Codimension one foliations with Bott-Morse singularities II

Abstract: We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among other things, a good topological understanding of the leaves one may have around a center-type component of the singular set, and also of the topology of its basin. The stability theorem further allows the description of the topology of the boundary of the basin and how the topology of the leaves changes when passing from inside to outside the basin. This is described via fiberwise Milnor-Wallace surgery. A key-point for this is to show that if the boundary of the basin of a center is non-empty, then it contains a saddle; in this case we say that the center and the saddle are paired. We then describe the possible pairings one may have in dimension three and use a construction motivated by the classical saddle-node bifurcation, that we call foliated surgery, that allows the reduction of certain pairings of singularities of a foliation. This is used together with our previous work on the topic to prove an extension for 3-manifolds of Reeb's sphere recognition theorem.


Gromov-Witten invariants for G/B and Pontryagin product for \Omega K

Abstract: We give an explicit formula for (T-equivariant) 3-pointed genus zero Gromov-Witten invariants for G/B. We derive it by finding an explicit formula for the equivariant Pontryagin product on the homology of the based loop group \Omega K.


Hom-bialgebras and comodule Hom-algebras

Abstract: We study Hom-bialgebras and objects admitting coactions by Hom-bialgebras. In particular, we construct a Hom-bialgebra M representing the functor of 2x2-matrices on Hom-associative algebras. Then we construct a Hom-algebra analogue of the affine plane and show that it is a comodule Hom-algebra over M in a suitable sense. It is also shown that the enveloping Hom-associative algebra of a Hom-Lie algebra is naturally a Hom-bialgebra.


On Systems of Equations over Free Partially Commutative Groups

Abstract: Version 2: Corrected Section 3.3: instead of lexicographical normal forms we now use a normal form due to V. Diekert and A. Muscholl. Consequent changes made and some misprints corrected. Using an analogue of Makanin-Razborov diagrams, we give an effective description of the solution set of systems of equations over a partially commutative group (right-angled Artin group) $G$. Equivalently, we give a parametrisation of $Hom(H, G)$, where $H$ is a finitely generated group.


Elementary equivalence of right-angled Coxeter groups and graph products of finite abelian groups

Abstract: We show that graph products of finite abelian groups are elementarily equivalent if and only if they are $\exists\forall$-equivalent if and only if they are isomorphic. In particular, two right-angled Coxeter groups are elementarily equivalent if and only if they are isomorphic.


Masaki Kashiwara and Algebraic Analysis

Abstract: This paper is a brief overview of the main contributions of Masaki Kashiwara in the domain of microlocal and algebraic analysis.


Shcherbina's Theorem for Finely Holomorphic Functions

Abstract: We prove an analogue of Sadullaev's theorem concerning the size of the set where a maximal totally real manifold can meet a pluripolar set. The manifold has to be of class C-1 only. This readily leads to a version of Shcherbina's theorem for C-1 functions f that are defined in a neighborhood of certain compact sets K in the complex plane. If the graph of f on K is pluripolar, then f satisfies the Cauchy Riemann equations in the closure of the fine interior of K.


The profile of bubbling solutions of a class of fourth order geometric equations on 4-manifolds

Abstract: We study a class of fourth order geometric equations defined on a 4-dimensional compact Riemannian manifold which includes the Q-curvature equation. We obtain sharp estimates on the difference near the blow-up points between a bubbling sequence of solutions and the standard bubble.


Orientability in Yang-Mills Theory over Nonorientable Surfaces

Abstract: In arXiv:math/0605587, the first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface. This space can be identified with the real locus of the space of connections on the pullback of this bundle over the orientable double cover of this nonorientable surface. In this context, the normal bundles to the Morse strata are real vector bundles. We show that these bundles, and their associated homotopy orbit bundles, are orientable for any n when the nonorientable surface is not homeomorphic to the Klein bottle, and for n<4 when the nonorientable surface is the Klein bottle. We also derive similar orientability results when the structure group is SU(n).


In search of a Lebesgue density theorem for R^\infty

Abstract: We look at a measure, $\lambda^\infty$, on the infinite-dimensional space, ${\mathbb R}^\infty$, for which we attempt to put forth an analogue of the Lebesgue density theorem. Although this measure allows us to find partial results, for example for continuous functions, we prove that it is impossible to give an analogous theorem in full generality. In particular, we proved that the Lebesgue density of probability density functions on ${\mathbb R}^\infty$ is zero almost everywhere.


Vanishing cycle sheaves of one-parameter smoothings and quasi-semistable degenerations

Abstract: We study the vanishing cycles of a one-parameter smoothing of a complex analytic space and show that the weight filtration on its perverse cohomology sheaf of the highest degree is quite close to the monodromy filtration so that its graded pieces have a modified Lefschetz decomposition. We describe its primitive part using the weight filtration on the perverse cohomology sheaves of the constant sheaves. As a corollary we show in the local complete intersection case that 1 is not an eigenvalue of the monodromy on the reduced Milnor cohomology at any points if and only if the total space and the singular fiber are both rational homology manifolds. Also we introduce quasi-semistable degenerations and calculate the limit mixed Hodge structure by constructing the weight spectral sequence. As a corollary we show non-triviality of the space of vanishing cycles of the Lefschetz pencil associated with a tensor product of any two very ample line bundles except for the case of even-dimensional projective space where two has to be replaced by three.


Asymptotics of multivariate sequences, part III: quadratic points

Abstract: We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deformations as well as Fourier-Laplace transforms of generalized functions. We apply the results of the theory to specific combinatorial problems, such as Aztec diamond tilings, cube groves, and multi-set permutations.


Klazar trees and perfect matchings

Abstract: Martin Klazar computed the total weight of ordered trees under 12 different notions of weight. The last and perhaps most interesting of these weights, w_{12}, led to a recurrence relation and an identity for which he requested combinatorial explanations. Here we provide such explanations. To do so, we introduce the notion of a "Klazar violator" vertex in an increasing ordered tree and observe that w_{12} counts what we call Klazar trees--increasing ordered trees with no Klazar violators. A highlight of the paper is a bijection from n-edge increasing ordered trees to perfect matchings of [2n]={1,2,...,2n} that sends Klazar violators to even numbers matched to a larger odd number. We find the distribution of the latter matches and, in particular, establish the one-summation explicit formula sum_{k=1}^{lfloor n/2 rfloor}(2k-1)!!^2 StirlingPartition{n+1}{2k+1} for the number of perfect matchings of [2n] with no even-to-larger-odd matches. The proofs are mostly bijective.


On the Study of Richard Tom Robert Identity

Abstract: In order to estimate the average speed of mosquitoes, a simple experiment was designed by Richard (Lu-Hsing Tsai), Tom (Po-Yu Tsai) and Robert (Hung-Ming Tsai). The result of the experiment was posted in the science exhibitions Taichung Taiwan 1993. The average speed of mosquitoes is inferred by the simple relation that is obtained easily. In this paper, we will show how the data generated by computer. Though the rigorous proof is not shown, a sketch proof will be shown in this paper. There are five figures one table and one fortran computer source program in the end of the paper.


Constructive proof of extended Kapranov theorem

Abstract: Kapranov Theorem is a well known generalization of Newton-Puiseux theorem for the case of several variables. This theorem is stated mainly in the context of tropical geometry. We present a new, constructive proof, that also characterizes the possible principal terms of points in a hypersurface contained in the algebraic torus $(K^*)^n$.


Generalized Demailly-Semple jet bundles and holomorphic mappings into complex manifolds

Abstract: Motivated by the Green-Griffiths conjecture, we study maximal rank holomorphic maps from $\C^p$ into complex manifolds. When $p>1$ such maps should in principle be more tractable than entire curves. We extend to this setting the jet-bundles techniques introduced by Semple, Green-Griffiths and Demailly. Our main application is the non-existence of maximal rank holomorphic maps from $\C^2$ into the very general degree $d$ hypersurface in $\bP^4$, as soon as $d\geq 93.$


Structured vector bundles define differential K-theory

Abstract: A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ring K, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f.Hopkins-Singer (2005), as well as a useful codification of vector bundles with connection.


Combinatorial models of expanding dynamical systems

Abstract: We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally defined "Julia set" of the generalized dynamical systems depends only on the associated iterated monodromy group. We show then that the Julia set of every expanding dynamical system is an inverse limit of simplicial complexes constructed by inductive cut-and-paste rules.


Low eigenvalues and one-dimensional collapse

Abstract: Our main result is that if a generic convex domain in $\R^n$ collapses to a domain in $\R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate the general case, we first prove the analogous result for triangular and polygonal domains. In so doing, we prove that the first two eigenvalues of triangular domains cannot be polyhomogeneous on the moduli space of triangles without blowing up a certain point. Our results show that the gap generically diverges under one dimensional collapse and is bounded only if the domain is sufficiently close to a rectangle in two dimensions or a cylinder in higher dimensions.


