Embeddings of k-connected n-manifolds into R^{2n-k-1}

Abstract: We obtain estimations for isotopy classes of embeddings of closed k-connected n-manifolds into R^{2n-k-1} for n>2k+5 and k\ge0. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of H_{k+1}(N;Z_2) on the set of embeddings. (For k\ne1 classification results were obtained by algebraic methods without direct construction of embeddings or homology invariants.) The proof involves reduction to classification of embeddings of punctured manifold and uses parametric connected sum of embeddings.


Une conjecture sur la torsion des classes de Chern des fibr\'es de Gauss-Manin

Abstract: Pour tout $t\in\mN$ nous d\'efinissons un certain entier positif $\N_t$ et nous conjecturons: si $H$ est un fibr\'e de Gauss-Manin d'une fibration semi-stable alors la $t$-\`eme classe de Chern de $H$ est annul\'ee par $\N_t$. Nous d\'emontrons diverses cons\'equences de cette conjecture. For any $t\in\mN$, we define a certain positive integer $\N_t$ and we conjecture: if $H$ is a Gauss-Manin bundle of a semi-stable fibration then the $t$-th Chern class of $H$ is kiled by $\N_t$. We prove various consequences of this conjecture.


Open strings, Lagrangian conductors and Floer functor

Abstract: We introduce a contravariant functor, called Floer functor, from the category of Lagrangian conductors of a symplectic manifold to the homotopy category of bounded chain complexes of open strings in this manifold. The latter two categories are defined for all symplectic manifolds, whereas Floer functor is defined for semipositive manifolds which are either closed or convex at infinity. We then prove that when the first Chern class of the symplectic manifold vanishes, Lagrangian spheres define Lagrangian conductors so that in particular their integral Floer cohomology is well defined. This requires the introduction of singular almost-complex structures given by symplectic field theory.


Stable ergodicity of dominated systems

Abstract: We provide a new approach to stable ergodicity of systems with dominated splittings, based on a geometrical analysis of global stable and unstable manifolds of hyperbolic points. Our method suggests that the lack of uniform size of Pesin's local stable and unstable manifolds - a notorious problem in the theory of non-uniform hyperbolicity - is often less severe than it appeas to be.


The projective McKay correspondence

Abstract: Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent bundle T*P^1 and modules over the preprojective algebra of an affine Dynkin quiver. These different equivalences are related by spherical twists, which take the place of the reflection functors for P^1.


Representing geometric morphisms using power locale monads

Abstract: It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with the double power locale monad and whose right adjoints preserve finite coproduct. They are also characterised as those adjunctions that preserve the order enrichment and commute with both the upper and the lower power locale monads.


Types are weak omega-groupoids

Abstract: We define a notion of weak omega-category internal to a model of Martin-L\"of type theory, and prove that each type bears a canonical weak omega-category structure obtained from the tower of iterated identity types over that type. We show that the omega-categories arising in this way are in fact omega-groupoids.


Vortex invariants and toric manifolds

Abstract: We consider the symplectic vortex equations for a linear Hamiltonian torus action. We show that the associated genus zero moduli space itself is homotopic (in the sense of a homotopy of regular G-moduli problems) to a toric manifold with combinatorial data directly obtained from the original torus action. This allows to view the wall crossing formula of Cieliebak and Salamon for the computation of vortex invariants as a consequence of a generalized Jeffrey-Kirwan localization formula for integrals over symplectic quotients.


Holomorphic factorization of mappings into SL_n(C)

Abstract: We solve Gromov's Vaserstein problem. Namely, we show that a null-homotopic holomorphic mapping from a finite dimensional reduced Stein space into SL_n(C) can be factored into a finite product of unipotent matrices with holomorphic entries.


Splitting Monoidal Stable Model Categories

Abstract: If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit S forms a commutative ring. An idempotent e of this ring will split the homotopy category. We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to the product of C localised at the object eS and C localised at the object (1-e)S. This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.


Higher Bers maps

Abstract: The Bers embebbing realizes the Teichm\"uller space of a Fuchsian group $G$ as a open, bounded and contractible subset of the complex Banach space of bounded quadratic differentials for $G$. It utilizes the schlicht model of Teichm\"uller space, where each point is represented by an injective holomorphic function on the disc, and the map is constructed via the Schwarzian differential operator. In this paper we prove that a certain class of differential operators acting on functions of the disc induce holomorphic mappings of Teichm\"uller spaces, and we also obtain a general formula for the differential of the induced mappings at the origin. The main focus of this work, however, is on two particular series of such mappings, dubbed higher Bers maps, because they are induced by so-called higher Schwarzians -- generalizations of the classical Schwarzian operator. For these maps, we prove several further results. The last section contains a discussion of possible applications, open questions and speculations.


Classifying Rational G-Spectra for Finite G

Abstract: We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model.


Polynomial parametrization of the solutions of Diophantine equations of genus 0

Abstract: Let f in Z[X,Y,Z] be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by f=0 has a function field isomorphic to the rational function field Q(t). We show that all integral solutions of the Diophantine equation f=0 (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a single triple of polynomials with integer coefficients.


Einstein manifolds with nonnegative isotropic curvature are locally symmetric

Abstract: Let (M,g) be an Einstein manifold of dimension n \geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric.


Uniform Time Average Consistency of Monte Carlo Particle Filters

Abstract: We prove that bootstrap type Monte Carlo particle filters approximate the optimal nonlinear filter in a time average sense uniformly with respect to the time horizon when the signal is ergodic and the particle system satisfies a tightness property. The latter is satisfied without further assumptions when the signal state space is compact, as well as in the noncompact setting when the signal is geometrically ergodic and the observations satisfy additional regularity assumptions.


Gross--Schoen Cycles and Dualising Sheaves

Abstract: The aim of this paper is to study the modified diagonal cycle in the triple product of a curve over a global field defined by Gross and Schoen. Our main result is an identity between the height of this cycle and the self-intersection of the relative dualising sheaf. We have some applications to the following problems in number theory and algebraic geometry.


Splitting and composition methods in the numerical integration of differential equations

Abstract: We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.


Qualitative Properties of Local Random Invariant Manifolds for SPDEs with Quadratic Nonlinearity

Abstract: The qualitative properties of local random invariant manifolds for stochastic partial differential equations with quadratic nonlinearities and multiplicative noise is studied by a cut off technique. By a detail estimates on the Perron fixed point equation describing the local random invariant manifold, the structure near a bifurcation is given.


Weak omega-categories from intensional type theory

Abstract: We show that for any type in Martin-L\"of Intensional Type Theory, the terms of that type and its higher identity types form a weak omega-category in the sense of Leinster. Precisely, we construct a contractible globular operad of definable composition laws, and give an action of this operad on the terms of any type and its identity types.


Which infinite abelian groups admit an almost maximally almost-periodic group topology?

Abstract: A topological group G is said to be almost maximally almost-periodic if its von Neumann radical is non-trivial, but finite. In this paper, we prove that every abelian group with an infinite torsion subgroup admits a (Hausdorff) almost maximally almost-periodic group topology. Some open problems are also formulated.


Buchsteiner loops: associators and constructions

Abstract: Let $Q$ be a Buchsteiner loop. We describe the associator calculus in three variables, and show that $|Q| \ge 32$ if $Q$ is not conjugacy closed. We also show that $|Q| \ge 64$ if there exists $x \in Q$ such that $x^2$ is not in the nucleus of $Q$. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that is nilpotency class 3 and possesses an abelian inner mapping group.


