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Given an oriented rational homology 3-sphere M, it is known how to associate to any Spin^c-structure \sigma on M two quadratic functions over the linking pairing. One quadratic function is derived from the reduction modulo 1 of the Reidemeister-Turaev torsion of (M,\sigma), while the other one can be defined using the intersection pairing of an appropriate compact oriented 4-manifold with boundary M. In this paper, using surgery presentations of the manifold M, we prove that those two quadratic functions coincide. Our proof relies on the comparison between two distinct combinatorial descriptions of Spin^c-structures on M Turaev's charges vs Chern vectors.
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arxiv:math/0301041
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Let $D$ be an integral domain with quotient field $K$. A star-operation $\star$ on $D$ is a closure operation $A \longmapsto A^\star$ on the set of nonzero fractional ideals, $F(D)$, of $D$ satisfying the properties: $(xD)^\star = xD$ and $(xA)^\star = xA^\star$ for all $x \in K^\ast$ and $A \in F(D)$. Let ${\M S}$ be a multiplicatively closed set of ideals of $D$. For $A \in F(D)$ define $A_{\M S} = \{x \in K \mid xI \subseteq{A}$, for some $I \in {\M S}\}$. Then $D_{\M S}$ is an overring of $D$ and $A_{\M S}$ is a fractional ideal of $D_{\M S}$. Let ${\M S}$ be a multiplicative set of finitely generated nonzero ideals of $D$ and $A \in F(D)$, then the map $A \longmapsto A_{\M S}$ is a finite character star-operation if and only if for each $I \in {\M S}$, $I_v = D$. We give an example to show that this result is not true if the ideals are not assumed to be finitely generated. In general, the map $A \longmapsto A_{\M S}$ is a star-operation if and only if $\bar {\M S}$, the saturation of ${\M S}$, is a localizing GV-system. We also discuss star-operations given of the form $A \longmapsto \cap AD_\alpha$, where $D = \cap D_\alpha$.
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arxiv:math/0301046
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Suppose that Fourier transform of a function f is zero on the interval [-a,a]. We prove that the lower density of sign changes of f is at least a/pi, provided that f is a locally integrable temperate distribution in the sense of Beurling, with non-quasianalytic weight. We construct an example showing that the last condition cannot be omitted.
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arxiv:math/0301060
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CR invariant differential operators on densities with leading part a power of the sub-Laplacian are derived. One family of such operators is constructed from the ``conformally invariant powers of the Laplacian'' via the Fefferman metric; the powers which arise for these operators are bounded in terms of the dimension. A second family is derived from a CR tractor calculus which is developed here; this family includes operators for every positive power of the sub-Laplacian. This result together with work of Cap, Slovak and Soucek imply in three dimensions the existence of a curved analogue of each such operator in flat space.
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arxiv:math/0301092
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For a graph G, we construct two algebras, whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the sandpile model.
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arxiv:math/0301110
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In his previous papers (Math. Res. Letters 7 (2000), 123--13; Math. Res. Letters 8 (2001), 429--435; Moscow Math. J. 2 (2002), issue 2, 403-431) the author proved that in characteristic $\ne 2$ the jacobian $J(C)$ of a hyperelliptic curve $C: y^2=f(x)$ has only trivial endomorphisms over an algebraic closure $K_a$ of the ground field $K$ if the Galois group $\Gal(f)$ of the irreducible polynomial $f(x) \in K[x]$ is either the symmetric group $\Sn$ or the alternating group $\A_n$. Here $n\ge 9$ is the degree of $f$. The goal of this paper is to extend this result to the case of certain ``smaller'' doubly transitive simple Galois groups. Namely, we treat the infinite series $n=2^m+1, \Gal(f)=\L_2(2^m):=\PSL_2(\F_{2^m})$, $n=2^{4m+2}+1, \Gal(f)=\Sz(2^{2m+1})= {^2\B_2}(2^{2m+1})$ and $n=2^{3m}+1, \Gal(f)=\U_3(2^m):=\PSU_3(\F_{2^m})$.
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arxiv:math/0301177
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We study ``forms of the Fermat equation'' over an arbitrary field $k$, i.e. homogenous equations of degree $m$ in $n$ unknowns that can be transformed into the Fermat equation $X_1^m+...+X_n^m$ by a suitable linear change of variables over an algebraic closure of $k$. Using the method of Galois descent, we classify all such forms. In the case that $k$ is a finite field of characteristic greater than $m$ that contains the $m$-th roots of unity, we compute the Galois representation on $l$-adic cohomology (and so in particular the zeta function) of the hypersurface associated to an arbitrary form of the Fermat equation.
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arxiv:math/0301186
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We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n >= 2 we supply n homotopic maps that each represent the (n-1)!-th power of a suitable generator of pi_2n(U(n)) = Z_{n!}. The product of these n commuting maps is the constant map to the identity matrix.
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arxiv:math/0301192
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For a new class of topological vector spaces, namely $\kappa $-normed spaces, and associated quasisemilinear topological preordered space is defined and investigated. This structure arise naturally from the consideration of a $\kappa $-norm, that is a distance function between a point and a $G_{\delta}$-subset. For it, analogs of the Hahn-Banach theorem are proved.
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arxiv:math/0301197
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We develop a purely set-theoretic formalism for binary trees and binary graphs. We define a category of binary automata, and display it as a fibred category over the category of binary graphs. We also relate the notion of binary graphs to transition systems, which arise in the theory of concurrent computing.
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arxiv:math/0301211
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It is well known since Stasheff's work that 1-fold loop spaces can be described in terms of the existence of higher homotopies for associativity (coherence conditions) or equivalently as algebras of contractible non-symmetric operads. The combinatorics of these higher homotopies is well understood and is extremely useful. For $n \ge 2$ the theory of symmetric operads encapsulated the corresponding higher homotopies, yet hid the combinatorics and it has remain a mystery for almost 40 years. However, the recent developments in many fields ranging from algebraic topology and algebraic geometry to mathematical physics and category theory show that this combinatorics in higher dimensions will be even more important than the one dimensional case. In this paper we are going to show that there exists a conceptual way to make these combinatorics explicit using the so called higher nonsymmetric $n$-operads.
