of $C^{*}$-algebras $\mathcal{O}\to{\mathfrak{A}}(\mathcal{O}):=\\{\,A\in{\mathfrak{A}}^{\textrm{vN}}(\mathcal{O})\,|\,\lim_{x\to 0}\|\alpha_{x}(A)-A\|=0\,\\},$ (3.5) and denote the corresponding quasilocal algebra by ${\mathfrak{A}}$. Then $({\mathfrak{A}},U)$ is a local net of $C^{*}$-algebras in the sense of Definition 2.1. We will call it the regular chiral net and refer to $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ as its chiral components. We note for future reference that if ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$, then $U$ is automatically the canonical representation of translations of this net (cf. Subsection 2.1). Another useful fact is the obvious inclusion $\bar{\cal A}_{1}\otimes_{\mathrm{alg}}\bar{\cal A}_{2}\subset{\mathfrak{A}},$ (3.6) where $\otimes_{\mathrm{alg}}$ is the algebraic tensor product. Let $({\mathfrak{A}}^{\textrm{vN}},U)$ be a chiral net, whose chiral components are $({\cal A}_{1},V_{1})$ and $({\cal A}_{2},V_{2})$. Let $\pi_{1}$, $\pi_{2}$ be locally normal, covariant, positive energy representations of the respective nets on $\mathbb{R}$. Then the chiral net of $(\pi_{1}({\cal A}_{1}),V_{\pi_{1}})$, $(\pi_{2}({\cal A}_{2}),V_{\pi_{2}})$ is a covariant, positive energy representation of $({\mathfrak{A}}^{\textrm{vN}},U)$, which will be denoted by $(\pi({\mathfrak{A}}^{\textrm{vN}}),U_{\pi})$, $\pi=\pi_{1}\otimes\pi_{2}$ and $\pi$ is called the product representation of $\pi_{1}$ and $\pi_{2}$. We note that $(\pi({\mathfrak{A}}),U_{\pi})$ is contained in the regular subnet of $(\pi({\mathfrak{A}}^{\textrm{vN}}),U_{\pi})$. For faithful $\pi$ these two nets coincide, due to Proposition 2.3.3 (2) of [17]. It is easily seen that $\pi$ is faithful (resp. irreducible), if $\pi_{1}$ and $\pi_{2}$ are faithful (resp. irreducible). (Cf. Theorems 5.2 and 5.9 from Chapter IV of [49]). ### 3.2 Vacuum representations and asymptotic completeness A regular chiral net $({\mathfrak{A}},U)$ is in a vacuum representation, with the vacuum vector $\Omega\in\mathcal{H}$, if and only if its chiral components $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ are in vacuum representations with the respective vacuum vectors $\Omega_{1}\in{\cal K}_{1}$, $\Omega_{2}\in{\cal K}_{2}$ s.t. $\Omega=\Omega_{1}\otimes\Omega_{2}$. (Cf. Proposition 3.5 below). In this subsection we show that any such regular chiral net has a complete particle interpretation in terms of non-interacting Wigner particles. These facts follow from our results in [25], but the argument below is more direct. We start from the observation that the asymptotic fields have a particularly simple form in chiral theories: ###### Proposition 3.2. Let $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ be two local nets of von Neumann algebras in vacuum representations, with the respective vacuum vectors $\Omega_{1}$, $\Omega_{2}$. Then, for any $A_{1}\in\bar{\cal A}_{1}$, $A_{2}\in\bar{\cal A}_{2}$ $\displaystyle\Phi_{+}^{\mathrm{out}/\mathrm{in}}(A_{1}\otimes A_{2})$ $\displaystyle=$ $\displaystyle A_{1}\otimes(\Omega_{2}|A_{2}\Omega_{2})I,$ (3.7) $\displaystyle\Phi_{-}^{\mathrm{out}/\mathrm{in}}(A_{1}\otimes A_{2})$ $\displaystyle=$ $\displaystyle(\Omega_{1}|A_{1}\Omega_{1})I\otimes A_{2}.$ (3.8) Proof. We consider only $\Phi_{+}^{\mathrm{out}}$, as the remaining cases are analogous. From the defining relation (2.6) and formula (3.4), we obtain $\Phi_{+}^{\mathrm{out}}(A_{1}\otimes A_{2})=\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;A_{1}\otimes\int dt\,h_{T}(t)\beta_{\sqrt{2}t}^{(2)}(A_{2}).$ (3.9) We set $A_{2}(h_{T}):=\int dt\,h_{T}(t)\beta_{\sqrt{2}t}^{(2)}(A_{2})$. This sequence has the following properties: $\displaystyle\lim_{T\to\infty}A_{2}(h_{T})\Omega_{2}$ $\displaystyle=$ $\displaystyle(\Omega_{2}|A_{2}\Omega_{2})\Omega_{2},$ (3.10) $\displaystyle\lim_{T\to\infty}\|[A_{2}(h_{T}),A]\|$ $\displaystyle=$ $\displaystyle 0,\textrm{ for any }A\in\bar{\cal A}_{2}.$ (3.11) The first identity above follows from the mean ergodic theorem and the fact that $\Omega_{2}$ is the only vector invariant under the action of $V_{2}$. The second equality is a consequence of the locality assumption from Definition 3.1. Since $\bar{\cal A}_{2}$ acts irreducibly, any $\Psi\in\mathcal{K}_{2}$ has the form $\Psi=A\Omega_{2}$ for some $A\in\bar{\cal A}_{2}$ [45]. Thus we obtain from (3.10), (3.11) $\underset{T\to\infty}{\mathrm{s}\textrm{-}\lim}\;A_{2}(h_{T})=(\Omega_{2}|A_{2}\Omega_{2})I,$ (3.12) which completes the proof. $\Box$ Now we can easily prove the main result of this subsection: ###### Theorem 3.3. Any regular chiral net $({\mathfrak{A}},U)$ in a vacuum representation is asymptotically complete. More precisely: $\displaystyle\mathcal{H}_{+}=\mathcal{K}_{1}\otimes{\mathbb{C}}\Omega_{2},$ (3.13) $\displaystyle\mathcal{H}_{-}={\mathbb{C}}\Omega_{1}\otimes\mathcal{K}_{2},$ (3.14) $\displaystyle\mathcal{H}_{+}\overset{\mathrm{out}}{\times}\mathcal{H}_{-}=\mathcal{H}_{+}\overset{\mathrm{in}}{\times}\mathcal{H}_{-}=\mathcal{H}.$ (3.15) Moreover, any such theory is non-interacting. ###### Remark 3.4. This result and Theorem 2.11 imply the convergence of the asymptotic functional approximants $\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$ for all $\Psi\in\mathcal{H}$ of bounded energy in any regular chiral net in a vacuum representation. Proof. Using formula (2.5) and the cyclicity of the vacuum $\Omega$ under the action of ${\mathfrak{A}}$, we obtain $\mathcal{H}_{\pm}=[\,\Phi_{\pm}^{\mathrm{out}}(F)\Omega\,|\,F\in{\mathfrak{A}}\,],$ (3.16) where $[\,\cdot\,]$ denotes the norm closure. Applying Proposition 3.2 and exploiting the cyclicity of $\Omega_{1/2}$ under the action of $\bar{\cal A}_{1/2}$, we obtain (3.13) and (3.14). The asymptotic completeness relation (3.15) also follows from Proposition 3.2: For any $A_{1}\in\bar{\cal A}_{1}$, $A_{2}\in\bar{\cal A}_{2}$ $\displaystyle\Phi_{+}^{\mathrm{out}}(A_{1}\otimes I)\Phi_{-}^{\mathrm{out}}(I\otimes A_{2})\Omega=\Phi_{+}^{\mathrm{in}}(A_{1}\otimes I)\Phi_{-}^{\mathrm{in}}(I\otimes A_{2})\Omega=A_{1}\Omega_{1}\otimes A_{2}\Omega_{2}.$ (3.17) Exploiting once again cyclicity of $\Omega_{1/2}$, we obtain that scattering states are dense in the Hilbert space. Now let us show the lack of interaction: Let $\Psi_{\pm}\in\mathcal{H}_{\pm}$. Then, by (3.13), (3.14) and the irreducibility of the action of $\bar{\cal A}_{1/2}$ on $\mathcal{K}_{1/2}$, there exist $A_{1}\in\bar{\cal A}_{1}$, $A_{2}\in\bar{\cal A}_{2}$ s.t. $\Psi_{+}=A_{1}\Omega_{1}\otimes\Omega_{2}$ and $\Psi_{-}=\Omega_{1}\otimes A_{2}\Omega_{2}$. Then $\displaystyle\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}=\Phi_{+}^{\mathrm{out}}(A_{1}\otimes I)\Phi_{-}^{\mathrm{out}}(I\otimes A_{2})\Omega=A_{1}\Omega_{1}\otimes A_{2}\Omega_{2}$ $\displaystyle=\Phi_{+}^{\mathrm{in}}(A_{1}\otimes I)\Phi_{-}^{\mathrm{in}}(I\otimes A_{2})\Omega=\Psi_{+}\overset{\mathrm{in}}{\times}\Psi_{-}.$ (3.18) Hence the scattering operator, defined in (2.10), equals the identity on $\mathcal{H}$. $\Box$ ### 3.3 Charged representations and infraparticles It is the goal of this subsection to clarify the particle content of chiral conformal field theories in charged representations. More detailed particle properties of such theories, e.g. superselection of direction of motion, will be studied in the next subsection. Let us first note the following simple relation between the single-particle subspaces of a regular chiral net and the invariant vectors of its chiral components. ###### Proposition 3.5. Let $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ be local nets of von Neumann algebras on $\mathbb{R}$. Then $V_{1}$ (resp. $V_{2}$) has a non- trivial invariant vector, if and only if the single-particle subspace $\mathcal{H}_{-}$ (resp. $\mathcal{H}_{+}$) of the corresponding regular chiral net $({\mathfrak{A}},U)$ is non-trivial. Proof. Suppose there exists a non-zero $\Omega_{1}\in\mathcal{K}_{1}$, invariant under the action of $V_{1}$. Then, for any $\Psi_{2}\in\mathcal{K}_{2}$, $\displaystyle U(t,-t)(\Omega_{1}\otimes\Psi_{2})$ $\displaystyle=$ $\displaystyle\Omega_{1}\otimes\Psi_{2},$ (3.19) for $t\in\mathbb{R}$. Hence the subspace $\mathcal{H}_{-}$ is non-trivial. Similarly, the existence of a non-zero $\Omega_{2}\in\mathcal{K}_{2}$, invariant under the action of $V_{2}$, implies the non-triviality of $\mathcal{H}_{+}$. Now suppose $\Psi\in\mathcal{H}_{-}$ and $V_{1}$ has no non-trivial, invariant vectors. Then, by the mean ergodic theorem, $\displaystyle\Psi=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\,U(t,-t)\Psi=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}dt\,\big{(}V_{1}(\sqrt{2}t)\otimes I\big{)}\Psi=0.$ (3.20) Thus we established that $\mathcal{H}_{-}=\\{0\\}$. Similarly, the absence of non-trivial, invariant vectors of $V_{2}$ implies that $\mathcal{H}_{+}=\\{0\\}$. $\Box$ Let $({\mathfrak{A}},U)$ be a regular chiral net in a charged irreducible (product) representation. That is ${\mathfrak{A}}$ acts irreducibly on a non- trivial Hilbert space, which has the tensor product structure, by our definition of chiral nets, and does not contain non-zero invariant vectors of $U$. The particle structure of such theories is described by the following theorem. ###### Theorem 3.6. Let $({\mathfrak{A}},U)$ be a regular chiral net in a charged irreducible (product) representation acting on a Hilbert space $\mathcal{H}$. Then: 1. (a) $\mathcal{H}_{+}=\\{0\\}$ or $\mathcal{H}_{-}=\\{0\\}$ i.e. the theory does not describe Wigner particles. 2. (b) For any non-zero vector $\Psi\in P_{E}\mathcal{H}$, $E\geq 0$, all the limit points of the net $\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$, given by (2.34), are different from zero. Hence $({\mathfrak{A}},U)$ describes infraparticles. Proof. Part (a) follows immediately from Proposition 3.5 and the absence of non-zero invariant vectors of $U$ in $\mathcal{H}$. As for part (b), since ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$, its chiral components $({\cal A}_{1/2},V_{1/2})$ act irreducibly on their respective Hilbert spaces ${\cal K}_{1/2}$. We note that for any non-zero vector $\Psi\in P_{E}\mathcal{H}$ we can find a sequence of vectors $\\{\Psi_{n}\\}_{n\in\mathbb{N}}$ from ${\cal K}_{1}$ s.t. $\Psi_{1}\neq 0$ and $(\Psi|(C\otimes I)\Psi)=\sum_{n\in\mathbb{N}}(\Psi_{n}|C\Psi_{n})$ (3.21) for all $C\in B({\cal K}_{1})$. Moreover, we can assume without loss of generality that ${\cal K}_{1}$ does not contain non-trivial invariant vectors of $V_{1}$. Then we obtain from Lemma A.1 (b) the existence of a local operator $A\in{\cal A}_{1}$ and $f\in S(\mathbb{R})$ s.t. $\mathrm{supp}\,\tilde{f}\cap\mathbb{R}_{+}=\varnothing$, which satisfy $A(f)\Psi_{1}\neq 0$. We note that any $B:=A(f)\otimes I$ is a non-zero element of ${\mathfrak{A}}$ which is almost local and energy decreasing. Consequently, $B^{*}B$ belongs to the algebra of detectors $\mathcal{C}$ of the net $({\mathfrak{A}},U)$. We consider the corresponding asymptotic functional approximants $\displaystyle\sigma_{\Psi}^{(T)}(B^{*}B)$ $\displaystyle=$ $\displaystyle\int dt\,h_{T}(t)\int d\boldsymbol{x}\,(\Psi|\alpha_{(t,\boldsymbol{x})}(B^{*}B)\Psi)$ (3.22) $\displaystyle=$ $\displaystyle\int dt\,h_{T}(t)\int d\boldsymbol{x}\,(\Psi|\beta_{(\sqrt{2})^{-1}(t-\boldsymbol{x})}^{(1)}(A(f)^{*}A(f))\otimes I)\Psi)$ $\displaystyle\geq$ $\displaystyle\int d\boldsymbol{x}\,(\Psi_{1}|(\beta^{(1)}_{(\sqrt{2})^{-1}\boldsymbol{x}}(A(f)^{*}A(f)))\Psi_{1})\neq 0,$ where in the last step we made use of (3.21). As the last expression is independent of $T$, all the limit points of $\\{\sigma_{\Psi}^{(T)}\\}_{T\in\mathbb{R}_{+}}$ are different from zero. $\Box$ ### 3.4 Infraparticles with superselected direction of motion In Theorem 3.6 above we have shown that any charged irreducible (product) representation of a chiral conformal field theory contains infraparticles. In this subsection we demonstrate that in a large class of examples these infraparticles have superselected direction of motion in the sense of Definition 2.12. Let $({\cal A},V)$ be a local net of von Neumann algebras on $\mathbb{R}$, acting on a Hilbert space $\mathcal{K}$. We assume that this net is in a vacuum representation, with the vacuum vector $\Omega_{0}\in\mathcal{K}$. Let $W$ be a unitary operator on $\mathcal{K}$ which implements a symmetry of this net i.e. $\displaystyle W{\cal A}({\cal I})W^{*}\subset{\cal A}({\cal I}),$ (3.23) $\displaystyle WV(t)W^{*}=V(t),$ (3.24) $\displaystyle W\Omega_{0}=\Omega_{0},$ (3.25) for any open, bounded interval ${\cal I}\subset\mathbb{R}$ and any $t\in\mathbb{R}$. We assume that $W$ gives rise to a non-trivial representation of the group $\mathbb{Z}_{2}$ i.e. ${\hbox{\rm Ad}}W\neq{\rm id}$ and $W^{2}=I$. We define the subspaces $\displaystyle{\cal A}_{\mathrm{ev}}({\cal I})$ $\displaystyle=$ $\displaystyle\\{\,A\in{\cal A}({\cal I})\,|\,WAW^{*}=A\,\\},$ (3.26) $\displaystyle{\cal A}_{\mathrm{odd}}({\cal I})$ $\displaystyle=$ $\displaystyle\\{\,A\in{\cal A}({\cal I})\,|\,WAW^{*}=-A\,\\}.$ (3.27) Let ${\cal A}_{\mathrm{ev}}$ (resp. ${\cal A}_{\mathrm{odd}}$) be the norm- closed linear span of all operators from some ${\cal A}_{\mathrm{ev}}({\cal I})$ (resp. ${\cal A}_{\mathrm{odd}}({\cal I})$), ${\cal I}\subset{\mathbb{R}}$. Clearly, $({\cal A}_{\mathrm{ev}},V)$ is again a local net of von Neumann algebras on the real line acting on $\mathcal{K}$. We introduce the subspaces $\mathcal{K}_{\mathrm{ev}}=[{\cal A}_{\mathrm{ev}}\Omega_{0}]$, $\mathcal{K}_{\mathrm{odd}}=[{\cal A}_{\mathrm{odd}}\Omega_{0}]$, where $[\,\cdot\,]$ denotes the closure, which are invariant under the action of ${\cal A}_{\mathrm{ev}}$ and $V$, and satisfy $\mathcal{K}=\mathcal{K}_{\mathrm{ev}}\oplus\mathcal{K}_{\mathrm{odd}}$. $\mathcal{K}_{\mathrm{odd}}$ gives rise to the representation $\displaystyle\pi_{\mathrm{odd}}(A)$ $\displaystyle=$ $\displaystyle A|_{\mathcal{K}_{\mathrm{odd}}},\quad A\in{\cal A}_{\mathrm{ev}},$ (3.28) $\displaystyle V_{\mathrm{odd}}(t)$ $\displaystyle=$ $\displaystyle V(t)|_{\mathcal{K}_{\mathrm{odd}}},\quad t\in\mathbb{R}.$ (3.29) Its relevant properties are summarized in the following lemma, which we prove in Appendix B. ###### Lemma 3.7. $(\pi_{\mathrm{odd}},\mathcal{K}_{\mathrm{odd}})$ is a covariant, positive energy representation of $({\cal A}_{\mathrm{ev}},V)$, in which the translation automorphisms are implemented by $V_{\mathrm{odd}}$. Moreover: 1. (a) $\pi_{\mathrm{odd}}$ is a locally normal, faithful and irreducible representation of ${\cal A}_{\mathrm{ev}}$. 2. (b) $V_{\mathrm{odd}}$ does not admit non-trivial invariant vectors. We set $\hat{\cal A}:=\pi_{\mathrm{odd}}({\cal A}_{\mathrm{ev}})$, $\hat{V}(t):=V_{\mathrm{odd}}(t)$. By the above lemma $(\hat{\cal A},\hat{V})$ is again a local net of von Neumann algebras on the real line. We define its representation on $\mathcal{K}_{\mathrm{ev}}$ $\displaystyle\pi_{\mathrm{ev}}(\hat{A})$ $\displaystyle=$ $\displaystyle\pi_{\mathrm{odd}}^{-1}(\hat{A})|_{\mathcal{K}_{\mathrm{ev}}},\quad\hat{A}\in\hat{\cal A},$ (3.30) $\displaystyle V_{\mathrm{ev}}(t)$ $\displaystyle=$ $\displaystyle V(t)|_{\mathcal{K}_{\mathrm{ev}}},\quad t\in\mathbb{R}$ (3.31) and state the following fact, whose proof is given in Appendix B. ###### Lemma 3.8. $(\pi_{\mathrm{ev}},\mathcal{K}_{\mathrm{ev}})$ is a covariant, positive energy representation of $(\hat{\cal A},\hat{V})$, in which the translation automorphisms are implemented by $V_{\mathrm{ev}}$. Moreover 1. (a) $\pi_{\mathrm{ev}}$ is a locally normal, faithful and irreducible representation of $\hat{\cal A}$. 2. (b) $V_{\mathrm{ev}}$ admits a unique (up to a phase) invariant vector, which is cyclic for any $\pi_{\mathrm{ev}}(\hat{\cal A}({\cal I}))$. We conclude that $(\pi_{\mathrm{ev}}(\hat{\cal A}),V_{\mathrm{ev}})$ is a local net of von Neumann algebras in a vacuum representation with the vacuum vector $\Omega_{0}\in\mathcal{K}_{\mathrm{ev}}$. We remark that the above abstract construction can be performed in a number of concrete cases. If a Möbius covariant net ${\cal I}\to{\cal A}({\cal I})$ on $S^{1}$, in a vacuum representation, admits an automorphism777An automorphism $\gamma$ of a net ${\cal A}$ is an automorphism of the quasilocal algebra ${\cal A}$ which preserves each local algebra ${\cal A}({\cal I})$. $\gamma$ of order 2 which preserves the vacuum state, then one can define $W$ by $WA\Omega_{0}=\gamma(A)\Omega_{0},\quad A\in{\cal A}({\cal I}).$ (3.32) This does not depend on the choice of the interval ${\cal I}$ and defines a unitary operator thanks to the invariance of the vacuum state. This $W$ automatically commutes with the action of the Möbius group (in particular with the action of translations) as a consequence of the Bisognano-Wichmann property [31]. Thus, upon restriction to the real line, we obtain a local net equipped with a unitary $W$ which satisfies (3.23)-(3.25). Non-trivial automorphisms $\gamma$ appear, in particular, in the $U(1)$-current net ($\gamma:J(z)\to-J(z)$) [15], in loop group nets of a compact group $G$ with a ${\mathbb{Z}}_{2}$-subgroup in $G$ [51] and in the tensor product net ${\cal A}\otimes{\cal A}$ for an arbitrary Möbius covariant net ${\cal A}$, where $\gamma$ is the flip symmetry. Coming back to the abstract setting, we introduce the class of two-dimensional theories, we are interested in: Let $({\cal A}_{1},V_{1})$, $({\cal A}_{2},V_{2})$ be two local nets of von Neumann algebras on $\mathbb{R}$, in vacuum representations, acting on Hilbert spaces $\mathcal{K}_{1}$, $\mathcal{K}_{2}$. We denote the respective vacuum vectors by $\Omega_{1}$, $\Omega_{2}$ and introduce the corresponding regular chiral net $({\mathfrak{A}},U)$. We assume the existence of unitaries $W_{1}$, $W_{2}$, which give rise to non-trivial representations of $\mathbb{Z}_{2}$ and implement symmetries of the respective nets on $\mathbb{R}$ as defined in (3.23)-(3.25). By the construction described above we obtain the nets $(\hat{\cal A}_{1},\hat{V}_{1})$, $(\hat{\cal A}_{2},\hat{V}_{2})$, acting on $\mathcal{K}_{1,\mathrm{odd}}$, $\mathcal{K}_{2,\mathrm{odd}}$. We denote by $(\hat{\mathfrak{A}}^{\textrm{vN}},\hat{U})$ the corresponding chiral net acting on $\hat{\mathcal{H}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}$ and by $(\hat{\mathfrak{A}},\hat{U})$ its regular subnet. Let us summarize its properties. ###### Proposition 3.9. The regular chiral net $(\hat{\mathfrak{A}},\hat{U})$, whose chiral components are $(\hat{\cal A}_{1},\hat{V}_{1})$, $(\hat{\cal A}_{2},\hat{V}_{2})$, has the following properties: 1. (a) $\hat{\mathfrak{A}}$ acts irreducibly on $\hat{\mathcal{H}}$. 2. (b) $(\hat{\mathfrak{A}},\hat{U})$ does not admit Wigner particles ($\hat{\mathcal{H}}_{\pm}=\\{0\\}$), but all the asymptotic functionals of the form $\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in P_{E}\hat{\mathcal{H}},\,E\geq 0\,\\}$ are non-zero. 3. (c) $\pi_{{\rm{R}}}=\iota_{1}\otimes\pi_{2,\mathrm{ev}}$ and $\pi_{{\rm{L}}}=\pi_{1,\mathrm{ev}}\otimes\iota_{2}$ are irreducible, faithful, covariant representations of $(\hat{\mathfrak{A}},\hat{U})$, acting on $\mathcal{H}_{\pi_{{\rm{R}}}}:=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$ and $\mathcal{H}_{\pi_{{\rm{L}}}}:=\mathcal{K}_{1,\mathrm{ev}}\otimes\mathcal{K}_{2,\mathrm{odd}}$, respectively. The respective (canonical) unitary representations of translations are given by $U_{\pi_{\rm{R}}}(x):=U(x)|_{\mathcal{H}_{\pi_{{\rm{R}}}}}$ and $U_{\pi_{\rm{L}}}(x):=U(x)|_{\mathcal{H}_{\pi_{{\rm{L}}}}}$. 4. (d) $\mathcal{H}_{\pi_{{\rm{R}}},-}=\\{0\\}$ and $\mathcal{H}_{\pi_{{\rm{R}}},+}\neq\\{0\\}$ while $\mathcal{H}_{\pi_{{\rm{L}}},-}\neq\\{0\\}$ and $\mathcal{H}_{\pi_{{\rm{L}}},+}=\\{0\\}$. Consequently, $\pi_{{\rm{R}}}$ is not unitarily equivalent to $\pi_{{\rm{L}}}$. In part (c) $\iota_{1/2}$ are the defining representations of $\hat{\cal A}_{1/2}$. Representations $\pi_{1/2,\mathrm{ev}}$ are defined as in (3.30), (3.31). Proof. Part (a) follows from the irreducibility of $\pi_{1/2,\mathrm{odd}}$, shown in Lemma 3.7. As for part (b), we obtain from Lemma 3.7 (b) and Proposition 3.5 that $\hat{\mathcal{H}}_{\pm}=\\{0\\}$. On the other hand, Theorem 3.6 ensures that the relevant asymptotic functionals are non-zero. Irreducibility and faithfulness of $\pi_{{\rm{R}}/{\rm{L}}}$ in part (c) follow from Lemma 3.8 (a) and Lemma 3.7 (a). Proceeding to part (d), we note that, by faithfulness of $\pi_{{\rm{R}}}$, the net $(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}),U_{\pi_{{\rm{R}}}})$ coincides with the regular chiral subnet of $(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}^{\textrm{vN}}),U_{\pi_{{\rm{R}}}})$, whose chiral components are $(\hat{\cal A}_{1},\hat{V}_{1})$ and $(\pi_{2,\mathrm{ev}}(\hat{\cal A}_{2}),V_{2,\mathrm{ev}})$. From Lemma 3.7 (b), Lemma 3.8 (b) and Proposition 3.5 we obtain that $\mathcal{H}_{\pi_{{\rm{R}}},-}=\\{0\\}$ and $\mathcal{H}_{\pi_{{\rm{R}}},+}\neq\\{0\\}$. An analogous reasoning, applied to $\pi_{{\rm{L}}}$, shows that $\mathcal{H}_{\pi_{{\rm{L}}},-}\neq\\{0\\}$ and $\mathcal{H}_{\pi_{{\rm{L}}},+}=\\{0\\}$. Hence, due to relation (2.3), the two nets are not unitarily equivalent. $\Box$ In view of part (b) of the above proposition, the theory $(\hat{\mathfrak{A}},\hat{U})$ describes infraparticles. In the following theorem, which is our main result, we show that these infraparticles have superselected direction of motion, in the sense of Definition 2.12. ###### Theorem 3.10. Consider the regular chiral net $(\hat{\mathfrak{A}},\hat{U})$, constructed above. Let $\psi\in\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\neq 0,\,\Psi\in P_{E}\hat{\mathcal{H}},\,E\geq 0\,\\}$ and let $\pi_{\psi}$ be its GNS representation. Then $\pi_{\psi}$ is a type I representation with atomic center. It contains both right-moving and left-moving irreducible subrepresentations which are unitarily equivalent to $\pi_{{\rm{R}}}$ and $\pi_{{\rm{L}}}$, respectively. Hence the theory describes infraparticles with superselected direction of motion. ###### Remark 3.11. Let us consider the regular chiral net $({\mathfrak{A}},U)$ in the vacuum representation. Then, similarly as in the theorem above, the GNS representation $\pi_{\psi}$ induced by any particle weight $\psi\in\\{\,\psi_{\Psi}^{\mathrm{out}}\,|\,\Psi\notin{\mathbb{C}}\Omega,\Psi\in P_{E}\mathcal{H},E\geq 0\,\\}$ is of type I with atomic center. However, any non-trivial irreducible subrepresentation of $\pi_{\psi}$ is unitarily equivalent to the defining vacuum representation i.e. $\psi$ is neutral. These facts are easily verified by modifying the proof below. Proof. Let us first consider the regular chiral net $({\mathfrak{A}},U)$ acting on $\mathcal{H}$. By Theorem 3.3, ${\cal K}_{+}:={\cal K}_{1,\mathrm{odd}}\otimes{\mathbb{C}}\Omega_{2}\subset\mathcal{H}_{+}$ and ${\cal K}_{-}:={\mathbb{C}}\Omega_{1}\otimes{\cal K}_{2,\mathrm{odd}}\subset\mathcal{H}_{-}$. Any vector $\Psi\in P_{E}({\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$, $E\geq 0$, gives rise to functionals $\rho_{\Psi,\pm}$, defined by (2.11), (2.12). They have the form $\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n\in\mathbb{N}}(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n}),$ (3.33) where $\Psi_{\pm,n}\in P_{E}{\cal K}_{\pm}$. (Cf. formula (2.13) and the subsequent discussion). Since $\Psi\neq 0$, we can assume that $\Psi_{+,1}\neq 0$ and $\Psi_{-,1}\neq 0$. We also note for future reference that ${\cal K}_{+}\subset\mathcal{H}_{\pi_{{\rm{R}}}}$ and ${\cal K}_{-}\subset\mathcal{H}_{\pi_{{\rm{L}}}}$. Let us now proceed to the net $(\hat{\mathfrak{A}},\hat{U})$, acting on $\hat{\mathcal{H}}={\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-}\subset\mathcal{H}$, and let $\hat{\mathcal{L}}$ be the left ideal of $\hat{\mathfrak{A}}$, given by definition (2.16). For any ${\hat{L}}\in\hat{\mathcal{L}}$ we define $L=(\pi_{1,\mathrm{odd}}^{-1}\otimes\pi_{2,\mathrm{odd}}^{-1})({\hat{L}})\in\mathcal{L},$ (3.34) where $\mathcal{L}$ is the corresponding left ideal of ${\mathfrak{A}}$. We note that such $L$ leaves the subspaces $\mathcal{H}_{\pi_{{\rm{R}}}}$ and $\mathcal{H}_{\pi_{{\rm{L}}}}$ invariant. Exploiting Theorem 2.11 and formula (2.13), we obtain $\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})=\sum_{n\in\mathbb{N}}\int d\boldsymbol{x}\,\\{(\Psi_{+,n}|(L_{1}^{*}L_{2})(\boldsymbol{x})\Psi_{+,n})+(\Psi_{-,n}|(L_{1}^{*}L_{2})(\boldsymbol{x})\Psi_{-,n})\\}.$ (3.35) It follows from Theorem 2.9 that for any $L$ given by (3.34) the Fourier transforms $\displaystyle L\tilde{\Psi}_{+,n}(\boldsymbol{p})$ $\displaystyle:=$ $\displaystyle(2\pi)^{-1/2}\int d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{\rm{R}}}(\boldsymbol{x})^{*}\Psi_{+,n},$ (3.36) $\displaystyle L\tilde{\Psi}_{-,n}(\boldsymbol{p})$ $\displaystyle:=$ $\displaystyle(2\pi)^{-1/2}\int d\boldsymbol{x}\,e^{i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{\rm{L}}}(\boldsymbol{x})^{*}\Psi_{-,n}$ (3.37) belong to $\mathcal{H}_{\pi_{{\rm{R}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ and $\mathcal{H}_{\pi_{{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ respectively. Since $\pi_{{\rm{R}}}(\hat{\mathfrak{A}})$ acts irreducibly on $\mathcal{H}_{\pi_{{\rm{R}}}}$ and $U_{\pi_{\rm{R}}}$ does not have non-zero invariant vectors, we obtain from Lemma A.1 (a) the existence of ${\hat{L}}_{+}\in\hat{\mathcal{L}}$ s.t. $L_{+}\Psi_{+,1}\neq 0$. Since $L_{+}U_{\pi_{\rm{R}}}(\boldsymbol{x})^{*}\Psi_{+,1}$ is a continuous function of $\boldsymbol{x}$, it is nonzero as a square-integrable function, hence $\\{L_{+}\tilde{\Psi}_{+,1}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\neq 0$. Analogously, we can find ${\hat{L}}_{-}\in\hat{\mathcal{L}}$ s.t. $\\{L_{-}\tilde{\Psi}_{-,1}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\neq 0$. For future reference, we note the equalities $\displaystyle\alpha_{x}(L)\tilde{\Psi}_{+,n}(\boldsymbol{p})$ $\displaystyle=$ $\displaystyle e^{-i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi_{\rm{R}}}(x)L\tilde{\Psi}_{+,n}(\boldsymbol{p}),$ (3.38) $\displaystyle\alpha_{x}(L)\tilde{\Psi}_{-,n}(\boldsymbol{p})$ $\displaystyle=$ $\displaystyle e^{-i(\boldsymbol{p},-\boldsymbol{p})x}U_{\pi_{\rm{L}}}(x)L\tilde{\Psi}_{-,n}(\boldsymbol{p}),$ (3.39) which hold in the sense of $\mathcal{H}_{\pi_{{\rm{R}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ and $\mathcal{H}_{\pi_{{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ respectively. These relations are easily verified for such $\Psi_{\pm,n}\in P_{E}{\cal K}_{\pm}$ that $\mathbb{R}\ni\boldsymbol{x}\to LU_{\pi_{{\rm{R}}/{\rm{L}}}}(\boldsymbol{x})^{*}\Psi_{\pm,n}$ decay rapidly in norm as $|\boldsymbol{x}|\to\infty$, since in this case the Fourier transform is pointwise defined. The general case follows from the fact that such vectors form a dense subspace in $P_{E}{\cal K}_{\pm}$ (cf. formula (3.53) below) and that the maps $P_{E}{\cal K}_{\pm}\ni\Psi\to\\{L\tilde{\Psi}(\boldsymbol{p})\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\in\mathcal{H}_{\pi_{{\rm{R}}/{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$ are norm-continuous. This latter fact is a consequence of Theorem 2.9 and the (Hilbert space valued) Plancherel theorem. After this preparation we study the structure of the GNS representation induced by $\psi$. Let us first consider the following auxiliary representation of $(\hat{\mathfrak{A}},\hat{U})$ $\pi_{1}(\,\cdot\,):=\bigoplus_{n\in\mathbb{N}}\big{(}\\{\pi_{{\rm{R}}}(\,\cdot\,)\otimes I\\}\oplus\\{\pi_{{\rm{L}}}(\,\cdot\,)\otimes I\\}\big{)},$ (3.40) acting on $\mathcal{H}_{\pi_{1}}:=\bigoplus_{n\in\mathbb{N}}\big{(}\\{\mathcal{H}_{\pi_{{\rm{R}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}\oplus\\{\mathcal{H}_{\pi_{{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}\big{)}$. From definition (3.40) and relation (2.22) we conclude that $\pi_{1}$ and its subrepresentations are of type I with atomic center. Moreover, $\pi_{1}$ is covariant and it is easily seen that the canonical representation of translations is given by $U^{\mathrm{can}}_{\pi_{1}}(x)=\bigoplus_{n\in\mathbb{N}}\big{(}\big{\\{}U_{\pi_{\rm{R}}}(x)\otimes I\big{\\}}\oplus\big{\\{}U_{\pi_{\rm{L}}}(x)\otimes I\big{\\}}).$ (3.41) We note that $\pi_{\psi}$ is unitarily equivalent to a subrepresentation of $\pi_{1}$. In fact, the map $W_{1}:\mathcal{H}_{\pi_{\psi}}\to\mathcal{H}_{\pi_{1}}$, given by $W_{1}:|{\hat{L}}\rangle\to\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,L\tilde{\Psi}_{+,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{\,L\tilde{\Psi}_{-,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)},$ (3.42) intertwines the two representations and is an isometry by formula (3.35) and the (Hilbert space valued) Plancherel theorem. It is easily checked that the canonical representation of translations $U_{\pi_{\psi}}^{\mathrm{can}}$ in the representation $\pi_{\psi}$ is given by the relation $W_{1}U^{\mathrm{can}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{1}}(x)W_{1}.$ (3.43) Recalling that $U^{\mathrm{char}}_{\pi_{\psi}}(x)=U^{\mathrm{can}}_{\pi_{\psi}}(x)U_{\pi_{\psi}}^{-1}(x)$, where $U_{\pi_{\psi}}$ is given by (2.20), we obtain $\displaystyle W_{1}U^{\mathrm{char}}_{\pi_{\psi}}(x)|{\hat{L}}\rangle$ $\displaystyle=$ $\displaystyle W_{1}U^{\mathrm{can}}_{\pi_{\psi}}(x)|\alpha_{-x}({\hat{L}})\rangle=U^{\mathrm{can}}_{\pi_{1}}(x)W_{1}|\alpha_{-x}({\hat{L}})\rangle$ (3.44) $\displaystyle=$ $\displaystyle\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,U_{\pi_{\rm{R}}}(x)\alpha_{-x}(L)\tilde{\Psi}_{+,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{\,U_{\pi_{\rm{L}}}(x)\alpha_{-x}(L)\tilde{\Psi}_{-,n}(\boldsymbol{p})\,\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)}$ $\displaystyle=$ $\displaystyle\bigoplus_{n\in\mathbb{N}}\big{(}\\{\,I\otimes e^{i(\boldsymbol{p},\boldsymbol{p})x}\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\oplus\\{I\otimes e^{i(\boldsymbol{p},-\boldsymbol{p})x}\\}_{\boldsymbol{p}\in\mathbb{R}_{+}}\big{)}W_{1}|{\hat{L}}\rangle,$ where in the last step we made use of relations (3.38), (3.39). Now let $\boldsymbol{Q}$ be the generator of space translations of $U^{\mathrm{char}}_{\pi_{\psi}}$ and let $\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ (resp. $\mathcal{H}_{\pi_{\psi},\mathrm{L}}$) be its spectral subspace corresponding to the interval $[0,\infty)$ (resp. $(-\infty,0)$). Then, by formula (3.44), $\displaystyle W_{1}\mathcal{H}_{\pi_{\psi},\mathrm{R}}=P_{{\rm{R}}}W_{1}\mathcal{H}_{\pi_{\psi}},$ (3.45) $\displaystyle W_{1}\mathcal{H}_{\pi_{\psi},\mathrm{L}}=P_{{\rm{L}}}W_{1}\mathcal{H}_{\pi_{\psi}},$ (3.46) where $P_{{\rm{R}}/{\rm{L}}}$ are the projections on the subspaces $\bigoplus_{n\in\mathbb{N}}\\{\mathcal{H}_{\pi_{{\rm{R}}/{\rm{L}}}}\otimes L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\\}$ in $\mathcal{H}_{\pi_{1}}$. From definition (3.42) and the remarks after formula (3.37) we conclude that $P_{{\rm{R}}}W_{1}\mathcal{H}_{\pi_{\psi}}\neq\\{0\\}$ and $P_{{\rm{L}}}W_{1}\mathcal{H}_{\pi_{\psi}}\neq\\{0\\}$. Consequently $\pi_{\psi}$ has both right-moving and left-moving irreducible subrepresentations. Let $\pi$ be an irreducible subrepresentation of $\pi_{\psi}$, acting on a non-trivial subspace ${\cal K}\subset\mathcal{H}_{\pi_{\psi},\mathrm{R}}$ (i.e. a right-moving subrepresentation). Then $W_{1}\pi(\,\cdot\,)W_{1}^{*}$ is an irreducible subrepresentation of $\pi_{\rm{R}}(\,\cdot\,)\otimes I$ acting on $\mathcal{H}_{\pi_{{\rm{R}}}}\otimes\big{(}\bigoplus_{n\in\mathbb{N}}L^{2}(\mathbb{R}_{+},d\boldsymbol{p})\big{)}$. By irreducibility, we conclude that $\pi$ is unitarily equivalent to $(\pi_{{\rm{R}}}(\,\cdot\,)\otimes I)|_{\mathcal{H}_{\pi_{{\rm{R}}}}\otimes{\mathbb{C}}e}$ for some non-zero $e\in\bigoplus_{n\in\mathbb{N}}L^{2}(\mathbb{R}_{+},d\boldsymbol{p})$. This latter representation can be identified with $\pi_{{\rm{R}}}$. An analogous argument shows that any left-moving irreducible subrepresentation of $\pi_{\psi}$ is unitarily equivalent to $\pi_{{\rm{L}}}$. Hence, by Proposition 3.9 (d), $(\hat{\mathfrak{A}},\hat{U})$ describes infraparticles with superselected direction of motion. $\Box$ Let us assume for a moment that $\hat{\mathcal{H}}$ is separable. Then we obtain from the above theorem and formula (2.29) that the GNS representation of any particle weight $\psi^{\mathrm{out}}_{\Psi}$ of the net $(\hat{\mathfrak{A}},\hat{U})$, where $\Psi\neq 0$ is a vector of bounded energy, has the form $\displaystyle\pi_{\psi^{\mathrm{out}}_{\Psi}}\simeq\int^{\oplus}_{Z_{{\rm{R}}}}d\mu_{{\rm{R}}}(z)\,\pi_{{\rm{R}},z}\oplus\int^{\oplus}_{Z_{{\rm{L}}}}d\mu_{{\rm{L}}}(z)\,\pi_{{\rm{L}},z}.$ (3.47) Here $(Z_{{\rm{R}}/{\rm{L}}},d\mu_{{\rm{R}}/{\rm{L}}})$ are some Borel spaces and $\pi_{{\rm{R}}/{\rm{L}},z}=\pi_{{\rm{R}}/{\rm{L}}}$ for all $z\in Z_{{\rm{R}}/{\rm{L}}}$. A decomposition of $\psi^{\mathrm{out}}_{\Psi}$ into pure particle weights, which induce the irreducible representations appearing in the decomposition of $\pi_{\psi^{\mathrm{out}}_{\Psi}}$, was obtained by Porrmann in [41, 42] (cf. formula (1.3) above). However, to apply Porrmann’s abstract argument, one has to restrict attention to countable (resp. separable) subsets of all the relevant objects and it is not guaranteed that the resulting (restricted) pure particle weights extend to the original domains. It is therefore worth pointing out that the theory $(\hat{\mathfrak{A}},\hat{U})$ admits a large class of particle weights, whose decomposition can be performed in the original framework. To our knowledge this is the first such decomposition in the presence of infraparticles. (See however [35] for some partial results on the Schroer model). These particle weights belong to the set $\\{\,\psi^{\mathrm{out}}_{\Psi}\,|\,\Psi\in\mathcal{D}\,\\}$, where $\mathcal{D}\subset\hat{\mathcal{H}}$ is a dense domain spanned by vectors of the form $\Psi=F_{1}\Omega_{1}\otimes F_{2}\Omega_{2},$ (3.48) where $F_{1}\in{\cal A}_{1,\mathrm{odd}},F_{2}\in{\cal A}_{2,\mathrm{odd}}$ are s.t. $F_{1}\otimes I,I\otimes F_{2}\in{\mathfrak{A}}$ are almost local and have compact energy-momentum transfer (see formula (2.15)). The proof of the following proposition exploits some ideas from [26]. ###### Proposition 3.12. Consider the regular chiral net $(\hat{\mathfrak{A}},\hat{U})$ constructed above. Denote by $\hat{\mathcal{L}}$ its left ideal, given by definition (2.16). Then, for any non-zero vector $\Psi\in\mathcal{D}$, there exist continuous fields of pure particle weights $\Delta_{{\rm{R}},n}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$ and $\Delta_{{\rm{L}},m}\ni\boldsymbol{p}\to\psi_{{\rm{L}},m,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$ s.t. for any ${\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}$ $\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})=\sum_{n\in C_{{\rm{R}}}}\int_{\Delta_{{\rm{R}},n}}d\boldsymbol{p}\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})+\sum_{m\in C_{{\rm{L}}}}\int_{\Delta_{{\rm{L}},m}}d\boldsymbol{p}\,\psi_{{\rm{L}},m,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}),$ (3.49) where $C_{{\rm{R}}},C_{{\rm{L}}}\subset\mathbb{N}$ are non-empty finite subsets and $\Delta_{{\rm{R}},n},\Delta_{{\rm{L}},m}\subset\mathbb{R}_{+}$ are non-empty, open subsets for any $n\in C_{{\rm{R}}}$, $m\in C_{{\rm{L}}}$. Moreover: 1. (a) The characteristic energy-momentum vectors of the weights $\psi_{{\rm{R}},n,\boldsymbol{p}}$ (resp. $\psi_{{\rm{L}},m,\boldsymbol{p}}$) are equal to $q_{{\rm{R}},n,\boldsymbol{p}}=(\boldsymbol{p},\boldsymbol{p})$ (resp. $q_{{\rm{L}},m,\boldsymbol{p}}=(\boldsymbol{p},-\boldsymbol{p})$). 2. (b) The GNS representation induced by any $\psi_{{\rm{R}},n,\boldsymbol{p}}$ (resp. $\psi_{{\rm{L}},m,\boldsymbol{p}}$) is unitarily equivalent to $\pi_{{\rm{R}}}$ (resp. $\pi_{{\rm{L}}}$). The representations $\pi_{{\rm{R}}/{\rm{L}}}$ appeared in Proposition 3.9. ###### Remark 3.13. Parts (b) and (d) of Proposition 3.9 show that spectral properties of the energy-momentum operators in the representations induced by the pure particle weights $\psi_{{\rm{R}}/{\rm{L}},n,\boldsymbol{p}}$ are different from those in the original representation: In the case of $U_{\pi_{{\rm{R}}}}$ the right branch of the lightcone contains the singularities characteristic for Wigner particles, while in the left branch such singularities are absent. (For $U_{\pi_{{\rm{L}}}}$ the opposite situation occurs). For infraparticles in physical spacetime a more radical version of this phenomenon may occur: There one expects isolated singularities at the characteristic energy-momentum values of the respective pure particle weights. (Cf. Section 2 (iii) of [16]). Proof. Any vector $\Psi\in\mathcal{D}$ has the form $\Psi=\sum_{k,l}c_{k,l}F_{{\rm{R}},k}\Omega_{1}\otimes F_{{\rm{L}},l}\Omega_{2},$ (3.50) where the sum is finite and $F_{{\rm{R}},k}$, $F_{{\rm{L}},l}$ have properties specified below formula (3.48). Applying the Gram-Schmidt procedure, we can ensure that the systems of vectors $\\{F_{{\rm{R}},k}\Omega_{1}\\}_{k=0}^{M}$, $\\{F_{{\rm{L}},l}\Omega_{2}\\}_{l=0}^{N}$ are orthonormal. Since $\hat{\mathcal{H}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{odd}}\subset\mathcal{K}_{1}\otimes\mathcal{K}_{2}=\mathcal{H}$, we can write $\Psi=\sum_{k,l}c_{k,l}\Phi^{\mathrm{out}}_{+}(F_{{\rm{R}},k}\otimes I)\Phi^{\mathrm{out}}_{-}(I\otimes F_{{\rm{L}},l})\Omega,$ (3.51) where we made use of Proposition 3.2 applied to the net $({\mathfrak{A}},U)$. For any ${\hat{L}}\in\hat{\mathcal{L}}$ we define $L=(\pi_{1,\mathrm{odd}}^{-1}\otimes\pi_{2,\mathrm{odd}}^{-1})({\hat{L}})\in\mathcal{L}$, where $\mathcal{L}$ is the left ideal of ${\mathfrak{A}}$, given by definition (2.16). In view of Theorem 2.11, we get $\displaystyle\psi_{\Psi}^{\mathrm{out}}({\hat{L}}_{1},{\hat{L}}_{2})$ $\displaystyle=$ $\displaystyle\sum_{n\in C_{{\rm{R}}}}\int d\boldsymbol{x}\,((G_{{\rm{R}},n}\otimes I)\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(G_{{\rm{R}},n}\otimes I)\Omega)$ (3.52) $\displaystyle+$ $\displaystyle\sum_{m\in C_{{\rm{L}}}}\int d\boldsymbol{x}\,((I\otimes{G}_{{\rm{L}},m})\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(I\otimes{G}_{{\rm{L}},m})\Omega),$ where $G_{{\rm{R}},n}=\sum_{k}c_{k,n}F_{{\rm{R}},k}$, $G_{{\rm{L}},m}=\sum_{l}c_{m,l}F_{{\rm{L}},l}$ and the sets $C_{{\rm{R}}}$ and $C_{{\rm{L}}}$ are chosen so that $\Psi_{{\rm{R}},n}:=(G_{{\rm{R}},n}\otimes I)\Omega\neq 0$ and $\Psi_{{\rm{L}},m}:=(I\otimes G_{{\rm{L}},m})\Omega\neq 0$ for $n\in C_{{\rm{R}}}$ and $m\in C_{{\rm{L}}}$. We note that both sets are non-empty, if $\Psi\neq 0$. (Cf. formula (2.13) and the subsequent remarks). Let us consider the first sum in (3.52) above: Any $L\in\mathcal{L}$ is a finite linear combination of operators of the form $AB$, where $A,B\in{\mathfrak{A}}$ and $B$ is almost local and energy decreasing. Since we assumed that $F_{{\rm{R}},k}\otimes I$ are almost local, the functions $\displaystyle\mathbb{R}\ni\boldsymbol{x}\to ABU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes I)\Omega=A[B,(G_{{\rm{R}},n}\otimes I)(-\boldsymbol{x})]\Omega$ (3.53) decrease in norm faster than any inverse power of $|\boldsymbol{x}|$. Consequently, the Fourier transform $\displaystyle L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}):=(2\pi)^{-1/2}\int d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}LU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes I)\Omega$ (3.54) is a norm-continuous function. It is compactly supported in $\mathbb{R}_{+}$ due to the spectrum condition and the fact that the energy-momentum transfer of each $G_{{\rm{R}},n}\otimes I$ is bounded. By the (Hilbert space valued) Plancherel theorem, we can write $\int d\boldsymbol{x}\,((G_{{\rm{R}},n}\otimes I)\Omega|(L_{1}^{*}L_{2})(\boldsymbol{x})(G_{{\rm{R}},n}\otimes I)\Omega)=\int d\boldsymbol{p}\,(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})).$ (3.55) We define $\displaystyle\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}):=(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})).$ (3.56) It is easy to see that non-zero $\psi_{{\rm{R}},n,\boldsymbol{p}}$ are particle weights in the sense of Definition 2.6: Positivity and property 1 are obvious. The continuity requirement in property 3 follows from the equality $\displaystyle\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2}(y)-{\hat{L}}_{2})$ $\displaystyle=(2\pi)^{-1/2}\int d\boldsymbol{x}\,e^{-i\boldsymbol{p}\boldsymbol{x}}(L_{1}\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})|[(L_{2}(y)-L_{2}),(G_{{\rm{R}},n}\otimes I)(-\boldsymbol{x})]\Omega)$ (3.57) and from the dominated convergence theorem. Invariance under translations (property 2) is a straightforward consequence of the formula $\displaystyle\alpha_{x}(L)\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=e^{-i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi_{{\rm{R}}}}(x)L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}),\quad x\in\mathbb{R}^{2}.$ (3.58) Making use of the above relation and the spectrum condition, it is easy to see that the distribution $\displaystyle\mathbb{R}^{2}\ni q\to(2\pi)^{-1}\int d^{2}x\,e^{-iqx}\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},\alpha_{x}({\hat{L}}_{2}))$ (3.59) is supported in $V_{+}-(\boldsymbol{p},\boldsymbol{p})$. Now let us show that any function $\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}(\,\cdot\,,\,\cdot\,)$, $n\in C_{{\rm{R}}}$, is non-zero on a non-empty open set $\Delta_{{\rm{R}},n}$: Since $(G_{{\rm{R}},n}\otimes I)\Omega\in\mathcal{H}_{\pi_{{\rm{R}}}}$ is different from zero, $U_{\pi_{{\rm{R}}}}$ does not have non-zero invariant vectors and $\pi_{{\rm{R}}}(\hat{\mathfrak{A}})$ acts irreducibly on $\mathcal{H}_{\pi_{{\rm{R}}}}$, Lemma A.1 ensures the existence of ${\hat{L}}\in\hat{\mathcal{L}}$ s.t. $L(G_{{\rm{R}},n}\otimes I)\Omega\neq 0$. Consequently, $\mathbb{R}\ni\boldsymbol{x}\to LU_{\pi_{{\rm{R}}}}(\boldsymbol{x})^{*}(G_{{\rm{R}},n}\otimes I)\Omega$ is a non-zero function and so is the norm of its Fourier transform $\mathbb{R}_{+}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}},{\hat{L}})$. Since the functions $\mathbb{R}_{+}\ni\boldsymbol{p}\to\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})$ are continuous for all ${\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}$, as we have shown above, the sets $\Delta_{{\rm{R}},n}:=\bigcup_{{\hat{L}}_{1},{\hat{L}}_{2}\in\hat{\mathcal{L}}}\\{\,\boldsymbol{p}\in\mathbb{R}_{+}\,|\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})\neq 0\,\\}$ (3.60) are open and non-empty. Let us now fix some $n\in C_{{\rm{R}}}$, $\boldsymbol{p}\in\Delta_{{\rm{R}},n}$ and consider the GNS representation $\pi$ induced by $\psi_{{\rm{R}},n,\boldsymbol{p}}$, acting on the Hilbert space $\mathcal{H}_{\pi}:=(\hat{\mathcal{L}}/\\{\,{\hat{L}}\in\hat{\mathcal{L}}\,|\,\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}},{\hat{L}})=0\,\\})^{\mathrm{cpl}}$. The equivalence class of ${\hat{L}}\in\hat{\mathcal{L}}$ is denoted by $|{\hat{L}}\rangle$ and the scalar product is given by $\langle{\hat{L}}_{1}|{\hat{L}}_{2}\rangle=\psi_{{\rm{R}},n,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})$. This GNS representation has the form $\displaystyle\pi(\hat{A})|{\hat{L}}\rangle$ $\displaystyle=$ $\displaystyle|\hat{A}{\hat{L}}\rangle,\quad\,\,\,\,\,\,{\hat{L}}\in\hat{\mathcal{L}},\,\hat{A}\in\hat{\mathfrak{A}},$ (3.61) $\displaystyle U_{\pi}(x)|{\hat{L}}\rangle$ $\displaystyle=$ $\displaystyle|\alpha_{x}({\hat{L}})\rangle,\quad{\hat{L}}\in\hat{\mathcal{L}},\,x\in\mathbb{R}^{2},$ (3.62) where $U_{\pi}$ is the standard representation of translations. We will show that $(\pi(\hat{\mathfrak{A}}),U_{\pi})$ is unitarily equivalent to $(\pi_{{\rm{R}}}(\hat{\mathfrak{A}}),U_{\pi_{{\rm{R}}}})$. To this end, we introduce the map $W_{{\rm{R}}}:\mathcal{H}_{\pi}\to\mathcal{H}_{\pi_{{\rm{R}}}}=\mathcal{K}_{1,\mathrm{odd}}\otimes\mathcal{K}_{2,\mathrm{ev}}$ given by $W_{{\rm{R}}}|{\hat{L}}\rangle=L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p}),\quad{\hat{L}}\in\hat{\mathcal{L}}.$ (3.63) This map is clearly an isometry. Since $\pi_{{\rm{R}}}$ acts irreducibly on $\mathcal{H}_{\pi_{{\rm{R}}}}$, we obtain that $W_{{\rm{R}}}$ has a dense range and hence it is a unitary operator. From the relation $W_{{\rm{R}}}\pi(\hat{A})|{\hat{L}}\rangle=\pi_{{\rm{R}}}(\hat{A})L\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=\pi_{{\rm{R}}}(\hat{A})W_{{\rm{R}}}|{\hat{L}}\rangle,\quad{\hat{L}}\in\hat{\mathcal{L}},\,\,\hat{A}\in\hat{\mathfrak{A}}$ (3.64) we conclude that $\pi$ and $\pi_{{\rm{R}}}$ are unitarily equivalent. Next, we obtain for any ${\hat{L}}\in\hat{\mathcal{L}}$ and $x\in\mathbb{R}^{2}$ $\displaystyle U_{\pi_{{\rm{R}}}}(x)W_{{\rm{R}}}|{\hat{L}}\rangle=e^{i(\boldsymbol{p},\boldsymbol{p})x}\alpha_{x}(L)\tilde{\Psi}_{{\rm{R}},n}(\boldsymbol{p})=e^{i(\boldsymbol{p},\boldsymbol{p})x}W_{{\rm{R}}}U_{\pi}(x)|{\hat{L}}\rangle,$ (3.65) where in the first step we made use of relation (3.58). We recall that the spectrum of $U_{\pi_{{\rm{R}}}}$ coincides with $V_{+}$ and note that $\pi(\hat{\mathfrak{A}})$ acts irreducibly on $\mathcal{H}_{\pi}$, by relation (3.64) and Proposition 3.9 (c). Thus, in view of equality (3.65), $U^{\mathrm{can}}_{\pi}(x)=e^{i(\boldsymbol{p},\boldsymbol{p})x}U_{\pi}(x),\quad x\in\mathbb{R}^{2}$ (3.66) is the canonical representation of translations in the GNS representation of $\psi_{{\rm{R}},n,\boldsymbol{p}}$. Relation (3.66) shows that $q_{{\rm{R}},n,\boldsymbol{p}}=(\boldsymbol{p},\boldsymbol{p})$. The analysis of the second term on the r.h.s. of (3.52) proceeds similarly: For any $m\in C_{{\rm{L}}}$ and ${\hat{L}}\in\hat{\mathcal{L}}$ one introduces vectors $L\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p}):=(2\pi)^{-1/2}\int d\boldsymbol{x}\,e^{i\boldsymbol{p}\boldsymbol{x}}LU(\boldsymbol{x})^{*}(I\otimes G_{{\rm{L}},m})\Omega$ (3.67) and functionals $\psi_{{\rm{L}},m,\boldsymbol{p}}({\hat{L}}_{1},{\hat{L}}_{2})=(L_{1}\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p})|L_{2}\tilde{\Psi}_{{\rm{L}},m}(\boldsymbol{p}))$. By an analogous reasoning as above one shows that for $\boldsymbol{p}$ in some non-empty, open set $\Delta_{{\rm{L}},m}\subset\mathbb{R}_{+}$ these functionals are particle weights with characteristic energy-momentum vectors $q_{{\rm{L}},m,\boldsymbol{p}}=(\boldsymbol{p},-\boldsymbol{p})$. Their GNS representations are unitarily equivalent to $\pi_{{\rm{L}}}$. $\Box$ ## 4 Conclusions and outlook In this work we carried out a systematic study of particle aspects of two- dimensional conformal field theories both in vacuum representations and in charged representations. In the former case we established a complete particle interpretation in terms of Wigner particles (or ‘waves’ in the terminology of [5]). In the latter case we proved the existence of infraparticles and verified superselection of their direction of motion in a large class of examples. We conclude that conformal field theories provide a valuable testing ground for fundamental concepts of scattering theory. An important question which remained outside of the scope of the present work is the problem of asymptotic completeness in the case of infraparticles. We remark that the theory of particle weights offers natural formulations of this property [9, 11] which can be adapted to the case of massless, two-dimensional theories. We conjecture that any charged representation of a chiral conformal field theory has a complete particle interpretation in terms of infraparticles. A more technical circle of problems concerns the decomposition of particle weights and their representations stated in formulas (1.2), (1.3). We recall that the general procedure of [41, 42] is not canonical: Firstly, it involves a choice of a maximal abelian subalgebra, acting on the representation space of the original weight. Secondly, it relies on a selection of countable subsets of all the objects involved. In view of these ambiguities it is not yet possible to associate a unique family of (infra-)particle types with any given quantum field theory. We feel that a satisfactory solution of these problems requires a systematic study of examples. A useful criterion for their classification is the type of representations induced by particle weights. Thus in the present paper we focused on representations of type I (with atomic center) which have a simple decomposition theory. Already in this elementary case we found a physically interesting phenomenon: superselection of direction of motion. It is a natural direction of further research to look for theories, whose asymptotic functionals induce representations which are not of type I with atomic center. We conjecture that such models exist and some of them describe infraparticles with superselected momentum, similar to the electron in QED. Acknowledgements. The authors would like to thank Prof. D. Buchholz and Prof. R. Longo for interesting discussions. ## Appendix A Proof of Theorem 2.11 ###### Lemma A.1. (a) Let $({\mathfrak{A}},U)$ be a local net of $C^{*}$-algebras on $\mathbb{R}^{2}$ in the sense of Definition 2.1, acting irreducibly on a Hilbert space $\mathcal{H}$ and let $U=U^{\mathrm{can}}$. Let $\Psi\in\mathcal{H}$ be s.t. $A(f)\Psi:=\int d^{2}x\,\alpha_{x}(A)f(x)\Psi=0$ (A.1) for all local operators $A\in{\mathfrak{A}}$ and all $f\in S(\mathbb{R}^{2})$ s.t. $\mathrm{supp}\,\tilde{f}$ is compact and $\mathrm{supp}\,\tilde{f}\cap V_{+}=\varnothing$. Then $\Psi$ is invariant under the action of $U$. (Here $\tilde{f}(p):=(2\pi)^{-1}\int d^{2}x\,e^{ipx}f(x)$ ). (b) Let $({\cal A},V)$ be a local net of von Neumann algebras on $\mathbb{R}$ in the sense of Definition 3.1, acting irreducibly on a Hilbert space ${\cal K}$. Let $\Psi\in{\cal K}$ be s.t. $A(f)\Psi:=\int ds\,\beta_{s}(A)f(s)\Psi=0$ (A.2) for all local operators $A\in{\cal A}$ and all $f\in S(\mathbb{R})$ s.t. $\mathrm{supp}\,\tilde{f}$ is compact and $\mathrm{supp}\,\tilde{f}\cap\mathbb{R}_{+}=\varnothing$. Then $\Psi$ is invariant under the action of $V$. (Here $\tilde{f}(\omega):=(2\pi)^{-\frac{1}{2}}\int ds\,e^{i\omega s}f(s)$ ). Proof. The argument exploits some ideas from the proof of Proposition 2.1 of [13]. As for part (a), suppose that $\Psi$ is not invariant under the action of $U$. Since the map $B(\mathcal{H})\ni A\to A(f)$ is $\sigma$-weakly continuous (cf. Lemma 5.3 of [41]) and ${\mathfrak{A}}$ acts irreducibly on $\mathcal{H}$, condition (A.1) implies that $P(\Delta_{1})A(f)P(\Delta_{2})\Psi=0,$ (A.3) where $P(\,\cdot\,)$ is the spectral measure of $U$ and $\Delta_{1},\Delta_{2}\subset\mathbb{R}^{2}$ are open bounded sets. Since the spectrum of $U$ has Lorentz invariant lower boundary and $\Psi$ is not invariant under translations, we can choose $\Delta_{1},\Delta_{2}$ s.t. $P(\Delta_{1})\neq 0$, $P(\Delta_{2})\Psi\neq 0$ and the closure of $(\Delta_{1}-\Delta_{2})$ does not intersect with $V_{+}$. Choosing $f\in S(\mathbb{R}^{2})$ s.t. $\mathrm{supp}\,\tilde{f}\cap V_{+}=\varnothing$ and $\tilde{f}(p)=1$ for $p$ in the closure of $(\Delta_{1}-\Delta_{2})$, we obtain that $P(\Delta_{1})AP(\Delta_{2})\Psi=0$ (A.4) for any $A\in{\mathfrak{A}}$. Exploiting irreducibility again, we obtain $P(\Delta_{1})=0$, which is a contradiction. The proof of part (b) is analogous. $\Box$ ###### Lemma A.2. Let ${\cal K}_{\pm}\subset\mathcal{H}_{\pm}$ be closed subspaces, invariant under the action of $U$. Let $\\{e_{+,m}\\}_{m\in\mathbb{I}}$ be a complete orthonormal basis in $(P_{E}{\cal K}_{+})$ and let $\\{e_{-,n}\\}_{n\in\mathbb{J}}$ be a complete orthonormal basis in $(P_{E}{\cal K}_{-})$ for some $E\geq 0$. Then any $\Psi\in P_{E}({\cal K}_{+}\overset{\mathrm{out}}{\times}{\cal K}_{-})$ can be expressed as $\Psi=\sum_{m,n}c_{m,n}e_{+,m}\overset{\mathrm{out}}{\times}e_{-,n},$ (A.5) where $\sum_{m,n}|c_{m,n}|^{2}<\infty$. Proof. First, we define a strongly continuous unitary representation of translations $U_{0}(x)(\Psi_{+}\otimes\Psi_{-})=(U(x)\Psi_{+})\otimes(U(x)\Psi_{-}),\quad\Psi_{\pm}\in{\cal K}_{\pm}$ (A.6) on ${\cal K}_{+}\otimes{\cal K}_{-}$. Then we obtain from Proposition 2.4 $\Omega^{\mathrm{out}}U_{0}(x)=U(x)\Omega^{\mathrm{out}}.$ (A.7) For $\Psi^{\prime}=(\Omega^{\mathrm{out}})^{-1}\Psi$ the above relation gives $P_{0,E}\Psi^{\prime}=\Psi^{\prime}$, where $P_{0,E}$ is the spectral projection of $U_{0}$ corresponding to the set $\\{\,(\omega,\boldsymbol{p})\in\mathbb{R}^{2}\,|\,\omega\leq E\,\\}$. By the functional calculus, we get $P_{0,E}=P_{0,E}(P_{E}\otimes P_{E})$. Hence $\Psi^{\prime}=(P_{E}\otimes P_{E})\Psi^{\prime}=\sum_{m,n}c_{m,n}e_{+,m}\otimes e_{-,n}.$ (A.8) By applying $\Omega^{\mathrm{out}}$, we obtain relation (A.5). $\Box$ ###### Lemma A.3. Let $\Psi^{\prime}\in\mathcal{H}$ be a vector of bounded energy, let $F_{1},F_{2}\in{\mathfrak{A}}$ be almost local and of compact energy-momentum transfer, and let $L=\sum_{k=1}^{n}A_{k}B_{k}$, where $A_{k},B_{k}\in{\mathfrak{A}}$ are almost local and $B_{k}$ are, in addition, energy decreasing. Then $\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)=0,$ (A.9) where $Q_{T}=\int dt\,h_{T}(t)\int d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x})$. (The above sequence is well defined by Theorem 2.9). Proof. First, we note that by Proposition 2.3 and Theorem 2.9 $\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)=\lim_{T\to\infty}(\Psi^{\prime}|[[Q_{T},F_{1,+}(h_{T})],F_{2,-}(h_{T})]\Omega),$ (A.10) if the limit on the r.h.s. exists. We introduce the auxiliary operators: $Q_{\pm,T}:=\int dt\,h_{T}(t)\int_{\mathbb{R}_{\pm}}d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x}).$ (A.11) As we show below, they satisfy $\lim_{T\to\infty}\|P_{E}[Q_{\pm,T},F_{\mp}(h_{T})]P_{E}\|=0$ (A.12) for any $E\geq 0$ and any $F\in{\mathfrak{A}}$ which is almost local and of compact energy-momentum transfer. Making use of this relation and the fact that $Q_{T}=Q_{+,T}+Q_{-,T}$, the proof is completed with the help of the Jacobi identity and Proposition 2.3 (c). Let us now verify (A.12). As the two cases are analogous, we focus on one of them and estimate the corresponding expression as follows. $\|P_{E}[Q_{-,T},F_{+}(h_{T})]P_{E}\|\leq\int dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\|[L^{*}L(t,\boldsymbol{x}),F(t_{1},t_{1})]\|.$ (A.13) Since $L^{*}L$ and $F$ are almost local, we can find sequences $C_{r},F_{r}\in{\mathfrak{A}}(\mathcal{O}_{r})$, s.t. for any $n\in\mathbb{N}$ there exist $C_{n},C^{\prime}_{n}$ s.t. $\displaystyle\|L^{*}L-C_{r}\|\leq\frac{C_{n}}{r^{n}},\quad\|F-F_{r}\|\leq\frac{C^{\prime}_{n}}{r^{n}}.$ (A.14) We choose $r=(1+\frac{1}{4}|\boldsymbol{x}|)^{\varepsilon}+T^{\varepsilon}$, where $0<\varepsilon<1$ appeared in the definition of $h_{T}$. We write $\displaystyle[(L^{*}L)(t,\boldsymbol{x}),F(t_{1},t_{1})]$ $\displaystyle=$ $\displaystyle[(L^{*}L-C_{r})(t,\boldsymbol{x}),F(t_{1},t_{1})]$ (A.15) $\displaystyle+$ $\displaystyle[C_{r}(t,\boldsymbol{x}),(F-F_{r})(t_{1},t_{1})]$ $\displaystyle+$ $\displaystyle[C_{r}(t,\boldsymbol{x}),F_{r}(t_{1},t_{1})].$ By estimates (A.14), the first two terms on the r.h.s. above give contributions to (A.13) which tend to zero in the limit $T\to\infty$. The contribution of the last term can be estimated as follows, exploiting locality, $\displaystyle\int dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\|[C_{r}(t,\boldsymbol{x}),F_{r}(t_{1},t_{1})]\|$ $\displaystyle\phantom{444444444444}\leq c\int dtdt_{1}\,h_{T}(t)h_{T}(t_{1})\int_{\mathbb{R}_{-}}d\boldsymbol{x}\chi(|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r),$ (A.16) where $\chi$ is the characteristic function of the respective set and $c$ is a constant independent of $T$. Let us now derive some inequalities which hold on the support of the integrand on the r.h.s. of (A.16). First, we note that $t,t_{1}\in\mathrm{supp}\,h_{T}$, if and only if $t,t_{1}\in T^{\varepsilon}\mathrm{supp}\,h+T$, in particular $|t-t_{1}|\leq c_{1}T^{\varepsilon}$ for some $c_{1}\geq 0$. Exploiting this fact, the inequality $|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r$ and the relation $r=(1+\frac{1}{4}|\boldsymbol{x}|)^{\varepsilon}+T^{\varepsilon}$, we find such $c_{2}\geq 0$ that $|\boldsymbol{x}|\leq c_{2}T$ and $r\leq c_{2}T^{\varepsilon}$, in particular the r.h.s. of (A.16) is finite for any $T\geq 1$. Making use of the inequalities $r\leq c_{2}T^{\varepsilon}$ and $|\boldsymbol{x}-t_{1}|\leq|t-t_{1}|+2r$, and of the fact that $t,t_{1}\in\mathrm{supp}\,h_{T}$, we obtain that $|\boldsymbol{x}-T|\leq c_{3}T^{\varepsilon}$ for some $c_{3}\geq 0$ which implies that $\boldsymbol{x}>0$ for sufficiently large $T$. As the region of integration in the $\boldsymbol{x}$ variable is restricted to $\mathbb{R}_{-}$, we conclude that the r.h.s. of (A.16) is zero for such $T$. $\Box$ Proof of Theorem 2.11: Let $Q_{T}=\int dt\,h_{T}(t)\int d\boldsymbol{x}\,(L^{*}L)(t,\boldsymbol{x})$, where $L=\sum_{k=1}^{n}A_{k}B_{k}$ is an element of the left ideal $\mathcal{L}$, $A_{k}$ are almost local and $B_{k}\in\mathcal{L}_{0}$. Moreover, we choose $\Psi=\Phi^{\mathrm{out}}_{+}(F_{1})\Phi^{\mathrm{out}}_{-}(F_{2})\Omega$ and $\Psi^{\prime}=\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega$, where $F_{1/2},F_{1/2}^{\prime}\in{\mathfrak{A}}$ are almost local and have compact energy-momentum transfer. Since $\Psi$ and $\Psi^{\prime}$ are vectors of bounded energy, we can write $\displaystyle(\Psi^{\prime}|Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Phi^{\mathrm{out}}_{-}(F_{2})\Omega)$ $\displaystyle=$ $\displaystyle(\Psi^{\prime}|[[Q_{T},\Phi^{\mathrm{out}}_{+}(F_{1})],\Phi^{\mathrm{out}}_{-}(F_{2})]\Omega)$ (A.17) $\displaystyle+$ $\displaystyle(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$ $\displaystyle+$ $\displaystyle(\Psi^{\prime}|\Phi^{\mathrm{out}}_{+}(F_{1})Q_{T}\Phi^{\mathrm{out}}_{-}(F_{2})\Omega).$ The term with the double commutator above vanishes as $T\to\infty$ due to Lemma A.3. The second term on the r.h.s. of relation (A.17) is treated as follows: $\displaystyle\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})Q_{T}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$ $\displaystyle\phantom{44444}=\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})\int h_{T}(t)e^{iHt}\int d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})e^{-i\boldsymbol{P}t}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$ $\displaystyle\phantom{444444}=\lim_{T\to\infty}(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})\int h_{T}(t)e^{i(H-\boldsymbol{P})t}\int d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$ $\displaystyle\phantom{4444444444444444444444}=(\Psi^{\prime}|\Phi^{\mathrm{out}}_{-}(F_{2})P_{+}\int d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega),$ (A.18) where in the first step we made use of the fact that $\Phi^{\mathrm{out}}_{+}(F_{1})\Omega=P_{+}\Phi^{\mathrm{out}}_{+}(F_{1})\Omega$, in the second step we exploited the invariance of $\int d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})$ under translations in space and in the last step we made use of the mean ergodic theorem as in the proof of Lemma 1 of [5]. Next, we obtain $\displaystyle(\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega|\Phi^{\mathrm{out}}_{-}(F_{2})P_{+}\int d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$ $\displaystyle\phantom{44}=(\Phi^{\mathrm{out}}_{-}(F_{2})^{*}\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})\Omega|\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})^{*}P_{+}\int d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega)$ $\displaystyle\phantom{44444}=(\Omega|\Phi^{\mathrm{out}}_{-}(F_{2}^{\prime})^{*}\Phi^{\mathrm{out}}_{-}(F_{2})\Omega)\,(\Omega|\Phi^{\mathrm{out}}_{+}(F_{1}^{\prime})^{*}\int d\boldsymbol{x}\,(L^{*}L)(\boldsymbol{x})\Phi^{\mathrm{out}}_{+}(F_{1})\Omega),$ (A.19) where we made use of the facts that $[\Phi^{\mathrm{out}}_{+}(F_{1}),\Phi^{\mathrm{out}}_{-}(F_{2})]=0$ and that $\mathcal{H}_{+}/{\mathbb{C}}\Omega$ is orthogonal to $\mathcal{H}_{-}/{\mathbb{C}}\Omega$ (as in the proof of Lemma 4 (a) of [5]). The last term on the r.h.s. of (A.17) is treated analogously. We note that any $\Psi_{\pm}\in P_{E}\mathcal{H}_{\pm}$ can be approximated by a sequence of vectors of the form $P_{\pm}F_{n}\Omega$, where $F_{n}\in{\mathfrak{A}}$ are quasilocal and have energy-momentum transfers in some fixed compact set. Hence, any $\Psi=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}$ has bounded energy. By the above considerations and Theorem 2.9, we obtain for any $\Psi=\Psi_{+}\overset{\mathrm{out}}{\times}\Psi_{-}$, $\Psi^{\prime}=\Psi_{+}^{\prime}\overset{\mathrm{out}}{\times}\Psi_{-}^{\prime}$, $\Psi_{\pm},\Psi_{\pm}^{\prime}\in P_{E}\mathcal{H}_{\pm}$, $\displaystyle\lim_{T\to\infty}(\Psi^{\prime}|Q_{T}\Psi)$ $\displaystyle=$ $\displaystyle(\Psi_{+}^{\prime}|\Psi_{+})\int d\boldsymbol{x}\,(\Psi_{-}^{\prime}|(L^{*}L)(\boldsymbol{x})\Psi_{-})$ (A.20) $\displaystyle+$ $\displaystyle(\Psi_{-}^{\prime}|\Psi_{-})\int d\boldsymbol{x}\,(\Psi_{+}^{\prime}|(L^{*}L)(\boldsymbol{x})\Psi_{+}).$ Now in view of Lemma A.2, any $\Psi\in P_{E}\mathcal{H}^{\mathrm{out}}$ has the form $\Psi=\sum_{m,n}c_{m,n}e_{+,m}\overset{\mathrm{out}}{\times}e_{-,n},$ (A.21) where $\\{e_{\pm,m}\\}_{m=0}^{\infty}$ are orthonormal systems in $\\{P_{E}\mathcal{H}_{\pm}\\}$, which we choose so that $e_{\pm,0}=\Omega$. Defining $\displaystyle\Psi_{+,n}=\sum_{m}c_{m,n}e_{+,m},\quad\Psi_{-,n}=\sum_{m}c_{n,m}e_{-,m},$ (A.22) we obtain $\rho_{\pm,\Psi}(\,\cdot\,)=\sum_{n}\,(\Psi_{\pm,n}|\,\cdot\,\Psi_{\pm,n})$. Relation (A.20) gives $\lim_{T\to\infty}(\Psi|Q_{T}\Psi)=\int d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L^{*}L)(\boldsymbol{x})\big{)}.$ (A.23) Exploiting the Cauchy-Schwarz inequality and the following bounds, valid for $L=AB$, $A\in\mathfrak{A}$, $B\in\mathcal{L}_{0}$, $\displaystyle|(\Psi|Q_{T}\Psi)|$ $\displaystyle\leq$ $\displaystyle\|P_{E}\int d\boldsymbol{x}\,(B^{*}B)(\boldsymbol{x})P_{E}\|\,\|A^{*}A\|,$ (A.24) $\displaystyle\int d\boldsymbol{x}\,(\rho_{+,\Psi}+\rho_{-,\Psi})\big{(}(L^{*}L)(\boldsymbol{x})\big{)}$ $\displaystyle\leq$ $\displaystyle 2\|P_{E}\int d\boldsymbol{x}\,(B^{*}B)(\boldsymbol{x})P_{E}\|\,\|A^{*}A\|,\,\,\,\,$ (A.25) one extends (A.23) to any $L\in\mathcal{L}$. Now formula (2.36) follows by a polarization argument. Let us now show that $\psi_{\Psi}^{\mathrm{out}}=0$ only if $\Psi\in{\mathbb{C}}\Omega$. By the above considerations we obtain, for any $B\in\mathcal{L}_{0}$, $\psi_{\Psi}^{\mathrm{out}}(B,B)=\sum_{n}\int d\boldsymbol{x}\,\big{\\{}(\Psi_{-,n}|(B^{*}B)(\boldsymbol{x})\Psi_{-,n})+(\Psi_{+,n}|(B^{*}B)(\boldsymbol{x})\Psi_{+,n})\big{\\}}.$ (A.26) If $\psi_{\Psi}^{\mathrm{out}}=0$, then $B\Psi_{\pm,n}=0$ for each $n$ and any such $B$. Thus, by Lemma A.1 (a), $\Psi_{\pm,n}$ are proportional to $\Omega$. Using definitions (A.21), (A.22) and the convention $e_{\pm,0}=\Omega$, it is easily seen that $\Psi$ is proportional to $\Omega$. $\Box$ ## Appendix B Proofs of Lemmas 3.7 and 3.8 Proof of Lemma 3.7: As for the main part of the lemma, it suffices to show that the spectrum of $V_{\mathrm{odd}}$ coincides with $\mathbb{R}_{+}$. It follows from the assumption ${\hbox{\rm Ad}}W\neq{\rm id}$ and the Reeh- Schlieder property of the net $({\cal A},V)$ that ${\cal A}_{\mathrm{odd}}({\cal I})\neq\\{0\\}$ and ${\cal K}_{\mathrm{odd}}=[{\cal A}_{\mathrm{odd}}({\cal I})\Omega_{0}]$ for any open, bounded subset ${\cal I}\subset\mathbb{R}$. Let $P(\,\cdot\,)$ be the spectral measure of $V$ and suppose that $P(\Delta){\cal K}_{\mathrm{odd}}=\\{0\\}$ for some open subset $\Delta\subset\mathbb{R}_{+}$. We fix a non-zero $A\in{\cal A}_{\mathrm{odd}}({\cal I})$. Then, for any $B\in{\cal A}({\cal I})$ the distribution $(\Omega_{0}|[B,\widetilde{A}(\omega)]\Omega_{0})=\frac{1}{\sqrt{2\pi}}\int dt\,e^{-i\omega t}(\Omega_{0}|[B,\beta_{t}(A)]\Omega_{0})$ (B.1) is supported outside of $\Delta\cup-\Delta$. Since, by locality, this distribution is a holomorphic function, it must be zero for all $\omega\in\mathbb{R}$. Thus for any $f\in S(\mathbb{R})$ s.t. $\tilde{f}$ is supported in the interior of $\mathbb{R}_{+}$ we obtain $(\Omega_{0}|BA(f)\Omega_{0})=(\Omega_{0}|B\tilde{f}(T)A\Omega_{0})=0$. Here $T\geq 0$ is the generator of $V$, $A(f)=\int dt\,\beta_{t}(A)f(t)$ and we made use of the fact that $A(f)^{*}\Omega_{0}=0$, due to the support property of $\tilde{f}$ and the spectrum condition. Approximating the characteristic function of the interior of $\mathbb{R}_{+}$ with such $\tilde{f}$ and making use of the fact that $(\Omega_{0}|A\Omega_{0})=0$, we conclude that $A\Omega_{0}=0$ and hence, by the Reeh-Schlieder property $A=0$, which contradicts our assumption. Consequently, $P(\Delta){\cal K}_{\mathrm{odd}}\neq\\{0\\}$ for any open subset $\Delta$ of $\mathbb{R}_{+}$, which means that the spectrum of $V_{\mathrm{odd}}$ coincides with $\mathbb{R}_{+}$. This fact can also be proven as follows: The representation of translations $V$ can be extended to a representation of the $ax+b$ group thanks to the Borchers theorem [27]. There is only one non-trivial, irreducible representation of this group which has positive energy [38] and its spectrum of translations is ${\mathbb{R}}_{+}$. Since ${\cal K}_{\mathrm{odd}}$ does not contain non-trivial invariant vectors of $V$, the spectrum of $V|_{{\cal K}_{\mathrm{odd}}}$ coincides with $\mathbb{R}_{+}$. Let us now proceed to part (a) of the lemma. To show the irreducibility of $\pi_{\mathrm{odd}}$, it suffices to check that any vector $\Psi\in\mathcal{K}_{\mathrm{odd}}$ is cyclic under the action of $\pi_{\mathrm{odd}}({\cal A}_{\mathrm{ev}})$. By contradiction, we suppose that there is $\Psi^{\prime}\in\mathcal{K}_{\mathrm{odd}}$ s.t. $(\Psi^{\prime}|A\Psi)=0$ for any $A\in{\cal A}_{\mathrm{ev}}$. But this implies that $(\Psi^{\prime}|B\Psi)=0$ for any $B\in{\cal A}$, which contradicts the irreducibility of the action of ${\cal A}$ on $\mathcal{K}$. Next we verify the faithfulness of $\pi_{\mathrm{odd}}$ restricted to a local algebra. Let $A\in{\cal A}_{\mathrm{ev}}({\cal I})$ be a positive local element which is zero upon restriction to $\mathcal{K}_{\mathrm{odd}}$. For any local odd element $B\in{\cal A}_{\mathrm{odd}}(\mathfrak{J})$ and for sufficiently large $s$ we obtain $0=(\Omega|\beta_{s}(B^{*})A\beta_{s}(B)\Omega)=(\Omega|\beta_{s}(B^{*}B)A\Omega)\to(\Omega|B^{*}B\Omega)\cdot(\Omega|A\Omega),$ (B.2) where in the last step we took the limit $s\to\infty$. By the Reeh-Schlieder property it follows that $A=0$. This implies that $\pi_{\mathrm{odd}}$ is faithful on ${\cal A}_{\mathrm{ev}}({\cal I})$ by Proposition 2.3.3 (3) of [17]. Now the faithfulness of $\pi_{\mathrm{odd}}$ on the quasilocal algebra ${\cal A}_{\mathrm{ev}}$ follows from Proposition 2.3.3 (2) of [17], which says that $\pi_{\mathrm{odd}}$ is faithful, if and only if $\|\pi_{\mathrm{odd}}(A)\|=\|A\|$ for any $A\in{\cal A}_{\mathrm{ev}}$. Local normality of $\pi_{\mathrm{odd}}$ is obvious, since $\pi_{\mathrm{odd}}$ acts by the restriction to a subspace. Indeed, making use of Lemma 2.4.19 from [17] and of the fact that $\pi_{\mathrm{odd}}$ preserves the norm, it is easy to check that $\mathrm{l.u.b.}\pi_{\mathrm{odd}}(A_{\alpha})=\pi_{\mathrm{odd}}(\mathrm{l.u.b.}\,A_{\alpha})$, where l.u.b denotes the least upper bound and $\\{A_{\alpha}\\}_{\alpha\in\mathbb{I}}$ is a uniformly bounded increasing net of positive operators from some ${\cal A}_{\mathrm{ev}}({\cal I})$. Part (b) of the lemma follows from the uniqueness of the invariant vector of $V$. $\Box$ Proof of Lemma 3.8: We know from Lemma 3.7 that ${\cal A}_{\mathrm{ev}}\neq{\mathbb{C}}I$, since it can be irreducibly represented on the infinite dimensional Hilbert space ${\cal K}_{\mathrm{odd}}$. Consequently, we can find a non-zero $A\in{\cal A}_{\mathrm{ev}}({\cal I})$, for some open, bounded ${\cal I}$, s.t. $(\Omega_{0}|A\Omega_{0})=0$. Proceeding identically as in the proof of the main part of Lemma 3.7, we conclude that the spectrum of $V_{\mathrm{ev}}$ coincides with $\mathbb{R}_{+}$. Part (b) follows trivially from the fact that the net $({\cal A},V)$ is in a vacuum representation. Irreducibility in part (a) follows from part (b). The remaining part of the statement is proven analogously as the corresponding part of Lemma 3.7. $\Box$ ## References * [1] H. Araki. _Mathematical theory of quantum fields_. Oxford University Press, Oxford, 1999. * [2] H. Araki and R. 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arxiv-papers
2011-01-29T15:36:25
2024-09-04T02:49:16.715014
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Wojciech Dybalski, Yoh Tanimoto", "submitter": "Yoh Tanimoto", "url": "https://arxiv.org/abs/1101.5700" }
1101.5785