Color Visualization of Blaschke Product Mappings

Abstract: A visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces. A working tool is the technique of simultaneous continuation we introduced in previous papers. Here, we are refining this technique for some particular types of Blaschke products for which coloring pre-images of annuli centered at the origin allow us to describe the mappings with a high degree of fidelity.


On the $\pd$- and $\barpd$-Operators of a Generalized Complex Structure

Abstract: In this note, we prove that the $\pd$- and $\barpd$-operators introduced by Gualtieri for a generalized complex structure coincide with the $\bdees$- and $\bdel$-operators introduced by Alekseev-Xu for Evens-Lu-Weinstein modules of a Lie bialgebroid.


A proximal method for composite minimization

Abstract: We consider minimization of functions that are compositions of convex or prox-regular functions (possibly extended-valued) with smooth vector functions. A wide variety of important optimization problems fall into this framework. We describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term. Properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem. Preliminary computational results on both convex and nonconvex examples are promising.


Quandle and hyperbolic volume

Abstract: We show that the hyperbolic volume of a hyperbolic knot is a quandle cocycle invariant. Further we show that it completely determines invertibility and positive/negative amphicheirality of hyperbolic knots.


On David type Siegel Disks of the Sine family

Abstract: In 2008 Petersen posed a list of questions on the application of trans-quasiconformal Siegel surgery developed by Zakeri and himself. In this paper we extend Petersen-Zakeri's idea so that the surgery can be applied to all the premodels which have no "free critical points". We explain how the idea is used in solving three of the questions posed by Petersen. To present the details of the idea, we focus on the solution of one of them: we prove that for typical rotation numbers $0< \theta < 1$, the boundary of the Siegel disk of $f_{\theta}(z) = e^{2 \pi i \theta} \sin (z)$ is a Jordan curve which passes through exactly two critical points $\pi/2$ and $-\pi/2$.


Mixed volume and an extension of intersection theory of divisors

Abstract: Let K(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L_1,..., L_n in K(X), we define an intersection index [L_1,..., L_n] as the number of solutions in X of a system of equations f_1 = ... = f_n = 0 where each f_i is a generic function from the space L_i. In counting the solutions, we neglect the solutions x at which all the functions in some space L_i vanish as well as the solutions at which at least one function from some subspace L_i has a pole. The collection K(X) is a commutative semigroup with respect to a natural multiplication. The intersection index [L_1,..., L_n] can be extended to the Grothendieck group of K(X). This gives an extension of the intersection theory of divisors. The extended theory is applicable even to non-complete varieties. We show that this intersection index enjoys all the main properties of the mixed volume of convex bodies. Our paper is inspired by the Bernstein-Kushnirenko theorem from the Newton polytope theory.


An algorithmic Littlewood-Richardson rule

Abstract: We introduce a Littlewood-Richardson rule based on an algorithmic deformation of skew Young diagrams and present a bijection with the classical rule. The result is a direct combinatorial interpretation and proof of the geometric rule presented by Coskun. We also present a corollary regarding the Specht modules of the intermediate diagrams.


Indecomposable Permutations, Hypermaps and Labeled Dyck Paths

Abstract: Hypermaps were introduced as an algebraic tool for the representation of embeddings of graphs on an orientable surface. Recently a bijection was given between hypermaps and indecomposable permutations; this sheds new light on the subject by connecting a hypermap to a simpler object. In this paper, a bijection between indecomposable permutations and labelled Dyck paths is proposed, from which a few enumerative results concerning hypermaps and maps follow. We obtain for instance an inductive formula for the number of hypermaps with n darts, p vertices and q hyper-edges; the latter is also the number of indecomposable permutations of with p cycles and q left-to-right maxima. The distribution of these parameters among all permutations is also considered.


Universally catenarian integral domains, strong S-domains and semistar operations

Abstract: Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over $D$. We introduce and investigate the notions of $\star$-universally catenarian and $\star$-stably strong S-domains and prove that, every $\star$-locally finite dimensional Pr\"{u}fer $\star$-multiplication domain is $\star$-universally catenarian, and this implies $\star$-stably strong S-domain. We also give new characterizations of $\star$-quasi-Pr\"{u}fer domains introduced recently by Chang and Fontana, in terms of these notions.


A bijectional attack on the Razumov-Stroganov conjecture

Abstract: We attempt to prove the Razumov-Stroganov conjecture using a bijectional approach. We have been unsuccessful but we believe the techniques we present can be used to prove the conjecture.


The existence of thick triangulations -- an "elementary" proof

Abstract: We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also emphasized.


Miyaoka-Yau inequality for minimal projective manifolds of general type

Abstract: In this short note, we prove the Miyaoka-Yau inequality for minimal projective $n$-manifolds of general type by using K\"ahler-Ricci flow.


Restricted involutions and Motzkin paths

Abstract: We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions.


On approximations by shifts of the Gaussian function

Abstract: The paper study the discrete sets of translations of the Gaussian function that span the spaces L1(R) and L2(R).


Operations in Milnor K-theory

Abstract: We show that operations in Milnor K-theory mod $p$ of a field are spanned by divided power operations. After giving an explicit formula for divided power operations and extending them to some new cases, we determine for all fields $k$ and all prime numbers $p$, all the operations $K^M_i/p \to K^M_j/p$ commuting with field extensions over the base field $k$. Moreover, the integral case is discussed and we determine the operations $K^M_i/p \to K^M_j/p$ for smooth schemes over a field.


Matricially free random variables

Abstract: We show that the operatorial framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a generalization of both freeness and monotone independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to Gaussian random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial generalizations of semicirle laws.


Building suitable sets for locally compact groups by means of continuous selections

Abstract: If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S \cup {1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.


Finite flat models of constant group schemes of rank two

Abstract: We calculate the number of the isomorphism class of the finite flat models over the ring of integers of an absolutely ramified $p$-adic field of constant group schemes of rank two over finite fields, by counting the rational points of a moduli space of finite flat models.


On a conjecture on exponential Diophantine equations

Abstract: We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.


On near optimal trajectories for a game associated with the \infty-Laplacian

Abstract: A two-player stochastic differential game representation has recently been obtained for solutions of the equation -\Delta_\infty u=h in a \calC^2 domain with Dirichlet boundary condition, where h is continuous and takes values in \RR\setminus\{0\}. Under appropriate assumptions, including smoothness of u, the vanishing \delta limit law of the state process, when both players play \delta-optimally, is identified as a diffusion process with coefficients given explicitly in terms of derivatives of the function u.


Zero-nonzero patterns for nilpotent matrices over finite fields

Abstract: Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper we initiate an investigation into which zero-nonzero patterns are potentially nilpotent over F, with a special emphasis on the case that F = Z_p is a finite field. As part of this investigation, we develop methods, using the tools of algebraic geometry and commutative algebra, to eliminate zero-nonzero patterns A as being potentially nilpotent over any field F. We then use these techniques to classify all irreducible zero-nonzero patterns of order two and three that are potentially nilpotent over Z_p for each prime p.