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arxiv:math/0301221
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We solve the following problem: to describe in geometric terms all differential operators of the second order with a given principal symbol. Initially the operators act on scalar functions. Operator pencils acting on densities of arbitrary weights appear naturally in the course of study. We show that for the algebra of densities it is possible to establish a one-to-one correspondence between operators and brackets generated by them. Everything is applicable to supermanifolds as well as to usual manifolds. In the super case the problem is closely connected with the geometry of the Batalin--Vilkovisky formalism in quantum field theory, namely the description of the generating operators for an odd bracket. We give a complete answer. This text is a concise outline of the main results. A detailed exposition is in \texttt{arXiv:math.DG/0212311}.
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arxiv:math/0301236
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The paper considers (a) Representations of measure preserving transformations (``rotations'') on Wiener space, and (b) The stochastic calculus of variations induced by parameterized rotations $\{T_\theta w, 0 \le \theta \le \eps\}$: ``Directional derivatives'' $(dF(T_\theta w)/d \theta)_{\theta=0}$, ``vector fields'' or ``tangent processes'' $(dT_\theta w /d\theta)_{\theta=0}$ and flows of rotations.
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arxiv:math/0301351
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Each labeled rooted tree is associated with a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests. These posets are used to define coalgebras, whose dual algebras are shown to have a simple presentation by generators and relations.
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arxiv:math/0301372
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We obtain Gaussian upper and lower bounds on the transition density q_t(x,y) of the continuous time simple random walk on a supercritical percolation cluster C_{\infty} in the Euclidean lattice. The bounds, analogous to Aronsen's bounds for uniformly elliptic divergence form diffusions, hold with constants c_i depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for q_t(x,\cdot) holds only for t\ge S_x(\omega), where the constant S_x(\omega) depends on the percolation configuration \omega.
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arxiv:math/0302004
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We study the geometric structure of Lorentzian spin manifolds, which admit imaginary Killing spinors. The discussion is based on the cone construction and a normal form classification of skew-adjoint operators in signature $(2,n-2)$. Derived geometries include Brinkmann spaces, Lorentzian Einstein-Sasaki spaces and certain warped product structures. Exceptional cases with decomposable holonomy of the cone are possible.
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arxiv:math/0302024
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I construct some smooth Calabi-Yau threefolds in characteristic two and three that do not lift to characteristic zero. These threefolds are pencils of supersingular K3-surfaces. The construction depends on Moret-Bailly's pencil of abelian surfaces and Katsura's analysis of generalized Kummer surfaces. The threefold in characteristic two turns out to be nonrigid.
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arxiv:math/0302064
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This article concerns the equations of motion of perfect incompressible fluids in a 3-D, smooth, bounded, simply connected domain. We suppose that the curl of the initial velocity field is a vortex patch, and examine the classical problems of the existence of a solution, either locally or globally in time, and of the persistence of the initial regularity.
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arxiv:math/0302066
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This article accompanies my ICM talk in August 2002. Three conjectural directions in Gromov-Witten theory are discussed: Gorenstein properties, BPS states, and Virasoro constraints. Each points to basic structures in the subject which are not yet understood.
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arxiv:math/0302077
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Three examples of free field constructions for the vertex operators of the elliptic quantum group ${\cal A}_{q,p}(\hat{sl}_2)$ are obtained. Two of these (for $p^{1/2}=\pm q^{3/2},p^{1/2}=-q^2$) are based on representation theories of the deformed Virasoro algebra, which correspond to the level 4 and level 2 $Z$-algebra of Lepowsky and Wilson. The third one ($p^{1/2}=q^{3}$) is constructed over a tensor product of a bosonic and a fermionic Fock spaces. The algebraic structure at $p^{1/2}=q^{3}$, however, is not related to the deformed Virasoro algebra. Using these free field constructions, an integral formula for the correlation functions of Baxter's eight-vertex model is obtained. This formula shows different structure compared with the one obtained by Lashkevich and Pugai.
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arxiv:math/0302097
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The aim of this paper is to study harmonic polynomials on the quantum Euclidean space E^N_q generated by elements x_i, i=1,2,...,N, on which the quantum group SO_q(N) acts. The harmonic polynomials are defined as solutions of the equation \Delta_q p=0, where p is a polynomial in x_i, i=1,2,...,N, and the q-Laplace operator \Delta_q is determined in terms of the differential operators on E^N_q. The projector H_m: {cal A}_m\to {\cal H}_{m} is constructed, where {\cal A}_{m} and {\cal H}_m are the spaces of homogeneous of degree m polynomials and homogeneous harmonic polynomials, respectively. By using these projectors, a q-analogue of the classical zonal polynomials and associated spherical polynomials with respect to the quantum subgroup SO_q(N-2) are constructed. The associated spherical polynomials constitute an orthogonal basis of {\cal H}_m. These polynomials are represented as products of polynomials depending on q-radii and x_j, x_{j'}, j'=N-j+1. This representation is in fact a q-analogue of the classical separation of variables. The dual pair (U_q(sl_2), U_q(so_n)) is related to the action of SO_q(N) on E^N_q. Decomposition into irreducible constituents of the representation of the algebra U_q(sl_2)\times U_q(so_n) defined by the action of this algebra on the space of all polynomials on E^N_q is given.