# Statistical Compressed Sensing of Gaussian Mixture Models Guoshen Yu ECE, University of Minnesota, Minneapolis, Minnesota, 55414, USA Guillermo Sapiro ECE, University of Minnesota, Minneapolis, Minnesota, 55414, USA ###### Abstract A novel framework of compressed sensing, namely statistical compressed sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution, and achieving accurate reconstruction on average, is introduced. SCS based on Gaussian models is investigated in depth. For signals that follow a single Gaussian model, with Gaussian or Bernoulli sensing matrices of $\mathcal{O}(k)$ measurements, considerably smaller than the $\mathcal{O}(k\log(N/k))$ required by conventional CS based on sparse models, where $N$ is the signal dimension, and with an optimal decoder implemented via linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is shown tightly upper bounded by a constant times the best $k$-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional sparsity-oriented CS. Stronger yet simpler results further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best $k$-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs), that assume multiple Gaussian distributions and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of sensing measurements. A maximum a posteriori expectation-maximization algorithm that iteratively estimates the Gaussian models parameters, the signals model selection, and decodes the signals, is presented for GMM-based SCS. In real image sensing applications, GMM-based SCS is shown to lead to improved results compared to conventional CS, at a considerably lower computational cost. ## I Introduction Compressed sensing (CS) aims at achieving accurate signal reconstruction while sampling signals at a low sampling rate, typically far smaller than that of Nyquist/Shannon. Let $\mathbf{x}\in\mathbb{R}^{N}$ be a signal of interest, $\Phi\in\mathbb{R}^{M\times N}$ a non-adaptive sensing matrix (encoder), consisting of $M\ll N$ measurements, $\mathbf{y}=\Phi\mathbf{x}\in\mathbb{R}^{M}$ a measured signal, and $\Delta$ a decoder used to reconstruct $\mathbf{x}$ from $\Phi\mathbf{x}$. CS develops encoder-decoder pairs $(\Phi,\Delta)$ such that a small reconstruction error $\mathbf{x}-\Delta(\Phi\mathbf{x})$ can be achieved. Reconstructing $\mathbf{x}$ from $\Phi\mathbf{x}$ is an ill-posed problem whose solution requires some prior information on the signal. Instead of the frequency band-limit signal model assumed in classic Shannon sampling theory, conventional CS adopts a sparse signal model, i.e., there exists a dictionary, typically an orthogonal basis $\Psi\in\mathbb{R}^{N\times N}$, a linear combination of whose columns generates an accurate approximation of the signal, $\mathbf{x}\approx\Psi\mathbf{a}$, the coefficients $\mathbf{a}[m]$, $1\leq m\leq N$, having their amplitude decay fast after being sorted. For signals following the sparse model, it has been shown that using some random sensing matrices such as Gaussian and Bernoulli matrices $\Phi$ with $M=\mathcal{O}(k\log(N/k))$ measurements, and an $l_{1}$ minimization or a greedy matching pursuit decoder $\Delta$ promoting sparsity, with high probability CS leads to accurate signal reconstruction: The obtained approximation error is tightly upper bounded by a constant times the best $k$-term approximation error, the minimum error that one may achieve by keeping the $k$ largest coefficients in $\mathbf{a}$ [12, 13, 17, 18]. Redundant and signal adaptive dictionaries that further improve the CS performance with respect to orthogonal bases have been investigated [11, 19, 31]. In addition to sparse models, manifold models have been considered for CS as well [6, 16]. The present paper introduces a novel framework of CS, namely statistical compressed sensing (SCS). As opposed to conventional CS that deals with one signal at a time, SCS aims at efficiently sampling a collection of signals and having accurate reconstruction on average. Instead of restricting to sparse models, SCS works with general Bayesian models. Assuming that the signals $\mathbf{x}$ follow a distribution with probability density function (pdf) $f(\mathbf{x})$, SCS designs encoder-decoder pairs $(\Phi,\Delta)$ so that the average error $E_{\mathbf{x}}\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}=\int\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}f(\mathbf{x})d\mathbf{x},\vspace{0ex}$ where $\|\cdot\|_{X}$ is a norm, is small. As an important example, SCS with Gaussian models is here shown to have improved performance (bounds) relative to conventional CS, the signal reconstruction calculated with an optimal decoder $\Delta$ implemented via a fast linear filtering. Moreover, for Gaussian mixture models (GMMs) that better describe most real signals, SCS with a piecewise linear decoder is investigated. The motivation of SCS with Gaussian models is twofold. First, controlling the average error over a collection of signals is useful in signal acquisition, not only because one is often interested in acquiring a collection of signals in real applications, but also because more effective processing of an individual signal, an image or a sound for example, is usually achieved by dividing the signal in (often overlapping) local subparts, patches (see Figure 10) or short-time windows for instance, so a signal can be regarded as a collection of subpart signals [2, 8, 35, 36]. In addition, Gaussian mixture models (GMMs), which model signals or subpart signals with a collection of Gaussians, assuming each signal drawn from one of them, have been shown effective in describing real signals, leading to state-of-the-art results in image inverse problems [36] and missing data estimation [24]. SCS based on a single Gaussian model is first developed in Section II. Following a similar mathematical approach as the one adopted in conventional CS performance analysis [17], it is shown that with the same random matrices as in conventional CS, but with a considerably reduced number $M=\mathcal{O}(k)$ of measurements, and with the optimal decoder implemented via linear filtering, significantly faster than the decoders applied in conventional CS, the average error of Gaussian SCS is tightly upper bounded by a constant times the best $k$-term approximation error with overwhelming probability, the failure probability being orders of magnitude smaller than that of conventional CS. Moreover, stronger yet simpler results further show that for any sensing matrix, the average error of Gaussian SCS is upper bounded by a constant times the best $k$-term approximation with probability one, and the bound constant can be efficiently calculated. Section III extends SCS to GMMs. A piecewise linear GMM-based SCS decoder, which essentially consists of estimating the signal using each Gaussian model included in the GMM and then selecting the best model, is introduced. The accuracy of the model selection, at the heart of the scheme, is analyzed in detail in terms of the properties of the Gaussian distributions and the number of sensing measurements. These results are then important in the general area of model selection from compressed measurements. Following [36], Section IV presents an maximum a posteriori expectation- maximization (MAP-EM) algorithm that iteratively estimates the Gaussian models and decodes the signals. GMM-based SCS calculated with the MAP-EM algorithm is applied in real image sensing, leading to improved results with respect to conventional CS, at a considerably lower computational cost. ## II Performance Bounds for a Single Gaussian Model This section analyzes the performance bounds of SCS based on a single Gaussian model. Perfect reconstruction of degenerated Gaussian signals is briefly discussed. After reviewing basic properties of linear approximation for Gaussian signals, the rest of the section shows that for Gaussian signals with fast eigenvalue decay, the average error of SCS using $k$ measurements and decoded by a linear estimator is tightly upper bounded by that of best $k$-term approximation. Signals $\mathbf{x}\in\mathbb{R}^{N}$ are assumed to follow a Gaussian distribution $\mathcal{N}(\mu,\Sigma)$ in this section. Principal Component Analysis (PCA) calculates a basis change $\mathbf{a}=\mathbf{B}^{T}(\mathbf{x}-\mu)$ of the data $\mathbf{x}$, with $\mathbf{B}$ the orthonormal PCA basis that diagonalizes the data covariance matrix $\Sigma=\mathbf{B}\mathbf{S}\mathbf{B}^{T},$ (1) where $\mathbf{S}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$ is a diagonal matrix whose diagonal elements $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N}$ are the sorted eigenvalues, and $\mathbf{a}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$ the PCA coefficient vector [27]. In this section, for most of the time we will assume without loss of generality that $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$ by looking in the PCA domain. For Gaussian and Bernoulli matrices that are known to be universal, analyzing CS in canonical basis or PCA basis is equivalent [4]. ### II-A Degenerated Gaussians Conventional CS is able to perfectly reconstruct $k$-sparse signals, i.e., $\mathbf{x}\in\mathbb{R}^{N}$ with at most $k$ non-zero entries (typically $k\ll N$), with $2k$ measurements [17]. Degenerated Gaussian distributions $\mathcal{N}(\mathbf{0},\mathbf{S}_{k})$, where $\mathbf{S}_{k}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{k},0,\ldots,0)$ with at most $k$ non-zero eigenvalues, give the counterpart of $k$-sparsity for the Gaussian signal models considered in this paper. Such signals belong to a linear subspace $\mathcal{S}_{k}=\\{\mathbf{x}|\mathbf{x}[m]=0,\forall k<m\leq N\\}$. The next lemma gives a condition for perfect reconstruction of signals in $\mathcal{S}_{k}$. ###### Lemma 1. If $\Phi$ is any $M\times N$ matrix and $k$ is a positive integer, then there is a decoder $\Delta$ such that $\Delta(\Phi\mathbf{x})=\mathbf{x}$, for all $\mathbf{x}\in\mathcal{S}_{k}$, if and only if $\mathcal{S}_{k}\cap\mathrm{Null}(\Phi)=\mathbf{0}$. ###### Proof. Suppose there is a decoder $\Delta$ such that $\Delta(\Phi\mathbf{x})=\mathbf{x}$, for all $\mathbf{x}\in\mathcal{S}_{k}$. Let $\mathbf{x}=\mathcal{S}_{k}\cap\mathrm{Null}(\Phi)$. We can write $\mathbf{x}=\mathbf{x}_{1}-\mathbf{x}_{2}$ where both $\mathbf{x}_{1},\mathbf{x}_{2}\in\mathcal{S}_{k}$. Since $\Phi\mathbf{x}=\mathbf{0}$, $\Phi\mathbf{x}_{1}=\Phi\mathbf{x}_{2}$. Plugging $\Phi\mathbf{x}_{1}$ and $\Phi\mathbf{x}_{2}$ into the decoder $\Delta$, we obtain $\mathbf{x}_{1}=\mathbf{x}_{2}$ and then $\mathbf{x}=\mathbf{x}_{1}-\mathbf{x}_{2}=\mathbf{0}$. Suppose $\mathcal{S}_{k}\cap\mathrm{Null}(\Phi)=\mathbf{0}$. If $\mathbf{x}_{1},\mathbf{x}_{2}\in\mathcal{S}_{k}$ with $\Phi\mathbf{x}_{1}=\Phi\mathbf{x}_{2}$, then $\mathbf{x}_{1}-\mathbf{x}_{2}\in\mathcal{S}_{k}\cap\mathrm{Null}(\Phi)$, so $\mathbf{x}_{1}=\mathbf{x}_{2}$. $\Phi$ is thus a one-to-one map. Therefore there must exist a decoder $\Delta$ such that $\Delta(\Phi\mathbf{x})=\mathbf{x}$. ∎ It is possible to construct matrices of size $M\times N$ with $M=k$ which satisfies the requirement of the Lemma. A trivial example is $[\mathbf{I}_{M\times M}|\mathbf{0}_{M\times(N-M)}]$, where $\mathbf{I}_{M\times M}$ is the identity matrix of size $M\times M$ and $\mathbf{0}_{M\times(N-M)}$ is a zero matrix of size $M\times(N-M)$. Comparing with conventional compressed sensing that requires $2k$ measurements for exact reconstruction of $k$-sparse signals, with only $k$ measurements signals in $\mathcal{S}_{k}$ can be exactly reconstructed. Indeed, in contrast to the $k$-sparse signals where the positions of the non-zero entries are unknown ($k$-sparse signals reside in multiple $k$-dimensional subspaces), with the degenerated Gaussian model $\mathcal{N}(\mathbf{0},\mathbf{S}_{k})$, the position of the non-zero coefficients are known a priori to be the first $k$ ones. $k$ measurements thus suffice for perfect reconstruction. In the following, we will concentrate on the more general case of non- degenerated Gaussian signals (i.e., Gaussians with full-rank covariance matrices $\Sigma$) with fast eigenvalue decay, in analogy to compressible signals for conventional CS. As mentioned before and will be further experimented in this paper, such simple models not only lead to improved theoretical bounds, but also provide state-of-the-art image reconstruction results. ### II-B Optimal Decoder To simplify the notation, we assume without loss of generality that the Gaussian has zero mean ${\mu}=\mathbf{0}$, as one can always center the signal with respect to the mean. It is well-known that the optimal decoders for Gaussian signals are calculated with linear filtering: ###### Theorem 1. [23] Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector with prior pdf $\mathcal{N}(\mathbf{0},\Sigma)$, and $\Phi\in\mathbb{R}^{M\times N}$, $M\leq N$, be a sensing matrix. From the measured signal $\mathbf{y}=\Phi\mathbf{x}\in\mathbb{R}^{M}$, the optimal decoder $\Delta$ that minimizes the mean square error (MSE) $E_{\mathbf{x}}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]=\min_{g}E_{\mathbf{x}}[\|\mathbf{x}-g(\Phi\mathbf{x})\|_{2}^{2}],$ as well as the mean absolute error (MAE) $E_{\mathbf{x}}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{1}]=\min_{g}E_{\mathbf{x}}[\|\mathbf{x}-g(\Phi\mathbf{x})\|_{1}],$ where $g:\mathbb{R}^{M}\rightarrow\mathbb{R}^{N}$, is obtained with a linear MAP estimator, $\Delta(\Phi\mathbf{x})=\arg\max_{\mathbf{x}}p(\mathbf{x}|\mathbf{y})=\underbrace{\Sigma\Phi^{T}(\Phi\Sigma\Phi^{T})^{-1}}_{\Delta}(\Phi\mathbf{x}),\vspace{0ex}$ (2) and the resulting error $\eta=\mathbf{x}-\Delta(\Phi\mathbf{x})$ is Gaussian with mean zero and with covariance matrix $\Sigma_{\eta}=E_{\mathbf{x}}[\eta\eta^{T}]=\Sigma-\Sigma\Phi^{T}(\Phi\Sigma\Phi^{T})^{-1}\Phi\Sigma,$ whose trace yields the MSE of SCS. $E_{\mathbf{x}}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]=Tr(\Sigma-\Sigma\Phi^{T}(\Phi\Sigma\Phi^{T})^{-1}\Phi\Sigma).\vspace{0ex}$ (3) In contrast to conventional CS, for which the $l_{1}$ minimization or greedy matching pursuit decoders, calculated with iterative procedures, have been shown optimal under certain conditions on $\Phi$ and the signal sparsity [12, 18], Gaussian SCS enjoys the advantage of having an optimal decoder (2) calculated fast via a closed-form linear filtering for any $\Phi$. ###### Corollary 1. If a random matrix $\Phi\in\mathbb{R}^{M\times N}$ is drawn independently to sense each $\mathbf{x}$, with all the other conditions as in Theorem 1, the MSE of SCS is $E_{\mathbf{x},\Phi}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]=E_{\Phi}[Tr(\Sigma-\Sigma\Phi^{T}(\Phi\Sigma\Phi^{T})^{-1}\Phi\Sigma)].\vspace{0ex}$ (4) Applying an independent random matrix realization to sense each signal has been considered in [17]. In real applications, these random sensing matrices need not to be stored, since the decoder can regenerate them itself given the random seed. Following a PCA basis change (1), it is equivalent to consider signals $\mathbf{x}\sim\mathcal{N}(\mu,\Sigma)$ and $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, where $\mathbf{S}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$ is a diagonal matrix whose diagonal elements $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N}$ are the sorted eigenvalues. Theorem 1 and Corollary 1 clearly hold for $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, with (2), (3), and (4) rewritten as $\displaystyle\Delta(\Phi\mathbf{x})$ $\displaystyle=$ $\displaystyle\underbrace{\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}}_{\Delta}(\Phi\mathbf{x}),$ (5) $\displaystyle E_{\mathbf{x}}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]$ $\displaystyle=$ $\displaystyle Tr(\mathbf{S}-\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{S}),$ (6) $\displaystyle E_{\mathbf{x},\Phi}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]$ $\displaystyle=$ $\displaystyle E_{\Phi}[Tr(\mathbf{S}-\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{S})].$ (7) Note that PCA bases, as sparsifying dictionaries, have been applied to do conventional CS based on sparse models [29], which is fundamentally different than the Gaussian models and SCS here studied. ### II-C Linear vs Nonlinear Approximation Before proceeding with the analysis of the SCS performance, let us make some comments on the relationship between linear and non-linear approximations for Gaussian signals. In particular, the following is observed via Monte Carlo simulations: For Gaussian signals $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, where $\mathbf{S}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$ whose eigenvalues $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N}$ decay fast, the best $k$-term linear approximation ${\mathbf{x}}^{l}_{k}(m)=\left\\{\begin{array}[]{cc}\mathbf{x}(m)&~{}~{}1\leq m\leq k,\\\ 0&~{}~{}k+1\leq m\leq N,\end{array}\right.$ (8) and the nonlinear approximation ${\mathbf{x}}^{n}_{k}=T_{k}(\mathbf{x}),$ (9) where $T_{k}$ is a thresholding operator that keeps the $k$ coefficients of largest amplitude and setting others to zero, lead to comparable approximation errors $\sigma^{l}_{k}(\\{\mathbf{x}\\})_{X}=E_{\mathbf{x}}[\|\mathbf{x}-{\mathbf{x}}^{l}_{k}\|_{X}]~{}~{}~{}\textrm{and}~{}~{}~{}\sigma^{n}_{k}(\\{\mathbf{x}\\})_{X}=E_{\mathbf{x}}[\|\mathbf{x}-{\mathbf{x}}^{n}_{k}\|_{X}].$ (10) Monte Carlo simulations are performed to test this. Assuming a power decay of the eigenvalues [27], $\lambda_{m}=m^{-\alpha},~{}~{}~{}1\leq m\leq N,\vspace{-1.5ex}$ (11) where $\alpha>0$ is the decay parameter, with $N=64$, Figure 1 plots the MSEs $\sigma^{l}_{k}(\\{\mathbf{x}\\})_{2}^{2}=E_{\mathbf{x}}[\|\mathbf{x}-{\mathbf{x}}^{l}_{k}\|_{2}^{2}]~{}~{}~{}\textrm{and}~{}~{}~{}\sigma^{n}_{k}(\\{\mathbf{x}\\})_{2}^{2}=E_{\mathbf{x}}[\|\mathbf{x}-{\mathbf{x}}^{n}_{k}\|_{2}^{2}],$ (12) normalized by the ideal signal energy $\|\mathbf{x}\|_{2}^{2}$, of best $k$-term linear and nonlinear approximations as a function of $\alpha$, with typical (for image patches of size $8\times 8$ for example) $k$ values $8$ and $16$ ($k/N=1/8$ and $1/4$). Both MSEs decrease as $\alpha$ increases, i.e., as the eigenvalues decay faster. With typical values $\alpha\approx 3$ (similar to the eigenvalue decay calculated with typical image patches) and $k=8$ or 16, both approximations are accurate and generate small and comparable MSEs, their difference being about $0.1\%$ of the signal energy and ratio about 2. | | ---|---|--- (a) | (b) | (c) Figure 1: (a). MSEs (normalized by the ideal signal energy) of best $k$-term linear and non-linear approximation, with $k=8$ and $16$ (signal dimension $N=64$). (b) and (c) Difference and ratio of normalized MSEs of best $k$-term linear and non-linear approximation shown in (a). Following this, the error of Gaussian SCS will be compared with that of best $k$-term linear approximation, which is comparable to that of best $k$-term nonlinear approximation. For simplicity, the best $k$-term linear approximation errors will be denoted as $\sigma_{k}(\\{\mathbf{x}\\})_{X}=\sigma^{l}_{k}(\\{\mathbf{x}\\})_{X}~{}~{}~{}\textrm{and}~{}~{}~{}\sigma_{k}(\\{\mathbf{x}\\})_{2}^{2}=\sigma^{l}_{k}(\\{\mathbf{x}\\})_{2}^{2}.$ (13) Note that $\sigma_{k}(\\{\mathbf{x}\\})_{2}^{2}=\sum_{m=k+1}^{N}\lambda_{m}$. ### II-D Performance of Gaussian SCS – A Numerical Analysis At First This section numerically evaluates the MSE of Gaussian SCS, and compares it with the minimal MSE generated by best $k$-term linear approximation, proceeding the theoretical bounds later developed. As before, a power decay of the eigenvalues (11), with $N=64$, is assumed in the Monte Carlo simulations. An independent random Gaussian matrix realization $\Phi$ is applied to sense each signal $\mathbf{x}$ [17]. Figures 2 (a) and (c)-top plot the MSE (normalized by the ideal signal energy) of SCS and that of the best $k$-term linear approximation, as well as their ratio as a function of $\alpha$, with $k$ fixed at typical values $8$ and $16$ ($k/N=1/8$ and $1/4$). As $\alpha$ increases, i.e., as the eigenvalues decay faster, the MSEs for both methods decrease. Their ratio increases almost linearly with $\alpha$. The same is plotted in figures 2 (b) and (c)-bottom, with eigenvalue decay parameter fixed at a typical value $\alpha=3$, and with $k$ varying from $5$ to $32$ ($k/N$ from $5/64$ to $1/2$). As $k$ increases, both MSEs decrease, their ratio being almost constant at about $3.7$. | | ---|---|--- (a) | (b) | (c) Figure 2: Comparison of the MSE of SCS and that of the best $k$-term linear approximation for Gaussian signals of dimension $N=64$. (a) and (c)-top. The MSE (normalized by the ideal signal energy) of SCS and that of best $k$-term linear approximation, as well as their ratio as a function of $\alpha$, with $k$ fixed at typical values $8$ and $16$. (b) and (c)-bottom. The same values, with eigenvalue decay parameter fixed at a typical value $\alpha=3$, and with $k$ varying from $5$ to $32$. These results indicate a good performance of Gaussian SCS, its MSE is only a small number of times larger than that of the best $k$-term linear approximation. 111Simulations using the same coefficient energy power decay model (11) show that the ratio between conventional CS based on sparse models, with $k$ measurements, and that of the best $k$-term nonlinear approximation, varies as a function of the decay parameter $\alpha$ and $k$. For typical values $\alpha=3$, the ratio is typically an order of magnitude larger than that between the MSE of SCS and that of the best $k$-term linear approximation. The next sections provide mathematical analysis of this performance. Let us notice that while the best $k$-term linear approximation decoding is feasible for signals following a single Gaussian distribution, it is impractical with GMMs (assuming multiple Gaussians and that each signal is generated from one of them with an unknown index), since the Gaussian index of the signal is unknown. SCS with GMMs, which describe real data considerably better than a single Gaussian model [36], will be described in sections III and IV. ### II-E Performance Bounds Following the analysis techniques in [17], this section shows that with Gaussian and Bernoulli random matrices of $\mathcal{O}(k)$ measurements, considerably smaller than the $\mathcal{O}(k\log(N/k))$ required by conventional CS, the average error of Gaussian SCS is tightly upper bounded by a constant times the best $k$-term linear approximation error with overwhelming probability, the failure probability being orders of magnitude smaller than that of conventional CS. We consider only the encoder-decoder pairs $(\Phi,\Delta)$ that preserve $\Phi\mathbf{x}$, i.e., $\Phi(\Delta(\Phi\mathbf{x}))=\Phi\mathbf{x}$, satisfied by the optimal $\Delta$ in (5) for Gaussian signals $\mathbf{x}$, $\forall\Phi$. #### II-E1 From Null Space Property to Instance Optimality The instance optimality in expectation bounds the average error of SCS with a constant times that of the best $k$-term linear approximation (13), defining the desired SCS performance: ###### Definition 1. Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain distribution. Let $K\subset\\{1,\ldots,N\\}$ be any subset of indices. We say that $(\Phi,\Delta)$ is instance optimal in expectation in $K$ in $\|\cdot\|_{X}$, with a constant $C_{0}$, if $E_{\mathbf{x},(\Phi)}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}]\leq C_{0}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{X}],\vspace{0ex}$ (14) where $\mathbf{x}_{K}$ is the signal $\mathbf{x}$ restricted to $K$ ($\mathbf{x}_{K}[n]=\mathbf{x}[n],~{}\forall~{}n\in K$, and $0$ otherwise), the expectation on the left side considered with respect to $\mathbf{x}$, and to $\Phi$ if one random $\Phi$ is drawn independently for each $\mathbf{x}$. Similarly, the MSE instance optimality in $K$ is defined as $E_{\mathbf{x},(\Phi)}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]\leq C_{0}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{2}^{2}].\vspace{0ex}$ (15) In particular, if $K=\\{1,\ldots,k\\}$, then we say that $(\Phi,\Delta)$ is instance optimal in expectation of order $k$ in $\|\cdot\|_{X}$, with a constant $C_{0}$, if $E_{\mathbf{x},(\Phi)}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}]\leq C_{0}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{X}]=C_{0}\sigma_{k}(\\{\mathbf{x}\\})_{X},\vspace{0ex}$ (16) and is instance optimal of order $k$ in MSE, with a constant $C_{0}$, if $E_{\mathbf{x},(\Phi)}[\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{2}^{2}]\leq C_{0}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{2}^{2}]=C_{0}\sigma_{k}(\\{\mathbf{x}\\})_{2}^{2}.\vspace{0ex}$ (17) The null space property in expectation defined next will be shown equivalent to the instance optimality in expectation. ###### Definition 2. Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain distribution. Let $K\subset\\{1,\ldots,N\\}$ be any subset of indices. We say that $\Phi$ in $(\Phi,\Delta)$ has the null space property in expectation in $K$ in $\|\cdot\|_{X}$, with constant $C$, if $E_{\mathbf{x},(\Phi)}[\|\eta\|_{X}]\leq CE_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}],~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}),$ (18) where $\eta_{K}$ is the signal $\eta$ restricted to $K$ ($\eta_{K}[n]=\eta[n],~{}\forall~{}n\in K$, and $0$ otherwise), the expectation considered on the left side with respect to $\mathbf{x}$, and to $\Phi$ if one random $\Phi$ is drawn independently for each $\mathbf{x}$. Note that $\eta\in\mathrm{Null}(\Phi)$. Similarly, the MSE null space property in $K$ is defined as $E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}\leq CE_{\mathbf{x}}[\|\eta-\eta_{K}\|_{2}^{2}],~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}).$ (19) In particular, if $K=\\{1,\ldots,k\\}$, with $1\leq k\leq N$, then we say that $\Phi$ in $(\Phi,\Delta)$ has the null space property in expectation of order $k$ in $\|\cdot\|_{X}$, with constant $C$, if $E_{\mathbf{x},(\Phi)}\|\eta\|_{X}\leq CE_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}]=C\sigma_{k}(\\{\eta\\})_{X},~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}),$ (20) and has the MSE null space property of order $k$, if $E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}\leq CE_{\mathbf{x}}[\|\eta-\eta_{K}\|_{2}^{2}]=C\sigma_{k}(\\{\eta\\})_{2}^{2},~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}).$ (21) ###### Theorem 2. Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain distribution. Given an $M\times N$ matrix $\Phi$, a norm $\|\cdot\|_{X}$, and a subset of indices $K\subset\\{1,\ldots,N\\}$, a sufficient condition that there exists a decoder $\Delta$ such that the instance optimality in expectation in $K$ in $\|\cdot\|_{X}$ (14) holds with constant $C_{0}$, is that the null space property in expectation (18) holds with $C=C_{0}/2$ for this $(\Phi,\Delta)$: $E_{\mathbf{x},(\Phi)}[\|\eta\|_{X}]\leq\frac{C_{0}}{2}E_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}],~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}).$ (22) A necessary condition is the null space property in expectation (18) with $C=C_{0}$: $E_{\mathbf{x},(\Phi)}[\|\eta\|_{X}]\leq C_{0}E_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}],~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}),$ (23) Similar results hold between the MSE instance optimality in $K$ (15) and the null space property (19), with the constant $C=C_{0}/4$ in the sufficient condition. In particular, if $K=\\{1,\ldots,k\\}$, with $1\leq k\leq N$, the same equivalence between the instance optimality in expectation of order $k$ in $\|\cdot\|_{X}$ (16) and the null space property in expectation (20), and that between the MSE instance optimality of order $k$ (17) and the null space property (21), hold as well. ###### Proof. To prove the sufficiency of (22), we consider the decoder $\Delta$ such that for all $\mathbf{y}=\Phi\mathbf{x}\in\mathbb{R}^{M}$, $\Delta(\mathbf{y}):=\arg\min_{\mathbf{z}}\|\mathbf{z}-\mathbf{z}_{K}\|_{X}~{}~{}~{}\textrm{s.t.}~{}~{}~{}\Phi\mathbf{z}=\mathbf{y}.$ (24) By (22), we have $\displaystyle E\|\mathbf{x}-\Delta(\Phi\mathbf{x})\|_{X}$ $\displaystyle\leq$ $\displaystyle\frac{C_{0}}{2}E_{\mathbf{x}}[\|(\mathbf{x}-\Delta(\Phi\mathbf{x}))-(\mathbf{x}_{K}-(\Delta\Phi\mathbf{x})_{K})\|_{X}]$ (25) $\displaystyle\leq$ $\displaystyle\frac{C_{0}}{2}(E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{X}]+E_{\mathbf{x}}[\|\Delta(\Phi\mathbf{x})-(\Delta\Phi\mathbf{x})_{K}\|_{X}])$ $\displaystyle\leq$ $\displaystyle{C_{0}}E_{\mathbf{x}}[\|\mathbf{x}-\mathbf{x}_{K}\|_{X}],$ where the second inequality uses the triangle inequality, and the last inequality follows from the choice of the decoder (24). To prove the necessity of (23), let $\Delta$ be any decoder for which (14) holds. Let $\eta=\mathbf{x}-\Delta(\Phi\mathbf{x})$ and let $\eta_{K}$ be the linear approximation of $\eta$ in $K$ ($\eta_{K}[n]=\eta[n],~{}\forall~{}n\in K$, and $0$ otherwise). Let $\eta_{K}=\eta_{1}+\eta_{2}$ be any splitting of $\eta_{K}$ into two vectors in the linear space $\mathcal{S}_{K}=\\{\mathbf{x}|\mathbf{x}[m]=0,\forall k\notin K\\}$. We can write $\eta=\eta_{1}+\eta_{2}+\eta_{3},$ with $\eta_{3}=\eta-\eta_{K}$. As the right side of (14) is equal to 0 for $\forall~{}\mathbf{x}\in\mathcal{S}_{K}$, we deduce $-\eta_{1}=\Delta(\Phi(-\eta_{1}))$. Since $\eta\in\mathrm{Null}(\Phi)$, we have $\Phi(-\eta_{1})=\Phi(\eta_{2}+\eta_{3})$, so that $-\eta_{1}=\Delta(\Phi(\eta_{2}+\eta_{3}))$. We derive $\displaystyle E[\|\eta\|_{X}]$ $\displaystyle=$ $\displaystyle E[\|\eta_{2}+\eta_{3}-\Delta\Phi(\eta_{2}+\eta_{3})\|_{X}]\leq C_{0}E_{\mathbf{x}}[\|(\eta_{2}+\eta_{3})-((\eta_{2})_{K}+(\eta_{3})_{K})\|_{X}]$ $\displaystyle=$ $\displaystyle C_{0}E_{\mathbf{x}}[\|\eta-\eta_{K}\|_{X}],$ where the inequality follows from (14), and the second and third equalities use the fact that $\eta=\eta_{1}+\eta_{2}+\eta_{3}$ and $\eta_{1}\in\mathcal{S}_{K}$. Thus we have obtained (23). A similar proof proceeds for MSE instance optimality and null space property. The second part of the theorem is a direct consequence of the first part. ∎ Comparing to conventional CS that requires the null space property to hold with the best $2k$-term nonlinear approximation error [17], the requirement for Gaussian SCS is relaxed to $k$, thanks to the linearity of the best $k$-term linear approximation for Gaussian signals. Theorem 2 proves the existence of the decoder $\Delta$ for which the instance optimality in expectation holds for $(\Phi,\Delta)$, given the null space property in expectation. However, it does not explain how such decoder is implemented. The following Corollary, a direct consequence of theorems 1 and 2, shows that for Gaussian signals the optimal decoder (5) leads to the instance optimality in expectation. ###### Corollary 2. For Gaussian signals $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, if an $M\times N$ sensing matrix $\Phi$ satisfies the null space property in expectation (20) of order $k$ in $\|\cdot\|_{1}$, with constant $C_{0}/2$, or the MSE null space property (21) of order $k$ with constant $C_{0}/4$, then the optimal and linear decoder $\Delta=\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}$ satisfies the instance optimality in expectation (16) in $\|\cdot\|_{1}$, or the MSE instance optimality (17). ###### Proof. It follows from Theorem 1 that the MAP decoder minimizes MAE and MSE among all the estimators for $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$. Therefore its MAE and MSE are smaller than the ones generated by the decoder considered in Theorem 2 (24). The latter satisfies the instance optimality, so is the former. ∎ #### II-E2 From RIP to Null Space Property The Restricted Isometry Property (RIP) of a matrix measures its ability to preserve distances, and is related to the null space property in conventional CS [14, 18]. The new linear RIP of order $k$ restricts the requirement of conventional RIP of order $k$ to a union of $k$-dimensional linear subspaces with consecutive supports: ###### Definition 3. Let $k\leq N$ be a positive integer. Let $\mathcal{K}_{1}$ define a linear subspace of functions with support in the first $k$ indices in $[1,N]$, $\mathcal{K}_{2}$ a linear subspace of functions with support in the next $k$ indices, and so on. The functions in the last linear subspace $\mathcal{K}_{J}$ defined this way may have support with less than $k$ indices. An $M\times N$ matrix $\Phi$ is said to have linear RIP of order $k$ with constant $\delta$ if $(1-\delta)\|\mathbf{x}\|_{2}\leq\|\Phi\mathbf{x}\|_{2}\leq(1+\delta)\|\mathbf{x}\|_{2},~{}~{}~{}\forall~{}\mathbf{x}\in\cup_{j=1}^{J}\mathcal{K}_{j}.\vspace{0ex}$ (27) The linear RIP is a special case of the block RIP [21], with block sparsity one and blocks having consecutive support of the same size. The following theorem relates the linear RIP (27) of a matrix $\Phi$ to its null space property in expectation (20). ###### Theorem 3. Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain distribution. Let $\Phi$ be an $M\times N$ matrix that satisfies the linear RIP of order $2k$ with $\delta<1$, and let $\Delta$ be a decoder. Let $\eta=\mathbf{x}-\Delta(\Phi\mathbf{x})$. Assume further that $E_{\mathbf{x},(\Phi)}|\eta[n]|$ decays in $n$: $E_{\mathbf{x},(\Phi)}|\eta[n+1]|<E_{\mathbf{x},(\Phi)}|\eta[n]|$, $\forall n<N-1$. Then $\Phi$ satisfies the null property in expectation of order $k$ in $\|\cdot\|_{1}$ (20), with constant $C_{0}=1+k^{1/2}\frac{1+\delta}{1-\delta}$. 222As in [17], the result here is in the $l_{1}$ norm, while in the next section we will consider a natural extension of the RIP for SCS which can be studied in the $l_{2}$ norm, something possible for conventional CS only in a probabilistic setting, with one random sensing matrix independently drawn for each signal [17]. ###### Proof. Let $K$ denote the set of first $k$ indices of the entries in $\eta$, $K_{1}$ the next $k$ indices, $K_{2}$ the next $k$ indices, etc. We have $\displaystyle\|\eta_{K}\|_{2}$ $\displaystyle\leq$ $\displaystyle\|\eta_{K\cup K_{1}}\|_{2}\leq(1-\delta)^{-1}\|\Phi\eta_{K\cup K_{1}}\|_{2}=(1-\delta)^{-1}\|\sum_{j=2}^{J}\Phi\eta_{K_{j}}\|_{2}$ $\displaystyle\leq$ $\displaystyle(1-\delta)^{-1}\sum_{j=2}^{J}\|\Phi\eta_{K_{j}}\|_{2}\leq(1+\delta)(1-\delta)^{-1}\sum_{j=2}^{J}\|\eta_{K_{j}}\|_{2},$ where the second and last inequalities follow the linear RIP property of $\Phi$, the third inequality follows from the triangle equality, and the equality holds since $\eta\in\mathrm{Null}(\Phi)$. Hence we have $E\|\eta_{K}\|_{2}\leq(1+\delta)(1-\delta)^{-1}\sum_{j=2}^{J}E\|\eta_{K_{j}}\|_{2}.$ (28) Since $E|\eta[n+k]|\leq E|\eta[n]|$, we derive $E\|\eta_{K_{j+1}}\|_{1}\leq E\|\eta_{K_{j}}\|_{1}$, so that $E\|\eta_{K_{j+1}}\|_{2}\leq E\|\eta_{K_{j+1}}\|_{1}\leq E\|\eta_{K_{j}}\|_{1},$ (29) where the first inequality follows from the fact that $\|\mathbf{x}\|_{2}\leq\|\mathbf{x}\|_{1},~{}\forall\mathbf{x}$. Inserting (29) into (28) gives $E\|\eta_{K}\|_{2}\leq(1+\delta)(1-\delta)^{-1}\sum_{j=1}^{J-1}E\|\eta_{K_{j}}\|_{1}\leq(1+\delta)(1-\delta)^{-1}E\|\eta_{{K}^{C}}\|_{1}.$ (30) By the Cauchy-Schwartz inequality $\|\eta_{K}\|_{1}\leq k^{1/2}\|\eta_{K}\|_{2}$, we therefore obtain $E\|\eta\|_{1}=E\|\eta_{K}\|_{1}+E\|\eta_{K^{C}}\|_{1}\leq\left(1+k^{1/2}\frac{1+\delta}{1-\delta}\right)E\|\eta_{K^{C}}\|_{1},$ (31) which verifies the null space property with constant $C_{0}$. ∎ For Gaussian signals $\mathbf{x}\in\mathcal{N}(\mathbf{0},\mathbf{S})$, with $\Phi$ Gaussian or Bernoulli matrices, one realization drawn independently for each $\mathbf{x}$, and with $\Delta$ the optimal decoder (5), the decay of $E_{\mathbf{x},\Phi}|\eta[n]|$ assumed in Theorem 3 is verified through Monte Carlo simulations. #### II-E3 From Random Matrices to Linear RIP The next Theorem shows that Gaussian and Bernoulli matrices satisfy the conventional RIP for one subspace with overwhelming probability. The linear RIP will be addressed after it. ###### Theorem 4. [1, 4] Let $\Phi$ be a random matrix of size $M\times N$ drawn according to any distribution that satisfies the concentration inequality $\textrm{Pr}(|\|\Phi\mathbf{x}\|_{2}^{2}-\|\mathbf{x}\|_{2}^{2}|\geq\epsilon\|\mathbf{x}\|_{2}^{2})\leq 2e^{-Mc_{0}(\delta/2)},~{}~{}~{}\forall~{}\mathbf{x}\in\mathbb{R}^{N},\vspace{0ex}$ (32) where $0<\delta<1$, and $c_{0}(\delta/2)>0$ is a constant depending only on $\delta/2$. Then for any set $K\subset\\{1,\ldots,N\\}$ with $|K|=k<M$, we have the conventional RIP condition $(1-\delta)\|\mathbf{x}\|_{2}\leq\|\Phi\mathbf{x}\|_{2}\leq(1+\delta)\|\mathbf{x}\|_{2},~{}~{}~{}\forall~{}\mathbf{x}\in\mathcal{X}_{K},\vspace{0ex}$ (33) where $\mathcal{X}_{K}$ is the set of all vectors in $\mathbb{R}^{N}$ that are zero outside of $K$, with probability greater than or equal to $1-2(12/\delta)^{k}e^{-c_{0}(\delta/2)M}.$ Gaussian and Bernoulli matrices satisfy the concentration inequality (32). The linear RIP of order $k$ (27) requires that (33) holds for $N/k\leq N$ subspaces. The next Theorem follows from Theorem 4 by simply multiplying by $N$ the probability that the RIP fails to hold for one subspace. ###### Theorem 5. Suppose that $M$, $N$ and $0<\delta<1$ are given. Let $\Phi$ be a random matrix of size $M\times N$ drawn according to any distribution that satisfies the concentration inequality (32). Then there exist constants $c_{1},c_{2}>0$ depending only on $\delta$ such that the linear RIP of order $k$ (27) holds with probability greater than or equal to $1-2Ne^{-c_{2}M}$ for $\Phi$ with the prescribed $\delta$ and $k\leq c_{1}M$. ###### Proof. Following Theorem 4, for a $k$-dimensional linear space $\mathcal{X}_{K}$, the matrix $\Phi$ will fail to satisfy (33) with probability $\leq 2(12/\delta)^{k}e^{-c_{0}(\delta/2)n}$. The linear RIP requires that (33) holds for at most $N$ such subspaces. Hence (33) will fail to hold with probability $\leq 2N(12/\delta)^{k}e^{-c_{0}(\delta/2)M}=2Ne^{-c_{0}(\delta/2)M+k\log(12/\delta)}.$ (34) Thus for a fixed $c_{1}>0$, whenever $k\leq c_{1}M$, the exponent in the exponential on the right side of (34) is $\leq c_{2}M$ provided that $c_{2}\leq c_{0}(\delta/2)-c_{1}(1+\log(12/\delta))$. We can always choose $c_{1}>0$ small enough to ensure $c_{2}>0$. This proves that with a probability $1-2Ne^{-c_{2}M}$, the matrix $\Phi$ will satisfy the linear RIP (27). ∎ Comparing with conventional CS, where the null space property requires that the RIP (33) holds for $\binom{N}{k}$ subspaces [4, 14, 18], the number of subspaces in the linear RIP (27) is sharply reduced to $N/k$ for Gaussian SCS, thanks to the coefficients pre-ordering and the linear estimation in consequence. Therefore with the same number of measurements $M$, the probability that a Gaussian or Bernoulli matrix $\Phi$ satisfies the linear RIP is substantially higher than that for the conventional RIP. Equivalently, given the same probability that $\Phi$ satisfies the linear RIP or the conventional RIP of order $k$, the required number of measurements for the linear RIP is $M\sim\mathcal{O}(k)$, substantially smaller than the $M\sim\mathcal{O}(k\log(N/k))$ required for the conventional RIP. Similar improvements have been obtained with model-based CS that assumes structured sparsity on the signals [5]. With the results above, we have shown that for Gaussian signals, with sensing matrices satisfying the linear RIP (27) of order $2k$, for example Gaussian or Bernoulli matrices with $\mathcal{O}(k)$ rows, with overwhelming probability, and with the optimal and linear decoder (5), SCS leads to the instance optimality in expectation of order $k$ in $\|\cdot\|_{1}$ (16), with constant $C_{0}=2(1+k^{1/2}\frac{1+\delta}{1-\delta})$. $k^{1/2}$ is typically small by the definition of CS. ### II-F Performance Bounds with RIP in Expectation This section shows that with an RIP in expectation, a matrix isometry property more adapted to SCS, the Gaussian SCS MSE instance optimality (17) of order $k$ and constant $C_{0}$, holds in the $l_{2}$ norm with probability one for any matrix. $C_{0}$ has a closed-form and can be easily computed numerically. ###### Definition 4. Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain distribution. Let $\Phi$ be an $M\times N$ sensing matrix and let $\Delta$ be a decoder. Let $\eta=\mathbf{x}-\Delta(\Phi\mathbf{x})$. $\Phi$ in $(\Phi,\Delta)$ is said to have RIP in expectation in $K$ with constant $c_{K}$ if ${E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K}\|_{2}^{2}}=c_{K}{E_{\mathbf{x},(\Phi)}\|\eta_{K}\|_{2}^{2}},~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta(\Phi\mathbf{x}),\vspace{0ex}$ (35) where $K\subset\\{1,\ldots,N\\}$, $\eta_{K}\in\mathbb{R}^{N}$ is the signal $\eta$ restricted to $K$ ($\eta_{K}[n]=\eta[n],~{}\forall~{}n\in K$, and $0$ otherwise), and the expectation is with respect to $\mathbf{x}$, and to $\Phi$ if one random $\Phi$ is drawn independently for each $\mathbf{x}$. The conventional RIP is known to be satisfied only by some random matrices, Gaussian and Bernoulli matrices for example, with high probability. For a given matrix, checking the RIP property is however NP-hard [4]. By contrast, the constant of the RIP in expectation (35) can be measured for any matrix via a fast Monte Carlo simulation, the quick convergence guaranteed by the concentration of measure [33]. The next proposition, directly following from (6) and (7), further shows that for Gaussian signals, the RIP in expectation has its constant in a closed form. ###### Proposition 1. Assume $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, $\Phi$ is an $M\times N$ sensing matrix and $\Delta$ is the optimal and linear decoder (5). Then $\Phi$ in $(\Phi,\Delta)$ satisfies the RIP in expectation in $K$, ${(E_{\Phi})\left[Tr\left(\Phi\mathbf{R}_{K}\mathbf{S}\mathbf{R}_{K}^{T}\Phi^{T}-\Phi\mathbf{R}_{K}\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{S}\mathbf{R}_{K}^{T}\Phi^{T}\right)\right]}\vspace{-1ex}$ $=c_{K}{(E_{\Phi})\left[Tr\left(\mathbf{R}_{K}\mathbf{S}\mathbf{R}_{K}^{T}-\mathbf{R}_{K}\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{S}\mathbf{R}_{K}^{T}\right)\right]}\vspace{0ex},$ (36) where $\mathbf{R}_{K}$ is an $N\times N$ extraction matrix giving $\eta_{K}=\mathbf{R}_{K}\eta$, i.e., $\mathbf{R}_{K}(i,i)=1$, $\forall i\in K$, all the other entries being zero. The expectation with respect to $\Phi$ is calculated if one random $\Phi$ is drawn independently for each $\mathbf{x}$. ###### Proof. Let $\eta=\mathbf{x}-\Delta\Phi\mathbf{x}=\mathbf{x}-\mathbf{S}\Phi^{T}(\Phi\mathbf{S}\Phi^{T})^{-1}\Phi\mathbf{x}$, which follows from the MAP estimation (5). (1) is derived by calculating the covariance matrices $\Sigma_{\Phi\eta_{K}}=E\left[\Phi\mathbf{R}_{K}\eta(\Phi\mathbf{R}_{K}\eta)^{T}\right]$ of $\Phi\eta_{K}=\Phi\mathbf{R}_{K}\eta$, and $\Sigma_{\eta_{K}}=E\left[\mathbf{R}_{K}\eta(\mathbf{R}_{K}\eta)^{T}\right]$ of $\eta_{K}=\mathbf{R}_{K}\eta$, and using the fact that the trace of a covariance matrix yields the average energy of the underlying random vector. ∎ The next Theorem shows that the RIP in expectation leads to the MSE null space property holding in equality. ###### Theorem 6. Let $\mathbf{x}\in\mathbb{R}^{N}$ be a random vector that follows a certain distribution, $\Phi$ an $M\times N$ sensing matrix, and $\Delta$ a decoder. Assume ${E_{\mathbf{x},(\Phi)}\|\eta_{K}\|_{2}^{2}}\neq 0$ and ${E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2}}\neq 0$, for some $K\subset\\{1,\ldots,N\\}$. Assume that $\Phi$ in $(\Phi,\Delta)$ has the RIP in expectation in $K$ with constant $a_{K}>0$, and in $K^{C}=\\{1,\ldots,N\\}\backslash K$ with constant $b_{K}>0$: $\frac{E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K}\|_{2}^{2}}{E_{\mathbf{x},(\Phi)}\|\eta_{K}\|_{2}^{2}}=a_{K},~{}~{}~{}\frac{E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K^{C}}\|_{2}^{2}}{E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2}}=b_{K},~{}\textrm{where}~{}\eta=\mathbf{x}-\Delta\Phi\mathbf{x},$ (37) where $K\subset\\{1,\ldots,N\\}$, and $\eta_{K}\in\mathbb{R}^{N}$ is the signal $\eta$ restricted to $K$ ($\eta_{K}[n]=\eta[n],~{}\forall~{}n\in K$, and $0$ otherwise). Then $\Phi$ satisfies $E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}=C_{0}E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2},\vspace{0ex}$ (38) where $C_{0}=1+{b_{K}}/{a_{K}}$. In particular, if $K=\\{1,\ldots,k\\}$, with $1\leq k\leq N$, then $\Phi$ satisfies the MSE null space property of order $k$, which holds with equality, $E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}=C_{0}\sigma_{k}(\\{\eta\\})_{2}^{2}.\vspace{0ex}$ (39) ###### Proof. We derive (38) by $\frac{E_{\mathbf{x},(\Phi)}\|\eta\|_{2}^{2}}{E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2}}=1+\frac{E_{\mathbf{x},(\Phi)}\|\eta_{K}\|_{2}^{2}}{E_{\mathbf{x},(\Phi)}\|\eta_{K^{C}}\|_{2}^{2}}=1+\frac{E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K}\|_{2}^{2}/a_{k}}{E_{\mathbf{x},(\Phi)}\|\Phi\eta_{K^{C}}\|_{2}^{2}/b_{k}}=1+\frac{b_{k}}{a_{k}},$ where the second equality follows from the RIP in expectation (37) and the last equality holds because $\Phi\eta_{K}=\Phi\eta_{K^{C}}$ since $\eta=\eta_{K}+\eta_{K^{C}}\in\mathrm{Null}(\Phi)$. (39) is obtained by inserting (13) in (38).∎ Following Corollary 2, the MSE null space property constant $C_{0}$ indicates the upper bound of the SCS reconstruction error relative to the best $k$-term linear approximation. Let us check $C_{0}$ of different sensing matrices in SCS for Gaussian signals $\mathbf{x}\in\mathbb{R}^{N}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$, assuming that the eigenvalues of $\mathbf{S}$ follow a power decay (11) with typical values $\alpha=3$ and $N=64$. Gaussian, Bernoulli and random subsampling matrices $\Phi$ of size $M\times N$ are considered, and the optimal and linear decoder $\Delta$ (5) is applied to reconstruct the signals. For each matrix distribution, a different random matrix realization $\Phi$ is applied to sense each signal $\mathbf{x}$. Note that since the random subsampling matrix $\Phi$, each row containing one entry with value 1 at a random position and 0 otherwise, has the maximal coherence with the canonical basis, this matrix is not suitable for directly sensing $\mathbf{x}$ [10], and is replaced by $\Phi\Psi$ in the simulation, with $\Psi$ a DCT basis having low coherence with $\Phi$. Monte Carlo simulations are performed to calculate the RIP constants $a_{K}$ and $b_{K}$ (37). Figure 3 (a) plots $C_{0}=1+{b_{K}}/{a_{K}}$, with a typical value $k=10$ ($k/N=5/32$), for different values of $M$. When the number $M$ of SCS measurements increases, the reconstruction error of SCS decreases, resulting in a smaller ratio over the best $k$-term linear approximation error with a fixed $k$. Gaussian and Bernoulli matrices lead to similar $C_{0}$ values, slightly smaller than that of random subsampling matrices. Figure 3 (b) plots $C_{0}$, as a function of $k$, with $M=k$. Gaussian and Bernoulli matrices lead to similar $C_{0}\approx 4.5$ that varies little with $k$, in line with the results obtained in Section II-D (Figure 2-(c)). For random subsampling matrices $C_{0}$ slowly increases, almost linearly, and is equal to $5.5$ for a typical value $k=10$, about 20% larger than that of Gaussian and Bernoulli matrices. The small $C_{0}$ values indicate that the SCS reconstruction error is tightly upper bounded by a constant times the best $k$-term approximation error. | ---|--- (a) | (b) Figure 3: The MSE null space property constant $C_{0}$ (39) of Gaussian, Bernoulli, and random subsampling matrices, as a function of $M$, with a fixed $k=10$ (left), and of $k$ with $M=k$ (right). The signal dimension is $N=64$. From Corollary 2 and Theorem 6, we obtain the next concluding Theorem, which shows that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best $k$-term linear approximation with probability one, and the bound constant can be efficiently calculated. ###### Theorem 7. Assume $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{S})$. Let $\Phi$ be an $M\times N$ sensing matrix and $\Delta$ the optimal and linear decoder (5). Then $\Phi$ satisfies the MSE instance optimality of order $k$ (17) with constant $C_{0}=4(1+{b_{K}}/{a_{K}})$, $a_{K}$ and $b_{K}$ given in (37), and $K=\\{1,\ldots,k\\}$. Theorem 7, together with the performance comparison of linear and nonlinear approximation for Gaussian signals described in Section II-C, show that for signals following a Gaussian distribution with fast eigenvalue decay, the average error of SCS using $k$ measurements is tightly upper bounded by that of the best $k$-term approximation. ## III Compressed Sensing Model Selection with GMMs Section II shows tight error bounds of SCS for signals following a Gaussian distribution with fast eigenvalue decay. A single Gaussian distribution, however, is too simplistic for modeling most real signals. Assuming multiple Gaussian distributions and that each signal follows one of them, Gaussian mixture models (GMMs) provide more precise signal descriptions. It has been shown that algorithms based on GMMs lead to results in the ballpark of the state-of-the-art in various signal inverse problems, for different types of real data including images and ranking score matrices [24, 36]. GMMs have also been used to model color distributions [32] and for clustering [20], among many satisfactory applications with these models. This section first introduces a piecewise linear decoder for GMM-based SCS, which essentially consists of estimating a signal using each Gaussian model included in the GMM and then selecting the best model. At the heart of the GMM-based SCS decoder is the model selection. The rest of the section analyzes the accuracy of the model selection in terms of the GMM properties and the number of the measurements. As correct Gaussian models are selected, the SCS performance bounds described in Section II apply. ### III-A Piecewise Linear Decoder GMMs describe signals with a mixture of Gaussian distributions. Assume there exist $J$ Gaussian distributions $\\{\mathcal{N}(\mu_{j},\Sigma_{j})\\}_{1\leq j\leq J}$, parametrized by their means $\mu_{j}$ and covariances $\Sigma_{j}$. To simplify the notation, we assume without loss of generality that the Gaussians have zero means ${\mu}_{j}=\mathbf{0}$, $\forall j$, as one can always center the signals with respect to the means. GMM assumes that each signal $\mathbf{x}\in\mathbb{R}^{N}$ is independently drawn from one of these Gaussians with an unknown index $j\in[1,J]$, whose probability density function is $f(\mathbf{x})=\frac{1}{(2\pi)^{N/2}|\Sigma_{j}|^{1/2}}\exp\left({-\frac{1}{2}\mathbf{x}^{T}\Sigma_{j}^{-1}\mathbf{x}}\right).$ (40) To decode a measured signal $\mathbf{y}=\Phi\mathbf{x}$, the GMM-based SCS decoder estimates the signal $\tilde{\mathbf{x}}$ and selects the Gaussian model $\tilde{j}$ by maximizing the log a-posteriori probability $(\tilde{\mathbf{x}},\tilde{j})=\arg\max_{\mathbf{x},j}\log f(\mathbf{x}|\mathbf{y},\Sigma_{j}).$ (41) (41) is calculated by first computing the linear MAP decoder (2) using each of the Gaussian models, $\tilde{\mathbf{x}}_{j}=\Delta_{j}(\Phi\mathbf{x})=\underbrace{\Sigma_{j}\Phi^{T}(\Phi\Sigma_{j}\Phi^{T})^{-1}}_{\Delta_{j}}(\Phi\mathbf{x}),~{}~{}~{}\forall 1\leq j\leq J,\vspace{0ex}$ (42) and then selecting a best model $\tilde{j}$ that maximizes the log a-posteriori probability among all the models [36] $\tilde{j}=\arg\max_{1\leq j\leq J}-\frac{1}{2}\left(\log|\Sigma_{j}|+\tilde{\mathbf{x}}_{j}^{T}\Sigma_{j}^{-1}\tilde{\mathbf{x}}_{j}\right),$ (43) whose corresponding decoder $\Delta_{\tilde{j}}$ implements a piecewise linear estimate: $\tilde{\mathbf{x}}=\tilde{\mathbf{x}}_{\tilde{j}}=\Delta_{\tilde{j}}(\Phi\mathbf{x}).$ (44) The model selection (43) is at the heart of the GMM-based SCS.333Correct model/class selection from compressed measurements is at the core of numerous applications beyond signal reconstruction, see for example [15] and references therein. To better understand it, we concentrate next in a simple case, where the GMM involves $J=2$ Gaussian distributions $\mathcal{N}(\mathbf{0},\Sigma_{1})$ and $\mathcal{N}(\mathbf{0},\Sigma_{2})$ that have the same “shape” and “size”, but different “orientation,” i.e., the two covariance matrices have the same eigenvalues, but different PCA bases: $\Sigma_{1}=\mathbf{B}_{1}\mathbf{S}\mathbf{B}_{1}^{T}~{}~{}~{}\textrm{and}~{}~{}~{}\Sigma_{2}=\mathbf{B}_{2}\mathbf{S}\mathbf{B}_{2}^{T},$ (45) with $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ the PCA bases of the two Gaussian distributions, and $\mathbf{S}=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{N})$ a diagonal matrix, whose diagonal elements $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{N}$ are the sorted eigenvalues. It follows directly that $|\Sigma_{1}|=|\Sigma_{2}|$. This will be used next. ### III-B Oracle Model Selection Let us first study the model selection in an oracle situation, where the underlying signals $\mathbf{x}$ are assumed to be known and, without loss of generality, to follow the first Gaussian distribution $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\Sigma_{1})$. Recall that $|\Sigma_{1}|=|\Sigma_{2}|$ is assumed. The probability of correct oracle model selection (43) that assigns $\mathbf{x}$ to the first Gaussian distribution $\mathcal{N}(\mathbf{0},\Sigma_{1})$, $P_{c}^{o}=\int_{\mathbf{x}^{T}\Sigma_{1}^{-1}\mathbf{x}<\mathbf{x}^{T}\Sigma_{2}^{-1}\mathbf{x}}f_{1}(\mathbf{x})d\mathbf{x}=\int\textrm{sign}\left(\mathbf{x}^{T}\Sigma_{2}^{-1}\mathbf{x}-\mathbf{x}^{T}\Sigma_{1}^{-1}\mathbf{x}\right)f_{1}(\mathbf{x})d\mathbf{x},$ (46) where $f_{1}(\mathbf{x})=\frac{1}{(2\pi)^{N/2}|\Sigma_{1}|^{1/2}}\exp\left({-\frac{1}{2}\mathbf{x}^{T}\Sigma_{1}^{-1}\mathbf{x}}\right)$, will be studied as a function of the relationship between $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$, the decay rate of the eigenvalues, and the signal dimension $N$. #### III-B1 KL Divergence To better understand (46), let us first check the Kullback-Leibler (KL) divergence from the first Gaussian distribution to the second $\displaystyle D_{KL}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int\left(\mathbf{x}^{T}\Sigma_{2}^{-1}\mathbf{x}-\mathbf{x}^{T}\Sigma_{1}^{-1}\mathbf{x}\right)f_{1}(\mathbf{x})d\mathbf{x}$ (47) $\displaystyle=$ $\displaystyle\frac{1}{2}\textrm{Tr}(\Sigma_{2}^{-1}\Sigma_{1}-\mathbf{I}_{N})=\frac{1}{2}(\textrm{Tr}(\Sigma_{2}^{-1}\Sigma_{1})-N),$ (48) where $\mathbf{I}_{N}$ denotes the $N\times N$ identify matrix, and the second equality holds since $E[\mathbf{x}^{T}\mathbf{A}\mathbf{x}]=\textrm{Tr}(\mathbf{A}\Sigma)$ if $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\Sigma)$ [30]. Comparing (47) and (46), we observe that $D_{KL}$ is monotonic relative to $P_{c}^{o}$. Analyzing the behavior of $D_{KL}$ as a function of the two Gaussians thus helps to understand that of $P_{c}^{o}$. Inserting (45) into (48) leads to $D_{KL}=\frac{1}{2}(\textrm{Tr}(\mathbf{B}_{2}\mathbf{S}^{-1}\mathbf{B}_{2}^{T}\mathbf{B}_{1}\mathbf{S}\mathbf{B}_{1}^{T})-N)=\frac{1}{2}(\textrm{Tr}(\mathbf{C}\mathbf{S}\mathbf{C}^{T}\mathbf{S}^{-1})-N),$ (49) where $\mathbf{C}=\mathbf{B}_{2}^{T}\mathbf{B}_{1}$, and the second equality follows from the cyclic permutation invariance property of the trace $\textrm{Tr}(\mathbf{A}\mathbf{B}\mathbf{C})=\textrm{Tr}(\mathbf{C}\mathbf{B}\mathbf{A})$. Note that $\mathbf{C}$ is an orthogonal matrix: $\mathbf{C}^{T}\mathbf{C}=\mathbf{I}_{N}$. Maximizing $D_{KL}$ with respect to $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ is therefore equivalent to maximizing $\textrm{Tr}(\mathbf{C}\mathbf{S}\mathbf{C}^{T}\mathbf{S}^{-1})$ with respect to $\mathbf{C}$. The following lemma shows that in dimension two, $D_{KL}$ is maximized when the first principal directions of the two Gaussians are orthogonal, and moreover, the maximum divergence increases as the Gaussians become more anisotropic. ###### Lemma 2. Let $\mathbf{B}_{1}$ and $\mathbf{B}_{2}$ be respectively the PCA bases ($\Sigma_{1}=\mathbf{B}_{1}\mathbf{S}\mathbf{B}_{1}^{T}$ and $\Sigma_{2}=\mathbf{B}_{2}\mathbf{S}\mathbf{B}_{2}^{T}$) of two centered 2D Gaussian distributions $\mathcal{N}(\mathbf{0},\Sigma_{1})$ and $\mathcal{N}(\mathbf{0},\Sigma_{2})$, and $\mathbf{S}=\left[\begin{array}[]{cc}\lambda_{1}&0\\\ 0&\lambda_{2}\\\ \end{array}\right]$, with $\lambda_{1}>0$ and $\lambda_{2}>0$ their common eigenvalues. The KL divergence from the first Gaussian distribution to the second (47) has a maximum value $D_{KL}^{\max}=\max_{\mathbf{B}_{1},\mathbf{B}_{2}}D_{KL}=\frac{1}{2}\left(\frac{\lambda_{2}}{\lambda_{1}}+\frac{\lambda_{1}}{\lambda_{2}}\right),$ (50) which is obtained when $\mathbf{B}_{2}^{T}\mathbf{B}_{1}=\left[\begin{array}[]{cc}0&1\\\ 1&0\\\ \end{array}\right]$. Let the determinant of the covariance matrices $|\Sigma_{1}|=|\Sigma_{2}|=\lambda_{1}\lambda_{2}$ further be assumed given. Then $D_{KL}^{\max}$ is minimized as $\lambda_{1}=\lambda_{2}$, and it increases as the ratio between $\lambda_{1}$ and $\lambda_{2}$ increases. ###### Proof. The first part of the lemma can be easily checked by maximizing $D_{KL}$ in (49) with respect to the 2D orthogonal matrix $\mathbf{C}=\mathbf{B}_{2}^{T}\mathbf{B}_{1}$ and writing $\mathbf{C}=\left[\begin{array}[]{cc}\cos\theta&-\sin\theta\\\ \sin\theta&\cos\theta\\\ \end{array}\right]$. The second part is verified via a direct observation of (50). ∎ Figure 4-(a) plots $D_{KL}$ as a function of the angle $\theta$ between the first principal components of the two 2D Gaussians going from $5^{\circ}$ to $90^{\circ}$, with different eigenvalue ratios $\lambda_{1}/\lambda_{2}$ from 5 to 100. As indicated by Lemma 2, given $\lambda_{1}/\lambda_{2}$, $D_{KL}$ increases as $\theta$ increases. At a given $\theta$, larger $\lambda_{1}/\lambda_{2}$ leads to larger $D_{KL}$. The analysis in higher dimension is more difficult, however, one can check via a greedy optimization that $\mathbf{C}=\mathbf{B}_{2}^{T}\mathbf{B}_{1}=\begin{bmatrix}0&\cdots&\cdots&0&1\\\ \vdots&\cdots&\iddots&1&0\\\ \vdots&\iddots&\iddots&\iddots&\vdots\\\ 0&1&\iddots&\cdots&\vdots\\\ 1&0&\cdots&\cdots&0\\\ \end{bmatrix},$ (51) with ones along the anti-diagonal, and zeros elsewhere, gives a local maximum of (49). In other words, the two Gaussians being “orthogonal” one another, i.e., the alignment of the first principal component of one Gaussian to the last principal component of the other, the second principal component of the former to the second to last principal component of the latter, and so on, leads to a local maximization of (49). This can be observed by inserting $\mathbf{C}=\begin{bmatrix}C_{11}&\ldots&C_{1N}\\\ \vdots&\ddots&\vdots\\\ C_{N1}&\ldots&C_{NN}\\\ \end{bmatrix}$ in (49), which gives $D_{KL}=\frac{1}{2}(\sum_{m=1}^{N}\frac{1}{\lambda_{m}}\sum_{n=1}^{N}\lambda_{n}C_{mn}^{2}-N).$ (52) A greedy maximization of (52) with respect to $\mathbf{C}$ is calculated by scanning $\mathbf{C}$ row by row from bottom to top, observing that $1/\lambda_{m}$ decreases as $m$ goes from $N$ to $1$, and at each $m$-th row scanning $C_{mn}$ from left to right, observing that $\lambda_{n}$ increases as $n$ goes from $1$ to $N$, taking into account the constraint $\mathbf{C}^{T}\mathbf{C}=\mathbf{I}_{N}$. A similar observation of (52) shows that when $D_{KL}$ is at the local maximum with $\mathbf{C}$ equal to (51), its value increases as the eigenvalues decay faster from $\lambda_{1}$ to $\lambda_{N}$. #### III-B2 Correct Model Seletion Probability The probability of correct oracle model selection $P_{c}^{o}$ (46) is now evaluated via Monte Carlo simulations. Figure 4-(b) plots $P_{c}^{o}$ as a function the angle $\theta$ between the first principal components of the two 2D Gaussians going from $5^{\circ}$ to $90^{\circ}$, with different eigenvalue ratios $\lambda_{1}/\lambda_{2}$ from 5 to 100. As illustrated in Figure 4, $P_{c}^{o}$ shows a behavior similar to the KL-divergence $D_{KL}$ as a function of $\theta$ and of $\lambda_{1}/\lambda_{2}$: Given $\lambda_{1}/\lambda_{2}$, $P_{c}^{o}$ increases as $\theta$ increases; at a given $\theta$, larger $\lambda_{1}/\lambda_{2}$ leads to larger $P_{c}^{o}$. In contrast to $D_{KL}$, whose value is roughly proportional to $\lambda_{1}/\lambda_{2}$ (as $\lambda_{1}\gg\lambda_{2}$), $P_{c}^{o}$ presents a saturation effect: $\lambda_{1}/\lambda_{2}$ values larger than about 40 lead to comparable $P_{c}^{o}$ that increases rapidly as a function of $\theta$, converging to a high value around 0.9; for $\lambda_{1}/\lambda_{2}$ smaller than about 40, on the other hand, $P_{c}^{o}$ reduces quickly as $\lambda_{1}/\lambda_{2}$ shrinks towards 1. | ---|--- (a) | (b) Figure 4: (a) The KL-divergence (47) between two 2D Gaussians, as a function the angle $\theta$ between the first principal components of the two Gaussians going from $5^{\circ}$ to $90^{\circ}$, with different eigenvalue ratios $\lambda_{1}/\lambda_{2}$ from 5 to 100. (b) The same for the probability of correct oracle model selection $P_{c}^{o}$ (46). Figure 5 shows the probability of correct oracle model selection $P_{c}^{o}$ (46) in higher dimensions, under the condition that (51) holds, i.e., the two Gaussians are “orthogonal.” A power decay of the eigenvalues (11) is assumed in the Monte Carlo simulations. In different signal dimensions $N$ from $2$ to $20$, $P_{c}^{o}$ as a function of the eigenvalue decay parameter $\alpha$ is plotted. For a given dimension, $P_{c}^{o}$ increases as $\alpha$ increases, i.e., as the eigenvalues decay faster so that the Gaussians are more anisotropic. It is important to remark that, with the same $\alpha$, $P_{c}^{o}$ rapidly increases as the signal dimension $N$ increases, which shows that anisotropic Gaussians with their energy concentrated in the first few dimensions are more separate in higher dimension. --- Figure 5: The probability of correct oracle model selection $P_{c}^{o}$ (46) between two Gaussians, as a function of the eigenvalue decay parameter $\alpha$ from 1 to 5, for different signal dimensions $N$ from 2 to 20. The two Gaussians satisfy (51). ### III-C Model Selection and Signal Reconstruction In SCS, the model selection (43) is calculated with the decoded signals (42) and not the ideal ones. Assume without loss of generality that the signals follow the first Gaussian distribution $\mathbf{x}~{}\sim\mathcal{N}(\mathbf{0},\Sigma_{1})$. This section checks via Monte Carlo simulations the probability of correct model selection (43) calculated with the decoded signals $\tilde{\mathbf{x}}_{1}=\Delta_{1}\Phi\mathbf{x}$ and $\tilde{\mathbf{x}}_{2}=\Delta_{2}\Phi\mathbf{x}$, $P_{c}=(E_{\Phi})\left(\int_{\tilde{\mathbf{x}}_{1}^{T}\Sigma_{1}^{-1}\tilde{\mathbf{x}}_{1}<\tilde{\mathbf{x}}_{2}^{T}\Sigma_{2}^{-1}\tilde{\mathbf{x}}_{2}}f_{1}(\mathbf{x})d\mathbf{x}\right)=(E_{\Phi})\left(\int\textrm{sign}\left(\tilde{\mathbf{x}}_{2}^{T}\Sigma_{2}^{-1}\tilde{\mathbf{x}}_{2}-\tilde{\mathbf{x}}_{1}^{T}\Sigma_{1}^{-1}\tilde{\mathbf{x}}_{1}\right)f_{1}(\mathbf{x})d\mathbf{x}\right),$ (53) where the expectation is with respect to $\Phi$ if one random $\Phi$ is independently drawn for each $\mathbf{x}$. We also investigate the MSE of the resulting signal reconstruction, $E_{\mathbf{x},(\Phi)}\|\mathbf{x}-\tilde{\mathbf{x}}\|^{2}_{2}=(E_{\Phi})\left(\int_{\tilde{\mathbf{x}}_{1}^{T}\Sigma_{1}^{-1}\tilde{\mathbf{x}}_{1}<\tilde{\mathbf{x}}_{2}^{T}\Sigma_{2}^{-1}\tilde{\mathbf{x}}_{2}}\|\mathbf{x}-\tilde{\mathbf{x}}_{1}\|^{2}_{2}f_{1}(\mathbf{x})d\mathbf{x}+\int_{\tilde{\mathbf{x}}_{1}^{T}\Sigma_{1}^{-1}\tilde{\mathbf{x}}_{1}\geq\tilde{\mathbf{x}}_{2}^{T}\Sigma_{2}^{-1}\tilde{\mathbf{x}}_{2}}\|\mathbf{x}-\tilde{\mathbf{x}}_{2}\|^{2}_{2}f_{1}(\mathbf{x})d\mathbf{x}\right),$ (54) as a function of the number of sensing measurements $M$ and the properties of the Gaussian distributions. Figure 6 shows the probability of correct model selection $P_{c}$ (53) and the MSE of signal reconstruction (54) as a function of the number of measurements $M$ and the signal dimension $N$. Figure 6-(a) plots $P_{c}$ as a function of $M$ going from $1$ to $N$, with different $N$ values from 2 to 15, assuming that (51) holds, i.e., the two Gaussians are “orthogonal.” A power decay model of the eigenvalues (11) with a typical decay parameter $\alpha=3$ is assumed in the simulations. A random Gaussian matrix realization $\Phi$ is drawn independently to sense each signal. As expected, $P_{c}$ increases as $M$ goes from 1 to $N$, i.e., as more measurements are dedicated. The signal dimension $N$ plays an important role. With only $M=1$ measurement, the model selection is uniformly random ($P_{c}\approx 0.5$), independent of the signal dimensions $N$. At an extremely low dimension $N=2$, even with $M=N$ measurements (which leads to perfect signal reconstruction, as if in the “oracle” case described in Section III-B), $P_{c}$ remains lower than 0.8. 444We observe that a mistake in the model selection will not necessarily lead to a mistake in the reconstruction, e.g., flat image patches can often be recovered by multiple different models. When $N$ goes higher, $P_{c}$ rapidly increases converging towards 1 as $M$ increases. After $N$ stands above a certain value (about 10 in this example, note that for the image examples in the next section $N=64$), $P_{c}$ converges very close to 1 as far as $M$ reaches a fixed value (about 8) independent of $N$. This indicates that accurate model selection can be achieved with very low sampling rates $M/N$, given that the energy of the signals is concentrated in the first few principal dimensions. In signal sampling, one is more interested in the signal reconstruction error than model selection. Figure 6-(b) similarly shows the MSE of the decoded signals (54) (normalized by the ideal signal energy). The MSE decreases as $M$ increases, and it goes to 0 as $M=N$. At high dimensions $N$ (over about 10), almost perfect signal reconstruction is obtained as far as $M$ reaches a fixed value (about 8). | ---|--- (a) | (b) Figure 6: (a) The probability of correct model selection (53) as a function the of the number of measurements $M$ from $1$ to the signal dimension $N$, with $N$ going from 2 to 15. (b) The same for MSE (54) (normalized by the ideal signal energy) of the decoded signals. Similarly, Figure 7 plots the probability of correct model selection $P_{c}$ (53) as well as the MSE of the decoded signals (54) (normalized by the ideal signal energy), as a function of the measurements $M$ going from 1 to the signal dimension $N=10$, with different eigenvalue decay parameter $\alpha$ from 1 to 5. As $\alpha$ increases, i.e., as the eigenvalues decay faster, $P_{c}$ and MSE respectively converge to 1 and 0 at a faster rate as $M$ goes from 1 to $N$. | ---|--- (a) | (b) Figure 7: (a) The probability of correct model selection (53), as a function of the number of measurements $M$ from $1$ to the signal dimension $N=10$, with different eigenvalue decay parameter $\alpha$ from 1 to 5. (b) The same for MSE (54) (normalized by the ideal signal energy) of the decoded signals. In summary, this section shows that the accuracy of the Gaussian model selection (43) in GMM-based SCS is influenced by a number of factors including the geometry of the Gaussian distributions in the GMM, the signal dimension, and the number of sensing measurements. More accurate model selection is obtained as the Gaussians distributions are more “orthogonal” one another, as each of the Gaussians is more anisotropic, as the signals are in a higher dimension given that the energy of the signals are concentrated in the first few dimensions, and as the number of sensing measurements increases. ## IV SCS with GMM – Algorithm and Experiments The GMM-based SCS decoder described in Section III-A assumes that the means and the covariances of the Gaussian distributions $\\{\mathcal{N}(\mu_{j},\Sigma_{j})\\}_{1\leq j\leq J}$ in the GMMs are known. However, in real sensing applications, these parameters are unavailable. Following [36], this ection presents a maximum a posteriori expectation- maximization (MAP-EM) algorithm [3] that iteratively estimates the Gaussian parameters and decodes the signals. GMM-based SCS calculated with the MAP-EM algorithm is applied in real signal sensing, and is compared with conventional CS based on sparse models. ### IV-A MAP-EM Algorithm The MAP-EM algorithm is an iterative procedure that alternates between two steps: #### IV-A1 E-step Assuming that the estimates of the Gaussian parameters $\\{(\tilde{\mu}_{j},\tilde{\Sigma}_{j})\\}_{1\leq j\leq J}$ are known (following the previous M-step), the E-step calculates the MAP signal estimation and model selection for all the signals, following (41)–(44) . #### IV-A2 M-step Assuming that the Gaussian model selection $\tilde{j}$ and the signal estimate $\tilde{\mathbf{x}}$ are known for all the signals (following the previous E-step), the M-step estimates (updates) the Gaussian models $\\{(\tilde{\mu}_{j},\tilde{\Sigma}_{j})\\}_{1\leq j\leq J}$. Let $\mathbf{x}_{i}$, $\mathbf{y}_{i}$, $\tilde{\mathbf{x}}_{i}$ and $\tilde{j}_{i}$ respectively denote the $i$-th signal in the collection, its coded version, its estimate, and its estimated Gaussian model index, $1\leq i\leq I$. Let $\mathcal{C}_{j}$ be the ensemble of the signal indices $i$ that are assigned to the $k$-th Gaussian model, i.e., $\mathcal{C}_{j}=\\{i:\tilde{j}_{i}=j\\}$, and let $|\mathcal{C}_{j}|$ be its cardinality. The parameters of each Gaussian model are estimated with the maximum likelihood estimate using all the signals assigned to that Gaussian model, $(\tilde{\mu}_{j},\tilde{\Sigma}_{j})=\arg\max_{\mu_{j},\Sigma_{j}}\log f(\\{\tilde{\mathbf{x}}_{i}\\}_{i\in\mathcal{C}_{j}}|\mu_{j},\Sigma_{j}).$ (55) With the Gaussian model (40) , it is well-known that the resulting estimate is the empirical estimate $\tilde{\mu}_{j}=\frac{1}{|\mathcal{C}_{j}|}\sum_{i\in\mathcal{C}_{j}}\tilde{\mathbf{x}}_{i}~{}~{}\textrm{and}~{}~{}\tilde{\Sigma}_{j}=\frac{1}{|\mathcal{C}_{j}|}\sum_{i\in\mathcal{C}_{j}}(\tilde{\mathbf{x}}_{i}-\tilde{\mu}_{j})(\tilde{\mathbf{x}}_{i}-\tilde{\mu}_{j})^{T}.$ (56) The computational complexity of the MAP-EM algorithm is dominated by the matrix inversion $(\Phi\Sigma_{j}\Phi^{T})^{-1}$ in (42) in the E-step. It can be implemented with $M^{3}/3$ flops through a Cholesky factorization [7]. With $J$ Gaussian models, the complexity per iteration is therefore dominated by $JM^{3}/3$ flops. As the MAP-EM algorithm described above iterates, the MAP probability of the observed signals $f(\\{\tilde{\mathbf{x}}_{i}\\}_{1\leq i\leq I}|\\{\mathbf{y}_{i}\\}_{1\leq i\leq I},\\{\tilde{\mu}_{j},\tilde{\Sigma}_{j}\\}_{1\leq j\leq J})$ always increases. This can be observed by interpreting the E- and M-steps as a coordinate descent optimization [22]. The algorithm initialization and the number $J$ of Gaussians in GMM can be selected according to the type of signals of interest. For sensing natural images, a geometry-motivated initialization as detailed in [36] will be applied in the experiments. ### IV-B Experiments The GMM-based SCS is applied in real image sensing, and is compared with conventional CS based on sparse models. Following standard practice, an image is decomposed into $\sqrt{N}\times\sqrt{N}=8\times 8$ local patches $\\{\mathbf{x}_{i}\\}_{1\leq i\leq I}$ (an image patch is reshaped to and considered as a vector) [2, 26, 36], which are assumed to follow a GMM [36]. SCS samples each patch $\mathbf{y}_{i}=\Phi_{i}\mathbf{x}_{i}$, with a possibly different $\Phi_{i}$ for each $\mathbf{x}_{i}$. The decoder is implemented with the MAP-EM algorithm, initialized with $J=19$ geometry- motivated Gaussian models, each capturing a local direction [36]. The algorithm typically converges within 3 iterations. No database is used, and all the parameters and reconstruction are learned from the compressed sensed image alone. The dictionary for conventional CS is learned with K-SVD [2] from 720,000 image patches, extracted from the entire standard Berkeley segmentation database containing 300 natural images [28]. In image estimation and sensing, learned dictionaries have been shown to produce better results than off-the- shelf ones [2, 19, 26]. The decoder is calculated with the $l_{1}$ minimization [34] implemented in [25]. Three standard images Lena ($512\times 512$), House ($256\times 256$), and Peppers ($512\times 512$), as illustrated in Figure 8, are used in the experiments. --- Figure 8: From left to right. Three standard images used for the experiments: Lena, House, and Peppers. | ---|--- (a) | (b) Figure 9: (a) PSNR (dB) vs sampling rate for SCS and CS using Gaussian and random subsampling sensing matrices on image patches extracted from Lena. (b) PSNR (dB) vs sampling rate for SCS and CS using Gaussian sensing matrices on image patches extracted from House and Peppers. Figure 9 (a) shows the sensing performance on about 260,000 (sliding) patches, regarded as signals $\mathbf{x}_{i}$, extracted from Lena. The PSNRs generated by SCS and CS using Gaussian and random subsampling sensing matrices, one independent realization for each patch, are plotted as a function of the sampling rate $M/N$. At the same sampling rate, SCS outperforms SC. The gain increases from about 0.5 dB at very low sampling rates ($M/N\approx 0.1$), learning a GMM from the poor-quality measured data being more challenging, to more than 3.5 dB at high sampling rates ($M/N\approx 0.5$). (SC using an “oracle” dictionary learned from the ideal Lena itself, undoable in practice, improves its performance from 0.2 dB at low sampling rates to 1.3 dB at high sample rates, still lower than SCS.) For both SCS and CS, Gaussian and random subsampling matrices lead to similar PSNRs at low sampling rates ($M/N<0.25$), and at higher sampling rates Gaussian sensing gains by about 0.5 dB. Recall that SCS is not just more accurate and significantly faster, but also uses only the compressed image, while conventional CS uses a pre-learned dictionary from a large database. Figure 9 (b) further compares SCS with CS on sliding patches, regarded as signals, extracted from Peppers (260,000 patches) and House (62,000 patches). One independent Gaussian matrix realization is applied to sense each patch. Similar results as on the patches from Lena are observed. At the same sampling rate, SCS outperforms SC. The gain is smaller (about 1 dB) at very low sampling rates ($M/N\approx 0.1$), and becomes substantial (about 3 dB) at high sampling rates ($M/N\approx 0.5$). Figure 10 illustrates some typical patches with geometry. The ground-truth patches are shown in the first row, and the patches reconstructed by conventional CS and SCS, all sensed with Gaussian matrices at a sampling rate $M/N=1/4$, are respectively illustrated in the second and the third row. Both CS and SCS lead to accurate reconstruction in uniform regions. SCS outperforms CS on the more geometrical parts, and the improvement is significant on the fine contours (the 2nd, 3rd and 7th patches). --- Figure 10: Some typical $8\times 8$ patches with geometry. First row: ground- truth patches. Second and third rows: patches reconstructed by conventional CS and SCS respectively, all sensed at a sampling rate $M/N=1/4$ with Gaussian matrices. In most image sensing applications, one is interested in reconstructing whole images instead of individual patches. Aggregating non-overlapped patches to a whole images produces block artifacts, as illustrated in Figure 11. It is well known that averaging overlapped reconstructed patches not only removes the block artifacts, but also considerably improves the image estimation [2, 26, 36]. However, compressed sensing only allows sensing non-overlapping patches, since sensing overlapping patches would dramatically increase the sampling rate. Nevertheless, overlapped reconstructed patches are computable if the sensing operators, performed on non-overlapped patches, are random subsampling matrices, which are diagonal operators (one non-zero entry per row). (The reconstruction is then equivalent to solving an inpainting problem [2, 36].) Figure 11 shows some typical regions in Lena. The overlapped reconstruction, which further supports the search for performance on average as in the proposed SCS, removes the block artifacts and significantly improves the reconstructed image. Figure 12 plots the PSNRs on the whole image Lena generated by SCS using random subsampling matrices and overlapped reconstruction are plotted, in comparison with those obtained using Gaussian sensing matrices and non-overlapped reconstruction, at different sampling rates. The former improves from about 3.5 dB, at low sampling rates, to 1.5 dB, at high sampling rates, at a cost of $N=64$ times computation. | | ---|---|--- Ground truth | No.-ovl. rec. 30.82 dB | Ovl. rec. 34.02 dB | | Ground truth | No.-ovl. rec. 24.72 dB | Ovl. rec. 27.87 dB Figure 11: From left to right. Zoomed crops from Lena, reconstructed images by SCS using Gaussian sensing matrices and non-overlapping reconstruction, and by SCS using subsampling random matrices and overlapping reconstruction. The image is sensed on non-overlapped patches at a sampling rate of $M/N=0.25$. Local PSNRs are reported. --- Figure 12: PSNR (dB) vs sampling rate (on the whole image Lena), for SCS using Gaussian sensing matrices with non-overlapping reconstruction, and subsampling random matrices with overlapping reconstruction. ## V Conclusion Statistical compressed sensing (SCS) based on statistical signal models has been introduced. As opposed to conventional compressed sensing that aims at efficiently sensing and accurately reconstructing one signal at a time, SCS deals simultaneously with a collection of signals. While CS assumes signal sparse models, SCS is based on a more general Bayesian assumption that signals follow a statistical distribution. SCS based on Gaussian models has been investigated in depth. It has been shown that based on a single Gaussian model, with Gaussian or Bernoulli sensing matrices of $\mathcal{O}(k)$ measurements, considerably smaller than the $\mathcal{O}(k\log(N/k))$ required by conventional CS, where $N$ is the signal dimension, and with an optimal decoder implemented with linear filtering, significantly faster than the pursuit decoders applied in conventional CS, the error of SCS is tightly upper bounded by a constant times the best $k$-term approximation error, with overwhelming probability. The failure probability is also significantly smaller than that of conventional CS. Stronger yet simpler results, derived from a new RIP in expectation property further show that for any sensing matrix, the error of Gaussian SCS is upper bounded by a constant times the best $k$-term approximation with probability one, and the bound constant can be efficiently calculated. For Gaussian mixture models (GMMs) that assume multiple Gaussian distributions, and that each signal follows one of them with an unknown index, a piecewise linear estimator is introduced to decode SCS. The accuracy of model selection, which is at the heart of the piecewise linear decoder, is analyzed in terms of the properties of the Gaussian distributions and the number of the sensing measurements. A MAP-EM algorithm that iteratively estimates the Gaussian models and decodes the compressed signals is presented for GMM-based SCS. Applications of GMM-based SCS in real image sensing has been shown. Comparing with conventional CS, SCS leads to improved results, at a considerably lower computational cost. This line of research opens numerous new questions in compressed sensing, from the formal development of bounds in the compressed domain model selection (see also [9]), to the study of model parameters estimation in the compressed domain and the extension of the results here reported to non-Gaussian distributions. The work here reported also shows that compressed sensing is significant beyond sparse signal models, generating the natural question of what type of models can benefit from such sensing scenario. Acknowledgements: Work partially supported by NSF, ONR, NGA, ARO, DARPA, and NSSEFF. The authors thank very much Stéphane Mallat for co-developing the GMM framework reported in [36] for solving inverse problems. ## References * [1] D. Achlioptas. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. 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arxiv-papers
2011-01-30T17:16:55