On sums of three squares

Abstract: We prove that a positive integer not of the form, 4^{k}(8m+7) can be expressible as a sum of three or fewer squares by using some results of Kane and Sun on mixed sums of squares and triangular numbers.


On the large genus asymptotics of Weil-Petersson volumes

Abstract: A relatively fast algorithm for evaluating Weil-Petersson volumes of moduli spaces of complex algebraic curves is proposed. On the basis of numerical data, a conjectural large genus asymptotics of the Weil-Petersson volumes is computed. Asymptotic formulas for the intersection numbers involving $\psi$-classes are conjectured as well. The accuracy of the formulas is high enough to believe that they are exact.


Metric and arithmetic properties of mediant-Rosen maps

Abstract: A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.


Flow invariants in the classification of Leavitt path algebras

Abstract: We analyze in the context of Leavitt path algebras some graph operations introduced in the context of symbolic dynamics by Williams, Parry and Sullivan, and Franks. We show that these operations induce Morita equivalence of the corresponding Leavitt path algebras. As a consequence we obtain our two main results: the first gives sufficient conditions for which the Leavitt path algebras in a certain class are Morita equivalent, while the second gives sufficient conditions which yield isomorphisms. We discuss a possible approach to establishing whether or not these conditions are also in fact necessary. In the final section we present many additional operations on graphs which preserve Morita equivalence (resp., isomorphism) of the corresponding Leavitt path algebras.


Index Theory for Boundary Value Problems via Continuous Fields of C*-algebras

Abstract: We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semigroupoid $\cT^-X$ generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field C*_r(\cT^-X) of C*-algebras over [0,1]. Its fiber in h=0, C*_r(T^-X), can be identified with the symbol algebra for Boutet de Monvel's calculus; for h\not=0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K_0(C*_r(T^-X))=K_0(C_0(T*X)) -> K_0(K)=Z. Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.


Statistical properties of intermittent maps with unbounded derivative

Abstract: We study the ergodic and statistical properties of a class of maps of the circle and of the interval of Lorenz type which present indifferent fixed points and points with unbounded derivative. These maps have been previously investigated in the physics literature. We prove in particular that correlations decay polynomially, and that suitable Limit Theorems (convergence to Stable Laws or Central Limit Theorem) hold for H\"older continuous observables. We moreover show that the return and hitting times are in the limit exponentially distributed.


Nets in groups, minimum length $g$-adic representations, and minimal additive complements

Abstract: The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems.


A case study of an Hamilton-Jacobi equation by the Adomian decompositional method

Abstract: We present a study of the Adomian's Decomposition Method (ADM) applied to the Hamilton-Jacobi equations ut + H (ux) = 0. We recall the well known characteristics methods in the case of this type of equations to justify the existence or not of solutions. This yields that the ADM gives efficient solutions in time only in ]0; T *[, where T * is the critical time of our equation.


A Minimal Poset Resolution of Stable Ideals

Abstract: We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal.


Degenerate Sklyanin algebras and Generalized Twisted Homogeneous Coordinate rings

Abstract: In this work, we introduce the point parameter ring B, a generalized twisted homogeneous coordinate ring associated to a degenerate version of the three-dimensional Sklyanin algebra. The surprising geometry of these algebras yields an analogue to a result of Artin-Tate-van den Bergh, namely that B is generated in degree one and thus is a factor of the corresponding degenerate Sklyanin algebra.


Simultaneous Continuation of Infinitely Many Sinks Near a Quadratic Homoclinic Tangency

Abstract: We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinksnear the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.


Unramified cohomology of finite groups of Lie type

Abstract: We prove vanishing results for unramified stable cohomology of finite groups of Lie type.


Fast approximation of solutions of SDE's with oblique reflection on an orthant

Abstract: We consider the discrete "fast" penalization scheme for SDE's driven by general semimartingale on orthant $\mathbb{R}_{+}^{d}$ with oblique reflection.


The mean curvature of cylindrically bounded submanifolds

Abstract: We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to a conjecture of Calabion complete minimal hypersurfaces cannot be proper. As another application of our method, we derive a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature.


A Frobenius theorem for Cartan geometries, with applications

Abstract: We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing the configuration of orbits for local Killing fields in a compact real-analytic Cartan geometry, and an open-dense theorem in the smooth case, which says that if there is a dense orbit, then there is an open, dense, locally homogeneus subset. Combining the Frobenius theorem with the embedding theorem of Bader, Frances, and the author gives a representation theorem that relates the fundamental group of the manifold with the automorphism group.


Giambelli, Pieri, and tableau formulas via raising operators

Abstract: We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall-Littlewood functions using Young's raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for Hall-Littlewood functions, theta polynomials, and related Stanley symmetric functions. Finally, we introduce the notion of a skew element w of the hyperoctahedral group and identify the set of reduced words for w with the set of standard k-tableaux on a skew Young diagram.


Combinatorial formulas for Le-coordinates in a totally nonnegative Grassmannian

Abstract: Postnikov constructed a decomposition of a totally nonnegative Grassmannian into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov's description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Pluecker coordinates as a totally positive base for the set of non-vanishing Pluecker coordinates for a given positroid cell.


An almost sure limit theorem for super-Brownian motion

Abstract: We establish an almost sure scaling limit theorem for super-Brownian motion on $\mathbb{R}^d$ associated with the semi-linear equation $u_t = {1/2}\Delta u +\beta u-\alpha u^2$, where $\alpha$ and $\beta$ are positive constants. In this case, the spectral theoretical assumptions that required in Chen et al (2008) are not satisfied. An example is given to show that the main results also hold for some sub-domains in $\mathbb{R}^d$.


Bass Numbers and Semidualizing Complexes

Abstract: Let R be a commutative local noetherian ring. We prove that the existence of a chain of semidualizing R-complexes of length (d+1) yields a degree-d polynomial lower bound for the Bass numbers of R. We also show how information about certain Bass numbers of R provide restrictions on the lengths of chains of semidualizing R-complexes. To make this article somewhat self-contained, we also include a survey of some of the basic properties of semidualizing modules, semidualizing complexes and derived categories.


Cluster-tilted algebras of type $D_n$

Abstract: Let $H$ be a hereditary algebra of Dynkin type $D_n$ over a field $k$ and $\mathscr{C}_H$ be the cluster category of $H$. Assume that $n\geq 5$ and that $T$ and $T'$ are tilting objects in $\mathscr{C}_H$. We prove that the cluster-tilted algebra $\Gamma=\mathrm{End}_{\mathscr{C}_H}(T)^{\rm op}$ is isomorphic to $\Gamma'=\mathrm{End}_{\mathscr{C}_H}(T')^{\rm op}$ if and only if $T=\tau^iT'$ or $T=\sigma\tau^jT'$ for some integers $i$ and $j$, where $\tau$ is the Auslander-Reiten translation and $\sigma$ is the automorphism of $\mathscr{C}_H$ defined in section 4.


Monodromy at infinity of $A$-hypergeometric functions and toric compactifications

Abstract: We study $A$-hypergeometric functions introduced by Gelfand-Kapranov-Zelevinsky and prove a formula for the eigenvalues of their monodromy automorphisms defined by the analytic continuaions along large loops contained in complex lines parallel to the coordinate axes. A method of toric compactifications will be used to prove our main theorem.