A general theory of Finite State Backward Stochastic Difference Equations

Abstract: By analogy with the theory of Backward Stochastic Differential Equations, we define Backward Stochastic Difference Equations on spaces related to discrete time, finite state processes. This paper considers these processes as constructions in their own right, not as approximations to the continuous case. We establish the existence and uniqueness of solutions under weaker assumptions than are needed in the continuous time setting, and also establish a comparison theorem for these solutions. The conditions of this theorem are shown to approximate those required in the continuous time setting. We also explore the relationship between the driver $F$ and the set of solutions; in particular, we determine under what conditions the driver is uniquely determined by the solution. Applications to the theory of nonlinear expectations are explored, including a representation result.


Fibrations of simplicial sets

Abstract: There are infinitely many variants of the notion of Kan fibration that, together with suitable choices of cofibrations and the usual notion of weak equivalence of simplicial sets, satisfy Quillen's axioms for a homotopy model category. The combinatorics underlying these fibrations is purely finitary and seems interesting both for its own sake and for its interaction with homotopy types. To show that these notions of fibration are indeed distinct, one needs to understand how iterates of Kan's Ex functor act on graphs and on nerves of small categories.


Linear Multistep Numerical Methods for Ordinary Differential Equations

Abstract: A review of the most popular Linear Multistep (LM) Methods for solving Ordinary Differential Equations numerically is presented. These methods are first derived from first principles, and are discussed in terms of their order, consistency, and various types of stability. Particular varieties of stability that may not be familiar, are briefly defined first. The methods that are included are the Adams-Bashforth Methods, Adams-Moulton Methods, and Backwards Differentiation Formulas. Advantages and disadvantages of these methods are also described. Not much prior knowledge of numerical methods or ordinary differential equations is required, although knowledge of basic topics from calculus is assumed.


Function Model of the Teichm\"uller space of a closed hyperbolic Riemann Surface

Abstract: We introduce a function model for the Teichm\"uller space of a closed hyperbolic Riemann surface. Then we introduce a new metric by using the maximum norm on the function space on the Teichm\"uller space. We prove that the identity map from the Teichm\"uller space equipped with the usual Teichm\"uller metric to the Teichm\"uller space equipped with this new metric is uniformly continuous. Furthermore, we also prove that the inverse of the identity, that is, the identity map from the Teichm\"uller space equipped with this new metric to the Teichm\"uller space equipped with the usual Teichm\"uller metric, is continuous. Therefore, the topology induced by the new metric is just the same as the topology induced by the usual Teichm\"uller metric on the Teichm\"uller space. We give a remark about the pressure metric and the Weil-Petersson metric.


Generic Uniqueness of Area Minimizing Disks for Extreme Curves

Abstract: We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.


Maximum Multiplicity of a Root of the Matching Polynomial of a Tree and Minimum Path Cover

Abstract: We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it.


Large deviations for Branching Processes in Random Environment

Abstract: A branching process in random environment $(Z_n, n \in \N)$ is a generalization of Galton Watson processes where at each generation the reproduction law is picked randomly. In this paper we give several results which belong to the class of {\it large deviations}. By contrast to the Galton-Watson case, here random environments and the branching process can conspire to achieve atypical events such as $Z_n \le e^{cn}$ when $c$ is smaller than the typical geometric growth rate $\bar L$ and $ Z_n \ge e^{cn}$ when $c > \bar L$. One way to obtain such an atypical rate of growth is to have a typical realization of the branching process in an atypical sequence of environments. This gives us a general lower bound for the rate of decrease of their probability. When each individual leaves at least one offspring in the next generation almost surely, we compute the exact rate function of these events and we show that conditionally on the large deviation event, the trajectory $t \mapsto \frac1n \log Z_{[nt]}, t\in [0,1]$ converges to a deterministic function $f_c :[0,1] \mapsto \R_+$ in probability in the sense of the uniform norm. The most interesting case is when $c < \bar L$ and we authorize individuals to have only one offspring in the next generation. In this situation, conditionally on $Z_n \le e^{cn}$, the population size stays fixed at 1 until a time $ \sim n t_c$. After time $n t_c$ an atypical sequence of environments let $Z_n$ grow with the appropriate rate ($\neq \bar L$) to reach $c.$ The corresponding map $f_c(t)$ is piecewise linear and is 0 on $[0,t_c]$ and $f_c(t) = c(t-t_c)/(1-t_c)$ on $[t_c,1].$


Newton polyhedra of discriminants of projections

Abstract: For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving mixed fiber polyhedra and Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results. The argument is based on a formula for the support function of a mixed fiber body, which also suggests a new proof for existence of mixed fiber bodies.


Cyclotomic double affine Hecke algebras and affine parabolic category O, I

Abstract: Using the orbifold KZ connection we construct a functor from an affine parabolic category O of type A to the category O of a cyclotomic rational double affine Hecke algebra. We give several results concerning this functor.


On Stepwise Control of the Generalized Familywise Error Rate

Abstract: A classical approach for dealing with the multiple testing problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of at least one false rejection. In many applications, one might be willing to tolerate more than one false rejection provided the number of such cases is controlled, thereby increasing the ability of the procedure to detect false null hypotheses. This suggests replacing control of the FWER by controlling the probability of $k$ or more false rejections, which is called the $k$-FWER. In this article, a unified approach is presented for deriving the $k$-FWER controlling procedures. We first generalize the well-known closure principle in the context of the FWER to the case of controlling the $k$-FWER. Then, we discuss how to derive the $k$-FWER controlling stepwise (stepdown or stepup) procedures based on marginal $p$-values using this principle. We show that, under certain conditions, generalized closed testing procedures can be reduced to stepwise procedures, and any stepwise procedure is equivalent to a generalized closed testing procedure. Finally, we generalize the well-known Hommel procedure in two directions, and show that any generalized Hommel procedure is equivalent to a generalized closed testing procedure with the same critical values.


On an elliptic system with symmetric potential possessing two global minima

Abstract: We consider the system {\Delta}u - W_u (u) = 0, for u: R^2 -> R^2, W: R^2 -> R, where W_u (u) is a smooth potential, symmetric with respect to the u_1, u_2 axes, possessing two global minima a^\pm := (\pma,0) and two connections e^\pm(x_1) connecting the minima. We prove that there exists an equivariant solution u(x_1, x_2) satisfying u(x_1, x_2) -> a^\pm, as x_1 -> \pminfiniti, and u(x_1, x_2) -> e^\pm(x_1), as x_2 -> \pminfiniti. The problem above was first studied by Alama, Bronsard, and Gui under related hypotheses to the ones introduced in the present paper. At the expense of one extra symmetry assumption, we avoid their considerations with the normalized energy and strengthen their result. We also provide examples for W.


On concrete models for local operator spaces

Abstract: In this short note, we propose a concrete analogue of the space $\cL(H)$ for local operator spaces, the multinormed $C^*$-algebra $\displaystyle\prod_{\alpha} \cL(H_{\alpha})$.


Mean Curvature flow in Higher Co-dimension

Abstract: We make several improvements on the results of M.-T. Wang in [8] and his joint paper with M.-P. Tsui [7] concerning the long time existence and convergence for solutions of mean curvature flow in higher co-dimension. Both the curvature condition and lower bound of $*\Omega$ are weakened. New applications are also obtained.


Contact pair structures and associated metrics

Abstract: We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector fields are geodesics and that the leaves of the Reeb action are totally geodesic. Mreover, we show that, in the case of a metric contact pair with decomposable endomorphism, the characteristic foliations are orthogonal and their leaves carry induced contact metric structures.


Jump-Diffusions in Hilbert Spaces: Existence, Stability and Numerics

Abstract: By means of an original approach, called "method of the moving frame", we establish existence, uniqueness and stability results for mild and weak solutions of stochastic partial differential equations (SPDEs) with path dependent coefficients driven by an infinite dimensional Wiener process and a compensated Poisson random measure. Our approach is based on a time-dependent coordinate transform, which reduces a wide class of SPDEs to a class of simpler SDE problems. We try to present the most general results, which we can obtain in our setting, within a self-contained framework to demonstrate our approach in all details. Also several numerical approaches to SPDEs in the spirit of this setting are presented.


Prescribing curvatures on three dimensional Riemannian manifolds with boundaries

Abstract: Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so that under these new metrics, $K$ is the scalar curvature and $c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $\bar M$, some of them may appear on $\partial M$. We describe the asymptotic behavior of the blowup solutions around each blowup point and derive an energy estimate as a consequence.