2024-09-04T02:49:16.728863
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Guoshen Yu and Guillermo Sapiro", "submitter": "Guoshen Yu", "url": "https://arxiv.org/abs/1101.5785" }
1101.5980
# Notes on a particular Weyl Algebra Giuseppe Iurato ###### Abstract By means of the notions of cross product algebras of the theory of quantum groups, in the context of classical Hopf algebra structures, we deduce some known structures of Weyl algebras type (as the Drinfeld quantum double, the restricted Heisenberg double, the generalized Schrödinger representation, and so on) that may be considered as a non-trivial examples of quantum groups having physical meaning, starting from a particular example of groupoid motivated by elementary quantum mechanics. 1\. Introduction In the paper [Iu], following a suggestion of Alain Connes (see [Co], I.1), it has been introduced a particular, simple groupoid, the so-called Heisenberg- Born-Jordan EBB-groupoid (or HBJ EBB-groupoid), whose physical motivations were, mainly, of spectroscopical nature. An E-groupoid (in the notations of [Iu]) is an algebraic system of the type $(G,G^{(0)},r,s,\star)$, with $G,G^{(0)}$ non-void sets, $G^{(0)}\subseteq G$, $G^{(0)}$ set of unities, $r,s:G\rightarrow G^{(0)}$ and $\star:G^{(2)}\rightarrow G$ partial groupoid law defined on $G^{(2)}=\\{(g_{1},g_{2})\in G\times G,s(g_{1})=r(g_{2})\\}$, satisfying the set of axioms described in [Iu], § 1. The HBJ EBB-groupoid is a particular E-groupoid that has been denoted with $\mathcal{G}_{HBJ}(\mathcal{F}_{I})=(\Delta\mathcal{F}_{I},\mathcal{F}_{I},r,s,\tilde{+})$, where $\mathcal{F}_{I}=\\{\nu_{i};\nu_{i}\in\mathbb{R}^{+},i\in I\subseteq\mathbb{N}\\}$ is the set of energy levels of a certain spectroscopic physical system, $\Delta\mathcal{F}_{I}=\\{\nu_{ij};\nu_{ij}=\nu_{i}-\nu_{j},i,j\in I\subseteq\mathbb{N}\\}$, $r:(i,j)\rightarrow i$ and $s:(i,j)\rightarrow j$ are the range and source maps, respectively, and $\nu_{ij}\tilde{+}\nu_{jk}=\nu_{ik}$ is the (partial) groupoid law as algebraic result of the Ritz-Rydberg combination principle. In [Iu], it has been only considered the structure of a no finitely generated groupoid algebra on $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, say $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=(\langle\Delta(\mathcal{F}_{I})\rangle,+,\cdot,\ast)$, respect to an arbitrary commutative field $\mathbb{K}$ and a non-commutative convolution product $\ast$; subsequently, it has been built up a (trivial) structure of braided non-commutative Hopf algebra on it. We claim that this last (albeit trivial) Hopf structure is the first and most natural possible one, on such a groupoid algebra, because of the no (algebraic) finiteness of this generated algebra (since card $I=\infty$, in general). Therefore, the main interest of the paper [Iu], must be searched in the physical construction of the EBJ EBB-groupoid. In this paper, we’ll try to build up other (less trivial) structures on this special HBJ EBB-groupoid, through adapted methods and tools of the theory of quantum groups, relative both to the infinite-dimensional case and finite- dimensional case111The most interesting case, from the physical view-point, is that finite-dimensional corresponding to card $I<\infty$, since any physical spectroscopic system has a finite number of energy levels.. Furthermore, these structures will be introduced taking into account eventual physical motivations. The (above mentioned) natural structure of Hopf algebra on $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ is given as follows: coproduct $\Delta(x)=x\otimes x$, counit $\varepsilon(x)=1$, and antipode the extended inversion map. As already said, the first, natural structure of a braided (or quasitriangular) non-commutative Hopf algebra on such an algebra, is trivially given by the universal R-matrix $R=1\otimes 1$, whence a (trivial) example of quantum group if one assume a braided (or quasitriangular) non-commutative Hopf algebra as definition of quantum group. Instead, a non-trivial example of quasitriangular Hopf algebra arises from Drinfeld quantum double constructions (see § 6.). There exists other definitions of a quantum group structure: for instance, if we consider a non-commutative and non-cocommutative Hopf algebra as quantum group, then a cross (or bicross, or double cross) product construction may provide examples of such a quantum group, whereas, if we consider as special ’quantum objects’ the result of a non-degenerate dual pairing of Hopf algebras, then a Heisenberg double may be taken as an example of quantum group. If one want to determine examples of these last structures starting from $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, it is necessary, at first, examines the possible dual structures of $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, taking into account the existence of some problems for this particular case study. The first problem (that we’ll sketch at the paragraph 3.) is related to dualization in the infinite-dimensional case, whereas the second problem222In a certain sense, preliminary to the first one., because of the infinity of $\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, is due to the tentative of giving a Hopf algebra structure to the $\mathbb{K}$-algebra of $\mathbb{K}$-valued functions defined on the HBJ EBB-groupoid $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, say $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$: in fact, on this algebra (that is strictly correlated to the first problem of dualization of $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, and viceversa) it is a problematic question to define the right comultiplication and counit, for the following reasons. For a group $(\mathcal{G},\cdot)$, the comultiplication question do not subsist in the finite-dimensional case, because of the natural identification $\mathcal{F}_{\mathbb{K}}(\mathcal{G})\otimes\mathcal{F}_{\mathbb{K}}(\mathcal{G})\cong\mathcal{F}_{\mathbb{K}}(\mathcal{G}\times\mathcal{G});$ in such a case, a natural structure of Hopf algebra on $\mathcal{F}_{\mathbb{K}}(\mathcal{G})$, is given by the following data : 1. 1. coproduct $\Delta:\mathcal{F}_{\mathbb{K}}(\mathcal{G})\rightarrow\mathcal{F}_{\mathbb{K}}(\mathcal{G}\times\mathcal{G})$, given by $\Delta(f)(g_{1},g_{2})=f(g_{1}\cdot g_{2})$, for all $g_{1},g_{2}\in\mathcal{G}$; 2. 2. counit $\varepsilon:\mathcal{F}_{\mathbb{K}}(\mathcal{G})\rightarrow\mathbb{K}$, with $\varepsilon(f)=1$; 3. 3. antipode $S:\mathcal{F}_{\mathbb{K}}(\mathcal{G})\rightarrow\mathcal{F}_{\mathbb{K}}(\mathcal{G})$, defined as $S(f)(g)=f(g^{-1})$ for all $g\in\mathcal{G}$, where the functional laws in 1. and 2. are well-defined since, respectively, the group law is totally defined in $\mathcal{G}$, and there exists a unique unit. Instead, if we consider a generic groupoid, these two questions remains unsolved, in the finite-dimensional case too, both for the partial definition of the groupoid law and for the existence of many unities: for these reasons, the initial definitions 1. and 2. of above, are ill-posed in this case. Nevertheless, it is possible to solve these last problems with some extensions in the above definitions, remaining in the context of classical Hopf algebra theory, but with minor usefulness of results. Instead, in the new realm of the extended Hopf algebra structures, this problem may be clarified and solved with fruitfulness, at least for the dual $\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\subseteq\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$. 2\. Cross product algebras The notions of cross product and bicrossproduct are important tools for an algebraic setting of some fundamental structures of Quantum Mechanics. In this paper, we’ll to do only with the notion of cross (or smash, or semidirect) product. Let $V_{\mathbb{K}}$ be a $\mathbb{K}$-linear space, $A$ a $\mathbb{K}$-algebra and $\psi:A\otimes V\rightarrow V$ a $\mathbb{K}$-linear map; if we pose $\psi(h\otimes v)=\psi_{h}(v)$, and if $\psi_{ab}(v)=\psi_{a}(\psi_{b}(v)),\ \ \psi_{1}(v)=v,\ \ \forall a,b\in A,\forall v\in V$, then $(A,V_{\mathbb{K}},\psi)$ is a left $A$-module on $V_{\mathbb{K}}$; we say that $(A,V_{\mathbb{K}},\psi)$ is a left action of $A$ on $V_{\mathbb{K}}$, or that $V_{\mathbb{K}}$ is a left $A$-module . Usually, we write $a\rhd v$ instead of $\psi_{a}(v)$, so that the action axioms are write as $(ab)\rhd v=a\rhd(b\rhd v)$ and $1\rhd v=v$. If $A$ is a Hopf algebra, $V_{\mathbb{K}}$ is an $A$-module algebra [coalgebra], and $a\rhd(vw)=(a_{(1)}\rhd v)(a_{(2)}\rhd w)$, $a\rhd 1_{V}=\varepsilon(h)1_{V}$ [$\Delta(a\rhd v)=(a_{(1)}\rhd v_{(1)})\otimes(a_{(2)}\rhd v_{(2)})$ (that is to say $\Delta(a\rhd v)=\Delta_{A}(a)\rhd\Delta(v)$), $\varepsilon(a\rhd v)=\varepsilon(a)\varepsilon(v)$] for all $v,w\in V_{\mathbb{K}}$ and $a\in A$, then $V_{\mathbb{K}}$ is said a left $A$-module algebra [coalgebra]. If $V$ is a Hopf algebra [bialgebra], then there exists the following two natural left actions on itself: the left regular action $L$, given by $L_{v}(w)=vw$, and the left adjoint action $Ad$, given by $Ad_{v}(w)=v_{(1)}wS(v_{(2)})$, for all $v,w\in V$. The left coregular action $R^{\ast}$ of a finite-dimensional Hopf algebra [bialgebra] $V$ on the dual $V^{\ast}$, is given by $R^{\ast}_{v}(\phi)=\phi_{(1)}\langle v,\phi_{(2)}\rangle$, whereas, in the infinite-dimensional case, we set $\langle R^{\ast}_{v}(\phi),w\rangle=\langle\phi,vw\rangle$, for all $v,w\in V$ and $\phi\in V^{\ast}$, being $\langle\ ,\ \rangle$ the dual pairing between $V$ and $V^{\ast}$ ($V^{o}$ in the infinite-dimensional case); furthermore, $R^{\ast}$ makes $V^{\ast}$ [$V^{o}$] into a $V$-module algebra. The left coadjoint action of a finite-dimensional Hopf algebra [bialgebra] $V$ on the dual $V^{\ast}$, is given by $Ad_{v}^{\ast}(\phi)=\phi_{(2)}\langle v,(S\phi_{(1)})\phi_{(2)}\rangle$, whereas, in the infinite-dimensional case, we put $\langle Ad_{v}^{\ast}(\phi),w\rangle=\langle\phi,(Sv_{(1)})wv_{(2)}\rangle$, for all $v,w\in V$ and $\phi\in V^{\ast}$, being $\langle\ ,\ \rangle$ the dual pairing between $V$ and $V^{\ast}$ ($V^{o}$ in the infinite-dimensional case); furthermore, $Ad^{\ast}$ makes $V^{\ast}$ [$V^{o}$] into a $V$-module coalgebra. The concept of $A$-module algebra generalizes333Because the structure of $A$-module (from commutative algebra) generalize the notion of representation. the notion of $G$-covariant algebra of the Physics: if $G$ is a symmetry group, given a $G$-covariant $\mathbb{K}$-algebra $V$, we construct the group algebra $\mathbb{K}G$ generated by $G$; then, the algebra generated by $\mathbb{K}G$ and $V$, with commutation relations given by $uv=(u\rhd v)u\ \ \forall v\in V,u\in G$, give rise to a semidirect, or cross, product algebra. Therefore, we have the following general structure. Given a Hopf algebra [bialgebra] $A$ and a left $A$-module algebra on $V$, then there exists a left cross product algebra on $V\otimes A$, with product $(v\otimes a)(w\otimes b)=v(a_{(1)}\rhd w)\otimes a_{(2)}b,\qquad v,w\in V,\ a,b\in A$ and unit element $1\otimes 1$. This algebra is denoted with444Or with $V\rtimes_{\psi}A$, if one want to specify the underling left action $\psi$. $V\rtimes A$. With obvious modifications, it is possible to have right actions, as follows. A right action of an algebra $A$ on the $\mathbb{K}$-linear space $V_{\mathbb{K}}$, is a linear map $V\otimes A\rightarrow V$, denoted by $v\otimes a\rightarrow v\lhd a$, such that $v\lhd(ab)=(v\lhd a)\lhd b$ and $v\lhd 1=v$ for all $a,b\in A,\ v\in V$. When $A$ is a Hopf algebra that acts at right on an algebra $V$, and $(ab)\lhd v=(a\lhd v_{(1)})(b\lhd v_{(2)})$, $1_{V}\lhd v=1_{V}\varepsilon(v)$, for all $a,b\in A$ and $v\in V$, then we say that $V$ is a right $A$-module algebra, and we write $(V,A,\lhd)$. A right $A$-coaction on a $\mathbb{K}$-linear space $V_{\mathbb{K}}$, is a linear map $\varphi:V\rightarrow V\otimes A$ such that $(\varphi\otimes\mbox{\rm id})\circ\varphi=(\mbox{\rm id}\otimes\Delta)\circ\varphi$ and $\mbox{\rm id}=(\mbox{\rm id}\otimes\varepsilon)\circ\varphi$; we say, also, that $V_{\mathbb{K}}$ is a right $A$-comodule. If we set $\varphi(v)=v^{(\bar{1})}\otimes v^{(\bar{2})}$ (in $V\otimes A$), then a coalgebra $V_{\mathbb{K}}$ is a right $A$-comodule coalgebra if $V_{\mathbb{K}}$ is a right $A$-comodule and $v_{(1)}^{(\bar{1})}\otimes v_{(2)}^{(\bar{1})}\otimes v^{(\bar{2})}=v_{(1)}^{(\bar{1})}\otimes v_{(2)}^{(\bar{1})}\otimes v_{(1)}^{(\bar{2})}v_{(2)}^{(\bar{2})},\qquad\varepsilon(v^{(\bar{1})})v^{(\bar{2})}=\varepsilon(v).$ If $V$ is a Hopf algebra [bialgebra], there are two natural right actions on itself: the right regular action R, given by $R_{v}(w)=wv$, and the right adjoint action Ad, given by $Ad_{v}(w)=(Sv_{(1)})wv_{(2)}$, for all $v,w\in V$. If $A$ is a Hopf algebra [bialgebra] and $V$ is a right $A$-comodule coalgebra, then there exists a right cross coproduct coalgebra structure on $A\otimes V$, given by $\Delta(a\otimes v)=a_{(1)}\otimes v_{(1)}^{(\bar{1})}\otimes a_{(2)}v_{(1)}^{(\bar{2})}\otimes v_{(2)},\quad\varepsilon(a\otimes v)=\varepsilon_{A}(a)\varepsilon(c),$ for all $a\in A,\ v\in V$; such a coalgebra, is denoted with555Or with $V\ltimes_{\psi}A$, if one want to specify the underling right action $\psi$. $A\ltimes V$. We do not discuss the notion of bicrossproduct algebra, introduced by S. Majid in connections with an important tentative of unifying Quantum mechanics and Gravity (see [Ma], Chap. 6), but we give only a simple example of what a bicrossproduct Hopf algebra is: let $G,M$ two subgroups that factorizes a given group, so that $G$ acts on $M$, and viceversa (for instance, $M$ may be the position space, while $G$ may be the momentum group); let $\mathbb{K}(M)$ the algebra of $\mathbb{K}$-valued functions on $M$, and let $\mathbb{K}G$ be the free algebra on $G$. Then, the following Hopf algebra $\mathbb{K}(M)\bowtie\mathbb{K}G=\left\\{\begin{array}[]{l}\mathbb{K}(M)\rtimes\mathbb{K}G\qquad\mbox{as\ algebra,}\\\ \mathbb{K}(M)\ltimes\mathbb{K}G\qquad\mbox{as\ coalgebra}\end{array}\right.$ is a first example of bicrossproduct algebra, whose dual Hopf algebra is $\mathbb{K}M\bowtie\mathbb{K}(G)$. Another notion strictly correlated to that of bicrossproduct, is the notion of double cross product (see [Ma]). The cross, bicross and double cross product constructions, provides a large class of quantum groups. 3\. The restricted Hopf algebra structure on $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ There exists various methods to define a classical Hopf algebra structure on $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, recalling that this algebra is infinite-dimensional. The main method, proceed as follows. In the case of the groupoid $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, that we recall to be a particular example of the general type $(G,G^{(0)},r,s,\star)$, the most natural modifications to the functional laws on the points 1. and 2. of the § 1, are the following (see [Va], § 2.2): 1’. coproduct: $\Delta(f)(g_{1},g_{2})=f(g_{1}\star g_{2})$ if $(g_{1},g_{2})\in G^{(2)}$, and $=0$ otherwise; 2’. counit: $\varepsilon(f)=\sum_{e\in G^{(0)}}f(e)$. The antipode definition 3., is the same also in this case. Besides the question relative to the functional laws, there exists the question related to their definition sets. Since, in the infinite-dimensional case we have $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\otimes\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\subseteq\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})\times\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ with $\Delta:\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rightarrow\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})\times\mathcal{G}_{HBJ}(\mathcal{F}_{I})),$ let $\mathcal{F}^{o}=\Delta^{-1}(\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\otimes\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})))\subseteq\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$; then, $\mathcal{F}^{o}$ is a Hopf algebra with the coalgebra structure given by 1’., 2’. and 3., although it is difficult to determine exactly its set- theoretic specificity. It is called the restricted Hopf algebra of $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, and is denoted with $\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$. For our purpose, in the finite-dimensional case, there exists the isomorphism (see [Ks], III.1; [Ma], Example 1.5.4) $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\cong\mathcal{A}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, so that we may construct a Hopf algebra structure on $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ via $\mathcal{A}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ by dual pairing; unfortunately, this isomorphism do not subsists in the infinite- dimensional case, and such a question will be at the basis of the discussion of § 6. Other methods for dualization (as Konstant duality, Cartier duality, Tannaka- Krein duality, Takeuki duality, Kadison-Szlachányi dual pairing, the weak antipode plus convolution-inverse method, Pontriyagin duality, and so on), may be found, for instance, in [Sw], [Sch1], [Sch2], [Sch3], [Ma]. However, in the context of the classical Hopf algebra structures, some of these methods do not lead to an explicit solution of the problem, while others provides complicated structures unadapted to the physical applications. But there exists different generalizations of the structure of Hopf algebra (for a recent survey of these, see [Ka]) as, for instance, the notions of weak Hopf algebra (or quantum groupoid) and Hopf algebroid (see [BNS], [NV]), through which it is possible to solve, more explicitly, the above problem, at least for the dual $\mathcal{A}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, in the context of weak Hopf algebras, and with more possibilities on the side of physical applications. Such a question, we’ll be the matter of a further paper. 4\. The restricted Heisenberg double $\mathcal{H}^{o}_{\mathcal{A}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ Hence, as regard what has been said above, we may consider the following dual pairing $\langle\ ,\ \rangle:\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\times\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rightarrow\mathbb{K}$ such that $\langle a_{1}\otimes a_{2},\Delta_{\mathcal{F}^{o}}(f)\rangle=\langle a_{1}a_{2},f\rangle,\qquad\langle\Delta_{\mathcal{A}}(a),f_{1}\otimes f_{2}\rangle=\langle a,f_{1}f_{2}\rangle$ $\langle 1_{\mathcal{A}},f\rangle=\varepsilon_{\mathcal{F}^{o}}(f),\qquad\ \langle 1_{\mathcal{F}^{o}},a\rangle=\varepsilon_{\mathcal{A}}(a)$ for all $f,f_{1},f_{2}\in\mathcal{F}^{o},$ and $a,a_{1},a_{2}\in\mathcal{A}$. It is know that it is always possible to consider, eventually quotienting, a non-degenerate dual pairing of this type. Therefore, if we consider the action $(b,a)\rightarrow b\rhd a=\langle b,a_{(1)}\rangle a_{(2)}\qquad\forall a\in\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),\ \forall b\in\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),$ it follows that it is possible to define the left cross product algebra $\mathcal{H}_{\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))}=\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),$ called the Heisenberg double of the pair $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}),\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$. This last construction may be repeated for the restricted dual $\mathcal{A}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))(\subseteq\mathcal{A}_{\mathbb{K}}^{*}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})))$ of $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, obtaining the so-called Heisenberg double of $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, that we denote, for simplicity, with $\mathcal{H}_{\mathcal{A}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$. 5\. The restricted Weyl algebra The notion of cross product lead to an algebraic formulation of some aspects of quantization. Let $V$ be a $A$-module algebra, with $A$ a Hopf algebra, and let $V\rtimes A$ be the corresponding left cross product. Hence, there exists a canonical representation on $V$ itself, given by $(v\otimes a)\rhd w=v(a\rhd w)$, called the generalized Schrödinger representation of $V$. The physical motivations to this terminology arise from the quantum meaning that such a representation has when applied to the bicrossproduct algebra $\mathbb{K}(M)\rtimes\mathbb{K}G$ of the end of paragraph 2 (see also [Ma], Chap. 6). Our interest is on infinite-dimensional case666But not only; for example, in the finite-dimensional case, we have $V^{o}=V^{\ast}$, and what follows holds also in this case, with obvious modifications., so let $V$ be a infinite- dimensional Hopf algebra, with restricted dual $V^{o}$; then, by the left coregular representation $R^{\ast}$ of $V$ on $V^{o}$ (that does holds also in the infinite-dimensional case, as seen at § 2.), $V^{o}$ is a $V$-module algebra, so that we may consider the left cross product algebra $V^{o}\rtimes V$. Nevertheless, we are interested to another type of left cross product algebra, built up as follows (see [Ma], § 6.1, for details). We consider the following action $\phi\rhd v=v_{(1)}\langle\phi,v_{(2)}\rangle$ for all $v\in V,\phi\in V^{o}$, making $V$ into a $V^{o}$-module algebra and that gives rise to the following product on $V\otimes V^{o}$ $(v\otimes\phi)(w\otimes\psi)=vw_{(1)}\otimes\langle w_{(2)},\phi_{(1)}\rangle\phi_{(2)}\psi,$ whence a structure of left cross product algebra on $V\otimes V^{o}$, namely $V\rtimes V^{o}$. Then, it is possible to prove that the related Schrödinger representation give rise to an isomorphism (of algebras) $\chi:V\rtimes V^{o}\rightarrow Lin(V)$, where $Lin(V)$ is the algebra of $\mathbb{K}$-endomorphisms of $V$, given by $\chi(v\otimes\psi)w=vw_{(1)}\langle\phi,w_{(2)}\rangle$. Therefore, we have the algebra isomorphism $\mathcal{W}(V)=V\rtimes V^{o}\cong Lin(V)$; we call $\mathcal{W}(V)$ the restricted Weyl algebra of the Hopf algebra $V$. This last construction is an algebraic generalization of the usual Weyl algebras of Quantum Mechanics on a group, whose finite-dimensional prototype is as follows. We consider the strict dual pair given by the $\mathbb{K}$-valued functions on $G$, say $\mathbb{K}(G)$, and the free algebra on $G$, say $\mathbb{K}G$; then, the left cross product algebra $\mathbb{K}(G)\rtimes\mathbb{K}G$ is given by the right action of $G$ on itself, namely $\psi_{u}(s)=su$, that induces a left regular representation of $G$ on $\mathbb{K}G$, hence a Schrödinger representation generated by this and by the action of $\mathbb{K}G$ on itself by pointwise product. Whence, if $V=\mathbb{K}(G)$, we obtain the left cross product algebra $\mathbb{K}(G)\rtimes\mathbb{K}G$, isomorphic to $Lin(\mathbb{K}(G))$ by Schrödinger representation, that formalizes the algebraic quantization of a particle moving on $G$ by translations. We may apply these well-known considerations (see [Ma]) to $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ when card $I<\infty$ (finite number of energy levels), taking into account that (in the finite-dimensional case) $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, in such a way that the Weyl algebra $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=$ $=\mathcal{W}(\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})))\cong Lin(\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})))$ represents the algebraic quantization of a particle moving on the groupoid $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ by translations777This remark may be think as the starting point for a quantum mechanics on a groupoid.. Instead, for the infinite-dimensional HBJ EBB-groupoid $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$, we obtain a particular restricted Weyl algebra of the following type $\mathcal{W}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ with $\mathcal{F}_{\mathbb{K}}^{o}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\neq\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ because of the no finitely generation of $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$; therefore, the above physical interpretation of the restricted Weyl algebra $\mathbb{K}(G)\rtimes\mathbb{K}G$, is no longer valid for $\mathcal{W}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$. However, as we’ll see in another place, the last lack of physical interpretation can be restored in the context of extended Hopf algebra structures. 6\. The Drinfeld quantum double If $V$ is a Hopf algebra, then through the left adjoint action (in infinite- dimensional setting) $Ad$ on itself, we have that $V$ is a $V$-module algebra, so that we may build the left cross product algebra $V\rtimes_{Ad}V$. We consider the right adjoint action of $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ on itself, given by $\psi_{g}(h)=g^{-1}\star h\star g$ if exists, $=0$ otherwise; such an action makes $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ into a $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$-module algebra. In the finite-dimensional case we have $\mathcal{A}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\cong\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, hence $\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))=\mathcal{A}_{\mathbb{K}}^{\ast\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\cong\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I})),$ so that $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ is also a $\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$-module algebra, whence the left cross product algebra $\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{F}_{\mathbb{K}}^{\ast}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\cong\mathcal{F}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))\rtimes\mathcal{A}_{\mathbb{K}}(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$ that the tensor product coalgebra makes into a Hopf algebra, called the quantum double of $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ and denoted with $D(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$; even in the finite-dimensional case, it represent the algebraic quantization of a particle constrained to move on conjugacy classes of $\mathcal{G}_{HBJ}(\mathcal{F}_{I})$ (quantization on homogeneous spaces over a groupoid). Besides, it was proved, for a finite group $G$, that this (Drinfeld) quantum double $D(G)$ has a quasitriangular structure (see [Ma], Chap. 6), given by $(\delta_{s}\otimes u)(\delta_{t}\otimes v)=\delta_{u^{-1}su,t}\delta_{t}\otimes uv,\qquad\Delta(\delta_{s}\otimes u)=\sum_{ab=s}\delta_{a}\otimes u\delta_{b}\otimes u,$ $\varepsilon(\delta_{s}\otimes u)=\delta_{s,e},\qquad S(\delta_{s}\otimes u)=\delta_{u^{-1}s^{-1}u}\otimes u^{-1},$ $R=\sum_{u\in G}\delta_{u}\otimes e\otimes 1\otimes u,$ where we have identifies the dual of $\mathbb{K}G$ with $\mathbb{K}(G)$ via the idempotents $p_{g},\ g\in G$ such that $p_{g}p_{h}=\delta_{g,h}p_{g}$ (see [NV], 2.5. and [Ma], 1.5.4); such a quantum double represents the algebra of quantum observables of a certain physical system. Hence, it is a natural question to ask if such structures may be extended to $D(\mathcal{G}_{HBJ}(\mathcal{F}_{I}))$, that is when we have a quantum double built on a groupoid. 7\. Conclusions. From what has been said above (and in [Iu]), it can be meaningful to study as the common known structures, just described above, may be extended when we consider a groupoid888Finite or not. instead of a finite group, both in the classical theory of Hopf algebras and in the new realm of the extended Hopf structures. Subsequently, the resulting structures must be interpreted from the physical view point, with a critical comparison respect to the physical meaning of the classical Hopf structures just seen in this paper. These questions are not trivial, because there are recent papers on a classical Physics on a groupoid (see, for instance, [CDMMM]), and many important works on the role of Hopf algebras in High Energy Physics999Perhaps, it should be interesting to apply the extended Hopf algebra structures to this context. (see, for instance, [Kr] and references therein). References. [BNS] G. Böhm, F. Nill, K. Szlachányi, Weak Hopf Algebra I. Integral Theory and $C^{\ast}$-structure, J. Algebra, 221 (1999) pp. 385-438. [Co] A. Connes, Noncommutative Geometry, Academic Press, New York, 1994. [CDMMM] J. Cortes, M. De Leon, J.C. Marrero, D. Martín de Diego, E. Martínez, A Survey of Lagrangian Mechanics and Control on Lie Algebroids and Groupoids, preprint e-arXiv: math.ph/0511009v1. [Iu] G. Iurato, A possible quantic motivation of the structure of quantum group, JP Journal of Algebra, Number Theory and Applications, 2010 (to appear). [Ka] G. Karaali, On the Hopf algebras and their generalizations, preprint e-arXiv: math.QA/0703441v2. [Ks] C. Kassel, Quantum Groups, Springer-Verlag, New York, 1995. [Kr] D. Kreimer, Algebraic structures in Quantum Field Theory, preprint e-arXiv: hep-th/1007.0341v1. [Ma] S. Majid, Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995. [NV] D. Nikshych, L. Vainerman, Finite quantum groupoids and their applications, in: New directions in Hopf algebras, S. Montgomery, H.J. Schneider eds., MSRI Publications, Vol. 43, (2002), pp. 211-262. [Sch1] P. Schauenburg, Tannaka Duality for arbitrary Hopf Algebras, Verlag Reinhard Fischer, Munich, 1992. [Sch2] P. Schauenburg, Duals and doubles of quantum groupoids ($\times_{R}$-bialgebras), in: New Trends in Hopf Algebra Theory, N. Andruskiewitsch, W.R. Ferrer Santos, and H.J. Schneider, Eds., vol. 267, Amer. Math. Soc., New York, 2000\. [Sch3] P. Schauenburg, Weak Hopf Algebras and Quantum Groupoids, preprint e-arXiv: math.QA/0204180v1. [Sw] M. Sweedler, Hopf Algebras, W.A. Benjamin Inc., New York, 1969. [Va] J.M. Vallin, Relative matched pairs of finite groups from depth two inclusions of Von Neumann algebras to quantum groupoid, preprint e-arXiv: math.OA/0703886v1.
arxiv-papers
2011-01-31T14:27:00
2024-09-04T02:49:16.739162
{ "license": "Public Domain", "authors": "Giuseppe Iurato", "submitter": "Giuseppe Iurato", "url": "https://arxiv.org/abs/1101.5980" }
1101.6040
# Generating GHZ state in $2m$-qubit spin network M. A. Jafarizadeha,d , R. Sufiania , F. Eghbalifama, M. Azimia, S. F. Taghavic and E. Baratie, aDepartment of Theoretical Physics and Astrophysics, University of Tabriz, Tabriz 51664, Iran. bCenter of excellence for photonic, University of Tabriz, Tabriz 51664, Iran. cInstitute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Iran. dResearch Institute for Fundamental Sciences, Tabriz 51664, Iran. eInstitute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warszawa, Poland. E-mail:jafarizadeh@tabrizu.ac.irE- mail:sofiani@tabrizu.ac.ir ###### Abstract We consider a pure $2m$-qubit initial state to evolve under a particular quantum mechanical spin Hamiltonian, which can be written in terms of the adjacency matrix of the Johnson network $J(2m,m)$. Then, by using some techniques such as spectral distribution and stratification associated with the graphs, employed in [1, 2], a maximally entangled $GHZ$ state is generated between the antipodes of the network. In fact, an explicit formula is given for the suitable coupling strengths of the hamiltonian, so that a maximally entangled state can be generated between antipodes of the network. By using some known multipartite entanglement measures, the amount of the entanglement of the final evolved state is calculated, and finally two examples of four qubit and six qubit states are considered in details. Keywords: maximal entanglement , GHZ states, Johnson network, Stratification, Spectral distribution PACs Index: 01.55.+b, 02.10.Yn ## 1 Introduction The idea to use quantum spin chains for short distance quantum communication was put forward by Bose [3]. After the work of Bose, the use of spin chains [4]-[18] and harmonic chains [19] as quantum wires have been proposed. In the previous work [1], the so called distance-regular graphs have been considered as spin networks (in the sense that with each vertex of a distance-regular graph, a qubit or a spin was associated) and perfect state transfer (PST) of a single qubit state over antipodes of these networks has been investigated. In that work, a procedure for finding suitable coupling constants in some particular spin Hamiltonians has been given so that perfect and optimal transfer of a quantum state between antipodes of the corresponding networks can be achieved, respectively. Entanglement is one of the other important tasks in quantum communication. Quantum entanglement in spin systems is an extensively-studied field in recent years [20,21,22,23], in the advent of growing realization that entanglement can be a resource for quantum information processing. Within this general field, entanglement of spin$1/2$ degrees of freedom, qubits, has been in focus for an obvious reason of their paramount importance for quantum computers, not to mention their well-known applicability in various condensed-matter systems, optics and other branches of physics. In [24], authors attempted to generate a Bell state between distant vertices in a permanently coupled spin network interacting via invariant stratification graphs (ISGs). At the first step, they established an upper bound over achievable entanglement between the reference site and the other vertices. Due to this upper bound they found that creation of a Bell state between the reference site and a vertex is possible if the stratum of that vertex is a single element, e.g. antipodal ISGs. The present work focuses on the provision of $GHZ$ state, by using a $2m$-qubit initial product state. To this end, we will consider the Johnson networks $J(2m,m)$ (which are distance-regular) as spin networks. Then, we use the algebraic properties of these networks in order to find suitable coupling constants in some particular spin Hamiltonians so that $2m$-qubit $GHZ$ state can be achieved. The organization of the paper is as follows: In section 2, we review some preliminary facts about graphs and their adjacency matrices, spectral distribution associated with them; In particular, some properties of the networks derived from symmetric group $S_{n}$ called also Johnson networks are reviewed. Section 3 is devoted to $2m$-qubit $GHZ$ state provision by using algebraic properties of Johnson network $J(2m,m)$, where a method for finding suitable coupling constants in particular spin Hamiltonians is given so that maximal entanglement in final state is possible. The paper is ended with a brief conclusion and two appendices. ## 2 preliminaries ### 2.1 Graphs and their adjacency matrices A graph is a pair $\Gamma=(V,E)$, where $V$ is a non-empty set called the vertex set and $E$ is a subset of $\\{(x,y):x,y\in V,x\neq y\\}$ called the edge set of the graph. Two vertices $x,y\in V$ are called adjacent if $(x,y)\in E$, and in that case we write $x\sim y$. For a graph $\Gamma=(V,E)$, the adjacency matrix $A$ is defined as $\bigl{(}A)_{\alpha,\beta}\;=\;\cases{1&if $\;\alpha\sim\beta$\cr 0&\mbox{otherwise}\cr}.$ (2-1) Conversely, for a non-empty set $V$, a graph structure is uniquely determined by such a matrix indexed by $V$. The degree or valency of a vertex $x\in V$ is defined by $\kappa(x)=|\\{y\in V:y\sim x\\}|$ (2-2) where, $|\cdot|$ denotes the cardinality. The graph is called regular if the degree of all of the vertices be the same. In this paper, we will assume that graphs under discussion are regular. A finite sequence $x_{0},x_{1},...,x_{n}\in V$ is called a walk of length $n$ (or of $n$ steps) if $x_{i-1}\sim x_{i}$ for all $i=1,2,...,n$. Let $l^{2}(V)$ denote the Hilbert space of $C$-valued square-summable functions on $V$. With each $\beta\in V$ we associate a vector $|\beta\rangle$ such that the $\beta$-th entry of it is $1$ and all of the other entries of it are zero. Then $\\{|\beta\rangle:\beta\in V\\}$ becomes a complete orthonormal basis of $l^{2}(V)$. The adjacency matrix is considered as an operator acting in $l^{2}(V)$ in such a way that $A|\beta\rangle=\sum_{\alpha\sim\beta}|\alpha\rangle.$ (2-3) ### 2.2 Spectral distribution associated with the graphs Now, we recall some preliminary facts about spectral techniques used in the paper, where more details have been given in Refs. [26,27,28,29] Actually the spectral analysis of operators is an important issue in quantum mechanics, operator theory and mathematical physics [30,31]. As an example $\mu(dx)=|\psi(x)|^{2}dx$ ($\mu(dp)=|\widetilde{\psi}(p)|^{2}dp$) is a spectral distribution which is assigned to the position (momentum) operator $\hat{X}(\hat{P})$. The mathematical techniques such as Hilbert space of the stratification and spectral techniques have been employed in [32,33] for investigating continuous time quantum walk on graphs. Moreover, in general quasi-distributions are the assigned spectral distributions of two hermitian non-commuting operators with a prescribed ordering. For example the Wigner distribution in phase space is the assigned spectral distribution for two non- commuting operators $\hat{X}$ (shift operator) and $\hat{P}$ (momentum operator) with Wyle-ordering among them [34, 35]. It is well known that, for any pair $(A,|\phi_{0}\rangle)$ of a matrix $A$ and a vector $|\phi_{0}\rangle$, one can assign a measure $\mu$ as follows $\mu(x)=\langle\phi_{0}|E(x)|\phi_{0}\rangle,$ (2-4) where $E(x)=\sum_{i}|u_{i}\rangle\langle u_{i}|$ is the operator of projection onto the eigenspace of $A$ corresponding to eigenvalue $x$, i.e., $A=\int xE(x)dx.$ (2-5) Then, for any polynomial $P(A)$ we have $P(A)=\int P(x)E(x)dx,$ (2-6) where for discrete spectrum the above integrals are replaced by summation. Therefore, using the relations (2-4) and (2-6), the expectation value of powers of adjacency matrix $A$ over reference vector $|\phi_{0}\rangle$ can be written as $\langle\phi_{0}|A^{m}|\phi_{0}\rangle=\int_{R}x^{m}\mu(dx),\;\;\;\;\ m=0,1,2,....$ (2-7) Obviously, the relation (2-7) implies an isomorphism from the Hilbert space of the stratification onto the closed linear span of the orthogonal polynomials with respect to the measure $\mu$. ### 2.3 Underlying networks derived from symmetric group $S_{n}$ Let $\lambda=(\lambda_{1},...,\lambda_{m})$ be a partition of $n$, i.e., $\lambda_{1}+...+\lambda_{m}=n$. We consider the subgroup $S_{m}\otimes S_{n-m}$ of $S_{n}$ with $m\leq[\frac{n}{2}]$. Then we assume the finite set $M^{\lambda}=\frac{S_{n}}{S_{m}\otimes S_{n-m}}$ with $|M^{\lambda}|=\frac{n!}{m!(n-m)!}$ as vertex set. In fact, $M^{\lambda}$ is the set of $(m-1)$-faces of $(n-1)$-simplex (recall that, the graph of an $(n-1)$-simplex is the complete graph with $n$ vertices denoted by $K_{n}$). If we denote the vertex $i$ by $m$-tuple $(i_{1},i_{2},...,i_{m})$, then the adjacency matrices $A_{k}$, $k=0,1,...,m$ are defined as $\bigl{(}A_{k})_{i,j}\;=\left\\{\begin{array}[]{c}1\quad\mathrm{if}\;\;\ \partial(i,j)=k,\\\ 0\quad\quad\mathrm{otherwise}\quad\quad\quad(i,j\in M^{\lambda})\\\ \end{array}\right.,\;\;\ k=0,1,...,m.$ (2-8) where, we mean by $\partial(i,j)$ the number of components that $i=(i_{1},...,i_{m})$ and $j=(j_{1},...,j_{m})$ are different (this is the same as Hamming distance which is defined in coding theory). The network with adjacency matrices defined by (2-8) is known also as the Johnson network $J(n,m)$ and has $m+1$ strata such that $\kappa_{0}=1,\;\ \kappa_{l}=\left(\begin{array}[]{c}m\\\ m-l\end{array}\right)\left(\begin{array}[]{c}n-m\\\ l\end{array}\right),\;\;\ l=1,2,...,m.$ (2-9) One should notice that for the purpose of maximal entanglement provision, we must have $\kappa_{m}=1$ which is fulfilled if $n=2m$, so we will consider the network $J(2m,m)$ (hereafter we will take $n=2m$ so that we have $\kappa_{m}=1$). If we stratify the network $J(2m,m)$ with respect to a given reference node $|\phi_{0}\rangle=|i_{1},i_{2},...,i_{m}\rangle$, where $|i_{1},i_{2},...,i_{m}\rangle\equiv|0...0\underbrace{1}_{i_{1}}0...0\underbrace{1}_{i_{2}}0...0\underbrace{1}_{i_{m}}0\rangle$ and $i_{1}\neq i_{2}\neq...\neq i_{m}$. The unit vectors $|\phi_{i}\rangle$, $i=1,...,m$ are defined as $|\phi_{1}\rangle=\frac{1}{\sqrt{\kappa_{1}}}(\sum_{i^{\prime}_{1}\neq i_{1}}|i^{\prime}_{1},i_{2},...,i_{m}\rangle+\sum_{i^{\prime}_{2}\neq i_{2}}|i_{1},i^{\prime}_{2},i_{3},...,i_{m}\rangle+...+\sum_{i^{\prime}_{m}\neq i_{m}}|i_{1},...,i_{m-1},i^{\prime}_{m}\rangle),$ $|\phi_{2}\rangle=\frac{1}{\sqrt{\kappa_{2}}}\sum_{k\neq l=1}^{m}\sum_{i^{\prime}_{l}\neq i_{l},i^{\prime}_{k}\neq i_{k}}|i_{1},...i_{l-1},i^{\prime}_{l},i_{l+1},...,i_{k-1},i^{\prime}_{k},i_{k+1}...,i_{m}\rangle,$ $\vdots$ $\hskip 14.22636pt|\phi_{j}\rangle=\frac{1}{\sqrt{\kappa_{j}}}\sum_{k_{1}\neq k_{2}\neq...\neq k_{j}=1}^{m}\sum_{i^{\prime}_{k_{1}}\neq i_{k_{1}},...,i^{\prime}_{k_{j}}\neq i_{k_{j}}}|i_{1},...,i_{k_{1}-1},i^{\prime}_{k_{1}},i_{k_{1}+1},...,i_{k-1},i^{\prime}_{k_{j}},i_{k_{j}+1}...,i_{m}\rangle,$ $\vdots$ $|\phi_{m}\rangle=\frac{1}{\sqrt{\kappa_{m}}}\sum_{i^{\prime}_{1}\neq i_{1},...,i^{\prime}_{m}\neq i_{m}}|i^{\prime}_{1},i^{\prime}_{2},...,i^{\prime}_{m}\rangle.$ (2-10) Since the network $J(2m,m)$ is distance-regular, the above stratification is independent of the choice of reference node. The intersection array of the network is given by $b_{l}=(m-l)^{2}\;\ ;\;\;\ c_{l}=l^{2}.$ (2-11) Then, by using the Eq. (B-49), the QD parameters $\alpha_{i}$ and $\omega_{i}$ are obtained as follows $\alpha_{l}=2l(m-l),\;\ l=0,1,...,m;\;\;\ \omega_{l}=l^{2}(m-l+1)^{2},\;\ l=1,2,...,m.$ (2-12) Then, one can show that [27] $A|\phi_{l}\rangle=(l+1)(m-l)|\phi_{l+1}\rangle+2l(m-l)|\phi_{l}\rangle+l(m-l+1)|\phi_{l-1}\rangle.$ (2-13) ## 3 $GHZ$ state generation by using quantum mechanical Hamiltonian in the network $J(2m,m)$ The model we consider is the distance-regular Johnson network $J(2m,m)$ consisting of $N=C^{2m}_{m}=\frac{(2m)!}{m!m!}$ sites labeled by $\\{1,2,...,N\\}$ and diameter $m$. Then, we stratify the network with respect to a chosen reference site, say 1 (the discussion about stratification has been given in appendix A; In these particular networks, the first and the last strata possess only one node, i.e., $|\phi_{0}\rangle=|1\rangle$ and $|\phi_{m}\rangle=|N\rangle$). At time $t=0$, a $2m$-qubit state is prepared in the first (reference) site of the network. We wish to provide a maximal quantum entanglement between the state of this site and the state of the $N$-th site after a well-defined period of time, in which the corresponding network is evolved under a particular Hamiltonian. If the network be assumed as a spin network, in which a spin-$1/2$ particle is attached to each vertex (node) of the network, the Hilbert space associated with the network is given by ${\mathcal{H}}=(C^{2})^{\otimes 2m}$. The standard basis for an individual qubit is chosen to be ${|0\rangle=|\downarrow\rangle,|1\rangle=|\uparrow\rangle}$. Then we consider the Hamiltonian $H_{s}=\frac{1}{2}\sum\limits_{1\leq i<j\leq 2m}H_{ij}$ (3-14) where, $H_{ij}=\sigma_{i}\cdot\sigma_{j}$ and $\sigma_{i}$ is a vector with familiar Pauli matrices $\sigma_{i}^{x},\sigma_{i}^{y}$ and $\sigma_{i}^{z}$. One can easily see that, the Hamiltoniaan (3.14) commutes with the total $z$ component of the spin, i.e., $[\sigma^{z}_{total},H_{s}]=0$, hence the Hilbert space ${\mathcal{H}}$ decompose into invariant subspaces, each of which is a distinct eigenspace of the operator $\sigma^{z}_{total}$. So the total number of up and down spins are invariant under action of Hamiltonian or time evolution operator. Now, we recall that the kets $|i_{1},i_{2},\ldots,i_{2m}\rangle$ with $i_{1},\ldots,i_{2m}\in\\{\uparrow,\downarrow\\}$ form an orthonormal basis for Hilbert space ${\mathcal{H}}$. Then, one can easily obtain $H_{ij}|...\underbrace{\uparrow}_{i}...\underbrace{\uparrow}_{j}...\rangle=|...\underbrace{\uparrow}_{i}...\underbrace{\uparrow}_{j}...\rangle$ and $H_{ij}|...\underbrace{\uparrow}_{i}...\underbrace{\downarrow}_{j}...\rangle=-|...\underbrace{\uparrow}_{i}...\underbrace{\downarrow}_{j}...\rangle+2|...\underbrace{\downarrow}_{i}...\underbrace{\uparrow}_{j}...\rangle.$ (3-15) Equation (3.15) implies that the action of $H_{ij}$ on the basis vectors is equivalent to the action of the operator $2P_{ij}-I$, i.e. we have $H_{ij}=2P_{ij}-I$ (3-16) where $P_{ij}$ is the permutation operator acting on sites $i$ and $j$. So $\frac{1}{2}\sum\limits_{1\leq i<j\leq 2m}\sigma_{i}\cdot\sigma_{j}=\sum\limits_{1\leq i<j\leq 2m}P_{ij}-\frac{1}{2}\left(\begin{array}[]{c}2m\\\ 2\\\ \end{array}\right)I,$ (3-17) In fact restriction of the operator $\sum_{1\leq i<j\leq 2m}P_{ij}$ on the $m$-particle subspace (subspace spanned by the states with $m$ spin up) which has dimension $C^{2m}_{m}$, is written as the adjacency matrix $A$ of the Johnson network $J(2m,m)$, as $\sum_{1\leq i<j\leq 2m}P_{ij}=A+m(m-1)I.$ (3-18) For more details see Ref.[1]. Then we stratify the network with respect to a chosen reference site, say $|\phi_{0}\rangle$. At time $t=0$, the state is prepared in the $2m$-qubit state $|\psi(t=0)\rangle=|\underbrace{11\ldots 1}_{m}\rangle|\underbrace{00\ldots 0}_{m}\rangle$. Now, we consider the dynamics of the system to be governed by the Hamiltonian $H=\sum_{k=0}^{m}J_{k}P_{k}(1/2\sum_{{}_{1\leq i<j\leq 2m}}{\mathbf{\sigma}}_{i}\cdot{\mathbf{\sigma}}_{j}+\frac{m}{2}I),$ (3-19) Then, by using (3.17)-(3.20), the Hamiltonian can be written as $H=\sum_{k=0}^{m}J_{k}P_{k}(A)$ (3-20) $J_{k}$ is the coupling strength between the reference site $|\phi_{0}\rangle$ and all of the sites belonging to the $k$-th stratum with respect to $|\phi_{0}\rangle$, and $P_{k}(A)$ are polynomials in terms of adjacency matrix of the Johnson network. Then, the total system is evolved under unitary evolution operator $U(t)=e^{-iHt}$ for a fixed time interval, say $t$. The final state becomes $|\psi(t)\rangle=\sum_{j=1}^{N}f_{jA}(t)|j\rangle$ (3-21) where, $N$ is the number of vertices, $|j\rangle$s have $2m$ entries inclusive $m$ entries equal to 1 and the other entries are 0 and $|A\rangle=|\underbrace{11\ldots 1}_{m}\underbrace{00\ldots 0}_{m}\rangle$ so that $f_{jA}(t):=\langle j|e^{-iHt}|A\rangle$. The evolution with the adjacency matrix $H=A\equiv A_{1}$ for distance-regular networks (see Appendix B) starting in $|\phi_{0}\rangle$, always remains in the stratification space. For distance-regular network $J(2m,m)$ for which the last stratum, i.e., $|\phi_{m}\rangle$ contains only one site, then maximal entanglement between the starting site $|\phi_{0}\rangle\equiv|A\rangle$ and the last stratum $|\phi_{m}\rangle$ (the corresponding antipodal node) is generated, by choosing suitable coupling constants $J_{k}$. In fact, for the purpose of a maximally entangled $GHZ$ state generation between the first and the last stratum of the network, we impose the constraints that the amplitudes $\langle\phi_{i}|e^{-iHt}|\phi_{0}\rangle$ be zero for all $i=1,...,m-1$ and $\langle\phi_{0}|e^{-iHt}|\phi_{0}\rangle=f$, $\langle\phi_{m}|e^{-iHt}|\phi_{0}\rangle=f^{\prime}$. Therefore, these amplitudes must be evaluated. To do so, we use the stratification and spectral distribution associated with the network $J(2m,m)$ to write $\langle\phi_{i}|e^{-iHt}|\phi_{0}\rangle=\langle\phi_{i}|e^{-it\sum_{l=0}^{m}J_{l}P_{l}(A)}|\phi_{0}\rangle=\frac{1}{\sqrt{\kappa_{i}}}\langle\phi_{0}|A_{i}e^{-it\sum_{l=0}^{m}J_{l}P_{l}(A)}|\phi_{0}\rangle$ Let the spectral distribution of the graph is $\mu(x)=\sum_{k=0}^{m}\gamma_{k}\delta(x-x_{k})$ (see Eq. (B-53)). The Johnson network is a kind of network with a highly regular structure that has a nice algebraic description; For example, the eigenvalues of this network can be computed exactly (see for example the notes by Chris Godsil on association schemes [39] for the details of this calculation). Indeed, the eigenvalues of the adjacency matrix of the network $J(2m,m)$ (that is $x_{k}$’s in $\mu(x)$) are given by $x_{k}=m^{2}-k(2m+1-k),\;\;\ k=0,1,\ldots,m.$ (3-22) Now, from the fact that for distance-regular graphs we have $A_{i}=\sqrt{\kappa_{i}}P_{i}(A)$ [27], $\langle\phi_{i}|e^{-iHt}|\phi_{0}\rangle=0$ implies that $\sum_{k=0}^{m}\gamma_{k}P_{i}(x_{k})e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}=0,\;\;\ i=1,...,m-1$ Denoting $e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}$ by $\eta_{k}$, the above constraints are rewritten as follows $\sum_{k=0}^{m}P_{i}(x_{k})\eta_{k}\gamma_{k}=0,\;\;\ i=1,...,m-1,$ $\sum_{k=0}^{m}P_{0}(x_{k})\eta_{k}\gamma_{k}=f$ $\sum_{k=0}^{m}P_{m}(x_{k})\eta_{k}\gamma_{k}=f^{\prime}.$ (3-23) From invertibility of the matrix ${\mathrm{P}}_{ik}=P_{i}(x_{k})$ (see Ref. [2]) one can rewrite the Eq. (3-23) as $\left(\begin{array}[]{c}\eta_{0}\gamma_{0}\\\ \eta_{1}\gamma_{1}\\\ \vdots\\\ \eta_{d-1}\gamma_{d-1}\\\ \eta_{d}\gamma_{d}\end{array}\right)=P^{-1}\left(\begin{array}[]{c}f\\\ 0\\\ \vdots\\\ 0\\\ f^{\prime}\end{array}\right).$ (3-24) The above equation implies that $\eta_{k}\gamma_{k}$ for $k=0,1,...,m$ are the same as the entries in the first column of the matrix ${\mathrm{P}}^{-1}=WP^{t}$ multiplied with $f$ and the entries in the last column multiplied with $f^{\prime}$, i.e., the following equations must be satisfied $\eta_{k}\gamma_{k}=\gamma_{k}e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}={(W{\mathrm{P}}^{t})}_{k0}f+{(W{\mathrm{P}}^{t})}_{km}f^{\prime}\;\ ,\;\;\ \mbox{for}\;\ k=0,1,...,m,$ (3-25) with $W:=diag(\gamma_{0},\gamma_{1},\ldots,\gamma_{m})$. By using the fact that $\gamma_{k}$ and ${(W{\mathrm{P}}^{t})}_{km}$ are real for $k=0,1,\ldots,m$, and so we have $\gamma_{k}=|{(W{\mathrm{P}}^{t})}_{km}|$ and $\gamma_{k}={(W{\mathrm{P}}^{t})}_{k0}$. The Eq. (3.26) can be rewritten as $\eta_{k}=e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}=f+\sigma(k)f^{\prime}$ (3-26) where $\sigma(k)$ is defined as $\sigma(k)=\;\cases{-1&for odd $k$\cr 1&\mbox{otherwise}\cr}.$ (3-27) Assuming $f=|f|e^{i\theta}$ and $f^{\prime}=|f^{\prime}|e^{i{\theta}^{\prime}}$, it should be considered ${\theta}^{\prime}=\theta\pm\frac{\pi}{2}$ then $e^{-it\sum_{l=0}^{m}J_{l}P_{l}(x_{k})}=e^{i\theta}(|f|\pm i\sigma(k)|f^{\prime}|)=e^{i(\theta\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})}+2c_{k}\pi)};\;\;\;c_{k}\in\mathcal{Z}$ (3-28) One should notice that, the Eq. (3.29) can be rewritten as $(J_{0},J_{1},\ldots,J_{m})=-\frac{1}{t}[\theta+2c_{0}\pi\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})},\theta+2c_{1}\pi\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})},$ $,\ldots,\theta+2c_{m}\pi\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})}](W{\mathrm{P}}^{t})$ (3-29) or $J_{k}=-\frac{1}{t}\sum_{j=0}^{m}[\theta+2c_{j}\pi\pm\arctan{(\frac{\sigma(k)|f^{\prime}|}{|f|})}](W{\mathrm{P}}^{t})_{jk}$ (3-30) where $c_{j}$ for $j=0,1,\ldots,m$ are integers. The result (3-30) gives an explicit formula for suitable coupling constants so that GHZ state in the final state can be achieved. The final state is as the form $|\psi(t)\rangle=f|11\ldots 100\ldots 0\rangle+f^{\prime}|00\ldots 011\ldots 1\rangle$ (3-31) One attempt to provide a computationally feasible and scalable quantification of entanglement in multipartite systems was made in Refs. [40,41,42]. For a pure $n$-qubit state $|\psi\rangle$, the so-called global entanglement is defined as $Q(|\psi\rangle)=2(1-\frac{1}{N}\sum\limits_{i=0}^{N-1}Tr[\rho_{i}^{2}])$ (3-32) where $\rho_{i}$ represents the density matrix of $i$th qubit after tracing out all other qubits. As seen from this definition, the global entanglement can be interpreted as the average over the (bipartite) entanglements of each qubit with the rest of the system. The global entanglement for state in Eq. (3.32) will be $Q(|\psi\rangle)=4|f|^{2}|f^{\prime}|^{2}$ (3-33) Also we introduce a simple multiqubit entanglement quantifier based on the idea of bipartition and the measure negativity (which is two times the absolute value of the sum of the negative eigenvalues of the corresponding partially transposed matrix of a state $\rho$) [43]. For an arbitrary $N$-qubit state $\rho_{s_{1}s_{2}...s_{N}}$ , a multiqubit entanglement measure can be formulated as [44] $\overline{\varrho}=\frac{N}{2}\sum\limits_{1}^{\frac{N}{2}}\varrho_{k|N-k}(\rho_{s_{1}s_{2}...s_{N}})$ (3-34) where $N$ is assumed even, otherwise $\frac{N}{2}$ should be replaced by $\frac{N-1}{2}$, and $\varrho_{k|N-k}(\rho_{s_{1}s_{2}...s_{N}})$ is the entanglement in terms of negativity between two blocks of a bipartition $k|N-k$ of the state $\rho_{s_{1}s_{2}...s_{N}}$. We can define the following partition-dependent residual entanglements (PREs) $\Pi_{q_{1}...q_{m}q_{m+1}...q_{k}|q_{k+1}...q_{n}q_{n+1}...q_{N}}={\varrho}^{2}_{q_{1}...q_{m}q_{m+1}...q_{k}|q_{k+1}...q_{n}q_{n+1}...q_{N}}$ $-\varrho^{2}_{q_{1}...q_{m}|q_{k+1}...q_{n}}-\varrho^{2}_{q_{1}...q_{m}|q_{n+1}...q_{N}}-\varrho^{2}_{q_{m+1}...q_{k}|q_{k+1}...q_{n}}-\varrho^{2}_{q_{m+1}...q_{k}|q_{n+1}...q_{N}}$ (3-35) and $\Pi^{\prime}_{q_{1}...q_{k}|q_{k+1}...q_{N}}=\varrho^{2}_{q_{1}...q_{k}|q_{k+1}...q_{N}}-\sum\limits_{i-1}^{k}\sum\limits_{j=k+1}^{N}\varrho^{2}_{q_{i}q_{j}}$ (3-36) For the state in Eq.(3.32), we have $\Pi_{q_{1}...q_{m}q_{m+1}...q_{k}|q_{k+1}...q_{n}q_{n+1}...q_{N}}=\Pi^{\prime}_{q_{1}...q_{k}|q_{k+1}...q_{N}}=\varrho^{2}_{q_{1}...q_{k}|q_{k+1}...q_{N}}=4|f|^{2}|f^{\prime}|^{2}$ (3-37) Another useful entanglement measure was introduced in Refs.[45,46] for $n$-qubit state $|\psi\rangle=\sum_{i=0}^{2^{n}-1}a_{i}|i\rangle$ with even $n$, as $\tau(\psi)=2|{\chi}^{*}(a,n)|$ (3-38) where ${\chi}^{*}(a,n)=\sum\limits_{i=0}^{2^{n-2}-1}sgn^{*}(n,i)(a_{2i}a_{(2^{n-1}-1)-2i}-a_{2i+1}a_{(2^{n-2}-2)-2i}),$ (3-39) $sgn^{*}(n,i)=\;\cases{{(-1)^{N(i)}}&$0\leq i\leq{2^{n-3}-1}$\cr{(-1)^{N(i)+n}}&$2^{n-3}\leq i\leq{2^{n-2}-1}$\cr}$ (3-40) where, $N(i)$ is the number of the occurrences of 1 in the $n$-bit binary representation of $i$ as $i_{n-1}...i_{1}i_{0}$ ( in binary representation, $i$ is written as $i=i_{n-1}2^{n-1}+...+i_{1}2^{1}+i_{0}2^{0}$). For the state Eq.(3.32), one can see that $\tau(\psi)=2|{\chi}^{*}(a,n)|=2|a_{2^{m}-1}a_{2^{2m}-2^{m}}|=2|ff^{\prime}|.$ (3-41) In order to achieve maximal entanglement ($GHZ$ state), we should have $|f|=|f^{\prime}|=\frac{1}{\sqrt{2}}$ (3-42) Then $Q(|\psi\rangle)=\Pi_{q_{1}...q_{m}q_{m+1}...q_{k}|q_{k+1}...q_{n}q_{n+1}...q_{N}}=\Pi^{\prime}_{q_{1}...q_{k}|q_{k+1}...q_{N}}=\tau(\psi)=1$. In the following we consider the four qubit state (the case $m=2$) $|\psi(t=0)\rangle=|1100\rangle$ and the six qubit state (the case $m=3$) $|\psi(t=0)\rangle=|111000\rangle$ in details: From Eq. (2-12), for $m=2$, the QD parameters are given by $\alpha_{1}=2,\;\ \alpha_{2}=0;\;\;\ \omega_{1}=\omega_{2}=4,$ Then by using the recursion relations (B-48) and (B-51), we obtain $Q_{2}^{(1)}(x)=x^{2}-2x-4,\;\;\ Q_{3}(x)=x(x-4)(x+2),$ so that the stieltjes function is given by $G_{\mu}(x)=\frac{Q_{2}^{(1)}(x)}{Q_{3}(x)}=\frac{x^{2}-2x-4}{x(x-4)(x+2)}.$ Then the corresponding spectral distribution is given by $\mu(x)=\sum_{l=0}^{2}\gamma_{l}\delta(x-x_{l})=\frac{1}{6}\\{3\delta(x)+\delta(x-4)+2\delta(x+2)\\},$ which indicates that $W=\left(\begin{array}[]{ccc}\gamma_{0}&0&0\\\ 0&\gamma_{1}&0\\\ 0&0&\gamma_{2}\\\ \end{array}\right)=\frac{1}{6}\left(\begin{array}[]{ccc}1&0&0\\\ 0&3&0\\\ 0&0&2\\\ \end{array}\right).$ In order to obtain the suitable coupling constants, we need also the eigenvalue matrix $P$ with entries $P_{ij}=P_{i}(x_{j})=\frac{1}{\sqrt{\omega_{1}\ldots\omega_{i}}}Q_{i}(x_{j})$. By using the recursion relations (B-48), one can obtain $P_{0}(x)=1,\;\ P_{1}(x)=\frac{x}{2}$ and $P_{2}(x)=\frac{1}{4}(x^{2}-2x-4)$, so that $P=\left(\begin{array}[]{ccc}1&1&1\\\ 2&0&-1\\\ 1&-1&1\\\ \end{array}\right).$ Then, Eq. (3-30) leads to $-t(J_{0}+2J_{1}+J_{2})=\theta\pm\frac{\pi}{4}\pm 2c_{0}\pi,$ $-t(J_{0}-J_{2})=\theta\mp\frac{\pi}{4}\pm 2c_{1}\pi,$ $-t(J_{0}-J_{1}+J_{2})=\theta\pm\frac{\pi}{4}\pm 2c_{2}\pi.$ Now, by considering $c_{0}=c_{1}=c_{2}=0$ we obtain $J_{0}=-\frac{\theta}{t},\;\ J_{1}=0,\;\ J_{2}=\mp\frac{\pi}{4t}.$ Also by considering $c_{0}=0,c_{1}=c_{2}=1$ the coupling constants will be $J_{0}=-\frac{3\theta\pm 5\pi}{3t},\;\ J_{1}=\pm\frac{2\pi}{3t},\;\ J_{2}=\pm\frac{\pi}{12t}$ and by considering $c_{0}=1,c_{1}=c_{2}=0$ $J_{0}=-\frac{3\theta\pm\pi}{3t},\;\ J_{1}=\mp\frac{2\pi}{3t},\;\ J_{2}=\mp\frac{7\pi}{12t}$ From Eq. (2-12), for $m=3$, the QD parameters are given by $\alpha_{1}=4,\;\ \alpha_{2}=4,\;\ \alpha_{3}=0;\;\;\ \omega_{1}=\omega_{3}=9,\;\omega_{2}=16$ Then by using the recursion relations (B-48) and (B-51), we obtain $Q_{3}^{(1)}(x)=x^{3}-8x^{2}-9x+36,\;\;\ Q_{4}(x)=(x^{2}-9)(x-9)(x+1),$ so that the stieltjes function is given by $G_{\mu}(x)=\frac{Q_{3}^{(1)}(x)}{Q_{4}(x)}=\frac{x^{3}-8x^{2}-9x+36}{(x^{2}-9)(x-9)(x+1)}.$ Then the corresponding spectral distribution is given by $\mu(x)=\sum_{l=0}^{3}\gamma_{l}\delta(x-x_{l})=\frac{1}{20}\\{\delta(x-9)+5\delta(x-3)+9\delta(x+1)+5\delta(x+3)\\},$ which indicates that $W=\left(\begin{array}[]{cccc}\gamma_{0}&0&0&0\\\ 0&\gamma_{1}&0&0\\\ 0&0&\gamma_{2}&0\\\ 0&0&0&\gamma_{3}\\\ \end{array}\right)=\frac{1}{20}\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&5&0&0\\\ 0&0&9&0\\\ 0&0&0&5\\\ \end{array}\right).$ By using the recursion relations (B-48), one can obtain $P_{0}(x)=1,\;\ P_{1}(x)=\frac{x}{3}$, $P_{2}(x)=\frac{1}{12}(x^{2}-4x-9)$ and $P_{3}(x)=\frac{1}{36}(x^{3}-8x^{2}-9x+36)$, so that $P=\left(\begin{array}[]{cccc}1&1&1&1\\\ 3&1&-\frac{1}{3}&-1\\\ 3&-1&-\frac{1}{3}&1\\\ 1&-1&1&-1\\\ \end{array}\right).$ Then, Eq. (3-30) gives $-t(J_{0}+3J_{1}+3J_{2}+J_{3})=\theta\pm\frac{\pi}{4}\pm 2c_{0}\pi,$ $-t(J_{0}+J_{1}-J_{2}-J_{3})=\theta\mp\frac{\pi}{4}\mp 2c_{1}\pi,$ $-t(J_{0}-\frac{1}{3}J_{1}-\frac{1}{3}J_{2}+J_{3})=\theta\pm\frac{\pi}{4}\pm 2c_{2}\pi,$ $-t(J_{0}-J_{1}+J_{2}-J_{3})=\theta\mp\frac{\pi}{4}\pm 2c_{3}\pi.$ Again, by considering $c_{0}=c_{1}=c_{2}=c_{3}=0$ we obtain $J_{0}=-\frac{\theta}{t},\;\ J_{1}=J_{2}=0,\;\ J_{3}=\frac{\mp\pi}{4t}.$ ## 4 Conclusion A $2m$-qubit initial state was prepared to evolve under a particular spin Hamiltonian, which could be written in terms of the adjacency matrix of the Johnson graph $J(2m,m)$. By using spectral analysis methods and employing algebraic structures of the Johnson networks, such as distance-regularity and stratification, a method for finding a suitable set of coupling constants in the Hamiltonians associated with the networks was given so that in the final state, the maximal entanglement of the form $GHZ$ state, could be generated. In this work we imposed a constraint so that all amplitudes in the final state were equal to zero except to two amplitudes corresponding to the first and the final strata (any pair of antinodes of the network), where for $J(2m,m)$ these strata contain only one vertex, then $GHZ$ state was generated. We hope to generalize this method to arbitrary Johnson networks $J(n,m)$ and other various graphs, in order to investigate the entanglement of such systems by using some multipartite entanglement measures. ## Appendix ## Appendix A Stratification technique In this section, we recall the notion of stratification for a given graph $\Gamma$. To this end, let $\partial(x,y)$ be the length of the shortest walk connecting $x$ and $y$ for $x\neq y$. By definition $\partial(x,x)=0$ for all $x\in V$. The graph becomes a metric space with the distance function $\partial$. Note that $\partial(x,y)=1$ if and only if $x\sim y$. We fix a vertex $o\in V$ as an origin of the graph, called the reference vertex. Then, the graph $\Gamma$ is stratified into a disjoint union of strata (with respect to the reference vertex $o$) as $V=\bigcup_{i=0}^{\infty}\Gamma_{i}(o),\;\ \Gamma_{i}(o):=\\{\alpha\in V:\partial(\alpha,o)=i\\}$ (A-43) Note that $\Gamma_{i}(o)=\emptyset$ may occur for some $i\geq 1$. In that case we have $\Gamma_{i}(o)=\Gamma_{i+1}(o)=...=\emptyset$. With each stratum $\Gamma_{i}(o)$ we associate a unit vector in $l^{2}(V)$ defined by $|\phi_{i}\rangle=\frac{1}{\sqrt{\kappa_{i}}}\sum_{\alpha\in\Gamma_{i}(o)}|\alpha\rangle,$ (A-44) where, $\kappa_{i}=|\Gamma_{i}(o)|$ is called the $i$-th valency of the graph ($\kappa_{i}:=|\\{\gamma:\partial(o,\gamma)=i\\}|=|\Gamma_{i}(o)|$). One should notice that, for distance regular graphs, the above stratification is independent of the choice of reference vertex and the vectors $|\phi_{i}\rangle,i=0,1,...,d-1$ form an orthonormal basis for the so called Krylov subspace $K_{d}(|\phi_{0}\rangle,A)$ defined as $K_{d}(|\phi_{0}\rangle,A)=\mathrm{span}\\{|\phi_{0}\rangle,A|\phi_{0}\rangle,\cdots,A^{d-1}|\phi_{0}\rangle\\}.$ (A-45) Then it can be shown that [25], the orthonormal basis $|\phi_{i}\rangle$ are written as $|\phi_{i}\rangle=P_{i}(A)|\phi_{0}\rangle,$ (A-46) where $P_{i}=a_{0}+a_{1}A+...+a_{i}A^{i}$ is a polynomial of degree $i$ in indeterminate $A$ (for more details see for example [25,26]). ## Appendix B Spectral distribution associated with the graphs In this section we recall some facts about spectral techniques used in the paper. From orthonormality of the unit vectors $|\phi_{i}\rangle$ given in Eq.(A-44) (with $|\phi_{0}\rangle$ as unit vector assigned to the reference node) we have $\delta_{ij}=\langle\phi_{i}|\phi_{j}\rangle=\int_{R}P_{i}(x)P_{j}(x)\mu(dx).$ (B-47) By rescaling $P_{k}$ as $Q_{k}=\sqrt{\omega_{1}\ldots\omega_{k}}P_{k}$, the spectral distribution $\mu$ under question will be characterized by the property of orthonormal polynomials $\\{Q_{k}\\}$ defined recurrently by $Q_{0}(x)=1,\;\;\;\;\;\ Q_{1}(x)=x,$ $xQ_{k}(x)=Q_{k+1}(x)+\alpha_{k}Q_{k}(x)+\omega_{k}Q_{k-1}(x),\;\;\ k\geq 1.$ (B-48) The parameters $\alpha_{k}$ and $\omega_{k}$ appearing in (B-48) are defined by $\alpha_{0}=0,\;\;\ \alpha_{k}=\kappa-b_{k}-c_{k},\;\;\;\;\ \omega_{k}\equiv\beta^{2}_{k}=b_{k-1}c_{k},\;\;\ k=1,...,d,$ (B-49) where, $\kappa\equiv\kappa_{1}$ is the degree of the networks and $b_{i}$’s and $c_{i}$’s are the corresponding intersection numbers. Following Refs. [34], we will refer to the parameters $\alpha_{k}$ and $\omega_{k}$ as $QD$ (Quantum Decomposition) parameters (see Refs. [26,27,28,34] for more details). If such a spectral distribution is unique, the spectral distribution $\mu$ is determined by the identity $G_{\mu}(x)=\int_{R}\frac{\mu(dy)}{x-y}=\frac{1}{x-\alpha_{0}-\frac{\omega_{1}}{x-\alpha_{1}-\frac{\omega_{2}}{x-\alpha_{2}-\frac{\omega_{3}}{x-\alpha_{3}-\cdots}}}}=\frac{Q_{d}^{(1)}(x)}{Q_{d+1}(x)}=\sum_{l=0}^{d}\frac{\gamma_{l}}{x-x_{l}},$ (B-50) where, $x_{l}$ are the roots of the polynomial $Q_{d+1}(x)$. $G_{\mu}(x)$ is called the Stieltjes/Hilbert transform of spectral distribution $\mu$ and polynomials $\\{Q_{k}^{(1)}\\}$ are defined recurrently as $Q_{0}^{(1)}(x)=1,\;\;\;\;\;\ Q_{1}^{(1)}(x)=x-\alpha_{1},$ $xQ_{k}^{(1)}(x)=Q_{k+1}^{(1)}(x)+\alpha_{k+1}Q_{k}^{(1)}(x)+\omega_{k+1}Q_{k-1}^{(1)}(x),\;\;\ k\geq 1,$ (B-51) respectively. The coefficients $\gamma_{l}$ appearing in (B-50) are calculated as $\gamma_{l}:=\lim_{x\rightarrow x_{l}}[(x-x_{l})G_{\mu}(x)]$ (B-52) Now let $G_{\mu}(z)$ is known, then the spectral distribution $\mu$ can be determined in terms of $x_{l},l=1,2,...$ and Gauss quadrature constants $\gamma_{l},l=1,2,...$ as $\mu=\sum_{l=0}^{d}\gamma_{l}\delta(x-x_{l})$ (B-53) (for more details see Refs. [35,36,37,38]). ## References * [1] M. A. Jafarizadeh and R. Sufiani, (2008), Phys. Rev. A 77, 022315\. * [2] M. A. Jafarizadeh, R. Sufiani, S. F. Taghavi and E. Barati, (2008), J. Phys. A: Math. Theor. 41, 475302. * [3] Sougato Bose, Phys. Rev. Lett. 91, 207901 (2003). * [4] V. Subrahmanyam, Phys. Rev. A 69, 034304 (2004). * [5] M. Christandl, N. Datta, A. Ekert and A. J. Landahl, Phys. Rev. Lett. 92, 187902 (2004). * [6] M. Christandl, N. Datta, T. C. Dorlas, A. Ekert, A. Kay and A. J. Landahl, (2005), Phys. Rev. A 71, 032312. * [7] C. Albanese, M. Christandl, N. Datta and A. Ekert, Phys. Rev. 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arxiv-papers
2011-01-31T17:54:06