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arxiv:math/0302119
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Let $W$ be a finite Weyl group of classical type which may not be irreducible, $F$ an algebraically closed field, $q$ an invertible element of $F$. We denote by $\mathcal H_W(q)$ the associated Hecke algebra. If $q=1$ then it is $FW$ and we know the representation type. Thus, we assume that $q\ne 1$. Let $P_W(x)$ be the Poincare polynomial of $W$. It is well-known that $\mathcal H_W(q)$ is semisimple if and only if $x-q$ does not divide $P_W(x)$. We show that the similar results hold for finiteness, tameness and wildness. In other words, the Poincare polynomial governs the representation type of $\mathcal H_W(q)$ completely. Note that the finiteness result was already given in the author's previous papers, some of which were written with Andrew Mathas. The proof uses the Fock space theory, which was developed for proving the LLT conjecture (see AMS Univ. Lec. Ser. 26), the Specht module theory, which was developed by Dipper, James and Murphy in this case, and results from the theory of finite dimensional algebras.
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arxiv:math/0302136
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We show that certain generating sets of Dykema and Radulescu for $L(F_r)$ have free Hausdorff dimension r and nondegenerate free Hausdorff r-entropy
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arxiv:math/0302179
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Z.-J. Ruan has shown that several amenability conditions are all equivalent in the case of discrete Kac algebras. In this paper, we extend this work to the case of discrete quantum groups. That is, we show that a discrete quantum group, where we do not assume its unimodularity, has an invariant mean if and only if it is strongly Voiculescu amenable.
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arxiv:math/0302222
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We prove the existence of a finite set of moves sufficient to relate any two representations of the same 3-manifold as a 4-fold simple branched covering of S^3. We also prove a stabilization result: after adding a fifth trivial sheet two local moves suffice. These results are analogous to results of Piergallini in degree 3 and can be viewed as a second step in a program to establish similar results for arbitrary degree coverings of S^3.
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arxiv:math/0302225
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We consider dynamical systems arising from substitutions over a finite alphabet. We prove that such a system is linearly repetitive if and only if it is minimal. Based on this characterization we extend various results from primitive substitutions to minimal substitutions. This includes applications to random Schr\"odinger operators and to number theory.
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arxiv:math/0302231
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We continue the study of scattering theory for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In a previous paper we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of solutions in the range of the wave operators, under a support condition on the asymptotic state required by the different propagation properties of the wave and Schr"odinger equations.Here we eliminate that condition by using an improved asymptotic form for the solutions.
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arxiv:math/0302247
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Let A be a separable unital nuclear purely infinite simple C*-algebra satisfying the Universal Coefficient Theorem, and such that the K_0-class of the identity is zero. We prove that every automorphism of order two of the K-theory of A is implemented by an automorphism of A of order two. As a consequence, we prove that every countable Z/2Z-graded module over the representation ring of Z/2Z is isomorphic to the equivariant K-theory for some action of Z/2Z on a separable unital nuclear purely infinite simple C*-algebra. Along the way, we prove that every not necessarily finitely generated module over the group ring of Z/2Z which is free as an abelian group has a direct sum decomposition with only three kinds of summands, namely the group ring itself and Z on which the nontrivial element of Z/2Z acts either trivially or by multiplication by -1.
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arxiv:math/0302273
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This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength $\epsilon$ of the perturbation is $\gg h$ (or sometimes only $\gg h^2$) and bounded from above by $h^{\delta}$ for some $\delta>0$. We get a complete asymptotic description of all eigenvalues in certain rectangles $[-1/C, 1/C]+ i\epsilon [F_0-1/C,F_0+1/C]$.
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arxiv:math/0302297
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Entringer, Jackson, and Schatz conjectured in 1974 that every infinite cubefree binary word contains arbitrarily long squares. In this paper we show this conjecture is false: there exist infinite cubefree binary words avoiding all squares xx with |x| >= 4, and the number 4 is best possible. However, the Entringer-Jackson-Schatz conjecture is true if "cubefree" is replaced with "overlap-free".
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arxiv:math/0302303
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P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus s, of the least conformal length of a nonseparating loop, is logarithmic in s. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic invariant of a 4-manifold X with indefinite intersection form. The estimate turns out to be polynomial, rather than logarithmic, in \chi(X), if the conjectured surjectivity of the period map is correct. Such surjectivity is targeted by the current work in gauge theory. The surjectivity allows one to insert suitable lattices with metric properties prescribed in advance, into the second de Rham cohomology group of X, as its integer lattice. The idea is to adapt the well-known Lorentzian construction of the Leech lattice, by replacing the Leech lattice by the Conway-Thompson unimodular lattices which define asymptotically dense packings. The final step can be described, in terms of the successive minima \lambda_i, as deforming a \lambda_2-bound into a \lambda_1-bound.
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arxiv:math/0302306
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We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not necessarily local) transformations called {\em flips}. This study allows us to formulate an exhaustive generation algorithm and a uniform random sampling algorithm. We finally extend these results to other types of tilings (calisson tilings, tilings with bicolored Wang tiles).
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arxiv:math/0302344
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Multiplier ideals, and the vanishing theorems they satisfy, have found many applications in recent years. In the global setting they have been used to study pluricanonical and other linear series on a projective variety. More recently, they have led to the discovery of some surprising uniform results in local algebra. The present notes aim to provide a gentle introduction to the algebraically-oriented local side of the theory. They follow closely a short course on multiplier ideals given in September 2002 at the Introductory Workshop of the program in commutative algebra at MSRI.
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arxiv:math/0302351
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Let G_R be a Lie group acting on an oriented manifold M, and let $\omega$ be an equivariantly closed form on M. If both G_R and M are compact, then the integral $\int_M \omega$ is given by the fixed point integral localization formula (Theorem 7.11 in [BGV]). Unfortunately, this formula fails when the acting Lie group G_R is not compact: there simply may not be enough fixed points present. A proposed remedy is to modify the action of G_R in such a way that all fixed points are accounted for. Let G_R be a real semisimple Lie group, possibly noncompact. One of the most important examples of equivariantly closed forms is the symplectic volume form $d\beta$ of a coadjoint orbit $\Omega$. Even if $\Omega$ is not compact, the integral $\int_{\Omega} d\beta$ exists as a distribution on the Lie algebra g_R. This distribution is called the Fourier transform of the coadjoint orbit. In this article we will apply the localization results described in [L1] and [L2] to get a geometric derivation of Harish-Chandra's formula (9) for the Fourier transforms of regular semisimple coadjoint orbits. Then we will make an explicit computation for the coadjoint orbits of elements of G_R* which are dual to regular semisimple elements lying in a maximally split Cartan subalgebra of g_R.