On the Calder\'on-Zygmund lemma for Sobolev functions

Abstract: We correct an inaccuracy in the original proof


Localisation de faisceaux caract\`eres

Abstract: Nous obtenons une formule pour les valeurs de la fonction caract\'eristique d'un faisceau caract\`ere en fonction de la th\'eorie des repr\'esentations de certains groupes finis, li\'es au groupe de Weyl. Cette formule, qui g\'en\'eralise des r\'esultats ant\'erieurs de M{\oe}glin et de Waldspurger, d\'epend de la connaissance de certains sous-groupes r\'eductifs admettant un faisceau caract\`ere cuspidal. Dans un second temps, afin de rendre la formule plus explicite dans le cas d'un groupe quasi-simple, nous d\'eterminons ces sous-groupes \`a conjugaison pr\`es. We obtain a formula for the values of the characteristic function of a character sheaf, in terms of the representation theory of certain finite groups related to the Weyl group. This formula, a generalization of previous results due to M{\oe}glin and Waldspurger, depends on knowledge of certain reductive subgroups that admit cuspidal character sheaves. For quasi-simple groups, we make the formula truly explicit by determining all such subgroups upto conjugation.


Addendum to "Maximal regularity and Hardy spaces"

Abstract: We correct an inaccuracy in a previous article [Auscher, Pascal; Bernicot, Fr\'ed\'eric; Zhao, Jiman. Maximal regularity and Hardy spaces. Collect. Math. 59 (2008), no. 1, 103-127.]


Uniqueness of Morava K-theory

Abstract: We show that there is an essentially unique S-algebra structure on the Morava K-theory spectrum K(n), while K(n) has uncountably many MU or \hE{n}-algebra structures. Here \hE{n} is the K(n)-localized Johnson-Wilson spectrum. To prove this we set up a spectral sequence computing the homotopy groups of the moduli space of A-infinity structures on a spectrum, and use the theory of S-algebra k-invariants for connective S-algebras due to Dugger and Shipley to show that all the uniqueness obstructions are hit by differentials.


Dehn twists have roots

Abstract: We construct nontrivial roots of Dehn twists about nonseparating curves.


Explicit birational geometry of 3-folds of general type, I

Abstract: Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=\text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0\leq 24$. A direct consequence is the birationality of the pluricanonical map $\varphi_m$ for all $m\geq 126$. Besides, the canonical volume $\text{Vol}(V)$ has a universal lower bound $\nu(3)\geq \frac{1}{63\cdot 126^2}$.


A characterization of sub-riemannian spaces as length dilatation structures constructed via coherent projections

Abstract: We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained from pairs of length dilatation structures, the first being a tempered one and the second obtained via a coherent projection. Thus we get an intrinsic, synthetic, axiomatic description of sub-riemannian geometry, which transforms the classical construction of a Carnot-Caratheodory distance on a regular sub-riemannian manifold into a model for this abstract sub-riemannian geometry.


On global H\"older estimates for optimal transportation

Abstract: We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation $T$ between uniformly log-concave probability measures. Let $T : \R^d \to \R^d$ be an optimal transportation pushing forward $\mu = e^{-V}dx$ to $\nu = e^{-W}dx$. Assume that 1) the second differential quotient of $V$ can be estimated from above by a power function, 2) modulus of convexity of $W$ can be estimated from below by $A_q |x|^{1+q}$, $q \ge 1$. Under these assumptions we show that $T$ is globally H\"older with a dimension-free coefficient. In addition, we study optimal transportation $T$ between $\mu$ and the uniform measure on a bounded convex set $K \subset \R^d$. We get estimates for the Lipschitz constant of $T$ in terms of $d$, ${diam(K)}$ and $D V, D^2 V$.


Explicit birational geometry of 3-folds of general type, II

Abstract: Let $V$ be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of $V$. We show that the $m$-canonical map of $V$ is birational for all $m\geq 73$ and that the canonical volume $\text{Vol}(V)\geq {1/2660}$. When $\chi(\mathcal{O}_V)\leq 1$, our result is $\text{Vol}(V)\geq {1/420}$, which is optimal. Other effective results are also included in the paper.


L2-Homogenization of Heat Equations on Tubular Neighborhoods

Abstract: We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold. We show that, as the tube radius decreases, the semigroup of a suitably rescaled and renormalized generator can be effectively described by a Hamiltonian on the submanifold with a potential that depends on the geometry of the submanifold and of the embedding.


Smooth Homogenization of Heat Equations on Tubular Neighborhoods

Abstract: We consider the heat equation with Dirichlet boundary conditions on the tubular neighborhood of a closed Riemannian submanifold. We show that, as the tube diameter tends to zero, a suitably rescaled and renormalized semigroup converges to a limit semigroup in Sobolev spaces of arbitrarily large Sobolev index.


Structural properties of semilinear SPDEs driven by cylindrical stable processes

Abstract: We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise.We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities.


Kirillov--Reshetikhin crystals for nonexceptional types

Abstract: We provide combinatorial models for all Kirillov--Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types D_n^(1), B_n^(1), A_{2n-1}^(2) we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type C_n^(1) we use a Dynkin diagram folding and for types A_{2n}^(2), D_{n+1}^(2) a similarity construction. We also show that for types C_n^(1) and D_{n+1}^(2) the analog of the Dynkin diagram automorphism exists on the level of crystals.


L^p Bernstein estimates and approximation by spherical basis functions

Abstract: The purpose of this paper is to establish L^p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L^p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L^p norm of the function itself. An important step in its proof involves measuring the L^p stability of functions in the approximating space in terms of the l^p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L^P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.


Demonstrative and non-demonstrative reasoning by analogy

Abstract: The paper analizes a set of issues related to analogy and analogical reasoning, namely: 1) The problem of analogy and its duplicity; 2) The role of analogy in demonstrative reasoning; 3) The role of analogy in non-demonstrative reasoning; 4) The limits of analogy; 5) The convergence, particularly in multiple analogical reasoning, of these two apparently distinct aspects and its methodological and philosophical consequences. The paper, using example from number theory, argues for an heuristc conception of analogy.


On some modular representations of the Borel subgroup of GL_2(Q_p)

Abstract: Colmez has given a recipe to associate a smooth modular representation Omega(W) of the Borel subgroup of GL_2(Q_p) to a F_p^bar-representation W of Gal(Qp^bar/Qp) by using Fontaine's theory of (phi,Gamma)-modules. We compute Omega(W) explicitly and we prove that if W is irreducible and dim(W)=2, then Omega(W) is the restriction to the Borel subgroup of GL_2(Q_p) of the supersingular representation associated to W in Breuil's correspondence.


On notions of harmonicity

Abstract: In this paper, we address the equivalence of the analytic and probabilistic notions of harmonicity in the context of general symmetric Hunt processes on locally compact separable metric spaces. Extensions to general symmetric right processes on Lusin spaces including infinite dimensional spaces are mentioned at the end of this paper.


On unique extension of time changed reflecting Brownian motions

Abstract: Let $D$ be an unbounded domain in $\RR^d$ with $d\geq 3$. We show that if $D$ contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on $\overline D$ is transient. Next assume that RBM $X$ on $\overline D$ is transient and let $Y$ be its time change by Revuz measure ${\bf 1}_D(x) m(x)dx$ for a strictly positive continuous integrable function $m$ on $\overline D$. We further show that if there is some $r>0$ so that $D\setminus \overline {B(0, r)}$ is an unbounded uniform domain, then $Y$ admits one and only one symmetric diffusion that genuinely extends it and admits no killings. In other words, in this case $X$ (or equivalently, $Y$) has a unique Martin boundary point at infinity.


Ein gesuchter, dennoch bislang unbekannter elementarer Satz

Abstract: If and only if each point of a set of the phase-space is in the topological hull of a trajectory running through any other point of this set, we call this set a quasiergodic set. But which are these so defined quasiergodic sets in the case of a given continuous dynamical system, which has piecewise differentiable trajectories in a finit-dimensional real phase-space, which is compact? Let its trajectories define a field of normed tangents, which is continuous in almost each point of the phase-space: Then the topological hulls of all trajectories of the phase-space form a partition of it. Thus the elements of this partition are the quasiergodic sets of the given continuous dynamical system. This is the important but rather trivial statement of the elementary theorem 1.1, which we present in this tractatus. We call this theorem elementary, because it is limited to finit-dimensional real phase-spaces. We also find, that there is a linear homogeneous partial differential equation of first order, describing the invariants of the system, which allow us the construction of the quasiergodic sets. Furthermore we show, that any quasiergodic set is a sensitive attractor, if it is neither a closed trajectory or a fixed point.


Generalizations of the Goulden-Jackson Cluster Method

Abstract: We give several modifications of the Goulden-Jackson Cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various 'weights' on words, including single letter probability distributions, double letter (i.e. pairwise) probability distributions, and triple letter probability distributions. We also describe an alternative, recursive approach to finding such generating functions. We describe Maple implementations of the various modifications. The accompanying Maple package is available at the website for this paper.