2024-09-04T02:49:16.745396
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. A. Jafarizadeh, R. Sufiani, F. Eghbalifam, M. Azimi, S. F. Taghavi\n and E. Barati", "submitter": "Mohamad Ali Jafarizadeh", "url": "https://arxiv.org/abs/1101.6040" }
1102.0048
∎ 11institutetext: S. Mottelet 22institutetext: Laboratoire de Mathématiques Appliquées Université de Technologie de Compiègne 60205 Compiègne France 22email: stephane.mottelet@utc.fr 33institutetext: L. de Saint Germain, O. Mondin44institutetext: Luxilon 21, rue du Calvaire 92210 Saint-Cloud France 44email: lsg@lsg-studio.com # Smart depth of field optimization applied to a robotised view camera Stéphane Mottelet Luc de Saint Germain Olivier Mondin ###### Abstract The great flexibility of a view camera allows the acquisition of high quality images that would not be possible any other way. Bringing a given object into focus is however a long and tedious task, although the underlying optical laws are known. A fundamental parameter is the aperture of the lens entrance pupil because it directly affects the depth of field. The smaller the aperture, the larger the depth of field. However a too small aperture destroys the sharpness of the image because of diffraction on the pupil edges. Hence, the desired optimal configuration of the camera is such that the object has the sharpest image with the greatest possible lens aperture. In this paper, we show that when the object is a convex polyhedron, an elegant solution to this problem can be found. The problem takes the form of a constrained optimization problem, for which theoretical and numerical results are given. ###### Keywords: Large format photography Computational photography Scheimpflug principle ## 1 Introduction --- (a) (b) Figure 1: The Sinar e (a) and its metering back (b). Since the registration of Theodor Scheimpflug’s patent in 1904 (see Scheimpflug ), and the book of Larmore in 1965 where a proof of the so-called Scheimpflug principle can be found (see (larmore, , p. 171-173)), very little has been written about the mathematical concepts used in modern view cameras, until the development of the Sinar e in 1988 (see Figure 1). A short description of this camera is given in (Tillmans, , p. 23): The Sinar e features an integrated electronic computer, and in the studio offers a maximum of convenience and optimum computerized image setting. The user-friendly software guides the photographer through the shot without technical confusion. The photographer selects the perspective (camera viewpoint) and the lens, and chooses the areas in the subject that are to be shown sharp with a probe. From these scattered points the Sinar e calculates the optimum position of the plane of focus, the working aperture needed, and informs the photographer of the settings needed Sinar sold a few models of this camera and discontinued its development in the early nineties. Surprisingly, there has been very little published user feedback about the camera itself. However many authors started to study (in fact, re-discover) the underlying mathematics (see e.g. Merklinger and the references therein). The most crucial aspect is the consideration of depth of field and the mathematical aspects of this precise point are now well understood. When the geometrical configuration of the view camera is precisely known, then the depth of field region (the region of space where objects have a sharp image) can be determined by using the laws of geometric optics. Unfortunately, these laws can only be used as a rule of thumb when operating by hand on a classical view camera. Moreover, the photographer is rather interested in the inverse problem: given an object which has to be rendered sharply, what is the optimal configuration of the view camera? A fundamental parameter of this configuration is the aperture of the camera lens. Decreasing the lens aperture diameter increases the depth of field but also increases the diffraction of light by the lens entrance pupil. Since diffraction decreases the sharpness of the image, the optimal configuration should be such that the object fits the depth of field region with the greatest aperture. This paper presents the mathematical tools used in the software of a computer controlled view camera solving this problem. Thanks to the high precision machining of its components, and to the known optical parameters of the lens and digital sensor, a reliable mathematical model of the view camera has been developed. This model allows the acquisition of 3D coordinates of the object to be photographed, as explained in Section 2. In Section 3 we study the depth of field optimization problem from a theoretical and numerical point of view. We conclude and briefly describe the architecture of the software in Section 4. ## 2 Basic mathematical modeling ### 2.1 Geometrical model of the View Camera --- (a) (b) Figure 2: Geometrical model (a) and robotised view camera (b). We consider the robotised view camera depicted in Figure 2(a) and its geometrical model in Figure 2(b). We use a global Euclidean coordinate system $(\mathbf{O},X_{1},X_{2},X_{3})$ attached to the camera’s tripod. The front standard, symbolized by its frame with center $L$ of global coordinates $\mathbf{L}=(L_{1},L_{2},L_{3})^{\top}$, can rotate along its tilt and swing axes with angles $\theta_{L}$ and $\phi_{L}$. Most camera lenses are in fact thick lenses and nodal points $H^{\prime}$ and $H$ have to be considered (see Ray p. 43-46). The rear nodal plane, which is parallel and rigidly fixed to the front standard, passes through the rear nodal point $H^{\prime}$. Since $L$ and $H^{\prime}$ do not necessarily coincide, the translation between these two points is denoted $\mathbf{t^{L}}$. The vector $\overrightarrow{HH^{\prime}}$ is supposed to be orthogonal to the rear nodal plane. The rear standard is symbolized by its frame with center $S$, whose global coordinates are given by $\mathbf{S}=(S_{1},S_{2},S_{3})^{\top}$. It can rotate along its tilt and swing axes with angles $\theta_{S}$ and $\phi_{S}$. The sensor plane is parallel and rigidly fixed to the rear standard. The eventual translation between $S$ and the center of the sensor is denoted by $\mathbf{t^{S}}$. The rear standard center $S$ can move in the three $X_{1},X_{2}$ and $X_{3}$ directions but the front standard center $L$ is fixed. The rotation matrices associated with the front and rear standard alt-azimuth mounts are respectively given by $\mathbf{R^{L}}=\mathbf{R}(\theta_{L},\phi_{L})$ and $\mathbf{R^{S}}=\mathbf{R}(\theta_{S},\phi_{S})$ where $\mathbf{R}(\theta,\phi)=\left(\begin{array}[]{ccc}\cos\phi&-\sin\phi\sin\theta&-\sin\phi\cos\theta\\\ 0&\cos\theta&-\sin\theta\\\ -\sin\phi&\cos\phi\sin\theta&\cos\phi\cos\theta\end{array}\right).$ The intrinsic parameters of the camera (focal length $f$, positions of the nodal points $H$, $H^{\prime}$, translations $\mathbf{t^{S}}$, $\mathbf{t^{L}}$, image sensor characteristics) are given by their respective manufacturers data-sheets. The extrinsic parameters of the camera are $\mathbf{S}$, $\mathbf{L}$, the global coordinate vectors of $S$ and $L$, and the four rotation angles $\theta_{S}$, $\phi_{S}$, $\theta_{L}$, $\phi_{L}$. The precise knowledge of the extrinsic parameters is possible thanks to the computer-aided design model used for manufacturing the camera components. In addition, translations and rotations of the rear and front standards are controlled by stepper motors whose positions can be precisely known. In the following, we will see that this precise geometrical model of the view camera allows one to solve various photographic problems. The first problem is the determination of coordinates of selected points of the object to be photographed. In the sequel, for sake of simplicity, we will give all algebraic details of the computations for a thin lens, i.e. when the two nodal points $H$, $H^{\prime}$ coincide. In this case, the nodal planes are coincident in a so- called lens plane. We will also consider that $\mathbf{t^{L}}=(0,0,0)^{\top}$ so that $L$ is the optical center of the lens. Finally, we also consider that $\mathbf{t}^{S}=(0,0,0)^{\top}$ so that $S$ coincides with the center of the sensor. ### 2.2 Acquisition of object points coordinates --- Figure 3: Graphical construction of the image $A$ of an object point $X$ in the optical coordinate system. Green rays and blue rays respectively lie in the $(L,x_{3},x_{2})$ and $(L,x_{3},x_{1})$ planes and $F$ is the focal point. --- Figure 4: Image formation when considering a lens with a pupil. Let us consider a point $X$ with global coordinates given by $\mathbf{X}=(X_{1},X_{2},X_{3})^{\top}$. The geometrical construction of the image $A$ of $X$ through the lens is depicted in Figure 3. We have considered a local optical coordinate system attached to the lens plane with origin $L$. The local coordinates of $X$ are given by $\mathbf{x}=(\mathbf{R^{L}})^{-1}(\mathbf{X}-\mathbf{L})$ and the focal point $F$ has local coordinates $(0,0,f)^{\top}$. Elementary geometrical optics (see Ray p. 35-42) allows one to conclude that if the local coordinates of $A$ are given by $\mathbf{a}=(a_{1},a_{2},a_{3})^{\top}$, then $a_{3}$, $x_{3}$ and $f$ are linked by the thin lens equation given in its Gaussian form by $-\frac{1}{a_{3}}+\frac{1}{x_{3}}=\frac{1}{f}.$ Since $A$ lies on the $(XL)$ line, the other coordinates are obtained by straightforward computations and we have the conjugate formulas $\displaystyle\mathbf{a}$ $\displaystyle=\frac{f}{f-x_{3}}\mathbf{x},$ (1) $\displaystyle\mathbf{x}$ $\displaystyle=\frac{f}{f+a_{3}}\mathbf{a}.$ (2) Bringing an object into focus is one of the main tasks of a photographer but it can also be used to calculate the coordinates of an object point. It is important to remember that all light rays emanating from $X$ converge to $A$ but pass through a pupil (or diaphragm) assumed to be circular, as depicted in Figure 4. Since all rays lie within the oblique circular cone of vertex $A$ and whose base is the pupil, the image of $X$ on the sensor will be in focus only if the sensor plane passes through $A$, otherwise its extended image will be a blur spot. By using the full aperture of the lens, the image will rapidly go out of focus if the sensor plane is not correctly placed, e.g. by translating $S$ into the $x_{3}$ direction. This is why auto-focus systems on classical cameras only work at near full aperture: the distance to an object is better determined when the depth of field is minimal. The uncertainty on the position of $S$ giving the best focus is related to the diameter of the so-called “circle of confusion”, i.e. the maximum diameter of a blur spot that is indistinguishable from a point. Hence, everything depends on the size of photosites on the sensor and on the precision of the focusing system (either manual or automatic). This uncertainty is acceptable and should be negligible compared to the uncertainty of intrinsic and extrinsic camera parameters. The previous analysis shows that the global coordinates of $X$ can be computed, given the position $(u,v)^{\top}$ of its image $A$ on the sensor plane. This idea has been already used on the Sinar e, where the acquisition of $(u,v)^{\top}$ was done by using a mechanical metering unit (see Figure 1 (b)). In the system we have developed, a mouse click in the live video window of the sensor is enough to indicate these coordinates. Once $(u,v)^{\top}$ is known, the coordinates of $A$ in the global coordinate system are given by $\mathbf{A}=\mathbf{S}+\mathbf{R^{S}}\left(\begin{array}[]{c}u\\\ v\\\ 0\end{array}\right),$ and its coordinates in the optical system by $\mathbf{a}=(\mathbf{R^{L}})^{-1}(\mathbf{A}-\mathbf{L}).$ Then the local coordinate vector $\mathbf{x}$ of the reciprocal image is computed with (2) and the global coordinate vector $\mathbf{X}$ is obtained by $\mathbf{X}=\mathbf{L}+\mathbf{R^{L}}\mathbf{x}.$ By iteratively focusing on different parts of the object, the photographer can obtain a set of points $\mathcal{X}=\\{{X}^{1},\dots,{X}^{n}\\}$, with $n\geq 3$, which can be used to determine the best configuration of the view camera, i.e. the positions of front and rear standards and their two rotations, in order to satisfy focus requirements. ## 3 Focus and depth of field optimization In classical digital single-lens reflex (DLSR) cameras, the sensor plane is always parallel to the lens plane and to the plane of focus. For example, bringing into focus a long and flat object which is not parallel to the sensor needs to decrease the aperture of the lens in order to extend the depth of field. On the contrary, view cameras with tilts and swings (or DLSR with a tilt/shift lens) allow to skew away the plane of focus from the parallel in any direction. Hence, bringing into focus the same long and flat object with a view camera can be done at full aperture. This focusing process is unfortunately very tedious. However, if a geometric model of the camera and the object are available, the adequate rotations can be estimated precisely. In the next sections, we will explain how to compute the rear standard position and the tilt and swing angles of both standards to solve two different problems: 1. 1. when the focus zone is roughly flat, and depth of field is not a critical issue, then the object plane is computed from the set of object points $\mathcal{X}$. If $n=3$ and the points are not aligned then this plane is uniquely defined. If $n>3$ and at least $3$ points are not aligned, we compute the best fitting plane minimizing the sum of squared orthogonal distances to points of $\mathcal{X}$. Then, we are able to bring this plane into sharp focus by acting on: 1. (a) the angles $\theta_{L}$ and $\phi_{L}$ of the front standard and the position of the rear standard, for arbitrary rotation angles $\theta_{S}$, $\phi_{S}$. 2. (b) the angles $\theta_{S}$, $\phi_{S}$ and position of the rear standard, for arbitrary rotation angles $\theta_{L}$, $\phi_{L}$ (in this case there is a perspective distortion). 2. 2. when the focus zone is not flat, then the tridimensional shape of the object has to be taken into account. The computations in case 1a are detailed in Section 3.1. In Section 3.3 a general algorithm is described that allows the computation of angles $\theta_{L}$ and $\phi_{L}$ of the front standard and the position of the rear standard such that all the object points are in the depth of field region with a maximum aperture. We give a theoretical result showing that the determination of the solution amounts to enumerate a finite number of configurations. ### 3.1 Placement of the plane of sharp focus by using tilt and swing angles Figure 5: Illustration of the Scheimpflug rule. Figure 6: Illustration of the Hinge rule. In this section we study the problem of computing the tilt and swing angles of front standard and the position of the rear standard for a given sharp focus plane. Although the underlying laws are well-known and are widely described (see Merklinger ; Wheeler ; Evens ), the detail of the computations is always done for the particular case where only the tilt angle $\theta$ is considered. Since we aim to consider the more general case where tilt and swing angles are used, we will describe the various objects (lines, planes) and the associated computations by using linear algebra tools. #### 3.1.1 The Scheimpflug and the Hinge rules In order to explain the Scheimpflug rule, we will refer to the diagram depicted in Figure 5. The Plane of sharp focus (abbreviated $\mathrm{SFP}$) is determined by a normal vector $\mathbf{n^{SF}}$ and a point $Y$. The position of the optical center $L$ and a vector $\mathbf{n^{S}}$ normal to the sensor plane (abbreviated $\mathrm{SP}$) are known. The unknowns are the position of the sensor center $S$ and a vector $\mathbf{n^{L}}$ normal to the lens plane (abbreviated $\mathrm{LP}$). The Scheimplug rule stipulates that if $\mathrm{SFP}$ is into focus, then $\mathrm{SP}$, $\mathrm{LP}$ and $\mathrm{SFP}$ necessarily intersect on a common line called the ”Scheimpflug Line” (abbreviated $\mathrm{SL}$). The diagram of Figure 5a should help the reader to see that this rule is not sufficient to uniquely determine $\mathbf{n^{L}}$ and $\mathrm{SP}$, as this plane can be translated toward $\mathbf{n^{S}}$ if $\mathbf{n^{L}}$ is changed accordingly. The missing constraints are provided by the Hinge rule, which is illustrated in Figure 6. This rule considers two complimentary planes: the front focal plane (abbreviated $\mathrm{FFP}$), which is parallel to $\mathrm{LP}$ and passes through the focal point $F$, and the parallel to sensor lens plane (abbreviated $\mathrm{PSLP}$), which is parallel to $\mathrm{SP}$ and passes through the optical center $L$. The Hinge Rule stipulates that $\mathrm{FFP}$, $\mathrm{PSLP}$ and $\mathrm{SFP}$ must intersect along a common line called the Hinge Line (abbreviated $\mathrm{HL}$). Since $\mathrm{HL}$ is uniquely determined as the intersection of $\mathrm{SFP}$ and $\mathrm{PSLP}$, this allows one to determine $\mathbf{n^{L}}$, or equivalently the tilt and swing angles, such that $\mathrm{FFP}$ passes through $\mathrm{HL}$ and $F$. Then $\mathrm{SL}$ is uniquely defined as the intersection of $\mathrm{LP}$ and $\mathrm{SFP}$ by the Scheimpflug rule (note that $\mathrm{SL}$ and $\mathrm{HL}$ are parallel by construction). Since $\mathbf{n^{S}}$ is already known, any point belonging to $\mathrm{SL}$ is sufficient to uniquely define $\mathrm{SP}$. Hence, the determination of tilt and swing angles and position of the rear standard can be summarized as follows: 1. 1. determination of $\mathrm{HL}$, intersection of $\mathrm{FFP}$ and $\mathrm{SFP}$, 2. 2. determination of tilt and swing angles such that $\mathrm{HL}$ belongs to $\mathrm{FFP}$, 3. 3. determination of $\mathrm{SL}$, intersection of $\mathrm{LP}$ and $\mathrm{SFP}$, 4. 4. translation of $S$ such that $\mathrm{SL}$ belongs to $\mathrm{SP}$. #### 3.1.2 Algebraic details of the computations In this section the origin of the coordinate system is the optical center $L$ and the inner product of two vectors $\mathbf{X}$ and $\mathbf{Y}$ is expressed by using the matrix notation ${\mathbf{X}}^{\top}{\mathbf{Y}}$. All planes are defined by a unit normal vector and a point in the plane as follows: $\displaystyle\mathrm{SP}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X-S})}^{\top}{\mathbf{n^{S}}}=0\right\\}$ $\displaystyle\mathrm{PSLP}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{S}}}=0\right\\},$ $\displaystyle\mathrm{LP}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{L}}}=0\right\\},$ $\displaystyle\mathrm{FFP}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{L}}}-f=0\right\\},$ $\displaystyle\mathrm{SFP}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{Y})}^{\top}{\mathbf{n^{SF}}}=0\right\\},$ where the equation of $\mathrm{FFP}$ takes this particular form because the distance between $L$ and $F$ is equal to the focal length $f$ and we have imposed that $n^{L}_{3}>0$. The computations are detailed in the following algorithm: ###### Algorithm 1 * Step 1 : compute the Hinge Line by considering its parametric equation $\displaystyle\mathrm{HL}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;\exists\,t\in\mathbb{R},\;\mathbf{X}=\mathbf{W}+t\mathbf{V}\right\\},$ where $\mathbf{V}$ is a direction vector and $\mathbf{W}$ is the coordinate vector of an arbitrary point of $\mathrm{HL}$. Since this line is the intersection of $\mathrm{PSLP}$ and $\mathrm{SFP}$, $\mathbf{V}$ is orthogonal to $\mathbf{n^{L}}$ and $\mathbf{n^{SF}}$. Hence, we can take $\mathbf{V}$ as the cross product $\mathbf{V}=\mathbf{n^{SF}}\times\mathbf{n^{S}}$ and $\mathbf{W}$ as a particular solution (e.g. the solution of minimum norm) of the overdetermined system of equations $\displaystyle{\mathbf{W}}^{\top}{\mathbf{n^{S}}}$ $\displaystyle=0,$ $\displaystyle{\mathbf{W}}^{\top}{\mathbf{n^{SF}}}$ $\displaystyle={\mathbf{Y}}^{\top}{\mathbf{n^{SF}}}.$ * Step 2 : since $\mathrm{HL}$ belongs to $\mathrm{FFP}$ we have ${(\mathbf{W}+t\mathbf{V})}^{\top}{\mathbf{n^{L}}}-f=0,\quad\forall~{}t\in\mathbb{R},$ hence $\mathbf{n^{L}}$ verifies the overdetermined system of equations $\displaystyle{\mathbf{W}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=f,$ (3) $\displaystyle{\mathbf{V}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=0.$ (4) with the constraint $\|\mathbf{n^{L}}\|^{2}=1$. The computation of $\mathbf{n^{L}}$ can be done by the following two steps: 1. 1. compute $\widetilde{\mathbf{W}}=\mathbf{V}\times\mathbf{W}$ and $\widetilde{\mathbf{V}}$ the minimum norm solution of system (3)-(4), which gives a parametrization $\mathbf{n^{L}}=\widetilde{\mathbf{V}}+t\widetilde{\mathbf{W}},$ of all its solutions, where $t$ is an arbitrary real. 2. 2. determination of $t$ such that $\|\mathbf{n^{L}}\|^{2}=1$: this is done by taking the solution $t$ of the second degree equation ${\widetilde{\mathbf{W}}}^{\top}{\widetilde{\mathbf{W}}}t^{2}+2{\widetilde{\mathbf{W}}}^{\top}{\widetilde{\mathbf{V}}}t+{\widetilde{\mathbf{V}}}^{\top}{\widetilde{\mathbf{V}}}-1=0,$ such that $n^{L}_{3}>0$. The tilt and swing angles are then obtained as $\theta_{L}=-\arcsin n_{2}^{L},\quad\phi_{L}=-\arcsin\frac{n_{1}^{L}}{\cos\theta_{L}}.$ * Step 3 : since $\mathrm{SL}$ is the intersection of $\mathrm{LP}$ and $\mathrm{SFP}$, the coordinate vector $\mathbf{U}$ of a particular point $U$ on $\mathrm{SL}$ is obtained as the minimum norm solution of the system $\displaystyle{\mathbf{U}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=0,$ $\displaystyle{\mathbf{U}}^{\top}{\mathbf{n^{SF}}}$ $\displaystyle={\mathbf{W}}^{\top}{\mathbf{n^{SF}}},$ where we have used the fact that $W\in\mathrm{SFP}$. * Step 4 : the translation of $S$ can be computed such that $U$ belongs to $\mathrm{SP}$, i.e. ${(\mathbf{U}-\mathbf{S})}^{\top}{\mathbf{n^{S}}}=0.$ If we only act on the third coordinate of $S$ and leave the two others unchanged, then $S_{3}$ can be computed as $S_{3}=\frac{{\mathbf{U}}^{\top}{\mathbf{n^{S}}}-S_{1}n^{S}_{1}-S_{2}n^{S}_{2}}{n^{S}_{3}}.$ ###### Remark 1 When we consider a true camera lens, the nodal points $H,H^{\prime}$ and the front standard center $L$ do not coincide. Hence, the tilt and swing rotations of the front standard modify the actual position of the $\mathrm{PSLP}$ plane. In this case, we use the following iterative fixed point scheme: 1. 1. The angles $\phi_{L}$ and $\theta_{L}$ are initialized with starting values $\phi_{L}^{0}$ and $\theta_{L}^{0}$. 2. 2. At iteration $k$, 1. (a) the position of $\mathrm{PSLP}$ is computed considering $\phi_{L}^{k}$ and $\theta_{L}^{k}$, 2. (b) the resulting Hinge Line is computed, then the position of $\mathrm{FFP}$ and the new values $\phi_{L}^{k+1}$ and $\theta_{L}^{k+1}$ are computed. Point 2 is repeated until convergence of $\phi_{L}^{k}$ and $\theta_{L}^{k}$ up to a given tolerance. Generally 3 iterations are sufficient to reach the machine precision. ### 3.2 Characterization of the depth of field region Figure 7: Position of planes $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ delimiting the depth of field region. As in the previous section, we consider that $L$ and the nodal points $H$ and $H^{\prime}$ coincide. Moreover, $L$ will be the origin of the global coordinates system. We consider the configuration depicted in Figure 7 where the sharp focus plane $\mathrm{SFP}$, the lens plane $\mathrm{LP}$ and the sensor plane $\mathrm{SP}$ are tied by the Scheimpflug and the Hinge rule. The depth of field can be defined as follows: ###### Definition 1 Let $X$ be a 3D point and $A$ its image through the lens. Let $\mathcal{C}$ be the disk in $\mathrm{LP}$ of center $L$ and diameter $f/N$, where $N$ is called the f-number. Let $K$ be the cone of base $\mathcal{C}$ and vertex $A$. The point $X$ is said to lie in the depth of field region if the diameter of the intersection of $\mathrm{SP}$ and $K$ is lower that $c$, the diameter of the so-called circle of confusion. The common values of $c$, which depend on the magnification from the sensor image to the final image and on its viewing conditions, lie typically between 0.2 mm and 0.01 mm. In the following the value of $c$ is not a degree of freedom but a given input. If the ellipticity of extended images is neglected, the depth of field region can be shown to be equal to the unbounded wedge delimited by $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ intersecting at $\mathrm{HL}$, where the corresponding sensor planes $\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$ are tied to $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ by the Scheimpflug rule. By mentally rotating $\mathrm{SFP}$ around $\mathrm{HL}$, it is easy to see that $\mathrm{SP}$ is translated through $\mathbf{n^{S}}$ and spans the region between $\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$. The position of $\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$, the f-number $N$ and the diameter of the circle of confusion $c$ are related by the formula $\frac{Nc}{f}=\frac{p_{1}-p_{2}}{p_{1}+p_{2}},$ (5) where $p_{1}$, respectively $p_{2}$, are the distances between the optical center $L$ and $\mathrm{SP_{1}}$, respectively $\mathrm{SP_{2}}$, both measured orthogonally to the optical plane. The distance $p$ between $\mathrm{SP}$ and L can be shown to be equal to $p=\frac{2p_{1}p_{2}}{p_{1}+p_{2}},$ (6) the harmonic mean of $p_{1}$ and $p_{2}$. This approximate definition of the depth of field region has been proposed by various authors (see Wheeler ; Bigler ) but when the ellipticity of images is taken into account a complete study can be found in Evens . For sake of completeness, we give the justification of formulas (5) and (6) in Appendix A.1. In most practical situations the necessary angle between $\mathrm{SP}$ and $\mathrm{LP}$ is small (less that 10 degrees), so that this approximation is correct. ###### Remark 2 The analysis in Appendix A.1 shows that the ratio $\frac{Nc}{f}$ in equation (5) does not depend on the direction used for measuring the distance between $\mathrm{SP}$, $\mathrm{SP_{1}}$, $\mathrm{SP_{2}}$ and $L$. The only condition, in order to take into account the case where $\mathrm{SP}$ and $\mathrm{LP}$ are parallel, is that this direction is not orthogonal to $\mathbf{n^{S}}$. Hence, by taking the direction given by $\mathbf{n^{S}}$, we can obtain an equivalent formula to (5). To compute the distances, we need the coordinate vector of two points $U_{1}$ and $U_{2}$ on $\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$ respectively. To this purpose we consider Step 3 of Algorithm 1 in section 3.1: if $\mathbf{W}$ is the coordinate vector of any point $W$ of $\mathrm{HL}$, each vector $\mathbf{U}^{i}$ can be obtained as a particular solution of the system $\displaystyle{\mathbf{U}^{i}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=0,$ $\displaystyle{\mathbf{U}^{i}}^{\top}{\mathbf{n}^{i}}$ $\displaystyle={\mathbf{W}}^{\top}{\mathbf{n}^{i}}.$ Since $\mathrm{SP_{i}}$ can be defined as $\mathrm{SP_{i}}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{U}^{i})}^{\top}{\mathbf{n^{S}}}=0\right\\},$ and $\|\mathbf{n^{S}}\|=1$, the signed distance from $L$ to $\mathrm{SP_{i}}$ is equal to $d(L,\mathrm{SP_{i}})={\mathbf{U_{i}}}^{\top}{\mathbf{n^{S}}}.$ So the equivalent formula giving the ratio $\frac{Nc}{f}$ is given by $\displaystyle\frac{Nc}{f}$ $\displaystyle=\left|\frac{d(L,\mathrm{SP_{1}})-d(L,\mathrm{SP_{2}})}{d(L,\mathrm{SP_{1}})+d(L,\mathrm{SP_{2}})}\right|,$ $\displaystyle=\left|\frac{{(\mathbf{U_{1}}-\mathbf{U_{2}})}^{\top}{\mathbf{n^{S}}}}{{(\mathbf{U_{1}}+\mathbf{U_{2}})}^{\top}{\mathbf{n^{S}}}}\right|.$ (7) The above considerations show that for a given orientation of the rear standard given by $\mathbf{n^{S}}$, if the depth of field wedge is given, then the needed f-number, the tilt and swing angles of the front standard and the translation of the sensor plane, can be determined. The related question that will be addressed in the following is the question: given a set of points $\mathcal{X}=\\{{X}^{1},\dots,{X}^{n}\\}$, how can we minimize the f-number such that all points of $\mathcal{X}$ lie in the depth of field region? ### 3.3 Depth of field optimization with respect to tilt angle We first study the depth of field optimization in two dimensions, because in this particular case all computations can be carried explicitly and a closed form expression is obtained, giving the f-number as a function of front standard tilt angle and of the slope of limiting planes. First, notice that $N$ has a natural upper bound, since (5) implies that $N\leq\frac{f}{c}.$ #### 3.3.1 Computation of f-number with respect to tilt angle and limiting planes Figure 8: The depth of field region when only a tilt angle $\theta$ is used. Without loss of generality, we consider that the sensor plane has the normal $\mathbf{n^{S}}=(0,0,1)^{\top}$. Let us denote by $\theta$ the tilt angle of the front standard and consider that the swing angle $\phi$ is zero. The lens plane is given by $\mathrm{LP}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{L}}}=0\right\\},$ where $\mathbf{n^{L}}=(0,-\sin\theta,\cos\theta)^{\top},$ and any collinear vector to $\mathbf{n^{L}}\times\mathbf{n^{S}}$ is a direction vector of the Hinge Line. Hence we can take, independently of $\theta$, $\mathbf{V}=(1,0,0)^{\top}$ and a parametric equation of $\mathrm{HL}$ is thus given by $\displaystyle\mathrm{HL}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;\exists\,t\in\mathbb{R},\;\mathbf{X}=\mathbf{W}(\theta)+t\mathbf{V}\right\\},$ where $\mathbf{W}(\theta)$ is the coordinate vector of a particular point ${W}(\theta)$ on $\mathrm{HL}$, obtained as the minimum norm solution of $\displaystyle{\mathbf{W}(\theta)}^{\top}{\mathbf{n^{S}}}$ $\displaystyle=0,$ $\displaystyle{\mathbf{W}(\theta)}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=f.$ Straightforward computations show that for $\theta\neq 0$ $\mathbf{W}(\theta)=\left(\begin{array}[]{c}0\\\ -\frac{f}{\sin\theta}\\\ 0\end{array}\right).$ Consider, as depicted in Figure 8, the two sharp focus planes $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ passing through $\mathrm{HL}$, with normals $\mathbf{n^{1}}=(0,-1,a_{1})^{\top}$ and $\mathbf{n^{2}}=(0,-1,a_{2})^{\top}$, $\displaystyle\mathrm{SFP_{1}}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{1}}}=0\right\\},$ $\displaystyle\mathrm{SFP_{2}}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{2}}}=0\right\\}.$ The two corresponding sensor planes $\mathrm{SP_{i}}$ are given by following Steps 3-4 in Algorithm 1 by $\displaystyle\mathrm{SFP_{i}}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;X_{3}=t_{i}\right\\},\;i=1,2,$ where $t_{i}=\frac{f}{a_{i}\sin\theta-\cos\theta},\;i=1,2.$ Using equation (7) the corresponding f-number is equal to $N(\theta,\mathbf{a})=\left|\frac{t_{1}-t_{2}}{t_{1}+t_{2}}\right|\left(\frac{f}{c}\right),$ (8) where we have used the notation $\mathbf{a}=(a_{1},a_{2})$. Finally we have, for $\theta\neq 0$ $N(\theta,\mathbf{a})=\operatorname{sign}\theta\frac{(a_{1}-a_{2})\sin\theta}{2\cos\theta-(a_{1}+a_{2})\sin\theta}\left(\frac{f}{c}\right).$ (9) ###### Remark 3 When $\theta=0$, then $\mathrm{SFP}$ does not intersect $\mathrm{PSLP}$ and the depth of field region is included between two parallel planes $\mathrm{SFP_{i}}$ given by $\displaystyle\mathrm{SFP_{i}}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;X_{3}=z_{i}\right\\},\;i=1,2,$ where $z_{1}$ and $z_{2}$ depend on the f-number and on the position of $\mathrm{SFP}$. One can show by using the thin lens equation and equation (5) that the corresponding f-number is equal to $N_{0}(z_{1},z_{2})=\frac{\left|\frac{1}{z_{2}}-\frac{1}{z_{1}}\right|}{\frac{1}{z_{1}}+\frac{1}{z_{2}}-\frac{2}{f}}\left(\frac{f}{c}\right).$ (10) #### 3.3.2 Theoretical results for the optimization problem Without loss of generality, we will consider a set of only three non-aligned points $\mathcal{X}=\\{{X}^{1},{X}^{2},{X}^{3}\\}$, which have to be within the depth of field region with minimal f-number and we denote by $\mathbf{X}^{1},\mathbf{X}^{2},\mathbf{X}^{3}$ their respective coordinate vectors. The corresponding optimization problem can be stated as follows: find $(\theta^{*},\mathbf{a}^{*})=\arg\min_{\begin{array}[]{c}\theta\in\mathbb{R}\\\ a\in\mathcal{A(\theta)}\end{array}}N(\theta,\mathbf{a}),$ (11) where for a given $\theta$ the set $\mathcal{A(\theta)}$ is defined by the inequalities $\displaystyle{(\mathbf{X}^{i}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{1}}}$ $\displaystyle\geq 0,~{}i=1,2,3,$ (12) $\displaystyle{(\mathbf{X}^{i}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{2}}}$ $\displaystyle\leq 0,~{}i=1,2,3,$ (13) meaning that ${X}^{1},{X}^{2},{X}^{3}$ are respectively under $\mathrm{SFP_{1}}$ and above $\mathrm{SFP_{2}}$, and by the inequalities $\displaystyle-{(\mathbf{X}^{i}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{L}}}+f$ $\displaystyle\leq 0,~{}i=1,2,3,$ (14) meaning that ${X}^{1},{X}^{2},{X}^{3}$ are in front of $\mathrm{FFP}$. ###### Remark 4 Points behind the focal plane cannot be in focus, and the constraints (14) are just expressing this practical impossibility. However, we have to notice that when one of these constraints is active, we can show that $a_{1}=\cot\theta$ or $a_{2}=\cot\theta$ so that $N(\mathbf{a},\theta)$ reaches its upper bound $\frac{f}{c}$. Moreover, we have to eliminate the degenerate case where the points ${X}^{i}$ are such that there exists two active constraints in (14): in this case, there exists a unique admissible pair $(\mathbf{a},\theta)$ and the problem has no practical interest. To this purpose, we can suppose that $X^{i}_{3}>f$ for $i=1,2,3$. For $\theta\neq 0$ the gradient of $N(\mathbf{a},\theta)$ is equal to $\mathbf{\nabla}(\mathbf{a},\theta)=\frac{2\operatorname{sign}\theta}{(2\cot\theta-(a_{1}+a_{2}))^{2}}\left(\begin{array}[]{c}\cot\theta- a_{2}\\\ -\cot\theta+a_{1}\\\ \frac{a_{1}-a_{2}}{\sin^{2}\theta}\end{array}\right)$ and cannot vanish since $a_{1}=a_{2}$ is not possible because it would mean that ${X}^{1},{X}^{2},{X}^{3}$ are aligned. This implies that $\mathbf{a}$ lies on the boundary of $\mathcal{A}(\theta)$ and we have the following intuitive result (the proof is given in Appendix A.2): ###### Proposition 1 Suppose that $X^{i}_{3}>f$, for $i=1,2,3$. Then when $N(\mathbf{a},\theta)$ reaches its minimum, there exists $i_{1}$, $i_{2}$ with $i_{1}\neq i_{2}$ such that $\displaystyle{(\mathbf{X}^{i_{1}}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{1}}}$ $\displaystyle=0,$ $\displaystyle{(\mathbf{X}^{i_{2}}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{2}}}$ $\displaystyle=0.$ ###### Remark 5 The above result shows that at least two points touch the depth of field limiting planes $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ when the f-number is minimal. In the following, we will show that the three points ${X}^{1},{X}^{2}$ and ${X}^{3}$ are necessarily in contact with one of the limiting planes (the proof is given in Appendix A.3): ###### Proposition 2 Suppose that the vertices $\\{{X}^{i}\\}_{i=1\dots 3}$ verify the condition $\frac{\|\mathbf{X}^{i}\times\mathbf{X}^{j}\|}{\|\mathbf{X}^{i}-\mathbf{X}^{j}\|}>f,~{}~{}i\neq j.$ (15) Then $N(\mathbf{a},\theta)$ reaches its minimum when all vertices are in contact with the limiting planes. ###### Remark 6 If $\theta$ is small, then $N(\theta,\mathbf{a})$ in (9) can be approximated by $\tilde{N}(\theta,\mathbf{a})=\operatorname{sign}\theta{(a_{1}-a_{2})\sin\theta}\left(\frac{f}{2c}\right),$ (16) and the proof of Proposition 2 is considerably simplified: the same result holds with the weaker condition $\|\mathbf{X}^{i}\times\mathbf{X}^{j}\|>0.$ (17) In fact, an approximate way of specifying the depth of field region using the hyperfocal distance, proposed in Merklinger , leads to the same approximation of $N(\theta,\mathbf{a})$, under the a priori hypothesis of small $\theta$ and distant objects, i.e. $X^{i}_{3}\gg f$. This remark is clarified in Appendix A.4. We will illustrate the theoretical result by considering sets of 3 points. For a set with more than 3 vertices (but being equal to the vertices of the convex hull of $\mathcal{X}$), the determination of the optimal solution is purely combinatorial, since it is enough to enumerate all admissible situations where two points are in contact with one plane, and a third one with the other. The value $\theta=0$ also has to be considered because it can be a critical value if the object has a vertical edge. We will also give an Example which violates condition (15) and where $N(\mathbf{a},\theta)$ reaches its minimum when only two vertices are in contact with the limiting planes. #### 3.3.3 Numerical results In this section, we will consider the following function, defined for $\theta\neq 0$ $n(\theta)=\min_{a\in\mathcal{A}(\theta)}N(\mathbf{a},\theta).$ Finding the minimum of this function allows one to solve the original constrained optimization problem, but considering the results of the previous section, $n(\theta)$ is non-differentiable. In fact, the values of $\theta$ for which $n(\theta)$ is not differentiable correspond to the situations where 3 points are in contact with the limiting planes. We extend $n(\theta)$ by continuity for $\theta=0$ by defining $n(0)=\frac{\frac{1}{z_{2}}-\frac{1}{z_{1}}}{\frac{1}{z_{1}}+\frac{1}{z_{2}}-\frac{2}{f}}\left(\frac{f}{c}\right),$ where $z_{1}=\max_{i=1,2,3}X^{i}_{3},~{}~{}z_{2}=\min_{i=1,2,3}X^{i}_{3}.$ This formula can be directly obtained by using conjugation formulas or by taking $\theta=0$ in equation (24). ###### Example 1 Figure 9: Enumeration of the 3 candidates configurations for Example 1. --- (a) (b) Figure 10: Example 1 graph of $n(\theta)$ for $\theta\in[-0.6,0.6]$ (a) and $\theta\in[0,0.06]$ (b). Labels give the different types of contact. We consider a lens with focal length $f=5.10^{-2}\mathrm{m}$ and a confusion circle $c=3.10^{-5}m$ (commonly used value for 24x36 cameras). The vertices have coordinates $\mathbf{X}^{1}=\left(\begin{array}[]{r}0\\\ -1\\\ 1\end{array}\right),~{}\mathbf{X}^{2}=\left(\begin{array}[]{r}0\\\ 3\\\ 1\end{array}\right),~{}\mathbf{X}^{3}=\left(\begin{array}[]{r}0\\\ 0\\\ 1.5\end{array}\right).$ Figure 9 shows the desired depth of field region (dashed zone) and the three candidates hinge lines corresponding to contacts $E_{ij}V_{k}$: * • $E_{12}V_{3}$, obtained when the edge $[{X}^{1},{X}^{2}]$ is in contact with $\mathrm{SFP_{1}}$ and vertex ${X}^{3}$ with $\mathrm{SFP_{2}}$. * • $E_{13}V_{2}$, obtained when the edge $[{X}^{1},{X}^{3}]$ is in contact with $\mathrm{SFP_{2}}$ and vertex ${X}^{2}$ with $\mathrm{SFP_{1}}$. * • $E_{23}V_{1}$, obtained when the edge $[{X}^{2},{X}^{3}]$ is in contact with $\mathrm{SFP_{1}}$ and vertex ${X}^{1}$ with $\mathrm{SFP_{2}}$. The associated values of $\theta$ and $n(\theta)$ are given in Table 1, Contact | $\theta$ | $n(\theta)$ ---|---|--- $E_{12}V_{3}$ | 0.0166674 | 9.52 $E_{13}V_{2}$ | 0.025002 | 7.16 $E_{23}V_{1}$ | 0.0500209 | 28.28 Table 1: Value of $\theta$ for each possible optimal contact and corresponding f-number $n(\theta)$ for Example 1. which shows that contact $E_{13}V_{2}$ seems to give the minimum f-number. Condition (15) is verified, since $\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{2}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{2}\|}=1,~{}\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{3}\|}=1.34,$ $\frac{\|\mathbf{X}^{2}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{2}-\mathbf{X}^{3}\|}=1.48.$ Hence, the derivative of $n(\theta)$ cannot vanish and the minimum f-number is necessarily reached for $\theta^{*}=0.025002$. We can confirm this by considering the graph of $n(\theta)$ depicted on Figure 10. In the zone of interest, the function $n(\theta)$ is almost piecewise affine. For each point in the interior of curved segments of the graph, the value of $\theta$ is such that the contact of $\mathcal{X}$ with the limiting planes is of type $V_{i}V_{j}$. Clearly, the derivative of $n(\theta)$ does not vanish. The possible optimal values of $\theta$, corresponding to contacts of type $E_{i}jV_{k}$, are the abscissa of angular points of the graph, marked with red dots. The graph confirms that the minimal value of $n(\theta)$ is reached for contact $E_{12}V_{3}$. The minimal f-number is equal to $n(\theta^{*})=7.16$. By comparison, the f-number without tilt optimization is $n(0)=28.74$. This example highlights the important gain in terms of f-number reduction with the optimized tilt angle. ###### Example 2 Contact | $\theta$ | $n(\theta)$ ---|---|--- $E_{12}V_{3}$ | 0.185269 | 29.49 $E_{23}V_{1}$ | 0.100419 | 83.67 $E_{13}V_{2}$ | 0.235825 | 47.04 Table 2: Value of $\theta$ for each possible optimal contact and corresponding f-number $n(\theta)$ for Example 2. --- (a) (b) Figure 11: Example 2 graph of $n(\theta)$ for $\theta\in[-0.6,0.6]$ (a) and $\theta\in[0,0.3]$ (b). Labels give the different types of contact. We consider the same lens and confusion circle as in Example 1 ($f=5.10^{-2}\mathrm{m}$, $c=3.10^{-5}m$) but the vertices have coordinates $\mathbf{X}^{1}=\left(\begin{array}[]{r}0\\\ -0.1\\\ 0.12\end{array}\right),~{}\mathbf{X}^{2}=\left(\begin{array}[]{r}0\\\ 0\\\ 0.19\end{array}\right),~{}\mathbf{X}^{3}=\left(\begin{array}[]{r}0\\\ -0.0525\\\ 0.17\end{array}\right).$ The object is almost ten times smaller than the object of the previous example (it has the size of a small pen), but it is also ten times closer: this is a typical close up configuration. The values of $\theta$ and $n(\theta)$ associated to contacts of type $E_{ij}V_{k}$ are given in Table 2 which shows that contact $E_{13}V_{2}$ seems to give the minimum f-number. We have $\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{2}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{2}\|}=0.16,~{}\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{3}\|}=0.16,$ $\frac{\|\mathbf{X}^{2}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{2}-\mathbf{X}^{3}\|}=0.1775527,$ showing that condition (15) is still verified, even if the values are smaller than the values of Example 1. Hence, the derivative of $n(\theta)$ cannot vanish and the minimum f-number is reached for $\theta^{*}=0.185269$. We can confirm this by considering the graph of $n(\theta)$ depicted on Figure 11. As in Example 1, the derivative of $n(\theta)$ does not vanish and the graph confirms that the minimal value of $n(\theta)$ is reached for contact $E_{12}V_{3}$. The minimal f-number is equal to $n(\theta^{*})=29.49$. By comparison, the f-number without tilt optimization is $n(0)=193.79$. Such a large value gives an aperture of diameter $0.26$mm, almost equivalent to a pin hole ! Since the maximum f-number of view camera lenses is never larger than 64, the object cannot be in focus without using tilt. This example also shows that Proposition 2 is still valid even if the object is close to the lens and the obtained optimal tilt angle $0.185269$ (10.61 degrees) is large. ###### Example 3 --- (a) (b) Figure 12: Example 3 graphs of $n(\theta)$ for $\theta\in[-1,1]$. In this example we consider an extremely flat triangle which is almost aligned with the optical center. From the photographer’s point of view, this is an unrealistic case. We consider the same optical parameters as before and consider the following vertices $\mathbf{X}^{1}=\left(\begin{array}[]{r}0\\\ 0\\\ 1\end{array}\right),~{}\mathbf{X}^{2}=\left(\begin{array}[]{r}0\\\ h\\\ 1.5\end{array}\right),~{}\mathbf{X}^{3}=\left(\begin{array}[]{r}0\\\ -h\\\ 2\end{array}\right),$ for $h=0.01$ and we have $\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{2}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{2}\|}=0.02,~{}\frac{\|\mathbf{X}^{1}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{1}-\mathbf{X}^{3}\|}=0.01,$ $\frac{\|\mathbf{X}^{2}\times\mathbf{X}^{3}\|}{\|\mathbf{X}^{2}-\mathbf{X}^{3}\|}=0.07,$ hence condition (15) is violated in configurations $V_{1}V_{2}$ and $V_{1}V_{3}$. Since this condition is sufficient, the minimum value of $n(\theta)$ could still occur for a contact of type $E_{ij}V_{k}$. However, we can see in Figure 12a that $n(\theta)$ has a minimum at a differentiable point in configuration $V_{1}V_{3}$. ###### Remark 7 Condition (15) is not necessary: in the proof of Proposition 2 (given in Appendix A.3), it can be seen that (15) is a condition ensuring that the polynomial $p(\theta)$ defined by equation (25) has no root in $[-\pi/2,\pi/2]$. However, the relevant interval is smaller because ${X}^{i_{1}}$ and ${X}^{i_{2}}$ do not stay in contact with the limiting planes for all values of $\theta$ in $[-\pi/2,\pi/2]$, and because all vertices must be in front of the focal plane. Anyway, it is always possible to construct absolutely unrealistic configurations. For example, when we consider the above vertices with $h=0.005$, then no edge is in contact with the limiting planes and vertices ${X}^{1}$ and ${X}^{3}$ stay in contact with the limiting planes for all admissible values of $\theta$. The corresponding graph of $n(\theta)$ is given in Figure 12b. ### 3.4 Depth of field optimization with respect to tilt and swing angles As in the previous section, we consider the case of a thin lens and where the optical and sensor centers coincide respectively with the front and rear standard rotation centers. Without loss of generality, we consider that the sensor plane has the normal $\mathbf{n^{S}}=(0,0,1)^{\top}$. The lens plane is given by $\mathrm{LP}=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{X}}^{\top}{\mathbf{n^{L}}}=0\right\\},$ where $\mathbf{n^{L}}=(-\sin\phi\cos\theta,-\sin\theta,\cos\phi\cos\theta)^{\top}.$ A parametric equation of $\mathrm{HL}$ is given by $\displaystyle\mathrm{HL}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;\exists\,t\in\mathbb{R},\;\mathbf{X}=\mathbf{W}(\theta,\phi)+t\mathbf{V}(\theta,\phi)\right\\},$ where the direction vector is given by $\mathbf{V}(\theta,\phi)=\mathbf{n^{L}}\times\mathbf{n^{S}}=(-\sin\theta,\sin\phi\cos\theta,0)^{\top},$ and $\mathbf{W}(\theta,\phi)$ is the coordinate vector of a particular point ${W}(\theta,\phi)$ on $\mathrm{HL}$, obtained as the minimum norm solution of $\displaystyle{\mathbf{W}(\theta)}^{\top}{\mathbf{n^{S}}}$ $\displaystyle=0,$ $\displaystyle{\mathbf{W}(\theta)}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=f.$ Consider, as depicted in Figure 7, the two planes of sharp focus $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ intersecting at HL, with normals $\mathbf{n^{1}}$ and $\mathbf{n^{1}}$ respectively. The point $W(\theta,\phi)$ belongs to $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ and any direction vector of $\mathrm{SFP_{1}}\cap\mathrm{SFP_{2}}$ is collinear to $\mathbf{V}(\theta,\phi)$. Hence, we have $\displaystyle\mathrm{SFP_{1}}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{W}(\theta,\phi))}^{\top}{\mathbf{n^{1}}}=0\right\\},$ $\displaystyle\mathrm{SFP_{2}}$ $\displaystyle=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{(\mathbf{X}-\mathbf{W}(\theta,\phi))}^{\top}{\mathbf{n^{2}}}=0\right\\},$ and $(\mathbf{n^{1}}\times\mathbf{n^{2}})\times\mathbf{V}(\theta,\phi)=0.$ (18) Using equation (7) the f-number is equal to $N(\theta,\phi,\mathbf{n^{1}},\mathbf{n^{2}})=\frac{f}{c}\left|\frac{{(\mathbf{U}^{1}-\mathbf{U}^{2})}^{\top}{\mathbf{n^{S}}}}{{(\mathbf{U}^{1}+\mathbf{U}^{2})}^{\top}{\mathbf{n^{S}}}}\right|,$ (19) where each coordinate vector $\mathbf{U}^{i}$ of point $U^{i}$, for $i=1,2$, is obtained as a particular solution of the following system: $\displaystyle{\mathbf{U}^{i}}^{\top}{\mathbf{n^{L}}}$ $\displaystyle=0,$ $\displaystyle{\mathbf{U}^{i}}^{\top}{\mathbf{n}^{i}}$ $\displaystyle={\mathbf{W}(\theta,\phi)}^{\top}{\mathbf{n}^{i}}.$ ###### Remark 8 Figure 13: Position of the hinge line when both tilt and swing are used and $\mathbf{n^{S}}=(0,0,1)^{\top}$. The LP and FFP planes have not been represented for reasons of readability. When $\mathbf{n^{S}}=(0,0,1)^{\top}$ the intersections of $\mathrm{HL}$ with the $(L,X_{1})$ and the $(L,X_{2})$ axes can be determined, as depicted in Figure 13. In this case, it is easy to show that the coordinates of the minimum norm $\mathbf{W}(\theta,\phi)$ are given by $\mathbf{W}(\theta,\phi)=-\frac{f}{\sin^{2}\phi\cos^{2}\theta+\sin^{2}\theta}(\sin\phi\cos\theta,\sin\theta,0)^{\top}.$ We note that up to a rotation of axis $(0,0,1)^{\top}$ and angle $\alpha$, we recover a configuration where only a tilt angle $\psi$ is used, where these two angles are respectively defined by $\displaystyle\sin\psi$ $\displaystyle=\operatorname{sign}\theta\sqrt{\sin^{2}\phi\cos^{2}\theta+\sin^{2}\theta},$ $\displaystyle\sin\alpha$ $\displaystyle=\frac{\sin\phi\cos\theta}{\sin\psi}.$ #### 3.4.1 Optimization problem Consider a set 4 non coplanar points $\mathcal{X}=\\{{X}^{i}\\}_{i=1\dots 4}$, which have to be within the depth of field region with minimal f-number and denote by $\\{\mathbf{X}^{i}\\}_{i=1\dots 4}$ their respective coordinate vectors. The optimization problem can be stated as follows: find $(\theta^{*},\phi^{*},\mathbf{n^{1}}^{*},\mathbf{n^{2}}^{*})=\arg\min_{\theta,\phi,\mathbf{n^{1}},\mathbf{n^{2}}}N(\theta,\phi,\mathbf{n^{1}},\mathbf{n^{2}}),$ (20) where the minimum is taken for $\theta,\phi,\mathbf{n^{1}}$ and $\mathbf{n^{2}}$ such that equation (18) is verified and such that the points $\\{{X}^{i}\\}_{i=1\dots 4}$ lie between $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$. This last set of constraints can be expressed in a way similar to equations (12)-(13). #### 3.4.2 Analysis of configurations Consider the function $n(\theta,\phi)$ defined by $n(\theta,\phi)=\min_{\mathbf{n^{1}},\mathbf{n^{2}}}N(\theta,\phi,\mathbf{n^{1}},\mathbf{n^{2}}),$ (21) where $\mathbf{n^{1}}$ and $\mathbf{n^{2}}$ are constrained as in the previous section. Consider the two limiting planes $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ with normals $\mathbf{n^{1}}$ and $\mathbf{n^{2}}$ satisfying the minimum in equation (21). Using the rotation argument of Remark 8 we can easily show that $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ are necessary in contact with at least two vertices. However, for each value of the pair $(\theta,\phi)$, we have three types of possible contact between the tetrahedron formed by points $\\{{X}^{i}\\}_{i=1\dots 4}$ and the limiting planes: vertex-vertex, edge-vertex, edge-edge or face-vertex. These configurations can be analyzed as follows: let us consider a pair $(\theta_{0},\phi_{0})$ and the corresponding type of contact: * • Vertex-vertex: each limiting plane is in contact with only one vertex, respectively $V_{i}$ and $V_{j}$. In this case, $n(\theta,\phi)$ is differentiable at $(\theta_{0},\phi_{0})$ and there exists a curve $\gamma_{vv}$ defined by $\gamma_{vv}(t)=(\theta(t),\phi(t)),~{}\gamma_{vv}(0)=(\theta_{0},\phi_{0}),$ such that $\frac{d}{dt}n(\theta(t),\phi(t))$ exists and does not vanish for $t=0$. This can be proved using again the rotation argument of Remark 8. Hence, $n(\theta_{0},\phi_{0})$ cannot be minimal. * • Edge-vertex: one of the two planes is in contact with edge $E_{ij}$ and the other one is in contact with vertex $V_{k}$. In this case there is still a degree of freedom since the plane in contact with $E_{ij}$ can rotate around this edge in either directions while keeping the other plane in contact with $V_{k}$ only. If the plane in contact with $E_{ij}$ is $\mathrm{SFP_{1}}$, its normal $\mathbf{n^{1}}$ can be parameterized by using a single scalar parameter $t$ and we obtain a family of planes defined by $\mathrm{SFP_{1}}(t)=\left\\{\mathbf{X}\in\mathbb{R}^{3},\;{\mathbf{n^{1}}(t)}^{\top}{(\mathbf{X}-\mathbf{X}^{i})}=0\right\\}.$ For each value of $t$, the intersection of $\mathrm{SFP_{1}}(t)$ with $\mathrm{PSLP}$ defines a Hinge Line and thus a pair $(\theta(t),\phi(t))$ of tilt and swing angles. Hence, there exists a parametric curve $\gamma_{ev}(t)=(\theta(t),\phi(t)),~{}\gamma_{ev}(0)=(\theta_{0},\phi_{0}),$ (22) along which $n(\theta,\phi)$ is differentiable. As we will see in the numerical results, the curve $\gamma_{ev}(t)$ is almost a straight line when $\theta$ and $\phi$ are small, and $\frac{d}{dt}n(\theta,\phi(t))$ does not vanish for $t=0$. * • Edge-edge: the limiting planes are respectively in contact with edges $E_{ij}$, $E_{kl}$ connecting, respectively, vertices $V_{i},V_{j}$ and vertices $V_{k},V_{l}$. There is no degree of freedom left since these edges cannot be parallel (otherwise all points would be coplanar). Hence, $n(\theta,\phi)$ is not differentiable at ($\theta_{0},\phi_{0}$). * • Face-vertex: the limiting planes are respectively in contact with vertex $V_{l}$ and with the face $F_{ijk}$ connecting vertices $V_{i},V_{j},V_{k}$. As in the previous case, there is no degree of freedom left and $n(\theta,\phi)$ is not differentiable at ($\theta_{0},\phi_{0}$). We can already speculate that the first two configurations are necessary suboptimal. Consequently we just have to compute the f-number associated with each one of the 7 possible configurations of type edge-edge of face-vertex. #### 3.4.3 Numerical results We have considered the flat object of Example 1, translated in plane $X_{1}=-0.5$, and a complimentary point in order to form a tetrahedron. The vertices have coordinates $\mathbf{X}^{1}=\left(\begin{array}[]{r}-0.5\\\ -1\\\ 1\end{array}\right),~{}\mathbf{X}^{2}=\left(\begin{array}[]{r}-0.5\\\ 3\\\ 1\end{array}\right),$ $\mathbf{X}^{3}=\left(\begin{array}[]{r}-0.5\\\ 0\\\ 1.5\end{array}\right),\mathbf{X}^{4}=\left(\begin{array}[]{r}1\\\ 1\\\ 1.5\end{array}\right).$ Contact | $\theta$ | $\phi$ | $n(\theta,\phi)$ ---|---|---|--- $E_{12}E_{34}$ | 0.021430 | -0.028582 | 12.35 $E_{23}E_{14}$ | 0.150568 | -0.203694 | 90.75 $E_{13}E_{24}$ | 0.030005 | -0.020010 | 8.64 $F_{123}V_{4}$ | 0 | 0.100167 | 88.18 $F_{243}V_{1}$ | 0.075070 | -0.050162 | 42.91 $F_{134}V_{2}$ | 0.033340 | -0.033358 | 9.64 $F_{124}V_{3}$ | 0.018751 | -0.012503 | 10.76 Table 3: Value of $\theta$ and $\phi$ for each possible optimal contact and corresponding f-number $n(\theta,\phi)$. All configurations of type edge-edge and face-vertex have been considered and the corresponding values of $\theta,\phi$ and $n(\theta,\phi)$ are given in Table 3. The $E_{13}E_{24}$ contact seems to give the minimum f-number. Figure 14: Graph of $n(\theta,\phi)$ for $(\theta,\phi)\in[-0.175,0.225]\times[-0.3225,0.2775]$. Figure 15: Level curves of $n(\theta,\phi)$ and types of contact for $(\theta,\phi)\in[-0.175,0.225]\times[-0.3225,0.2775]$. Figure 16: Level curves and types of of $n(\theta,\phi)$ contact for $(\theta,\phi)\in[-0.015,0.035]\times[-0.0375,-0.0075]$. Figure 14 gives the graph of $n(\theta,\phi)$ and its level curves in the vicinity of the minimum are depicted in Figures 15 and 16. In the interior of each different shaded region, the $(\theta,\phi)$ pair is such that the contact of $\mathcal{X}$ with the limiting planes is of type $V_{i}V_{j}$. The possible optimal $(\theta,\phi)$ pairs, corresponding to contacts of type $E_{ij}E_{kl}$ of $F_{ijk}V_{l}$, are marked with red dots. Notice that the graph of $n(\theta,\phi)$ is almost polyhedral, i.e. in the interior of regions of type $V_{i}V_{j}$, the gradient is almost constant and does not vanish, as seen on the level curves. If confirms that the minimum cannot occur in these regions, as announced in Section 3.4.2. The frontiers between regions of type $V_{i}V_{j}$ are curves corresponding to contacts of type $E_{ij}V_{k}$ and defined by Equation (22). The extremities of these curves are $(\theta,\phi)$ pairs corresponding to contacts of type $E_{ij}E_{kl}$ or $F_{ijk}V_{l}$. For example, in Figure 16, the $(\theta,\phi)$ pairs on the curve separating $V_{2}V_{3}$ and $V_{3}V_{4}$ regions correspond to the $E_{24}V_{3}$ contact. The extremities of this curve are the two $(\theta,\phi)$ pairs corresponding to contacts $F_{124}V_{3}$ and $E_{13}E_{24}$. Along this curve, $n(\theta,\phi)$ is strictly monotone as shown by its level curves. Finally, the convergence of its level curves in Figure 16 confirms that the minimum of $n(\theta,\phi)$ is reached for the $E_{13}E_{24}$ contact. Hence, the optimal angles are $(\theta^{*},\phi^{*})=(0.030005,-0.020010)$ and the minimal f-number is equal to $n(\theta^{*},\phi^{*})=8.64$. By comparison, the f-number without tilt and swing optimization is $n(0,0)=28.74$. This example highlights again the important gain in terms of f-number reduction with the optimized tilt and swing angles. In our experience, the optimal configuration for general polyhedrons can be of type edge-edge or face-vertex. ## 4 Trends and conclusion In this paper, we have given the optimal solution of the most challenging issue in view camera photography: bring an object of arbitrary shape into focus and at the same time minimize the f-number. This problem takes the form of a continuous optimization problem where the objective function (the f-number) and the constraints are non-linear with respect to the design variables. When the object is a convex polyhedron, we have shown that this optimization problem does not need to be solved by classical methods. Under realistic hypotheses, the optimal solution always occurs when the maximum number of constraints are saturated. Such a situation corresponds to a small number of configurations (seven when the object is a tetrahedron). Hence, the exact solution is found by comparing the value of the f-number for each configuration. The linear algebra framework allowed us to efficiently implement the algorithms in a numerical computer algebra software. The camera software is able to interact with a robotised view camera prototype, which is actually used by our partner photographer. With the robotised camera, the time elapsed in the focusing process is often orders of magnitude smaller than the systematic trial and error technique. The client/server architecture of the software allows us to rapidly develop new problem solvers by validating them first on a virtual camera before implementing them on the prototype. We are currently working on the fine calibration of some extrinsic parameters of the camera, in order to improve the precision of the acquisition of 3D points of the object. ###### Acknowledgements. This work has been partly funded by the Innovation and Technological Transfer Center of Région Ile de France. ## Appendix A Appendix ### A.1 Computation of the depth of field region Figure 17: Construction of four approximate intersections of cones with $\mathrm{SP}$ (a) (b) Figure 18: (a) close-up of a particular intersection exhibiting the vertices of cones for a given directrix $\mathcal{D}$. (b) geometrical construction allowing to derive the depth of field formula. In order to explain the kind of approximation used, we have represented in Figure 17 the geometric construction of the image space limits corresponding to the depth of field region. Let us consider the cones whose base is the pupil and having an intersection with $\mathrm{SP}$ of diameter $c$. The image space limits are the locus of the vertex of such cones. The key point, suggested in Bigler and Evens , is the way the diameter of the intersection is measured. For a given line $\mathcal{D}$ passing through $L$ and a circle $\mathcal{C}$ of center $L$ in $\mathrm{LP}$ let us call $\mathcal{K}(\mathcal{C},\mathcal{D})$ the set of cones with directrix $\mathcal{D}$ and base $\mathcal{C}$. For a given directrix $\mathcal{D}$ let us call $A$ its intersection with $SP$, as depicted in Figure 18a. Instead of considering the intersection of cones of directrix $\mathcal{D}$ with $\mathrm{SP}$, we consider their intersections with the plane passing through $A$ and parallel to $\mathrm{LP}$. By construction, all intersections are circles, and there exists only two cones $K_{1}$ and $K_{2}$ in $\mathcal{K}(\mathcal{C},\mathcal{D})$ such that this intersection has a diameter equal to $c$, with their respective vertices $A_{1},A_{2}$ on each side of $\mathrm{SP}$, respectively marked in Figure 18a by a red and a green spot. Moreover, for all cones in $\mathcal{K}(\mathcal{C},\mathcal{D})$ only those with a vertex lying on the segment $[A_{1},A_{2}]$ have an ”approximate” intersection of diameter less that $c$. The classical laws of homothety show that for any directrix $\mathcal{D}$, the locus of the vertices of cones $K_{1}$ and $K_{2}$ will be on two parallel planes located in front of and behind $\mathrm{SP}$, as illustrated by a red and a green frame in Figure 17a. Hence, the depth of field region in the object space is the reciprocal image of the region between parallel planes $\mathrm{SP_{1}}$ and $\mathrm{SP_{2}}$ as depicted in Figure 7. Formulas (5) and (6) are obtained by considering the directrix that is orthogonal to $\mathrm{LP}$, as depicted in Figure 18b. If we note $p=AL$, $p_{1}=A_{1}L$, $p_{2}=A_{2}L$, by considering similar triangles, we have $\displaystyle\frac{p_{1}-p}{p_{1}}=\frac{p-p_{2}}{p_{2}}=\frac{Nc}{f},$ (23) which gives immediately $p=\frac{2p_{1}p_{2}}{p_{1}+p_{2}},$ and by substituting $p$ in (23), we obtain $\frac{p_{1}-p_{2}}{p_{1}+p_{2}}=\frac{Nc}{f},$ which allows to obtain (5). ### A.2 Proof of Proposition 1 Without loss of generality, we consider that the optimal $\theta$ is positive. Suppose now that only one constraint is active in (12). Then there exists $i_{1}$ such that ${(\mathbf{X}^{i_{1}}-\mathbf{W})}^{\top}{\mathbf{n^{1}}}=0$ and the first order optimality condition is verified: if we define $g(\mathbf{a},\theta)=-{(\mathbf{X}^{i}-\mathbf{W}(\theta))}^{\top}{\mathbf{n^{1}}}=X^{i_{1}}_{2}-a_{1}X^{i_{1}}_{3}+\frac{f}{\sin\theta},$ there exists $\lambda_{1}\geq 0$ such that the Kuhn and Tucker condition $\nabla N(\mathbf{a},\theta)+\lambda_{1}\nabla g(\mathbf{a},\theta)=0,$ is verified. Hence, we have $\frac{2}{(2\cot\theta-(a_{1}+a_{2}))^{2}}\left(\begin{array}[]{c}\cot\theta- a_{2}\\\ -\cot\theta+a_{1}\\\ \frac{a_{1}-a_{2}}{\sin^{2}\theta}\end{array}\right)+\lambda_{1}\left(\begin{array}[]{c}-X^{i_{1}}_{3}\\\ 0\\\ -\frac{\cos\theta}{\sin^{2}\theta}\end{array}\right),$ and necessarily, $a_{1}=\cot\theta$ so that $N(a,\theta)$ reaches its upper bound and thus is not minimal. We obtain the same contradiction when only a constraint in (13) is active, or only two constraints in (12), or only two constraints in (13).∎ ### A.3 Proof of Proposition 2 Without loss of generality we suppose that $\theta\geq 0$. Suppose that the minimum of $N(\mathbf{a},\theta)$ is reached with only vertices $i_{1}$ and $i_{2}$ respectively in contact with limiting planes $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$. The values of $a_{1}$ and $a_{2}$ can be determined as the following functions of $\theta$ $a_{1}(\theta)=\frac{X_{2}^{i_{1}}+\frac{f}{\sin\theta}}{X_{3}^{i_{1}}},~{}a_{1}(\theta)=\frac{X_{2}^{i_{2}}+\frac{f}{\sin\theta}}{X_{3}^{i_{2}}},$ and straightforward computations give $N(\mathbf{a}(\theta),\theta)=\frac{\left(\frac{X^{i_{1}}_{2}}{X^{i_{1}}_{3}}-\frac{X^{i_{2}}_{2}}{X^{i_{2}}_{3}}\right)\sin\theta+\left(\frac{f}{X^{i_{1}}_{3}}-\frac{f}{X^{i_{2}}_{3}}\right)}{2\cos\theta-\left(\frac{X^{i_{1}}_{2}}{X^{i_{1}}_{3}}+\frac{X^{i_{2}}_{2}}{X^{i_{2}}_{3}}\right)\sin\theta-\left(\frac{f}{X^{i_{1}}_{3}}+\frac{f}{X^{i_{2}}_{3}}\right)}\left(\frac{f}{c}\right).$ (24) In order to prove the result, we just have to check that the derivative of $N(\mathbf{a}(\theta),\theta)$ with respect to $\theta$ cannot vanish for $\theta\in[0,\frac{\pi}{2}]$. The total derivative of $N(\mathbf{a}(\theta),\theta))$ with respect to $\theta$ is given by $\frac{d}{d\theta}N(\mathbf{a}(\theta),\theta)=\\\ 2\frac{\left(\frac{X^{i_{1}}_{2}}{X^{i_{1}}_{3}}-\frac{X^{i_{2}}_{2}}{X^{i_{2}}_{3}}\right)-f\left(\frac{X^{i_{1}}_{2}-X^{i_{2}}_{2}}{X^{i_{1}}_{3}X^{i_{2}}_{3}}\right)\cos\theta+\left(\frac{f}{X^{i_{1}}_{3}}-\frac{f}{X^{i_{2}}_{3}}\right)\sin\theta}{\left(2\cos\theta-\left(\frac{X^{i_{1}}_{2}}{X^{i_{1}}_{3}}+\frac{X^{i_{2}}_{2}}{X^{i_{2}}_{3}}\right)\sin\theta-\left(\frac{f}{X^{i_{1}}_{3}}+\frac{f}{X^{i_{2}}_{3}}\right)\right)^{2}}\left(\frac{f}{c}\right)$ and its numerator is proportional to the trigonometrical polynomial $p(\theta)=b_{0}+b_{1}\cos\theta+b_{2}\sin\theta,$ (25) where $b_{0}=X^{i_{1}}_{2}X^{i_{2}}_{3}-X^{i_{2}}_{2}X^{i_{1}}_{3}$, $b_{1}=-f\left(X^{i_{2}}_{2}-X^{i_{1}}_{2}\right)$, $b_{2}=f(X^{i_{2}}_{3}-X^{i_{1}}_{3})$. It can be easily shown by using the Schwartz inequality that $p(\theta)$ does not vanish provided that $b_{0}^{2}>b_{1}^{2}+b_{2}^{2}.$ (26) Since $X_{1}^{i_{2}}=X_{1}^{i_{1}}=0$, whe have $b_{0}^{2}=\|X_{1}^{i_{1}}\times X_{1}^{i_{1}}\|^{2}$ and $b_{1}^{2}+b_{2}^{2}=f^{2}\|X_{1}^{i_{1}}-X_{1}^{i_{1}}\|^{2}$. Hence (26) is equivalent to condition (15), this ends the proof.∎ ### A.4 Depth of field region approximation used by A. Merklinger Figure 19: Depth of field region approximation using the hyperfocal. In his book (Merklinger , Chapter 7) A. Merklinger has proposed the following approximation based on the assumption of distant objects and small tilt angles: if $h$ is the distance from $\mathrm{SFP_{1}}$ to $\mathrm{SFP_{2}}$, measured in a direction parallel to the sensor plane at distance $z$ from the lens plane, as depicted in Figure 19, we have $\frac{h}{2z}\approx\frac{f}{H\sin\theta},$ (27) where $H$ is the hyperfocal distance (for a definition see Ray p. 221), related to the f-number $N$ by the formula $H=\frac{f^{2}}{Nc},$ and $c$ is the diameter of the circle of confusion. Using (27), the f-number $N$ can be approximated by $\tilde{N}=\sin\theta\frac{h}{2z}\frac{f}{c}.$ Since the slopes of $\mathrm{SFP_{1}}$ and $\mathrm{SFP_{2}}$ are respectively given by $a_{1}$ and $a_{2}$, we have $\frac{h}{z}={a_{1}-a_{2}},$ and we obtain immediately $\tilde{N}=(a_{1}-a_{2})\sin\theta\left(\frac{f}{2c}\right),$ which is the same as (16). ## References * (1) E. Bigler. Depth of field and Scheimpflug rule : a minimalist geometrical approach. http://www.galerie-photo.com/profondeur-de-champ-scheimpflug-english.html, 2002. * (2) L. Evens. View camera geometry. http://www.math.northwestern.edu/~len/photos/pages/vc.pdf, 2008\. * (3) L. Larmore. Introduction to Photographic Principles. Dover Publication Inc., New York, 1965. * (4) A. Merklinger. Focusing the view camera. Bedford, Nova Scotia, 1996. http://www.trenholm.org/hmmerk/FVC161.pdf. * (5) S. F. Ray. Applied Photographic Optics. Focal Press, 2002. third edition. * (6) T. Scheimpflug. Improved Method and Apparatus for the Systematic Alteration or Distortion of Plane Pictures and Images by Means of Lenses and Mirrors for Photography and for other purposes. GB Patent No. 1196, 1904. * (7) U. Tillmans. Creative Large Format: Basics and Applications. Sinar AG. Feuerthalen, Switzerland, 1997. * (8) R. Wheeler. Notes on view camera geometry, 2003. http://www.bobwheeler.com/photo/ViewCam.pdf.