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arxiv:math/0302352
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We study the asymptotic behavior of the simple random walk on oriented version of $\mathbb{Z}^2$. The considered latticesare not directed on the vertical axis but unidirectional on the horizontal one, with symmetric random orientations which are positively correlated. We prove that the simple random walk is transient and also prove a functionnal limit theorem in the space of cadlag functions, with an unconventional normalization.
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arxiv:math/0303063
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We show that untwisted respectively twisted conjugacy classes of a compact and simply connected Lie group which satisfy a certain integrality condition correspond naturally to irreducible highest weight representations of the corresponding affine Lie algebra. Along the way, review the classification of twisted conjugacy classes of a simply connected compact Lie group $G$ and give a description of their stabilizers in terms of the Dynkin diagram of the corresponding twisted affine Lie algebra.
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arxiv:math/0303118
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We study a one parameter family of discrete Loewner evolutions driven by a random walk on the real line. We show that it converges to the stochastic Loewner evolution (SLE) under rescaling. We show that the discrete Loewner evolution satisfies Markovian-type and symmetry properties analogous to SLE, and establish a phase transition property for the discrete Loewner evolution when the parameter equals 4.
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arxiv:math/0303119
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We mainly study 3-dimensional complete gradient Ricci solitons with positive sectional curvature, whose scalar curvature attains its maximum at some point. In section 2, we estimate the area growth of level sets and the volume growth of sublevel sets of a Ricci potential. In section 3, we show that the scalar curvature of such solitons approaches zero at infinity. In section 4, we investigate the geometry of such solitons at infinity, e.g., the tangent cone, the asymptotic behavior, etc.
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arxiv:math/0303135
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On non-K\"ahler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is {\it not} in divergence form. The case of noncompact complete preimage and target manifolds is considered. We give conditions for existence and uniqueness of Hermitian-harmonic maps and solutions of the corresponding parabolic system, which observe the non-divergence form of the underlying equations. Numerous examples illustrate the theoretical results and the fundamental difference to harmonic maps.
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arxiv:math/0303137
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In this paper we study the solitary waves for the coupled Schr\"odinger - Maxwell equations in three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed $L^2$ norm. We study the asymptotic behavior and the smoothness of these solutions. We show also the fact that the eigenvalues are negative and the first one is isolated.
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arxiv:math/0303142
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Under the continuum hypothesis, there is a compact homogeneous strong S-space.
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arxiv:math/0303239
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A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We compute this formulas and discuss reductions and refinements for modular categories related with SU(N).Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer.We introduce the notion of a spin modular category, and give the proof of the decomposition theorem in this general context.
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arxiv:math/0303240
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We take some first steps in providing a synthetic theory of distributions. In particular, we are interested in the use of distribution theory as foundation, not just as tool, in the study of the wave equation.
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arxiv:math/0303297
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This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We give numerous examples and then discuss what happens in the singular case. We also describe rational maps to smooth toric varieties.
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arxiv:math/0303316
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In this note we shall improve some congruences of D.F. Bailey [Two p^3 variations of Lucas' Theorem, JNT 35(1990), pp. 208-215] to higher prime power moduli, by studying the relation between irregular pairs of the form (p,p-3) and refined version of Wolstenholme's theorem.
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arxiv:math/0303332
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A class K of structures is controlled if for all cardinals lambda, the relation of L_{infty,lambda}-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled.
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arxiv:math/0303345
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Necessary and sufficient conditions are investigated for the existence of local bases in which the components of derivations of tensor algebras over differentiable manifold vanish in a neighborhood or only at a single point. The problem when these bases are holonomic or anholonomic is considered. Attention is paid to the case of linear connections. Relations of these problems with the equivalence principle are pointed out.
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arxiv:math/0303373
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We consider, following the work of S. Kerov, random walks which are continuous-space generalizations of the Hook Walks defined by Greene-Nijenhuis-Wilf, performed under the graph of a continual Young diagram. The limiting point of these walks is a point on the graph of the diagram. We present several explicit formulas giving the probability densities of these limiting points in terms of the shape of the diagram. This partially resolves a conjecture of Kerov concerning an explicit formula for the so-called Markov transform. We also present two inverse formulas, reconstructing the shape of the diagram in terms of the densities of the limiting point of the walks. One of these two formulas can be interepreted as an inverse formula for the Markov transform. As a corollary, some new integration identities are derived.
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arxiv:math/0303376
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The proof by Ullmo and Zhang of Bogomolov's conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for ``small points'' in the associated complex abelian variety. We study the analogous equidistribution property at $p$-adic places. Our results can be conveniently stated within the framework of the analytic spaces defined by Berkovich. The first one is valid in any dimension but is restricted to ``algebraic metrics'', the second one is valid for curves, but allows for more general metrics, in particular to the normalized heights with respect to dynamical systems.
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arxiv:math/0304023
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We classify the connected components of the space of representations of the fundamental group of a closed oriented surface of genus $\geq 2$ in $Sp(4,{\mathbf R})$. We prove that this is equivalent to classifying the connected components of the moduli space of representations (which consists of conjugacy classes of reductive representations). We then continue the analysis initiated by Gothen in math/9904114, relying on: (1) the relation between our moduli space of representations and the moduli space of certain (polystable) analogues of Higgs bundles (which follows essentially from work of Hitchin, Simpson and Corlette) and (2) the existence of a proper function on the moduli space whose set of local minima can be identified with the moduli space $N$ of (polystable) quadratic bundles of rank 2; we finally deduce our result from the classification of the connected components of $N$.