Ensemble Control of Finite Dimensional Time-Varying Linear Systems

Abstract: In this article, we investigate the problem of simultaneously steering an uncountable family of finite dimensional time-varying linear systems. We call this class of control problems Ensemble Control, a notion coming from the study of spin dynamics in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI). This subject involves controlling a continuum of parameterized dynamical systems with the same open-loop control input. From a viewpoint of mathematical control theory, this class of problems is challenging because it requires steering a continuum of dynamical systems between points of interest in an infinite dimensional state space by use of the same control function. The existence of such a control raises fundamental questions of ensemble controllability. We derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for ensemble control of time-varying linear systems. We show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics. In addition, we study the problem of optimal ensemble control of harmonic oscillators to demonstrate our main results. We show that the optimal solutions are pertinent to the study of time-frequency limited signals and prolate spheroidal wave functions. A systematic study of ensemble control systems has immediate applications to systems with parameter uncertainties as well as to broad areas of quantum control systems as arising in coherent spectroscopy and quantum information processing. The new mathematical structures appearing in such problems are an excellent motivation for new developments in control theory.


A non-negative expansion for small Jensen-Shannon Divergences

Abstract: In this report, we derive a non-negative series expansion for the Jensen-Shannon divergence (JSD) between two probability distributions. This series expansion is shown to be useful for numerical calculations of the JSD, when the probability distributions are nearly equal, and for which, consequently, small numerical errors dominate evaluation.


A priori estimate for a family of semi-linear elliptic equations with critical nonlinearity

Abstract: We consider positive solutions of $\Delta u-\mu u+Ku^{\frac{n+2}{n-2}}=0$ on $B_1$ ($n\ge 5$) where $\mu $ and $K>0$ are smooth functions on $B_1$. If $K$ is very sub-harmonic at each critical point of $K$ in $B_{2/3}$ and the maximum of $u$ in $\bar B_{1/3}$ is comparable to its maximum over $\bar B_1$, then all positive solutions are uniformly bounded on $\bar B_{1/3}$. As an application, a priori estimate for solutions of equations defined on $\mathbb S^n$ is derived.


Iterated bar complexes of E-infinity algebras and homology theories

Abstract: We proved in a previous article that the bar complex of an E-infinity algebra inherits a natural E-infinity algebra structure. As a consequence, a well-defined iterated bar construction B^n(A) can be associated to any algebra over an E-infinity operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E-infinity algebras. We use this effective definition to prove that the n-fold bar complex B^n(A) admits an extension to categories of algebras over E_n-operads. Then we prove that the n-fold bar complex determines the homology theory associated to a category of algebras over E_n-operads. For n infinite, we obtain an isomorphism between the homology of an infinite bar construction and the usual Gamma-homology with trivial coefficients.


Combinatorial bases of Feigin-Stoyanovsky's type subspaces of higher-level standard $\tilde{\gsl}(\ell+1,\C)$-modules

Abstract: Let $\tilde{\mathfrak g}$ be an affine Lie algebra of the type $A_\ell^{(1)}$. We find a combinatorial basis of Feigin-Stoyanovsky's type subspace $W(\Lambda)$ given in terms of difference and initial conditions. Linear independence of the generating set is proved inductively by using coefficients of intertwining operators. A basis of $L(\Lambda)$ is obtained as an "inductive limit" of the basis of $W(\Lambda)$.


A construction for coisotropic subalgebras of Lie bialgebras

Abstract: Given a Lie bialgebra (g,g*), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g*. We write down families of examples for the case that g is a classical complex simple Lie algebra.


Some computations about Kazhdan-Lusztig cells in affine Weyl groups of rank 2

Abstract: In the last section of the paper "Generalized induction of Kazhdan-Lusztig cells" and in "Kazhdan-Lusztig cells in affine Weyl groups of rank 2" the author described the partition into Kazhdan-Lusztig cells of the affine Weyl groups of rank 2 for all choices of parameters. The proof of these results relies on some explicit computations with GAP. In these notes we give some details of these computations.


A generalization of the Collatz problem and conjecture

Abstract: We introduce an infinite set of integer mappings that generalize the well-known Collatz-Ulam mapping and we conjecture that an infinite subset of these mappings feature the remarkable property of the Collatz conjecture, namely that they converge to unity irrespective of which positive integer is chosen initially.


Action and Index Spectra and Periodic Orbits in Hamiltonian Dynamics

Abstract: The main theme of this paper is the connection between the existence of infinitely many periodic orbits for a Hamiltonian system and the behavior of its action or index spectrum under iterations. We use the action and index spectra to show that any Hamiltonian diffeomorphism of a closed, rational manifold with zero first Chern class has infinitely many periodic orbits and that, for a general rational manifold, the number of geometrically distinct periodic orbits is bounded from below by the ratio of the minimal Chern number and half of the dimension. These generalizations of the Conley conjecture follow from another result proved here asserting that a Hamiltonian diffeomorphism with a symplectically degenerate maximum on a closed rational manifold has infinitely many periodic orbits. We also show that for a broad class of manifolds and/or Hamiltonian diffeomorphisms the minimal action--index gap remains bounded for some infinite sequence of iterations and, as a consequence, whenever a Hamiltonian diffeomorphism has finitely many periodic orbits, the actions and mean indices of these orbits must satisfy a certain relation. Furthermore, for Hamiltonian diffeomorphisms of the n-dimensional complex projective space with exactly n+1 periodic orbits a stronger result holds. Namely, for such a Hamiltonian diffeomorphism, the difference of the action and the mean index on a periodic orbit is independent of the orbit, provided that the symplectic structure on the projective space is normalized to be in the same cohomology class as the first Chern class.


Derived Smooth Manifolds

Abstract: We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local $C^\infty$-rings that is obtained by patching together homotopy zero-sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection $A\cap B$ of submanifolds $A,B\subset X$ exists on the categorical level in our theory, and a cup product formula $$[A]\smile[B]=[A\cap B]$$ holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a {\em categorification} of intersection theory.


Global Existence Of Smooth Solutions Of A 3D Loglog Energy-Supercritical Wave Equation

Abstract: We prove global existence of smooth solutions of the 3D loglog energy-supercritical wave equation $\partial_{tt} u - \triangle u = -u^{5} \log^{c} (log(10+u^{2})) $ with $0 < c < {8/225}$ and smooth initial data $(u(0)=u_{0}, \partial_{t} u(0)=u_{1})$. First we control the $L_{t}^{4} L_{x}^{12}$ norm of the solution on an arbitrary size time interval by an expression depending on the energy and an \textit{a priori} upper bound of its $L_{t}^{\infty} \tilde{H}^{2}(\mathbb{R}^{3})$ norm, with $\tilde{H}^{2}(\mathbb{R}^{3}):=\dot{H}^{2}(\mathbb{R}^{3}) \cap \dot{H}^{1}(\mathbb{R}^{3})$. The proof of this long time estimate relies upon the use of some potential decay estimates \cite{bahger, shatstruwe} and a modification of an argument in \cite{taolog}. Then we find an \textit{a posteriori} upper bound of the $L_{t}^{\infty} \tilde{H}^{2}(\mathbb{R}^{3})$ norm of the solution by combining the long time estimate with an induction on time of the Strichartz estimates.


The Modular number, Congruence number, and Multiplicity One

Abstract: Let $N$ be a positive integer and let $f$ be a newform of weight 2 on $\Gamma_0(N)$. In earlier joint work with K. Ribet and W. Stein, we introduced the notions of the modular number and the congruence number of the quotient abelian variety $A_f$ of $J_0(N)$ associated to the newform $f$. These invariants are analogs of the notions of the modular degree and congruence primes respectively associated to elliptic curves. We show that if $p$ is a prime such that every maximal ideal of the Hecke algebra of characteristic $p$ that contains the annihilator ideal of $f$ satisfies multiplicity one, then the modular number and the congruence number have the same $p$-adic valuation.


A visible factor for analytic rank one

Abstract: Let $E$ be an optimal elliptic curve of conductor $N$, such that the $L$-function of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ also vanishes to order one at $s=1$. In view of the Gross-Zagier theorem, the second part of the Birch and Swinnerton-Dyer conjecture says that the index in $E(K)$ of the subgroup generated by the Heegner point is equal to the product of the Manin constant of $E$, the Tamagawa numbers of $E$, and the square root of the order of the Shafarevich-Tate group of $E$ (over $K$). We extract an integer factor from the index mentioned above and relate this factor to certain congruences of the newform associated to $E$ with eigenforms of analytic rank bigger than one. We use the theory of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime $q$ divides this factor, then $q$ divides the order of the Shafarevich-Tate group or the order of an arithmetic component group of $E$, as predicted by the second part of the Birch and Swinnerton-Dyer conjecture.