arxiv-papers
2011-02-01T01:05:10
2024-09-04T02:49:16.755599
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "St\\'ephane Mottelet and Luc de Saint Germain and Olivier Mondin", "submitter": "St\\'ephane Mottelet", "url": "https://arxiv.org/abs/1102.0048" }
1102.0097
Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space T. P. Shestakova Department of Theoretical and Computational Physics, Southern Federal University, Sorge St. 5, Rostov-on-Don 344090, Russia E-mail: shestakova@sfedu.ru Abstract A starting point for the present work was the statement recently discussed in the literature that two Hamiltonian formulations for the theory of gravity, the one proposed by Dirac and the other by Arnowitt – Deser – Misner, may not be related by a canonical transformation. In its turn, it raises a question about the equivalence of these two Hamiltonian formulations and their equivalence to the original formulation of General Relativity. We argue that, since the transformation from components of metric tensor to the ADM variables touches gauge degrees of freedom, which are non-canonical from the point of view of Dirac, the problem cannot be resolved in the limits of the Dirac approach. The proposed solution requires the extension of phase space by treating gauge degrees of freedom on an equal footing with other variables and introducing missing velocities into the Lagrangian by means of gauge conditions in differential form. We illustrate with a simple cosmological model the features of Hamiltonian dynamics in extended phase space. Then, we give a clear proof for the full gravitational theory that the ADM-like transformation is canonical in extended phase space in a wide enough class of possible parametrizations. ## 1\. Introduction It is generally accepted that the problem of formulating Hamiltonian dynamics for systems with constraints has been solved by Dirac in his seminal papers [1, 2]. It was Dirac who pointed to the importance of Hamiltonian formulation for any dynamical theory before its quantization [3]. Other approaches, such as the Batalin – Fradkin – Vilkovisky (BFV) path integral approach [4, 5, 6] follow the Dirac one in what concerns the rule of constructing a Hamiltonian and the role of constraints as generators of transformations in phase space. It is believed that Dirac generalized Hamiltonian dynamics is equivalent to Lagrangian dynamics of original theory. However, even for electrodynamics the constraints do not generate a correct transformation for zero component of vector potential, $A_{0}$. The same situation we face in General Relativity, since gravitational constraints cannot produce correct transformations for $g_{00}$, $g_{0\mu}$ components of metric tensor. In fact, it means that the group of transformations generated by constraints differs from the group of gauge transformations of the original theory. Some authors have tried to remedy this shortcoming by modifying the Dirac approach and proposing some special prescriptions how the generator should be constructed (see, for example, [7, 8]). Until now this problem has not attracted much attention mainly because that it touches only transformations of gauge variables which, according to conventional viewpoint, are redundant and must not affect the physical content of the theory. It will be demonstrated in this paper that the role of gauge degrees of freedom may be more significant that it is usually thought, and the difference in the groups of transformations is the first indication to the inconsistence of the theory. Historically, while constructing Hamiltonian dynamics for gravitational field theorists used various parametrizations of gravitational variables. Dirac dealt with original variables, which are components of metric tensor [3], whereas the most famous parametrization is probably that of Arnowitt – Deser – Misner (ADM) [9], who expressed $g_{00}$, $g_{0\mu}$ through the lapse and shift functions. To give another example, let us mention the work by Faddeev [10] where quite specific variables $\lambda^{0}=1/h^{00}+1$, $\lambda^{i}=h^{0i}/h^{00}$, $q^{ij}=h^{0i}h^{0j}-h^{00}h^{ij}$, $h^{\mu\nu}=\sqrt{-g}g^{\mu\nu}$ were introduced. From the point of view of Lagrangian formalism, all the parametrizations are rightful, and the correspondent formulations are equivalent. Meanwhile, it has been shown in [11] that components of metric tensor and the ADM variables are not related by a canonical transformation. In other words, it implies that the Dirac Hamiltonian formulation for gravitation and the ADM one are not equivalent, though it is believed that each of them is equivalent to the Einstein (Lagrangian) formulation. There exists the contradiction that again witnesses about the incompleteness of the theoretical foundation. The purpose of the present paper is to demonstrate that this contradiction can be resolved if one treats gauge gravitational degrees of freedom on an equal footing with physical variables in extended phase space. The idea of extended phase space was put forward by Batalin, Fradkin and Vilkovisky [4, 5, 6] who included integration over gauge and ghost degrees of freedom in their definition of path integral. However, in their approach gauge variables were still considered as non-physical, secondary degrees of freedom playing just an auxiliary role in the theory. To construct Hamiltonian dynamics for a constrained system which would be completely equivalent to Lagrangian formulation, we need to take yet another step: we should introduce into the Lagrangian missing velocities corresponding to gauge variables by means of special (differential) gauge conditions. It actually extends the phase space of physical degrees of freedom. In Section 2 a mathematical formulation of the problem will be given. We shall see that non-equivalence of Hamiltonian formulations for different parametrizations prevents from constructing a generator of transformation in phase space which would produce correct transformations for any parametrizations. These ideas will be illustrated in Section 3 for a simple model with finite number of degrees of freedom. The mentioned above algorithms [7, 8] work correctly only for some parametrizations. One possible point of view (advocated, in particular, in [11]) is that only these parametrizations should be allowed while all other, not related with the first ones by canonical transformations, should be prohibited, including the ADM parametrization. However, imposing any limitations on admissible parametrizations or transformations does not seem to be a true solution to the problem. In Section 4 the outline of Hamiltonian dynamics in extended phase space will be presented, and in Section 5 it will be demonstrated for the full gravitational theory that different parametrizations from a wide enough class are related by canonical transformations. In particular, it will restore a legitimate status of the ADM parametrization. We shall discuss the results and future problems in Section 6. ## 2\. Canonical transformations in phase space It is generally known that for a system without constraints Lagrangian as well as Hamiltonian equations maintain their form under transformations to a new set of generalized coordinates $q^{a}=v^{a}(Q),$ (1) where $v^{a}(Q)$ are invertible functions of their arguments. It is easy to see that any transformation (2.1) correspond to a canonical transformation in phase space. Indeed, consider a quadratic in velocities Lagrangian $L=\frac{1}{2}\;\Gamma_{ab}(q)\dot{q}^{a}\dot{q}^{b}-U(q).$ (2) After the transformation (2.1) the Lagrangian (2.2) would read $L=\frac{1}{2}\;\Gamma_{cd}(Q)\frac{\partial v^{c}}{\partial Q^{a}}\frac{\partial v^{d}}{\partial Q^{b}}\dot{Q}^{a}\dot{Q}^{b}-U(Q).$ (3) New momenta $\\{P_{a}\\}$ are expressed through old momenta $\\{p_{a}\\}$ by relations $P_{a}=p_{b}\frac{\partial v^{b}}{\partial Q^{a}}.$ (4) The transformation (2.1), (2.4) is canonical with the generating function which depends on new coordinates and old momenta, $\Phi(Q,\,p)=-p_{a}v^{a}(Q).$ (5) The equations $q^{a}=-\frac{\partial\Phi}{\partial p_{a}};\qquad P^{a}=-\frac{\partial\Phi}{\partial Q^{a}}$ (6) reproduce exactly the transformation (2.1), (2.4). It is also easy to check that the transformation (2.1), (2.4) maintains the Poisson brackets $\\{Q^{a},\,Q^{b}\\}=0,\qquad\\{P_{a},\,P_{b}\\}=0,\qquad\\{Q^{a},\,P_{b}\\}=\delta^{a}_{b}.$ (7) For a system with constraints, gauge variables (i.e. the variables whose velocities cannot be expressed in terms of conjugate momenta) do not enter into the Lagrangian quadratically, and a general transformation like (2.1) may not be canonical. An example can be found in the theory of gravity by the transformation from components of metric tensor to the ADM variables, $g_{00}=\gamma_{ij}N^{i}N^{j}-N^{2},\qquad g_{0i}=\gamma_{ij}N^{j},\qquad g_{ij}=\gamma_{ij}.$ (8) This transformation concerns gauge degrees of freedom which, from the viewpoint of Dirac, are not canonical variables at all. To pose the question, if the transformation (2.8) is canonical, one should formally extend the original phase space including into it gauge degrees of freedom and their momenta. In order to prove non-canonicity of (2.8) it is enough to check that some of the relations (2.7) are broken. Using the transformation inverted to (2.8), one can see that $\\{N,\,\Pi^{ij}\\}\neq 0$, where $\Pi^{ij}$ are the momenta conjugate to $\gamma_{ij}$ (see Equation (152) in [11]). More generally, let us consider the ADM-like transformation $N_{\mu}=V_{\mu}(g_{0\nu},\,g_{ij}),\qquad\gamma_{ij}=g_{ij}.$ (9) Here $V_{\mu}$ are some functions of components of metric tensor (but $N_{\mu}$ ought not to form 4-vector). A feature of this transformation is that space components of metric tensor remain unchanged, and so do their conjugate momenta: $\Pi^{ij}=p^{ij}$. Then $\left.\\{N_{\mu},\,\Pi^{ij}\\}\right|_{g_{\nu\lambda},p^{\rho\sigma}}=\frac{\partial V_{\mu}}{\partial g_{ij}}.$ (10) It is equal to zero if only the functions $V_{\mu}$ do not depend on $g_{ij}$. This is quite a trivial case when old gauge variables are expressed through some new gauge variables only, and the ADM transformation (2.8) does not belong to this class. One could pose the question: Is it worth considering the equivalence of different formulations in extended phase space? Would not it better to restrict ourself by transformations in phase space of original canonical variables in the sense of Dirac? In the second case, we can prove the equivalence of equation of motion in Lagrangian and Hamiltonian formalism, however, we have to fix a form of gravitational constraints by forbidding any reparametrizations of gauge variables. Determination of the constraints’ form is of importance for a subsequent procedure of quantization which gives rise to the problem of parametrization noninvariance (see, for example, [12]). From the viewpoint of subsequent quantization, the ADM parametrization is more preferable, since the constraints do not depend on gauge variables in this case. I would like to emphasize that there are no solid grounds for fixing the form of the constraints, and, as we shall see in this paper, extension of phase space enables us to solve the problem of equivalence of Lagrangian and Hamiltonian formalism for gravity without any restriction on parametrizations. As it has been already mentioned, the constraints, being considered as generators of transformations in phase space, do not produce correct transformation for all gravitational variables. To ensure the full equivalence of two formulations one has to modify the Dirac prescription, according to which the generator must be a linear combination of constraints, and replace it by a more sophisticated algorithm. The known algorithms, firstly, are relied upon algebra of constraints and, secondly, require extension of phase space. Indeed, a transformation for a variable $q^{a}$ produced by any generator $G$ in phase space reads $\delta q^{a}=\\{q^{a},\,G\\}.$ (11) So, to generate correct transformations for gauge variables the Poisson brackets should be defined in extended phase space. Again, the dependence of the algorithm on the algebra of constraints together with non-canonicity of the transformations like (2.9) leads to the fact that the algorithm would work only for a limited class of parametrizations. Thus, non-equivalence of Hamiltonian formulations for different parametrizations, resulting in different algebra of constraints, prevents from constructing the generator which would produce correct transformations for any parametrizations. In the next section we shall illustrate it making use of the algorithm [7], for a simple model with finite number of degrees of freedom. ## 3\. The generator of gauge transformation: a simple example Now we shall consider a closed isotropic cosmological model with the Lagrangian $L_{1}=-\frac{1}{2}\frac{a\dot{a}^{2}}{N}+\frac{1}{2}Na.$ (1) This model is traditionally described in the ADM variables ($N$ is the lapse function, $a$ is the scale factor). For our purpose, it is more convenient to go to a new variable $\mu=N^{2}$ which corresponds to $g_{00}$. So the Lagrangian is $L_{2}=-\frac{1}{2}\frac{a\dot{a}^{2}}{\sqrt{\mu}}+\frac{1}{2}\sqrt{\mu}\,a.$ (2) The canonical Hamiltonian constructed according to the rule $H=p_{a}\dot{q}^{a}-L$, where $\\{p_{a},\;q^{a}\\}$ are pairs of variables called canonical in the sense that all the velocities $\dot{q}^{a}$ can be expressed through conjugate momenta, for our model is $H_{C}=p\dot{a}-L_{2}=-\frac{1}{2}\frac{\sqrt{\mu}}{a}\;p^{2}-\frac{1}{2}\sqrt{\mu}\,a$ (3) ($p$ is the momentum conjugate to the scale factor). However, some authors include into the form $p_{a}\dot{q}^{a}$ also gauge variables and their momenta which are non-canonical variables in the above sense. Then we have the so-called total Hamiltonian which for our model takes the form $H_{T}=\pi\dot{\mu}+p\dot{a}-L_{2}=\pi\dot{\mu}-\frac{1}{2}\frac{\sqrt{\mu}}{a}\;p^{2}-\frac{1}{2}\sqrt{\mu}\,a$ (4) ($\pi$ is the momentum conjugate to the gauge variable $\mu$). Making use of the total Hamiltonian implies a mixed formalism in which the Hamiltonian is written in terms of canonical coordinates and momenta but as well of velocities that cannot be expressed through the momenta. Nevertheless, this very Hamiltonian plays the central role in the algorithm suggested in [7] while the canonical Hamiltonian (3.3) will not lead to the correct result. In [7] the generator of gauge transformations is sought in the form $G=\sum\limits_{n}\theta_{\mu}^{(n)}G_{n}^{\mu},$ (5) where $G_{n}^{\mu}$ are first class constraints, $\theta_{\mu}^{(n)}$ are the $n$th order time derivatives of the gauge parameters $\theta_{\mu}$. In the theory of gravity the variations of $g_{\mu\nu}$ involve first order derivatives of gauge parameters, thus the generator is $G=\theta_{\mu}G_{0}^{\mu}+\dot{\theta}_{\mu}G_{1}^{\mu}.$ (6) $G_{n}^{\mu}$ satisfy the following conditions that were derived from the requirement of invariance of motion equations under transformations in phase space: $G_{1}^{\mu}\quad{\rm are\;primary\;constraints};$ (7) $G_{0}^{\mu}+\left\\{G_{1}^{\mu},\;H\right\\}\quad{\rm are\;primary\;constraints};$ (8) $\left\\{G_{0}^{\mu},\;H\right\\}\quad{\rm are\;primary\;constraints}.$ (9) In our case $\pi=0$ is the only primary constraint of the model, so that $G_{1}=\pi$. The secondary constraint is $\dot{\pi}=\left\\{\pi,\;H_{T}\right\\}=-\frac{\partial H_{T}}{\partial\mu}=\frac{1}{4}\frac{1}{a\sqrt{\mu}}\;p^{2}+\frac{1}{4}\frac{a}{\sqrt{\mu}}=T.$ (10) The canonical Hamiltonian (3.3) appears to be proportional to the secondary constraint $T$, $H_{C}=-2\mu T$. The condition (3.8) becomes $G_{0}+\left\\{\pi,\;H_{T}\right\\}=\alpha\pi;$ (11) $G_{0}=-T+\alpha\pi,$ (12) $\alpha$ is a coefficient that can be found from the requirement (3.9): $\left\\{G_{0},\;H_{T}\right\\}=\beta\pi;$ (13) $\displaystyle\left\\{G_{0},\;H_{T}\right\\}$ $\displaystyle=$ $\displaystyle-\left\\{T,\;H_{T}\right\\}+\alpha\left\\{\pi,\;H_{T}\right\\}=-\left\\{T,\;\pi\dot{\mu}-2\mu T\right\\}+\alpha T$ (14) $\displaystyle=$ $\displaystyle-\left\\{T,\;\pi\right\\}\dot{\mu}+\alpha T=\frac{1}{2\mu}\;\dot{\mu}T+\alpha T;$ $\beta=0;\qquad\alpha=-\frac{1}{2\mu}\;\dot{\mu};$ (15) $G_{0}=-\frac{1}{2\mu}\;\dot{\mu}\pi-T.$ (16) The full generator $G$ (3.6) can be written as $G=\left(-\frac{1}{2\mu}\;\dot{\mu}\pi-T\right)\theta+\pi\dot{\theta}.$ (17) The transformation of the variable $\mu$ is $\delta\mu=\left\\{\mu,\;G\right\\}=-\frac{1}{2\mu}\;\dot{\mu}\theta+\dot{\theta}.$ (18) The same expression (up to the multiplier being equal to 2) can be obtained from general transformations of the metric tensor, $\delta g_{\mu\nu}=\theta^{\lambda}\partial_{\lambda}g_{\mu\nu}+g_{\mu\lambda}\partial_{\nu}\theta^{\lambda}+g_{\nu\lambda}\partial_{\mu}\theta^{\lambda};$ (19) $\delta g_{00}=\dot{g}_{00}\theta^{0}+2g_{00}\dot{\theta}^{0},$ (20) if one keeps in mind that $g_{00}=\mu$ and in the above formulas $\theta=\theta_{0}=g_{00}\theta^{0}$. Ir is easy to see that the correct expression (3.18) is entirely due to the replacement of the canonical Hamiltonian (3.3) by the total Hamiltonian (3.4), otherwise one would miss the contribution from the Poisson bracket $\\{T,\;\pi\\}$ to the generator (3.17) (see the second line of (3.14)). On the other hand, making use of the total Hamiltonian may not lead to a correct result for another parametrization. Let us return to the Lagrangian (3.1). Now the total Hamiltonian is $H_{T}=\pi\dot{N}-\frac{1}{2}\frac{N}{a}\;p^{2}-\frac{1}{2}\;N\,a$ (21) Again, $\pi$ is the momentum conjugate to the gauge variable $N$, and $\pi=0$ is the only primary constraint. The secondary constraint does not depend on $N$ in this case: $\dot{\pi}=\left\\{\pi,\;H_{T}\right\\}=-\frac{\partial H_{T}}{\partial N}=\frac{1}{2a}\;p^{2}+\frac{1}{2}\;a=T,$ (22) therefore, the Poisson bracket $\left\\{T,\;\pi\right\\}$ in (3.14) is equal to zero, and one would obtain an incorrect expression for the generator, $G=-T\theta+\pi\dot{\theta}.$ (23) It cannot produce the correct variation of $N$, that reads $\delta N=-\dot{N}\theta-N\dot{\theta}.$ (24) As we can see, this algorithm fails to produce correct results for an arbitrary parametrization. In the next section we shall construct Hamiltonian dynamics in extended phase space and discuss its features and advantages. ## 4\. Extended phase space: the isotropic model We shall consider the effective action including gauge and ghost sectors as it appears in the path integral approach to gauge field theories, $S=\int dt\left(L_{(grav)}+L_{(gauge)}+L_{(ghost)}\right)$ (1) As was mentioned above, it is not enough just to extend pase space by including formally gauge degrees of freedom in it. One should also introduce missing velocities into the Lagrangian. It can be done by means of special (differential) gauge conditions that actually extends the phase space and enables one to avoid the mixed formalism. For our model (3.1) the equation $N=f(a)$ determines in a general form a relation between the only gauge variable $N$ and the scale factor $a$. The differential form of this relation is $\dot{N}=\frac{df}{da}\;\dot{a}.$ (2) The ghost sector of the model reads $L_{(ghost)}=\dot{\bar{\theta}}N\dot{\theta}+\dot{\bar{\theta}}\left(\dot{N}-\frac{df}{da}\;\dot{a}\right)\theta,$ (3) so that $\displaystyle L$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\frac{a\dot{a}^{2}}{N}+\frac{1}{2}Na+\lambda\left(\dot{N}-\frac{df}{da}\;\dot{a}\right)+\dot{\bar{\theta}}\left(\dot{N}-\frac{df}{da}\;\dot{a}\right)\theta+\dot{\bar{\theta}}N\dot{\theta}=$ (4) $\displaystyle=$ $\displaystyle-\frac{1}{2}\frac{a\dot{a}^{2}}{N}+\frac{1}{2}Na+\pi\left(\dot{N}-\frac{df}{da}\;\dot{a}\right)+\dot{\bar{\theta}}N\dot{\theta}.$ The conjugate momenta are: $\pi=\lambda+\dot{\bar{\theta}}\theta;\quad p=-\frac{a\dot{a}}{N}-\pi\frac{df}{da};\quad\bar{\cal P}=N\dot{\bar{\theta}};\quad{\cal P}=N\dot{\theta}.$ (5) Let us now go to a new variable $N=v(\tilde{N},a).$ (6) At the same time, the rest variables are unchanged: $a=\tilde{a};\quad\theta=\tilde{\theta};\quad\bar{\theta}=\tilde{\bar{\theta}}.$ (7) It is the analog of the transformation from the original gravitational variables $g_{\mu\nu}$ to the ADM variables. Indeed, in the both cases only gauge variables are transformed while the rest variables remain unchanged. After the change (4.6) the Lagrangian is written as (below we shall omit the tilde over $a$ and ghost variables which remain unchanged) $L=-\frac{1}{2}\;\frac{a\dot{a}^{2}}{v(\tilde{N},a)}+\frac{1}{2}\;v(\tilde{N},a)\;a+\pi\left(\frac{\partial v}{\partial\tilde{N}}\;\dot{\tilde{N}}+\frac{\partial v}{\partial a}\;\dot{a}-\frac{df}{da}\;\dot{a}\right)+v(\tilde{N},a)\;\dot{\bar{\theta}}\dot{\theta}.$ (8) The new momenta are: $\tilde{\pi}=\pi\frac{\partial v}{\partial\tilde{N}};\qquad\tilde{p}=-\frac{a\dot{a}}{v(\tilde{N},a)}+\pi\frac{\partial v}{\partial a}-\pi\frac{df}{da}=p+\pi\frac{\partial v}{\partial a};$ (9) $\tilde{\bar{\cal P}}=v(\tilde{N},a)\;\dot{\bar{\theta}}=\bar{\cal P};\qquad\tilde{\cal P}=v(\tilde{N},a)\;\dot{\theta}={\cal P}.$ (10) It is easy to demonstrate that the transformations (4.6), (4.7), (4.9), (4.10) are canonical in extended phase space. The generating function will depend on new coordinates and old momenta, $\Phi\left(\tilde{N},\;\tilde{a},\;\tilde{\bar{\theta}},\;\tilde{\theta},\;\pi,\;p,\;\bar{\cal P},\;{\cal P}\right)=-\pi\,v(\tilde{N},\tilde{a})-p\,\tilde{a}-\bar{\cal P}\,\tilde{\theta}-\tilde{\bar{\theta}}\,{\cal P}.$ (11) One can see that the generating function has the same form as in (2.5). The relations $N=-\frac{\partial\Phi}{\partial\pi};\qquad a=-\frac{\partial\Phi}{\partial p};\qquad\tilde{\pi}=-\frac{\partial\Phi}{\partial\tilde{N}};\qquad\tilde{p}=-\frac{\partial\Phi}{\partial\tilde{a}};$ (12) $\theta=-\frac{\partial\Phi}{\partial\bar{\cal P}\vphantom{\sqrt{N}}};\qquad\bar{\theta}=-\frac{\partial\Phi}{\partial{\cal P}};\qquad\tilde{\cal P}=-\frac{\partial\Phi}{\partial\tilde{\bar{\theta}}};\qquad\tilde{\bar{\cal P}}=-\frac{\partial\Phi}{\partial\tilde{\theta}}$ (13) give exactly the transformation (4.6), (4.7), (4.9), (4.10). On the other hand, one can check that Poisson brackets among all phase variables maintain their canonical form. Now we are going to write down equations of motion in extended phase space. Firstly, we rewrite the Lagrangian (4.8) through the momentum $\tilde{\pi}$. $\displaystyle L$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\;\frac{a\dot{a}^{2}}{v(\tilde{N},a)}+\frac{1}{2}\;v(\tilde{N},a)\;a$ (14) $\displaystyle+$ $\displaystyle\tilde{\pi}\left[\dot{\tilde{N}}+\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\;\dot{a}-\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}\;\dot{a}\right]+v(\tilde{N},a)\;\dot{\bar{\theta}}\dot{\theta}.$ The variation of (4.14) gives, accordingly, the equation of motion (4.15), the constraint (4.16), the gauge condition (4.17) and the ghost equations (4.18) – (4.19): $\displaystyle\frac{a\ddot{a}}{v(\tilde{N},a)}$ $\displaystyle+$ $\displaystyle\frac{1}{2}\;\frac{\dot{a}^{2}}{v(\tilde{N},a)}-\frac{1}{2}\;\frac{a\dot{a}^{2}}{v^{2}(\tilde{N},a)}\;\frac{\partial v}{\partial a}-\frac{a\dot{a}}{v^{2}(\tilde{N},a)}\;\frac{\partial v}{\partial\tilde{N}}\dot{\tilde{N}}$ (15) $\displaystyle+$ $\displaystyle\frac{1}{2}\;\frac{\partial v}{\partial a}\;a+\frac{1}{2}v(\tilde{N},a)-\dot{\tilde{\pi}}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}+\dot{\tilde{\pi}}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}$ $\displaystyle+$ $\displaystyle\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{\partial v}{\partial a}\;\dot{\tilde{N}}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial^{2}v}{\partial\tilde{N}\partial a}\;\dot{\tilde{N}}$ $\displaystyle-$ $\displaystyle\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{df}{da}\;\dot{\tilde{N}}+\frac{\partial v}{\partial a}\;\dot{\bar{\theta}}\dot{\theta}=0;$ $\displaystyle\frac{1}{2}\;\frac{a\dot{a}^{2}}{v^{2}(\tilde{N},a)}\;\frac{\partial v}{\partial\tilde{N}}$ $\displaystyle+$ $\displaystyle\frac{1}{2}\;\frac{\partial v}{\partial\tilde{N}}\;a-\dot{\tilde{\pi}}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{\partial v}{\partial a}\;\dot{a}+\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial^{2}v}{\partial\tilde{N}\partial a}\;\dot{a}$ (16) $\displaystyle+$ $\displaystyle\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{df}{da}\;\dot{a}+\frac{\partial v}{\partial\tilde{N}}\;\dot{\bar{\theta}}\dot{\theta}=0;$ $\frac{\partial v}{\partial\tilde{N}}\;\dot{\tilde{N}}+\frac{\partial v}{\partial a}\;\dot{a}-\frac{df}{da}\;\dot{a}=0;$ (17) $v(\tilde{N},\;a)\;\ddot{\theta}+\frac{\partial v}{\partial\tilde{N}}\;\dot{\tilde{N}}\dot{\theta}+\frac{\partial v}{\partial a}\;\dot{a}\dot{\theta}=0;$ (18) $v(\tilde{N},\;a)\;\ddot{\bar{\theta}}+\frac{\partial v}{\partial\tilde{N}}\;\dot{\tilde{N}}\dot{\bar{\theta}}+\frac{\partial v}{\partial a}\;\dot{a}\dot{\bar{\theta}}=0.$ (19) The Hamiltonian in extended phase space looks like $\displaystyle H$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\;\frac{v(\tilde{N},a)}{a}\left[\tilde{p}^{2}+2\tilde{p}\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}+\tilde{\pi}^{2}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\left(\frac{df}{da}\right)^{2}\right.$ (20) $\displaystyle-$ $\displaystyle\left.2\tilde{p}\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}-2\tilde{\pi}^{2}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial v}{\partial a}\;\frac{df}{da}+\tilde{\pi}^{2}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\left(\frac{\partial v}{\partial a}\right)^{2}\right]$ $\displaystyle-$ $\displaystyle\frac{1}{2}\;v(\tilde{N},a)\;a+\frac{1}{v(\tilde{N},a)}\;\bar{\cal P}{\cal P}.$ The Hamiltonian equations in extended phase space are: $\displaystyle\dot{\tilde{p}}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[\frac{1}{a}\frac{\partial v}{\partial a}-\frac{v(\tilde{N},a)}{a^{2}}\right]\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]^{2}$ (21) $\displaystyle-$ $\displaystyle\frac{v(\tilde{N},a)}{a}\left[\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}\partial a}\;\frac{df}{da}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{d^{2}f}{da^{2}}\right.$ $\displaystyle-$ $\displaystyle\left.\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}\partial a}\;\frac{\partial v}{\partial a}+\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial^{2}v}{\partial a^{2}}\right]$ $\displaystyle\times$ $\displaystyle\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]$ $\displaystyle+$ $\displaystyle\frac{1}{2}\;\frac{\partial v}{\partial a}\;a+\frac{1}{2}\;v(\tilde{N},a)+\frac{1}{v^{2}(\tilde{N},a)}\;\bar{\cal P}{\cal P};$ $\displaystyle\dot{a}$ $\displaystyle=$ $\displaystyle-\frac{v(\tilde{N},a)}{a}\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right];$ (22) $\displaystyle\dot{\tilde{\pi}}$ $\displaystyle=$ $\displaystyle\frac{1}{2a}\;\frac{\partial v}{\partial\tilde{N}}\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]^{2}$ (23) $\displaystyle-$ $\displaystyle\frac{v(\tilde{N},a)}{a}\left[\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{df}{da}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-2}\frac{\partial^{2}v}{\partial\tilde{N}^{2}}\;\frac{\partial v}{\partial a}\right.$ $\displaystyle+$ $\displaystyle\left.\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial^{2}v}{\partial\tilde{N}\partial a}\right]\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]$ $\displaystyle+$ $\displaystyle\frac{1}{2}\;\frac{\partial v}{\partial\tilde{N}}\;a+\frac{1}{v^{2}(\tilde{N},a)}\;\frac{\partial v}{\partial\tilde{N}}\;\bar{\cal P}{\cal P};$ $\displaystyle\dot{\tilde{N}}$ $\displaystyle=$ $\displaystyle-\frac{v(\tilde{N},a)}{a}\left[\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right]$ (24) $\displaystyle\times$ $\displaystyle\left[\tilde{p}+\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{df}{da}-\tilde{\pi}\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\frac{\partial v}{\partial a}\right];$ $\displaystyle\dot{\bar{\cal P}}$ $\displaystyle=$ $\displaystyle 0;$ (25) $\displaystyle\dot{\theta}$ $\displaystyle=$ $\displaystyle\frac{1}{v(\tilde{N},a)}\;{\cal P};$ (26) $\displaystyle\dot{\cal P}$ $\displaystyle=$ $\displaystyle 0;$ (27) $\displaystyle\dot{\bar{\theta}}$ $\displaystyle=$ $\displaystyle\frac{1}{v(\tilde{N},a)}\;\bar{\cal P}.$ (28) One can check that the Hamiltonian equations (4.21) – (4.28) are completely equivalent to the Lagrangian equations (4.15) – (4.19), the constraint (4.23) and the gauge condition (4.24) being true Hamiltonian equations. The Hamiltonian equations (4.21) – (4.28) in extended phase space, as well as the equations (4.15) – (4.19), include gauge-dependent terms. In this connection one can object that the equations are not equivalent to the original Einstein equation, which are known to be gauge-invariant. However, we remember that any solution to the gauge-invariant Einstein equation is determined up to arbitrary functions which have to be fix by a choice of a reference frame (a state of the observer). It is usually done on the final stage of solving the Einstein equations. It is important that one cannot avoid fixing a reference frame to obtain a final form of the solution. By varying the gauged action (4.1), in fact, we deal with a generalized mathematical problem, its generalization has come from the development of quantum field theory. In the case of the extended set of equations (4.21) – (4.28) (or, (4.15) – (4.19)) one can keep the function $f(a)$ non-fixed up to the final stage of their resolution. Further, under the conditions $\bar{\pi}=0$, $\theta=0$, $\bar{\theta}=0$ all gauge-dependent terms are excluded, and the extended set of equations is reduced to gauge-invariant equations, therefore, any solution of the Einstein equations can be found among solutions of the extended set. Solutions with non-trivial $\bar{\pi}$, $\theta$, $\bar{\theta}$ should be considered and physically interpreted separately. One can also reveal that there exists a quantity conserved if the Hamiltonian (or, equivalently, Lagrangian) equations hold. It plays the role of the BRST generator for our model: $\Omega=-H\theta-\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}\tilde{\pi}{\cal P}.$ (29) It generates correct transformations for the variables $a$, $\theta$, $\bar{\theta}$ and for any gauge variable $\tilde{N}$ given by the relation (4.6), $\delta\tilde{N}=-\frac{\partial H}{\partial\tilde{\pi}}\;\theta-\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}{\cal P}=-\dot{\tilde{N}}\theta-\left(\frac{\partial v}{\partial\tilde{N}}\right)^{-1}v(\tilde{N},a)\;\dot{\theta}.$ (30) In particular, for the original variable $N$ one gets the transformation (3.24). ## 5\. The canonicity of transformations in extended phase space for the full gravitational theory In this section we shall demonstrate for the full gravitational theory that different parametrizations from a wide enough class (2.9) are related by canonical transformations. Again, we shall start from the gauged action $S=\int d^{4}x\left({\cal L}_{(grav)}+{\cal L}_{(gauge)}+{\cal L}_{(ghost)}\right)$ (1) We shall use a gauge condition in a general form, $f^{\mu}(g_{\nu\lambda})=0$. The differential form of this gauge condition introduces the missing velocities and actually extends phase space, $\frac{d}{dt}f^{\mu}(g_{\nu\lambda})=0,\qquad\frac{\partial f^{\mu}}{\partial g_{00}}\dot{g}_{00}+2\frac{\partial f^{\mu}}{\partial g_{0i}}\dot{g}_{0i}+\frac{\partial f^{\mu}}{\partial g_{ij}}\dot{g}_{ij}=0.$ (2) Then, ${\cal L}_{(gauge)}=\lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial g_{00}}\dot{g}_{00}+2\frac{\partial f^{\mu}}{\partial g_{0i}}\dot{g}_{0i}+\frac{\partial f^{\mu}}{\partial g_{ij}}\dot{g}_{ij}\right).$ (3) Taking into account the gauge transformations, $\delta g_{\mu\nu}=\partial_{\lambda}g_{\mu\nu}\theta^{\lambda}+g_{\mu\lambda}\partial_{\nu}\theta^{\lambda}+g_{\nu\lambda}\partial_{\mu}\theta^{\lambda},$ (4) one can write the ghost sector: ${\cal L}_{(ghost)}=\bar{\theta}_{\mu}\frac{d}{dt}\left[\frac{\partial f^{\mu}}{\partial g_{\nu\lambda}}\left(\partial_{\rho}g_{\nu\lambda}\theta^{\rho}+g_{\lambda\rho}\partial_{\nu}\theta^{\rho}+g_{\nu\rho}\partial_{\lambda}\theta^{\rho}\right)\right].$ (5) It is convenient to write down the action (5.1), (5.3), (5.5) in the form $\displaystyle S$ $\displaystyle=$ $\displaystyle\int d^{4}x\left[{\cal L}_{(grav)}+\Lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial g_{00}}\dot{g}_{00}+2\frac{\partial f^{\mu}}{\partial g_{0i}}\dot{g}_{0i}+\frac{\partial f^{\mu}}{\partial g_{ij}}\dot{g}_{ij}\right)\right.$ (6) $\displaystyle-$ $\displaystyle\dot{\bar{\theta_{\mu}}}\left(\frac{\partial f^{\mu}}{\partial g_{00}}\left(\partial_{i}g_{00}\theta^{i}+2g_{0\nu}\dot{\theta}^{\nu}\right)+2\frac{\partial f^{\mu}}{\partial g_{0i}}\left(\partial_{j}g_{0i}\theta^{j}+g_{0\nu}\partial_{i}\theta^{\nu}+g_{i\nu}\dot{\theta}^{\nu}\right)\right.$ $\displaystyle+$ $\displaystyle\left.\left.\frac{\partial f^{\mu}}{\partial g_{ij}}\left(\partial_{k}g_{ij}\theta^{k}+g_{i\nu}\partial_{j}\theta^{\nu}+g_{j\nu}\partial_{i}\theta^{\nu}\right)\right)\right].$ Here $\Lambda_{\mu}=\lambda_{\mu}-\dot{\bar{\theta_{\mu}}}\theta^{0}$. One can see that the generalized velocities enter into the bracket multiplied by $\Lambda_{\mu}$, in addition to the gravitational part ${\cal L}_{(grav)}$. This very circumstance will ensure the canonicity of the transformation to new variables. Our goal now is to introduce new variables by $g_{0\mu}=v_{\mu}\left(N_{\nu},g_{ij}\right).\qquad g_{ij}=\gamma_{ij};\qquad\theta^{\mu}=\tilde{\theta}^{\mu};\qquad\bar{\theta}_{\mu}=\tilde{\bar{\theta}_{\mu}}.$ (7) This is the inverse transformation for (2.9) and concerns only $g_{0\mu}$ metric components. After the transformation the action will read $\displaystyle S$ $\displaystyle=$ $\displaystyle\int d^{4}x\left[{\cal L^{\prime}}_{(grav)}+\Lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial N_{\nu}}\;\dot{N}_{\nu}+\frac{\partial f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial g_{ij}}\;\dot{g}_{ij}\right.\right.$ (8) $\displaystyle+$ $\displaystyle\left.2\;\frac{\partial f^{\mu}}{\partial g_{0i}}\;\frac{\partial v_{i}}{\partial N_{\nu}}\;\dot{N}_{\nu}+2\;\frac{\partial f^{\mu}}{\partial g_{0k}}\;\frac{\partial v_{k}}{\partial g_{ij}}\;\dot{g}_{ij}+\frac{\partial f^{\mu}}{\partial g_{ij}}\;\dot{g}_{ij}\right)$ $\displaystyle-$ $\displaystyle\dot{\bar{\theta_{\mu}}}\left(\frac{\partial f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial N_{\nu}}\;\partial_{i}N_{\nu}\theta^{i}+\frac{\partial f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial g_{ij}}\;\partial_{k}g_{ij}\theta^{k}+2\;\frac{\partial f^{\mu}}{\partial g_{00}}\;v_{\nu}(N_{\lambda},g_{ij})\;\dot{\theta}^{\nu}\right.$ $\displaystyle+$ $\displaystyle 2\;\frac{\partial f^{\mu}}{\partial g_{0i}}\;\frac{\partial v_{i}}{\partial N_{\nu}}\;\partial_{j}N_{\nu}\theta^{j}+2\;\frac{\partial f^{\mu}}{\partial g_{0i}}\;\frac{\partial v_{i}}{\partial g_{jk}}\;\partial_{l}g_{jk}\theta^{l}$ $\displaystyle+$ $\displaystyle 2\;\frac{\partial f^{\mu}}{\partial g_{0i}}\left[v_{\nu}(N_{\lambda},g_{jk})\;\partial_{i}\theta^{\nu}+v_{i}(N_{\lambda},g_{jk})\;\dot{\theta}^{0}+g_{ij}\dot{\theta}^{j}\right]$ $\displaystyle+$ $\displaystyle\frac{\partial f^{\mu}}{\partial g_{ij}}\left[\partial_{k}g_{ij}\theta^{k}+v_{i}(N_{\lambda},g_{kl})\;\partial_{j}\theta^{0}+g_{ik}\partial_{j}\theta^{k}\right.$ $\displaystyle+$ $\displaystyle\left.\left.\left.v_{j}(N_{\lambda},g_{kl})\;\partial_{i}\theta^{0}+g_{jk}\partial_{i}\theta^{k}\right]\right)\right]$ We can write down the “old” momenta, $\displaystyle\pi^{ij}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal L}_{(grav)}}{\partial\dot{g}_{ij}}+\Lambda_{\mu}\;\frac{\partial f^{\mu}}{\partial g_{ij}};$ (9) $\displaystyle\pi^{0}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal L}_{(grav)}}{\partial\dot{g}_{00}}+\Lambda_{\mu}\;\frac{\partial f^{\mu}}{\partial g_{00}};$ (10) $\displaystyle\pi^{i}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal L}_{(grav)}}{\partial\dot{g}_{0i}}+2\Lambda_{\mu}\;\frac{\partial f^{\mu}}{\partial g_{0i}},$ (11) and the “new” momenta are: $\displaystyle\Pi^{ij}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal L^{\prime}}_{(grav)}}{\partial\dot{g}_{ij}}+\Lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial g_{ij}}+2\;\frac{\partial f^{\mu}}{\partial g_{0k}}\;\frac{\partial v_{k}}{\partial g_{ij}}+\frac{\partial f^{\mu}}{\partial g_{ij}}\right);$ (12) $\displaystyle\Pi^{0}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal L^{\prime}}_{(grav)}}{\partial\dot{N}_{0}}+\Lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial N_{0}}+2\;\frac{\partial f^{\mu}}{\partial g_{0i}}\;\frac{\partial v_{i}}{\partial N_{0}}\right);$ (13) $\displaystyle\Pi^{i}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal L^{\prime}}_{(grav)}}{\partial\dot{N}_{i}}+\Lambda_{\mu}\left(\frac{\partial f^{\mu}}{\partial g_{00}}\;\frac{\partial v_{0}}{\partial N_{i}}+2\;\frac{\partial f^{\mu}}{\partial g_{0j}}\;\frac{\partial v_{j}}{\partial N_{i}}\right).$ (14) The relations between the “old” and “new” momenta: $\displaystyle\Pi^{ij}$ $\displaystyle=$ $\displaystyle\pi^{ij}+\left(\pi^{\mu}-\frac{\partial{\cal L}_{(grav)}}{\partial\dot{g}_{0\mu}}\right)\frac{\partial v_{\mu}}{\partial g_{ij}};$ (15) $\displaystyle\Pi^{\mu}$ $\displaystyle=$ $\displaystyle\frac{\partial{\cal L^{\prime}}_{(grav)}}{\partial\dot{N}_{\mu}}+\left(\pi^{\nu}-\frac{\partial{\cal L}_{(grav)}}{\partial\dot{g}_{0\nu}}\right)\frac{\partial v_{\nu}}{\partial N_{\mu}}.$ (16) It is easy to check that the momenta conjugate to ghosts remain unchanged, $\tilde{\cal P}^{\mu}={\cal P}^{\mu}$, $\tilde{\bar{\cal P}}_{\mu}=\bar{\cal P}_{\mu}$. As any Lagrangian is determined up to total derivatives, the gravitational Lagrangian density ${\cal L}_{(grav)}$ can be modified in such a way for the primary constraints to take the form $\pi^{\mu}=0$, where $\pi^{\mu}$ are the momenta conjugate to gauge variables $g_{0\mu}$. This change of the Lagrangian density does not affect the equation of motion. It was made by Dirac [3] to simplify the calculations. A similar change of the Lagrangian density by omitting a divergence and a total time derivative was made also in the ADM paper [9]. Therefore, one can put $\frac{\partial{\cal L}_{(grav)}}{\partial\dot{g}_{0\mu}}=0,\qquad\frac{\partial{\cal L^{\prime}}_{(grav)}}{\partial\dot{N}_{\mu}}=0.$ (17) Then, the relations (5.15) – (5.16) would become simpler and take the form $\Pi^{ij}=\pi^{ij}+\pi^{\mu}\frac{\partial v_{\mu}}{\partial g_{ij}};\qquad\Pi^{\mu}=\pi^{\nu}\frac{\partial v_{\nu}}{\partial N_{\mu}}.$ (18) It is easy to demonstrate that the transformations (2.9), (5.18) are canonical in extended phase space. The generating function again depends on new coordinates and old momenta and has the same form as for a non-constrained system (see (2.5), compare also with (4.11)), $\Phi\left(N_{\mu},\;\gamma_{ij},\;\tilde{\theta}^{\mu},\;\tilde{\bar{\theta}}_{\mu},\;\pi^{\mu},\;\pi^{ij},\;\bar{\cal P}_{\mu},\;{\cal P}^{\mu}\right)=-\pi^{\mu}v_{\mu}(N_{\nu},\gamma_{ij})-\pi^{ij}\gamma_{ij}-\bar{\cal P}_{\mu}\tilde{\theta}^{\mu}-\tilde{\bar{\theta}}_{\mu}{\cal P}^{\mu}.$ (19) Then the following relations take place $g_{0\mu}=-\frac{\partial\Phi}{\partial\pi^{\mu}};\qquad g_{ij}=-\frac{\partial\Phi}{\partial\pi^{ij}};\qquad\theta^{\mu}=-\frac{\partial\Phi}{\partial\bar{\cal P}\vphantom{\sqrt{N}}_{\mu}};\qquad\bar{\theta}_{\mu}=-\frac{\partial\Phi}{\partial{\cal P}^{\mu}};$ (20) $\Pi^{\mu}=-\frac{\partial\Phi}{\partial N_{\mu}};\qquad\Pi^{ij}=-\frac{\partial\Phi}{\partial\gamma_{ij}};\qquad\tilde{\bar{\cal P}}_{\mu}=-\frac{\partial\Phi}{\partial\tilde{\theta}^{\mu}};\qquad\tilde{\cal P}^{\mu}=-\frac{\partial\Phi}{\partial\tilde{\bar{\theta}}\vphantom{\sqrt{N}}_{\mu}},$ (21) that give exactly the transformations $g_{0\mu}=v_{\mu}(N_{\nu},\gamma_{ij});\qquad g_{ij}=\gamma_{ij};\qquad\qquad\qquad\qquad\theta^{\mu}=\tilde{\theta}^{\mu};\qquad\bar{\theta}_{\mu}=\tilde{\bar{\theta}}_{\mu};$ (22) $\Pi^{\mu}=\pi^{\nu}\frac{\partial v_{\nu}}{\partial N_{\mu}};\qquad\qquad\Pi^{ij}=\pi^{ij}+\pi^{\mu}\frac{\partial v_{\mu}}{\partial g_{ij}};\qquad\quad\tilde{\bar{\cal P}}_{\mu}=\bar{\cal P}_{\mu};\qquad\tilde{\cal P}^{\mu}={\cal P}^{\mu}.$ (23) We can now check if the Poisson brackets maintain their form. Differentiating the first relation in (2.9) with respect to $g_{ij}$ one gets $\frac{\partial V_{\mu}}{\partial g_{ij}}+\frac{\partial V_{\mu}}{\partial g_{0\lambda}}\frac{\partial v_{\lambda}}{\partial g_{ij}}=0,$ (24) Similarly, differentiating the same relation with respect to $N_{\nu}$ gives $\delta_{\mu}^{\nu}-\frac{\partial V_{\mu}}{\partial g_{0\lambda}}\frac{\partial v_{\lambda}}{\partial N_{\nu}}=0.$ (25) Making use of (5.24), (5.25), it is not difficult to calculate the Poisson brackets. So, we can recalculate the bracket (2.10) to see that it will be zero in our extended phase space formalism, $\displaystyle\left.\\{N_{\mu},\,\Pi^{ij}\\}\right|_{g_{\nu\lambda},p^{\rho\sigma}}$ $\displaystyle=$ $\displaystyle\frac{\partial N_{\mu}}{\partial g_{0\rho}}\;\frac{\partial\Pi^{ij}}{\partial\pi^{\rho}}+\frac{\partial N_{\mu}}{\partial g_{kl}}\;\frac{\partial\Pi^{ij}}{\partial\pi^{kl}}=\left\\{V_{\mu}(g_{0\nu},g_{kl}),\;\pi^{ij}+\pi^{\lambda}\frac{\partial v_{\lambda}}{\partial g_{ij}}\right\\}$ (26) $\displaystyle=$ $\displaystyle\frac{\partial V_{\mu}}{\partial g_{0\rho}}\;\frac{\partial v_{\lambda}}{\partial g_{ij}}\delta_{\rho}^{\lambda}+\frac{\partial V_{\mu}}{\partial g_{kl}}\;\frac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{j}\delta_{k}^{i}\right)=\frac{\partial V_{\mu}}{\partial g_{0\lambda}}\;\frac{\partial v_{\lambda}}{\partial g_{ij}}+\frac{\partial V_{\mu}}{\partial g_{ij}}=0.$ To give another example, let us check the following bracket: $\left.\\{N_{\mu},\,\Pi^{\nu}\\}\right|_{g_{\lambda\rho},p^{\sigma\tau}}=\frac{\partial N_{\mu}}{\partial g_{0\rho}}\;\frac{\partial\Pi^{\nu}}{\partial\pi^{\rho}}=\left\\{V_{\mu}(g_{0\rho},g_{ij}),\;\pi^{\lambda}\frac{\partial v_{\lambda}}{\partial N_{\nu}}\right\\}=\frac{\partial V_{\mu}}{\partial g_{0\rho}}\;\frac{\partial v_{\rho}}{\partial N_{\nu}}=\delta_{\mu}^{\nu}.$ (27) The rest of the brackets can be checked by analogy. This completes the proof of canonicity of the transformation (2.9) for the full gravitational theory. ## 6\. Discussion A starting point for the present investigation was the paper [11] and the statement made by its authors that components of metric tensor and the ADM variables are not related by a canonical transformation. However, it is misunderstanding to pose the question about canonicity of the transformation (2.8) which involves, from the viewpoint of the Dirac approach, non-canonical variables. Let us remind that Dirac himself consider these variables, $g_{0\mu}$ (along with the zero component of vector potential of electromagnetic field $A_{0}$) as playing the role of Lagrange multipliers while the phase space in his approach includes pairs of generalized coordinates and momenta for which corresponding velocities can be expressed through the momenta. We should remember also that the Einstein equations were originally formulated in Lagrangian formalism. Dirac’s Hamiltonian formulation for gravity is equivalent to Einstein’s formulation at the level of equations. It means that Hamiltonian equations for canonical variables (in Dirac’s sense) are equivalent to the ($ij$) Einstein equations, and the gravitational constraints are equivalent to the ($0\mu$) Einstein equations. On the other hand, it implies that a group of transformations in Hamiltonian formalism must involve the full group of gauge transformations of the original theory. However, in the limits of the Dirac approach we fail to construct a generator that would produce correct transformations for all variables. We inevitably have to modify the Dirac scheme, and attempts to do it were presented yet in [7, 8]. Therefore, we cannot consider the Dirac approach as fundamental and undoubted. The ADM formulation of Hamiltonian dynamics for gravity is, first of all, the choice of parametrization, which is preferable because of its geometrical interpretation. There is no any special ”ADM procedure”: Arnowitt, Deser and Misner constructed the Hamiltonian dynamics following exactly the Dirac scheme, just making use of another variables. The fact that two Hamiltonian formulations (both according to the Dirac scheme, but the one for original variables and the other for the ADM variables) are not related by canonical transformations, should not lead to any bad-grounded conclusions like the one made in [11], p. 68, that the gravitational Lagrangian used by Dirac and the ADM Lagrangian are not equivalent. At the Lagrangian level, the transition to the ADM variables is nothing more as a change of variables in the Einstein equations, and there are no mathematical rules that would prohibit such change of variables. It is the Lagrangian formulation of General Relativity which is original and fundamental while its Hamiltonian formulation still remains questionable, in spite of fifty years passed after Dirac’s paper [3]. The extended phase space approach treating all degrees of freedom on an equal footing may be a real alternative to the Dirac generalization of Hamiltonian dynamics. The example considered in Section 4 shows that the BRST charge can play the role of a sought generator in extended phase space. Nevertheless, the algorithm suggested by BFV for constructing the BRST charge again relies upon the algebra of constraints. Even for the model from Section 4 it would not lead to the correct result (4.30). Another way is to construct the BRST charge as a conserved quantity based on BRST-invariance of the action and making use of the first Noether theorem. This method works satisfactory for simple models with a given symmetry. Below we mentioned that the gravitational Lagrangian density can be modified for the primary constraints to take the simplest form $\pi^{\mu}=0$ without affecting the equation of motion. However, after this modification the full action may not be BRST-invariant. Some authors (see, for example, [12, 13]) use some boundary conditions to exclude total derivatives and ensure BRST-invariance. The boundary conditions (as a rule, these are trivial boundary conditions for ghosts and $\pi^{\mu}$) correspond to asymptotic states and are well-grounded in ordinary quantum field theory. This way does not seem to be general enough, and for gravitational field the justification of the boundary conditions, as well as the control of BRST- invariance of the action, requires special study. In [3] Dirac pointed out that “any dynamical theory must first be put in the Hamiltonian form before one can quantize it”. Based upon Hamiltonian dynamics in extended phase space, a new approach to quantum theory of gravity has been proposed in [14, 15]. Ir was argued that it is impossible to construct a mathematically consistent quantum theory of gravity without taking into account the role of gauge degrees of freedom in description of quantum gravitational phenomena from the point of view of different observers. The present paper show that even at the classical level gauge degrees of freedom cannot be excluded from consideration. As we have seen, the extension of phase space by introducing the missing velocities changes the relations between the “old” and “new” momenta (see (5.18)). As a consequence, the transformation (2.9) is canonical. In that way, we consider extended phase space not just as an auxiliary construction which enables one to compensate residual degrees of freedom and regularize a path integral, as it was in the Batalin – Fradkin – Vilkovisky approach [4, 5, 6], but rather as a structure that ensures equivalence of Hamiltonian dynamics for a constrained system and Lagrangian formulation of the original theory. ## Acknowledgements I would like to thank Giovanni Montani and Francesco Cianfrani for attracting my attention to the paper [11] and discussions. ## References * [1] P. A. M. Dirac, Can. J. Math. 2 (1950), P. 129–148. * [2] P. A. M. Dirac, Proc. Roy. Soc. A246 (1958), P. 326–332. * [3] P. A. M. Dirac, Proc. Roy. Soc. A246 (1958), P. 333–343. * [4] E. S. Fradkin and G. A. Vilkovisky, Phys. Lett B55 (1975), P. 224–226. * [5] I. A. Batalin and G. A. Vilkovisky, Phys. Lett B69 (1977), P. 309–312. * [6] E. S. Fradkin and T. E. Fradkina, Phys. Lett B72 (1978), P. 343–348. * [7] L. Castellani, Ann. Phys. 143 (1982), P. 357–371. * [8] R. Banerjee, H. J. Rothe and K. D. Rothe, Phys. Lett. B463 (1999), P. 248–251. * [9] R. Arnowitt, S. Deser and C. W. Misner, “The Dynamics of General Relativity”, in: Gravitation: an Introduction to Current Research, ed. by L. Witten, John Wiley & Sons, New York (1962), P. 227–265. * [10] L. D. Faddeev, Usp. Fiz. Nauk 136 (1982), P. 435–457 [Sov. Phys. Usp. 25 (1982). P. 130–142]. * [11] N. Kiriushcheva and S. V. Kuzmin, “The Hamiltonian formulation of General Relativity: myth and reality”, E-print arXiv: gr-qc/0809.0097. * [12] J. J. Halliwell, Phys. Rev. D38 (1988), P. 2468–2481. * [13] M. Hennaux, Phys. Rep. 126 (1985), P. 1–66. * [14] V. A. Savchenko, T. P. Shestakova and G. M. Vereshkov, Gravitation & Cosmology 7 (2001), P. 18–28. * [15] V. A. Savchenko, T. P. Shestakova and G. M. Vereshkov, Gravitation & Cosmology 7 (2001), P. 102–116.