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arxiv:math/0304027
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Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V \subset \Gamma(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds with a Lie structure at infinity" there -- was studied from an axiomatic point of view. In this paper, we define and study the algebra $\Psi_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at infinity $V \subset\Gamma(TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to $\Psi_{1,0,V}^\infty(M_0)$. We also consider the algebra $\DiffV{*}(M_0)$ of differential operators on $M_0$ generated by $V$ and $\CI(M)$, and show that $\Psi_{1,0,V}^\infty(M_0)$ is a ``microlocalization'' of $\DiffV{*}(M_0)$. Finally, we introduce and study semi-classical and ``suspended'' versions of the algebra $\Psi_{1,0,V}^\infty(M_0)$. Our construction solves a problem posed by Melrose in his talk at the ICM in Kyoto.
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arxiv:math/0304044
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Let f be an orientation-preserving homeomorphism of the disk D, P a finite invariant subset and [f] the isotopy class of f in D\P. We give a non trivial lower bound of the topological entropy for maps in [f], using the spectral radius of some specializations in GL(n,C) of the Burau matrix associated with [f] and we discuss some examples.
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arxiv:math/0304105
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Together with F. Morel, we have constructed in \cite{CR, Cobord1, Cobord2} a theory of {\em algebraic cobordism}, an algebro-geometric version of the topological theory of complex cobordism. In this paper, we give a survey of the construction and main results of this theory; in the final section, we propose a candidate for a theory of higher algebraic cobordism, which hopefully agrees with the cohomology theory represented by the $\P^1$-spectrum $MGL$ in the Morel-Voevodsky stable homotopy category.
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arxiv:math/0304206
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The purpose of this paper is to compare two spectral sequences converging to the cohomology of a configuration space. The collapsing of these spectral sequences is established, in some cases, using Massey products.
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arxiv:math/0304226
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Some companions of Gruss inequality in inner product spaces and applications for integrals are given.
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arxiv:math/0304239
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In this paper we survey several intersection and non-intersection phenomena appearing in the realm of symplectic topology. We discuss their implications and finally outline some new relations of the subject to algebraic geometry.
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arxiv:math/0304260
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We generalise the concepts introduced by Baez and Dolan to define opetopes constructed from symmetric operads with a category, rather than a set, of objects. We describe the category of 1-level generalised multicategories, a special case of the concept introduced by Hermida, Makkai and Power, and exhibit a full embedding of this category in the category of symmetric operads with a category of objects. As an analogy to the Baez-Dolan slice construction, we exhibit a certain multicategory of function replacement as a slice construction in the multitopic setting, and use it to construct multitopes. We give an explicit description of the relationship between opetopes and multitopes.
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arxiv:math/0304277
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We continue our previous modifications of the Baez-Dolan theory of opetopes to modify the Baez-Dolan definition of universality, and thereby the category of opetopic n-categories and lax functors. For the case n=2 we exhibit an equivalence between this category and the category of bicategories and lax functors. We examine notions of strictness in the opetopic theory.
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arxiv:math/0304285
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Let $G$ be a reductive group in the Harish-Chandra class e.g. a connected semisimple Lie group with finite center, or the group of real points of a connected reductive algebraic group defined over $\R$. Let $\sigma$ be an involution of the Lie group $G$, $H$ an open subgroup of the subgroup of fixed points of $\sigma$. One decomposes the elements of $L^2(G/H)$ with the help of joint eigenfunctions under the algebra of left invariant differential operators under $G$ on $G/H$.
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arxiv:math/0304322
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We study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations consisting of a single line and three spheres for which there are infinitely many lines tangent to the three spheres that also meet the given line. All such configurations are degenerate. The path to this result involves the interplay of some beautiful and intricate geometry of real surfaces in 3-space, complex algebraic geometry, explicit computation and graphics.
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arxiv:math/0304346
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In our previous papers \cite{doz1,doz2} we studied Laguerre functions and polynomials on symmetric cones $\Omega=H/L$. The Laguerre functions $\ell^{\nu}_{\mathbf{n}}$, $\mathbf{n}\in\mathbf{\Lambda}$, form an orthogonal basis in $L^{2}(\Omega,d\mu_{\nu})^{L}$ and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations $(\pi_{\nu}, \mathcal{H}_{\nu})$ of the automorphism group $G$ corresponding to a tube domain $T(\Omega)$. In this article we consider the case where $\Omega$ is the space of positive definite Hermitian matrices and $G=\mathrm{SU}(n,n)$. We describe the Lie algebraic realization of $\pi_{\nu}$ acting in $L^{2}(\Omega,d\mu_{\nu})$ and use that to determine explicit differential equations and recurrence relations for the Laguerre functions.
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arxiv:math/0304357
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We survey the recent mathematical results about aging in certain simple disordered models. We start by the Bouchaud trap model. We then survey the results obtained for simple models of spin-glass dynamics, like the REM (the Random Energy Model, which is well approximated by the Bouchaud model on the complete graph), then the spherical Sherrington-Kirkpatrick model. We will insist on the differences in phenomenology for different types of aging in different time scales and different models. This talk is based on joint works with A.Bovier, J.Cerny, A.Dembo, V.Gayrard, A.Guionnet, as well as works by C.Newman, R.Fontes, M.Isopi, D.Stein.
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arxiv:math/0304364
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Resonances, or scattering poles, are complex numbers which mathematically describe meta-stable states: the real part of a resonance gives the rest energy, and its imaginary part, the rate of decay of a meta-stable state. This description emphasizes the quantum mechanical aspects of this concept but similar models appear in many branches of physics, chemistry and mathematics, from molecular dynamics to automorphic forms. In this article we will will describe the recent progress in the study of resonances based on the theory of partial differential equations.
