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Local exact controllability of the 1D NLS (subject to zero boundary conditions) with distributed control is shown to hold in a $H^1$--neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to J.-L. Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail.
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arxiv:math/0608135
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It is shown that the standard Kolmogorov model for probability spaces cannot in general allow the elimination but of only a small amount of probabilistic redundancy. This issue, a purely theoretical weakness, not necessarily related to empirical reality, appears not to have received enough attention in foundational studies of Probability Theory.
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arxiv:math/0608141
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We show that on Kahler manifolds M with c_1(M)=0 the Calabi flow converges to a constant scalar curvature metric if the initial Calabi energy is sufficiently small. We prove a similar result on manifolds with c_1(M)<0 if the Kahler class is close to the canonical class.
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arxiv:math/0608154
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(1) We prove that, provided n>=4, a permutably reducible n-ary quasigroup is uniquely specified by its values on the n-ples containing zero. (2) We observe that for each n,k>=2 and r<=[k/2] there exists a reducible n-ary quasigroup of order k with an n-ary subquasigroup of order r. As corollaries, we have the following: (3) For each k>=4 and n>=3 we can construct a permutably irreducible n-ary quasigroup of order k. (4) The number of n-ary quasigroups of order k>3 has double-exponential growth as n tends to infinity; it is greater than exp exp(n ln[k/3]) if k>=6, and exp exp(n (ln 3)/3 - 0.44) if k=5.
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arxiv:math/0608269
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We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.
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arxiv:math/0608278
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For arrays $(S_{i,j})_{1\leq i\leq j}$ of random variables that are stationary in an appropriate sense, we show that the fluctuations of the process $(S_{1,n})_{n=1}^{\infty}$ can be bounded in terms of a measure of the ``mean subadditivity'' of the process $(S_{i,j})_{1\leq i\leq j}$. We derive universal upcrossing inequalities with exponential decay for Kingman's subadditive ergodic theorem, the Shannon--MacMillan--Breiman theorem and for the convergence of the Kolmogorov complexity of a stationary sample.
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arxiv:math/0608311
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Kempf [1976] studied proper, G-equivariant maps from equivariant vector bundles over flag manifolds to G-representations V, which he called _collapsings_. We give a simple formula for the G-equivariant cohomology class on V, or_multidegree_, associated to the image of a collapsing: apply a certain sequence of divided difference operators to a certain product of linear polynomials, then divide by the number of components in a general fiber. When that number of components is 1, we construct a desingularization of the image of the collapsing. If in addition the image has rational singularities, we can use the desingularization to give also a formula for the G-equivariant K-class of the image, whose leading term is the multidegree. Our application is to quiver loci and quiver polynomials. Let Q be a quiver of finite type (A, D, or E, in arbitrary orientation), and assign a vector space to each vertex. Let \Hom denote the (linear) space of representations of Q with these vector spaces. This carries an action of GL, the product of the general linear groups of the individual vector spaces. A_quiver locus_ \Omega is the closure in \Hom of a GL-orbit, and its multidegree is the corresponding _quiver polynomial_. Reineke [2004] proved that every ADE quiver locus is the image of a birational Kempf collapsing (giving a desingularization directly). Using Reineke's collapsings, we give formulae for ADE quiver polynomials, previously only computed in type A (though in this case, our formulae are new). In the A and D cases quiver loci are known to have rational singularities [Bobi\'nski-Zwara 2002], so we also get formulae for their K-classes, which had previously only been computed in equioriented type A (and again our formulae are new).
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arxiv:math/0608327
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The purpose of this note is to point to a gap in an argument in our paper "Stabilization for the automorphisms of free groups with boundaries", and explain how to fill it.
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arxiv:math/0608333
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Let $\widetilde{\cal J}(S^{2n})$ be the set of orthogonal complex structures on $TS^{2n}$. We show that the twistor space $\widetilde{\cal J}(S^{2n})$ is a Kaehler manifold. Then we show that an orthogonal almost complex structure $J_f$ on $S^{2n}$ is integrable if and only if the corresponding section $f\colon\; S^{2n}\to \widetilde{\cal J}(S^{2n}) $ is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere $S^{2n}$ for $n>1$. We also show that there is no complex structure in a neighborhood of the space $\widetilde{\cal J}(S^{2n})$. The method is to study the first Chern class of $T^{(1,0)}S^{2n}$.
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arxiv:math/0608368
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We study the formation of singularities for the mean curvature flow of monotone Lagrangians in $\C^n$. More precisely, we show that if singularities happen before a critical time then the tangent flow can be decomposed into a finite union of area-minimizing Lagrangian cones (Slag cones). When $n=2$, we can improve this result by showing that connected components of the rescaled flow converge to an area-minimizing cone, as opposed to possible non-area minimizing union of Slag cones. In the last section, we give specific examples for which such singularity formation occurs.
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arxiv:math/0608401
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In this article we prove derived invariance of Hochschild-Mitchell homology and cohomology and we extend to $k$-linear categories a result by Barot and Lenzing concerning derived equivalences and one-point extensions. We also prove the existence of a Happel long exact sequence and we give a generalization of this result which provides an alternative approach.
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arxiv:math/0608444
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We present a single operation for constructing skew diagrams whose corresponding skew Schur functions are equal. This combinatorial operation naturally generalises and unifies all results of this type to date. Moreover, our operation suggests a closely related condition that we conjecture is necessary and sufficient for skew diagrams to yield equal skew Schur functions.
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arxiv:math/0608446
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A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.