Global dimensions of endomorphism algebras for generator-cogenerators over $m$-replicated algebras

Abstract: Let $A$ be a hereditary artin algebra and $A^{(m)}$ be the $m$-replicated algebra of $A$. We investigate the possibilities for the global dimensions of the endomorphism algebras of generator-cogenerators over $A^{(m)}$.


Black Boxes

Abstract: We shall deal comprehensively with Black Boxes, the intention being that provably in ZFC we have a sequence of guesses of extra structure on small subsets, where the guesses are pairwise almost disjoint; by this we mean they have quite little interaction, and are far apart but together are dense. We first deal with the simplest case, where the existence comes from winning a game by just writing down the opponent's moves. We show how it helps when instead of orders we have trees with boundedly many levels, having freedom in the last. After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them.


Askey--Wilson polynomials, quadratic harnesses and martingales

Abstract: We use orthogonality measures of Askey--Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey--Wilson polynomials are orthogonal martingale polynomials for these processes.


Kazhdan's Property T for Discrete Quantum Groups

Abstract: We give a simple definition of property T for discrete quantum groups. We prove the basic expected properties: discrete quantum groups with property T are finitely generated and unimodular. Moreover we show that, for "I.C.C." discrete quantum groups, property T is equivalent to Connes' property T for the dual von Neumann algebra. This allows us to give the first example of a property T discrete quantum group which is not a group using the twisting construction.


Periodicities of T-systems and Y-systems

Abstract: The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).


Group actions on geodesic Ptolemy spaces

Abstract: In this paper we study geodesic Ptolemy metric spaces $X$ which allow proper and cocompact isometric actions of crystallographic or, more generally, virtual polycyclic groups. We show that $X$ is equivariantly rough isometric to a Euclidean space.


Classification de modules aux diff\'erences filtr\'es isogradu\'es

Abstract: The local analytic classification of irregular linear q-difference equations (Ramis-Sauloy-Zhang) involves the classfication of filtered q-difference modules with a prescribed associated graded module. We prove in a more general setting the existence for this problem of a moduli scheme which is an affine space. ----- La classification analytique locale des equations aux q-differences irregulieres se ramene a la classification de modules aux q-differences filtres a gradue fixe. Nous degageons ici des hypotheses generales qui assurent l'existence d'un schema de modules pour ce probleme, qui soit de plus un espace affine.


A generic multiplication in quantised Schur algebras

Abstract: We define a generic multiplication in quantised Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantised Schur algebras, defined by A. A. Beilinson, G. Lusztig and R. MacPherson, a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied by M. Reineke. We also prove that the subalgebra of the new algebra gives a geometric realisation of a positive part of 0-Schur algebras. Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras.


Natural Boundary Conditions in the Calculus of Variations

Abstract: We prove necessary optimality conditions for problems of the calculus of variations on time scales with a Lagrangian depending on the free end-point.


Notes on Cohomologies of Ternary Algebras of Associative Type

Abstract: The aim of this paper is to investigate the cohomologies for ternary algebras of associative type. We study in particular the cases of partially associative ternary algebras and weak totally associative ternary algebras. Also, we consider the Takhtajan's construction, which was used to construct a cohomology of ternary Nambu-Lie algebras using Chevalley-Eilenberg cohomology of Lie algebras, and discuss it in the case of ternary algebras of associative type. One of the main results of this paper states that a deformation cohomology does not exist for partially associative ternary algebras which implies that their operad is not a Koszul operad.


Zeros of the hypergeometric polynomial F(-n,b;c;z)

Abstract: Our interest lies in describing the zero behaviour of Gauss hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are arbitrary parameters. In general, this problem has not been solved and even when $b$ and $c$ are both real, the only cases that have been fully analyzed impose additional restrictions on $b$ and $c$. We review recent results that have been proved for the zeros of several classes of hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are real. We show that the number of real zeros of $F(-n,b; c; z)$ for arbitrary real values of the parameters $b$ and $c$, as well as the intervals in which these zeros (if any) lie, can be deduced from corresponding results for Jacobi polynomials.


Free joinings of C*-dynamical systems

Abstract: Joinings of C*-dynamical systems are defined in terms of free products of C*-algebras, as an analogue of joinings of classical dynamical systems. We then consider disjointness in this context, in particular for ergodic versus identity systems. Lastly we show how multi-time correlation functions appearing in quantum statistical mechanics naturally fit into this joining framework.


Holomorphic Extension of Eigenfunctions

Abstract: Let X be a Riemannian symmetric space of non-compact type. We prove a theorem of holomorphic extension for eigenfunctions of the Laplace-Beltrami operator on X, by techniques from the theory of partial differential equations.


Convexity of the zeros of some orthogonal polynomials and related functions

Abstract: We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.


Note on the X_(1)-Jacobi orthogonal polynomials

Abstract: This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Jacobi polynomials which are orthogonal in a weighted Hilbert function space on the the interval (-1,+1) of the real line. These polynomials are generated by a second-order ordinary linear differential equation with a spectral parameter. Some additional information on the Sturm-Liouville form of this equation is given in this note, together with details of the singular differential operators generated in the weighted Hilbert function space. In particular, structured boundary conditions are given to determine the special self-adjoint operator, whose discrete spectrum and associated eigenvectors yield the X_(1)-Jacobi polynomials.


Zeros of linear combinations of Laguerre polynomials from different sequences

Abstract: We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely $R_n=L_n^{\alpha}+aL_{n}^{\alpha'}$ and $S_n=L_n^{\alpha}+bL_{n-1}^{\alpha'}$. Proofs and numerical counterexamples are given in situations where the zeros of $R_n$, and $S_n$, respectively, interlace (or do not in general) with the zeros of $L_k^{\alpha}$, $L_k^{\alpha'}$, $k=n$ or $n-1$. The results we prove hold for continuous, as well as integral, shifts of the parameter $\alpha$.


Nonhomogeneous parking functions and noncrossing partitions

Abstract: For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haiman's notion of a parking function symmetric function.


A Limit Relation for Dunkl-Bessel Functions of Type A and B

Abstract: We prove a limit relation for the Dunkl-Bessel function of type $B_N$ with multiplicity parameters $k_1$ on the roots $\pm e_i$ and $k_2$ on $\pm e_i\pm e_j$ where $k_1$ tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type $A_{N-1}$ with multiplicity $k_2$. For certain values of $k_2$ an improved estimate is obtained from a corresponding limit relation for Bessel functions on matrix cones.


Relations between slices and quotients of the algebraic cobordism spectrum

Abstract: We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.


About Very Perfect Numbers

Abstract: In this short paper we prove that the square of an odd prime number cannot be a very perfect number.


Unipotent elements in small characteristic, III

Abstract: We give a uniform description of the decomposition of the unipotent variety of a classical group in arbitrary characteristic into pieces (considered in a non-uniform way in the earlier parts of this paper).


Eulerian quasisymmetric functions

Abstract: We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising $q$-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This $q$-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain $q$-analogs, $(q,p)$-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Bj\"orner and Welker.


Cyclic Homologies of Crossed Modules of Algebras

Abstract: The Hochschild and (cotriple) cyclic homologies of crossed modules of (not-necessarily-unital) associative algebras are investigated. Wodzicki's excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cyclic homology exact sequence.