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arxiv:math/0304400
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We propose a definition of varieties over the field with one element. These have extensions of scalars to the ring of integers which are varieties in the usual sense. We show that toric varieties can be defined over the field with one element. We also discuss zeta functions for such objects. We give a motivic interpretation of the image of the J-homomorphism defined by Adams. ~ ~ ~ ~
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arxiv:math/0304444
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A graph is {\em perfect} if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called {\em basic}. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak perfect graph conjecture, which states that a graph is perfect if and only if its complement is perfect, was proved in 1972 by Lov\'asz. This result is now known as the perfect graph theorem. The strong perfect graph conjecture (SPGC) states that a graph is perfect if and only if it does not contain an odd hole or its complement. The SPGC has attracted a lot of attention. It was proved recently (May 2002) in a remarkable sequence of results by Chudnovsky, Robertson, Seymour and Thomas. The proof is difficult and, as of this writing, they are still checking the details. Here we give a flavor of the proof.
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arxiv:math/0304464
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Time local well-posedness for the Maxwell-Schr\"odinger equation in the coulomb gauge is studied in Sobolev spaces by the contraction mapping principle. The Lorentz gauge and the temporal gauge cases are also treated by the gauge transform.
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arxiv:math/0304486
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We study the absolute Galois group by looking for invariants and orbits of its faithful action on Grothendieck's dessins d'enfants. We define a class of functions called Belyi-extending maps, which we use to construct new Galois invariants of dessins from previously known invariants. Belyi-extending maps are the source of the ``new-type'' relations on the injection of the absolute Galois group into the Grothendieck-Teichmuller group. We make explicit how to get from a general Belyi-extending map to formula for its associated invariant which can be implemented in a computer algebra package. We give an example of a new invariant differing on two dessins which have the same values for the other readily computable invariants.
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arxiv:math/0304489
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In this paper, we introduce a concept of B-minimal sub-manifolds and discuss the stability of such a sub-manifold in a Riemannian manifold $(M,g)$. Assume $B(x)$ is a smooth function on $M$. By definition, we call a sub-manifold $\Sigma$ {\em B-minimal} in $(M,g)$ if the product sub-manifold $\Sigma\times S^1$ is a {\em minimal} sub-manifold in a warped product Riemannian manifold $(M\times S^1, g+e^{2B(x)}dt^2)$, so its stability is closely related to the stability of solitons of mean curvature flows as noted earlier by G. Huisken, S. Angenent, and K. Smoczyk. We can show that the "grim reaper" in the curve-shortening problem is stable in the sense of "symmetric stable" defined by K. Smoczyk. We also discuss the graphic B-minimal sub-manifold in $R^{n+k}$.
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arxiv:math/0304493
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Our goal is to find accurate and efficient algorithms, when they exist, for evaluating rational expressions containing floating point numbers, and for computing matrix factorizations (like LU and the SVD) of matrices with rational expressions as entries. More precisely, {\em accuracy} means the relative error in the output must be less than one (no matter how tiny the output is), and {\em efficiency} means that the algorithm runs in polynomial time. Our goal is challenging because our accuracy demand is much stricter than usual.
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arxiv:math/0305004
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The nervous system displays a variety of rhythms in both waking and sleep. These rhythms have been closely associated with different behavioral and cognitive states, but it is still unknown how the nervous system makes use of these rhythms to perform functionally important tasks. To address those questions, it is first useful to understood in a mechanistic way the origin of the rhythms, their interactions, the signals which create the transitions among rhythms, and the ways in which rhythms filter the signals to a network of neurons. This talk discusses how dynamical systems have been used to investigate the origin, properties and interactions of rhythms in the nervous system. It focuses on how the underlying physiology of the cells and synapses of the networks shape the dynamics of the network in different contexts, allowing the variety of dynamical behaviors to be displayed by the same network. The work is presented using a series of related case studies on different rhythms. These case studies are chosen to highlight mathematical issues, and suggest further mathematical work to be done. The topics include: different roles of excitation and inhibition in creating synchronous assemblies of cells, different kinds of building blocks for neural oscillations, and transitions among rhythms. The mathematical issues include reduction of large networks to low dimensional maps, role of noise, global bifurcations, use of probabilistic formulations.
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arxiv:math/0305013
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I describe a new Markov chain method for sampling from the distribution of the state sequences in a non-linear state space model, given the observation sequence. This method updates all states in the sequence simultaneously using an embedded Hidden Markov model (HMM). An update begins with the creation of a ``pool'' of K states at each time, by applying some Markov chain update to the current state. These pools define an embedded HMM whose states are indexes within this pool. Using the forward-backward dynamic programming algorithm, we can then efficiently choose a state sequence at random with the appropriate probabilities from the exponentially large number of state sequences that pass through states in these pools. I show empirically that when states at nearby times are strongly dependent, embedded HMM sampling can perform better than Metropolis methods that update one state at a time.
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arxiv:math/0305039
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Let M be a 1-motive defined over a field of characteristic 0. To M we can associate its motivic Galois group, G_mot(M), which is the geometrical interpretation of the Munford-Tate group of M. We prove that the unipotent radical of the Lie algebra of G_mot(M) is the semi-abelian variety defined by the adjoint action of the semi-simplified of the Lie algebra of G_mot(M) on itself.
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arxiv:math/0305046
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We solve an asymptotic problem in the geometry of numbers, where we count the number of singular $n\times n$ matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as $T \to \infty $, the number is asymptotic to $ \frac{(n-1)u_n}{\zeta (n) \zeta(n-1)^{n}}T^{n^{2}-n}\log (T)$ for $n \ge 3$. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. Schmidt.
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arxiv:math/0305066
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In this paper, we consider the principal eigenvalue problem for Hormander's laplacian on $R^n$. We also study a related semi-linear sub-elliptic equation in the whole $R^n$ and prove that under a suitable condition, we have infinite many positive solutions of the problem.