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arxiv:math/0608450
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We show that the space of long knots in an euclidean space of dimension larger than three is a double loop space, proving a conjecture by Sinha. We construct also a double loop space structure on framed long knots, and show that the map forgetting the framing is not a double loop map in odd dimension. However there is always such a map in the reverse direction expressing the double loop space of framed long knots as a semidirect product. A similar compatible decomposition holds for the homotopy fiber of the inclusion of long knots into immersions. We show also via string topology that the space of closed knots in a sphere, suitably desuspended, admits an action of the little 2-discs operad in the category of spectra. A fundamental tool is the McClure-Smith cosimplicial machinery, that produces double loop spaces out of topological operads with multiplication.
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arxiv:math/0608490
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Suppose that S is a surface of genus two or more, with exactly one boundary component. Then the curve complex of S has one end.
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arxiv:math/0608505
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In this paper we study the solutions of the $Q$-curvature equation on a 4-dimensional Riemannian manifold $(M,g)$ with $\int_MQdV_g=8\pi^2$, proving some sufficient conditions for the existence.
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arxiv:math/0608543
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The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_t u(t,z)=\Delta u(t,z)+\xi(z) u(t,z)$ on $(0,\infty)\times {\mathbb Z}^d$ with random potential $(\xi(z) \colon z\in {\mathbb Z}^d)$. We consider independent and identically distributed potential variables, such that Prob$(\xi(z)>x)$ decays polynomially as $x\uparrow\infty$. If $u$ is initially localised in the origin, i.e. if $u(0,x)=\one_0(x)$, we show that, at any large time $t$, the solution is completely localised in a single point with high probability. More precisely, we find a random process $(Z_t \colon t\ge 0)$ with values in $\Z^d$ such that $\lim_{t \uparrow\infty} u(t,Z_t)/\sum_{z\in\Z^d} u(t,z) =1,$ in probability. We also identify the asymptotic behaviour of $Z_t$ in terms of a weak limit theorem.
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arxiv:math/0608544
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Let $E_G$ be a stable principal $G$--bundle over a compact connected Kaehler manifold, where $G$ is a connected reductive linear algebraic group defined over the complex numbers. Let $H\subset G$ be a complex reductive subgroup which is not necessarily connected, and let $E_H\subset E_G$ be a holomorphic reduction of structure group. We prove that $E_H$ is preserved by the Einstein-Hermitian connection on $E_G$. Using this we show that if $E_H$ is a minimal reductive reduction in the sense that there is no complex reductive proper subgroup of $H$ to which $E_H$ admits a holomorphic reduction of structure group, then $E_H$ is unique in the following sense: For any other minimal reductive reduction $(H', E_{H'})$ of $E_G$, there is some element $g$ of $G$ such that $H'= g^{-1}Hg$ and $E_{H'}= E_Hg$. As an application, we give an affirmative answer to a question of Balaji and Koll\'ar.
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arxiv:math/0608569
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We give a definition of the Clifford algebra of an antiautomorphism of a central simple algebra, and compute it for the algebras of degree 2.
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arxiv:math/0608625
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We give a necessary and sufficient condition for two Hopf algebras presented as central extensions to be isomorphic, in a suitable setting. We then study the question of isomorphism between the Hopf algebras constructed in 0707.0070v1 as quantum subgroups of quantum groups at roots of 1. Finally, we apply the first general result to show the existence of infinitely many non-isomorphic Hopf algebras of the same dimension, presented as extensions of finite quantum groups by finite groups.
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arxiv:math/0608647
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The purpose of this paper is to construct q-Euler numbers and polynomials by using p-adic q-integral equations on Zp. Finally, we will give some interesting formulae related to these q-Euler numbers and polynomials.
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arxiv:math/0608649
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We present a simple solution to a question posed by Candes, Romberg and Tao on the uniform uncertainty principle for Bernoulli random matrices. More precisely, we show that a rectangular k*n random subgaussian matrix (with k < n) has the property that by arbitrarily extracting any m (with m < k) columns, the resulting submatrices are arbitrarily close to (multiples of) isometries of a Euclidean space. We obtain the optimal estimate for m as a function of k,n and the degree of "closeness" to an isometry. We also give a short and self-contained solution of the reconstruction problem for sparse vectors.
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arxiv:math/0608665
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In this paper analysis of the concept of {\it associated homogeneous distributions} (generalized functions) is given, and some problems related to these distributions are solved. It is proved that (in the one-dimensional case) there exist {\it only} {\it associated homogeneous distributions} of order $k=1$. Next, we introduce a definition of {\it quasi associated homogeneous distributions} and provide a mathematical description of all quasi associated homogeneous distributions and their Fourier transform. It is proved that the class of {\it quasi associated homogeneous distributions} coincides with the class of distributions introduced by Gel$'$fand and Shilov \cite[Ch.I,\S 4.]{G-Sh} as the class of {\it associated homogeneous distributions}. For the multidimensional case it is proved that $f$ is a {\it quasi associated homogeneous distribution} if and only if it satisfies the Euler type system of differential equations. A new type of $\Gamma$-functions generated by quasi associated homogeneous distributions is defined.
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arxiv:math/0608669
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Recently it has been noticed that many interesting combinatorial objects belong to a class of semigroups called left regular bands, and that random walks on these semigroups encode several well-known random walks. For example, the set of faces of a hyperplane arrangement is endowed with a left regular band structure. This paper studies the module structure of the semigroup algebra of an arbitrary left regular band, extending results for the semigroup algebra of the faces of a hyperplane arrangement. In particular, a description of the quiver of the semigroup algebra is given and the Cartan invariants are computed. These are used to compute the quiver of the face semigroup algebra of a hyperplane arrangement and to show that the semigroup algebra of the free left regular band is isomorphic to the path algebra of its quiver.