On the Jacobson element and generators of the Lie algebra $\mathfrak{grt}$ in nonzero characteristic

Abstract: We state a conjecture (due to M. Duflo) analogous to the Kashiwara--Vergne conjecture in the case of a characteristic $p>2$, where the role of the Campbell--Hausdorff series is played by the Jacobson element. We prove a simpler version of this conjecture using Vergne's explicit rational solution of the Kashiwara--Vergne problem. Our result is related to the structure of the Grothendieck--Teichm\"{u}ller Lie algebra $\mathfrak{grt}$ in characteristic $p$: we conjecture existence of a generator of $\mathfrak{grt}$ in degree $p-1$, and we provide this generator for $p=3$ and $p=5$.


Separating Solution of a Quadratic Recurrent Equation

Abstract: In this paper we consider the recurrent equation $$\Lambda_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda_{q}\Lambda_{p+1-q}$$ for $p\ge 1$ with $f\in C[0,1]$ and $\Lambda_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence $\Lambda_p$ with $\Lambda_1=y^{(0)}$ tends to a finite positive limit as $p\to \infty$.


Nef divisors on $\bar{M}_{0,n}$ from GIT

Abstract: We introduce and study the GIT CONE of $\bar{M}_{0,n}$, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients $(\mathbb P^1)^n//SL(2)$. We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone. As one application, we prove unconditionally that the log canonical models of $\bar{M}_{0,n}$ with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson arXiv:0709.4037. (Cf. also a different proof by Fedorchuk and Smyth arXiv:0810.1677)


Poset homology of Rees products, and $q$-Eulerian polynomials

Abstract: The notion of Rees product of posets was introduced by Bj\"orner and Welker, where they study connections between poset topology and commutative algebra. Bj\"orner and Welker conjectured and Jonsson proved that the dimension of the top homology of the Rees product of the truncated Boolean algebra $B_n \setminus \{0\}$ and the $n$-chain $C_n$ is equal to the number of derangements in the symmetric group $\mathfrak S_n$. Here we prove a refinement of this result, which involves the Eulerian numbers, and a $q$-analog of both the refinement and the original conjecture, which comes from replacing the Boolean algebra by the lattice of subspaces of the $n$-dimensional vector space over the $q$ element field, and involves the $(\maj,\exc)$-$q$-Eulerian polynomials studied in previous papers of the authors. Equivariant versions of the refinement and the original conjecture are also proved, as are type BC versions (in the sense of Coxeter groups) of the original conjecture and its $q$-analog.


Cyclotomic Units and Class Groups in Z_p extensions of real abelian fields

Abstract: For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.


How long does it take to catch a wild kangaroo?

Abstract: We develop probabilistic tools for upper and lower bounding the expected time until two independent random walks on $\ZZ$ intersect each other. This leads to the first sharp analysis of a non-trivial Birthday attack, proving that Pollard's Kangaroo method solves the discrete logarithm problem $g^x=h$ on a cyclic group in expected time $(2+o(1))\sqrt{b-a}$ for an average $x\in_{uar}[a,b]$. Our methods also resolve a conjecture of Pollard's, by showing that the same bound holds when step sizes are generalized from powers of 2 to powers of any fixed $n$.


Increasing and Decreasing Sequences of Length Two in 01-Fillings of Moon Polyominoes

Abstract: We put recent results on the symmetry of the joint distribution of the numbers of crossings and nestings of two edges over matchings, set partitions and linked partitions, in the larger context of the enumeration of increasing and decreasing chains of length 2 in fillings of moon polyominoes.


Phi-entropy inequalities for diffusion semigroups

Abstract: We obtain and study new $\Phi$-entropy inequalities for diffusion semigroups, with Poincar\'e or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker-Plank type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The $\Gamma_2$ criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms.


Affine T-varieties of complexity one and locally nilpotent derivations

Abstract: Let X=spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus T of dimension n. Let also D be a homogeneous locally nilpotent derivation on the normal affine Z^n-graded domain A, so that D generates a k_+-action on X that is normalized by the T-action. We provide a complete classification of pairs (X,D) in two cases: for toric varieties (n=\dim X) and in the case where n=\dim X-1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we compute the homogeneous Makar-Limanov invariant of such varieties. In particular we exhibit a family of non-rational varieties with trivial Makar-Limanov invariant.


Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models

Abstract: We study the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients. Our motivation comes from the comparison between experimental tumor growth curves in mice endowed with intact or disrupted circadian clocks, known to exert their influence on the cell division cycle. We compare the growth rate of the model controlled by a time-periodic control on its coefficients with the growth rate of stationary models of the same nature, but with averaged coefficients. We firstly derive a delay differential equation which allows us to prove several inequalities and equalities on the growth rates. We also discuss about the necessity to take into account the structure of the cell division cycle for chronotherapy modeling. Numerical simulations illustrate the results.


Poisson boundary of the discrete quantum group A_u(F)^

Abstract: We identify the Poisson boundary of the dual of the universal compact quantum group A_u(F) with a measurable field of ITPFI factors.


The Vanishing Approach for the Average Continuous Control of Piecewise Deterministic Markov Processes

Abstract: The main goal of this paper is to derive sufficient conditions for the existence of an optimal control strategy for the long run average continuous control problem of piecewise deterministic Markov processes (PDMP's) taking values in a general Borel space and with compact action space depending on the state variable. In order to do that we apply the so-called vanishing discount approach to obtain a solution to an average cost optimality inequality associated to the long run average cost problem. Our main assumptions are written in terms of some integro-differential inequalities related to the so-called expected growth condition, and geometric convergence of the post-jump location kernel associated to the PDMP.


Branching of Representations to Symmetric Subgroups

Abstract: Let $\gg$ be the Lie algebra of a compact Lie group and let $\theta$ be any automorphism of $\gg$. Let $\gk$ denote the fixed point subalgebra $\gg^\theta$. In this paper we present LiE programs that, for any finite dimensional complex representation $\pi$ of $\gg$, give the explicit branching $\pi|_\gk$ of $\pi$ on $\gk$. Cases of special interest include the cases where $\theta$ has order 2 (corresponding to compact riemannian symmetric spaces $G/K$), where $\theta$ has order 3 (corresponding to compact nearly--kaehler homogeneous spaces $G/K$), where $\theta$ has order 5 (which include the fascinating 5--symmetric space $E_8/A_4A_4$), and the cases where $\gk$ is the centralizer of a toral subalgebra of $\gg$.


Duality, a-invariants and canonical modules of rings arising from linear optimization problems

Abstract: The aim of this paper is to study integer rounding properties of various systems of linear inequalities to gain insight about the algebraic properties of Rees algebras of monomial ideals and monomial subrings. We study the normality and Gorenstein property--as well as the canonical module and the a-invariant--of Rees algebras and subrings arising from systems with the integer rounding property. We relate the algebraic properties of Rees algebras and monomial subrings with integer rounding properties and present a duality theorem.


Degree bounds for type-A weight rings and Gelfand--Tsetlin semigroups

Abstract: A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in $\C^n$ modulo a twisted action of the maximal torus in $\SL(n,\C)$. We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worst $O(n^2)$. On the other hand, we show that the associated semigroup of Gelfand--Tsetlin patterns can have an essential generator of degree exponential in $n$.