Squareness in the special L-value and special L-values of twists

Abstract: Let N be a prime and let A be a quotient of J_0(N) over Q associated to a newform such that the special L-value of A (at s=1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of (N-1)/12. Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the special L-value of A and of the order of the Shafarevich-Tate group are both positive even numbers. Under a certain mod q non-vanishing hypothesis on special L-values of twists of A, we show that the q-adic valuations of the algebraic part of the special L-value of A and of the Birch and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group of A are both positive even numbers. We also give a formula for the algebraic part of the special L-value of A over quadratic imaginary fields K in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic N (equivalently, over conjugacy classes of maximal orders in the definite quaternion algebra over Q ramified at N and infinity) which shows that this algebraic part is a perfect square up to powers of the prime two and of primes dividing the discriminant of K. Finally, for an optimal elliptic curve E, we give a formula for the special L-value of the twist E_D of E by a negative fundamental discriminant -D, which shows that this special L-value is an integer up to a power of 2, under some hypotheses. In view of the second part of the Birch and Swinnerton-Dyer conjecture, this leads us to the surprising conjecture that the square of the order of the torsion subgroup of E_D divides the product of the order of the Shafarevich-Tate group of E_D and the orders of the arithmetic component groups of E_D, under certain mild hypotheses.


Rational torsion in elliptic curves and the cuspidal subgroup

Abstract: Let $A$ be an elliptic curve over $\Q$ of square free conductor $N$. We prove that if $A$ has a rational torsion point of prime order $r$ such that $r$ does not divide $6N$, then $r$ divides the order of the cuspidal subgroup of $J_0(N)$.


Representation dimension of $m$-replicated algebras

Abstract: Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field and $A^{(m)}$ the $m$-replicated algebra of $A$. We prove that the representation dimension of $A^{(m)}$ is at most three, and that the dominant dimension of $A^{(m)}$ is at least $m$.


Symmetric Schroder paths and restricted involutions

Abstract: Let $A_k$ be the set of permutations in the symmetric group $S_k$ with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns $A_k$. We present a bijection between symmetric Schroder paths of length $2n$ and involutions of length $n+1$ avoiding $\mathcal{A}_4$. Statistics such as the number of right-to-left maxima and fixed points of the involution correspond to the number of steps in the symmetric Schroder path of a particular type. For each $k> 2$ we determine the generating function for the number of involutions avoiding the subsequences in $A_k$, according to length, first entry and number of fixed points.


Partial tilting modules over $m$-replicated algebras

Abstract: Let $A$ be a hereditary algebra over an algebraically closed field $k$ and $A^{(m)}$ be the $m$-replicated algebra of $A$. Given an $A^{(m)}$-module $T$, we denote by $\delta (T)$ the number of non isomorphic indecomposable summands of $T$. In this paper, we prove that a partial tilting $A^{(m)}$-module $T$ is a tilting $A^{(m)}$-module if and only if $\delta (T)=\delta (A^{(m)})$, and that every partial tilting $A^{(m)}$-module has complements. As an application, we deduce that the tilting quiver $\mathscr{K}_{A^{(m)}}$ of $A^{(m)}$ is connected. Moreover, we investigate the number of complements to almost tilting modules over duplicated algebras.


Complete bounded holomorphic curves immersed in C^2 with arbitrary genus

Abstract: In the previous paper, the authors constructed a complete holomorphic immersion of the unit disk D into C^2 whose image is bounded. In this paper, we shall prove existence of complete holomorphic null immersions of Riemann surfaces with arbitrary genus and finite topology, whose image is bounded in C^2. To construct such immersions, we apply the method used by F. J. Lopez to perturb the genus zero example changing its genus. As an analogue the above construction, we also give a new method to construct complete bounded minimal immersions (resp. weakly complete maximal surface) with arbitrary genus and finite topology in Euclidean 3-space (resp. Lorentz-Minkowski 3-spacetime).


Near optimal thresholding estimation of a Poisson intensity on the real line

Abstract: The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $\tilde{f}_{n,\gamma}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a possible logarithmic term. Then, minimax properties of $\tilde{f}_{n,\gamma}$ on Besov spaces ${\cal B}^{\ensuremath \alpha}_{p,q}$ are established. Under mild assumptions, we prove that $$\sup_{f\in B^{\ensuremath \alpha}_{p,q}\cap \ensuremath \mathbb {L}_{\infty}} \ensuremath \mathbb {E}(\ensuremath | | \tilde{f}_{n,\gamma}-f| |_2^2)\leq C(\frac{\log n}{n})^{\frac{\ensuremath \alpha}{\ensuremath \alpha+{1/2}+({1/2}-\frac{1}{p})_+}}$$ and the lower bound of the minimax risk for ${\cal B}^{\ensuremath \alpha}_{p,q}\cap \ensuremath \mathbb {L}_{\infty}$ coincides with the previous upper bound up to the logarithmic term. This new result has two consequences. First, it establishes that the minimax rate of Besov spaces ${\cal B}^{\ensuremath \alpha}_{p,q}$ with $p\leq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. When $p>2$, the rate exponent, which depends on $p$, deteriorates when $p$ increases, which means that the support plays a harmful role in this case. Furthermore, $\tilde{f}_{n,\gamma}$ is adaptive minimax up to a logarithmic term.


Algorithms for Producing and Ordering Lexical and Nonlexical Sequences out of one Element

Abstract: This paper deals with algorithms for producing and ordering lexical and nonlexical sequences of a given degree. The notion of "elementary operations" on positive integral sequences is introduced. Our main theorem answers the question of when two lexical sequences are adjacent.


Signal Acquisition from Measurements via Non-Linear Models

Abstract: We consider the problem of reconstruction of a non-linear finite-parametric model $M=M_p(x),$ with $p=(p_1,...,p_r)$ a set of parameters, from a set of measurements $\mu_j(M)$. In this paper $\mu_j(M)$ are always the moments $m_j(M)=\int x^jM_p(x)dx$. This problem is a central one in Signal Processing, Statistics, and in many other applications. We concentrate on a direct (and somewhat "naive") approach to the above problem: we simply substitute the model function $M_p(x)$ into the measurements $\mu_j$ and compute explicitly the resulting "symbolic" expressions of $\mu_j(M_p)$ in terms of the parameters $p$. Equating these "symbolic" expressions to the actual measurement results, we produce a system of nonlinear equations on the parameters $p$, which we consequently try to solve. The aim of this paper is to review some recent results (mostly of \cite{Vet5,Mil1,Mil2,Put1,Vet4,Vet3,Mil3,Mil4,Vet2}) in this direction, stressing the algebraic structure of the arising systems and mathematical tools required for their solutions. In particular, we discuss the relation of the reconstruction problem above with the recent results of \cite{bfy,bry,chr,pak1,pak2,pak3,pry,ry} on the vanishing problem of generalized polynomial moments and on the Cauchy-type integrals of algebraic functions. The accompanying paper \cite{Kis1} (this volume) provides a solution method for a wide class of reconstruction problems as above, based on the study of linear differential equations with rational coefficient, which are satisfied by the moment generating function of the problem.


Morasses and finite support iterations

Abstract: We introduce a method of constructing a forcing along a simplified $(\kappa,1)$-morass such that the forcing satisfies the $\kappa$-chain condition. Alternatively, this may be seen as a method to thin out a larger forcing to get a chain condition. As an application, we construct a ccc forcing that adds an $\omega_2$-Suslin tree. Related methods are Shelah's historic forcing and Todorcevic's $\rho$-functions.


Limiting Distribution of Frobenius Numbers for $n=3$

Abstract: The purpose of this paper is to give a complete derivation of the limiting distribution of large Frobenius numbers outlined in earlier work of J. Bourgain and Ya. Sinai and fill some gaps formulated there as hypotheses.


Substitution Tilings and Separated Nets with Similarities to the Integer Lattice

Abstract: We show that any primitive substitution tiling of the plane creates a separated net which is biLipschitz to the integer lattice. Then we show that if H is a primitive Pisot substitution in an Euclidean space, for every separated net Y, that corresponds to some tiling of the tiling space, there exists a bijection F between Y and the integer lattice that translate every element of Y a bounded distance. As a corollary we get that we have such an F for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.


On the representation ring of the polynomial algebra over a perfect field

Abstract: We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type A is provided.


On General Balance Laws with Boundary

Abstract: This paper is devoted to general balance laws (with a possibly non local source term) with a non-characteristic boundary. Basic well posedness results are obtained, trying to provide sharp estimates. In particular, bounds tend to blow up as the boundary tends to be characteristic. New uniqueness results for the solutions to conservation and/or balance laws with boundary are also provided.


The classification of complete stable area-stationary surfaces in the Heisenberg group $\mathbb{H}^1$

Abstract: We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.


Spectral geometry, link complements and surgery diagrams

Abstract: We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require 'complicated' surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width for knots and 3-manifolds.


The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution

Abstract: In 1976 Procesi and Schacher developed an Artin-Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.


Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

Abstract: We compute the Szeg\"o kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in $\CC^n$ for Grassmannian manifolds of higher ranks. In particular they provide an infinite family of smoothly bounded strictly pseudo-convex domains on complex manifolds for which the log terms in the Fefferman expansion of the Szeg\"o kernel vanish and which are not diffeomorphic to the sphere. The analogous results for the Bergman kernel are also obtained.


Radon, cosine and sine transforms on Grassmannian manifolds

Abstract: Let $G_{n,r}(\bbK)$ be the Grassmannian manifold of $k$-dimensional $\bbK$-subspaces in $\bbK^n$ where $\bbK=\mathbb R, \mathbb C, \mathbb H$ is the field of real, complex or quaternionic numbers. We consider the Radon, cosine and sine transforms, $\mathcal R_{r^\prime, r}$, $\mathcal C_{r^\prime, r}$ and $\mathcal S_{r^\prime, r}$, from the $L^2$ space $L^2(G_{n,r}(\bbK))$ to the space $L^2(G_{n,r^\prime}(\bbK))$, for $r, r^\prime \le n-1$. The $L^2$ spaces are decomposed into irreducible representations of $G$ with multiplicity free. We compute the spectral symbols of the transforms under the decomposition. For that purpose we prove two Bernstein-Sato type formulas on general root systems of type BC for the sine and cosine type functions on the compact torus $\mathbb R^r/{2\pi Q^\vee}$ generalizing our recent results for the hyperbolic sine and cosine functions on the non-compact space $\mathbb R^r$. We find then also a characterization of the images of the transforms. Our results generalize those of Alesker-Bernstein and Grinberg. We prove further that the Knapp-Stein intertwining operator for certain induced representations is given by the sine transform and we give the unitary structure of the Stein's complementary series in the compact picture.


Degenerate p-Laplacian operators on H-type groups and applications to Hardy type inequalities

Abstract: Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \text{div}_X (|\na_{X} u|^{p-2} \na_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$, for $k=2$ and $\mathbb G$ being the Heisenberg group they are introduced by Greiner \cite{Greiner-cjm79}. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.


Second order structures for sprays and connections on Frechet manifolds

Abstract: Ambrose, Palais and Singer \cite{Ambrose} introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath \cite{Kumar} extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Frechet manifolds. It is proved that the existence of Christoffel and Hessian structures, connections, sprays and dissections are equivalent on those Frechet manifolds which can be considered as projective limits of Banach manifolds. These concepts provide also an alternative way for the study of ordinary differential equations on non-Banach infinite dimensional manifolds. Concrete examples of the structures are provided using direct and flat connections.


On the hyperbolic automorphisms of the 2-torus and their Markov partitions

Abstract: In the paper we give an introduction to Anosov diffeomorphisms, ways to represent their chaotic properties and some historical remarks on this subject. A complete classification of hyperbolic linear automorphisms of 2-torus is presented. We introduce a notion of pre-Markov partition for such automorphisms and give their classification and an algorithm for their construction.


Choice of neighbor order in nearest-neighbor classification

Abstract: The $k$th-nearest neighbor rule is arguably the simplest and most intuitively appealing nonparametric classification procedure. However, application of this method is inhibited by lack of knowledge about its properties, in particular, about the manner in which it is influenced by the value of $k$; and by the absence of techniques for empirical choice of $k$. In the present paper we detail the way in which the value of $k$ determines the misclassification error. We consider two models, Poisson and Binomial, for the training samples. Under the first model, data are recorded in a Poisson stream and are "assigned" to one or other of the two populations in accordance with the prior probabilities. In particular, the total number of data in both training samples is a Poisson-distributed random variable. Under the Binomial model, however, the total number of data in the training samples is fixed, although again each data value is assigned in a random way. Although the values of risk and regret associated with the Poisson and Binomial models are different, they are asymptotically equivalent to first order, and also to the risks associated with kernel-based classifiers that are tailored to the case of two derivatives. These properties motivate new methods for choosing the value of $k$.


On the structure of some moduli spaces of finite flat group schemes

Abstract: We consider the moduli space, in the sense of Kisin, of finite flat models of a 2-dimensional representation with values in a finite field of the absolute Galois group of a totally ramified extension of $mathbb{Q}_p$. We determine the connected components of this space and describe its irreducible components. These results prove a modified version of a conjecture of Kisin.


Automated computation of robust normal forms of planar analytic vector fields

Abstract: We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighborhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method.


On weak generalized stability and (c,d)-pseudostable random variables via functional equations

Abstract: In this paper we give a first attempt to define and study stable distributions with respect to the weak generalized convolution, focusing our attention on the symmetric weakly stable distribution. As in the case of the classical convolution, characterization of distributions stable in the sense of the weak generalized convolution depends on solving some functional equations in the class of characteristic functions.


Aggregation of penalized empirical risk minimizers in regression

Abstract: We give a general result concerning the rates of convergence of penalized empirical risk minimizers (PERM) in the regression model. Then, we consider the problem of agnostic learning of the regression, and give in this context an oracle inequality and a lower bound for PERM over a finite class. These results hold for a general multivariate random design, the only assumption being the compactness of the support of its law (allowing discrete distributions for instance). Then, using these results, we construct adaptive estimators. We consider as examples adaptive estimation over anisotropic Besov spaces or reproductive kernel Hilbert spaces. Finally, we provide an empirical evidence that aggregation leads to more stable estimators than more standard cross-validation or generalized cross-validation methods for the selection of the smoothing parameter, when the number of observation is small.


Poincar\'e lemma and global homotopy formulas with sharp anisotropic H\"older estimates in q-concave CR manifolds

Abstract: We prove sharp anisotropic H\"older estimates for the local solutions of the tangential Cauchy-Riemann equation in q-concave CR manifolds and we derive the same kind of estimates for global solutions when the manifold is compact.


Ten colours in quasiperiodic and regular hyperbolic tilings

Abstract: Colour symmetries with ten colours are presented for different tilings. In many cases, the existence of these colourings were predicted by group theoretical methods. Only in a few cases explicit constructions were known, sometimes using combination of two-colour and five-colour symmetries. Here we present explicit constructions of several of the predicted colourings for the first time, and discuss them in contrast to already known colourings with ten colours.


Liouville type theorems for conformal Gaussian curvature equation

Abstract: In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -\Delta u=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.


A class of R\'{e}nyi information estimators for multidimensional densities

Abstract: A class of estimators of the R\'{e}nyi and Tsallis entropies of an unknown distribution $f$ in $\mathbb{R}^m$ is presented. These estimators are based on the $k$th nearest-neighbor distances computed from a sample of $N$ i.i.d. vectors with distribution $f$. We show that entropies of any order $q$, including Shannon's entropy, can be estimated consistently with minimal assumptions on $f$. Moreover, we show that it is straightforward to extend the nearest-neighbor method to estimate the statistical distance between two distributions using one i.i.d. sample from each. (Wit Correction.)


Duality between Hyperbolic and de Sitter Geometry

Abstract: In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar triangles from spherical geometry into the Minkowski space, we relate hyperbolic with de Sitter triangles such that the proof of the hyperbolic law of cosines for angles becomes much clearer and easier than it is traditionally. Furthermore, polar triangles turn out to be a powerful tool for describing de Sitter trigonometry.


Burnside-Brauer Theorem and Character Products in Table Algebras

Abstract: In this paper, we first show that the irreducible characters of a quotient table algebra modulo a normal closed subset can be viewed as the irreducible characters of the table algebra itself. Furthermore, we define the character products for table algebras and give a condition in which the products of two characters are characters. Thereafter, as a main result we state and prove the Burnside-Brauer Theorem on finite groups for table algebras.


Hermitian modular forms congruent to 1 modulo p

Abstract: For any natural number $\ell $ and any prime $p\equiv 1 \pmod{4}$ not dividing $\ell $ there is a Hermitian modular form of arbitrary genus $n$ over $L:=\Q [\sqrt{-\ell}]$ that is congruent to 1 modulo $p$ which is a Hermitian theta series of an $O_L$-lattice of rank $p-1$ admitting a fixed point free automorphism of order $p$. It is shown that also for non-free lattices such theta series are modular forms.


Coactions on Hochschild Homology of Hopf-Galois Extensions and Their Coinvariants

Abstract: Let A be an H-Galois extension of B. If M is a Hopf bimodule then HH.(A,M), the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C:=H/[H,H]. Given an injective left C-comodule V, we denote the cotensor product of M and V by N. Our aim is to investigate the relationship between the cotensor product of HH.(A,M) and V, on the one hand, and HH.(B,N) on the other hand. The roots of this problem can be found in Lorenz's work on the descent of Hochschild homology of centrally Galois extensions, where HH.(A)^G, the subspace of invariant cycles with respect to the action of the Galois group, and HH.(B) are shown to be isomorphic.