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arxiv:math/0608698
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If F(x) = e^G(x), where F(x) = \sum f(n)x^n and $G(x) = \sum g(n)x^n, with 0 \le g(n) = O(n^{theta n}/n!),theta in (0,1), and gcd(n : g(n) > 0)=1, then f(n) = o(f(n-1)). This gives an answer to Compton's request in Question 8.3 for an ``easily verifiable sufficient condition'' to show that an adequate class of structures has a labelled first-order 0-1 law, namely it suffices to show that the labelled component count function is O(n^{theta n}) for some theta in (0,1). It also provides the means to recursively construct an adequate class of structures with a labelled 0-1 law but not an unlabelled 0-1 law, answering Compton's Question 8.4.
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arxiv:math/0608735
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The fundamental unit of $\Z[\sqrt{N}]$ for square-free $N=5 mod 8$ is either $\epsilon$ or $\epsilon^3$ where $\epsilon$ denotes the fundamental unit of the maximal order of $\Q(\sqrt{N})$. We give infinitely many examples for each case.
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arxiv:math/0608741
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We construct new families of conformally invariant differential operators acting on densities. We introduce a simple, direct approach which shows that all such operators arise via this construction when the degree is bounded by the dimension. The method relies on a study of well-known transformation laws and on Weyl's theory regarding identities holding ``formally'' vs. ``by substitution''. We also illustrate how this new method can strengthen existing results in the parabolic invariant theory for conformal geometries.
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arxiv:math/0608771
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The long-time asymptotics is analyzed for all finite energy solutions to a model U(1)-invariant nonlinear Klein-Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as time goes to plus or minus infinity to the set of all ``nonlinear eigenfunctions'' of the form $\psi(x)e\sp{-i\omega t}$. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation. We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time-spectrum in the spectral gap [-m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh Convolution Theorem reduces the spectrum of each omega-limit trajectory to a single harmonic in [-m,m]. The research is inspired by Bohr's postulate on quantum transitions and Schroedinger's identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled U(1)-invariant Maxwell-Schroedinger and Maxwell-Dirac equations.
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arxiv:math/0609013
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Set-coloring a graph means giving each vertex a subset of a fixed color set so that no two adjacent subsets have the same cardinality. When the graph is complete one gets a new distribution problem with an interesting generating function. We explore examples and generalizations.
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arxiv:math/0609049
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The goal of this article is to give a positive answer to Rockafellar's maximality of the sum conjecture in the linear multi-valued operator case.
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arxiv:math/0609069
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Although the phenomenon of chirality appears in many investigations of maps and hypermaps no detailed study of chirality seems to have been carried out. Chirality of maps and hypermaps is not merely a binary invariant but can be quantified by two new invariants -- the chirality group and the chirality index, the latter being the size of the chirality group. A detailed investigation of the chirality groups of maps and hypermaps will be the main objective of this paper. The most extreme type of chirality arises when the chirality group coincides with the monodromy group. Such hypermaps are called totally chiral. Examples of them are constructed by considering appropriate ``asymmetric'' pairs of generators for some non-abelian simple groups. We also show that every finite abelian group is the chirality group of some hypermap, whereas many non-abelian groups, including symmetric and dihedral groups, cannot arise as chirality groups.
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arxiv:math/0609070
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We calculate the microstates free entropy dimension of natural generators in an amalgamated free product of certain von Neumann algebras, with amalgamation over a hyperfinite subalgebra. In particular, some `exotic' Popa algebra generators of free group factors are shown to have the expected free entropy dimension. We also show that microstates and non--microstates free entropy dimension agree for generating sets of many groups. In the appendix by Wolfgang Lueck, the first L^2-Betti number for certain amalgamated free products of groups is calculated.
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arxiv:math/0609080
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We introduce a ZFC method that enables us to build spaces (in fact special dense subspaces of certain Cantor cubes) in which we have "full control" over all dense subsets. Using this method we are able to construct, in ZFC, for each uncountable regular cardinal $\lambda$ a 0-dimensional $T_2$, hence Tychonov, space which is $\mu$-resolvable for all $\mu < \lambda$ but not $\lambda$-resolvable. This yields the final (negative) solution of a celebrated problem of Ceder and Pearson raised in 1967: Are $\omega$-resolvable spaces maximally resolvable? This method enables us to solve several other open problems concerning resolvability as well.
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arxiv:math/0609090
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We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (even a stronger version of) Zariski's Conjecture about the multiplicity for such a singularity.
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arxiv:math/0609093
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Let mu(G) and mu_min(G) be the largest and smallest eigenvalues of the adjacency matricx of a graph G. We refine quantitatively the following two results on graph spectra. (i) if H is a proper subgraph of a connected graph G, then mu(G)>mu(H). (ii) if G is a connected nonbipartite graph, then mu(G)>-mu_min(G).
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arxiv:math/0609111
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We consider the problem of finding 4 rational squares, such that the product of any two plus the sum of the same two always gives a square. We give some historical background and exhibit one such quadruple.