arxiv-papers
2011-02-01T08:35:34
2024-09-04T02:49:16.763869
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "T. P. Shestakova", "submitter": "Tatyana P. Shestakova", "url": "https://arxiv.org/abs/1102.0097" }
1102.0098
eurm10 msam10 119–126 # Disentangle plume-induced anisotropy in the velocity field in buoyancy- driven turbulence Quan ZHOU1,2 and Ke-Qing XIA2 Email address for correspondence: qzhou@shu.edu.cnEmail address for correspondence: kxia@phy.cuhk.edu.hk 1Shanghai Key Laboratory of Mechanics in Energy and Environment Engineering, Shanghai Institute of Applied Mathematics and Mechanics, E-Institutes of Shanghai Universities, Shanghai University, Shanghai 200072, China 2Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China (1996; ?? and in revised form ??) ###### Abstract We present a method of disentangling the anisotropies produced by the cliff structures in turbulent velocity field and test it in the system of turbulent Rayleigh-Bénard (RB) convection. It is found that in the RB system the cliff structures in the velocity field are generated by thermal plumes. These cliff structures induce asymmetry in the velocity increments, which leads us to consider the plus and minus velocity structure functions (VSF). The plus velocity increments exclude cliff structures, while the minus ones include them. Our results show that the scaling exponents of the plus VSFs are in excellent agreement with those predicted for homogeneous and isotropic turbulence (HIT), whereas those of the minus VSFs exhibit significant deviations from HIT expectations in places where thermal plumes abound. These results demonstrate that plus and minus VSFs can be used to quantitatively study the effect of cliff structures in the velocity field and to effectively disentangle the associated anisotropies caused by these structures. ###### keywords: Cliff structures, structure functions, thermal plumes, turbulent thermal convection ††volume: 538 ## 1 Introduction A paradigm for studying turbulent flows is the so-called homogeneous and isotropic turbulence (HIT) Monin & Yaglom (1975). Studying this idealized model allows one to focus on the essential physics of small-scale turbulence in the simplest possible case and use it as a first step to understand more complicated turbulence problems. However, in almost all flow systems existing in nature, anisotropy is always present and unavoidable. How to disentangle the effects of anisotropy in the experimentally or numerically measured physical quantities has been a major focus in turbulence research in recent years and several methods, such as the SO(3) group decomposition, have been put forward to separate the isotropic and anisotropic contributions in turbulent flows (Arad _et al._ 1998, 1999; Grossmann, VON DER Heydt $\&$ Lohse 2001; Biferale _et al._ 2002). Buoyancy-driven turbulent flows occur widely in geophysical and astrophysical systems and in numerous engineering applications. In buoyancy-driven thermal turbulence, buoyant structures, such as thermal plumes, are predominant coherent structures that transport heat and drive the flow (Shang _et al._ 2003; Xia, Sun $\&$ Zhou 2003). Earlier visualization experiments (Moses, Zocchi $\&$ Libchaber 1993; Xi, Lam $\&$ Xia 2004) have shown that these structures consist of a cap with sharp temperature gradient and a stem that is relatively diffusive and hence would generate cliff-ramp-like structures in temperature time series when passing a thermal probe Belmonte & Libchaber (1996); Zhou & Xia (2002). It is well known that the so-called cliff-ramp structures would induce strong anisotropic effects. This has been widely studied in passive scalars Warhaft (2000), but the study on the effects of buoyancy-induced anisotropy is very limited. Here, we use turbulent Rayleigh- Bénard (RB) convection, a fluid layer heated from below and cooled on the top, as an example to study the anisotropic effects induced by buoyancy. In the past few decades, turbulent RB convection has become a model system for studying the phenomena and the generic physics associated with turbulent flows driven by buoyancy (Ahlers, Grossmann $\&$ Lohse 2009; Lohse $\&$ Xia 2010). In the field of turbulence, the velocity structure functions (VSF), namely $S_{p}(r)=\langle|\delta_{r}v|^{p}\rangle,$ (1) are usually used to characterize the turbulent kinetic energy cascades, and hence they are of prime importance and have been the central focus in the study of fluid turbulence (Sreenivasan $\&$ Antonia 1997; Ishihara, Gotoh $\&$ Kaneda 2009). Here, $\delta_{r}v=v(x+r)-v(x)$ is defined as the velocity increment over a separation $r$ and $\langle\cdots\rangle$ denotes an ensemble average. In particular, the situation for thermally-driven turbulence is more complicated. Bolgiano (1959) and Obukhov (1959), denoted hereafter as BO59 for short, have long argued that within the so-called inertial range and above a certain buoyant scale, i.e. the Bolgiano length scale $\ell_{B}$, buoyant forces drive the cascade processes and scale as $S_{p}(r)\sim r^{3p/5}$. However, experimentally whether the BO59 scaling exists in turbulent RB system remains unsettled (for a recent review, see Lohse & Xia, 2010). The maturing of spatial-velocity-field measuring techniques, e.g. the particle image velocimetry (PIV), has provided a great impetus for our understanding of such issues (Sun, Zhou $\&$ Xia 2006; Zhou, Sun $\&$ Xia 2008; Kunnen _et al._ 2008; Zhou $\&$ Xia 2010). In particular, the work of Sun et al. (2006) have shown that, at the center of the convection cell, the velocity field exhibits the same scaling behavior that one would find in HIT, whereas, near the cell sidewall, a “new scaling” was found. While results at both places imply no BO59, the observed scaling behavior near sidewall remains unexplained by any existing theoretical models. We note that in a closed RB cell, the spatial distribution of the buoyancy-driven thermal plumes is highly inhomogeneous, they abound near the sidewall but are scarcely found at the center Qiu & Tong (2001); Shang et al. (2003, 2004); Xi et al. (2004). These give us a clue that plumes may be responsible for the different scaling behavior observed at different places of the system and suggest that buoyancy plays an important role in the cascade process, but how this comes about is still missing. The objective of the present study is to address such a question, i.e., how buoyant forces influence the cascade properties in turbulent RB system? ## 2 Experimental setup and parameters The convection cell has been described in detail elsewhere Zhou & Xia (2010); Zhou et al. (2011) and here we give only its main features. It is a vertical cylinder of height $H=50$ cm and inner diameter $D=50$ cm and thus of unity aspect ratio. The top and bottom plates are made of 1.5 cm thick pure copper with nickel-plated fluid-contact surface and the sidewall is made of a plexiglas tube of 5 mm in wall thickness. Deionized and degassed water was used as the convecting fluid. A square-shaped jacket made of flat plexiglas plates and filled with water is fitted to the outside of the sidewall, which greatly reduced the distortion effect to the PIV images caused by the curvature of the cylindrical sidewall. Two series of measurements of the spatial velocity field were carried out using the PIV technique. In the first series, the measuring positions were fixed near the cell sidewall and the experiments covered the range $5.9\times 10^{9}\lesssim Ra\lesssim 1.1\times 10^{11}$ of the Rayleigh number $Ra=\alpha g\Delta TH^{3}/\nu\kappa$, with $g$ being the gravitational acceleration, $\Delta T$ the temperature difference across the fluid layer, and $\alpha$, $\nu$ and $\kappa$ being, respectively, the thermal expansion coefficient, the kinematic viscosity, and the thermal diffusivity of water, whereas in the second series, the measurements were made from near the cell sidewall to the cell center at fixed Rayleigh number $Ra=4.0\times 10^{10}$. For both series, the cell was tilted by a small angle of about 0.5∘ so that both series were made within the vertical plane of the large-scale circulation and at midheight of the cell. During the experiment the entire cell was wrapped by several layers of Styrofoam and the mean temperature of the convecting fluid was kept at $29^{\circ}$, corresponding to a Prandtl number $Pr=\nu/\kappa=5.5$. The details of the PIV measurement could be found elsewhere (Xia _et al._ 2003; Sun, Xia $\&$ Tong 2005), here we give only its main features. Hollow glass spheres of 10 $\mu$m in diameter were chosen as seed particles and the thickness of the laser lightsheet was $\sim 0.5$ mm. The spatial resolution of the measured velocity field is 0.59 mm, which is much smaller than the lower end of the inertial range and hence is sufficient to reveal the scaling properties in the inertial range. In each measurement, the measuring region has an area of $4.7\times 3.7$ cm2 (see the left panel of figure 1), corresponding to $79\times 63$ velocity vectors, and the experiment lasted 3 hours in which a total of 25000 two-dimensional vector maps were acquired with a sampling rate $\sim 2.3$ Hz. Since buoyant forces are exerted on the fluid in the vertical direction, we focus our attention mainly on the longitudinal vertical velocity increments $\delta_{r}w=w(x,z+r)-w(x,z)$, where $w(x,z)$ is the vertical velocity component obtained at position $(x,z)$. To acquire the accurate statistics, both temporal average and spatial average of moments of velocity increments within an area of $1.8\times 3.7$ cm2, i.e. $31\times 63$ velocity vectors, were used when calculating VSFs, as the flow is approximately locally homogeneous in turbulent RB system Sun et al. (2006); Zhou et al. (2008). Figure 1: Left: A snapshot of the instantaneous vertical velocity field $w(x,z)$ near the cell sidewall for $Ra=4.0\times 10^{10}$. Positive is defined for the upward motion and color coding is in cm/s. Right: Vertical slices of the vertical velocity component $w(x,z)$ through the left figure. The red curves mark the extracted cliff structures, corresponding to the regions of plume fronts. ## 3 Results and discussions ### 3.1 Cliff structures in the vertical velocity field The left panel of figure 1 shows a typical snapshot of the vertical velocity field $w(x,z)$ near the cell sidewall, where $x$ is the horizontal distance from the wall. The arrow in the figure indicates the region with large vertical velocity, which is typically caused by a group of hot plumes passing by. The plumes are generally believed to be detached thermal boundary layers by buoyant forces. As introduced in $\S$ 1, thermal plumes usually generate cliff structures in the temperature field. Here, the snapshot of $w(x,z)$ and the associated slices through $w(x,z)$ in the vertical direction (the right panel of figure 1) further show that in addition to temperature, plume fronts can also generate cliff structures in the vertical velocity field. The formation of cliff structures in the velocity field may be understood as follows: Under the action of buoyant forces thermal plumes possess a higher speed in the vertical direction in comparison to the background fluids, which would deform the plumes such that they are compressed in the vertical direction and stretched in the horizontal directions. This deformation shortens the distance between the (relatively) high speed plume front and the low-speed downstream fluids, therefore resulting a steep velocity gradient, i.e. a cliff structure. Figure 2: (_a_) PDFs of the normalized increments of vertical velocity components at $x/D=0.04$. From the inner to the outer PDF, $r/\eta=16.6$ (dark-green up-triangles), 33.3 (pink squares), 57.1 (blue down-triangles), and 107 (red circles), all within the inertial range. (_b_) Skewness of $\delta_{r}w$ as a function of $r/\eta$ for different measuring positions $x/D$ from near the cell sidewall to the cell center. The data were obtained at $Ra=4.0\times 10^{10}$. (_c_) Skewness of $\delta_{r}w$ as a function of $r/\eta$ for five different $Ra$ obtained near the cell sidewall ($x/D=0.04$). One can expect that the persistence of such cliff structures would violate the local isotropy of turbulent flows. Isotropy is the central hypothesis for most theories and models of small-scale turbulence. Although a number of investigations in the atmosphere and in laboratory flows have shown that the skewness of velocity derivative is nontrivial Sreenivasan & Antonia (1997), implying local anisotropy at small scales, the presence of cliffs would enhance the degree of this small-scale anisotropy. This can be reflected by the probability density functions (PDF) of velocity increments over different scales. Figure 2(_a_) shows PDFs of $\delta_{r}w$ normalized by the standard deviation of $w$, $\sigma_{w}=\sqrt{\langle(w-\langle w\rangle)^{2}\rangle}$, over several different length scales within the inertial range. The data were obtained near the cell sidewall ($x/D=0.04$) at $Ra=4.0\times 10^{10}$. The asymmetry of the distributions with long left tails is clear, especially for the (relatively) larger scales, which signifies the anisotropy at these scales. To quantify this anisotropy, we plot in figure 2(_b_) the skewness of $\delta_{r}w$ as a function of $r/\eta$ 111The Kolmogorov length scale $\eta$ is estimated from $\eta=(\nu^{3}/\varepsilon)^{1/4}$, where $\varepsilon=\sigma_{w}^{3}/L$ is the energy dissipation rate per unit mass and $L$ is the largest length scale of the turbulence. It is found that $\eta$ changes from 0.27 mm near the sidewall to 0.26 mm at cell center for $Ra=4.0\times 10^{10}$. for different measuring positions ranging from $x/D=0.04$ (near the sidewall) to $x/D=0.50$ (at cell center). It is seen that all skewness are negative, but their magnitudes decrease continuously when moving away from the wall and appear to remain invariant for $x/D\gtrsim 0.4$. If the nonvanishing skewness observed here is indeed induced by cliff structures of thermal plumes, the behaviors of the skewness may then be understood from the inhomogeneous spatial distributions of thermal plumes in the convection cell, i.e., plumes abound near the sidewall but are scarce in the central region Qiu & Tong (2001); Shang et al. (2003, 2004); Xi et al. (2004). Figure 2(_c_) shows the skewness of $\delta_{r}w$ as a function of $r/\eta$ for five different $Ra$ obtained near the cell sidewall ($x/D=0.04$). Again, all skewnesses are found to be negative. Another noticeable feature is that the magnitudes of the skewness decrease with increasing $Ra$. This may be understood as the cliff structures are being smoothed out by the increased turbulent fluctuations. As the cliff structures of the velocity field are generated by buoyancy, this result suggests that temperature becomes more passive when the convective flow becomes more turbulent. Note that previous experimental study has shown that temperature may become progressively passive for $Ra>10^{10}$ Belmonte & Libchaber (1996); Zhou & Xia (2002), which is qualitatively consistent with the picture obtained in the present study. ### 3.2 Plus and minus velocity structure functions Figure 3: (_a_) Compensated third-order VSFs $S_{3}^{+}(r)/r$, $S_{3}(r)/r$, and $S_{3}^{-}(r)/r$ measured near the cell sidewall. The arrow marks $r=1.5\ell_{B}$ for reference. (_b_) Comparison of VSF exponents $\zeta_{p}^{+}$, $\zeta_{p}$, and $\zeta_{p}^{-}$ with various model predictions. The data were measured at $x/D=0.04$ for $Ra=4.0\times 10^{10}$. (_c_) The third-order VSF exponents $\zeta_{3}^{+}$, $\zeta_{3}$, and $\zeta_{3}^{-}$ as a function of the normalized distance from the cell sidewall $x/D$ for $Ra=4.0\times 10^{10}$. (_d_) $Ra$-dependence of $\zeta_{3}^{+}$, $\zeta_{3}$, and $\zeta_{3}^{-}$ obtained near the cell sidewall ($x/D=0.04$). The fact that plumes can induce anisotropy suggests that the cascades of velocity field would possess different dynamics in different directions, i.e., the positive and negative velocity increments may have different scaling behaviors. To study this quantitatively, we examine the statistical properties of the plus and minus longitudinal VSFs Vainshtein & Sreenivasan (1994); Sreenivasan et al. (1996), defined as $S_{p}^{\pm}(r)=\langle[(|\delta_{r}w|\pm\delta_{r}w)/2]^{p}\rangle.$ (2) From this definition, it is clear that cliff structures such as those shown in the right panel of figure 1) will be excluded from $S_{p}^{+}(r)$ but included in $S_{p}^{-}(r)$. This enables one to study separately the contributions of positive and negative velocity increments to the corresponding structure functions. A similar analysis has been performed previously in a turbulent channel flow and it was found that the plus VSFs are less affected by the presence of the wall Onorato & Iuso (2001). Figure 3(_a_) plots in log-log scale the compensated third-order VSFs $S_{3}^{-}(r)/r$ (triangles), $S_{3}(r)/r$ (circles), and $S_{3}^{+}(r)/r$ (squares) vs $r/\eta$ near the cell sidewall, which exhibit slightly different scaling ranges. The compensated plot shows a flat range for $S_{3}^{+}(r)/r$, i.e. $S_{3}^{+}(r)\sim r$, suggesting that $S_{3}^{+}(r)$ possesses the same scaling behavior as that for HIT, whereas both $S_{3}(r)$ and $S_{3}^{-}(r)$ exhibit much steeper scalings. To reveal this more clearly, we show in figure 3(_b_) the measured scaling exponents $\zeta_{p}^{+}$, $\zeta_{p}$, and $\zeta_{p}^{-}$ of the VSFs of orders $p=1$ to 8. When comparing $\zeta_{p}^{+}$ with the predictions of the hierarchy models of She & Leveque (1994) (SL94) for HIT, we find excellent agreement. These results suggest that the scaling behaviors of $S_{p}^{+}(r)$ are consistent with what one would expect for HIT. Moreover, because cliff structures contribute mainly to the negative velocity increments, $\zeta_{p}^{-}$ should exhibit some deviation from the K41-type scaling. This is indeed observed. Figure 3(_b_) shows that $\zeta_{p}^{-}$ and $\zeta_{p}$ are both much larger than the predictions of SL94. Therefore, the plus and minus structures functions are effective means to study the effects of cliff structures in the velocity field and can be used to effectively disentangle the associated anisotropies caused by these structures. These cliff structures are produced by thermal plumes in the present case, but in general they can be produced by coherent structures in other types of flows. Let’s move on now to the location- and $Ra$-dependencies of the VSF scaling exponents. We focus mainly on the third-order because $\zeta_{3}=1$ is an exact result for HIT. Figure 3(_c_) shows the exponents $\zeta_{3}^{+}$, $\zeta_{3}$, and $\zeta_{3}^{-}$ as a function of the measuring position $x/D$. It is seen that both $\zeta_{3}$ and $\zeta_{3}^{-}$ are much larger than the K41-value of 1 near the cell sidewall and drop from the wall. For $x/D>0.3$ $\zeta_{3}$ and $\zeta_{3}^{-}$ are both essentially 1. On the other hand, apart from some data scatter, $\zeta_{3}^{+}$ assumes essentially the K41 value for all $x/D$. The decreases of $\zeta_{3}$ and $\zeta_{3}^{-}$ correspond to the reduced anisotropy associated to the cliff structures, i.e., the number of plumes decreases as the measuring position moves away from the cell sidewall, confirming quantitatively the results shown in figure 2(_b_). For $x/D>0.3$, i.e. the cell’s central region, thermal plumes are scarce and hence it is not surprising to obtain approximately the K41 scaling for all three VSFs. Figure 3(_d_) shows the measured scaling exponents versus $Ra$. Again, $\zeta_{3}^{+}$ is seen to remain nearly constant around the value of 1, but $\zeta_{3}$ and $\zeta_{3}^{-}$, despite certain scatter, show an overall decreasing trend with increasing $Ra$. Here, the decrease of $\zeta_{3}^{-}$ implies that the influence of buoyancy on the cascade processes becomes weaker when the flow becomes more turbulent, which could also be reflected by the behavior of the skewness of $\delta_{r}w$, whose magnitude is found to increase with decreasing $Ra$ as shown in figure 2(_c_). Figure 4: Comparison of ESS VSF exponents $\zeta_{p,3}^{+}$, $\zeta_{p,3}^{-}$, and the SL94 scaling exponents. It should be noted that near the sidewall the result $\zeta_{3}^{-}\gtrsim\zeta_{3}^{+}$, shown in figure 3, does not imply that the positive velocity increments are more intermittent than the negative ones. To illustrate this, we examine the scaling behavior of VSFs via the extended self-similarity (ESS) method Benzi et al. (1993), i.e., $S_{p}(r)$ is plotted against $S_{3}(r)$, instead of $r$, in a log-log scale. Figure 4 shows the measured ESS (relative) scaling exponents $\zeta_{p,3}^{+}$ and $\zeta_{p,3}^{-}$ for the plus and minus VSFs, respectively. One sees that $\zeta_{p,3}^{-}$ are slightly smaller than $\zeta_{p,3}^{+}$. This result suggests that it is the minus velocity increments that possesses a higher degree of intermittency, which should be attributed to the persistence of cliff structures. A detailed analysis of such cliff structures would therefore be helpful for understanding the present results. To do this, we use a criterion to identify cliff structures in the vertical slices of $w(x,z)$ which is similar to those used for passive and active scalars Moisy et al. (2001); Zhou & Xia (2002): a cliff is identified when $-\partial w/\partial z>\sigma_{w}/\ell_{B}$, where $\ell_{B}$ is based on the global quantities Sun et al. (2006). When such a cliff is found, we define its position $z_{0}$ as the point maximizing $|\partial w/\partial z|$, and its width $\lambda_{C}$ as the separation in space between the two extrema of $w$ surrounding $z_{0}$. Applying this procedure, over 50 000 cliffs were identified near the sidewall ($x/D=0.04$) for each $Ra$. Three examples of the extracted cliff structures are shown as red curves in the right panel of figure 1. Figure 5 shows the mean cliffs’ width $\langle\lambda_{C}\rangle$, normalized by $\ell_{B}$. One sees that $\langle\lambda_{C}\rangle=(1.5\pm 0.2)\ell_{B}$ is nearly independent of $Ra$. We further note that the cliff structures mainly occur at the scales near the upper end of the VSF scaling range [see figure 3(_a_)]. This could also be reflected qualitatively from figure 2(_a_), which shows that the distributions of $\delta_{r}w$ over large scales seem to be more asymmetric than those over relatively small scales, and quantitatively from figure 2(_b_), from which one sees that within the inertial range ($16\lesssim r/\eta\lesssim 110$) the magnitude of the skewness of $\delta_{r}w$ increases with the scale $r$ for the near wall data ($x/D=0.04$, dark-green up- triangles). Cliff structures contain large velocity variations and would enhance the magnitude of velocity increments over the scales around their mean width $\langle\lambda_{C}\rangle$, i.e. the scale near the upper end of the VSF scaling range. This can also be seen from figure 3(_a_) that $S_{p}^{+}$ and $S_{p}^{-}$ differ the most around $1.5\ell_{B}$, signifying that scale is representative of the typical size of a cliff structure. The cliffs’ influences to the scales near the lower end of the scaling range, however, are much weaker [see figure 3(_a_)]. Therefore, under the influences of cliff structures, the value of $S_{p}^{-}(r)$ near the upper end of the VSF scaling range would increase, while those near the lower end do not change significant. As a result, the measured scaling exponents of $S_{p}^{-}(r)$ would increase. With decreasing buoyancy effects, the scaling exponents are also expected to drop, which is indeed observed in figure 3(_d_). Figure 5: Mean cliffs’ width $\langle\lambda_{C}\rangle$, normalized by the Bolgiano length scale $\ell_{B}$, as a function of $Ra$ obtained near the cell sidewall ($x/D=0.04$). The error bars mark the standard deviation of $\lambda_{C}$. ## 4 Conclusion To summarize, we have demonstrated that as a manifestation of buoyancy effects thermal plumes can generate cliff structures in the vertical velocity field, which would in turn generate asymmetry in the velocity increments. We further show that such effects can be quantified by examining respectively the plus and minus velocity increments. For the plus increments, which largely exclude the cliff structures and hence removing the buoyancy effects, the scaling of their moments are the same as those expected of homogeneous and isotropic turbulence (HIT). For the minus increments, the scaling is found to be much steeper than that for HIT, which is caused by the presence of cliff structures. Such effects of buoyant forces are found to vanish gradually when moving away from the cell sidewall, owing to the inhomogeneous distributions of thermal plumes in a closed convection cell. It is also shown that, as a result of cliff structures being smoothed out by the increased turbulent fluctuations, the effect of buoyancy on the velocity field decreases with increasing $Ra$. Despite its simpleness, the analysis presented here provides a useful method to quantitatively study the effect of buoyancy and disentangle its contribution in the velocity field in buoyancy-driven turbulence. More generally, it may be used to quantify the effect of anisotropy in other complex phenomena when sharp fronts exist in the related random fields. ###### Acknowledgements. This work was supported in part by Natural Science Foundation of China (No. 11002085), “Pu Jiang” project of Shanghai (No. 10PJ1404000), “Cheng Guang” project of Shanghai (No. 09CG41), and E-Institutes of Shanghai Municipal Education Commission (Q.Z.) and by the Research Grants Council of Hong Kong SAR (Grant No. CUHK403806 and No. CUHK403807) (K.Q.X.). ## References * Ahlers et al. (2009) Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys. 81, 503–537. * Arad et al. (1999) Arad, I., Biferale, L., Mazzitelli, I. & Procaccia, I. 1999 Disentangling scaling properties in anisotropic and inhomogeneous turbulence. Phys. Rev. Lett. 82, 5040–43. * Arad et al. (1998) Arad, I., Dhruva, B., Kurien, S., L’vov, V. S., Procaccia, I. & Sreenivasan, K. R. 1998 Extraction of anisotropic contributions in turbulent flows. Phys. Rev. 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arxiv-papers
2011-02-01T08:41:52
2024-09-04T02:49:16.770152
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Quan Zhou and Ke-Qing Xia", "submitter": "Quan Zhou", "url": "https://arxiv.org/abs/1102.0098" }
1102.0099
Automatic Network Fingerprinting Through Single-Node Motifs Christoph Echtermeyer1, Luciano da Fontoura Costa2, Francisco A. Rodrigues3, Marcus Kaiser1,4,5,∗ 1 School of Computing Science, Claremont Tower, Newcastle University, Newcastle-upon-Tyne NE1 7RU, UK 2 Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, PO Box 369, 13560-970 São Carlos, São Paulo, Brazil 3 Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, PO Box 668, 13560-970 São Carlos, São Paulo, Brazil 4 Institute of Neuroscience, The Medical School, Framlington Place, Newcastle University, Newcastle-upon-Tyne NE2 4HH, UK 5 Department of Brain and Cognitive Sciences, Seoul National University, Seoul 151-746, Korea $\ast$ E-mail: m.kaiser@newcastle.ac.uk ## Abstract Complex networks have been characterised by their specific connectivity patterns (network motifs), but their building blocks can also be identified and described by node-motifs—a combination of local network features. One technique to identify single node-motifs has been presented by Costa et al. (L. D. F. Costa, F. A. Rodrigues, C. C. Hilgetag, and M. Kaiser, Europhys. Lett., 87, 1, 2009). Here, we first suggest improvements to the method including how its parameters can be determined automatically. Such automatic routines make high-throughput studies of many networks feasible. Second, the new routines are validated in different network-series. Third, we provide an example of how the method can be used to analyse network time-series. In conclusion, we provide a robust method for systematically discovering and classifying characteristic nodes of a network. In contrast to classical motif analysis, our approach can identify individual components (here: nodes) that are specific to a network. Such special nodes, as hubs before, might be found to play critical roles in real-world networks. ## Introduction Networks appear in a variety of real-world systems ranging from biology to engineering [1, 2]. Examples include neural [3, 4, 5], social [6, 7, 8], and computer networks [9, 10] to name but a few. Networks have been used to study the emergence of cooperative behaviour [11, 12, 13]; to address epidemiological questions [14, 15] especially in scale free networks [16, 17]; and to investigate the causes of cascade effects [18, 19] for a more complete understanding of why networks differ in robustness against error and attack [20, 21]. Attempts to classify network-topologies [22] were accompanied by detailed studies of scale-free [23] and small-world networks [24, 25]—properties that were identified in many real networks. Additional to investigations of concrete structures, theoretical studies of random networks collected valuable information about large classes of networks [26, 27, 28]. Mapping complex systems to networks revealed that some nodes are remarkably different from other nodes of the same network. For instance, hubs, characterized by a high number of connections (a high node degree), often play a fundamental role in protein-protein interaction networks and their removal can be lethal for an organism [29, 30]. Hubs are similarly important for socio-economic systems, where defective hubs can cause cooperation to decline [31]. Also, in engineered systems like the Internet, hubs are important to maintain the communication between autonomous systems [20]. These outlier nodes have been identified since the introduction of complex network theory, e.g. in the World Wide Web [9] and the Internet [10], but hubs are outliers only in terms of their degree; other network properties can also define special nodes. For instance, Internet topology has been shown decompose onion- like into different shells around a relatively small core network [32]. The closer a node’s layer is to the core, the higher is the node’s shell-index (coreness) [33]. Nodes with high coreness are not necessarily hubs, which one might suspect to be the most efficient spreaders of information. Instead, the position of a node close to the network-core has more impact on successful dissemination than having a high degree [34]. In networks where hubs are not present, as in most geographical networks, nodes whose neighbours are also connected to each other are special (high local clustering coefficient). Further examples of outlier nodes can be found with different measures some of which examine more than the direct neighbourhood of a node [26, 35], such that they specify rather global (network specific) than local (node specific) characteristics. Global measures, such as characteristic path length or clustering coefficient [24], summarise the whole network in a single value. Local measurements, on the other hand, analyse each node or edge individually, yielding a more fine-grained picture of the network. Nodes that express common features and outliers that are different can be identified with pattern recognition approaches, which group nodes of similar characteristics. Corresponding techniques have been proposed recently [36, 37, 38] and revealed important network properties. For example, in protein-protein interaction networks the relative number of outliers tends to decrease with the complexity of organism, i.e. proteins in more complex species show higher homogeneity in their interplay [38]. This demonstrates that, by considering multiple node- features jointly, pattern recognition based methods can point out exceptional network components. Networks can describe complex systems whose interactivity between dynamical components changes over time. Altered connections between the elements (represented by nodes) may in turn feed back on the dynamics, such that the dynamical process and the network topology evolve in an adaptive fashion [39]. In the context of game theory, corresponding coevolution of behaviour and connectivity has been studied in socio-economic systems [12, 13]. In complex scenarios like these, analysing a single network is often insufficient and several networks must be compared to gain insights. Further examples for the need of network-comparisons are families of protein-protein interaction networks, brain connectivity networks in patient- and control-populations, or time-dependent (developing or declining) networks [40]. Comparing such sets of networks requires consistent approaches, which are often non-trivial, because networks differ in size (number of nodes or edges) or they comprise a disjoint sets of nodes (some nodes occur in one network but not in others). Direct comparisons between structures may thus be ruled out. Based on outlier- detection as described above, we previously proposed motif-regions for which the relative frequencies of outliers falling into one of them yields a network specific fingerprint [41]. Relating different networks to each other has thereby become as easy as comparing bar-graphs. Nevertheless, although this methodology has been demonstrated to be suitable and accurate for outlier identification as well as for network comparisons, it suffers from several limitations, which we address in this paper. Here, we describe a novel workflow for detecting characteristic single-node motifs and for using fingerprints for network comparison. Improvements compared to the previous approach include (a) automatic parameter determination, which facilitates high throughput analysis without user interaction, and (b) replacing the k-means clustering algorithm with a deterministic method to simplify the workflow and to improve robustness of results. In addition, we provide (c) a validation of our method and (d) an application to networks where the topology changes over time (addition or deletion of nodes or edges). ### Previous work The application of single measures to complex networks has revealed important insights in many cases. However, as Newman and Leicht recognised [36], detecting exceptions is limited to network features that are quantified by the measures in use. Otherwise, if the chosen characteristics do not reflect the properties that are specific for a network or its components, important features remain unnoticed. To solve this problem, two complementary approaches have been suggested. The first approach by Newman and Leicht groups nodes based on their connectivity without any further prior information [36]. By fitting the parameters of a mixture model (using an expectation-maximization algorithm), each node is assigned a probability of belonging to any one group that has been identified. The probabilistic nature of this approach has the advantage that nodes that can not be unambiguously categorised are not crudely assigned to one particular group, but the conflict becomes evident, such that it can be dealt with. The structure of networks can thereby be investigated without requiring any other parameter than the number of groups that are to be created. This elegant method has been examined thoroughly and improvements to it have been suggested [42, 37]. Analyses with focus on only one particular aspect of a network at a time might fail to detect irregularities or similarities in structure. The second approach is to avoid single measures and to use a combination of multiple ones [41]. Instead of reducing network components down to one dimension, joint measures map it into a multi-dimensional feature space [43]; each vector-point in that space corresponds to a combination of node-characteristics and statistical methods are used to identify motif regions, such that each vertex falls into one of them: A node is either classified regular—showing features like the majority of nodes—or singular, i.e. its features deviate by following a particular single node-motif. The term motif refers to the concept of network motifs, i.e. patterns incorporating multiple nodes [44]. Each of these two approaches to identify patterns in complex networks has its drawbacks and advantages. The Newman and Leicht algorithm (NLA) does not depend on one or few network measures, but it works on network links directly. Networks are not restricted to undirected ones, but directed links and even weighted ones can be considered. The NLA requires the number of node-groups to be specified; this is also true for the approach by Costa et al. [Beyond the Average (BtA)], where the number of motif regions needs to be chosen a priori [41]. Unfortunately, for real-world networks this number is often unknown. The BtA-workflow requires two additional parameters to control which nodes will constitute individual motif regions. Both methods differ in their output, as BtA not only provides a grouping of nodes, but also a network-fingerprint, which can be used to compare networks from different domains. Most importantly, however, is the conceptional distinction between NLA and BtA, as they rely on local edge connectivity and local node measures, respectively. BtA will fail to pinpoint features of the network, if the chosen set of measures can not formulate a corresponding motif. Similarly NLA can fail, as it only takes into account direct connections between nodes: NLA does not consider how the neighbours of a node are connected, for example, but BtA can deal with such information (by evaluating the local clustering coefficient). Indeed, the extensibility concerning features to assess is the biggest advantage of BtA; (un-)directed and weighted links can be processed likewise and in spatial networks the location of nodes can be taken into account. In conclusion, NLA is readily applicable to a broad variety of network domains; however, considering direct connections only is a weakness. BtA can be nicely adopted to these cases, but care has to be taken at all times to ensure the set of measures is diverse enough to cover as many patterns that might occur in networks as possible. In the next section we suggest several improvements to the BtA-workflow (Fig. 1), which can be sketched as follows: Initially, multiple local network measures are applied to each node, which yields a multi-dimensional characterisation in form of a feature vector. Correlation between different measures is accounted for by principal component analysis (PCA), which is used to map feature vectors of all nodes to two dimensional space [45, Chapter 8]. Next, nodes are assigned probabilities in order to distinguish nodes with common and rare features. The required probability density function (PDF) is gained by smoothing over points in the two dimensional PCA-plane (Parzen window approach [46, 47, Chapter 4.3]). Now, the least probable nodes, i.e those with uncommon features, can be identified from the PDF. These singular nodes are then clustered in order to distinguish different motif-groups. Each of the two dimensional motif-groups corresponds to a higher dimensional motif- region into which the feature vectors split up and the distribution of feature vectors among the different motif-regions is the fingerprint of the network. Apart from the initial decision on which measures to use, the user needs to choose the bandwidth of the smoothing kernel, the number of singular nodes $w$, and the number of motif-groups $k$, respectively (steps 3, 4, and 5 in Fig. 1). Additionally, when comparing multiple networks, a limit must be specified below which motif regions are considered too close to each other to constitute different motifs (join threshold; Step 7). So far, these settings had to be chosen manually, but here we suggest how to determine all three parameters (bandwidth, $w$, and $k$) automatically. The last setting (join threshold), however, is not considered for automation: So far we could not identify a procedure that yields results as good as manual selection by the user. We thus concentrated our efforts on the parameters that need to be set for every network (bandwidth, $w$, and $k$), such that high-throughput applications become possible. Automating the setting of the main parameters is thus of higher benefit than for the threshold that determines Voronoi cells to be joined; this needs to be chosen only once, when all networks are compared to each other at the end. ## Results In this paper we propose how to choose all relevant parameters of the BtA- workflow automatically (see Methods section), which allows for the analysis of many networks without the need for human interaction. The only remaining limiting factor for high throughput analyses are the computational costs of the analysis, which predominantly depend on the measures that are chosen to characterise each node. Using implementations of common local measures (see supporting information), the estimated run-time scales linearly to cubic with network size (Fig. S1). Costs are thus comparatively cheap considering methods that identify specific connectivity patterns by counting occurrences of particular sub-graphs (e.g. [48, 49, 44, 50, 51, 52]); such motif-counts also scale at least linearly in network size, but they show exponentially growing costs as the size of the motif-pattern increases [50]. In practice this often means that counts can not be determined for patterns involving 10 nodes or more [53], which renders some domains computationally intractable for this approach, but eventually not for BtA. However, before processing huge networks or many different structures with BtA, we first need to verify that parameters are indeed chosen adequately, which is confirmed in the next section. ### Method Verification The first validation is on a network that is small enough to confirm BtA- results by eye: We use a family-tree from The Simpsons [54] to create a network with nodes representing characters and directed links pointing to their offspring (Fig. 2a). Nodes that have a sparsely connected and homogeneous neighbourhood are suitably highlighted as outliers by BtA. With these reassuring results from a single network, we proceed by testing BtA on whole series: We generate structures with both regular components and exceptional ones, which BtA has to identify. In our first series we compose networks of two components: a regular ring lattice and a smaller Erdős-Rényi (ER) [55] random network (Fig. 2b). While the ring lattice remains unchanged, the size of the random module increases throughout the series, such that its proportion of the full network grows gradually111 The ring lattice is comprised of 100 nodes, each of which is connected to its four closest neighbours (Fig. 2b). ER-random networks ($n=1,\ldots,50$ nodes) have an average edge-density of 25%. . Composed networks are analysed with BtA: Of all outlier-nodes less than 2% are missed while over 96% are classified correctly, if the random component contributes less than 25% of nodes to the network. Beyond that limit, the number of nodes in the random-part does no longer match the number of identified outliers $w$. But this does not imply a mis- classification by BtA: The larger a random network, the more likely it is that a few nodes are connected regularly (or close to that). Quantifying these nodes with local network measures yields the same values as (or similar to) those of the ring lattice, which is why it would be incorrect to consider them singular. Additional to regular connection patterns in large random networks, other local motifs can also be frequent enough, such that they constitute a common rather than an exceptional feature of the network. Thus, network components that seem clearly separable at first may actually be very similar or—although intended to form outliers—they may contain common elements, due to random effects. Together this explains the observed deviations in numbers of outliers for growing ER-components in this test-series. Finally, we reverse the nature of the networks: The major component is set to a random network [ER, Barabási and Albert (BA) [56], or Watts-Strogatz (WS) [24] model] in which we embed a small, but highly regular structure (Fig. 2c). The inserted structure was chosen, such that its nodes are highly clustered (both on level 1 and 2); the six outer nodes further show significant variability in their neighbours’ degrees. These characteristics are rarely observed in our random networks, which is why BtA should identify these nodes (alongside with other outliers that might emerge). We confirm this in a series of networks with varying sparseness222Random networks ($n=100$ nodes) are generated according to the ER, BA, and WS model; edge-density is gradually increased from 1% to 50% (step-size 1%). The regular structure (7 nodes) illustrated in Fig. 2c is added to each random network before BtA is applied. : The 6 outer nodes of the regular structure are classified singular in over 97% of all networks. Additionally, the inner node (with less extreme features) is regarded uncommon in 81% of all cases. In conclusion, the automatic parameter determination gives very satisfying results, which yield confidence in BtA’s ability to identify outliers in complex networks autonomously. ### Network Time-Series: A Small-World Emerging Large complex networks are challenging to analyse; time-series of such are even more so. We attempt to approach this challenge by first condensing networks to a compact representation—mapping a series of changing structures to a uniform representation benefits the identification of trends and changes of such. Therefore, all networks have to be characterised, which we do using single node-motifs. These are identified with BtA using six common local measures: (1) the normalised average degree $r$, (2) the coefficient of variation of the degrees of the immediate neighbours of a node $cv$, (3) the clustering coefficient $cc$ [24, 57], (4) the locality index $loc$, (5) the hierarchical clustering coefficient of level two $cc_{2}$ [58], and (6) the normalised node degree $K$. (For definitions of these measures see Methods section.) Next, we describe the time-series of 600 networks and the results found with BtA. Similar to random graphs, small-world networks have a small characteristic path length, but at the same time they exhibit a high degree of clustering, as regular ring lattices, for example. It has been discovered early that the combination of short paths plus grouping is inherent to social networks; a phenomenon that became known as six degrees of separation [59, 60, 61]. Today it is known that small-world networks can be found in many other domains (e.g. [26, 27, 2, 28]). We thus created a network-time series in which structures gradually change from a completely regular ring lattice to a small-world network (see Methods section, Fig. S2). In total we identified 5 single node-motifs, which differ in characteristics, frequency, and time of emergence (Fig. 3): A node according to motif 1 has relatively few connections in contrast to its well connected neighbourhood. Different from that, nodes corresponding to motif 2 are signified by many connections to a rather sparsely connected neighbourhood. Motif 3-nodes have relatively few connections and nodes in their neighbourhood are similar in number of links and corresponding targets. Motif 4 describes rarely connected nodes whose neighbours have a diverse number of connections; but instead of being linked between each other, neighbours share other common targets. The final motif 5 can be best characterised by its relation to the rest of the network, which shows a higher degree of connectivity than any node involved in the motif. Neighbours of the motif-node further vary in their number of connections and do not link to each other. Motifs 2, 3 and 5 appear right from the beginning of the rewiring process; motifs 2 and 5 gradually become more common over time, whereas 3 levels out after a transient peak. The remaining motifs 1 and especially 4 only become apparent at later stages towards which both become more frequent. Together, BtA reveals the increasing irregularity in network structure and it also provides details on the characteristic connectivity patterns at different times. Both would be valuable information if real networks were analysed; here, with precise knowledge about the network-changing process, the temporally dependent motif expression levels yield another validation of the technique (detailed discussion in supporting information). Overall, results are very satisfying and we are confident that BtA could be successfully applied to real networks using the automatic parameter determination. ## Discussion In this paper we presented a method to detect single node-motifs automatically. The main parameters of the previous routine [41]—the smoothing kernel bandwidth plus the number of singular nodes and motif groups—are now selected based on the data. We further proposed a deterministic replacement for the k-means algorithm, which is used to form the different motif-groups. In contrast to k-means, our alternative approach can determine the number of motifs itself and due to the lack of random elements, clustering results are robust over multiple repetitions. Despite our improvements to BtA certain issues and room for further advancements remain. For example, reducing feature vectors in dimension inevitably leads to a loss of information, but which has to be kept withing reasonable bounds. In other words, although 6-dimensional feature vectors were suitably represented in the 2-dimensional plane so far [41], different networks may require the use of more than just the first 2 principal components in order to ensure that network characteristics are represented properly. Thus, if the chosen number of principal components does not account for at least 80% of the variance, their number should be increased (Kaiser’s rule). The degree to which feature vectors can be reduced thereby depends on the correlation between measured values, which is specific to the analysed network. In cases where feature vectors can not be suitably represented in 2 dimensions, their display becomes more complicated and verifying a good fit of the estimated probability density function (PDF) is challenging. However, a good PDF estimate is needed in the BtA workflow to determine outlier nodes. Problems that might arise in these situations could possibly be circumvented by a major change to the workflow: The use of PCA to compact information offers the possibility to replace both the PDF estimation and the subsequent outlier selection with a more direct and non-parametric standard technique, which is Hotelling’s $T^{2}$ (a generalisation of Student’s t-statistic). This modification would allow to identify outliers without the need to estimate a PDF, but the exploration of the resulting workflow will be addressed in another publication. Considering the BtA workflow as presented in this paper, the technique can be easily adapted by including different local network measures in the analysis. Measures that take spacial aspects of the network into account, for instance, or those including link-weights can increase quality of the analysis. Finally, interest might not only lie on motifs formed by outlier nodes, but on all single node-motifs occurring in the network. In this case regular and singular nodes are not distinguished, but all of them have to be included in the network fingerprint. BtA-fingerprinting of many networks has so far been prevented by the need to choose parameters during the analysis manually. With the improvements presented in this paper, however, it is now possible to process large numbers of networks fully unsupervised. Identified outliers are characteristic nodes that can provide a fingerprint of a network; fingerprinting networks from numerous domains allows easy characterisation and comparisons. As already demonstrated [41], such studies can reveal important characteristics and differences between network domains. Additionally, the example on an emerging small-world network in this paper showed that BtA can also be used to analyse time-series of networks. To encourage the use of the BtA methodology by other researchers, we provide our implementation of the workflow including the automatic parameter determination for download (http://www.biological-networks.org/). Two versions of the code exist: The first one requires Matlab (Mathworks Inc, Natick, USA) and allows the user to apply the workflow using a graphical user interface (Fig. S3). The other one is a command line utility that either requires Matlab or the free alternative Octave [62] and it can be easily used to batch process many networks. In conclusion, we provide a robust method for systematically discovering and classifying characteristic nodes of a network. The distribution of node- classes results in a fingerprint, which in turn can give a classification of whole networks, as for network motifs of multiple nodes [63]. In contrast to classical motif analysis, our approach can identify the individual components that are specific to a network. Such special nodes, as hubs before, might be found to play critical roles in real-world networks. ## Methods #### Local Network Measures Network nodes were characterised with six common local measures whose definitions are given in the following. Therefore, let $A=(a_{ij})$ denote the adjacency matrix of the network, i.e. $a_{ij}=1$, if a link from node $i$ to node $j$ exists, and otherwise $a_{ij}=0$. Row- and column-sums of $A$ correspond to the in- and out-degrees of nodes, respectively. In undirected networks, in- and out-degree are equal and either of them can be used as a node’s degree. If links are directed, the degree is the sum of in- and out- degree. Dividing a node’s degree by the number of all links in the network yields the normalised node degree $K$. The normalised average degree $r_{i}$ of a node $i$ is the average over all its neighbours’ degrees. (Nodes that are directly linked to node $i$ are called neighbours.) Likewise, the coefficient of variation $cv$ of the degrees of the immediate neighbours of a node can be calculated. The neighbours’ connectivity with each other is quantified by the clustering coefficient $cc_{i}$, which is the proportion of existing connections between node $i$’s neighbours to the number of all possible links between them [24, 57]. The clustering coefficient thus reflects the relative number of triangle-shaped paths a node has—a concept that is extended to connections between neighbours’ neighbours (further away node node $i$) by the hierarchical clustering coefficient of level two $cc_{2}$ [58]. Whereas the cluster coefficients quantify connectivity within a node’s neighbourhood, the locality index $loc_{i}$, which is based on the matching index (e.g. [64]), is the fraction of neighbours’ links that connect to the same node (not necessarily a neighbour of node $i$). Further details and measures can be found in the literature [26, 27, 28, 35]. In the following sections we describe how appropriate settings for the parameters of the BtA-workflow can be found automatically. Kernel-bandwidth, the number of singular nodes $w$, and the number of motif regions $k$ are discussed separately below. #### Kernel-Bandwidth In step 3 of the workflow (Fig. 1), the Parzen window approach is used to estimate a probability density function (PDF) over all nodes [46, 47, Chapter 4.3]. This is achieved by smoothing the overall arrangement of reduced feature vectors, which were obtained using principal component analysis (PCA) [45, Chapter 8] in the previous step 2. The dimensions of the smoothing kernel, i.e. the width and breadth of the Gaussian function $\mathcal{N}_{2}(\mu,\,\Sigma)$ can be controlled through its covariance matrix $\Sigma=(\sigma_{ij})$. (Mean vectors $\mu$ are fixed to equal the data-points.) The original publication made use of the fact that the absolute covariance values ($\forall k:\,\sigma_{kk}=0.05$) do not matter for the estimated PDF. However, their values relative to each other do matter and we therefore scale them according to the standard deviation along each principal component (PC) axis. Variability-based re-shaping of the kernel function improves the overall fit of the PDF to the points (Fig. S4). A further refinement would be to tilt the Gaussian in order to account for correlation between axes (Fig. S5); however, the PCs are expected to show weak correlation only, which is why we chose un-tilted kernels (for which the covariance matrix $\Sigma$ is zero except for the variances on the diagonal). #### Number of Singular Nodes $w$ After assigning probabilities to all nodes (Step 3), nodes with an exceptionally low probability come into focus: These outliers correspond to points in the PCA-plane that are spatially separated from larger clusters; and this separation corresponds to abnormalities of measured features. Due to their uncommon characteristics, these nodes are considered singular. For humans it is usually straightforward to identify these non-regular nodes, if interactive visual aids are provided; we therefore implemented a graphical user interface for the whole workflow (Fig. S3). In the following, however, we discuss how the number of singular nodes $w$ can be adjusted without interaction. To determine singular nodes, automated methods can query the PDF that has been estimated earlier (Step 3). For example, for a fixed number $w$ of singular nodes, the $w$ least probable ones can be selected easily. Alternatively, a probability cut-off can be set, e.g. at 1% or 5%, to separate nodes into regular and singular ones. Both these simple methods involve constants, but which have to be chosen depending on network size to yield sensible results333Choosing one fixed number of singular nodes $w$ for differently sized networks can render the majority of nodes non-regular in comparatively small networks; vice versa, $w$ may be too small compared to the number of exceptional nodes in large networks. A fixed probability cut-off does not circumvent this problem, because the nodes’ absolute probability values are dependent on network size.. In the following, we therefore propose a flexible probability-threshold: The cut-off does not occur at a fixed pre-defined level, but where it yields the best separation between singular and regular nodes. A necessary condition for a node being considered singular is a sufficiently low probability compared to other nodes. Additionally, it is desirable that singular nodes appear somewhat separated from the regular ones, which renders their classification non-arbitrary. We therefore suggest to set the borderline between regular and singular nodes where the steepest increase in probability among the low probability nodes appears. Nodes with a probability below mean $\bar{p}$ minus one standard deviation $\sigma(p)$ of all nodes’ probabilities $p=(p_{k})_{k=1,\ldots,n}$ are potentially singular. Given that the probabilities $p=(p_{k})_{k=1,\ldots,n}$ are sorted increasingly, the number of singular nodes $w$ is then chosen as $w=\arg\max_{k\ :\ p_{k}<\ \bar{p}-\sigma(p)}{p_{k+1}-p_{k}}\ ,$ (1) or $w=0$, if probabilities undershoot the mean only minimally (i.e. $\nexists\ k:p_{k}<\bar{p}-\sigma(p)$). #### Number of Motif Groups $k$ Once nodes are classified as either regular or singular (Step 4), clusters of singular nodes (motif-groups) are identified using $k$-means [65, Chapter 20.1]. The $k$-means clustering algorithm requires the number of clusters $k$ to be chosen a priori; the actual procedure then determines $k$ centroids and assigns each node to the closest one of them. Choosing $k$ too low results in clustering errors, because multiple motif-groups are falsely considered as one. Conversely, too many clusters split motif-groups into non-existing sub- groups. Determining a suitable $k$ is thus crucial for automating the workflow and we come back to this issue later. Even if $k$ is chosen adequately, clustering results are not guaranteed to be satisfactory when using k-means: The algorithm initially chooses the cluster-centroids at random, but their actual distribution impacts on the quality of clustering results [66]. Attempts to optimise the centroid initialisation have been made (e.g. the $k$++-algorithm [67]), but random effects still remain; we therefore suggest a deterministic replacement for $k$-means. Optimal groupings of singular nodes consider well separated nodes to be in different clusters, whereas relatively close ones are grouped together. The standard deviations along each PC-axis can serve as a threshold for closeness and we consider each of the singular nodes to occupy a certain volume in the PCA-plane, i.e. an ellipse-shaped area centred on it. All ellipses have the same dimensions, which equal the standard deviations along the two axes. Nodes are then assigned to the same motif-group if all their ellipses constitute a connected area (Fig. 4). Practically, this idea can be implemented in 3 steps: 1. 1. Similar to an adjacency matrix, create a binary overlap-matrix $O=(o_{ij})$ in which nodes are connected if their ellipses overlap; otherwise they are not. For two nodes $i$ and $j$ let $x=(x_{1},x_{2})$ and $y=(y_{1},y_{2})$ denote their corresponding points on the PCA-plane, i.e. the centres of their ellipses with dimensions $\sigma_{1}$ and $\sigma_{2}$. Using the rescaled centres $c_{x}=(x_{1}/\sigma_{1},x_{2}/\sigma_{2})$ and $c_{y}=(y_{1}/\sigma_{1},y_{2}/\sigma_{2})$ the entry of the overlap-matrix is defined by $o_{ij}=\begin{cases}1,&d_{2}(c_{x},c_{y})<1\ ,\\\ 0,&\text{otherwise}\ ,\end{cases}$ (2) where $d_{2}(\cdot,\cdot)$ is the Euclidean distance. 2. 2. Determine a corresponding clique-matrix $C=(c_{ij})$ that specifies whether a path—a connected area of ellipses—between any two nodes exists or not. Paths or cliques can be determined through powers $O^{k}=\left(o_{ij}^{(k)}\right)$ of the overlap-matrix $O$ via $c_{ij}=\begin{cases}1,&\exists\ k\in\mathbb{N}:o_{ij}^{(k)}>0\ ,\\\ 0,&\text{otherwise}\ .\end{cases}$ (3) 3. 3. Colour all cliques differently, which finally yields the motif-groups. Note that this procedure has no parameter controlling the number of motif- groups, but these are identified automatically. Instead of using this method to actually group nodes it might also serve as a pre-processing step in order to determine the number of clusters $k$ for k-means. The drawback of this simple approach is that long elongated clusters can result when nodes are widely distributed, but connected by a chain of nodes that are just less than one standard deviation apart from each other. However, we have not observed such formation in practical applications. ### Generation of Small-World Networks The prevalence of small-world networks has risen questions about their generating mechanisms and different explanatory models have been proposed [24, 68]. We use one of them here in order to generate a series of networks: Watts and Strogatz described a rewiring procedure by which a regular ring-lattice is randomly rewired by which it becomes a small-world network [24]. This is a step-wise process, which allows to sample a network at each intermediate stage. Starting with a completely regular structure, over time, networks become increasingly perturbed (Fig. S2). In total, we sampled 600 networks (à 200 nodes), which were then analysed with BtA, to determine the single node- motifs that evolve over time. ## Acknowledgments Marcus Kaiser and Christoph Echtermeyer were supported by EPSRC (EP/G03950X/1) and the CARMEN e-science project (http://www.carmen.org.uk) funded by EPSRC (EP/E002331/1). Marcus Kaiser also acknowledges support by the WCU program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R32-10142). Luciano da F. Costa is grateful to CNPq (301303/06-1) and FAPESP (05/00587-5) for financial support. Francisco A. Rodrigues is grateful to FAPESP (2007/50633-9) for sponsorship. ## References * 1. Bornholdt S, Schuster HG, editors (2003) Handbook of graphs and networks: from the Genome to the Internet. John Wiley and Sons, 1st edition. * 2. 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MacKay DJ (2003) Information theory, inference, and learning algorithms. Cambridge University Press, 1st edition. * 66. Jain AK, Murty MN, Flynn PJ (1999) Data clustering: a review. ACM Computing Surveys 31: 264–323. * 67. Arthur D, Vassilvitskii S (2007) k-means++: The advantages of careful seeding. In: Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms. Society for Industrial and Applied Mathematics, pp. 1027–35. * 68. Ozik J, Hunt BR, Ott E (2004) Growing networks with geographical attachment preference: Emergence of small worlds. Physical Review E 69: 026108(5). * 69. Mahalanobis PC (1936) On the generalized distance in statistics. Proceedings of the National Institute of Science of India 2: 49–55. ## Figures Figure 1: Analysis work-flow to identify global singular nodes from local features [41]. Step 1: Choose set of local measures to characterise network nodes [35]. Calculate local measurements for all nodes in the network (feature vectors). Step 2: Map each node’s feature vector to lower dimensional space using principal component analysis (PCA plane) [45, Chapter 8]. Step 3: Estimate each node’s probability using the Parzen window approach (PDF) [46, 47, Chapter 4.3]. Step 4: Query PDF to identify least probable nodes (singular nodes). Step 5: Cluster singular nodes in PCA plane using k-means (motif groups) [65, Chapter 20.1]. Step 6: Determine Voronoi cells for grouped nodes using a modified Mahalanobis distance (potential motif regions) [69]. Step 7: Join potential motif regions that are close to each other (motif regions). Step 8: Calculate relative frequencies of nodes falling into motif-regions (A–F) or non-motif region (NO) (fingerprint). Figure 2: Network types used for testing BtA: a Network derived from The Simpsons family-tree [54]. Nodes in very regular parts of the network were identified singular (shaded grey) because of two characteristics: Their neighbours’ degrees are comparatively low and show no variation (values $r$ and $cv$ significantly below average). b Schematic of large regular ring lattice combined with a minor ER-random component (shaded grey). c A small regular structure (white nodes) embedded into a large random network (ER, BA, or WS model). Figure 3: Single node-motifs in emerging small-world network (Fig. S2). Vertical axes in subfigures a–c correspond to number of outlier nodes $w$, number of single node-motifs $k$, and their frequencies, respectively. a Number of identified outliers $w$ rising from 0 to 54. b Diversity of node- motifs $k$ quickly rising during the 1st re-wiring round; less increase during 2nd round; and stable during the 3rd. c Proportions of nodes expressing identified motifs (motif frequencies). Nodes classified regular not shown. d Schematics of identified single node-motifs and their distinguishing characteristics. Figure 4: Example of 2 clusters (left, right) with 3 points each (1–3, 4–6). Ellipses are centred on each point with dimensions corresponding to standard deviations $\sigma$ along PC-axes. A set of points is considered a clique, if the area of all their ellipses is connected (e.g. {1}, {1, 2}, or {1, 2, 3}; but not $\\{2,3\\}$). A maximal clique is called a cluster (i.e. {1,2,3} or {4,5,6}) and is used to define a distinct motif-group.
arxiv-papers
2011-02-01T08:46:22
2024-09-04T02:49:16.776026
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Christoph Echtermeyer, Luciano da Fontoura Costa, Francisco A.\n Rodrigues, Marcus Kaiser", "submitter": "Christoph Echtermeyer", "url": "https://arxiv.org/abs/1102.0099" }
1102.0128
# Adiabatic Conditions and the Uncertainty Relation Qian-Heng Duan, Ping-Xing Chen pxchen@nudt.edu.cn Wei Wu Department of Physics, National University of Defense Technology, Changsha, 410073, China pxchen@nudt.edu.cn Department of Physics, National University of Defense Technology, Changsha 410073, P. R. China ###### Abstract The condition for adiabatic approximation are of basic importance for the applications of the adiabatic theorem. The traditional quantitative condition was found to be necessary but not sufficient, but we do not know its physical meaning and the reason why it is necessary from the physical point of view. In this work, we relate the adiabatic theorem to the uncertainty relation, and present a clear physical picture of the traditional quantitative condition. It is shown that the quantitative condition is just the amplitude of the probability of transition between two levels in the time interval which is of the order of the time uncertainty of the system. We also present a new sufficient condition with clear physical picture. ###### pacs: 03.65.Ta, 03.65.Ca, 03.67.Lx The adiabatic theorem s1 ; s2 is one of the basic results in quantum theory and has applications in many fields, for example, in quantum field theory s3 , geometric phase s4 as well as in quantum control and adiabatic quantum computation s5 . As described in many publications, the traditional adiabatic theorem s6 ; s7 states that if a quantum system with a time-dependent Hamiltonian $\hat{H}(\mathbf{t})$ is initially in the $n\\_th$ instantaneous eigenstate of $\hat{H}(0)$, $\hat{H}(\mathbf{t})$ evolves slowly enough and the energy levels don’t cross in the evolution process, then the state of the system will stay at the $n\\_th$ instantaneous eigenstate of $\hat{H}(\mathbf{t})$ up to a phase factor at a later time. But the application of the theorem depends on the criterion of the “slowness”. Usually, the “slowness” is described as follows s8 $\left|{\frac{{\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}}{{E_{m}\left(t\right)-E_{n}\left(t\right)}}}\right|\ll 1,m\neq n,t\in\left[{0,T}\right]$ (1) where $E_{n}\left(t\right)$ and $\left|{E_{n}\left(t\right)}\right\rangle$ are the instantaneous eigenvalues and eigenstates of $\hat{H}\left(t\right)$, and $T$ is the total evolution time. In recent years, many doubts have been raised in the traditional criterion s8 ; s9 ; s10 ; s11 ; s12 . It was first shown by Marzlin and Sanders s8 and then by Tong et al s9 that if two systems which we call system $S^{A}$ and $S^{B}$ are related though $\hat{H}^{B}\left(t\right)=-\hat{U}^{A+}\left(t\right)\hat{H}^{A}\left(t\right)\hat{U}^{A}\left(t\right)$ (2) The two systems can’t have an adiabatic evolution at the same time unless $\left|{\left\langle{E_{n}^{A}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}^{A}\left(t\right)}{E_{n}^{A}\left(0\right)}}}\right.\kern-1.2pt}{{E_{n}^{A}\left(0\right)}}\right\rangle}\right|\approx 1$, even if both of the system satisfy condition (1). Many authors investigated the reasons of the insufficiency s18 ; s19 ; s20 ; s22 . Recently, Amin pointed out that the violations of the traditional criterion all arise from resonant transitions between energy levels s11 . At the same time, some authors proposed some new alternative criterions s13 ; s16 ; s14 ; s15 ; s23 ; s24 ; s25 ; s26 . In 2008, Du et al experimentally examined the traditional criterion s10 . However, the physical pictures of the criterions proposed before are not clear. Even though Tong proved that the traditional condition (1) is necessary in guaranteeing the validity of the adiabatic approximation s12 , we still do not know the reason why it is necessary from the physical point of view. It is foundmentally important to find a new condition with clear physics picture or probe the physical meaning of the existed conditions. In this letter, we relate the adiabatic condition to the uncertainty relation. We first propose a new sufficient condition for adiabatic process, and then give clear physical pictures of the new condition and the necessary condition (1) in terms of the uncertainty relation. It is shown that the state of a system cannot be appreciably modified by an evolution until a least evolution time has elapsed, and $\left|{\frac{{\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}}{{E_{n}\left(t\right)-E_{m}\left(t\right)}}}\right|$ in Eq. (1) is just the amplitude of the probability of the transition between $\left|{E_{n}\left(t\right)}\right\rangle$ and $\left|{E_{m}\left(t\right)}\right\rangle$ in the least evolution time. The least evolution time is of the order of the time-uncertainty of the system. In an adiabatic process, if the system is initially in the $n\\_th$ instantaneous eigenstate ${\left|{E_{n}\left(0\right)}\right\rangle}$, then at the end of the adiabatic evolution process the state $\left|{\psi\left(t\right)}\right\rangle$ fulfills $\left|{\left\langle{E_{n}\left(T\right)}\mathrel{\left|{\vphantom{{E_{n}\left(T\right)}{\psi\left(T\right)}}}\right.\kern-1.2pt}{{\psi\left(T\right)}}\right\rangle}\right|^{2}\approx 1.$ (3) Let $\left|{\psi\left(t\right)}\right\rangle=\sum\limits_{m}{a_{m}\left(t\right)\left|{E_{m}\left(t\right)}\right\rangle,}$ (4) where $a_{m}\left(t\right)$ is a complex number. Substituting the above equation into the Schrödinger equation, we obtain $\frac{d}{{dt}}a_{n}\left(t\right)=-\sum\limits_{m}{a_{m}\left(t\right)\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}-\frac{i}{\hbar}a_{n}\left(t\right)E_{n}\left(t\right)$ (5) and $\frac{d}{{dt}}a_{n}^{\ast}\left(t\right)=-\sum\limits_{m}{a_{m}^{\ast}\left(t\right)\left\langle{\dot{E}_{m}\left(t\right)}\mathrel{\left|{\vphantom{{\dot{E}_{m}\left(t\right)}{E_{n}\left(t\right)}}}\right.\kern-1.2pt}{{E_{n}\left(t\right)}}\right\rangle}+\frac{i}{\hbar}a_{n}^{\ast}\left(t\right)E_{n}\left(t\right)$ (6) Using equations (5) and (6), and denoting $P_{n}(t)={a_{n}\left(t\right)a_{n}^{\ast}\left(t\right)}$, we have $\displaystyle\frac{d}{{dt}}P_{n}\left(t\right)$ $\displaystyle=$ $\displaystyle a_{n}\left(t\right)\frac{d}{{dt}}a_{n}^{\ast}\left(t\right)+a_{n}^{\ast}\left(t\right)\frac{d}{{dt}}a_{n}\left(t\right)$ (7) $\displaystyle=$ $\displaystyle-\sum\limits_{m}{a_{n}\left(t\right)a_{m}^{\ast}\left(t\right)\left\langle{\dot{E}_{m}\left(t\right)}\mathrel{\left|{\vphantom{{\dot{E}_{m}\left(t\right)}{E_{n}\left(t\right)}}}\right.\kern-1.2pt}{{E_{n}\left(t\right)}}\right\rangle}$ $\displaystyle-\sum\limits_{m}{a_{n}^{\ast}\left(t\right)a_{m}\left(t\right)\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}$ $\displaystyle=$ $\displaystyle-2\sum\limits_{m}\mathit{{Re}\left\\{{a_{n}^{\ast}\left(t\right)a_{m}\left(t\right)\chi_{nm}}\right\\}}$ where $\chi_{nm}=\left\langle{E_{n}\left(t\right)}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{\dot{E}_{m}\left(t\right)}\right\rangle,$ and we use a gauge in which $\chi_{nn}=0$. Integrating equation (7), we get $\displaystyle P_{n}\left(T\right)$ $\displaystyle=$ $\displaystyle 1-2\sum\limits_{m}{\int_{0}^{T}\mathit{{Re}{\left({a_{n}\left(t\right)a_{m}^{\ast}\left(t\right)\chi_{nm}}\right)dt}}}$ (8) $\displaystyle\geq$ $\displaystyle 1-2\sum\limits_{m\neq n}{\int_{0}^{T}{\left|{\chi_{nm}}\right|dt}}$ $\displaystyle\geq$ $\displaystyle 1-2\sum\limits_{m\neq n}{T\max\left\\{{\left|{\chi_{nm}}\right|}\right\\}}$ When the dimension of the system is finite, if we have $2T\max\left\\{{\left|{\left\langle{E_{m}\left(t\right)}\mathrel{\left|{\vphantom{{E_{m}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left(t\right)}}\right\rangle}\right|}\right\\}\ll 1,$ (9) the sum in the equation (8) can always be a small number so that $P_{n}\left(T\right)\approx 1,$ which means that condition (9) is a sufficient condition for adiabatic theorem. Condition (9) can sufficiently guarantee the validity of the adiabatic approximation, but we do not understand its physical meaning clearly, just as we do with the necessary condition (1). Especially, condition (9) means seemingly that only if $T$ is small enough and $\max\left\\{{\left|{\left\langle{E_{m}\left(t\right)}\mathrel{\left|{\vphantom{{E_{m}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left(t\right)}}\right\rangle}\right|}\right\\}$ is finite, it can always be fulfilled and the adiabatic approximation can be guaranteed. This conflicts seemingly with condition (1) in which the time $T$ seems be not involved. How to attemper this conflict? Let’s go to the central purpose of this letter, we will present the clear physical pictures of conditions (1) and (9). From these pictures conditions (1) and (9) are consistent. Interestingly, the uncertainty relation plays a key role here. We first show that the evolution time must be more than the least evolution time to get an obvious state change, and the least evolution time is in the order of the time uncertainty of the system. For simplicity, we consider a two level system. The Hamiltonian $\hat{H}\left(t\right)$ has two eigenstates $\left|{E_{k}\left(t\right)}\right\rangle$ and $\left|{E_{n}\left(t\right)}\right\rangle$ which satisfy the following equation $\hat{H}\left(t\right)\left|{E_{n,k}\left(t\right)}\right\rangle=E_{n,k}\left(t\right)\left|{E_{n,k}\left(t\right)}\right\rangle.$ (10) The state of the system at time $t,$ $\left|{\psi\left(t\right)}\right\rangle,$ can be expanded as $\left|{\psi\left(t\right)}\right\rangle=\sum\limits_{n}{a_{n}\left(t\right)e^{i\beta_{n}\left(t\right)}\left|{E_{n}\left(t\right)}\right\rangle}$ (11) where $a_{n}\left(t\right)$ and $\beta_{n}\left(t\right)$ are real, and the phase $\beta_{n}\left(t\right)$ can be expressed as s17 $\beta_{n}\left(t\right)\mathrm{\ =}-\frac{1}{\hbar}\int_{0}^{t}{E_{n}\left({t^{\prime}}\right)}dt^{\prime}+i\int_{0}^{t}{\left\langle{{E_{n}\left({t^{\prime}}\right)}}\mathrel{\left|{\vphantom{{E_{n}\left({t^{\prime}}\right)}{\dot{E}_{n}\left({t^{\prime}}\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left({t^{\prime}}\right)}}\right\rangle dt^{\prime}.}$ (12) Substituting equations (11) and (12) into the Schrödinger equation, we obtain $\frac{{da_{k}\left(t\right)}}{{dt}}=-{a_{n}\left(t\right)e^{i\beta_{nk}\left(t\right)}\left\langle{E_{k}\left(t\right)}\mathrel{\left|{\vphantom{{E_{k}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left(t\right)}}\right\rangle}$ (13) where $\beta_{nk}\left(t\right)=\beta_{n}\left(t\right)-\beta_{k}\left(t\right)$. Let us consider two systems $S^{A}$ and $S^{B}$, the Hamiltonian of which are related though Eq. (2) as shown in s8 ; s9 . The instantaneous eigenvalues and eigenstates of the two system satisfy s9 $\begin{array}[]{l}E_{{}_{n}}^{B}\left(t\right)=-E_{{}_{n}}^{A}\left(t\right)\\\ \left|{E_{{}_{n}}^{B}\left(t\right)}\right\rangle=\hat{U}^{A+}\left(t\right)\left|{E_{{}_{n}}^{A}\left(t\right)}\right\rangle\end{array}$ (14) and their evolution operator $\hat{U}^{B}\left(t\right)=\hat{U}^{A+}\left(t\right).$ (15) From Eqs. (14) and (15) we have $\left\langle{{E_{k}^{B}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{k}^{B}\left(t\right)}{\dot{E}_{n}^{B}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{B}\left(t\right)}}\right\rangle=\frac{i}{\hbar}E_{n}^{A}\left(t\right)\delta_{nk}+\left\langle{{E_{k}^{A}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{k}^{A}\left(t\right)}{\dot{E}_{n}^{A}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{A}\left(t\right)}}\right\rangle$ (16) Since $a_{k}\left(t\right)$ is real, from Eqs. (13) and (16) we can get that $\beta_{nk}^{A}\left(t\right)+\widetilde{\omega}_{nk}^{A}=q^{A}\pi;$ $\beta_{nk}^{B}\left(t\right)+\widetilde{\omega}_{nk}^{B}=q^{B}\pi,$ and then $\beta_{{}_{nk}}^{B}\left(t\right)=\beta_{{}_{nk}}^{A}\left(t\right)+q\pi$ (17) where $q^{A},q^{B},q$ are integer, and $\widetilde{\omega}_{nk}^{A}=\widetilde{\omega}_{nk}^{B}$ are the phases of ${\left\langle{E_{k}^{A}\left(t\right)}\mathrel{\left|{\vphantom{{E_{k}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{A}\left(t\right)}}\right\rangle}$ and ${\left\langle{E_{k}^{B}\left(t\right)}\mathrel{\left|{\vphantom{{E_{k}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{B}\left(t\right)}}\right\rangle}$. From Eqs. (12), (14) and (16), we obtain $\displaystyle\beta_{{}_{nk}}^{B}\left(t\right)=-\frac{1}{\hbar}\int_{0}^{t}{\left({E_{{}_{n}}^{B}-E_{{}_{k}}^{B}}\right)dt^{\prime}}$ (18) $\displaystyle+i\int_{0}^{t}{\left\\{{\left\langle{E_{n}^{B}}\mathrel{\left|{\vphantom{{E_{n}^{B}}{\dot{E}_{n}^{B}}}}\right.\kern-1.2pt}{{\dot{E}_{n}^{B}}}\right\rangle-\left\langle{E_{k}^{B}}\mathrel{\left|{\vphantom{{E_{k}^{B}}{\dot{E}_{k}^{B}}}}\right.\kern-1.2pt}{{\dot{E}_{k}^{B}}}\right\rangle}\right\\}dt^{\prime}}$ $\displaystyle=$ $\displaystyle\beta_{{}_{nk}}^{A}\left(t\right)+\frac{1}{\hbar}\int_{0}^{t}{\left({E_{{}_{n}}^{A}-E_{{}_{k}}^{A}}\right)dt^{{}^{\prime}}.}$ By Eqs. (17) and (18), we get $\frac{1}{\hbar}\int_{0}^{t}{\left({E_{{}_{n}}^{A}-E_{{}_{k}}^{A}}\right)dt^{\prime}}=q\pi.$ (19) Eq. (19) is very interesting since it shows the relation between the evolution time and the instantaneous eigenvalues of the Hamiltonian. For any arbitrary system $S^{A}$ one can always find a corresponding system $S^{B}$ satisfying Eq. (2), so Eq. (19) is only a result of the Schrödinger equation. If we denote $\overline{\bigtriangleup E_{nk}}\equiv\frac{1}{t}\int_{0}^{t}{\left({E_{{}_{n}}^{A}-E_{{}_{k}}^{A}}\right)dt^{{}^{\prime}}}$ as the average of the ${E_{{}_{n}}^{A}-E_{{}_{k}}^{A}}$ in the time interval $[0,t],$ Eq. (19) can be expressed as $t=\frac{q\pi\hbar}{\overline{\bigtriangleup E_{nk}}}.$ (20) Eq. (20) means the least evolution time is $\frac{\pi\hbar}{\overline{\bigtriangleup E_{nk}}}$(i.e.,$q=1$). Furthermore, if we regard $\overline{\bigtriangleup E_{nk}}$ as the energy uncertainty of the system, according to the uncertainty relation $\overline{\bigtriangleup E_{nk}}t\sim h$, the time uncertainty is $\frac{h}{\overline{\bigtriangleup E_{nk}}}$ which is in the order of the leat evolution time. In fact, if the system undergoes a quantum transition between $\left|{E_{n}\left(t\right)}\right\rangle$ and $\left|{E_{k}\left(t\right)}\right\rangle$ by the evolution according to the Schrödinger equation, the energy of the system has uncertainty of $E_{k}\left(t\right)-E_{n}\left(t\right)$ This can be explained as follows. Suppose the system is in the state $\left|{E_{n}\left(t^{\prime}\right)}\right\rangle$ in the time $t^{\prime}$, after a evolution from $t^{\prime}$ to $t$ the system’s state becomes $\left|{\psi\left(t\right)}\right\rangle\ $which is a superposition of the instantaneous eigenstates $\left|{E_{n}\left(t\right)}\right\rangle$ and $\left|{E_{k}\left(t\right)}\right\rangle$(in this case, there is a quantum transition between $\left|{E_{n}\left(t\right)}\right\rangle$ and $\left|{E_{k}\left(t\right)}\right\rangle).$ According to quantum mechanics theory, when the system is in the superposition state $\left|{\psi\left(t\right)}\right\rangle$ one cannot distinguish whether the system is in the state $\left|{E_{n}\left(t\right)}\right\rangle$ or $\left|{E_{k}\left(t\right)}\right\rangle.$ So we can say the system has energy uncertainty ${E_{k}\left(t\right)}-{E_{n}\left(t\right)}.$ Owing to the uncertainty relation the corresponding time-uncertainty is $\frac{1}{{E_{k}\left(t\right)}-{E_{n}\left(t\right)}}$ (We let $h=1$). How to understand that the least evolution time is the order of time- uncertainty? We can say that any evolution in the time much less than the time-uncertainty $\frac{1}{{E_{k}\left(t\right)}-{E_{n}\left(t\right)}}$ will be negligible, namely, the evolution time must not be much less than $\frac{1}{{E_{k}\left(t\right)}-{E_{n}\left(t\right)}}$ to produce an effective evolution. Otherwise, we can determinate time parameter with precision more than the time-uncertainty by distinguishing the difference between the states before and after the effective evolution sc , which violates the uncertainty relation. A similar conclusion can also be reached from a different point of view s7 . Let $\left|{\psi\left(0\right)}\right\rangle$ and $\left|{\psi\left(t\right)}\right\rangle=u(t)\left|{\psi\left(0\right)}\right\rangle$ denote the initial state and the state at time $t$ of the system, where $u(t)$ is the evolution operator. The expansion of the $u(t)$ is $\displaystyle u(t)$ $\displaystyle=$ $\displaystyle 1-i\int_{0}^{t}{H(t}_{1}{)dt}_{1}$ (21) $\displaystyle+\frac{(-i)^{2}}{2}\int_{0}^{t}{{dt}_{1}}\int_{0}^{t_{1}}{dt}_{2}{H(t}_{1}{)H(t}_{2}{)+\cdots.}$ Since $t$ is small, we can keep only the first order approximation. let $\overline{H}\equiv\frac{1}{t}\int_{0}^{t}{H(t}_{1}{)dt}_{1},$ then at time $t$ the probability $p$ of finding the system not being in the initial state $\left|{\psi\left(0\right)}\right\rangle$ is $\displaystyle p$ $\displaystyle=$ $\displaystyle\left\langle{\psi\left(0\right)}\right|u(t)^{+}[I-\left|{\psi\left(0\right)}\right\rangle\left\langle{\psi\left(0\right)]}\right|u(t)\left|{\psi\left(0\right)}\right\rangle$ (22) $\displaystyle\approx$ $\displaystyle\left\langle{\psi\left(0\right)}\right|(1+it\overline{H})[I-\left|{\psi\left(0\right)}\right\rangle\left\langle{\psi\left(0\right)]}\right|(1-it\overline{H})\left|{\psi\left(0\right)}\right\rangle$ $\displaystyle=$ $\displaystyle\left\langle{\psi\left(0\right)}\right|\overline{H}^{2}\left|{\psi\left(0\right)}\right\rangle t^{2}-\left\langle{\psi\left(0\right)}\right|\overline{H}\left|{\psi\left(0\right)}\right\rangle^{2}t^{2}$ $\displaystyle\equiv$ $\displaystyle(\bigtriangleup\overline{H})^{2}t^{2},$ where $\bigtriangleup\overline{H}$, the root mean square deviation of the energy, is the average uncertainty of the energy of the system in the time interval $[0,t]$, its inversion $\frac{1}{\bigtriangleup\overline{H}}$ is the uncertainty of the time. If evolution time $t\ll\frac{1}{\bigtriangleup\overline{H}},$ then $p\ll 1.$ Namely, if the evolution time is much less than the time-uncertainty, the system will stay in the initial state. From the discussion above, we can conclude that any system has the least effective evolution time (LEET) which is the order of time-uncertainty. $E_{k}\left(t\right)-E_{n}\left(t\right)$ can be regarded as the energy- uncertainty when the system undergoes a transition between the two states $\left|{E_{n}\left(t\right)}\right\rangle$ and $\left|{E_{k}\left(t\right)}\right\rangle.$ So the time $\frac{1}{{E_{k}\left(t\right)}-{E_{n}\left(t\right)}}$ can be regarded roughly as the least effective evolution time which we denote as $T_{LEET}$. With those in mind, we can discuss the physical pictures of conditions (1) and (9) easily. By the basic meaning of the inner product of two vectors in a Hilbert space, we know that $\left\langle{{E_{n}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle$ is proportional to the amplitude of the probability of the transition from $\left|{E_{m}\left(t\right)}\right\rangle$ to $\left|{E_{n}\left(t\right)}\right\rangle$ in an unit time interval. By equation (8) we know ${\int_{0}^{T}\mathit{{Re}{\left({a_{n}\left(t\right)a_{m}^{\ast}\left(t\right)}\left\langle{{E_{n}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle\right)dt}}}$ is proportional to the probability of the transition from $\left|{E_{n}\left(t\right)}\right\rangle$ to $\left|{E_{m}\left(t\right)}\right\rangle)$ in the time interval $[0,T].$ And then $2T\max\left\\{{\left|{\left\langle{E_{m}\left(t\right)}\mathrel{\left|{\vphantom{{E_{m}\left(t\right)}{\dot{E}_{n}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{n}\left(t\right)}}\right\rangle}\right|}\right\\}$ is the maximal probability of the transition from $\left|{E_{n}\left(t\right)}\right\rangle$ to $\left|{E_{m}\left(t\right)}\right\rangle$ in the time interval $[0,T].$ Condition (9) means just that the transition between $\left|{E_{n}\left(t\right)}\right\rangle$ and $\left|{E_{m}\left(t\right)}\right\rangle$ is very small and can be neglected in the whole time interval $[0,T]$. So it is sufficient to assure adiabatic process. In condition (1), $\frac{\left\langle{{E_{n}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}{{E_{m}\left(t\right)}-{E_{n}\left(t\right)}}$ is nothing else but the amplitude of the average probability of the transition between $\left|{E_{n}\left(t\right)}\right\rangle$ and $\left|{E_{m}\left(t\right)}\right\rangle$ in one LEET. The condition $\frac{\left\langle{{E_{n}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{n}\left(t\right)}{\dot{E}_{m}\left(t\right)}}}\right.\kern-1.2pt}{{\dot{E}_{m}\left(t\right)}}\right\rangle}{{E_{m}\left(t\right)}-{E_{n}\left(t\right)}}<<1$ for each LEET in the whole time interval $[0,T]$ is necessary for adiabatic process, otherwise, it is possible for the system has an obvious transition between $\left|{E_{n}\left(t\right)}\right\rangle$ and $\left|{E_{m}\left(t\right)}\right\rangle$ in a LEET. To make the pictures of the necessary condition (1) and the sufficient condition (9) more clear we discuss when the necessary condition becomes sufficient, we investigate the effect of the phases of ${a_{n}\left(t\right)\ }$and ${\chi_{nm}(t).}$ Let ${a_{n}\left(t\right)=}\left|{a_{n}\left(t\right)}\right|e^{-i\int_{0}^{t}{E_{n}\left(t^{\prime}\right)dt}^{\prime}};$ (23) ${\chi_{nm}(t)=}\left|{\chi_{nm}(t)}\right|{e}^{i\omega(t)dt},$ (24) from Eq. (8), the probability of the transition from the $\left|{E_{n}\left(t\right)}\right\rangle$ to $\left|{E_{m}\left(t\right)}\right\rangle$ is proportional to $\epsilon_{nm}.$ $\displaystyle\epsilon_{nm}$ $\displaystyle\equiv$ $\displaystyle{\int_{0}^{T}\mathit{{Re}{\left({a_{n}\left(t\right)a_{m}^{\ast}\left(t\right)\chi_{nm}}\right)dt}}}$ $\displaystyle=$ $\displaystyle{\int_{0}^{T}\mathit{{Re}{\left(\left|{a_{n}\left(t\right)}\right|\left|{a_{m}^{\ast}\left(t\right)}\right|{e}^{-i\omega_{nm}(t)t}\left|{\chi_{nm}}\right|{e}^{i\omega(t)t}\right)dt}}}$ $\displaystyle=$ $\displaystyle{\int_{0}^{T}{\left|{a_{n}\left(t\right)}\right|\left|{a_{m}^{\ast}\left(t\right)}\right|\left|{\chi_{nm}}\right|\cos((\omega(t)-\omega_{nm}(t))t)dt}}$ where ${{\omega_{nm}(t)\equiv}}\frac{1}{t}\int_{0}^{t}{E_{n}-E_{m}\left(t^{\prime}\right)dt}^{\prime}.$ As shown in s11 in the presence of resonant oscillation, i.e., ${{\omega(t)=\omega_{nm}(t),}}$ $\displaystyle\epsilon_{nm}$ $\displaystyle=$ $\displaystyle{\int_{0}^{T}{\left|{a_{n}\left(t\right)}\right|\left|{a_{m}^{\ast}\left(t\right)}\right|\left|{\chi_{nm}}\right|dt}}$ $\displaystyle\leq$ $\displaystyle T{\max_{t\in[0,T]}{\left|{\chi_{nm}}\right|.}}$ Suppose that $T$ includes $M$ LEET, i.e., $M\approx\frac{T}{T_{LEET}},$ then $\displaystyle\epsilon_{nm}$ $\displaystyle\leq$ $\displaystyle T{\max_{t\in[0,T]}{\left|{\chi_{nm}}\right|=}}\sum_{i=1}^{M}{\max{\left|{\chi_{nm}}\right|T}}_{LEET}^{i}$ (27) $\displaystyle=$ $\displaystyle\sum_{i=1}^{M}{\max}\frac{{{\left|{\chi_{nm}}\right|}}^{i}}{{E_{m}\left(t\right)}^{i}-{E_{n}\left(t\right)}^{i}},$ where ${T}_{LEET}^{i}=\frac{{1}}{{E_{m}\left(t\right)}^{i}-{E_{n}\left(t\right)}^{i}}$ is the $i\\_th$ LEET. The conditions (1), which means ${\max}\frac{{{\left|{\chi_{nm}}\right|}}^{i}}{{E_{m}\left(t\right)}^{i}-{E_{n}\left(t\right)}^{i}}\ll 1$ for each LEET, cannot assure the error of the whole process is small since $M$ may increase as the time $T$ does. But the condition (9) means that $\sum_{i=1}^{M}{\max{\left|{\chi_{nm}}\right|T}}_{LEET}^{i}\ll 1$, i.e., the error of the whole process is small, so it is sufficient. In the absence of resonant oscillation, i.e., ${{\omega(t)-\omega_{nm}(t)\equiv\omega}}^{\prime}\neq 0,$ $\epsilon_{nm}={\int_{0}^{T}{\left|{a_{n}\left(t\right)}\right|\left|{a_{m}^{\ast}\left(t\right)}\right|\left|{\chi_{nm}}\right|\cos{\omega}^{\prime}tdt}}$. This means $\epsilon_{nm}$ may not increase as $T$ owing to the different sign of ${\cos\omega}^{\prime}t$ in the different LEET. In this case, the adiabatic opproximation holds under condition (1). It should be noted that if the evolution time $T$ is of the order $T_{LEET}$, the error of the whole process is small and adiabatic approximation is valid in many cases. For example, consider a simple two-state system as used by Amin s11 . The Hamiltonian of the system is $H\left(t\right)=-\varepsilon\frac{{\sigma_{z}}}{2}-V\sin\left({\omega_{0}t}\right)\sigma_{x}$ (28) and $V$ is a small positive number. The system’s exact instantaneous eigenvalues and eigenstates are $\displaystyle E_{0,1}$ $\displaystyle\mathrm{=}$ $\displaystyle\pm\frac{1}{2}\Omega;$ (29) $\displaystyle\left|{E_{0,1}}\right\rangle$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{l}\alpha^{\pm}\\\ \pm\alpha^{\mp}\end{array}\right)$ (32) where $\Omega=\sqrt{\varepsilon^{2}+4V^{2}\sin^{2}\left({\omega_{0}t}\right)},\alpha^{\pm}=\sqrt{{\raise 3.01385pt\hbox{${\left({\Omega\pm\varepsilon}\right)}$}\\!\mathord{\left/{\vphantom{{\left({\Omega\pm\varepsilon}\right)}{2\Omega}}}\right.\kern-1.2pt}\\!\lower 3.01385pt\hbox{${2\Omega}$}}}.$ If $\varepsilon\approx\omega_{0}$, and the system starts at its ground state, then at time $T$, the probability of the system ends at the ground state is $P_{0}\left(t\right)=\left|{\left\langle{{E_{0}\left(t\right)}}\mathrel{\left|{\vphantom{{E_{0}\left(t\right)}{\psi\left(t\right)}}}\right.\kern-1.2pt}{{\psi\left(t\right)}}\right\rangle}\right|^{2}\approx\frac{{\left({\cos Vt+1}\right)}}{2}.$ (33) $E_{0}-E_{1}=\Omega\approx\varepsilon\approx\omega_{0}$, so the $T_{LEET}\approx\frac{1}{\omega_{0}}.$ If the evolution time $T$ is of order $\frac{1}{\omega_{0}},$ then $VT\ll 1$ and $P_{0}\left(t\right)\approx\frac{{\left({\cos Vt+1}\right)}}{2}\approx 1.$ That means adiabatic approximatoin is valid even in the presence of fast driven oscillations. In conclusion, we have shown that the evolution time must not be much less than a lower bound which is in the order of the time uncertainty of the system to get an obvious change of the state of the system. The quantitative condition has a clear physical picture: the amplitude of the probability of transition between two levels in each of the least evolution time is small. We also present a new sufficient condition with clear physical meaning. Our results are helpful to clarify the physical images of the some existing conditions for adiabatic approximation and remove the previous doubts on the quantitative condition. A possible interesting topic in the further is: what is the role of the uncertainty relation in the evolution of a quantum system. We thank Prof. Chengzu Li for helpful discussions. P.-X Chen is very grateful for friendly help of Prof. Ian Walmsley, Dr. Lijian Zhang and the other members in the Walmsley’s group when he visited in physics department of Oxford university. This work was supported by NSFC (no:10774192) and FANEDD in China (no 200524). ## References * (1) P. Ehrenfest, Ann. Phys. 51, 327 (1916). * (2) M. Born, V. Fock, Z. Phys. 51, 165 (1928). * (3) M. Gell-Mann and F. Low, Phys. Rev. 84, 350 (1951). * (4) M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984). * (5) E. Farhi, et al., Science 292, 472 (2001). * (6) T. Kato, J. Phys. Soc. Jpn. 5, 435 (1950). * (7) A. Messiah, Quantum Mechanics $\left(Dover\right)$, New York, (1999). * (8) K. P. Marzlin, B. C. Sanders, Phys. Rev. Lett. 93, 160408 (2004). * (9) D. M. Tong, et al., Phys. Rev. Lett. 95, 110407 (2005). * (10) J. Du, L. Hu, Y. Wang, J. Wu, M. Zhao, and D. Suter, Phys. Rev. Lett. 101, 060403 (2008). * (11) M.H.S Amin, Phys. Rev. Lett. 102, 220401 (2009). * (12) D. M. Tong, Phys. Rev. Lett. 104, 120401 (2010). * (13) J. Ma, Y.P. Zhang, E. G. Wang and B. Wu, Phys. Rev. Lett. 97, 128902 (2006). * (14) S. Duki, H. Mathur and O. Narayan, Phys. Rev. Lett. 97, 128901 (2006). * (15) M. Y. Ye, X. F. Zhou, Y. S. Zhang and G. C. Guo, Phys. Lett. A. 368, 18 (2007). * (16) Y. Zhao, Phys. Rev. A. 77, 032109 (2008). * (17) R. MacKenzie, E. Marcotte, and H. Paquette, Phys. Rev. A 73, 042104 (2006); R. MacKenzie et al, Phys. Rev. A. 76, 044102 (2007). * (18) D. M. Tong, K. Singh, L. C. Kwek, and C. H. Oh, Phys. Rev. Lett. 98, 150402 (2007). * (19) J.-D. Wu, M.-S. Zhao, J.-L. Chen, and Y. D. Zhang, Phys. Rev. A 77, 06214 (2008) * (20) J.-L. Chen, M.-S. Zhao, J.-D. Wu, and Y. D. Zhang, arXiv:quant-ph/07060299. * (21) Z. H. Wei and M. S. Ying, Phys. Rev. A. 76, 024304 (2007). * (22) M. Maamache and Y. Saadi, Phys. Rev. Lett. 101, 150407 (2008). * (23) V. I. Yukalov, Phys. Rev. A. 79, 052117 (2009). * (24) X. L. Huang and X. X. Yi, Phys. Rev. A. 80, 032108 (2009). * (25) J. Samuel, R. Bhandari, Phys. Rev. Lett. 60, 2339 (1988). * (26) Samuel L. Braunstein and Carlton M. Caves, Phys. Rev. Lett. 72, 3439 (1984).