Euler-Mahonian distributions of type $B_n$

Abstract: Adin, Brenti, and Roichman introduced the pairs of statistics $(\ndes, \nmaj)$ and $(\fdes, \fmaj)$. They showed that these pairs are equidistributed over the hyperoctahedral group $B_n$, and can be considered "Euler-Mahonian" in that they generalize the Carlitz identity. Further, they asked whether there exists a bijective proof of the equidistribution of their statistics. We give such a bijection, along with a new proof of the generalized Carlitz identity.


The Algorithmic Behaviour of the F5 Algorithm

Abstract: This paper has been withdrawn due to an error in the proof of the termination of the algorithm. Moreover, some ideas of this paper are a joint work with John Perry and it is inappropriate to mention only me as an author. John Perry was in spite of the statement in the acknowledgment of previous versions not proofreading this paper. Recently we have found a new solution for ensuring F5's termination, which can be found here : arXiv:1006.0318.


Interface evolution: water waves in 2-D

Abstract: We study the free boundary evolution between two irrotational, incompressible and inviscid fluids in 2-D without surface tension. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition. The well-posedness of the full water wave problem was first obtained by Wu \cite{Wu}. The methods introduced in this paper allows us to consider multiple cases: with or without gravity, but also a closed boundary or a periodic boundary with the fluids placed above and below it. It is assumed that the initial interface does not touch itself, being a part of the evolution problem to check that such property prevails for a short time, as well as it does the Rayleigh-Taylor condition, depending conveniently upon the initial data. The addition of the pressure equality to the contour dynamic equations is obtained as a mathematical consequence, and not as a physical assumption, from the mere fact that we are dealing with weak solutions of Euler's equation in the whole space.


Orthogonally additive holomorphic functions of bounded type over $C(K)$

Abstract: It is known that all $k$-homogeneous orthogonally additive polynomials $P$ over $C(K)$ are of the form $$ P(x)=\int_K x^k d\mu . $$ Thus $x\mapsto x^k$ factors all orthogonally additive polynomials through some linear form $\mu$. We show that no such linearization is possible without homogeneity. However, we also show that every orthogonally additive holomorphic functions of bounded type $f$ over $C(K)$ is of the form $$ f(x)=\int_K h(x) d\mu $$ for some $\mu$ and holomorphic $h\colon C(K) \to L^1(\mu)$ of bounded type.


Convergent and divergent numbers games for certain collections of edge-weighted graphs

Abstract: The numbers game is a one-player game played on a finite simple graph with certain "amplitudes" assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. Here, the edge amplitudes will be negative integers. Combinatorial methods are used to investigate the convergence and divergence of numbers games played on certain such graphs. The results obtained here provide support for results in a companion paper.


A note on spectral triples and quasidiagonality

Abstract: Spectral triples (of compact type) are constructed on arbitrary separable quasidiagonal C*-algebras. On the other hand an example of a spectral triple on a non-quasidiagonal algebra is presented.


The C$^*$-envelope of a semicrossed product and Nest Representations

Abstract: Let $X$ be compact Hausdorff, and $\phi: X \to X$ a continuous surjection. Let $\mathcal{A}$ be the semicrossed product algebra corresponding to the relation fU = Uf\circ \phi$ or to the relation $Uf = f\circ \phi U.$ Then the C$^*$-envelope of $\mathcal{A}$ is the crossed product of a commutative C$^*$-algebra which contains $C(X)$ as a subalgebra, with respect to a homeomorphism which we construct. We also show there are"sufficiently many" nest representations.


On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters

Abstract: We determine the unitary dual of the geometric graded Hecke algebras with {unequal} parameters which appear in Lusztig's classification of unipotent representations for {exceptional} $p$-adic groups. The largest such algebra is of type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the $p$-adic groups. In order for this correspondence to be applicable here, we show (following Lusztig's geometric classification, and Barbasch and Moy's original argument) that the set of tempered modules with real central character for a geometric graded Hecke algebra is linearly independent when restricted to the Weyl group.


The numbers game and Dynkin diagram classification results

Abstract: The numbers game is a one-player game played on a finite simple graph with certain "amplitudes" assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. Combinatorial reasoning is used to show that those connected graphs with negative integer amplitudes for which the numbers game meets a certain finiteness requirement are precisely the Dynkin diagrams associated with the finite-dimensional complex simple Lie algebras. This strengthens a result originally due to the second author. A more general result is obtained when certain real number amplitudes are allowed. The resulting graphs are in families, each family corresponding to a finite irreducible Coxeter group. These results are used to demonstrate that the only generalized Cartan matrices for which there exist finite edge-colored ranked posets enjoying a certain structure property are the Cartan matrices for the finite-dimensional complex semisimple Lie algebras. In this setting, classifications of the finite-dimensional Kac--Moody algebras and of the finite Coxeter and Weyl groups are re-derived.


Divisors in the moduli spaces of curves

Abstract: In this mostly expository paper we review several known results about the cohomology of moduli spaces of smooth and stable curves, focusing in particular on low degree cohomology. We also give a new proof of Harer's theorem describing the second cohomology group of the moduli space of smooth n-pointed curves of given genus


Median structures on asymptotic cones and homomorphisms into mapping class groups

Abstract: The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median subspace. One of the applications is that a group with Kazhdan's property (T) can have only finitely many pairwise non-conjugate homomorphisms into a mapping class group. We also give a new proof of the rank conjecture of Brock and Farb (previously proved by Behrstock and Minsky, and independently by Hamenstaedt).


The critical dimension for a fourth order elliptic problem with singular nonlinearity

Abstract: We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\f{\lambda}{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u=\partial_\nu u=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0<u_\la<1$ exists for $\la\in (0,\la^*)$, while there is none of any kind when $\la>\la^*$. Our main result asserts that the extremal solution $u_{\lambda^*}$ is regular $(\sup_B u_{\lambda^*} <1)$ provided $ N \le 8$ while $u_{\lambda^*} $ is singular ($\sup_B u_{\lambda^*} =1$) for $N \ge 17$, in which case $1-C_0|x|^{4/3}\leq u_{\lambda^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $ C_0:= <(\frac{\lambda^*}{\bar{\lambda}}>)^{1/3}$ and $ \bar{\lambda}:= \frac{8 (N-{2/3}) (N- {8/3})}{9}$. The singular character of the extremal solution for the remaining cases (i.e., when $9\leq N\leq 16$) requires a computer assisted proof and will not be addressed in this paper.


Iwahori-Hecke type algebras associated with the Lie superalgebras A(m,n), B(m,n), C(n) and D(m,n)

Abstract: In this paper we give Iwahori-Hecke type algebras H_q(g) associated with the Lie superalgebras g=A(m,n), B(m,n), C(n) and D(m,n). We classify the irreducible representations of H_q(g) for generic q.


Geodesic Webs on a Two-Dimensional Manifold and Euler Equations

Abstract: We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar results are obtained for planar d-webs, d > 4, provided that additional d-4 second-order invariants vanish.


Ising vectors in the vertex operator algebra $V_{\Lambda}^+$ associated with the Leech lattice $\Lambda$

Abstract: In this article, we study the Ising vectors in the vertex operator algebra $V_\Lambda^+$ associated with the Leech lattice $\Lambda$. The main result is a characterization of the Ising vectors in $V_\Lambda^+$. We show that for any Ising vector $e$ in $V_\Lambda^+$, there is a sublattice $E\cong \sqrt{2}E_8$ of $\Lambda$ such that $e\in V_E^+$. Some properties about their corresponding $\tau$-involutions in the moonshine vertex operator algebra $V^\natural$ are also discussed. We show that there is no Ising vector of $\sigma$-type in $V^\natural$. Moreover, we compute the centralizer $C_{\aut V^\natural}(z, \tau_e)$ for any Ising vector $e\in V_\Lambda^+$, where $z$ is a 2B element in $\aut V^\natural$ which fixes $V_\Lambda^+$. Based on this result, we also obtain an explanation for the 1A case of an observation by Glauberman-Norton (2001), which describes some mysterious relations between the centralizer of $z$ and some 2A elements commuting $z$ in the Monster and the Weyl groups of certain sublattices of the root lattice of type $E_8$ .


Geodesic Webs and PDE Systems of Euler Equations

Abstract: We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d > n) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler equations, and for flat connections we give an explicit construction of their solutions.