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arxiv:math/0609127
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The problem of linear and circular permutations of n identical objects in m boxes, where a limit l is imposed on the number of objects in a box, is considered. In the linear case, where the boxes are arranged as a row, two methods of solution are described. The first uses the partition diagram that is modified with prescribed number of zeros, and the second applies a direct combinatorial approach. Subject to constraints set on the relation between n and l, results of the combinatorial approach are used to solve the problem where there are more than one type of objects. These solutions are then extended for the case of boxes comprising subgroups that are characterized by different values of l. In the circular case, where the boxes are arranged in the form of a circle, no direct combinatorial solution in close form is available. Consequently, in the latter case, the solution relies on the use of the modified partition diagram. This solution, which involves the application of multinomial permutations on a circle, is also extended to the case of two kinds of objects.
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arxiv:math/0609178
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We study the existence of positive solutions to the quasilinear elliptic problem -\epsilon \Delta u+V(x)u-\epsilon k(\Del(|u|^{2}))u=g(u), \quad u>0, x \in R^N, where g has superlinear growth at infinity without any restriction from above on its growth. Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schr\"{o}dinger equations. Schr\"{o}dinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.
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arxiv:math/0609191
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Kasparov $KK$-groups $KK(A,B)$ are represented as homotopy groups of the Pedersen-Weibel nonconnective algebraic $K$-theory spectrum of the additive category of Fredholm $(A,B)$-bimodules for $A$ and $B$, respectively, a separable and $\sigma$-unital trivially graded real or complex $C^*$-algebra acted upon by a fixed compact metrizable group.
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arxiv:math/0609208
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Bidirected graphs (earlier studied by Edmonds, Johnson and, in equivalent terms of skew-symmetric graphs, by Tutte, Goldberg, Karzanov, and others) proved to be a useful unifying language for describing both flow and matching problems. In this paper we extend the notion of ear decomposition to the class of strongly connected bidirected graphs. In particular, our results imply Two Ear Theorem on matching covered graphs of Lov\'asz and Plummer. The proofs given here are self-contained except for standard Barrier Theorem on skew-symmetric graphs.
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arxiv:math/0609221
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A set A of positive integers is called a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence of this new approach, we prove that there exists a perfect difference set A such that A(x) >> x^{\sqrt{2}-1-o(1)}. We also prove that there exists a perfect difference set A such that limsup_{x\to \infty}A(x)/\sqrt x\geq 1/\sqrt 2.
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arxiv:math/0609244
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The paper introduces a number of new techniques to handle minimal hyersurface singularities. In particular, they allow to extend the obstruction theory for postive scalr curvature to any dimension.
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arxiv:math/0609338
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Several thoughts are presented on the long ongoing difficulties both students and academics face related to Calculus 101. Some of these thoughts may have a more general interest.
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arxiv:math/0609343
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We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place. As an application, we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v.
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arxiv:math/0609387
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This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by the Floer homologies of powers of the symplectomorphism, the other algebraic invariants of symplectic field theory provide natural generalizations of symplectic Floer homology. For symplectically aspherical manifolds and Hamiltonian symplectomorphisms we study the moduli spaces of rational curves and prove a transversality result, which does not need the polyfold theory by Hofer, Wysocki and Zehnder. Besides that our result shows that one does not get nontrivial operations on Floer homology from symplectic field theory, we use it to compute the full contact homology of the corresponding Hamiltonian mapping torus.
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arxiv:math/0609406
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We characterize the $L^p$-range, $1 < p < +\infty$, of the Poisson transform on the Shilov boundary for non-tube bounded symmetric domains. We prove that this range is a Hua-Hardy type space for harmonic functions satisfying a Hua system.
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arxiv:math/0609445
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Let $(X,0)$ be an isolated complete intersection complex singularity ($X$ can also be smooth at 0). Let $K$ be its link, $\cal X$ its canonical contact structure and $\D_X$ the complex vector bundle associated to $\cal X$. We prove that the bundle $\D_X$ is trivial if and only if the Milnor number of $X$ satisfies $\mu(X,0) \equiv (-1)^{n-1}$ modulo $(n-1)!$. This follows from a general theorem stating that the complex orthogonal complement of a vector field in $X$ with an isolated singularity at 0 is trivial iff the GSV-index of $v$ is a multiple of $(n-1)!$. We have also an application to foliation theory: a holomorphic foliation $\cal F$ in a ball $\B_r$ around the origin in $\C^3$, with an isolated singularity at 0, admits a $C^\infty$ normal section (away from 0) iff its multiplicity (or local index) is even, and this happens iff its normal bundle in $\B_r \setminus \{0\}$ is topologically trivial.
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arxiv:math/0609448
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The authors use Riemann-Hilbert methods to compute the constant that arises in the asymptotic behavior of the Airy-kernel determinant of random matrix theory.
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arxiv:math/0609451
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Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how the Einstein metrics fit into the classification of general toric anti-self-dual metrics given in an earlier paper (math.DG/0602423). The results complement the work of Calderbank-Pedersen (math.DG/0105263), who describe where the Einstein metrics appear amongst the Joyce spaces, leading to a different classification. Taking the twistor transform of our result gives a new proof of their theorem.
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arxiv:math/0609487
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Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space $Hol (\D,\Omega)$ of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.
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arxiv:math/0609499
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The undecidability of the additive theory of primes (with identity) as well as the theory Th(N,+, n -> p\_n), where p\_n denotes the (n+1)-th prime, are open questions. As a possible approach, we extend the latter theory by adding some extra function. In this direction we show the undecidability of the existential part of the theory Th(N, +, n -> p\_n, n -> r\_n), where r\_n is the remainder of p\_n divided by n in the euclidian division.