arxiv-papers
2011-02-01T11:11:31
2024-09-04T02:49:16.783258
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Qian-Heng Duan, Ping-Xing Chen, Wei Wu", "submitter": "Chen Ping Xing", "url": "https://arxiv.org/abs/1102.0128" }
1102.0406
# Threshold Saturation on Channels with Memory via Spatial Coupling Shrinivas Kudekar1 and Kenta Kasai2 1 New Mexico Consortium and Center for Non-linear Studies, Los Alamos National Laboratory, NM, USA Email: skudekar@lanl.gov 2 Dept. of Communications and Integrated Systems, Tokyo Institute of Technology, 152-8550 Tokyo, Japan. Email: kenta@comm.ss.titech.ac.jp ###### Abstract We consider spatially coupled code ensembles. A particular instance are convolutional LDPC ensembles. It was recently shown that, for transmission over the memoryless binary erasure channel, this coupling increases the belief propagation threshold of the ensemble to the maximum a-posteriori threshold of the underlying component ensemble. This paved the way for a new class of capacity achieving low-density parity check codes. It was also shown empirically that the same threshold saturation occurs when we consider transmission over general binary input memoryless channels. In this work, we report on empirical evidence which suggests that the same phenomenon also occurs when transmission takes place over a class of channels with memory. This is confirmed both by simulations as well as by computing EXIT curves. ## I Introduction It has long been known that convolutional LDPC (or spatially coupled) ensembles, introduced by Felström and Zigangirov [1], have excellent thresholds when transmitting over general binary-input memoryless symmetric- output (BMS) channels. The fundamental reason underlying this good performance was recently discussed in detail in [2] for the case when transmission takes place over the binary erasure channel (BEC). In particular, it was shown in [2] that the BP threshold of the spatially coupled ensemble (see the last paragraph of this section for a definition) is essentially equal to the MAP threshold of the underlying component ensemble. It was also shown that for long chains the MAP performance of the chain cannot be substantially larger than the MAP threshold of the component ensemble. In this sense, the BP threshold of the chain is increased to its maximal possible value. This is the reason why they call this phenomena threshold saturation via spatial coupling. In a recent paper [3], Lentmaier and Fettweis independently formulated the same statement as conjecture. They attribute the observation of the equality of the two thresholds to G. Liva. The phenomena of threshold saturation seems not to be restricted to the BEC. It was also shown recently in [4] that the same phenomena manifests itself when we consider transmission over more general BMS channels. The principle which underlies the good performance of spatially coupled ensembles is very broad. It has been shown to apply to many other problems in communications, and more generally computer science. To mention just a few, the threshold saturation effect (dynamical threshold of the system being equal to the static or condensation threshold) of coupled graphical models has recently been shown to occur for compressed sensing [5], and a variety of graphical models in statistical physics and computer science like the so- called $K$-SAT problem, random graph coloring, or the Curie-Weiss model [6]. Other communication scenarios where the spatially coupled codes have found immediate application is to achieve the whole rate-equivocation region of the BEC wiretap channel [7]. It is tempting to conjecture that the same phenomenon occurs for transmission over general channels with memory. We provide some empirical evidence that this is indeed the case. In particular, we compute EXIT curves for transmission over a class of channels with memory known as the Dicode Erasure Channel (DEC). We show that these curves behave in an identical fashion to the ones when transmission takes place over the memoryless BEC. We also compute fixed points (FPs) of the spatial configuration and we demonstrate again empirically that these FPs have properties identical to the ones in the BEC case. For a review on the literature on convolutional LDPC ensembles we refer the reader to [2] and the references therein. As discussed in [2], there are many basic variants of coupled ensembles. For the sake of convenience of the reader, we quickly review the ensemble $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$. This is the ensemble we use throughout the paper as it is the simplest to analyze. ### I-A $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ Ensemble [2] We assume that the variable nodes are at sections $[-L,L]$, $L\in\mathbb{N}$. At each section there are $M$ variable nodes, $M\in\mathbb{N}$. Conceptually we think of the check nodes to be located at all integer positions from $[-\infty,\infty]$. Only some of these positions actually interact with the variable nodes. At each position there are $\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}M$ check nodes. It remains to describe how the connections are chosen. We assume that each of the $d_{\mathrm{l}}$ connections of a variable node at position $i$ is uniformly and independently chosen from the range $[i,\dots,i+w-1]$, where $w$ is a “smoothing” parameter. In the same way, we assume that each of the $d_{\mathrm{r}}$ connections of a check node at position $i$ is independently chosen from the range $[i-w+1,\dots,i]$. A discussion on the above ensemble and a proof of the following lemma can be found in [2]. ###### Lemma 1 (Design Rate) The design rate of the ensemble $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$, with $w\leq 2L$, is given by $\displaystyle R(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ $\displaystyle=(1-\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}})-\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}\frac{w+1-2\sum_{i=0}^{w}\bigl{(}\frac{i}{w}\bigr{)}^{d_{\mathrm{r}}}}{2L+1}.$ In the next section we provide the channel model and the joint iterative decoder. We also present the density evolution analysis of the joint iterative decoder when we consider $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensembles. In the section on main results, we demonstrate the threshold saturation phenomena by using spatially coupled codes. ## II Channels with Memory: The Dicode Erasure Channel The particular class of channel with memory that we consider is the Dicode Erasure Channel (DEC). The DEC is a binary-input channel defined as follows. The output of a binary-input linear filter $(1-D)$ ($D$ is the delay element) is erased with probability $\epsilon$ and transmitted perfectly with probability $1-\epsilon$. For this channel we will be interested in the symmetric information rate (SIR), i.e., the capacity assuming i.i.d Bern(1/2) signalling. In this case, the Shannon threshold for a given rate $r$ is given by $\frac{1-r}{4}+\frac{1}{4}\sqrt{(1-r)^{2}+8(1-r)}$. The details on the definition of the channel and the analytical formula for the SIR can be found in the thesis of Pfister [8] and in [9]. ### II-A Joint Iterative Decoder, Density Evolution and the Extended BP Fixed Points We use the joint iterative decoder (JIT) of Pfister and Siegel [9]. More precisely, we consider a turbo equalization system, which performs one channel iteration (BCJR step) for each iteration over the LDPC code. As a result, in every iteration, first the channel detector uses the extrinsic information provided by the LDPC code to compute its extrinsic erasure fraction. This is then fed to the LDPC decoder which then again computes the usual variable node and check node erasure messages. The simplicity of the DEC gives an analytical formula for the erasure fraction of the message which is passed from the channel detector to the LDPC code (see [9] for a derivation). This is given by $f(x)=\frac{4\epsilon^{2}}{(2-x(1-\epsilon))^{2}},$ where $x$ represents the fraction of erasures entering the channel detector from the LDPC code. $f(.)$ represents the extrinsic erasure information provided by the channel detector. To summarize: the density evolution111See [9] for a rigorous justification of the density evolution analysis. (DE) equation for the case of $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble is given by $\displaystyle x=f((1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}})(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}.$ Note that the term inside the brackets in $f(.)$ represents the probability that a variable node is in erasure as given by the LDPC code. Also it is not hard to see that $f(x)\leq 1$ for any $x$. ###### Example 2 Consider JIT decoding of the DEC with $(5,15)$-regular LDPC ensemble. The design rate of this code is $2/3$. Using the SIR formula ($=1-2\epsilon^{2}/(1+\epsilon)$) from [9] we get that the Shannon threshold at rate=2/3 is given by $\epsilon^{\text{Sh}}_{\text{\tiny DEC}}=0.5$. Figure 1 shows the performance of the JIT decoder. We see that the threshold is given by $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(5,15)\approx 0.363471$, which is far away from the capacity. Throughout the paper we will use $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$ to denote the threshold of the JIT decoder when we use $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble and transmit over the DEC. (50,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(36,0)(0,32)[l] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$$0.0$$\epsilon$ $(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}}$ Figure 1: The BP curve for the $(d_{\mathrm{l}}=5,d_{\mathrm{r}}=15)$-regular ensemble and transmission over the DEC. The threshold of the JIT decoder is given by $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(5,15)\approx 0.363471$. #### The EXIT curve The EXIT curve222To be very precise, we should call the curves we plot as EXIT-like curves. The reason being that we do not provide any operation interpretation of these curves, like the Area theorem [10] in this work. The curves serve only to illustrate the capacity achieving nature of coupled- codes. plots all the fixed-points of the DE equation. The curve is given by the parametric curve $\\{(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}},\epsilon(x)\\}$. We obtain $\epsilon(x)$ by solving for $\epsilon$ in the DE equation. As an example, we plot the EXIT curve for various $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensembles as shown in Figure 2. The JIT threshold is got by dropping a vertical line from the leftmost point on any given curve. We note that for every $\epsilon>\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$, there are exactly 3 fixed-points. One of them being the trivial 0 fixed-point. This “C” shape of the EXIT curve is also what we observe when we transmit through a memoryless BEC using $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensemble. Also we remark that as the degrees increase, keeping the design rate fixed, the JIT threshold keeps on decreasing. This is also the case for transmission over memoryless BEC. In fact, for memoryless BEC case, the BP threshold goes to zero as we increase the degrees. (0,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(-14,0)(0,32)[l] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$$0.0$$\epsilon$ $(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}}$ $(3,9)$ $(5,15)$ $(7,21)$ $(10,30)$ $(30,90)$ Figure 2: The EXIT curve for regular LDPC ensembles with $(d_{\mathrm{l}},d_{\mathrm{r}})$ given by $(3,9)$, $(5,15)$, $(7,21)$, $(10,30)$, $(30,90)$, and transmission over the DEC. We observe that the JIT threshold moves to the left and eventually will go to zero as degrees go to infinity. We can also show the same result for the DEC. More precisely, we have ###### Lemma 3 (JIT Threshold Goes to Zero) For any $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble we have $\displaystyle\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})\leq\sqrt{\frac{1}{\sqrt{d_{\mathrm{r}}-1}(1-(d_{\mathrm{l}}-1)e^{-\sqrt{d_{\mathrm{r}}-1}})}}.$ ###### Proof: We claim that the necessary condition for the JIT decoder to succeed is given by $\epsilon^{2}(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}<x,$ for all $x\in(0,1]$. Indeed, suppose on the contrary that there exists a $c\in(0,1]$ such that the above inequality is violated. Thus we have $\epsilon^{2}(1-(1-c)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}\geq c.$ Since $f(x)\geq\epsilon^{2}$ for all $x\in[0,1]$ we get $f(c)(1-(1-c)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}\geq c.$ This implies that there exists a FP of DE for the DEC for some value in $[c,1]$. It is not hard to see that this implies the JIT decoder will get stuck at this FP, resulting in unsuccessful decoding. Thus we must have that for all $x\in(0,1]$ $\epsilon^{2}(1-(1-x)^{d_{\mathrm{r}}-1})^{d_{\mathrm{l}}-1}<x.$ For the choice of $x=\frac{1}{\sqrt{d_{\mathrm{r}}-1}}$ we get the statement of the lemma. To see this computation first write $(1-x)^{d_{\mathrm{r}}-1}$ as $e^{(d_{\mathrm{r}}-1)\log(1-x)}$. Then use $\log(1-x)\leq-x$ and $x=\frac{1}{\sqrt{d_{\mathrm{r}}-1}}$ to get $(1-x)^{d_{\mathrm{r}}-1}\leq e^{-\sqrt{d_{\mathrm{r}}-1}}$. After this use $\displaystyle(1-e^{-\sqrt{d_{\mathrm{r}}-1}})^{d_{\mathrm{l}}-1}$ $\displaystyle=1-(1-(1-e^{-\sqrt{d_{\mathrm{r}}-1}})^{d_{\mathrm{l}}-1})$ $\displaystyle\geq 1-(d_{\mathrm{l}}-1)e^{-\sqrt{d_{\mathrm{r}}-1}},$ to complete the argument. ∎ As a consequence of Lemma 3 we get that, with the ratio $d_{\mathrm{l}}/d_{\mathrm{r}}$ kept fixed, $\lim_{d_{\mathrm{l}}\to\infty}\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})=0$. ## III Main Results In this section we show, empirically, that spatially coupled-codes achieve the Shannon capacity of the DEC. We recall that we are consider SIR which is give by the formula SIR$=1-2\epsilon^{2}/(1+\epsilon)$. For the sake of exposition, we demonstrate our results only for rate equals $2/3$. The Shannon threshold for this rate is given by $\epsilon^{\text{Sh}}_{\text{\tiny DEC}}=0.5$. For other rates similar results can be observed. From the preceding section we see that standard $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular LDPC ensembles do not saturate the JIT threshold (to the Shannon threshold). We begin by writing down the DE equation for the coupled-codes. ### III-A Density Evolution Consider the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble. Recall that there are $2L+1$ sections of variable nodes. Each section has $M$ variable nodes. We transmit variable nodes sectionwise over the DEC. More precisely, the variable nodes in section $-L$ are transmitted first, followed by variable nodes in section $-L+1$ and so on so forth till we finally transmit all the variable node in section $L$. As a consequence we have a channel detected factor graph sitting on top of each section of the coupled-code. To perform the DE analysis, we already take the limit $M\to\infty$. As a result of this limit, one can ignore the boundary effects of the channel detector and treat the channel detectors as disconnected333Another way to think about this is to imagine that we transmit a known sequence of bits of length equal to the memory of the channel after we transmit all the variable nodes in each section. Since the channel memory is finite, this induces a rate loss going to zero as $M\to\infty$. Now the known sequence is the initial state for each of the channel detectors and hence we can consider them disconnected.. Let $x_{i}$, $i\in\mathbb{Z}$, denote the average erasure probability which is emitted by variable nodes at position $i$. For $i\not\in[-L,L]$ we set $x_{i}=0$. For $i\in[-L,L]$ the DE is given by $\displaystyle x_{i}$ $\displaystyle=\epsilon_{i}\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}-1},$ (1) where $\epsilon_{i}$ is given by $\displaystyle\epsilon_{i}=f\Big{(}\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}}\Big{)},$ (2) where recall that $f(\cdot)$ is the channel extrinsic transfer function. We will use the notation $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ to denote the threshold of the JIT decoder when we use the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble for transmission. As a shorthand we use $g(x_{i-w+1},\dots,x_{i+w-1})$ to denote $\Bigl{(}1-\frac{1}{w}\sum_{j=0}^{w-1}\bigl{(}1-\frac{1}{w}\sum_{k=0}^{w-1}x_{i+j-k}\bigr{)}^{d_{\mathrm{r}}-1}\Bigr{)}^{d_{\mathrm{l}}-1}.$ ###### Definition 4 (FPs of Density Evolution) Consider DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble. Let $\underline{x}=(x_{-L},\dots,{x}_{L})$. We call $\underline{x}$ the constellation. We say that $\underline{x}$ forms a FP of DE with channel $\epsilon$ if $\underline{x}$ fulfills (1) for $i\in[-L,L]$. As a shorthand we then say that $(\epsilon,\underline{x})$ is a FP. We say that $(\epsilon,\underline{x})$ is a non-trivial FP if $\underline{x}$ is not identically equal to $0\,\forall\,i$. Again, for $i\notin[-L,L]$, $x_{i}=0$. ∎ ###### Definition 5 (Forward DE and Admissible Schedules) Consider forward DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble. More precisely, pick a channel $\epsilon$. Initialize $\underline{x}^{(0)}=(1,\dots,1)$. Let $\underline{x}^{(\ell)}$ be the result of $\ell$ rounds of DE. More precisely, $\underline{x}^{(\ell+1)}$ is generated from $\underline{x}^{(\ell)}$ by applying the DE equation (1) to each section $i\in[-L,L]$, $\displaystyle x_{i}^{(\ell+1)}$ $\displaystyle=\epsilon_{i}g(x_{i-w+1}^{(\ell)},\dots,x_{i+w-1}^{(\ell)}).$ We call this the parallel schedule. The important difference with the memoryless BEC case is that the channel $\epsilon_{i}$ is not fixed for the DEC and decreases with increasing iterations according to (2). More generally, consider a schedule in which in each step $\ell$ an arbitrary subset of the sections is updated, constrained only by the fact that every section is updated in infinitely many steps. We call such a schedule admissible. Again, we call $\underline{x}^{(\ell)}$ the resulting sequence of constellations. ∎ One can show that if we perform forward DE under any admissible schedule, then the constellation $\underline{x}^{(\ell)}$ converges to a FP of DE and this FP is independent of schedule. This statement can be proved similar to the one in [2]. ### III-B Forward DE – Simulation Results We consider forward DE for the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ ensemble. More precisely, we fix an $\epsilon$ and initialize all $x_{i}$ for $i\in[-L,L]$ to 1. Then we run the DE given by (1) till we reach a fixed- point. We fix $L=250$. For $d_{\mathrm{l}}=3$ and $d_{\mathrm{r}}=9$, we have that $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(3,9,300,3)\approx 0.49815$. If we increase the degrees we get $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(5,15,300,5)\approx 0.49995$, $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(7,21,300,7)\approx 0.499989$ and $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(9,27,300,9)\approx 0.499996$. We observe that for increasing the degrees the threshold approaches the Shannon threshold of $0.5$. ### III-C The EXIT Curve for Coupled Ensembles We now come to the key point of the paper, the computation of the EXIT curve. Before we do this, we define the entropy of a constellation $\underline{x}=(x_{-L},\dots,x_{L})$ as $\displaystyle\chi=\frac{1}{2L+1}\sum_{i=-L}^{L}x_{i}.$ To plot the EXIT curve we first fix $\chi\in[0,1]$ and then run DE such that the resulting FP constellation has entropy equal to $\chi$. This is the reverse DE procedure as described in [11]. We remark that $f(x)$ is an increasing function of $\epsilon$, hence in the reverse DE procedure one can easily find an appropriate $\epsilon$ by the bisection method. Figure 3 shows the plot of the EXIT curve for the $(5,15,L,5)$ ensemble with $L=2,4,8,16,32,64,128,256,512$. We see that the curves look very similar to the curves when transmitting over a BMS channel. For very small values of $L$, the curves are far to the right due to significant rate loss that is incurred at the boundary. As $L$ increases the rate loss diminishes and the JIT threshold is very close to the Shannon threshold. This picture strongly suggests that the same threshold saturation effect ($\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\approx\epsilon^{\text{\tiny MAP}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$) also occurs for the DEC as it was shown analytically in [2]. $L\\!=\\!2$ $L\\!=\\!4$ $L\\!=\\!8$ $L\\!=\\!16$ (0,-8)(32,0)[cb] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$(-16,0)(0,32)[l] ,$0.2$,$0.4$,$0.6$,$0.8$,$1.0$ EXIT Figure 3: The EXIT curve for the $(d_{\mathrm{l}}=5,d_{\mathrm{r}}=15,L,5)$ ensemble and transmission over the DEC for $L=2,4,8,16,32,64,128,256,512$. The curves keep moving to the left as $L$ increases similar to the curves when transmitting over BMS. The “vertical” drop in the EXIT curves occurs at $\approx 0.5$ for $L\geq 32$. Also shown in light gray is the BP exit curve for the uncoupled $(5,15)$-regular ensemble. ### III-D Shape of Fixed Point of Density Evolution We plot the constellation representing the unstable FP of DE. This FP cannot be reached via forward DE and is obtained via reverse DE procedure. We recall that this FP played a key role in proving the threshold saturation phenomena when transmitting over the BEC. Let us describe the (empirically observed) crucial properties of this constellation. * (i) The constellation is symmetric around $i=0$ and is unimodal. The constellation has $\epsilon\approx 0.49995$. * (ii) Let $x_{\text{s}}(\epsilon)$ denote a stable FP of DE. The value in the flat part in the middle is $\approx 0.4434$ which is very close to the stable FP of DE for the underlying uncoupled $(5,15)$-regular ensemble. * (iii) The transition from close to zero to close to $x_{\text{s}}(\epsilon)$ is very quick. (6,0)(14.4,0)[b]$\text{-}16$,$\text{-}14$,$\text{-}12$,$\text{-}10$,$\text{-}8$,$\text{-}6$,$\text{-}4$,$\text{-}2$,0,2,4,6,8,10,12,14,16 Figure 4: The constellation representing FP of DE for $(5,15,33,5)$ ensemble and entropy fixed to $\chi=0.2$. This is an unstable FP constellation. The constellation is very similar to any unstable FP constellation when transmitting over memoryless BEC. The constellation is unimodal. There is a long tail of zeros followed by a sharp transition and then a long flat part with values close to $x_{\text{s}}(\epsilon)$. The constellation has $\epsilon\approx 0.49995$. ## IV A Possible Proof Approach Till now we gave empirical evidence of the threshold saturation phenomena when transmitting over the DEC using coupled-codes. Before we proceed to give the proof idea for the threshold saturation, we first show that coupling indeed helps. More precisely we have the following lemma, ###### Lemma 6 (Spatial Coupling Helps) For $d_{\mathrm{l}},d_{\mathrm{r}}\to\infty$ with the ratio $d_{\mathrm{l}}/d_{\mathrm{r}}$ kept fixed, we have $\displaystyle\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\geq\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}.$ ###### Proof: Since $\epsilon_{i}$ is an increasing function of $x_{i-w+1},\dots,x_{i+w-1}$, we have $\epsilon_{i}\leq f(1)\leq\frac{4\epsilon^{2}}{(1+\epsilon)^{2}}\leq\epsilon.$ Combining this with the DE equation for the coupled-codes, we get $\displaystyle x_{i}\leq\epsilon g(x_{i-w+1},\dots,x_{i+w-1}),$ for all $i\in[-L,L]$. But we know from Theorem 10 in [2] that $\lim_{d_{\mathrm{l}}\to\infty}\epsilon^{\text{\tiny BP}}_{\text{BEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\to\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}$. Thus for $\epsilon<\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}$ the right-hand-side of the above inequality goes to zero. Hence the lemma. ∎ As an example, consider the $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble with $d_{\mathrm{l}}/d_{\mathrm{r}}=1/3$ (rate equal to $2/3$) . For $L\to\infty$, the rate of the $(d_{\mathrm{l}},d_{\mathrm{r}},L,w)$ goes to $2/3$. From Lemma 3 we have that $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})\to 0$ and from Lemma 6 we have that $\epsilon^{\text{\tiny JIT}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}},L,w)\geq\frac{d_{\mathrm{l}}}{d_{\mathrm{r}}}=\frac{1}{3}$. Thus spatial coupling indeed boosts the JIT threshold. However the empirical evidence suggests that the boost is all the way up to the Shannon threshold (which is $0.5$ in this case). Since there is ample similarity between the DEC and the BEC, the guideline for a proof is similar to when we are transmitting over the BEC. (i) Existence of FP: A key ingredient in proving the result for the BEC was to show the existence of a special FP of DE $(\underline{x},\epsilon^{*})$. In principle, the BEC proof should extend. The only difference is that instead of a constant channel $\epsilon$, we have a channel value which depends on the FP constellation itself. However, since the functions involved are rational, this should not be a big hurdle. (ii) Shape of the constellation and the transition length: The next task is to show that the FP guaranteed by the above theorem has the properties as given in Section III-D. Proving this would first involve showing that the underlying regular ensemble has a “C” shaped EXIT curve. Intuitively, this means that the FP constellation (of the coupled-code) can only hover around the stable FPs of DE (of the underlying regular ensemble), implying that it has either a large tail of zeros or a large flat part with values close to $x_{\text{s}}(\epsilon^{*})$. (iii) Construction of the EXIT curve and the Area Theorem: Another key part of the BEC proof was to construct a family of FPs (not necessarily stable FPs) using the special FP guaranteed by the Existence theorem. The EXIT curve plus the fast transition would allow us to show that this special FP must have an associated channel parameter, $\epsilon^{*}$, very close to the Shannon threshold (for large degrees.)444For finite degrees, $\epsilon^{*}$ should be very close to the MAP threshold of the $(d_{\mathrm{l}},d_{\mathrm{r}})$-regular ensemble. One should be able to prove this by formulating an appropriate Area theorem (see Section 3.20 in [10]). Operational interpretation: The proof would be completed by providing an operation meaning to the EXIT curve. Loosely speaking, the EXIT constructed above would have a vertical drop at $\epsilon\approx\epsilon^{\text{Sh}}(d_{\mathrm{l}},d_{\mathrm{r}})$ (cf. Figure 3). This would help to show that for any $\epsilon<\epsilon^{\text{Sh}}(d_{\mathrm{l}},d_{\mathrm{r}})$, the JIT decoder will go to the trivial FP. ## V Conclusions In this paper we show that empirically coupled-codes saturate the JIT threshold on the DEC. For the channel extrinsic transfer function we consider the case when there is no precoding. We list below some comments and open questions. * (i) An obvious future direction is to complete the proof of threshold saturation. The guidelines provided above serve as a starting point. Following this route, in principle, it should be possible to prove the capacity achieving nature of these codes on the DEC. * (ii) Another interesting question is that whether the threshold saturation phenomena can be shown to be true for all channel extrinsic transfer functions $f(.)$ which are non-decreasing both in $\epsilon$ and $x$ (threshold saturation holds when $f(.)$ represents precoding). * (iii) A proof of the threshold saturation phenomena should also pave the way for the justification of the Maxwell construction to determine $\epsilon^{\text{\tiny MAP}}_{\text{\tiny DEC}}(d_{\mathrm{l}},d_{\mathrm{r}})$ for the DEC. * (iv) Recently, it was observed that coupled MacKay-Neal (MN) codes with bounded degree exhibit the BP threshold very close to the Shannon threshold over the BEC [12]. It is interesting to see if the coupled MN codes have the JIT threshold close to the SIR over the DEC. ## VI Acknowledgments SK acknowledges support of NMC via the NSF collaborative grant CCF-0829945 on “Harnessing Statistical Physics for Computing and Communications.” SK would also like to thank Rüdiger Urbanke, Misha Chertkov and Henry Pfister for their encouragement. ## References * [1] A. J. Felström and K. S. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix,” _IEEE Trans. Inform. Theory_ , vol. 45, no. 5, pp. 2181–2190, Sept. 1999. * [2] S. Kudekar, T. Richardson, and R. Urbanke, “Threshold saturation via spatial coupling: Why convolutional LDPC ensembles perform so well over the BEC,” 2010, e-print: http://arxiv.org/abs/1001.1826. * [3] M. Lentmaier and G. P. Fettweis, “On the thresholds of generalized LDPC convolutional codes based on protographs,” in _Proc. of the IEEE Int. Symposium on Inform. Theory_ , Austing, TX, USA, June 2010, pp. 709–713. * [4] S. Kudekar, C. Méasson, T. Richardson, and R. Urbanke, “Threshold saturation on BMS channels via spatial coupling,” Apr. 2010, e-print: http://arxiv.org/abs/1004.3742. * [5] S. Kudekar and H. D. Pfister, “The effect of spatial coupling on compressive sensing,” in _Proc. of the Allerton Conf. on Commun., Control, and Computing_ , Monticello, IL, USA, 2010. * [6] S. H. Hassani, N. Macris, and R. Urbanke, “Coupled graphical models and their thresholds,” in _Proc. of the IEEE Inform. Theory Workshop_ , Dublin, Ireland, Sept. 2010. * [7] V. Rathi, R. Urbanke, M. Andersson, and M. Skoglund, “Rate-equivocation optimally spatially coupled LDPC codes for the BEC wiretap channel,” 2010, e-print: http://arxiv.org/abs/1010.1669. * [8] H. D. Pfister, “On the capacity of finite state channels and the analysis of convolutional accumulate-$m$ codes,” Ph.D. dissertation, UCSD, San Diego, CA, USA, 2003. * [9] H. D. Pfister and P. H. Siegel, “Joint iterative decoding of LDPC codes for channels with memory and erasure noise,” _IEEE J. Sel. Area. Commun._ , vol. 26, no. 2, pp. 320–337, Feb. 2008. * [10] T. Richardson and R. Urbanke, _Modern Coding Theory_. Cambridge University Press, 2008. * [11] C. Méasson, A. Montanari, T. Richardson, and R. Urbanke, “The generalized area theorem and some of its consequences,” _IEEE Trans. Inform. Theory_ , vol. 55, no. 11, pp. 4793–4821, Nov. 2009. * [12] K. Kasai and K. Sakaniwa, “Spatially-coupled bounded-density capacity-achieving codes,” in _Proc. Symp. on Inf. Theory and its Applications_ , Dec. 2010, pp. 1–6, (in Japanese).
arxiv-papers
2011-02-02T11:26:03
2024-09-04T02:49:16.794722
{ "license": "Public Domain", "authors": "Shrinivas Kudekar and Kenta Kasai", "submitter": "Kenta Kasai", "url": "https://arxiv.org/abs/1102.0406" }
1102.0432
11institutetext: Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Svientsitsky str.1, 79011 Lviv, Ukraine; 11email: pavlenko@mailaps.org 22institutetext: Institute for Applied Mathematics and Fundamental Sciences, Lviv Technical University, Ustyianowycha str. 10, 79013 Lviv, Ukraine # Interstitial Fe-Cr alloys: Tuning of magnetism by nanoscale structural control and by implantation of nonmagnetic atoms Interstitial Fe-Cr alloys N. Pavlenko 11 N. Shcherbovskikh 22 and Z.A. Duriagina 22 ###### Abstract Using the density functional theory, we perform a full atomic relaxation of the bulk ferrite with $12.5\%$-concentration of monoatomic interstitial Cr periodically located at the edges of the bcc Feα cell. We show that structural relaxation in such artificially engineered alloys leads to significant atomic displacements and results in the formation of novel highly stable configurations with parallel chains of octahedrically arranged Fe. The enhanced magnetic polarization in the low-symmetry metallic state of this type of alloys can be externally controlled by additional inclusion of nonmagnetic impurities like nitrogen. We discuss possible applications of generated interstitial alloys in spintronic devices and propose to consider them as a basis of novel durable types of stainless steels. ## 1 Introduction Last years demonstrate increased activities in the search for novel materials exhibiting controlled modification of electronic properties by inclusion or implantation of different atoms or ionic groups. A prominent example of the implantation-altered systems is the stainless steel. In the steels, the implantation of chromium, molibdenium, nitrogen and other chemical elements substantially changes the microstructure of subsurface layers and modify their corrosion resistance and hardness steels . In the development of novel efficient multifunctional materials for technological applications in the long-term devices, the properties like hardness, corrosion, heat resistance and other types of mechanical and chemical durability are of central interest mai ; yokokawa . It frequently appears in science and technology that well known materials doped by different chemical elements exhibit unexpected physical properties not revealed previously. As an example of such a new unexpected behavior, in the present work we consider an alloy Fe-Cr. The alloys of Fe and Cr, doped by C, Ni and by other elements, are widely used as basic components for ferritic and martensitic steels. Substitutional alloys of Fe and Cr have attracted much attention of theory and experiment due to their rich magnetic properties characterized by local antiferromagnetism in the proximity of Cr atoms implanted into ferromagnetic iron victora ; paxton ; paduani ; davies . Due to small differences between the atomic radii of iron and chromium, the modification of the substitutional alloy properties is limited to the local magnetic transformation due to local changes in the electronic orbital occupancies, without significant structural modifications. In contrast to the substitutional structural configurations, the interstitial Fe-Cr alloys considered in the present work contain Cr impurities which are located in the interstitial positions of the bcc lattice of Feα. In the recent theoretical studies of the Cr intersitials in Fe-Cr alloys, different types of interstitial configurations were analyzed. Among them, a pair configuration $\langle 111\rangle$ dumbbell is considered as the most energetically favourable which requires about 4.2eV for its formation under irradiation klaver ; olsson . In the present work, we consider a novel monoatomic interstitial configuration which contains single Cr atoms positioned in the centers of the edges of the bcc ferrite. In contrast to the substitutional alloys, the significant forces due to the interstitial atoms induce substantial structural optimization which enhances the volume due to modified lattice constants and leads to the relaxation of the atomic positions in the unit cell. We find that the relaxation of the initial bcc unit cell results in significant atomic distortions and in the formation of atomic chain-like structures. As appears in the density-functional-theory (DFT) calculations of the optimized structures, the energy gain achieved due to the structural relaxation of the considered interstitial alloy can approach 6.17 eV which makes this type of systems highly stable and durable. In the present work, we propose to consider these artificially generated alloys as candidates for novel types of stainless steels. The fundamental difference between the industrial alloys and the alloys studied in the present work is the ordered and periodic character of the latters. In the industrial steels, the amorphic character of the systems is related to the random distribution of the impurities. The hardening of the steels proceeds through the surface treatment and is accompanied by formation of granular microstructure with the spatially inhomogeneous impurity concentration and modified subsurface properties afm . In the studies of the subsurface Cr-doped alloyed ferrite, we consider the supercells containing periodically located Cr atoms in the cubic lattice of Feα. The interstitial Cr induces significant atomic reconstruction with consequent break of initial cubic symmetry and stabilization of a new lower-symmetry state. The appearing structural transformation has a character of a phase transition which occurs due to nanoscale tailoring of cubic Fe by interstitial inclusion of Cr atoms, the effect which can be experimentally verified by the means of modern methods like AFM spectroscopy . Using the DFT-based structural optimization, we obtain the optimized atomic microstructure of a chain-like character where the chains of octahedrically arranged Fe atoms are formed along the (001)-axis. We find that the competing ferromagnetic and antiferromagnetic interactions lead to spatially inhomogeneous spin polarization. The magnetization of the structurally relaxed system is significantly enhanced as compared to the pure ferrite without Cr inclusions. The obtained enhancement makes the generated alloys perspective candidates for spin polarizers in spintronic applications. In the generated chain-like structures, the relaxation is accompanied by the formation of spatial channels with extremely low carrier density. We suggest that these channels can be considered as paths for the low-barrier-migration of light impurities like H, N, Li or C. As an example of a light atom in the interstitial alloy, we study of the migration paths of nonmagnetic nitrogen and calculate the energy barriers along the migration paths. We obtain a strong influence of the nonmagnetic N on the alloy magnetization. Our findings show that the structural modifications due to possible nanoscale tuning of Cr impurities on the edges of bcc cubic cells of iron can play a central role in the control of their electronic properties. ## 2 Structural relaxation of interstitial alloy Feα-Cr The present studies of the electronic properties of the considered interstitial Fe-Cr alloy are based on the DFT calculations of the electronic structure of the systems generated by periodic translation of specially chosen supercells. The initial supercell shown in Fig. 1 contains the doubled $2\times 2$ cubic bcc cell of ferrite (Feα) and a single Cr atom centered in one of the edges of the Feα cubic unit cell with the lattice constant $a=3.85$ Å. The obtained structure is described by a chemical formula Fe8Cr and determines an interstitial Fe-Cr alloy with the Cr concentration $n=0.125$ which is typical for stainless steels. The presence of interstitial Cr leads to significant local forces acting on the neighbouring Fe atoms. To minimize the forces, the coordinates of all atoms have been relaxed. In the present studies, the optimization of the supercell has been performed by employing the DFT approach implemented withing the linearized augmented plane wave (LAPW) scheme in the full potential Wien2k code wien2k . To study the role of the spin polarization in the structural relaxation, two different relaxation procedures have been employed. In the first procedure, the atomic optimal positions are calculated in the local density approximation (LDA) on a $2\times 2\times 5$ k-points grid. To explore the role of spin degrees of freedom in the relaxation, in the second procedure the local spin density approximation (LSDA) has been used in the optimization of the structure. The results of both methods of the structural relaxation are presented in Fig. 2. Figure 1: Schematic view of unrelaxed 2$\times$2 Feα cell which contain 12.5$\%$ of edge-centered interstitial Cr. A central common feature which characterizes both (LDA- and LSDA-relaxed) structures is the clusterization of the sublattice of the iron atoms. In the LDA-optimized structure (Fig. 2(a)), the relaxation results in formation of a high-symmetry clusterized network. This network consists of the Fe6-octahedra which form the square plaquettes in the $(x,y)$ ($(a,b)$) plane with Cr atoms located in the center of each plaquette. The distance from the centered Cr to each nearest iron octaherda amounts $1.9$ Å. Despite the significant displacements of the iron atoms from their initial positions, the net electric polarization of the cell is zero due to high structural symmetry C4/m obtained after the relaxation. The formation energy of the relaxed Fe8Cr-configuration can be expressed as $\displaystyle E_{f}({\rm LDA})=E_{{\rm tot}}({\rm Fe}_{8}{\rm Cr})-8E_{{\rm tot}}({\rm Fe})-E_{{\rm tot}}({\rm Cr}),$ where the last two terms identify the total energies of the bulk bcc Feα and Cr, respectively. To determine $E_{{\rm tot}}({\rm Fe})$, we have calculated the total energy value of the bulk Feα in the ferromagnetic state. As the LSDA-calculation of the spin-polarized configurations of the bulk Cr are converged to the paramagnetic state, we consider the total energy $E_{{\rm tot}}({\rm Cr})$ for the paramagnetic Cr. With these values, we find that $E_{f}({\rm LDA})=4.82$eV. To analyze the role of the relaxation, we have also calculated the energy $E_{f}({\rm unrel})$ of the formation of initial unrelaxed configuration which is equal to 5.02eV. As a consequence, the significant energy gain due to the structural relaxation $\displaystyle\Delta E({\rm LDA})=E_{f}({\rm unrel})-E_{f}({\rm LDA})=0.196eV,$ shows a central importance of the atomic displacements for the stability of the considered systems. The optimization procedure based on the LSDA approach accounts for additional corrections due to spin polarization and produces new ordered structural patterns presented in Fig. 2(b) and Fig. 2(c) for two different (unrelaxed $a=b=2.86$Å and relaxed $a=b=3$Å) lattice constants. The volume-optimized structure (c) is signified by the 13%th increase of the unit cell volume due to the insertion of the interstitial Cr. The LSDA-optimized structural pattern is characterized by the chains of atomic Fe-groups along the $x$($a$)-direction, each group containing six Fe-atoms. The nearest chains are separated by a distance about $4$Å and are connected to each other by the Fe- Cr bonds of the length about $2.4$Å for the structure (b) with $a=2.86$Å, and $2.7$Å for the structure (c) with the optimized $a=3.0$Å. The local antiferromagnetic ordering in the vicinity of Cr is characterized by the magnetic moments $\mu_{Cr}=-0.72$ $\mu_{B}$ and $\mu_{5}=2.4$ $\mu_{B}$ and $\mu_{6}=1.25$ $\mu_{B}$ of the neighbouring atoms Fe5 and Fe6, respectively. The magnetic moments of more distant iron atoms have the values around 2.5 $\mu_{B}$, which is close to results obtained for substitutional alloys and in pure Feα klaver . As compared to the tetragonal structure of the LDA-optimized system, the chain-like structure of the LSDA-relaxed supercell is characterized by substantially lower crystal symmetry and by the absence of the inversion center. In contrast to the LDA-based configurations, the formation energy of the LSDA-relaxed Fe8Cr configuration $E_{f}({\rm LSDA})=E_{{\rm tot}}({\rm Fe}_{8}{\rm Cr})-8E_{{\rm tot}}({\rm Fe})-E_{{\rm tot}}(Cr)=-1.15$eV is negative which implies its high stability. We can also calculate the energy gain due to the structural relaxation by the LSDA approach $\displaystyle\Delta E({\rm LSDA})=E_{f}({\rm unrel})-E_{f}({\rm LSDA})=6.17eV,$ which also demonstrates the high stability of the relaxed spin-polarized structure and a necessity to account for a spin polarization in the structural optimization of the systems with strong magnetoelastic effect. ### 2.1 Magnetic properties In the considered systems, we have also analyzed modification of the local magnetic properties due to the relaxation of the interatomic distances. To see how the atomic displacements influence the spin polarization of the surrounding atoms, in Fig. 3 we present the dependences of the local moments of Cr and of two nearest neighboring Fe on the Cr displacement along the bond [Fe5-Cr-Fe6] $\displaystyle\Delta=[x({\rm Cr})-x({\rm Fe5})]-[x({\rm Cr})-x({\rm Fe5})]^{0},$ where $[x({\rm Cr})-x({\rm Fe5})]^{0}$ is the optimized [Fe5-Cr]-bond length. The increase of $\Delta$ leads to the change of $\mu_{\rm Cr}$ from -0.7 $\mu_{B}$ to the value about -0.73 $\mu_{B}$. In addition, the larger $\Delta$ implies the elongation of the [Fe5-Cr] bond and lead to the reduced $\mu_{5}=2.39$ $\mu_{B}$ due to the tendency for a suppression of antiferromagnetism in the vicinity of Fe5. The increase of $\Delta$ also produces an enhancement of $\mu_{6}$ from 1.25 $\mu_{B}$ to the values about $1.28-1.3$ $\mu_{B}$, an opposite trend which occurs due to the shortening of the bond between Cr and Fe6. In Fig. 3, the $\Delta$-dependences of the atomic magnetic moments are highly asymmetric with respect to $\Delta$. Consequently, the obtained magnetoelastic coupling produces an anisotropy of the magnetic moments and is accompanied by the loss of the inversion center due to the atomic displacements, the effect which can be observed in Fig. 2(b) and (c). In Fig. 2(c), the low-symmetry structure corresponds to the minimum of the total energy. As a conclusion, the neglect of the magnetoelastic coupling in the electronic structure calculations does not allow to achieve a full optimization in this type of interstitial alloys. Figure 2: Relaxed structure of Fe with 12.5$\%$ of Cr: (a) LDA calculations, (b) spin-polarized LSDA calculations in the structure with $a=b=2.86$ Å and (c) spin-polarized LSDA calculations in the structure with $a=b=3.0$ Å. The path 1 and path 2 identify possible pathes for diffusion through the channels formed due to atomic relaxation. ### 2.2 Electronic structure Fig. 4 shows the $3d$ spin-polarized electronic density contours of the LSDA- optimized structure in the ($x$, $z$) plane. One can see that the majority $3d$ spin-up states of Fe are highly occupied by the electrons whereas the electron concentration of Cr spin up states is substantially lower. In contrast to this, the spin-down (minority) electrons are characterized by high electron occupation of Cr and lower electron density on Fe. In Fig. 4, the chain-like structures Fe-Cr in the $z$-direction are characterized by strong hybridization between the intra-chain $3d$ spin-down orbitals of Fe and Cr. The last feature leads to the spatial charge redistribution and to higher charge densities on the bonds between spin-down Cr and Fe. In the LSDA- optimized system, the structural optimization produces areas with low charge density in the $y$ ($b$)-direction, where each area can be identified between the chains of Fe-octahedra. As can be seen in Fig. 4, these areas are almost free of the charge and can be considered as channels for the migration of light atoms like H, Li or N. Similarly to the contours in Fig. 4, the electron density of the majority Fe and Cr orbitals and on the bonds between Cr and Fe calculated for the LDA-optimized structure (Fig. 5) is substantially lower than the charge density on the spin-down contours, although the spatial charge distribution is more homogeneous as compared to that in Fig. 4. Figure 3: Local magnetic moments (in $\mu_{B}$) of the atoms in Fe5-Cr-Fe6 triad versus the displacement $\Delta$=[Fe5-Cr]-[Fe5-Cr]0 of Cr along the (100) axis. Here [Fe5-Cr]0 is the equilibrium distance between Fe6 and Cr. Figure 4: Contours of electron density maps in the ($x$, $z$)-plane ($y/b=0.25$, $x$ and $z$ given in Å) obtained by integration of electronic states in the energy window $E$ between $-3$ eV below the Fermi level and the Fermi level. The results obtained by the structural optimization using the LSDA approximation. Figure 5: Contours of electron density maps in the ($x$, $z$)-plane ($y/b=0.25$, $x$ and $z$ given in Å) calculated by integration of electron states in the energy window $E$ between $-3$ eV below the Fermi level and the Fermi level. The LSDA results obtained in the initially LDA-relaxed structure. For the LDA-relaxed structure, the density of states is characterized by strong suppression of the majority spin-up DOS at the Fermi level (Fig. 6(a)), whereas the minority DOS at the Fermi level remains significant. Similar, although much stronger, suppression of majority DOS is typically observed in half-metallic systems where the electric current is conducted by the electrons with the same direction of spin park . In contrast to the half-metallic-like features of the LDA-relaxed structure, the DOS of the LSDA-optimized system (Fig. 6(b)) demonstrates substantial values at the Fermi level for both spin directions which implies an enhancement of the metallic state for the majority electrons. In transition metal oxides, the metallic state obtained in the LDA approach is usually strongly influenced by additional account for the local Coulomb corrections for the $d$-electronic states anisimov ; czyzyk ; pavlenko3 ; pavlenko4 . In our work, the Coulomb corrections are incorporated within the SIC-variant of the LSDA+$U$ approximation introduced in Ref. anisimov . The results are presented in Fig. 7 for two different values of $U=2$ eV and $U=4.5$ eV estimated and employed in Ref. bandyopadhyay ; zhang ; korotin to account for the electron repulsion of 3d electrons of Fe and Cr. Fig. 7 shows the finite density of states at the Fermi ($E=0$) level, although larger $U$ leads to a significant suppression of the majority DOS at $E_{F}$ which suggests a prevailing tendency towards a half-metallic behavior. In the LSDA-optimized structure, we find that the cell magnetic moment $M_{\rm LSDA}=3.84$ $\mu_{B}$ is larger then the magnetic moment $M_{\rm LDA}=2.88$ $\mu_{B}$ in the LDA-optimized cell. Such an enhancement of the magnetic polarization is connected with the substantial distortions $\Delta\boldsymbol{R}_{i}$ in the range $0.2-0.84$ Å and can be considered as a direct evidence of significant magnetoelastic effect. The local Coulomb corrections in the LSDA+$U$-calculations result in enhanced spin polarization. Specifically, we obtain $M_{\rm LSDA}=4.07$ $\mu_{B}$ for $U=2$ eV, and $M_{\rm LSDA}=4.66$ $\mu_{B}$ for $U=4.5$ eV. It is remarkable that in the substitutional alloy Fe-Cr with 12.5$\%$ of Cr, the LSDA approach gives the value 3.8 $\mu_{B}$ for the cell magnetic moment which is slightly lower than the magnetic moment for the considered LSDA-relaxed substitutional alloy. Figure 6: Total density of states (in eV-1) for structures optimized using (a) LDA approach and (b) spin-polarized LSDA approximation. The Fermi level corresponds to $E=0$. Figure 7: Total densities of states (in eV-1) for the LSDA-optimized structure calculated by the LSDA+$U$ method with the local Coulomb corrections for the 3d-orbitals of Fe and Cr $U=2$ eV (black curves) and 4 eV (blue curves). The Fermi level corresponds to $E=0$. The obtained high spin polarization of the considered interstitial alloys Fe- Cr allows us to suggest these materials as possible candidates for spin polarizers in the spintronic devices. Possible technological applications of artificially generated interstitial alloys would be related to the thin films produced for the needs of modern electronic industry. In such artificial systems, a central question is related to the methods of implantation and positioning of Cr in centers of the edges of cubic bcc lattice of bulk Fe. With the current state of the art, such structural nanoscale manipulation can be confined to the first subsurface layers of ferrite films by the using for instance the methods of optical trapping by lasers optic or by atomic force microscopy afm . To estimate the stability of ultra-thin films of iron- chromium alloys which can be also considered as a basis for stainless steels, we have also extended our calculations to the nanoscale two-monolayers-thick iron films containing one interstitial Cr atom per each 20-25 subsurface Fe atoms pavlenko2 . For such type of films, we have performed the calculations of the surface formation energy and of the electronic work functions, which were also compared to the corresponding quantities in the films of standard substitutional Fe-Cr alloys. In the interstitial Fe-Cr films, we find that the energy of the surface formation is about 1.69eV and the electronic work function amounts to 1eV, whereas for the substitutional Fe-Cr films we obtain 2.56eV for the surface energy and 0.57eV for the work function. These results allow us to expect high stability and durability of films generated on the basis of nanoscale-manipulated interstitial Fe-Cr alloys, as compared to the iron films with substitutional Cr impurities. ## 3 Migration paths of N in interstitial alloys Feα-Cr A central question related to the stability of the considered interstitial alloys is how various atomic impurities can modify the electronic properties. In the structurally relaxed alloys, the chains of atomic Fe-groups are separated by 4Å-wide atomic empty channels, which are expected to contain pathways for light impurity atoms like H, N or Li. To explore a possibility of the migration of the impurities, we consider possible migration paths of a single nitrogen atom in the vicinity of the atomic Fe-chains in the interstitial Feα-Cr. In the studies of the migration paths of N impurities, we employed the nudged- elastic-band (NEB) method implemented in the Quantum-Espresso (QE) Package for the DFT calculations with the use of plane-wave basis sets and pseudopotentials qe ; jonsson . In these calculations, for the atomic cores of Fe and Cr we employ the Perdew-Burke-Ernzerhof (PBE) norm-conserving pseudopotentials pbe . For each stage of the nitrogen transport, the NEB method involves a relaxation of the atomic positions and of the distances between the different atoms in the supercell until the forces acting on the atoms reach their minima. In these calculations, we use the plane-wave cutoff 680 eV and the energy cutoff for charge and potential given by 1360 eV. In the NEB-approach, the relaxation of the atomic positions along the nitrogen migration path is performed by the minimization of the total energy of each intermediate configuration (image). These images correspond to different positions of N on the migration path and they are produced by the optimization of a specially generated object functional (action) with the consequent minimization of the spring forces perpendicular to the path. In our calculations, the convergence criteria for the norm of the force orthogonal to the path is achieved at the values below 0.05eV/Å. As the initial atomic configuration, the supercell Fe-Cr relaxed by the full-potential LSDA-approach wien2k has been considered. Recent studies of the migration paths of single hydrogen atoms by the pseudopotential NEB method demonstrate a good agreement of the obtained transport mechanisms and energy barriers with the experimental measurements pavlenko . Similarly to Ref. pavlenko , in the present studies, each NEB- generated configuration has been modified by the introduction of the N atom and the obtained in this way extended supercell has been fully structurally optimized. To study the migration of N, we consider two different migration paths across the atomic Fe-chains in the Fe-Cr supercell. The first path (path (a)) describes the migration of N from the initial position inside the cell ($z/c=0.5$) near the chain (1) across the channel to the chain (2) schematically presented in Fig. 8(a). In distinction to the path (a), the second path (b) reflects path2 in Fig. 2(c) and contains additional migration step of N from the supercell boundary ($z\approx 0$) along the $c$-direction inside the supercell, with the further relocation through the channel to the atomic Fe-chain (2) indicated in Fig. 8(b). Fig. 9 shows the profiles of the total energy calculated along the N migration paths (a) and (b). The path (a) is characterized by the high energy barriers about $0.8$ eV in the path coordinate range ($0\leq r/R_{N}\leq 0.4$) and ($0.8\leq r/R_{N}\leq 1$) which corresponds to the migration of N within the two Fe-chains (1) and (2). Here $R_{N}$ denotes the maximal length of the N path in the supercell which reaches about 1 nm for the path (a). The interchain motion inside the channel is signified by a low energy barrier about $0.2$ eV ($0.4\leq r/R_{N}\leq 0.7$ in Fig. 9(a)). In contrast to this, the energy profile for the migration path (b) (Fig. 9(b)) contains a plateau-like region at $0.1\leq r/R_{N}\leq 0.4$. This miration step indicates the intracell replacement of N near the Fe-chain (1) along the [001]-direction demonstrated in Fig. 4, with a further relocation between the atomic chains across the atomic-empty channel with a low energy barrier about $0.2$ eV. As a conclusion, we can note that the possible migration paths of the light atoms in the considered interstitial Fe-Cr alloys contain a combination of the motion (i) within the atomic-empty channels and (ii) along the $c$-direction along to the Fe-contained atomic chains. Figure 8: Two different migration paths of nitrogen through the channel of the optimized crystal cell of Fe-Cr. The top picture (a) represents the interchain migration of N inside the cell with the coordinate $z/c$ near $0.5$. The bottom picture (b) corresponds to the migration of N from the cell boundary ($z=0$) along the $z$ direction with the further migration between two neighbouring Fe-chains. The symbols (1) and (2) denote the different atomic chains; (i) and (f) correspond to the initial and final positions of nitrogen in the migration paths. Figure 9: Total energy profiles of the system along the migration paths of N. Here $E_{0}$ denotes the energy of the system in the initial position of N and $R_{N}$ is the N coordinate in the final position of the path. Figure 10: Cell ($M_{\rm tot}$) and atomic magnetic moments (in $\mu_{B}$) along the migration paths of N. The red arrows identify the maximal cell magnetic moments approached upon the minimization of the distance [N-Cr] along the N migration paths. The question which arise due to the inclusion of N into the magnetic Fe-Cr alloy is how the N impurities modify the magnetic properties of the system. In the work of I. Mazinmazin , a comparison of the degrees of spin-polarization (DSP) calculated for Fe in the static limit through the density of electronic states, via the current densities and in the ballistic limit is presented. It is shown that all three definitions of the DSP give very similar behavior for Fe due to strong hybridization of the $sp$ and $d$ states at the Fermi level. Thus we expect that in the considered Fe alloy with relatively low concentration of Cr it is sufficient to study the static spin polarization in order to capture the main properties of the alloy. Fig. 10 presents the change of the cell and atomic magnetic moments at the migration of N along the path (a) and path (b). Although the nitrogen is initially nonmagnetic in the bulk, it becomes weakly magnetic inside the Cr-Fe alloy with a small magnetic moment $-0.04$ $\mu_{B}$ induced by the magnetism of the surrounding. It is noteworthy that the cell magnetic moment is increased to 4 $\mu_{B}$ as N approaches Cr and the distance [N-Cr] becomes about 1.95 Å. Such an enhancement of $M_{tot}$ is explained by the strong atomic distortions in the range between $0.04$ Å(Fe7) up to $0.2$ Å(Fe3) caused by the replacement of N and by the consequentmagnetoelastic effect. In Fig. 10, the increase of the distance from N to Cr suppresses the magnetic moment of N and decreases the cell magnetic polarization to the typical values about $3.5-3.8$ $\mu_{B}$ obtained in LSDA-calculations for the artificial Fe-Cr alloys. The obtained drastic change of the magnetic polarization clearly demonstrates a crucial importance of the location of nonmagnetic impurities like $N$ for the electronic properties of alloy. As follows from our findings, a control of the location of N, for example by external electric field, can lead to externally tuned changes of the magnetic polarization, a feature which is of central importance for possible spintronic devices based on the artificial Fe-Cr alloys. ## 4 Conclusion We have performed the DFT studies of the bulk ferrite with $12.5\%$-concentration of monoatomic interstitial Cr periodically located at the edges of the bcc Feα cell. We have shown that the full atomic relaxation of the obtained interstitial Fe-Cr stabilizes a new chain-like low-symmetry structure. In this structure, the monoatomic Cr at the edges of ferrite bcc cells leads to the local atomic distortions and results in the formation of parallel chains of Fe6-ochahedra, which are connected by the interchain Fe-Cr bonds. The significant energy gain caused by such a structural relaxation approaches 6.17 eV which makes this type of interstitial alloy highly stable and energetically favorable with the negative formation energy approaching $-1.15$ eV. The novel electronic state of the system can be characterized as metallic, where the metallic properties is the result of strong Fe-Cr hybridization of the structurally relaxed alloy. In the investigations of the magnetic state of the generated relaxed structures, we have obtained a local antiferromagnetic order in the close proximity of Cr atoms, whereas the more distant Fe atoms are coupled ferromagnetically. We also find that the nonmagnetic impurities like nitrogen can substantially modify the magnetic properties of the interstitial alloy which can be considered as an additional manifestation of the strong magnetoelastic effect in this type of alloys. We propose to consider the generated interstitial alloys as perspective candidates for fabrication of novel highly durable stainless steels and for possible applications in spintronic and multifunctional devices. ## 5 Acknowledgements This work has beed partially supported through the project ”Models of quantum statistical description of catalytic processes on metallic substrates” of the Ministry of Education and Sciences of Ukraine and the grant 0108U002091 of the National Academy of Science of Ukraine. A grant of computer time from the Ukrainian Academic Greed is acknowledged. ## References * (1) D. Peckner and I.M. Bernstein. Handbook on Stainless Steels, McGraw-Hill Book Co., New York, 1977. * (2) A. Mai, V.A.C. Haanappel, S. 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Luitz, WIEN2K, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, ISBN 3-9501031-1-2 (TU Wien, Austria, 2001). * (14) J.-H. Park et al., Nature 392, 794 (1998). * (15) V.I. Anisimov, I.V. Solovyov, M.A. Korotin, M.T. Czyzyk, and G.A. Sawatzky, Phys.Rev. B 48, 16929 (1993). * (16) M.T. Czyzyk and G.A. Sawatzky, Phys.Rev. B 49, 14211 (1994). * (17) N. Pavlenko, Phys. Rev. B 80 075105 (2009). * (18) N. Pavlenko, I. Elfimov, T. Kopp, and G.A. Sawatzky, Phys. Rev. B 75, 140512(R) (2007). * (19) T. Bandyopadhyay and D.D. Sarma, Phys.Rev. B 39, 3517 (1989). * (20) Ze Zhang and S. Satpathy, Phys.Rev. B 44, 13319 (1991). * (21) M.A. Korotin, V.I. Anisimov, D.I. Khomskii, and G.A. Sawatzky, Phys.Rev.Lett. 80, 4305 (1998). * (22) A. Ashkin. Optical trapping and manipulation of neutral particles using lasers, World Scientific Pub., Singapore, 2006. * (23) P. Giannozzi et al., J. Phys. Cond. Matter 21, 395502(2009). * (24) H. Jonsson, G. Mills, and K.W. Jacobsen, Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions in Classical and Quantum Dynamics in Condensed Phase Simulations in Classical and Quantum Dynamics in Condensed Phase Simulations, ed. by B.J. Berne, G. Ciccoti, and D.F. Coker (Singapore: World Scientific, 1998). * (25) J.P. Perdew, S. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865(1996). * (26) N. Pavlenko, A. Pietraszko, A. Pawlowski, M. Polomska, I.V.Stasyuk, and B. Hilczer, Phys. Rev. B 84, 064303 (2011). * (27) I. Mazin, Phys. Rev. Lett. 83, 1427(1999).