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arxiv:math/0305068
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We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display good behavior in terms of the coefficients of the form, and greatly improve upon previous estimates in this regard.
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arxiv:math/0305078
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A quantum group of type A is defined as a Hopf algebra associated to a Hecke symmetry. We show the homology of a Koszul complex associated to the Hecke symmetry is one dimensional and determines a group-like element in the Hopf algebra. This group-like element can be interpreted as a homological determinant as suggested by Yu. Manin.
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arxiv:math/0305115
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Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases. We prove the conjecture for type G_2, and also verify a consequence of the conjecture in the general case.
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arxiv:math/0305175
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We study in this paper some connections between the Fraisse theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures.
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arxiv:math/0305241
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We answer an open question of Grigorchuk and Zuk about amenability using random walks. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.
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arxiv:math/0305262
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We show that the equivariant chain complex associated to a minimal CW-structure X on the complement M(A) of a hyperplane arrangement A, is independent of X. When A is a sufficiently general linear section of an aspheric arrangement, we explain a new way for computing the twisted homology of M(A).
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arxiv:math/0305266
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We present a modification to the Prikry on Extenders forcing notion allowing the blow up of the power set of a large cardinal, change its cofinality to omega without adding bounded subsets, working directly from arbitrary extender (e.g., n-huge extender). Using this forcing, starting from a superstrong cardinal, we construct a model in which the added Prikry sequences are a scale in the normal Prikry sequence.
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arxiv:math/0305269
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We outline the proof of a theorem of M. Baker and A. Tamagawa that gives a complete description of the torsion points on a modular curve embedded in its Jacobian using the notion of an `almost rational torsion point.'
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arxiv:math/0305281
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We generalise the theory of Cuntz-Krieger families and graph algebras to the class of finitely aligned $k$-graphs. This class contains in particular all row-finite $k$-graphs. The Cuntz-Krieger relations for non-row-finite $k$-graphs look significantly different from the usual ones, and this substantially complicates the analysis of the graph algebra. We prove a gauge-invariant uniqueness theorem and a Cuntz-Krieger uniqueness theorem for the $C^*$-algebras of finitely aligned $k$-graphs.
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arxiv:math/0305370
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There has recently been much interest in the $C^*$-algebras of directed graphs. Here we consider product systems $E$ of directed graphs over semigroups and associated $C^*$-algebras $C^*(E)$ and $\mathcal{T}C^*(E)$ which generalise the higher-rank graph algebras of Kumjian-Pask and their Toeplitz analogues. We study these algebras by constructing from $E$ a product system $X(E)$ of Hilbert bimodules, and applying recent results of Fowler about the Toeplitz algebras of such systems. Fowler's hypotheses turn out to be very interesting graph-theoretically, and indicate new relations which will have to be added to the usual Cuntz-Krieger relations to obtain a satisfactory theory of Cuntz-Krieger algebras for product systems of graphs; our algebras $C^*(E)$ and $\mathcal{T}C^*(E)$ are universal for families of partial isometries satisfying these relations. Our main result is a uniqueness theorem for $\mathcal{T}C^*(E)$ which has particularly interesting implications for the $C^*$-algebras of non-row-finite higher-rank graphs. This theorem is apparently beyond the reach of Fowler's theory, and our proof requires a detailed analysis of the expectation onto the diagonal in $\mathcal{T}C^*(E)$.
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arxiv:math/0305371
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We introduce a new concept of mixed representations of quivers that is a generalization of ordinary representations of quivers and orthogonal (symplectic) representations of symmetric quivers introduced recently by Derksen and Weyman. We describe the generating invariants of mixed representations of quivers (First Fundamental Theorem) and prove additional results that allow us to describe the defining relations between them in the second article.
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arxiv:math/0305386
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Natural q analogues of classical statistics on the symmetric groups $S_n$ are introduced; parameters like: the q-length, the q-inversion number, the q-descent number and the q-major index. MacMahon's theorem about the equi-distribution of the inversion number and the reverse major index is generalized to all positive integers q. It is also shown that the q-inversion number and the q-reverse major index are equi-distributed over subsets of permutations avoiding certain patterns. Natural q analogues of the Bell and the Stirling numbers are related to these q statistics -- through the counting of the above pattern-avoiding permutations.
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arxiv:math/0305393
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Let I_n(\pi) denote the number of involutions in the symmetric group S_n which avoid the permutation \pi. We say that two permutations \alpha,\beta\in\S{j} may be exchanged if for every n, k, and ordering \tau of j+1,...,k, we have I_n(\alpha\tau)=I_n(\beta\tau). Here we prove that 12 and 21 may be exchanged and that 123 and 321 may be exchanged. The ability to exchange 123 and 321 implies a conjecture of Guibert, thus completing the classification of S_4 with respect to pattern avoidance by involutions; both of these results also have consequences for longer patterns. Pattern avoidance by involutions may be generalized to rook placements on Ferrers boards which satisfy certain symmetry conditions. Here we provide sufficient conditions for the corresponding generalization of the ability to exchange two prefixes and show that these conditions are satisfied by 12 and 21 and by 123 and 321. Our results and approach parallel work by Babson and West on analogous problems for pattern avoidance by general (not necessarily involutive) permutations, with some modifications required by the symmetry of the current problem.
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arxiv:math/0306002
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We present two examples in toric geometry concerning the relationship between toric and quasitoric manifolds, and provide the sufficient conditions on the base polytope and characteristic map so that the resulting quasitoric manifold is almost complex.