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arxiv:math/0609554
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Let I=(x^{v_1},...,x^{v_q} be a square-free monomial ideal of a polynomial ring K[x_1,...,x_n] over an arbitrary field K and let A be the incidence matrix with column vectors {v_1},...,{v_q}. We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedrons and clutters associated to A and I respectively. Some applications to Rees algebras and combinatorial optimization are presented. We study a conjecture of Conforti and Cornu\'ejols using an algebraic approach.
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arxiv:math/0609609
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We show that a wreath product of two finitely generated abelian groups is LERF. Consequently the free metabelian groups are LERF.
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arxiv:math/0609611
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The braid monodromy factorization of the branch curve of a surface of general type is known to be an invariant that completely determines the diffeomorphism type of the surface. Calculating this factorization is of high technical complexity; computing the braid monodromy factorization of branch curves of surfaces uncovers new facts and invariants of the surfaces. Since finding the branch curve of a surface is very difficult, we degenerate the surface into a union of planes. Thus, we can find the braid monodromy of the branch curve of the degenerated surface, which is a union of lines. The regeneration of the singularities of the branch curve, studied locally, leads us to find the global braid monodromy factorization of the branch curve of the original surface. So far, only the regeneration of the BMF of 3,4 and 6-point (a singular point which is the intersection of 3 / 4 / 6 planes) were done. In this paper, we fill the gap and find the braid monodromy of the regeneration of a 5-point. This is of great importance to the understanding of the BMT (braid monodromy type) of surfaces. This braid monodromy will be used to find the global braid monodromy factorization of different surfaces; in particular - the monodromy of the branch curve of the Hirzebruch surface $F_{2,(2,2)}$.
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arxiv:math/0609643
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We extend the notion of thin multiple Heegaard splittings of a link in a 3-manifold to take into consideration not only compressing disks but also cut-disks for the Heegaard surfaces. We prove that if H is a c-strongly compressible bridge surface for a link K contained in a closed orientable irreducible 3-manifold M then one of the following is satisfied: H is stabilized, H is meridionally stabilized, H is perturbed, a component of K is removable, M contains an essential meridional surface.
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arxiv:math/0609674
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In this short note we construct Calabi-Yau threefolds with nonabelian fundamental groups of order 64 as quotients of the small resolutions of certain complete intersections of quadrics in $\PP^7$ that were first considered by M. Gross and S. Popescu.
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arxiv:math/0609728
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The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $u_t - \Delta u + |\nabla u|^q = 0$ in the whole space $R^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a rescaled self-similar solution of the linear heat equation is shown, the rescaling factor being $(\log(t))^{-(N+1)}$. The proof relies on the construction of a one-dimensional invariant manifold for a suitable truncation of the equation written in self-similar variables.
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arxiv:math/0609750
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In this paper we look into the structure of finite-dimensional graded superalgebras of various types such as associative, Lie and Jordan over an algebraically closed field of characteristic zero.
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arxiv:math/0609760
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The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give parametrizations of the cells. We present several different combinatorial descriptions of the cells, study the partial order on the cells, and describe how they are glued to each other.
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arxiv:math/0609764
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We prove an inequality related to questions in Approximation Theory, Probability Theory, and to Irregularities of Distribution. Let $h_R$ denote an $L ^{\infty}$ normalized Haar function adapted to a dyadic rectangle $R\subset [0,1] ^{3}$. We show that there is a postive $\eta$ so that for all integers $n$, and coefficients $ \alpha (R)$ we have 2 ^{-n} \sum_{\abs{R}=2 ^{-n}} \abs{\alpha(R)} {}\lesssim{} n ^{1 - \eta} \NOrm \sum_{\abs{R}=2 ^{-n}} \alpha(R) h_R >.\infty . This is an improvement over the `trivial' estimate by an amount of $n ^{- \eta}$, and the optimal value of $\eta$ (which we do not prove) would be $ \eta =\frac12$. There is a corresponding lower bound on the $L ^{\infty}$ norm of the Discrepancy function of an arbitary distribution of a finite number of points in the unit cube in three dimensions. The prior result, in dimension 3, is that of J{\'o}zsef Beck \cite{MR1032337}, in which the improvement over the trivial estimate was logarithmic in $n$. We find several simplifications and extensions of Beck's argument to prove the result above.
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arxiv:math/0609815
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We consider the symmetric non-local Dirichlet form $(E, F)$ given by \[ E (f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$ the closure of the set of $C^1$ functions on $R^d$ with compact support with respect to $E_1$, where $E_1 (f, f):=E (f, f)+\int_{R^d} f(x)^2 dx$, and where the jump kernel $J$ satisfies \[ \kappa_1|y-x|^{-d-\alpha} \leq J(x,y) \leq \kappa_2|y-x|^{-d-\beta} \] for $0<\alpha< \beta <2, |x-y|<1$. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to $(E, F)$. We prove a parabolic Harnack inequality for nonnegative functions that solve the heat equation with respect to $E$. Finally we construct an example where the corresponding harmonic functions need not be continuous.
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arxiv:math/0609842
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We develop a Belyi type theory that applies to Klein surfaces, i.e. (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. In particular we extend Belyi's famous theorem from Riemann surfaces to Klein surfaces.
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arxiv:math/0609844
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We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain and with initial condition in $\FF L^{\alpha,p}$ spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.
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arxiv:math/0610006
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We prove that the degree of Fano threefolds with terminal Q-factorial singularities and Picard number one is at most 125/2 and the bound is sharp.
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arxiv:math/0610026
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The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.