arxiv-papers
2011-02-02T14:05:22
2024-09-04T02:49:16.800618
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "N.Pavlenko, N.Shcherbovskikh, and Z.A.Duriagina", "submitter": "Natalia Pavlenko", "url": "https://arxiv.org/abs/1102.0432" }
1102.0452
author subject XV International Conference on Gravitational Microlensing Conference Book January 20–22, 2011 University of Salerno, Italy Editors V. Bozza, S. Calchi Novati, L. Mancini, G. Scarpetta (Local Organizing Committee) Organized by the Astrophysics Group of the Physics Department of the University of Salerno Sponsored by IIASS (International Institute for Advanced Scientific Studies), INFN (Istituto Nazionale di Fisica Nucelare), International Ph.D. in Astrophysics, and University of Salerno http://smc2011.physics.unisa.it Foreword This volume collects the abstracts in extended format of the $15^{\mathrm{th}}$ Microlensing Conference held in the University of Salerno on January 20-22, 2011. The Conference has gathered 68 scientists from 17 countries confirming microlensing as a mature and established tool of research over a broad range of astrophysical issues, from dark matter searches to the detection of new extrasolar planets of very low mass, down to Earth-size or below. The 3-days Conference has been preceeded by a 2-days school on “Modelling Planetary Microlensing Events” focused on this last issue. The abstracts collected here offer an updated snapshot of the current researches in this field. The topics include: the status of current surveys, planetary events, dark matter searches, cosmological microlensing, theoretical investigations and an outlook towards the future, with in particular a discussion on the possible role to be played by microlensing searches for exoplanets in the forecoming space missions, WFIRST and EUCLID. Finally, the conference has been enriched by a series of topical speeches on related issues from a non-microlensing point of view: the physics of giant planet accretion and evolution, “new physics” and dark matter in the LHC era and an update on the GAIA mission and its potential to characterize planetary systems with high-precision astrometry. Scientific Rationale 25 years after the seminal intuition by Bohdan Paczynski, microlensing has rapidly evolved into a new promise for modern astrophysics. Indeed, the detection of celestial bodies by their gravitational effects on the light of background sources has proved to be a very powerful tool for the study of many aspects of our Galaxy and beyond. The original proposal of using microlensing to estimate the amount of baryonic dark matter in the form of compact substellar bodies is still topical. The observation of microlensing events toward nearby galaxies still represents a hard challenge for present observational facilities. Very strong efforts are under way for upgrading the current strategies in particular toward the galaxy M31. Hundreds of microlensing events are discovered toward the Galactic Bulge every year. This considerable amount of statistics can be used to characterize the stellar populations of the bulge and the disc of our Galaxy. After several years of observations, the significance of the microlensing sample in the characterization of our Galaxy is growing more and more. Yet, the most intriguing perspective offered by microlensing is represented by its power to discover new extrasolar planets of very low mass, down to Earth- size or below. Several planetary events have already been detected and studied, while many more are expected as soon as future dedicated telescopes become operational. Interestingly, microlensing is already being used to estimate the abundance of planets around stars in the disc of our Galaxy and to characterize their distributions in distance and mass. As the planetary anomalies typically last only a few hours, the cooperation among all observing groups is mandatory, in order to characterize the events properly and maximize the scientific achievements. Even amateur astronomers are now giving their fundamental contribution. In this respect, microlensing stands as a perfect example of how science can unite the whole mankind in a common path toward pure knowledge. Salerno Microlensing Conference 2011 will gather all people active in this field, providing the state of the art of microlensing searches and the perspectives opened by new methodologies and new observational and computational facilities. Colloquia on dark matter searches and planet formation theories are also foreseen as a central part of the conference. The three-days conference will be preceded by a school dedicated to the delicate issue of efficient modelling of planetary microlensing events, which requires major efforts and new ideas from new talents in order to get access to the precious physical information hidden in microlensing light curves. For one week in January 2011, Salerno will thus be the place in which the present and the future of microlensing will be unveiled. Valerio Bozza --- Local Organizing Committee Welcome address On behalf of the Rector of Salerno University, professor Raimondo Pasquino, the Dean of the Science Faculty, professor Mariella Transirico, the Decan of the Department of Physics, professor Ferdinando Mancini and the Organizing Committee, I welcome all the participants to this exciting Conference. This University has the peculiar status to be at the same time one of the youngest Universities in Italy and the oldest one. The re-establishment of Salerno University in the Modern Era dates back to about forty years ago, and as you can see, in a very short time it has grown in the Irno valley as a university campus, with about forty thousand students, more than one thousand professors and researchers, distributed in ten faculties. Five years ago the Campus was completed with the Medical Science Faculty, realizing the ideal link with one of the oldest Medical Schools of the West, the so called “Scuola Medica Salernitana”. Salerno was not the seat of the first Medieval University, but the Medical School was the first to organize studies and diffuse culture on international scale during the medieval period. According to the legend, the “Scuola Medica Salernitana” traces its origins to four erudite: the Greek Pontus, the Jew Helinus, the Arab Abdela, and the Latin Salernus. Thanks to the encounter of these four cultures the Medical School established its knowledge divulgating the Greek, Jewish, Arab and Latin medical knowledge. The divulgation in the West of the Islamic and Greek Medical Science is certainly due to Constantino the African (XI century), thanks to his translations into Latin of the most important Arabic and Greek medical treatises. I like to read some words extracted from the nobel lecture of Abdus Salam (1979) “Scientific thought and its creation is the common and shared heritage of mankind. In this respect, the history of science, like the history of all civilization, has gone through cycles. Perhaps I can illustrate this with an actual example. Seven hundred and sixty years ago, a young Scotsman left his native glens to travel south to Toledo in Spain. His name was Michael, […] Michael reached Toledo in 1217 AD. […] From Toledo, Michael travelled to Sicily, to the Court of Emperor Frederick II. Visiting the medical school at Salerno, chartered by Frederick in 1231, Michael met the Danish physician, Henrik Harpestraeng […] Henrik had come to Salerno to compose his treatise on blood-letting and surgery. Henrik’s sources were the medical canons of the great clinicians of Islam, Al-Razi and Avicenna, which only Michael the Scot could translate for him. Toledo’s and Salerno’s schools, representing as they did the finest synthesis of Arabic, Greek, Latin and Hebrew scholarship, were some of the most memorable of international assays in scientific collaboration”. The “Scuola Medica Salernitana” was at the height of its fame in the XIV century and carried on its teaching for nine centuries, until Giocchino Murat’s decree, dated the 25th of January 1812, prescribed the closure. After two centuries, we all, coming here in the Salerno University from east and west countries, renew in some sense the old tradition, establishing an ideal link with the cultural heritage of the “Scuola Medica Salernitana” . I hope you will enjoy your stay in Salerno and I wish you good and useful work. Gaetano Scarpetta --- Chair of the Local Organizing Committee Local Organizing Committee V. Bozza | ---|--- S. Calchi Novati | L. Mancini | G. Scarpetta (Chair) | Scientific Organizing Committee --- J.-P. Beaulieu | France D. Bennett | USA I. Bond | New Zealand S. Calchi Novati (Chair) | Italy M. Dominik | UK A. Gould | USA C. Han | South Korea Ph. Jetzer | Switzerland E. Kerins | UK M. Moniez | France T. Sumi | Japan Y. Tsapras | UK A. Udalski | Poland L. Wyrzykowski | UK Scientific Secretary --- O. De Pasquale V. Di Marino T. Nappi S. Russo CONTENTS Foreword iii Scientific Rationale v welcome vi Organization ix Participants xvi School Program xix Conference Program xx School on Modelling Planetary Microlensing Events 1 Introduction to microlensing 2 Philippe Jetzer From microlensing observations to science 3 Martin Dominik The theory and phenomenology of planetary microlensing 4 Scott Gaudi From raw images to lightcurves: how to make sense of your data 5 Yiannis Tsapras The efficient modeling of planetary microlensing events 6 David Bennett Contour integration and downhill fitting 7 Valerio Bozza Microlensing modelling and high performance computing 8 Ian Bond Conference topical speeches 9 Giant planet accretion and dynamical evolution: considerations on systems around small-mass stars 10 Alessandro Morbidelli The dark matter - LHC endeavour to unveil TeV new physics 14 Antonio Masiero Characterization of planetary systems with high-precision astrometry: the Gaia potential 15 Alessandro Sozzetti Status of current surveys 18 Status of the OGLE-IV survey 19 Andrzej Udalski MOA-II observation in 2010 season 20 Takahiro Sumi The RoboNet 2010 season 22 Yiannis Tsapras Cosmological Microlensing 25 Dark matter determinations from Chandra observations of quadruply lensed quasars 26 David Pooley Cosmic equation of state from strong gravitational lensing systems 27 Marek Biesiada & Beata Malec Galactic Microlensing: the dark matter search 31 PAndromeda - the Pan-STARRS M31 survey for dark matter 32 Arno Riffeser, Stella Seitz, Ralf Bender, C.-H. Lee, Johannes Koppenhoefer Final OGLE-II and OGLE-III results on microlensing towards the LMC and SMC 34 Lukasz Wyrzykowski Analysis of microlensing events towards the LMC 37 Luigi Mancini & Sebastiano Calchi Novati Simulation of short time scale pixel lensing towards the Virgo cluster 39 Sedighe Sajadian & Sohrab Rahvar M31 pixel lensing and the PLAN project 40 Sebastiano Calchi Novati Planetary events 44 MOA-2009-BLG-266LB: the first cold Neptune with a measured mass 45 David Bennett Increasing the detection rate of low-mass planets in high-magnification events and MOA-2006-BLG-130 46 Julie Baudry & Philip Yock The complete orbital solution for OGLE-2008-BLG-513 49 Jennifer Yee Planetary microlensing event MOA-2010-BLG-328 51 Kei Furusawa Binary microlensing event OGLE-2009-BLG-020 gives orbit predictions verifiable by follow-up observations 53 Jan Skowron Theoretical investigations 59 Microlensing and planet populations - What do we know, and how could we learn more 60 Martin Dominik The Frequency of extrasolar planet detections with microlensing simulations 65 Rieul Gendron & Shude Mao A semi-analytical model for gravitational microlensing events 66 Denis Sullivan, Paul Chote, Michael Miller GPU-assisted contouring for modeling binary microlensing events 70 Markus Hundertmark, Frederic V. Hessman, Stefan Dreizler Red noise effect in space-based microlensing observations 74 Achille Nucita, Daniele Vetrugno, Francesco De Paolis, Gabriele Ingrosso, Berlinda M. T. Maiolo, Stefania Carpano Light curve errors introduced by limb-darkening models 76 David Heyrovsky Isolated, stellar-mass black holes through microlensing 77 Kailash Sahu, Howard E. Bond, Jay Anderson, Martin Dominik, Andrzej Udalski, Philip Yock The observability of isolated compact remnants with microlensing 80 Nicola Sartore & Aldo Treves Gravitational microlensing by the Ellis wormhole 82 Fumio Abe The deflection of light ray in strong field: a material medium approach 100 Asoke Kumar Sen Rapidly rotating lenses - repeating orbital motion features in close binary microlensing 104 Matthew Penny, Eamonn Kerins, Shude Mao Towards the future: new facilities/instrumentation/procedures 107 Microlensing with the SONG global network 108 Uffe G. Jørgensen, Kennet B. W. Harpsøe, Per K. Rasmussen, Michael I. Andersen, Anton N. Sørensen, Jørgen Christensen-Dalsgaard, Søren Frandsen, Frank Grundahl, Hans Kjeldsen Next-generation microlensing pilot planet search and the frequency of planetary systems 112 Dan Maoz & Yossi Shvartzvald Kohyama Astronomical Observatory: current status 116 Atsunori Yonehara, Mizuki Isogai, Akira Arai, Hiroki Tohyama Optimal imaging for gravitational microlensing 118 Kennet B.W. Harpsøe, Uffe G. Jørgensen, Per K. Rasmussen, Michael I. Andersen, Anton N. Sørensen, Jørgen Christensen-Dalsgaard, Søren Frandsen, Frank Grundahl, Hans Kjeldsen IPAC s role as the science center for NASA s WFIRST mission 121 Kaspar von Braun EUCLID microlensing planet hunt 122 Jean-Philippe Beaulieu & Matthew Penny Space-based microlensing exoplanet survey: WFIRST and/or Euclid 124 David Bennett Microlensing with Gaia satellite 125 Lukasz Wyrzykowski Poster session 127 Critical curve topology in special triple lens configurations 128 Kamil Danek PAndromeda - a dedicated deep survey of M31 with Pan-STARRS 1 129 Chien-Hsiu Lee, Arno Riffeser, Stella Seitz, Ralf Bender, Johannes Koppenhoefer SMC2011 Participants Abe, Fumio | Nagoya University, Japan ---|--- Baudry, Julie | University of Orsay, France Bachelet, Etienne | University of Toulouse, France Beaulieu, Jean-Philippe | Institut d’Astrophysique de Paris, France Bennett, David | University of Notre Dame, USA Bond, Ian | Massey University, New Zealand Bonino, Donata | INAF - Turin Astronomical Observatory, Italy Bozza, Valerio | University of Salerno, Italy Browne, Paul | University of St Andrews, UK Calchi Novati, Sebastiano | University of Salerno, Italy Danek, Kamil | Charles University, Czech Republic De Paolis, Francesco | University of Salento, Italy Dominik, Martin | University of St Andrews, UK Dominis Prester, Dijana | University of Rijeka, Croatia Fouqu$\acute{\mathrm{e}}$, Pascal | University of Toulouse, France Furusawa, Kei | Nagoya University, Japan Gardiol, Daniele | INAF - Turin Astronomical Observatory, Italy Gaudi, Scott | Ohio State University, USA Gendron, Riel | University of Manchester, UK Gould, Andrew | Ohio State University, USA Harpsøe, Kennet | Niels Bohr Institute, Denmark Harris, Pauline | Victoria University of Wellington, New Zealand Henderson, Calen | Ohio State University, USA Heyrovsky, David | Charles University, Czech Republic Horne, Keith | University of St Andrews, UK Hundertmark, Markus | University of Göttingen, Germany Jetzer, Philippe | University of Zurich, Switzerland Jørgensen, Uffe G. | Niels Bohr Institute, Denmark Lambiase, Gaetano | University of Salerno, Italy Lee, Chien-Hsiu | University Observatory Munich Liebig, Christine | University of St Andrews Lubini, Mario | University of Zurich, Switzerland Malec, Beata | Copernicus Center for Interdisciplinary Studies, Poland Mancini, Luigi | University of Salerno, Italy Maoz, Dan | Tel-Aviv University, Israel Masiero, Antonio | University of Padova, Italy Mirzoyan, Sergey | University of Salerno, Italy Miyake, Noriyuki | Nagoya University, Japan Morbidelli, Alessandro | Observatoire de la Cote d’Azur, France Muraki, Yasushi | Konan University, Japan Nucita, Achille | University of Salento, Italy Orio, Marina | INAF - Padova Astronomical Observatory, Italy Paulin-Henriksson, St$\acute{\mathrm{e}}$phane | CEA - Paris, France Payandeh, Farrin | Payame Noor University Tabriz, Iran Penny, Matthew | University of Manchester, UK Pooley, David | Eureka Scientific, USA Retana Montenegro, Edwin F. | Universidad de Costa Rica, Costa Rica Riffeser, Arno | Max Planck Institute for Extraterrestrial Physics, Germany Sahu, Kailash | Space Telescope Science Institute, USA Sajadian, Sedighe | Sharif University of Technology, Iran Sartore, Nicola | INAF - IASF Milano, Italy Scarpetta, Gaetano | University of Salerno, Italy Sen, Asoke Kumar | Sen Assam University, India Shvartzvald, Yossi | Tel-Aviv University, Israel Skowron, Jan | Ohio State University, USA Sozzetti, Alessandro | INAF - Turin Astronomical Observatory, Italy Sullivan, Denis | Victoria University of Wellington, New Zealand Sumi, Takahiro | Nagoya University, Japan Tortora, Crescenzo | University of Zurich, Switzerland Tsapras, Yiannis | Queen Mary University, UK Udalski, Andrzej | Warsaw University Observatory, Poland Vetrugno, Daniele | University of Salento, Italy Vilasi, Gaetano | University of Salerno, Italy von Braun, Kaspar | California Institute of Technology, USA Yee, Jennifer | Ohio State University, USA Yock, Philip | University of Auckland, New Zealand Yonehara, Atsunori | Kyoto Sangyo University, Japan Wyrzykowski, Lukasz | University of Cambridge, UK School on Modelling Planetary Microlensing Events Program Tuesday, January 18, 2011 --- 09:00 | Introduction to microlensing | Philippe Jetzer 10:50 | Coffee Break 11:15 | From microlensing observations to science | Martin Dominik 13:00 | Lunch Break 14:45 | The theory and phenomenology of planetary | Scott Gaudi | microlensing 16:45 | Coffee Break 17:15 | From raw images to lightcurves: how to make | Yiannis Tsapras | sense of your data Wednesday, January 19, 2011 09:00 | The efficient modeling of planetary microlensing | David Bennett | events 10:50 | Coffee Break 11:15 | Contour integration and downhill fitting | Valerio Bozza 13:00 | Lunch Break 14:45 | Microlensing modelling and high performance | Ian Bond | computing 16:45 | Coffee Break SMC2011 Program Thursday, January 20, 2011 --- 09:00 | Registration 09:30 | Greetings 09:40 | Communications | | | Chair: Gaetano Scarpetta Status of current Surveys I | | 09:50 | Status of the OGLE-IV survey | Andrzej Udalski 10:20 | MOA-II observation in 2010 season | Takahiro Sumi 10:50 | Coffee Break Topical Speech I | | 11:20 | Formation and evolution of our | Alessandro Morbidelli | solar system Theoretical investigations I | | 12:10 | Microlensing and planet populations: | Martin Dominik | what do we know, and how could we | learn more? 12:40 | The frequency of extrasolar planet | Rieul Gendron | detections with microlensing simulations 13:00 | Lunch Break | | Chair: Philippe Jetzer Dark Matter Search | | 14:40 | PAndromeda - the Pan-STARRS M31 | Arno Riffeser | survey for dark matter 15:10 | Final OGLE-II and OGLE-III results on | Lukasz Wyrzykowski | microlensing towards the LMC and SMC 15:30 | Analysis of microlensing events towards the LMC | Luigi Mancini 15:50 | Simulation of short time scale pixel lensing | Sedighe Sajadian | towards the Virgo cluster 16:10 | M31 pixel lensing and the PLAN project | Sebastiano Calchi Novati 16:30 | Coffee Break Towards the future I | | 17:00 | Microlensing with the SONG global network | Uffe G. Jørgensen 17:20 | Next-generation microlensing pilot planet search | Dan Maoz | and the frequency of planetary systems 17:40 | Kohyama Astronomical Observatory: | Atsunori Yonehara | current status 18:00 | The lucky imaging technique for microlensing | Kennet Harpsøe | observations Friday, January 21, 2011 | | Chair: P. Yock Status of current Surveys II | | 09:00 | The 2010 MicroFUN season | Jennifer Yee 09:30 | The RoboNet 2010 season | Yiannis Tsapras Theoretical investigations II | | 10:00 | A semi-analytical model for gravitational | Denis Sullivan | microlensing events 10:20 | GPU-assisted contouring for modeling | Markus Hundertmark | binary microlensing events 10:40 | Rapidly rotating lenses - repeating orbital motion | Matthew Penny | features in close binary microlensing 11:00 | Coffee Break Topical Speech II | | 11:30 | The dark matter - LHC endeavour to unveil | Antonio Masiero | TeV new physics Cosmological microlensing | | 12:20 | Dark matter determinations from Chandra | David Pooley | observations of quadruply lensed quasars 12:40 | Cosmic equation of state from strong | Beata Malec | gravitational lensing systems 13:00 | Lunch Break | | Chair: Pascal Fouqu$\acute{\mathrm{{\bf e}}}$ Planetary events | | 14:30 | MOA-2009-BLG-266LB: the first cold Neptune | David Bennett | with a measured mass 14:50 | Increasing the detection rate of low-mass | Julie Baudry | planets in high-magnification events and | MOA-2006-BLG-130 15:10 | The complete orbital solution for | Jennifer Yee | OGLE-2008-BLG-513 15:30 | Planetary microlensing event MOA-2010-BLG-328 | Kei Furusawa 15:50 | Binary microlensing event OGLE-2009-BLG-020 | Jan Skowron | gives orbit predictions verifiable by follow-up | observations 16:10 | Coffee Break 17:30 | Tour of the old town of Salerno 20:00 | Social dinner Saturday, January 22, 2011 | | Chair: Francesco De Paolis Towards the future II | | 09:00 | IPAC s role as the science center for | Kaspar von Braun | NASA s WFIRST mission 09:30 | EUCLID microlensing planet hunt | Jean-Philippe Beaulieu 09:50 | Simulating the planet hunting capability of Euclid | Matthew Penny 10:10 | Space-based microlensing exoplanet survey: | David Bennett | WFIRST and/or Euclid 10:40 | Coffee Break Topical Speech III | | 11:10 | Characterization of planetary systems with | Alessandro Sozzetti | high-precision astrometry: the Gaia Potential 12:00 | Microlensing with Gaia satellite | Lukasz Wyrzykowski Theoretical investigations III | | 12:20 | Red noise effect in space-based microlensing | Achille Nucita | observations 12:40 | Light curve errors introduced by limb-darkening | David Heyrovsky | models 13:00 | Lunch Break | | Chair: Scott Gaudi 14:40 | Isolated, stellar-mass black holes through | Kailash Sahu | microlensing 15:00 | The observability of isolated compact | Nicola Sartore | remnants with microlensing 15:20 | Gravitational microlensing by the Ellis wormhole | Fumio Abe 15:40 | The deflection of light ray in strong field: | Asoke Kumar Sen | a material medium approach 16:00 | How to stop a runaway (Monte Carlo Markov) | Keith Horne 16:20 | Coffee Break 16:50 | Open session | Poster Session | Critical curve topology in special triple lens | Kamil Danek | configurations | PAndromeda - a dedicated deep survey of M31 | Chien-Hsiu Lee | with Pan-STARRS 1
arxiv-papers
2011-02-02T14:53:01
2024-09-04T02:49:16.808138
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Valerio Bozza, Sebastiano Calchi Novati, Luigi Mancini, Gaetano\n Scarpetta (Eds.)", "submitter": "Luigi Mancini", "url": "https://arxiv.org/abs/1102.0452" }
1102.0509
# Considerations on the subgroup commutativity degree and related notions Francesco G. Russo Dipartimento di Matematica e Informatica, Universitá di Palermo, Via Archirafi 34, 90123, Palermo, Italy. francescog.russo@yahoo.com ###### Abstract. The concept of subgroup commutativity degree of a finite group $G$ is arising interest in several areas of group theory in the last years, since it gives a measure of the probability that a randomly picked pair $(H,K)$ of subgroups of $G$ satisfies the condition $HK=KH$. In this paper, a stronger notion is studied and relations with the commutativity degree are found. ###### Key words and phrases: Subgroup commutativity degree, permutable subgroups, centralizers, subgroup lattices. ###### 2010 Mathematics Subject Classification: Primary: 20D60, 20P05; Secondary: 20D08. ## 1\. Introduction In the present paper we deal only with finite group, even if there is a recent interest to the subject in the context of infinite groups [1, 11, 10, 17, 25]. The commutativity degree of a group $G$, given by (1.1) $d(G)=\frac{|\\{(x,y)\in G\times G\ |\ [x,y]=1\\}|}{|G|^{2}}=\frac{1}{|G|^{2}}\sum_{x\in G}|\\{y\in G\ |\ y^{-1}xy=x\\}|$ $=\frac{1}{|G|^{2}}\sum_{x\in G}|C_{G}(x)|,$ was studied extensively in [2, 4, 6, 9, 12, 16, 18, 19, 20, 21, 22, 23, 26] an generalized in various ways. Its importance is testified in the theory of the groups of prime power orders in [5, Chapter 2], where it is called measure of commutativity by Y. Berkovich in order to emphasize the fact that it really gives a measure of how far is the group from being abelian. In [7, 8, 9] it was introduced the following variation, (1.2) $d(H,K)=\frac{|\\{(h,k)\in H\times K\ |\ [h,k]=1\\}|}{|H||K|}=\frac{1}{|H||K|}\underset{h\in H}{\sum}|C_{K}(h)|,$ where $H$ and $K$ are two arbitrary subgroups of $G$. Of course, $d(G,G)=d(G)$, whenever $H=K=G$, and, consequently, the bounds known in literature for $d(G)$ may be sharpened by examining $d(H,K)$. In recent years, there is an increasing interest in studying the problem from the point of view of the lattice theory (see [13, 14, 15, 27, 28]). Tǎrnǎuceanu [30, 31] has introduced the subgroup commutativity degree of a finite group, that is, the ratio (1.3) $sd(G)=\frac{|\\{(H,K)\in\mathcal{L}(G)\times\mathcal{L}(G)\ |\ HK=KH\\}|}{|\mathcal{L}(G)|^{2}},$ where $\mathcal{L}(G)$ denote the subgroup lattice of $G$. It turns out that (1.4) $sd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}\underset{H\in\mathcal{L}(G)}{\sum}|\mathcal{C}(H)|,$ where (1.5) $\mathcal{C}(H)=\\{K\in\mathcal{L}(G)\ |\ HK=KH\\}.$ Variations on this theme have been considered in [3, 13, 14, 15, 24, 27, 28], involving weaker notions of permutability among subgroups. Of course, if $[H,K]=1$, then $HK=KH$, where $[H,K]=\langle[h,k]\ |\ h\in H,k\in K\rangle$. Conversely, $HK=KH$ does not imply that $[H,K]=1$. In fact, the equality among the sets $\\{hk\ |\ h\in H,k\in K\\}$ and $\\{kh\ |\ k\in K,h\in H\\}$ does not imply, in general, that all the elements of $H$ permute with all elements of $K$. Many examples can be given. Therefore it is meaningful to define the following ratio (1.6) $ssd(G)=\frac{|\\{(H,K)\in\mathcal{L}(G)^{2}\ |\ [H,K]=1\\}|}{|\mathcal{L}(G)|^{2}},$ which we will call strong subgroup commutativity degree of $G$. It is easy to see that (1.7) $ssd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}\underset{H\in\mathcal{L}(G)}{\sum}|Comm_{G}(H)|,$ where (1.8) $Comm_{G}(H)=\\{K\in\mathcal{L}(G)\ |\ [H,K]=1\\},$ and that $ssd(G)$ is the probability that the subgroup $[H,K]$ of an arbitrarily chosen pair of subgroups $H,K$ of a group $G$ is equal to the trivial subgroup of $G$. Equivalently, $ssd(G)$ expresses the probability that, randomly picked two subgroups of $G$, the subgroup generated by their commutators is trivial, and, in particular, the two subgroups are permutable. The present paper is devoted to study this notion, showing that it is related to the previous investigations in the area of the measure theory of finite groups. ## 2\. Some basic properties There are some considerations which come by default with the strong subgroup commutativity degree. A group $G$ is quasihamiltonian, if all pairs of its subgroups are permutable. $G$ is called modular, if $\mathcal{L}(G)$ satisfies the well–known modular law (see [29]). Quasihamiltonian groups were classified by Iwasawa (see [5, Chapter 6] or [29, Chapter 2]), who proved that they are nilpotent and modular. This is equivalent to say that a group $G$ is quasihamiltonian if and only if all its Sylow $p$-subgroups are modular (see [29, Exercise 3 at p.87]), being $p$ a prime. Therefore the knowledge of quasihamiltonian groups may be reduced to that of modular $p$-groups. In literature, for $m\geq 3$ the groups (2.1) $M(p^{m})=\langle x,y\ |\ x^{p^{m-1}}=y^{p}=1,y^{-1}xy=x^{p^{m-2}+1}\rangle=\langle y\rangle\ltimes\langle x\rangle,$ are nonabelian modular $p$-groups and their properties have interested the researches of many authors in various contexts (see [5, 29, 30]). An immediate observation is the following. If $G=M(p^{m})$, then $[\langle x\rangle,\langle y\rangle]\not=1$ and consequently $sd(G)=1$ but $ssd(G)\not=1$. In this sense, it is important to know when the strong subgroup commutativity degree is trivial. ###### Proposition 2.1. A group $G$ has $ssd(G)=1$ if and only if it is abelian. ###### Proof. We have that $ssd(G)=1$ if, and only if, $[H,K]=1$ for all subgroups $H$ and $K$ of $G$, if, and only if, $[h,k]=1$ for all $h\in H$, $k\in K$ and for all $H$ and $K$ in $G$. This implies, in particular, that $[h,k]=1$ for all $h,k\in G$, that is, $G$ is abelian. Conversely, if $G$ is abelian, then it is clear that $ssd(G)=1$. ∎ On another hand, the following relation shows that $ssd(G)$ is related to $d(H,K)$ in a deep way. ###### Theorem 2.2. Let $H$ and $K$ be two subgroups of a group $G$. Then $ssd(G)<\frac{|G|^{2}}{|\mathcal{L}(G)|^{2}}\sum_{H,K\in\mathcal{L}(G)}d(H,K).$ ###### Proof. We claim that (2.2) $\bigcup_{K\in\mathcal{L}(G)}C_{K}(H)=Comm_{G}(H).$ Let $T={\underset{K\in\mathcal{L}(G)}{\bigcup}}C_{K}(H)$ and $t\in T$. Then there exists a $K_{t}\in\mathcal{L}(G)$ containing $t$ such that $t\in C_{K_{t}}(H)$, that is, $[t,H]=1$, which means that $t$ permutes with all elements of $H$. In particular, the powers of $t$ permutes with all elements of $H$ and so $[\langle t\rangle,H]=1$, which means $\langle t\rangle$ is in $Comm_{G}(H)$. We conclude that $T\subseteq Comm_{G}(H)$. Conversely, if $K\in\mathcal{L}(G)$ is in $Comm_{G}(H)$, then $[K,H]=1$ and so $K\subseteq C_{G}(H)$, then $K\subseteq T$. The claim follows. Therefore (2.3) $|\mathcal{L}(G)|^{2}\ ssd(G)=\sum_{H\in\mathcal{L}(G)}|Comm_{G}(H)|=\sum_{H\in\mathcal{L}(G)}\left|\bigcup_{K\in\mathcal{L}(G)}C_{K}(H)\right|$ $<\sum_{K\in\mathcal{L}(G)}\sum_{H\in\mathcal{L}(G)}|C_{K}(H)|$ and we note that the equality cannot occur here as the identity $1\in C_{K}(H)$ for all $H$ and $K$ in $\mathcal{L}(G)$. Since $C_{K}(H)\subseteq C_{K}(h)$ whenever $h\in H$, we may continue, finding the following upper bound (2.4) $\leq\sum_{K\in\mathcal{L}(G)}{\underset{H\in\mathcal{L}(G)}{\underset{h\in H}{\sum}}}|C_{K}(h)|=\sum_{H,K\in\mathcal{L}(G)}\Big{(}\sum_{h\in H}|C_{K}(h)|\Big{)}$ $=\sum_{H,K\in\mathcal{L}(G)}d(H,K)\ |H|\ |K|\leq|G|^{2}\ \sum_{H,K\in\mathcal{L}(G)}d(H,K).$ ∎ ###### Remark 2.3. We want just to illustrate two points of views which allow us to decide whether a group $G$ is abelian or not. The first deals with the subgroups: from Proposition 2.1 $G$ is abelian if and only if $ssd(G)$ is trivial. The second deals with the elements: $G$ is abelian if and only if $d(G)$ is trivial. Theorem 2.2 is relevant, because it correlates $d(G)$ with $ssd(G)$. This is very helpful, because we have literature on $d(G)$ but few is known about $ssd(G)$ and $sd(G)$. In virtue of the previous remark, the following result is significative and answers partially some open questions in [31]. We will see, concretely, that the argument of Theorem 2.2 is very general and can be adapted to the context of $sd(G)$. ###### Theorem 2.4. Let $H$ and $K$ be two subgroups of a group $G$. Then $sd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\sum_{H\in\mathcal{L}(G)}\left|\bigcap_{h\in H}C_{K}(h)\right|$ with $\sum_{H,K\in\mathcal{L}(G)}d(H,K)\ |H|\ |K|\geq\sum_{H,K\in\mathcal{L}(G)}\left|\bigcap_{h\in H}C_{K}(h)\right|.$ ###### Proof. From Theorem 2.2 (more precisely from (3.18)), we may restrict to prove only the first inequality. In order to do this, we claim that (2.5) $C_{K}(H)\subseteq\bigcup_{K\in\mathcal{L}(G)}C_{K}(H)\subseteq\mathcal{C}(H).$ The first inclusion is trivial. Let $S={\underset{K\in\mathcal{L}(G)}{\bigcup}}C_{K}(H)$ and $s\in S$. Then there exists a $K_{s}\in\mathcal{L}(G)$ containing $s$ such that $s\in C_{K_{s}}(H)$, that is, $[s,H]=1$, which means that $s$ permutes with all elements of $H$. In particular, $[\langle s\rangle,H]=1$ then $\langle s\rangle H=H\langle s\rangle$, which means $\langle s\rangle\in\mathcal{C}(H)$. We conclude that $S\subseteq\mathcal{C}(H)$. Therefore (2.6) $|\mathcal{L}(G)|^{2}\ sd(G)=\sum_{H\in\mathcal{L}(G)}|\mathcal{C}(H)|\geq\sum_{H\in\mathcal{L}(G)}\left|\bigcup_{K\in\mathcal{L}(G)}C_{K}(H)\right|\geq\sum_{H\in\mathcal{L}(G)}|C_{K}(H)|$ but we observe that in general the following is true (2.7) ${\underset{h\in H}{\bigcap}}C_{K}(h)=C_{K}(H)$ so that (2.8) $=\sum_{H\in\mathcal{L}(G)}\left|\bigcap_{h\in H}C_{K}(h)\right|.$ On another hand, we note that (2.9) $\sum_{H,K\in\mathcal{L}(G)}d(H,K)\ |H|\ |K|=\sum_{H,K\in\mathcal{L}(G)}\ \Big{(}\sum_{h\in H}|C_{K}(h)|\Big{)}$ $=\sum_{K\in\mathcal{L}(G)}\ \Big{(}{\underset{H\in\mathcal{L}(G)}{\underset{h\in H}{\sum}}}|C_{K}(h)|\Big{)}\geq\sum_{K\in\mathcal{L}(G)}\ \Big{(}\sum_{H\in\mathcal{L}(G)}\left|\bigcap_{h\in H}C_{K}(h)\right|\Big{)}.$ ∎ In the rest of this section we reformulate $ssd(G)$ in terms of arithmetic functions. It is possible to rewrite $ssd(G)$ in the following form: (2.10) $ssd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}{\underset{X,Y\in\mathcal{L}(G)}{\sum}\varphi(X,Y)},$ where $\varphi:\mathcal{L}(G)^{2}\rightarrow\\{0,1\\}$ is the function defined by (2.11) $\varphi(X,Y)=\left\\{\begin{array}[]{lcl}1,&&\mathrm{if}\ [X,Y]=1,\\\ 0,&&\mathrm{if}\ [X,Y]\not=1.\end{array}\right.$ The reader may note that $\varphi(X,Y)=\varphi(Y,X)$, that is, $\varphi$ is symmetric in the variables $X$ and $Y$. There is a corresponding property of symmetry for the subgroup commutativity degree in [30, Section 2], but, in general, this property depends on the permutability which we are going to study. For instance, this does not happen for weaker forms of permutability with respect to the maximal sugroups, as shown in [24]. However, the introduction of the function $\varphi$ allows us to simplify the notations. In fact, if $Z$ is a given subgroup of $G$ and we consider the sets $\mathcal{B}_{1}=\\{(X\in\mathcal{L}(G):Z\subseteq X\\}$ and $\mathcal{B}_{2}=\\{X\in\mathcal{L}(G):X\subset Z\\},$ then $\mathcal{B}_{1}\cup\mathcal{B}_{2}\subseteq\mathcal{L}(G)$ and so (2.12) $|\mathcal{L}(G)|^{2}\ ssd(G)\geq\sum_{X,Y\in\mathcal{B}_{1}\cup\mathcal{B}_{2}}\varphi(X,Y)$ $=\sum_{X,Y\in\mathcal{B}_{1}}\varphi(X,Y)+\sum_{X,Y\in\mathcal{B}_{2}}\varphi(X,Y)+2\sum_{X\in\mathcal{B}_{1}}\sum_{Y\in\mathcal{B}_{2}}\varphi(X,Y).$ A consequence of this equation is examined below and overlaps a similar situation for $sd(G)$ in [30]. ###### Proposition 2.5. Let $G$ be a group and $N$ be a normal subgroup of $G$. Then $ssd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\ \Big{(}\Big{(}|\mathcal{L}(N)|+|\mathcal{L}(G/N)|-1\Big{)}^{2}$ $+(ssd(N)-1)|\mathcal{L}(N)|^{2}+(ssd(G/N)-1)|\mathcal{L}(G/N)|^{2}\Big{)}.$ ###### Proof. We are going to rewrite more properly the terms in the left side of (2.12). (2.13) $|\mathcal{L}(G/N)|^{2}\ ssd(G/N)={\underset{X,Y\in\mathcal{B}_{1}}{\sum}}\varphi(X,Y);$ (2.14) $|\mathcal{L}(N)|^{2}\ ssd(G/N)-2|\mathcal{L}(N)|+1={\underset{X,Y\in\mathcal{B}_{2}\cup\\{N\\}}{\sum}}\varphi(X,Y)$ $-2{\underset{X\in\mathcal{B}_{2}\cup\\{N\\}}{\sum}}\varphi(X,N)+1={\underset{X,Y\in\mathcal{B}_{2}}{\sum}}\varphi(X,Y);$ (2.15) $2|\mathcal{L}(G/N)|(|\mathcal{L}(N)|-1)=2|\mathcal{B}_{1}||\mathcal{B}_{2}|=2{\underset{X\in\mathcal{B}_{1}}{\sum}}{\underset{Y\in\mathcal{B}_{2}}{\sum}}\varphi(X,Y).$ Replacing these expressions in (2.12), the result follows. ∎ We list three consequences of Proposition 2.5, overlapping similar situations for $sd(G)$ in [30]. Their proof is omitted, since it is enough to note that for a normal abelian subgroup $N$ of $G$ we have $ssd(G/N)=1$ by Proposition 2.5, and, if it is of prime index in $G$, then $|\mathcal{L}(G/N)|=2$ . ###### Corollary 2.6. Let $G$ be a group and $N$ be a normal subgroup of $G$ such that $G/N$ and $N$ are abelian. Then $ssd(G)\geq\frac{1}{|\mathcal{L}(G)|}\Big{(}|\mathcal{L}(N)|+|\mathcal{L}(G/N)|-1\Big{)}^{2}.$ ###### Corollary 2.7. Let $G$ be a group and $N$ be a normal subgroup of $G$ of prime index. Then $ssd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\Big{(}ssd(N)|\mathcal{L}(N)|^{2}+2|\mathcal{L}(N)|+1\Big{)}.$ ###### Corollary 2.8. A nonabelian solvable group $G$ has $ssd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\Big{(}ssd(G^{\prime})|\mathcal{L}(G^{\prime})|^{2}+2|\mathcal{L}(G^{\prime})|+1\Big{)}.$ In particular, if $G$ is metabelian, then $ssd(G)\geq\frac{1}{|\mathcal{L}(G)|^{2}}\Big{(}|\mathcal{L}(G^{\prime})|^{2}+2|\mathcal{L}(G^{\prime})|+1\Big{)}.$ Now we list some general bounds, related to subgroups and quotients. In a different context, these relations have been found in [24]. ###### Theorem 2.9. Let $H$ be a subgroup of a group $G$. Then $\frac{|\mathcal{L}(H)|^{2}}{|\mathcal{L}(G)|^{2}}\ ssd(H)\leq ssd(G)$ and for all subgroups $L$ and $M$ of $H$ $\frac{1}{|\mathcal{L}(G)|^{2}}\sum_{L\in\mathcal{L}(H)}\left|\bigcap_{l\in L}C_{M}(l)\right|\leq sd(H)\leq sd(G).$ ###### Proof. We proceed to prove the first inequality. The result is obviously true for $H=G$ and then we may assume $H\not=G$. Since $\mathcal{L}(H)\subseteq\mathcal{L}(G)$, (2.16) $|\mathcal{L}(H)|^{2}\ ssd(H)={\underset{X,Y\in\mathcal{L}(H)}{\sum}}\varphi(X,Y)\leq{\underset{X,Y\in\mathcal{L}(G)}{\sum}}\varphi(X,Y)=|\mathcal{L}(G)|^{2}\ ssd(G).$ The inequality follows. Now we proceed to prove the remaining part. When we consider the corresponding function $\psi$, related to $sd(G)$ (details can be found in [30, 31]), instead of $\varphi$, we may overlap the previous argument and find that $\frac{|\mathcal{L}(H)|^{2}}{|\mathcal{L}(G)|^{2}}sd(H)\leq sd(G)$. From Theorem 2.4, it follows that (2.17) $\frac{1}{|\mathcal{L}(H)|^{2}}\sum_{L\in\mathcal{L}(H)}\left|\bigcap_{l\in L}C_{M}(l)\right|\leq\ sd(H)$ then (2.18) $\frac{|\mathcal{L}(H)|^{2}}{|\mathcal{L}(G)|^{2}}\ \Big{(}\frac{1}{|\mathcal{L}(H)|^{2}}\sum_{L\in\mathcal{L}(H)}\left|\bigcap_{l\in L}C_{M}(l)\right|\Big{)}\leq sd(H)$ and the result follows. ∎ In [29, Chapter 1], it is shown that $\mathcal{L}(G_{1}\times G_{2})\neq\mathcal{L}(G_{1})\times\mathcal{L}(G_{2})$ in general, but if $G_{1}$ and $G_{2}$ have coprime orders then it is true. This motivates our assumption in the following proposition. ###### Proposition 2.10. For two groups $G_{1}$ and $G_{2}$ of coprime orders, $ssd(G_{1}\times G_{2})=ssd(G_{1})\cdot ssd(G_{2}).$ ###### Proof. We have $\mathcal{L}(G_{1}\times G_{2})=\mathcal{L}(G_{1})\times\mathcal{L}(G_{2})$, because $G_{1}$ and $G_{2}$ have coprime orders. Therefore, with obvious meaning of symbols, (2.19) $ssd(G_{1}\times G_{2})=\frac{1}{|\mathcal{L}(G_{1}\times G_{2})|^{2}}\underset{A_{1}\times A_{2}\in\mathcal{L}(G_{1}\times G_{2})}{\sum}|Comm_{G_{1}\times G_{2}}(A_{1}\times A_{2})|$ $=\frac{1}{|\mathcal{L}(G_{1})\times\mathcal{L}(G_{2})|^{2}}\underset{A_{1}\times A_{2}\in\mathcal{L}(G_{1})\times\mathcal{L}(G_{2})}{\sum}|Comm_{G_{1}}(A_{1})\times Comm_{G_{2}}(A_{2})|$ $=\left(\frac{1}{|\mathcal{L}(G_{1})|^{2}}\underset{A_{1}\in\mathcal{L}(G_{1})}{\sum}|Comm_{G_{1}}(A_{1})|\right)\left(\frac{1}{|\mathcal{L}(G_{2})|^{2}}\underset{A_{2}\in\mathcal{L}(G_{2})}{\sum}|Comm_{G_{2}}(A_{2})|\right)$ $=ssd(G_{1})\cdot ssd(G_{2}).$ Hence the proposition follows. ∎ ###### Corollary 2.11. Proposition 2.10 is still true for finitely many factors. ###### Proof. We can mimick the proof of Proposition 2.10. ∎ ## 3\. Multiple strong subgroup commutativity degree In analogy with $d^{(n)}(H,G)$ ($n\geq 1$), introduced in [12], the notion of strong subgroup commutativity degree, given in Section 1, can be further generalized in the following way: (3.1) $ssd^{(n)}(H,G)=\frac{|\\{(L_{1},\dots,L_{n},K)\in\mathcal{L}(H)^{n}\times\mathcal{L}(G)\ |\ [L_{1},\dots,L_{n},K]=1\\}|}{|\mathcal{L}(H)|^{n}\ |\mathcal{L}(G)|}.$ In particular, if $n=1$ and $H=G$, then $ssd^{(1)}(G,G)=ssd(G)$. Briefly, $ssd^{(n)}(H)$ denotes (3.2) $ssd^{(n)}(H,H)=\frac{|\\{(L_{1},\dots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n+1}\ |\ [L_{1},\dots,L_{n},L_{n+1}]=1\\}|}{|\mathcal{L}(H)|^{n+1}}.$ On another hand, we note that (3.3) $[L_{1},\dots,L_{n},K]=[[L_{1},\dots,L_{n}],K]=\ldots=[[\ldots[[L_{1},L_{2}],L_{3}]\ldots L_{n}],K]=1$ and so (3.4) $Comm_{G}(L_{1},\dots,L_{n})=\\{K\in\mathcal{L}(G)\ |\ [L_{1},\dots,L_{n},K]=1\\},$ (3.5) $Comm_{H\times G}(L_{1},\dots,L_{n-1})=\\{(L_{n},K)\in\mathcal{L}(H)\times\mathcal{L}(G)\ |\ [[[L_{1},\dots,L_{n-1}],L_{n}],K]=1\\}$ $\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots$ $Comm_{H^{n-1}\times G}(L_{1})=\\{(L_{2},L_{3},\ldots,L_{n},K)\in\mathcal{L}(H)^{n-1}\times\mathcal{L}(G)\ |$ $\ [\ldots[L_{1},L_{2}],\ldots,L_{n}],K]=1\\}.$ Of course, all these sets are nonempty, since they contain at least the trivial subgroup. By construction, $Comm_{H^{n-1}\times G}(L_{1})\subseteq Comm_{H^{n-2}\times G}(L_{1},L_{2})\subseteq\ldots\subseteq Comm_{H\times G}(L_{1},\dots,L_{n-1})\subseteq Comm_{G}(L_{1},\dots,L_{n}).$ From the above inclusions we observe that for $n$ which is growing the $Comm_{H^{n-1}\times G}(L_{1})$ is getting to the trivial subgroup. Therefore (3.6) $|\mathcal{L}(H)|^{n}\ |\mathcal{L}(G)|\ ssd^{(n)}(H,G)={\underset{L_{1},\dots,L_{n}\in\mathcal{L}(H)}{\sum}}|Comm_{G}(L_{1},\dots,L_{n})|$ $={\underset{L_{1},\dots,L_{n}\in\mathcal{L}(H)}{\sum}}|Comm_{H^{n-1}\times G}(L_{1})|$ and to the extreme case we have (3.7) $\lim_{n\rightarrow\infty}\ ssd^{(n)}(H,G)=\lim_{n\rightarrow\infty}\frac{1}{|\mathcal{L}(H)|^{n}\ |\mathcal{L}(G)|}\ \cdot\ \lim_{n\rightarrow\infty}{\underset{L_{1},\dots,L_{n}\in\mathcal{L}(H)}{\sum}}|Comm_{H^{n-1}\times G}(H_{1})|$ $=\frac{1}{|\mathcal{L}(G)|}\ \cdot\ \lim_{n\rightarrow\infty}\frac{1}{|\mathcal{L}(H)|^{n}}\ \cdot\ 1=0.$ This is a qualitative argument which shows that it is meaningful to consider values of probabilities of $ssd^{(n)}(H,G)$ for a small number of commuting subgroups. At the same time, the above construction shows that $ssd^{(n)}(H,G)$ is a strictly decreasing sequence of numbers in $[0,1]$ in the variable $n$. Namely, (3.8) $ssd^{(1)}(H,G)\geq ssd^{(2)}(H,G)\geq\ldots\geq ssd^{(n)}(H,G)\geq ssd^{(n+1)}(H,G)\geq\ldots$ We want to point out that a similar treatment can be done for $sd(G)$, as proposed in a series of opens problems in [31], where the corresponding version of $ssd^{(n)}(H,G)$ is called relative subgroup commutativity degree. As done in Section 2, we may rewrite $ssd^{(n)}(H,G)$ in the following form: (3.9) $ssd^{(n)}(H,G)=\frac{1}{|\mathcal{L}(H)|^{n}\ |\mathcal{L}(G)|}{\underset{\underset{Y\in\mathcal{L}(G)}{X_{1},\ldots,X_{n}\in\mathcal{L}(H)}}{\sum}\varphi_{n}(X_{1},\ldots,X_{n},Y)},$ where $\varphi_{n}:\mathcal{L}(H)^{n}\times\mathcal{L}(G)\rightarrow\\{0,1\\}$ is the function defined by (3.10) $\varphi_{n}(X_{1},\ldots,X_{n},Y)=\left\\{\begin{array}[]{lcl}1,&&\mathrm{if}\ [X_{1},\ldots,X_{n},Y]=1,\\\ 0,&&\mathrm{if}\ [X_{1},\ldots,X_{n},Y]\not=1\end{array}\right.$ and continues to be symmetric. ###### Proposition 3.1. Given subgroup $H$ of a group $G$, $ssd^{(n)}(H,G)\leq ssd^{(n)}(G,G)\leq ssd(G)\leq sd(G).$ ###### Proof. We begin to prove the first inequality. Since $\mathcal{L}(H)\subseteq\mathcal{L}(G)$, (3.11) $ssd^{(n)}(H,G)\leq|\mathcal{L}(H)|^{n}\ |\mathcal{L}(G)|\ ssd^{(n)}(H,G)={\underset{\underset{Y\in\mathcal{L}(G)}{X_{1},\ldots,X_{n}\in\mathcal{L}(H)}}{\sum}\varphi_{n}(X_{1},\ldots,X_{n},Y)}$ (3.12) $\leq\sum_{X_{1},\ldots,X_{n},Y\in\mathcal{L}(G)}\varphi_{n}(X_{1},\ldots,X_{n},Y)=|\mathcal{L}(G)|^{n}\ |\mathcal{L}(G)|\ ssd^{(n)}(G,G).$ The second inequality follows once we note that $ssd^{(n)}(H,G)$ is a decreasing sequence. Therefore, if we fix $H=G$, then $ssd(G)=ssd^{(1)}(G,G)\geq ssd^{(2)}(G,G)\geq\ldots\geq ssd^{(n)}(G,G)\geq\ldots$. The last inequality follows once we note that $Comm_{G}(H)\subseteq\mathcal{C}(H)$ and that (3.13) $ssd(G)=\frac{1}{|\mathcal{L}(G)|^{2}}\sum_{H\in\mathcal{L}(G)}|Comm_{G}(H)|\leq\frac{1}{|\mathcal{L}(G)|^{2}}\sum_{H\in\mathcal{L}(G)}|\mathcal{C}(H)|=sd(G).$ ∎ ###### Proposition 3.2. For two groups $C$ and $D$ of coprime orders and two subgroups $A\leq C$ and $B\leq D$, $ssd^{(n)}(A\times B,C\times D)=ssd^{(n)}(A,C)\cdot ssd^{(n)}(B,D).$ ###### Proof. (3.14) $ssd^{(n)}(A\times B,C\times D)$ $=\frac{1}{|\mathcal{L}(A\times B)|^{n}\ |\mathcal{L}(C\times D)|}\underset{A_{1}\times B_{1},\ldots,A_{n}\times B_{n}\in\mathcal{L}(A\times B)}{\sum}|Comm_{A\times B}(A_{1}\times B_{1},\dots,A_{n}\times B_{n})|$ $=\frac{1}{|\mathcal{L}(A)|^{n}\cdot|\mathcal{L}(B)|^{n}\cdot|\mathcal{L}(C)|\cdot|\mathcal{L}(D)|}\Big{(}\underset{A_{1}\times B_{1},\ldots,A_{n}\times B_{n}\in\mathcal{L}(A\times B)}{\sum}|Comm_{A}(A_{1},\ldots,A_{n})|$ $\cdot|Comm_{B}(B_{1},\ldots,B_{n})|\Big{)}=\frac{1}{|\mathcal{L}(A)|^{n}\cdot|\mathcal{L}(B)|^{n}\cdot|\mathcal{L}(C)|\cdot\mathcal{L}(D)|}$ $=\Big{(}\underset{A_{1},\ldots,An\in\mathcal{L}(A)}{\sum}|Comm_{A}(A_{1},\dots,A_{n})|\Big{)}\cdot\Big{(}\underset{B_{1},\ldots,B_{n}\in\mathcal{L}(B)}{\sum}|Comm_{B}(B_{1},\ldots,B_{n})|\Big{)}$ $=\frac{1}{|\mathcal{L}(A)|^{n}\ |\mathcal{L}(C)|}\Big{(}\underset{A_{1},\ldots,An\in\mathcal{L}(A)}{\sum}|Comm_{A}(A_{1},\dots,A_{n})|\Big{)}$ $\cdot\frac{1}{|\mathcal{L}(B)|^{n}\ |\mathcal{L}(D)|}\Big{(}\underset{B_{1},\ldots,B_{n}\in\mathcal{L}(B)}{\sum}|Comm_{B}(B_{1},\ldots,B_{n})|\Big{)}$ $=ssd^{(n)}(A,C)\cdot ssd^{(n)}(B,D).$ ∎ We note that Proposition 2.10 follows from Proposition 3.2, when $n=1$, $A=C=G_{1}$, $B=D=G_{2}$. ###### Corollary 3.3. Proposition 3.2 is still true for finitely many factors. ###### Proof. We can mimick the proof of Proposition 3.2. ∎ We end with a variation on the theme of Theorems 2.2 and 2.4. ###### Theorem 3.4. Let $H$ and $K$ be two subgroups of a group $G$. Then for all $n\geq 1$ $ssd^{(n)}(H,H)<\frac{|H|^{n+1}}{|\mathcal{L}(H)|^{n+1}}\ \sum_{K\in\mathcal{L}(H)}d^{(n)}(K,K).$ ###### Proof. Overlapping the argument in the proof of Theorem 2.2,we firstly prove that (3.15) $\bigcup_{(L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n}}C_{H^{n}}(L_{1})=Comm_{H^{n}}(L_{1}),$ where (3.16) $Comm_{H^{n}}(L_{1})=Comm_{H^{n-1}\times H}(L_{1})$ $=\\{(L_{2},L_{3},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n-1}\times\mathcal{L}(H)\ |\ [\ldots[L_{1},L_{2}],\ldots,L_{n}],L_{n+1}]=1\\}$ and then (3.17) $|\mathcal{L}(H)|^{n+1}\ ssd^{(n)}(H,H)=\sum_{L_{1}\in\mathcal{L}(H)}|Comm_{H^{n}}(L_{1})|$ $=\sum_{L_{1}\in\mathcal{L}(H)}\left|\bigcup_{(L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n}}C_{H^{n}}(L_{1})\right|$ $<\sum_{(L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n}}\sum_{L_{1}\in\mathcal{L}(H)^{n}}|C_{H^{n}}(L_{1})|$ and we note that the equality must be strict for the same motivation of the corresponding step in the proof of Theorem 2.2. Since $C_{H^{n}}(L_{1})\subseteq C_{H^{n}}(l_{1})$ whenever $l_{1}\in L_{1}$, we may continue, finding that (3.18) $\leq\sum_{(L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n}}{\underset{L_{1}\in\mathcal{L}(H)}{\underset{l_{1}\in L_{1}}{\sum}}}|C_{H^{n}}(l_{1})|$ $=\sum_{(L_{1},L_{2},\ldots,L_{n},L_{n+1})\in\mathcal{L}(H)^{n+1}}\Big{(}\sum_{l_{1}\in L_{1}}|C_{H^{n}}(l_{1})|\Big{)}$ $=\sum_{K\in\mathcal{L}(H)}d^{(n)}(K,K)\ |K|^{n+1}\leq|H|^{n+1}\ \sum_{K\in\mathcal{L}(H)}d^{(n)}(K,K).$ ∎ Roughly speaking, in the proof of Theorem 2.9 we may replace the role of $\varphi=\varphi_{2}$ with that of $\varphi_{n}$ for $n>2$. We will find the following generalization of Theorem 2.9, whose proof is easy to check and so it is omitted. ###### Theorem 3.5. Let $H$ be a subgroup of a group $G$. Then for all $n\geq 1$ $\frac{|\mathcal{L}(H)|^{n+1}}{|\mathcal{L}(G)|^{n+1}}\ ssd^{(n)}(H)\leq ssd^{(n)}(G).$ We note that a similar treatment can be done for the relative subgroup commutativity degree in [31], since the arguments involve only combinatorial properties and set theory. This fact motivates to conjecture that the context of infinite compact groups, once a suitable Haar measure is replaced with $ssd(G)$ or with $sd(G)$, may be subject to an analogous treatment. ## 4\. Two applications Here we illustrate an application to the theory of characters and another to the dihedral groups. Relations with the theory of characters are due to the fact that in a group $G$ (4.1) $d(G)=\frac{|\mathrm{Irr}(G)|}{|G|},$ where $\mathrm{Irr}(G)$ denotes the set of all irreducible complex characters of $G$. For an element $g$ of $G$, let (4.2) $\xi(g)=|(X,Y)\in\mathcal{L}(\langle g\rangle)\times\mathcal{L}(G)\ |\ [X,Y]=1\\}|.$ Thus, (4.3) $ssd(\langle g\rangle,G)=\frac{\xi(g)}{|\mathcal{L}(\langle g\rangle)||\mathcal{L}(G)|}.$ ###### Lemma 4.1. $\xi(g)$ is a class function. ###### Proof. It is enough to note that, for each $a\in G$, the map (4.4) $f:(X,Y)\mapsto f(X,Y)=(aXa^{-1},aYa^{-1})$ defines a one to one correspondence between the sets $\\{(X,Y)\in\mathcal{L}(\langle g\rangle)\times\mathcal{L}(G)\ |\ [X,Y]=1\\}$ and $\\{(X,Y)\in\mathcal{L}(\langle aga^{-1}\rangle)\times\mathcal{L}(G)\ |\ [X,Y]=1\\}$. ∎ Thus, it is meaningful to write (4.5) $\xi(g)=\underset{\chi\in\mathrm{Irr}(G)}{\sum}[\xi,\chi]\chi(g)$ where $[\,,\,]$ denotes the usual inner product of characters, defined by (4.6) $[\xi,\chi]=\dfrac{1}{|G|}\sum_{g\in G}\xi(g)\overline{\chi(g)}=\dfrac{1}{|G|}\sum_{g\in G}\xi(g)\chi(g^{-1}).$ We recall that a class function defined on a finite group $G$ is said to be an $R$–generalized character of $G$, for any ring $\mathbb{Z}\subseteq R\subseteq\mathbb{C}$, if it is an $R$–linear combination of irreducible complex characters of $G$. ###### Theorem 4.2. $\xi$ is a $\mathbb{Q}$-generalized character of $G$. ###### Proof. Clearly, if two elements $x$ and $y$ of $G$ generate the same cyclic group then $\xi(x)=\xi(y)$. Suppose that $o(x)=o(y)=n$. Let $\varepsilon$ be a primitive $n$th root of unity. We have $y=x^{m}$ for some $m$ with $(m,n)=1$ and thus $\varepsilon^{m}$ is a primitive $n$th root of unity. As usual, $\mathrm{Gal}(\mathbb{Q}[\varepsilon]/\mathbb{Q})$ denotes the Galois group, related to the algebraic extension $\mathbb{Q}[\varepsilon]$ over $\mathbb{Q}$, obtained adding $\varepsilon$. Therefore, for any $\sigma\in\mathrm{Gal}(\mathbb{Q}[\varepsilon]/\mathbb{Q})$ we have (4.7) $\chi(x)^{\sigma}=\sum{\epsilon_{i}}^{\sigma}=\sum{\epsilon_{i}}^{m}=\chi(x^{m}).$ Thus for any $\chi\in\mathrm{Irr}(G)$ and $g\in G$, (4.8) $\chi(g)^{\sigma}=\chi(g^{m})$ and hence $\left(\delta(g)\chi(g^{-1})\right)^{\sigma}=\delta(g^{m})\chi(g^{-m})$. Hence $\sigma$ fixes $\sum_{g\in G}\delta(g)\chi(g^{-1})$ and this completes the proof. ∎ ###### Corollary 4.3. $|G|[\xi,\chi]$ is an integer for all $\chi\in\mathrm{Irr}(G)$. ###### Proof. Since $\chi(g)$ is an algebraic integer the result follows from Lemma 4.1 and Theorem 4.2. ∎ For the second application, the dihedral group (4.9) $D_{2n}=\langle x,y\ |\ x^{2}=y^{n}=1,x^{-1}yx=y^{-1}\rangle$ of symmetries of a regular polygon with $n\geq 1$ edges has order $2n$ and a well–known de- scription of $|\mathcal{L}(D_{2n})|$ can be found in [29, 30, 31]. For instance, it is easy to see that $D_{2n}\simeq C_{2}\ltimes C_{n}$ is the semidirect product of a cyclic group $C_{2}$ of order 2 acting by inversion on a cyclic group $C_{n}$ of order $n$. For every divisor $r$ of $n$, $D_{2n}$ has a subgroup isomorphic to $C_{r}$ , namely $\langle x^{\frac{n}{r}}\rangle$, and $\frac{n}{r}$ subgroups isomorphic to $D_{2r}$, namely $\langle x^{\frac{n}{r}},x^{i-1},y\rangle$ for $i=1,2,\ldots,\frac{n}{r}$. Then (4.10) $|\mathcal{L}(D_{2n})|=\sigma(n)+\tau(n),$ where $\sigma(n)$ and $\tau(n)$ are the sum and the number of all divisors of $n$, respectively. The next result generalizes the above considerations, when we have a group with a structure very close to that of $D_{2n}$. ###### Corollary 4.4. Assume that $G$ is a metabelian group of even order. If $|\mathcal{L}(G)|=\sigma(\frac{|G|}{2})+\tau(\frac{|G|}{2})$ and $G^{\prime}$ is cyclic, then $\frac{(\tau(G^{\prime})+1)^{2}}{\Big{(}\sigma\Big{(}\frac{|G|}{2}\Big{)}+\tau\Big{(}\frac{|G|}{2}\Big{)}\Big{)}^{2}}\leq\sum_{H,K\in\mathcal{L}(G)}\varphi(H,K)\leq\frac{|G|^{2}}{\Big{(}\sigma\Big{(}\frac{|G|}{2}\Big{)}+\tau\Big{(}\frac{|G|}{2}\Big{)}\Big{)}^{2}}\sum_{H,K\in\mathcal{L}(G)}d(H,K).$ ###### Proof. Since $G^{\prime}$ is cyclic, $|\mathcal{L}(G^{\prime})|=\tau(G^{\prime})$. Then the lower bound follows from Corollary 2.8, specifying the numerical values of the subgroup lattices. From Theorem 2.2, we get the upper bound, adapted to our case. The result follows. ∎ Corollary 4.4 is a counting formula for the number of permuting subgroups via $\varphi$, or, equivalently, via the strong subgroup commutativity degree and the commutativity degree. This observation is important in virtue of the fact that we know explicitly $d(H,K)$ by results in [2, 7, 8, 9, 12, 18, 19]. ## References * [1] A.M. Alghamdi, D.E. Otera and F.G. Russo, On some recent investigations of probability in group theory, Boll. Mat. Pura Appl. 3 (2010), 87–96. * [2] A.M. Alghamdi and F.G. Russo, A generalization of the probability that the commutator of two group elements is equal to a given element, preprint,Cornell University Library, 2010, arXiv: 1004.0934. * [3] R. Barman, Quasinormality degrees of subgroups of a finite group and a class function, preprint, 2011. * [4] F. Barry, D. MacHale and Á. Ní Shé, Some supersolvability conditions for finite groups, Math. Proc. Royal Irish Acad. 106 A (2) (2006), 163–177. * [5] Y. Berkovich, Groups of prime power order Vol. I, de Gruyter, Berlin, 2008. * [6] K. Chiti, M.R.R. Moghaddam and A. Salemkar, $n$-isoclinism classes and $n$-nilpotency degree of finite groups, Algebra Colloq. 12 (2005), 255–261. * [7] A.K. Das and R.K. Nath, On the generalized relative commutative degree of a finite group, Int. Electr. J. Algebra 7 (2010), 140–151. * [8] A.K. Das and R.K. Nath, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo 59 (2010), 137–142. * [9] A. Erfanian, P. Lescot and R. Rezaei, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra 35 (2007), 4183–4197. * [10] A. Erfanian and F.G. Russo, Probability of mutually commuting $n$-tuples in some classes of compact groups, Bull. Iran. Math. Soc. 34 (2008), 27–37. * [11] A. Erfanian and R. Rezaei, On the commutativity degree of compact groups, Arch. Math. (Basel) 93 (2009), 345–356. * [12] A. Erfanian, R. Rezaei and F.G. Russo, Relative $n$-isoclinism classes and relative $n$-th nilpotency degree of finite groups, preprint, Cornell University Library, 2010, arXiv: 1003.2306. * [13] M. Farrokhi, H. Jafari and F. Saeedi, Subgroup normality degree of finite groups I, Arch. Math. (Basel) 96 (2011), 215–224. * [14] M. Farrokhi and F. Saeedi, Subgroup normality degree of finite groups II, preprint, 2011. * [15] M. Farrokhi, Finite groups with two subgroup normality degrees, preprint, 2011. * [16] R.M. Guralnick and G.R. Robinson, On the commuting probability in finite groups, J. Algebra 300 (2006), 509–528. * [17] K.H. Hofmann and F.G. Russo, The probability that $x$ and $y$ commute in a compact group, preprint, Cornell University Library, 2010, arXiv:1001.4856. * [18] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177 (1995), 847–869. * [19] P. Lescot, Central extensions and commutativity degree, Comm. Algebra 29 (2001), 4451–4460. * [20] H. Mohammadzadeh, A. Salemkar and H. Tavallaee, A remark on the commuting probability in finite groups, Southeast Asian Bull. Math. 34 (2010), 755–763, * [21] P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335–343. * [22] P. Niroomand and R. Rezaei, The exterior degree of a pair of finite groups, preprint, Cornell University Library, arXiv:1101.4312v1, 2011. * [23] P. Niroomand, R. Rezaei and F.G. Russo, Commuting powers and exterior degree of finite groups, preprint, Cornell University Library, arXiv:1102.2304, 2011. * [24] D.E. Otera and F.G. Russo, Subgroup $S$-commutativity degree of finite groups, preprint, Cornell University Library, 2010, arXiv:1009.2171. * [25] R. Rezaei and F.G. Russo, $n$-th relative nilpotency degree and relative $n$-isoclinism classes, Carpathian J. Math. 27 (2011), 123–130. * [26] D.J. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82 (1979), 237–247. * [27] F.G. Russo, A probabilistic meaning of certain quasinormal subgroups, Int. J. Algebra 1 (2007), 385–392. * [28] F.G. Russo, The generalized commutativity degree in a finite group, Acta Univ. Apulensis Math. Inform. 18 (2009), 161–167. * [29] R. Schmidt, Subgroup Lattices of Groups, de Gruyter, Berlin, 1994. * [30] M. Tǎrnǎuceanu, Subgroup commutativity degrees of finite groups, J. Algebra 321 (2009), 2508–2520. * [31] M. Tǎrnǎuceanu, Addendum [Subgroup commutativity degrees of finite groups], J. Algebra (2011), in press.
arxiv-papers
2011-02-02T17:32:29
2024-09-04T02:49:16.813414
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Francesco G. Russo (Universita' degli Studi di Palermo, Palermo,\n Italy)", "submitter": "Francesco G. Russo", "url": "https://arxiv.org/abs/1102.0509" }
1102.0739
# The Renormalization Group and the Effective Action D.G.C. McKeon Department of Applied Mathematics, The University of Western Ontario, London, ON N6A 5B7, Canada ###### Abstract The renormalization group is used to sum the leading-log (LL) contributions to the effective action for a large constant external gauge field in terms of the one-loop renormalization group (RG) function $\beta$, the next-to-leading-log (NLL) contributions in terms of the two-loop RG function etc. The log independent pieces are not determined by the RG equation, but can be fixed by considering the anomaly in the trace of the energy-momentum tensor. Similar considerations can be applied to the effective potential $V$ for a scalar field $\phi$; here the log independent pieces are fixed by the condition $V^{\prime}(\phi=v)=0$. 11footnotetext: Email: dgmckeo2@uwo.ca The effective Lagrangian for a constant external gauge field has been considered in a number of papers [1-7]. In the limit of a strong external field strength, these lead to logarithmic corrections to the classical Lagrangian $-\frac{1}{4}F_{\mu\nu}^{2}\equiv-\frac{1}{4}\Phi$. (We can regard $F$ as being either the electromagnetic field strength in QED or a non-Abelian field strength which may be coupled to matter.) A systematic summation of these effects by using the RG equation has been discussed in ref. [8]; the summation of logarithmic effects arising due to radiative processes in other contents has been considered in [9]. In this note we show that the RG equation, when applied to the effective Lagrangian $L$ for a strong external gauge field, can be rewritten as a sequence of coupled ordinary differential equations for functions $S_{n}$ with $S_{0}$ giving the LL contribution to $L$, $S_{1}$ the NLL contribution to $L$, etc. The boundary conditions for these equations are the log-independent contributions to $L$. These can be fixed by examining the anomaly in the energy momentum tensor [10] as this anomaly can be used to find a formal expression for $L$ [11]. The approach used is similar to one employed with the effective potential when there is a fundamental scalar field in the model [12]. If $\mu$ is the renormalization scale in a model, $F_{\mu\nu}$ is the constant external field strength and $\lambda$ the gauge coupling, then the effective Lagrangian $L(F_{\mu\nu},\lambda,\mu)$ must be independent of $\mu$ and hence the RG equation follows, $\frac{dL}{d\mu}=\left(\mu\frac{\partial}{\partial\mu}+\beta(\lambda)\frac{\partial}{\partial\lambda}+\gamma(\lambda)F_{\mu\nu}\frac{\partial}{\partial F_{\mu\nu}}\right)L=0.$ (1) Since $\lambda F_{\mu\nu}$ is not renormalized, $\beta(\lambda)=-\lambda\gamma(\lambda)$ [13] and so if $\Phi=F^{\mu\nu}F_{\mu\nu}$, eq. (1) becomes $\left[\mu\frac{\partial}{\partial\mu}+\beta(\lambda)\left(\frac{\partial}{\partial\lambda}-\frac{2}{\lambda}\Phi\frac{\partial}{\partial\Phi}\right)\right]L=0.$ (2) With $t=\frac{1}{4}\ln\left(\frac{\lambda^{2}\Phi}{\mu^{4}}\right)$, the form of $L$ when $\lambda\Phi>>\mu^{2}$ is [8] $L=\sum_{n=0}^{\infty}\sum_{m=0}^{n}\,T_{n,m}\lambda^{2n}t^{m}\Phi=\sum_{n=0}^{\infty}S_{n}(\lambda^{2}t)\lambda^{2n}\Phi$ (3) where $S_{n}(\lambda^{2}t)=\displaystyle{\sum_{m=0}^{\infty}}T_{n+m,m}(\lambda^{2}t)^{m}$. If $\beta(\lambda)=\displaystyle{\sum_{n=1}^{\infty}}b_{2n+1}\lambda^{2n+1}$, eq. (2) is satisfied at progressively higher orders in $\lambda$ provided these functions satisfy a set of coupled ordinary differential equations, $\displaystyle wS_{0}^{\prime}(w)\\!\\!$ $\displaystyle-$ $\displaystyle\\!\\!S_{0}=0$ (4a) $\displaystyle b_{3}wS_{1}^{\prime}(w)\\!\\!$ $\displaystyle-$ $\displaystyle\\!\\!b_{5}S_{0}(w)+(1+w)b_{5}S_{0}^{\prime}(w)=0$ (4b) $\displaystyle-b_{3}S_{2}^{\prime}+b_{3}S_{2}\\!\\!$ $\displaystyle-$ $\displaystyle\\!\\!b_{7}S_{0}+(1+w)(b_{7}S_{0}^{\prime}+b_{5}S_{1}^{\prime}+b_{3}S_{2}^{\prime})=0$ (4c) etc. with $w=-1+2b_{3}(\lambda^{2}t)$. In general $S_{n}(\xi)$ can be found once $S_{0}\ldots S_{n-1}$ have been determined provided $b_{3}\ldots b_{2n+3}$ are known and the boundary conditions $S_{n}(\lambda^{2}t=0)=T_{n0}$ have been specified. In particular, $S_{0}=T_{00}w$, $S_{1}=-\frac{T_{00}b_{5}}{b_{3}}\ln|w|+T_{10}$ and $S_{2}=\left[\left(\frac{b_{5}}{b_{3}}\right)^{2}-\frac{b_{7}}{b_{3}}\right]T_{00}\ln|w|-\left(\frac{b_{3}}{b_{5}}\right)^{2}T_{00}\left(\frac{1}{w}-1\right)+T_{20}$. To find these boundary conditions, an extra condition must be found. To do this, we reexpress $L$ in eq. (3) as $L=\displaystyle{\sum_{n=0}^{\infty}}A_{n}(\lambda)t^{n}\Phi$ where$A_{n}=\displaystyle{\sum_{m=n}^{\infty}}T_{m,n}\lambda^{2m}$. Eq. (2) is now satisfied at each order in $t$ provided $\frac{1}{\lambda^{2}}A_{n+1}(\lambda)=\frac{1}{n+1}\beta(\lambda)\frac{d}{d\lambda}\left(\frac{1}{\lambda^{2}}A_{n}(\lambda)\right).$ (5) If now $A_{n}(\lambda)=\lambda^{2}\overline{A}_{n}(\lambda)$ and $\eta=\displaystyle{\int_{\lambda_{0}}^{\lambda(\eta)}}\frac{dx}{\beta(x)}$ then $\overline{A}_{n+1}(\lambda(\eta))=\frac{1}{(n+1)!}\,\frac{d^{n+1}}{d\eta^{n+1}}\overline{A}_{0}(\lambda(\eta))$ (6) so that $L=\lambda^{2}(\eta)\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\,\frac{d^{n}}{d\eta^{n}}\overline{A}_{0}(\lambda(\eta))\Phi=\lambda^{2}(\eta)\overline{A}_{0}(\lambda(\eta+t))\Phi=\frac{\lambda^{2}(\eta)}{\lambda^{2}(\eta+t)}A_{0}(\lambda(\eta+t))\Phi\,.$ (7) Since $A_{0}$ is determined by the $T_{n0}$, we see from eq. (7) that again the log independent contributions to $L$ fix the log dependent contributions once $\beta$ is known. When $\eta=0$, we take the value of the function $\lambda(\eta)$ to be $\lambda_{0}$. We now recall that the trace anomaly of the energy momentum tensor [10] $\left\langle\theta^{\mu}_{\,\,\mu}\right\rangle=\frac{\beta(\overline{\lambda})}{2\overline{\lambda}(t)}\,\frac{\lambda^{2}_{0}}{\overline{\lambda}^{2}(t)}\Phi$ (8) where $\frac{d\overline{\lambda}(t)}{dt}=\beta(\overline{\lambda}(t))\;\;\left(t=\int_{\lambda_{0}}^{\overline{\lambda}(t)}\,\frac{dx}{\beta(x)}\right)$ (9) leads to [11] $L=-\frac{1}{4}\frac{\lambda^{2}_{0}}{\overline{\lambda}^{2}(t)}\Phi$ (10) since $\left\langle\theta^{\mu\nu}\right\rangle=-\eta^{\mu\nu}L+2\frac{\partial L}{\partial\eta_{\mu\nu}}$. (It can be verified that eq. (10) satisfies eq. (2).) The usual “running coupling function” $\overline{\lambda}(t)$ has the boundary condition $\overline{\lambda}(0)=\lambda_{0}$ with $\lambda_{0}$ also being equal to $\lambda(\eta=0)$. It is now apparent that eqs. (7) and (10) are identical provided $\eta=0$, and so we now have the boundary conditions for eq. (4) $T_{n0}=-\frac{1}{4}\,\delta_{n0}.$ (11) Upon equating $L$ in eqs. (10) and (3) we see that $\frac{1}{\overline{\lambda}^{2}(t)}=\frac{-4}{\lambda_{0}^{2}}\left[\displaystyle{\sum_{n=0}^{\infty}}S_{n}(\lambda_{0}^{2}t)\lambda_{0}^{2n}\right]$, which is a novel expression for the running coupling in terms of $\beta(\lambda)$. Consequently the log-independent contributions to $L$ are fixed by the trace anomaly. The effective potential $V$ for a massless scalar field with the classical potential $V_{C1}=\lambda\phi^{4}$ can be treated in an analogous fashion. We will now review how [12] the RG equation can be used to express the log- dependent part of $V$ in terms of the log-independent parts, and how these log-independent parts can be determined by considering an extra condition (which in this case is $V^{\prime}(\phi=v)=0)$. The expansion $V=\sum_{n=0}^{\infty}\,\sum_{m=0}^{n}\lambda^{n+1}T_{n,m}L^{m}\phi^{4}\quad\left(L=\log\frac{\phi}{\mu}\right)$ (12) when expressed as $V=\sum_{n=0}^{\infty}A_{n}(\lambda)L^{n}\phi^{4}$ (13) (where $A_{n}=\displaystyle{\sum_{m=n}^{\infty}}T_{m,n}\lambda^{m+1}$) satisfies the RG equation $\left(\mu\frac{\partial}{\partial\mu}+\beta(\lambda)\frac{\partial}{\partial\lambda}+\gamma(\lambda)\phi\frac{\partial}{\partial\phi}\right)V=0$ (14) provided $A_{n+1}(\lambda)=\frac{1}{n+1}\left(\hat{\beta}\frac{\partial}{\partial\lambda}+4\hat{\gamma}\right)A_{n}(\lambda)$ (15) where $\hat{\beta}=\beta/(1-\gamma)$ and $\hat{\gamma}=\gamma/(1-\gamma)$. If now $\eta=\int_{\lambda_{0}}^{\lambda(\eta)}\,\frac{dx}{\hat{\beta}(x)}$ (16) and $\hat{A}_{n}(\lambda)=A_{n}(\lambda)\exp\left(4\int_{\lambda_{0}}^{\lambda}\frac{\hat{\gamma}(x)}{\hat{\beta}(x)}dx\right)$ (17) then by eq. (15) $\hat{A}_{n+1}(\lambda(\eta))=\frac{1}{n+1}\frac{d}{d\eta}\hat{A}_{n}(\lambda(\eta))=\frac{1}{(n+1)!}\frac{d^{n+1}}{d\eta^{n+1}}\hat{A}_{0}(\lambda(\eta)).$ (18) The sum of eq. (13) now leads to $V=A_{0}(\lambda(\eta+L))\exp\left(4\int_{\lambda(\eta)}^{\lambda(\eta+L)}\frac{\gamma(x)}{\beta(x)}dx\right)\phi^{4}.$ (19) As with eq. (7), eq. (19) shows that effective potential is determined by its log-independent contributions and the RG functions. To fix these log-independent contributions to $V$, we need a second condition. The trace of the energy-momentum tensor does not help us to do this. However, we can invoke the condition $\frac{dV(\phi)}{d\phi}\left|{}_{\phi=v}=0\right.$ (20) where $v$ is the vacuum expectation value of $V$. If the renormalization scale parameter $\mu$ is chosen to be equal to $v$, then by eqs. (13) and (20) $[A_{1}(\lambda)+4A_{0}(\lambda)]v^{3}=0.$ (21) This equation has been derived for a particular value of $\mu$, but as $\lambda$ at this value of $\mu$ is not fixed, eq. (21) implies the functional relation $A_{1}(\lambda)=-4A_{0}(\lambda)$ (22) provided $v\neq 0$. Eq. (22) and eq. (15) with $n=0$ together lead to $\left[\hat{\beta}\frac{d}{d\lambda}+4(1+\hat{\gamma})\right]A_{0}=0$ (23) so that $A_{0}(\lambda)=A_{0}(\lambda_{0})\exp\left(-4\int_{\lambda{{}_{0}}}^{\lambda}\frac{dx}{\beta(x)}\right),$ (24) and hence eq. (19) becomes $\displaystyle V$ $\displaystyle=A_{0}(\lambda_{0})\exp\left(-4\int_{\lambda_{0}}^{\lambda(\eta+L)}\,\frac{dx}{\beta(x)}\right)\exp\left(4\int_{\lambda_{(\eta)}}^{\lambda(\eta+L)}\,\frac{\gamma(x)}{\beta(x)}dx\right)\phi^{4}$ $\displaystyle=A_{0}(\lambda_{0})\exp\left(-4\int_{\lambda{{}_{0}}}^{\lambda}\frac{dx}{\beta(x)}\right)\mu^{4},$ (25) upon using eq. (16). Consequently, $V$ is independent of $\phi$ provided $v\neq 0$; either there is no spontaneous symmetry breakdown or the potential is “flat”. (Of course, this flatness does not preclude spontaneous symmetry breaking.) We thus see that the effective Lagrangian for a constant gauge field and the effective potential for a massless scalar field are completely determined by the RG functions when the RG equation is supplemented by a suitable extra condition. In the case of the effective action for an external electromagnetic field this extra condition is provided by the anomalous trace of the energy momentum tensor. For the effective potential $V(\phi)$ in a massless $\phi^{4}$ model, it is the fact that $V^{\prime}(0)$ disappears when $\phi=v$ which determines $V(\phi)$ completely in terms of the RG functions. ## 1 Acknowledgments The author would like to thank F.T. Brandt, F. Chishtie, T. Hanif, J. Jia, T.N. Sherry and C. Schubert for discussions. R. 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arxiv-papers
2011-02-03T17:31:31
2024-09-04T02:49:16.824464
{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "D.G.C. McKeon", "submitter": "Gerry McKeon", "url": "https://arxiv.org/abs/1102.0739" }