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arxiv:math/0306029
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We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type $\rm{I}\sb{\infty}.$
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arxiv:math/0306061
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We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor.In particular, we show that the module of syzygies of polynomials $f_1,...,f_n\in R[X_1,...,X_N]$ with coefficients in a Pr\"ufer domain $R$ can be generated by elements whose degrees are bounded by a number only depending on $N$, $n$ and the degree of the $f_j$. This implies that if $R$ is a B\'ezout domain, then the generators can be parametrized in terms of the coefficients of $f_1,...,f_n$ using the ring operations and a certain division function, uniformly in $R$.
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arxiv:math/0306240
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We consider an elliptic PDE in two variables. As one parameter approaches zero, this PDE collapses to a parabolic one, that is forward parabolic in a part of the domain and backward parabolic in the remainder. Such problems arise naturally in various stochastic models, such as fluid models for data-handling systems and Markov-modulated queues. We employ singular perturbation methods to study the problem for small values of the parameter.
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arxiv:math/0306261
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Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. We discuss a correspondence between Leonard pairs and a class of orthogonal polynomials consisting of the $q$-Racah polynomials and some related polynomials of the Askey scheme. For the polynomials in this class we obtain the 3-term recurrence, difference equation, Askey-Wilson duality, and orthogonality in a uniform manner using the corresponding Leonard pair.
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arxiv:math/0306301
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Let $U_\epsilon(\mathfrak g)$ be the simply connected quantized enveloping algebra associated to a finite-dimensional complex simple Lie algebra $\mathfrak g$ at the roots of unity. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of $U_\epsilon(\mathfrak g)$ is proved for representations corresponding to the spherical conjugacy classes of the simply connected algebraic group $G$ with Lie algebra $\mathfrak g$. We achieve this result by means of a new characterization of the spherical conjugacy classes of $G$ in terms of elements of the Weyl group.
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arxiv:math/0306321
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We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulas of the length function in some special cases.
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arxiv:math/0306327
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A model problem of the form -i\epsilon y''+q(x)y=\lambda y, y(-1)=y(1)=0, is associated with well-known in hydrodynamics Orr--Sommerfeld operator. Here (\lambda) is the spectral parameter, (\epsilon) is the small parameter which is proportional to the viscocity of the liquid and to the reciprocal of the Reynolds number, and (q(x)) is the velocity of the stationary flow of the liquid in the channel (|x|\leqslant 1). We study the behaviour of the spectrum of the corresponding model operator as (\epsilon\to 0) with monotonous analytic functions. We assert that the sets of the accumulation points of the spectra (the limit spectral graphs) of the model and the corresponding Orr--Sommerfeld operators coincide as well as the main terms of the counting eigenvalue functions along the curves of the graphs. We prove the estimate from below for the resolvent of the operator (L(\epsilon)) associated with the model problem. It turns out that the resolvent grows exponentially with respect to (\epsilon) provided that the spectral parameter varies at some compacts belonging to the numerical range of the operator.
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arxiv:math/0306342
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In two articles by Barthel, Brasselet, Fieseler and Kaup, and, Bressler and Lunts, a combinatorial theory of intersection cohomology and perverse sheaves has been developed on fans. In the first one, one tried to present everything on an elementary level,using only some commutative algebra and no derived categories. There remained two major gaps: First of all the Hard Lefschetz Theorem was only conjectured and secondly the intersection product seemed to depend on some non-canonical choices. Meanwhile the Hard Lefschetz theorem has been proved by Karu. The proof relies heavily on the intersection product, since what finally has to be shown are the Hodge-Riemann relations. In fact here again choices enter: The intersection product is induced from the intersection product on some simplicial subdivision via a direct embedding of the corresponding intersection cohomology sheaves, a fact, which makes the argumentation quite involved. In a recent paper by Bressler and Lunts, one shows by a detailed analysis that eventually all possible choices do not affect the definition of the pairing. Our goal here is the same, but we shall try to follow the spirit of the first cited paper avoiding the formalism of derived categories. For perverse sheaves we define their dual sheaf and check that the intersection cohomology sheaf is self- dual in a natural way.
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arxiv:math/0306344
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For an almost simple complex algebraic group $G$ with affine Grassmannian $Gr_G= G(C((t)))/G(C[[t]])$ we consider the equivariant homology $H^{G(C[[t]])}(Gr_G)$, and $K$-theory $K^{G(C[[t]])}(Gr_G)$. They both have a commutative ring structure, with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group $\check G$, and we relate the spectrum of $K$-homology ring to the universal group-group centralizer of $G$ and of $\check G$. If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the ($K$)-homology ring, and thus a Poisson structure on its spectrum. We identify this structure with the standard one on the universal centralizer. The commutative subring of $G(C[[t]])$-equivariant homology of the point gives rise to a polarization which is related to Kostant's Toda lattice integrable system. We also compute the equivariant $K$-ring of the affine Grassmannian Steinberg variety. The equivariant $K$-homology of $Gr_G$ is equipped with a canonical base formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the Feigin-Loktev fusion product of $G(C[[t]])$-modules.
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arxiv:math/0306413
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In order to have a better description of homogenization for parabolic partial differential equations with periodic coefficients, we define the notion of parametric two-scale convergence. A compactness theorem is proved to justify this notion.
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arxiv:math/0306419
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Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. We give an overview of the theory of Leonard pairs.
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arxiv:math/0307063
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Recently, a framework for controller design of sampled-data nonlinear systems via their approximate discrete-time models has been proposed in the literature. In this paper we develop novel tools that can be used within this framework and that are very useful for tracking problems. In particular, results for stability analysis of parameterized time-varying discrete-time cascaded systems are given. This class of models arises naturally when one uses an approximate discrete-time model to design a stabilizing or tracking controller for a sampled-data plant. While some of our results parallel their continuous-time counterparts, the stability properties that are considered, the conditions that are imposed and the the proof techniques that are used are tailored for approximate discrete-time systems and are technically different from those in the continuous-time context. We illustrate the utility of our results in the case study of the tracking control of a mobile robot. This application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete-time parameterized systems.
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arxiv:math/0307167
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