A proof of Lens Rigidity in the category of Analytic Metrics

Abstract: Consider a compact Riemannian manifold with boundary. Assume all maximally extended geodesics intersect the boundary at both ends. Then to each maximal geodesic segment one can form a triple consisting of the initial and final vectors of the segment and the length of the segment. The collection of all such triples comprises the lens data. In this paper, it is shown that in the category of analytic Riemannian manifolds, the lens data uniquely determine the metric up to isometry. There are no convexity assumptions on the boundary, and conjugate points are allowed, but with some restriction.


Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations

Abstract: In this paper, we first study the existence-uniqueness and large deviation estimate of solutions for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then, we apply them to a large class of semilinear stochastic partial differential equations (SPDE) driven by Brownian motions as well as by fractional Brownian motions, and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of Euclidean space, second order SPDEs on complete Riemannian manifolds, as well as stochastic Navier-Stokes equations are investigated.


Expansions of the real field by open sets: definability versus interpretability

Abstract: An open set U of the real numbers R is produced such that the expansion (R,+,x,U) of the real field by U defines a Borel isomorph of (R,+,x,N) but does not define N. It follows that (R,+,x,U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (R,+,x). In particular, there is a Cantor subset K of R such that for every exponentially bounded o-minimal expansion M of (R,+,x), every subset of R definable in (M,K) either has interior or is Hausdorff null.


Efficiency and influence function of estimators for ARCH models

Abstract: This paper proposes a closed-form optimal estimator based on the theory of estimating functions for a class of linear ARCH models. The estimating function (EF) estimator has the advantage over the widely used maximum likelihood (ML) and quasi-maximum likelihood (QML) estimators that (i) it can be easily implemented, (ii) it does not depend on a distributional assumption for the innovation, and (iii) it does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance equation. In the case of normality, the asymptotic distribution of the ML and QML estimators naturally turn out to be identical and, hence, coincides with ours. Moreover, a robustness property of the EF estimator is derived by means of influence function. Simulation results show that the efficiency benefits of our estimator relative to the ML and QML estimators are substantial for some ARCH innovation distributions.


Testing the equality of error distributions from k independent GARCH models

Abstract: In this paper we study the problem of testing the null hypothesis that errors from k independent parametrically specified generalized autoregressive conditional heteroskedasticity (GARCH) models have the same distribution versus a general alternative. First we establish the asymptotic validity of a class of linear test statistics derived from the k residual-based empirical distribution functions. A distinctive feature is that the asymptotic distribution of the test statistics involves terms depending on the distributions of errors and the parameters of the models, and weight functions providing the flexibility to choose scores for investigating power performance. A Monte Carlo study assesses the asymptotic performance in terms of empirical size and power of the three-sample test based on the Wilcoxon and Van der Waerden score generating functions in finite samples. The results demonstrate that the two proposed tests have overall reasonable size and their power is particularly high when the assumption of Gaussian errors is violated. As an illustrative example, the tests are applied to daily individual stock returns of the New York Stock Exchange data.


Total positivity in loop groups I: whirls and curls

Abstract: This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GL_n(\R[t,t^{-1}]), and for the formal loop group GL_n(\R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeroes of the Edrei-Thoma theorem classifying totally positive functions (corresponding to our case n=1). We give a solution to the ``factorization problem'' using limits of ratios of minors. This is in a similar spirit to the Berenstein-Fomin-Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi-Yamada's study of discrete Painlev\'{e} dynamical systems and Berenstein-Kazhdan's study of geometric crystals.


Withdraw a paper entitled "On the growth rate of solutions for 2D incompressible Euler equations"

Abstract: This paper has been withdrawn by the author due to a crucial definition error of Triebel space.


Algebraic Entropy and the Action of Mapping Class Groups on Character Varieties

Abstract: We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov mapping class. Our calculations are compatible with all known calculations of the topological entropy of this action.


The length of unknotting tunnels

Abstract: We show there exist tunnel number one hyperbolic 3-manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams.


Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes (extended abstract)

Abstract: We show that the Kronecker coefficients (the Clebsch-Gordan coefficients of the symmetric group) indexed by two two-row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients.


Table of minimum ranks of graphs of order at most 7 and selected optimal matrices

Abstract: The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. We have also established several additional strategies for computation of minimum rank. These techniques have been used to determine the minimum ranks of all graphs of order 7. This paper contains a list of minimum ranks for all graphs of order at most 7. We also present selected optimal matrices.


Maximums and minimums of overall survival functions with fixed marginal distributions and transmission of technology

Abstract: Title: An unlikely result Authors: T.M. Other Comments: This paper has been withdrawn Abstract: This paper has been withdrawn by the author due to the fact that some of the results turned out to be known.


Two-Parameter Heavy-Traffic Limits for Infinite-Server Queues

Abstract: In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables $Q^e(t,y)$ and $Q^r(t,y)$ representing the number of customers in the system at time $t$ that have elapsed service times less than or equal to time $y$, or residual service times strictly greater than $y$. We also consider $W^r(t,y)$ representing the total amount of work in service time remaining to be done at time $t+y$ for customers in the system at time $t$. The two-parameter stochastic-process limits in the space $D([0,\infty),D)$ of $D$-valued functions in $D$ draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f.


Distribution and asymptotics under beta random scaling

Abstract: Let X,Y,B be three independent random variables such that $X$ has the same distribution function as Y B. Assume that B is a Beta random variable with positive parameters a,b and Y has distribution function H. Pakes and Navarro (2007) show under some mild conditions that the distribution function H_{a,b} of X determines H. Based on that result we derive in this paper a recursive formula for calculation of H, if H_{a,b} is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H_{a,b}, respectively, and the conditional limiting distribution of bivariate elliptical distributions.


An operator extension of Bohr's inequality

Abstract: We establish an operator extension of the following generalization of Bohr's inequality, due to M.P. Vasi\'c and D.J. Ke\v{c}ki\'{c}: $$|\sum_{i=1}^n z_i|^r \leq (\sum_{i=1}^n \alpha_i^{1/(1-r)})^{r-1}\sum_{i=1}^n \alpha_i|z_i|^r \quad (r>1, z_i \in{\mathbb C}, \alpha_i>0, 1 \leq i \leq n) .$$ We also present some norm inequalities related to our noncommutative generalization of Bohr's inequality.


A characterization of Hilbert $C^*$-modules over finite dimensional $C^*$-algebras

Abstract: We show that the unit ball of a full Hilbert $C^*$-module is sequentially compact in a certain weak topology if and only if the underlying $C^*$-algebra is finite dimensional. This provides an answer to the question posed in J. Chmieli\'nski et al [Perturbation of the Wigner equation in inner product $C^*$-modules, J. Math. Phys. 49 (2008), no. 3, 033519; arXiv:0801.2726].


Boxicity and Cubicity of Asteroidal Triple free graphs

Abstract: An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denoted as $\cub(G)$, is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-cubes. Let $S(m)$ denote a star graph on $m+1$ nodes. We define {\it claw number} of a graph $G$ as the largest positive integer $k$ such that $S(k)$ is an induced subgraph of $G$ and denote it as $\claw$. Let $G$ be an AT-free graph with chromatic number $\chi(G)$ and claw number $\claw$. In this paper we will show that $\boxi(G) \leq \chi(G)$ and this bound is tight. We also show that $\cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2)$ $\leq$ $\chi(G)(\ceil{\log_2 \claw} +2)$. If $G$ is an AT-free graph having girth at least 5 then $\boxi(G) \leq 2$ and therefore $\cub(G) \leq 2\ceil{\log_2 \claw} +4$.


Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization

Abstract: Let $T$ be an underlying space with a non-atomic measure $\sigma$ on it (e.g. $T=\mathbb R^d$ and $\sigma$ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of $T$, with freely independent values. Such a process (field), $\omega=\omega(t)$, $t\in T$, is given a rigorous meaning through smearing out with test functions on $T$, with $\int_T \sigma(dt)f(t)\omega(t)$ being a (bounded) linear operator in a full Fock space. We define a set $\mathbf{CP}$ of all continuous polynomials of $\omega$, and then define a con-commutative $L^2$-space $L^2(\tau)$ by taking the closure of $\mathbf{CP}$ in the norm $\|P\|_{L^2(\tau)}:=\|P\Omega\|$, where $\Omega$ is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between $L^2(\tau)$ and a (Fock-space-type) Hilbert space $\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,\gamma_n)$, with explicitly given measures $\gamma_n$. We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set $\mathbf {CP}$ invariant. (Note that, in the general case, the projection of a continuous monomial of oder $n$ onto the $n$-th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions $\lambda$ and $\eta\ge0$ on $T$, such that, in the $\mathbb F$ space, $\omega$ has representation $\omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag\di^2_t$, where $\di_t^\dag$ and $\di_t$ are the usual creation and annihilation operators at point $t$.


Lowering and raising operators for the free Meixner class of orthogonal polynomials

Abstract: We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line.


Ergodic properties of linked-twist maps

Abstract: We study a class of homeomorphisms of surfaces collectively known as linked-twist maps. We introduce an abstract definition which enables us to give a precise characterisation of a property observed by other authors, namely that such maps fall into one of two classes termed co- and counter-twisting. We single out three specific linked-twist maps, one each on the two-torus, in the plane and on the two-sphere and for each prove a theorem concerning its ergodic properties with respect to the invariant Lebesgue measure. For the map on the torus we prove that there is an invariant, zero-measure Cantor set on which the dynamics are topologically conjugate to a full shift on the space of symbol sequences. Such features are commonly known as topological horseshoes. For the map in the plane we prove that there is a set of full measure on which the dynamics are measure-theoretically isomorphic to a full shift on the space of symbol sequences. This is commonly known as the Bernoulli property and verifies, under certain conditions, a conjecture of Wojtkowski's. We introduce the map on the sphere and prove that it too has the Bernoulli property. We conclude with some conjectures, drawn from our experience, concerning how one might extend the results we have for specific linked-twist maps to the abstract linked-twist maps we have defined.


Gantmakher-Krein theorem for 2-totally nonnegative operators in ideal spaces

Abstract: The tensor and exterior squares of a completely continuous non-negative linear operator $A$ acting in the ideal space $X(\Omega)$ are studied. The theorem representing the point spectrum (except, probably, zero) of the tensor square $A \otimes A$ in the terms of the spectrum of the initial operator $A$ is proved. The existence of the second (according to the module) positive eigenvalue $\lambda_2$, or a pair of complex adjoint eigenvalues of a completely continuous non-negative operator $A$ is proved under the additional condition, that its exterior square $A\wedge A$ is also nonnegative.


On local compactness in quasilinear elliptic problems

Abstract: One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work \cite{BN}, Br\'ezis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relatively compact.


Extremal functions for the anisotropic Sobolev inequalities

Abstract: The existence of multiple nonnegative solutions to the anisotropic critical problem - \sum_{i=1}^{N} \frac{\partial}{\partial x_i} (| \frac{\partial u}{\partial x_i} |^{p_i-2} \frac{\partial u}{\partial x_i}) = |u|^{p^*-2} u {in} \mathbb{R}^N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions $\calS$ is included in $L^\infty(\R^N)$ and is located outside of a ball of radius $\tau >0$ in $L^{p^*}(\R^N)$.


The Goldbach conjecture resulting from global-local cuspidal representations and deformations of Galois representations

Abstract: In the basic general frame of the Langlands global program, a local p-adic elliptic semimodule corresponding to a local (left) cuspidal form is constructed from its global equivalent covered by p-th roots. In the same context, global and local bilinear deformations of Galois representations inducing the invariance of their respective residue fields are introduced as well as global and local bilinear quantum deformations leaving invariant the orders of the inertia subgroups. More particularly, the inverse quantum deformation of a closed curve responsible for its splitting directly leads to the Goldbach conjecture.


Th\'eor\`emes d'\'equidistribution pour les syst\`emes dynamiques d'origine arithm\'etique

Abstract: This survey article is about algebraic dynamics. It is mainly concerned by the arithmetic equidistribution theorems featured by dynamical systems. The contents are: - heights - algebraic dynamics, conjectures - equidistribution theorem on the projective line (after Bilu and Baker). It was written for the sesion 2006 of \'Etats de la recherche, Syst\`emes dynamiques polynomiaux.


Lectures on height zeta functions: At the confluence of algebraic geometry, algebraic number theory, and analysis

Abstract: This is a survey on the theory of height zeta functions, written on the occasion of a French-Japanese winter school, held in Miura (Kanagawa, Japan) in Jan. 2008. It does not presuppose much knowledge in algebraic geometry. The last chapter of the survey explains recent results obtained in collaboration with Yuri Tschinkel concerning asymptotics of volumes of height balls in analytic geometry over local fields, or in adelic spaces.


Asymptotic Behaviour and Artinian Property of Graded Local Cohomology Modules

Abstract: In this paper, considering the difference between the finiteness dimension and cohomological dimension for a finitely generated module, we investigate the asymptotic behavior of grades of components of graded local cohomology modules with respect to irrelevant ideal; as long as we study some artinian and tameness property of such modules.


Symmetric identities for Euler polynomials

Abstract: In this paper we establish two symmetric identities on sums of products of Euler polynomials.


The two uniform infinite quadrangulations of the plane have the same law

Abstract: We prove that the uniform infinite random quadrangulations defined respectively by Chassaing-Durhuus and Krikun have the same distribution.


Quantum Giambelli formulas for isotropic Grassmannians

Abstract: Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the cohomological Giambelli formulas of arXiv:0811.2781.


Regularity results for stable-like operators

Abstract: For $\alpha\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d} [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}}$$ and for $\alpha\in (0,1)$ we consider the same operator but where the $\nabla f$ term is omitted. We prove, under appropriate conditions on $A(x,h)$, that the solution $u$ to $L u=f$ will be in $C^{\alpha+\beta}$ if $f\in C^\beta$.


Controlling Multiparticle System on the Line, II - Periodic case

Abstract: As in arXiv: math. 0809.2365 we consider classical system of interacting particles $\mathcal{P}_1, ..., \mathcal{P}_n$ on the line with only neighboring particles involved in interaction. On the contrast to arXiv: math. 0809.2365 now {\it periodic boundary conditions} are imposed onto the system, i.e. $\mathcal{P}_1$ and $\mathcal{P}_n$ are considered neighboring. Periodic Toda lattice would be a typical example. We study possibility to control periodic multiparticle systems by means of forces applied to just few of its particles; mainly we study system controlled by single force. The free dynamics of multiparticle systems in periodic and nonperiodic case differ substantially. We see that also the controlled periodic multiparticle system does not mimic its non-periodic counterpart. Main result established is global controllability by means of single controlling force of the multiparticle system with ageneric potential of interaction. We study the nongeneric potentials for which controllability and accessibility properties may lack. Results are formulated and proven in Sections~2,3.