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arxiv:math/0610076
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For a smooth curve of genus $g$ embedded by a line bundle of degree at least $2g+3$ we show that the ideal sheaf of the secant variety is 5-regular. This bound is sharp with respect to both the degree of the embedding and the bound on the regularity. Further, we show that the secant variety is projectively normal for the generic embedding of degree at least $2g+3$. We also give a conjectural description of the resolutions of the ideals of higher secant varieties.
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arxiv:math/0610081
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We consider the Derivative NLS equation with general quadratic nonlinearities. In \cite{be2} the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension $n = 2$. Here we prove a similar result for large initial data in all dimensions $n \geq 2$.
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arxiv:math/0610092
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Recently, Bocklandt proved a conjecture by Van den Bergh in its graded version, stating that a graded quiver algebra (with relations) which is Calabi-Yau of dimension 3 is defined from a homogeneous potential W. In this paper, we prove that if we add to W any potential of smaller degree, we get a Poincare-Birkhoff-Witt deformation of A. Such PBW deformations are Calabi-Yau and are characterised among all the PBW deformations of A. Various examples are presented.
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arxiv:math/0610112
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For orbifolds admitting a crepant resolution and satisfying a hard Lefschetz condition, we formulate a conjectural equivalence between the Gromov-Witten theories of the orbifold and the resolution. We prove the conjecture for the equivariant Gromov-Witten theories of the nth symmetric product of the complex plane and the Hilbert scheme of n points in the plane.
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arxiv:math/0610129
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C. Andre and N. Yan introduced the idea of a supercharacter theory to give a tractable substitute for character theory in wild groups such as the unipotent uppertriangular group $U_n(F_q)$. In this theory superclasses are certain unions of conjugacy classes, and supercharacters are a set of characters which are constant on superclasses. This paper gives a character formula for a supercharacter evaluated at a superclass for pattern groups and more generally for algebra groups.
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arxiv:math/0610161
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We study the quickest detection problem of a sudden change in the arrival rate of a Poisson process from a known value to an unknown and unobservable value at an unknown and unobservable disorder time. Our objective is to design an alarm time which is adapted to the history of the arrival process and detects the disorder time as soon as possible. In previous solvable versions of the Poisson disorder problem, the arrival rate after the disorder has been assumed a known constant. In reality, however, we may at most have some prior information about the likely values of the new arrival rate before the disorder actually happens, and insufficient estimates of the new rate after the disorder happens. Consequently, we assume in this paper that the new arrival rate after the disorder is a random variable. The detection problem is shown to admit a finite-dimensional Markovian sufficient statistic, if the new rate has a discrete distribution with finitely many atoms. Furthermore, the detection problem is cast as a discounted optimal stopping problem with running cost for a finite-dimensional piecewise-deterministic Markov process. This optimal stopping problem is studied in detail in the special case where the new arrival rate has Bernoulli distribution. This is a nontrivial optimal stopping problem for a two-dimensional piecewise-deterministic Markov process driven by the same point process. Using a suitable single-jump operator, we solve it fully, describe the analytic properties of the value function and the stopping region, and present methods for their numerical calculation. We provide a concrete example where the value function does not satisfy the smooth-fit principle on a proper subset of the connected, continuously differentiable optimal stopping boundary, whereas it does on the complement of this set.
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arxiv:math/0610184
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Competition is a major force in structuring ecological communities. The strength of competition can be measured using the concept of a niche. A niche comprises the set of requirements of an organism in terms of habitat, environment and functional role. The more niches overlap, the stronger competition is. The niche breadth is a measure of specialization: the smaller the niche space of an organism, the more specialized the organism is. It follows that, everything else being equal, generalists tend to be more competitive than specialists. In this paper, we compare the outcome of competition among generalists and specialists in a spatial versus a nonspatial habitat in a heterogeneous environment. Generalists can utilize the entire habitat, whereas specialists are restricted to their preferred habitat type. We find that although competitiveness decreases with specialization, specialists are more competitive in a spatial than in a nonspatial habitat as patchiness increases.
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arxiv:math/0610227
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This paper concerns regular connections on trivial algebraic G-principal fiber bundles over the infinitesimal punctured disc, where G is a connected reductive linear algebraic group over an algebraically closed field of characteristic zero. We show that the pull-back of every regular connection to an appropriate covering of the infinitesimal punctured disc is gauge equivalent to a connection of the form X z^{-1}dz for some X in the Lie algebra of G. We may even arrange that the only rational eigenvalue of ad X is zero. Our results allow a classification of regular SL_n-connections up to gauge equivalence.
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arxiv:math/0610250
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Due to a printing error the above mentioned article [Annals of Applied Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous equations appearing incorrectly in the print version of this paper. The entire article follows as it should have appeared. IMS apologizes to the author and the readers for this error. A recent paper by Harrison and Van Mieghem explained in general mathematical terms how one forms an ``equivalent workload formulation'' of a Brownian network model. Denoting by $Z(t)$ the state vector of the original Brownian network, one has a lower dimensional state descriptor $W(t)=MZ(t)$ in the equivalent workload formulation, where $M$ can be chosen as any basis matrix for a particular linear space. This paper considers Brownian models for a very general class of open processing networks, and in that context develops a more extensive interpretation of the equivalent workload formulation, thus extending earlier work by Laws on alternate routing problems. A linear program called the static planning problem is introduced to articulate the notion of ``heavy traffic'' for a general open network, and the dual of that linear program is used to define a canonical choice of the basis matrix $M$. To be specific, rows of the canonical $M$ are alternative basic optimal solutions of the dual linear program. If the network data satisfy a natural monotonicity condition, the canonical matrix $M$ is shown to be nonnegative, and another natural condition is identified which ensures that $M$ admits a factorization related to the notion of resource pooling.