Calculation of Coefficients of the Optimal Quadrature Formulas in the $W_2^{m,m-1}(0,1)$ Space

Abstract: In this paper problem of construction of optimal quadrature formulas in $W_2^{(m,m-1)}(0,1)$ space is considered. Here by using Sobolev's algorithm when $m=1,2$ we find the optimal coefficients of the quadrature formulas of the form $$ \int\limits_0^1\phi(x)dx\cong \sum\limits_{\beta=0}^NC_{\beta}\phi(x_{\beta}). $$


Properties of Discrete Analogue of the Differential Operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$

Abstract: In the paper properties of the discrete analogue $D_m(h\beta)$ of the differential operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$ are studied. It is known, that zeros of differential operator $\frac{d^{2m}}{dx^{2m}}-\frac{d^{2m-2}}{dx^{2m-2}}$ are functions $e^x$, $e^{-x}$ and $P_{2m-3}(x)$. It is proved that discrete analogue $D_m(h\beta)$ of this differential operator also have similar properties.


On the cohomology groups of holomorphic Banach bundles

Abstract: We consider a compact complex manifold $M$, and introduce the notion of two holomorphic Banach bundles $E,F$ over $M$ being compact perturbations of one another. Given two such bundles we show that if the cohomology groups $H^q(M,E)$ are finite dimensional then so are the cohomology groups $H^q(M,F)$; as well as a more precise result in the same spirit.


How hot can a heat bath get?

Abstract: We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for 'extreme' non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence / non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is 'too stiff', then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds.


A note on Talagrand's transportation inequality and logarithmic Sobolev inequality

Abstract: We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand's transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient conditions work even in the case where the Bakry-Emery curvature is not lower bounded. Several new examples are provided.


Projective pairs of profinite groups

Abstract: We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projective pair can be realized as absolute Galois groups of a PAC extension of a PAC field. Using this characterization we construct new examples of PAC extensions of relatively small fields, e.g., unbounded abelian extensions of the rational numbers.


Vector product algebras

Abstract: Vector products can be defined on spaces of dimensions 0, 1, 3 and 7 only, and their isomorphism types are determined entirely by their adherent symmetric bilinear forms. We present a short and elementary proof for this classical result.


Age-dependent equations with non-linear diffusion

Abstract: We consider the well-posedness of models involving age structure and non-linear diffusion. Such problems arise in the study of population dynamics. It is shown how diffusion and age boundary conditions can be treated that depend non-linearly and possibly non-locally on the density itself. The abstract approach is depicted with examples.


Approximating the marginal likelihood using copula

Abstract: Model selection is an important activity in modern data analysis and the conventional Bayesian approach to this problem involves calculation of marginal likelihoods for different models, together with diagnostics which examine specific aspects of model fit. Calculating the marginal likelihood is a difficult computational problem. Our article proposes some extensions of the Laplace approximation for this task that are related to copula models and which are easy to apply. Variations which can be used both with and without simulation from the posterior distribution are considered, as well as use of the approximations with bridge sampling and in random effects models with a large number of latent variables. The use of a t-copula to obtain higher accuracy when multivariate dependence is not well captured by a Gaussian copula is also discussed.


Complexity of PL-manifolds

Abstract: We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant). Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with non-positive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed 4-manifolds of complexity zero (manifolds without 3-handles, doubles of 2-handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group).


Arithmetic Fujita approximation

Abstract: We prove an arithmetic analogue of Fujita's approximation theorem in Arakelov geometry, conjectured by Moriwaki, by using slope method and measures associated to $\mathbb R$-filtrations.


Numerably Contractible Spaces

Abstract: Numerably contractible spaces play an important role in the theory of homotopy pushouts and pullbacks. The corresponding results imply that a number of well known weak homotopy equivalences are genuine ones if numerably contractible spaces are involved. In this paper we give a first systematic investigation of numerably contractible spaces. We list the elementary properties of the category of these spaces. We then study simplicial objects in this category. In particular, we show that the topological realization functor preserves fibration sequences if the base is path-connected and numerably contractible in each dimension. Consequently, the loop space functor commutes with realization up to homotopy. We give simple conditions which assure that free algebras over a topological operad are numerably contractible.


A differential Chevalley theorem

Abstract: We prove a differential analog of a theorem of Chevalley on extending homomorphisms for rings with commuting derivations, generalizing a theorem of Kac. As a corollary, we establish that, under suitable hypotheses, the image of a differential scheme under a finite morphism is a constructible set. We also obtain a new algebraic characterization of differentially closed fields. We show that similar results hold for differentially closed fields that are saturated, in the sense of model theory. In characteristic p > 0, we obtain related results and establish a differential Nullstellensatz. Analogs of these theorems for difference fields are also considered.


Remarks on missing faces and generalized lower bounds on face numbers

Abstract: We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented.


The Frolicher-Nijenhuis Calculus in Synthetic Differential Geometry

Abstract: Just as the Jacobi identity of vector fields is a natural consequence of the general Jacobi identity of microcubes in synthetic differential geometry, it is to be shown in this paper that the graded Jacobi identity of the Frolicher-Nijenhuis bracket is also a natural consequence of the general Jacobi identity.


Geometric construction of crystal bases for quantum generalized Kac-Moody algebras

Abstract: We provide a geometric realization of the crystal $B(\infty)$ for quantum generalized Kac-Moody algebras in terms of the irreducible components of certain Lagrangian subvarieties in the representation spaces of a quiver.


A problem in the Kourovka notebook concerning the number of conjugacy classes of a finite group

Abstract: In this paper, we consider Problem 14.44 in the Kourovka notebook, which is a conjecture about the number of conjugacy classes of a finite group. While elementary, this conjecture is still open and appears to elude any straightforward proof, even in the soluble case. However, we do prove that a minimal soluble counterexample must have certain properties, in particular that it must have Fitting height at least 3 and order at least 2000.


Two-dimensional quantum random walk

Abstract: We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. Our methods are based on analysis of the space-time generating function, following the methods of Pemantle and Wilson (2002).


Neighboring ternary cyclotomic coefficients differ by at most one

Abstract: A cyclotomic polynomial Phi_n(x) is said to be ternary if n=pqr with p,q and r distinct odd prime factors. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Eli Leher showed in 2007 that neighboring ternary cyclotomic coefficients differ by at most four. We show that, in fact, they differ by at most one. Consequently, the set of coefficients occurring in a ternary cyclotomic polynomial consists of consecutive integers. As an application we reprove in a simpler way a result of Bachman from 2004 on ternary cyclotomic polynomials with an optimally large set of coefficients.


Hydrodynamics, probability and the geometry of the diffeomorphisms group

Abstract: We characterize the solution of Navier-Stokes equation as a stochastic geodesic on the diffeomorphisms group, thus generalizing Arnold's description of the Euler flow.


Alvis-Curtis duality, central characters, and real-valued characters

Abstract: We prove that Alvis-Curtis duality preserves the Frobenius-Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius-Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius-Schur indicators of certain real-valued, irreducible, Frobenius-invariant Deligne-Lusztig characters, and the Frobenius-Schur indicators of real-valued regular and semisimple characters of finite unitary groups.


Representations of cyclic groups acting on complete simplicial fans

Abstract: Let $sigma$ be a complete simplicial fan in finite dimensional real Euclidean space $V$, and let $G$ be a cyclic subgroup of $GL(V)$ which acts properly on $\sigma$. We show that the representation of $G$ carried by the cohomology of $X_{sigma}$, the toric variety associated to $sigma$, is a permutation representation.


Introduction to Graph-Link Theory

Abstract: The present paper is an introduction to a combinatorial theory arising as a natural generalisation of classical and virtual knot theory. There is a way to encode links by a class of `realisable' graphs. When passing to generic graphs with the same equivalence relations we get `graph-links'. On one hand graph-links generalise the notion of virtual link, on the other hand they do not feel link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalisation of the Kauffman-Murasugi-Thistlethwaite theorem on `minmal diagrams' for graph-links


Boxicity of Circular Arc Graphs

Abstract: A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-dimensional boxes: that is two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. Let $G$ be a circular arc graph with maximum degree $\Delta$. We show that if $\Delta <\lfloor \frac{n(\alpha-1)}{2\alpha}\rfloor$, $\alpha \in \mathbb{N}$, $\alpha \geq 2$ then $box(G) \leq \alpha$. We also demonstrate a graph with boxicity $> \alpha$ but with $\Delta=n\frac{(\alpha-1)}{2\alpha}+\frac{n}{2\alpha(\alpha+1)}+(\alpha+2)$. So the result cannot be improved substantially when $\alpha$ is large. Let $r_{inf}$ be minimum number of arcs passing through any point on the circle with respect to some circular arc representation of $G$. We also show that for any circular arc graph $G$, $box(G) \leq r_{inf} + 1$ and this bound is tight. Given a family of arcs $F$ on the circle, the circular cover number $L(F)$ is the cardinality of the smallest subset $F'$ of $F$ such that the arcs in $F'$ can cover the circle. Maximum circular cover number $L_{max}(G)$ is defined as the maximum value of $L(F)$ obtained over all possible family of arcs $F$ that can represent $G$. We will show that if $G$ is a circular arc graph with $L_{max}(G)> 4$ then $box(G) \leq 3$.