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arxiv:math/0610352
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In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt-Thomason definability theorem in terms of the well established first-order topological language $L_t$.
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arxiv:math/0610357
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This article studies a Newton-like method already used by several authors but which has not been thouroughly studied yet. We call it the robust-variance scoring (RVS) algorithm because the main version of the algorithm that we consider replaces minus the Hessian of the loglikelihood used in the Newton-Raphson algorithm by a matrix $G$ which is an estimate of the variance of the score under the true probability, which uses only the individual scores. Thus an iteration of this algorithm requires much less computations than an iteration of the Newton-Raphson algorithm. Moreover this estimate of the variance of the score estimates the information matrix at maximum. We have also studied a convergence criterion which has the nice interpretation of estimating the ratio of the approximation error over the statistical error; thus it can be used for stopping the iterative process whatever the problem. A simulation study confirms that the RVS algorithm is faster than the Marquardt algorithm (a robust version of the Newton-Raphson algorithm); this happens because the number of iterations needed by the RVS algorithm is barely larger than that needed by the Marquardt algorithm while the computation time for each iteration is much shorter. Also the coverage rates using the matrix $G$ are satisfactory.
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arxiv:math/0610402
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We introduce a class of perverse sheaves on a partial flag manifold of a connected reductive group G defined over a finite field which are equivariant under the action of the group of rational points of G. The definition of this class is similar to the definition of parabolic character sheaves.
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arxiv:math/0610406
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We develop explicit formulas for Hecke operators of higher genus in terms of spherical coordinates. Applications are given to summation of various generating series with coefficients in local Hecke algebra and in a tensor product of such algebras. In particular, we formulate and prove Rankin's lemma in genus two. An application to a lifting from (GSp2 \times GSp2) to GSp4 is given using Ikeda-Miyawaki constructions.
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arxiv:math/0610417
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It is proved that, in certain subgroups of direct products of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. In particular, these properties coincide in all Abelian groups.
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arxiv:math/0610430
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We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schr\"odinger operator) $H=-d^2/dx^2+V$ to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential $V$. In the course of our analysis we also establish a functional model for periodic Schr\"odinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion. The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.
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arxiv:math/0610502
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The work studies some Difference equations, which are connected with Mejer's function.
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arxiv:math/0610545
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In this paper, we prove that there exists an equivalence between 2-category of smooth Deligne-Mumford stacks with torus-embeddings and actions, and the 1-category of stacky fans. For this purpose, we obtain two main results. The first is to investigate a combinatorial aspect of the 2-category of toric algebraic stacks defined in \cite{I2}. We establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second is to give a geometric characterization theorem for toric algebraic stacks.
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arxiv:math/0610548
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By a covering of a group G we mean an epimorphism from a group F to G. Introducing the notion of strong covering as a covering pi:F-->G such that every automorphism of G is a projection via pi of an automorphism of F, the main aim of this paper is to characterise double coverings which are strong. This is done in details for metacyclic groups, rotary platonic groups and some finite simple groups.
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arxiv:math/0610556
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In this paper we describe the intersection between the balls of maximal symplectic packings of $\P^2$. This analysis shows the existence of singular points for maximal packings of $\P^2$ by more than three equal balls. It also yields a construction of a class of very regular examples of maximal packings by five balls.
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arxiv:math/0610677
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Each packing of R^d by translates of the unit cube [0,1)^d admits a decomposition into at most two parts such that if a translate of the unit cube is covered by one of them, then it also belongs to such a part.
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arxiv:math/0610693
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We carry out the first main step towards the construction of new examples of complete embedded self-similar surfaces under mean curvature flow. An approximate solution is obtained by taking two known examples of self-similar surfaces and desingularizing the intersection circle using an appropriately modified singly periodic Scherk surface, called the core. Using an inverse function theorem, we show that for small boundary conditions on the core, there is an embedded surface close to the core that is a solution of the equation for self-similar surfaces. This provides us with an adequate central piece to substitute for the intersection.
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arxiv:math/0610695
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We find a necessary and sufficient conditions for the simplicity and uniqueness of trace for reduced free products of finite families of finite dimensional $C^*$-algebras with specified traces on them.
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arxiv:math/0610713
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We investigate a limit theorem on traversable length inside semi-cylinder in the 2-dimensional supercritical Bernoulli bond percolation, which gives an extension of Theorem 2 in Grimmett(1981). This type of limit theorems was originally studied for the extinction time for the 1-dimensional contact process on a finite interval in Wagner and Anantharam(2005). Actually, our main result Theorem 2.1 is stated under a rather general 2-dimensional bond percolation setting.
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arxiv:math/0610744
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In this paper, we define a new algebro-geometric invariant of 3-manifolds resulting from the Dehn surgery along a hyperbolic knot complement in S^3. We establish a Casson type invariant for these 3-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications, as well as the computation of our new invariants for some 3-manifolds resulting from the Dehn surgery along the figure-eight knot.
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arxiv:math/0610752
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We give an introduction to a theory of b-functions, i.e. Bernstein-Sato polynomials. After reviewing some facts from D-modules, we introduce b-functions including the one for arbitrary ideals of the structure sheaf. We explain the relation with singularities, multiplier ideals, etc., and calculate the b-functions of monomial ideals and also of hyperplane arrangements in certain cases.
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arxiv:math/0610783
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We give a short proof of the free analogue of the Talagrand inequality for the transportation cost to the semicircular which was originally proved by Biane and Voiculescu. The proof is based on a convexity argument and is in the spirit of the original Talagrand's proof. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of beta ensembles, which gives also yet another proof of the convergence of the eigenvalue distribution to the semicircle law.
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arxiv:math/0610826
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Likelihood ratio tests are intuitively appealing. Nevertheless, a number of examples are known in which they perform very poorly. The present paper discusses a large class of situations in which this is the case, and analyzes just how intuition misleads us; it also presents an alternative approach which in these situations is optimal.
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arxiv:math/0610835
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Fix a prime N, and consider the action of the Hecke operator T_N on the space M_k(SL(2,Z)) of modular forms of full level and varying weight k. The coefficients of the matrix of T_N with respect to the basis {E_4^i E_6^j | 4i + 6j = k} for M_k(SL(2,Z)) can be combined for varying k into a generating function F_N. We show that this generating function is a rational function for all N, and present a systematic method for computing F_N. We carry out the computations for N = 2, 3, 5, and indicate and discuss generalizations to spaces of modular forms of arbitrary level.
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arxiv:math/0610962
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In this work we introduce a new combinatorial notion of boundary $\Re C$ of an $\omega$-dimensional cubing $C$. $\Re C$ is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of $C$, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When $C$ arises as the dual of a cubulation -- or discrete system of halfspaces -- $\HH$ of a CAT(0) space $X$ (for example, the Niblo-Reeves cubulation of the Davis-Moussong complex of a finite rank Coxeter group), we show how $\HH$ induces a function $\rho:\bd X\to\Re C$. We develop a notion of uniformness for $\HH$, generalizing the parallel walls property enjoyed by Coxeter groups, and show that, if the pair $(X,\HH)$ admits a geometric action by a group $G$, then the fibers of $\rho$ form a stratification of $\bd X$ graded by the order structure of $\Re C$. We also show how this structure computes the components of the Tits boundary of $X$. Finally, using our result from another paper, that the uniformness of a cubulation as above implies the local finiteness of $C$, we give a condition for the co-compactness of the action of $G$ on $C$ in terms of $\rho$, generalizing a result of Williams, previously known only for Coxeter groups.
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arxiv:math/0611006
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Lueck expressed the Gromov norm of a knot complement in terms of an infinite series that can be computed from a presentation of the fundamental group of the knot complement. In this note we show that Lueck's formula, applied to torus knots, yields surprising power series expansions for the logarithm function. This generalizes an infinite series of Lehmer for the natural logarithm of 4.
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arxiv:math/0611027
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We consider the problem of isometric embedding of metric spaces to the Banach spaces; and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space has this property. We give a criterion of linear rigidity of a metric space, which allows us to give a simple proof of the linear rigidity of the Urysohn space and some other metric spaces. The various properties of linearly rigid spaces and related spaces are considered.
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arxiv:math/0611049
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Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein's view for the Euclidean Geometry, we defined a relation "<=" as: S<=T whenever there is a rigid motion f so that f(S) is a subset of T. This relation is not an order because there are figures (subsets of the plane) so that S<=T, T<=S and S, T not geometrically equal. Our goal is to avoid this paradox and to track down non-trivial classes of figures where the relation "<=" becomes, at least, a partial order. Such a class will be called a good class of figures. A reasonable question is whether the figures forming a good class have certain properties and whether the algebra of these figures is also a good class. Therefore we classified the figures into those that cause the paradox mentioned above and those that never cause it. The last ones are called good figures. Although simple, the definition of the good figure was difficult to handle, therefore we introduced a more technical, but intrinsic and handy definition, that of the strongly good figure. With these tools we constructed a new context, where we expanded our perspective about the geometric comparison not only in the Euclidean but also in the Hyperbolic and in the Elliptic Geometry. Eventually, there are still some open and quite challenging issues, which we present them at the last part of the paper.
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arxiv:math/0611062
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For compact CR manifolds of hypersurface type which embed in complex projective space, we show that for all k large enough there exist linear systems of ${\mathcal{O}}(k)$ which when restricted to the CR manifold are generic in a suitable sense. These systems are constructed using approximately holomorphic geometry.
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arxiv:math/0611125
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We construct (generalized) logarithmic derivatives for general n-dimensional local fields K of mixed characteristics (0,p) in which p is not necessarily a prime element with residue field k such that [k:k^p]=p^{n-1}. For the construction of the logarithmic derivative map, we define n-dimensional rings of overconvergent series and show that - as in the 1-dimensional case - they can be interpreted as functions converging on some annulus of the open unit p-adic disc. Using the generalized logarithmic derivative, we give a new construction of Kato's n-dimensional dual exponential map.
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arxiv:math/0611136
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A (smooth) K3 surface X defined over a field k of characteristic 0 is called singular if the N\'eron-Severi lattice NS (X) of X over the algebraic closure of k is of rank 20. Let X be a singular K3 surface defined over a number field F. For each embedding \sigma of F into the complex number field, we denote by T(X^\sigma) the transcendental lattice of the complex K3 surface X^\sigma obtained from X by \sigma. For each prime ideal P of F at which X has a supersingular reduction X_P, we define L(X, P) to be the orthogonal complement of NS(X) in NS(X_P). We investigate the relation between these lattices T(X^\sigma) and L(X, P). As an application, we give a lower bound of the degree of a number field over which a singular K3 surface with a given transcendental lattice can be defined.
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arxiv:math/0611208
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