A numerical study of the self-similar solutions of the Schroedinger Map

Abstract: We present a numerical study of the self-similar solutions of the Localized Induction Approximation of a vortex filament. These self-similar solutions, which constitute a one-parameter family, develop a singularity at finite time. We study a number of boundary conditions that allow us reproduce the mechanism of singularity formation. Some related questions are also considered.


Sharp error terms for return time statistics under mixing conditions

Abstract: We describe the statistics of repetition times of a string of symbols in a stochastic process. Denote by T(A) the time elapsed until the process spells the finite string A and by S(A) the number of consecutive repetitions of A. We prove that, if the length of the string grows unbondedly, (1) the distribution of T(A), when the process starts with A, is well aproximated by a certain mixture of the point measure at the origin and an exponential law, and (2) S(A) is approximately geometrically distributed. We provide sharp error terms for each of these approximations. The errors we obtain are point-wise and allow to get also approximations for all the moments of T(A) and S(A). To obtain (1) we assume that the process is phi-mixing while to obtain (2) we assume the convergence of certain contidional probabilities.


Mistuning-based Control Design to Improve Closed-Loop Stability of Vehicular Platoons

Abstract: We consider a decentralized bidirectional control of a platoon of N identical vehicles moving in a straight line. The control objective is for each vehicle to maintain a constant velocity and inter-vehicular separation using only the local information from itself and its two nearest neighbors. Each vehicle is modeled as a double integrator. To aid the analysis, we use continuous approximation to derive a partial differential equation (PDE) approximation of the discrete platoon dynamics. The PDE model is used to explain the progressive loss of closed-loop stability with increasing number of vehicles, and to devise ways to combat this loss of stability. If every vehicle uses the same controller, we show that the least stable closed-loop eigenvalue approaches zero as O(1/N^2) in the limit of a large number (N) of vehicles. We then show how to ameliorate this loss of stability by small amounts of "mistuning", i.e., changing the controller gains from their nominal values. We prove that with arbitrary small amounts of mistuning, the asymptotic behavior of the least stable closed loop eigenvalue can be improved to O(1/N) All the conclusions drawn from analysis of the PDE model are corroborated via numerical calculations of the state-space platoon model.


Lipschitz Characterisation of Polytopal Hilbert Geometries

Abstract: We prove that the Hilbert Geometry of a convex set is bi-lipschitz equivalent to a normed vector space if and only if the convex is a polytope.


Density of isoperimetric spectra

Abstract: We show that the set of k-dimensional isoperimetric exponents of finitely presented groups is dense in the interval [1, \infty) for k > 1. Hence there is no higher-dimensional analogue of Gromov's gap (1,2) in the isoperimetric spectrum.


On the holomorphic point of view in the theory of quantum knot invariants

Abstract: In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of flat SU(2)-connections on the tours. Two results are emphasized: the reconstruction from Weyl quantization of the restriction to the torus of the modular functor, and a description of a basis of the space of quantum observables on the torus in terms of colored curves, which answers a question related to quantum computing.


Algebraic Methods in Discrete Analogs of the Kakeya Problem

Abstract: We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, there are at most $O(N^{{3 \over 2}})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines in ${\Bbb R}^3$ so that no $N$ lines lie in the same plane and so that each line intersects a set $P$ of points in at least $N$ points then the cardinality of the set of points is $\Omega(N^3)$. Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.


Graph Minors and Minimum Degree

Abstract: Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\widehat{\mathcal{D}}_k$. This paper discusses $\widehat{\mathcal{D}}_k$ and related topics. We obtain four main results: We prove that every $(k+1)$-regular graph with less than ${4/3}(k+2)$ vertices is in $\widehat{\mathcal{D}}_k$, and this bound is best possible. We characterise the graphs in $\widehat{\mathcal{D}}_{k+1}$ that can be obtained from a graph in $\widehat{\mathcal{D}}_k$ by adding one new vertex. For $k\leq 3$ every graph in $\widehat{\mathcal{D}}_k$ is $(k+1)$-connected, but for large $k$, we exhibit graphs in $\widehat{\mathcal{D}}_k$ with connectivity 1. In fact, we construct graphs in $\mathcal{D}_k$ with arbitrary block structure. We characterise the complete multipartite graphs in $\widehat{\mathcal{D}}_k$, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth.


Open Orbits and Augmentations of Dynkin Diagrams

Abstract: Given any representation V of a complex linear reductive Lie group G_0, we show that a larger semi-simple Lie group G with g=g_0 + V + V* + ..., exists precisely when V has a finite number of G_0-orbits. In particular, V admits an open G_0-orbit. Furthermore, this corresponds to an augmentation of the Dynkin diagram of g_0. The representation theory of g should be useful in describing the geometry of manifolds with stable forms as studied by Hitchin.


Hilbert geometry of polytopes

Abstract: It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension.


Both necessary and sufficient conditions for Bayesian exponential consistency

Abstract: The last decade has seen a remarkable development in the theory of asymptotics of Bayesian nonparametric procedures. Exponential consistency has played an important role in this area. It is known that the condition of $f_0$ being in the Kullback-Leibler support of the prior cannot ensure exponential consistency of posteriors. Many authors have obtained additional sufficient conditions for exponential consistency of posteriors, see, for instance, Schwartz (1965), Barron, Schervish and Wasserman (1999), Ghosal, Ghosh and Ramamoorthi (1999), Walker (2004), Xing and Ranneby (2008). However, given the Kullback-Leibler support condition, less is known about both necessary and sufficient conditions. In this paper we give one type of both necessary and sufficient conditions. As a consequence we derive a simple sufficient condition on Bayesian exponential consistency, which is weaker than the previous sufficient conditions.


Embedding solenoids in foliations

Abstract: In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H^1(L;R)= 0 and if the foliation is a product foliation in some saturated open neighbourhood U of L, then there exists a foliation F' on M which is C^1-close to F, and F' has an uncountable set of solenoidal minimal sets contained in U that are pair wise non-homeomorphic. If H^1(L;R) is not 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighbourhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg.


Invariant Measures on Stationary Bratteli Diagrams

Abstract: We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we explicitly describe all ergodic probability measures invariant with respect to the tail equivalence relation (or the Vershik map). These measures are completely described by the incidence matrix of the diagram. Since such diagrams correspond to substitution dynamical systems, this description gives an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.


Chebyshev Knots

Abstract: A Chebyshev knot is a knot which admits a parametrization of the form $ x(t)=T_a(t); \ y(t)=T_b(t) ; \ z(t)= T_c(t + \phi), $ where $a,b,c$ are pairwise coprime, $T_n(t)$ is the Chebyshev polynomial of degree $n,$ and $\phi \in \RR .$ Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with $\phi = 0.$ We also show that every knot is a Chebyshev knot.


Highest weight categories arising from Khovanov's diagram algebra III: category O

Abstract: We prove that integral blocks of parabolic category O associated to the subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of 2-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type A.


Determination of the biquaternion divisors of zero, including the idempotents and nilpotents

Abstract: The biquaternion (complexified quaternion) algebra contains idempotents (elements whose square remains unchanged) and nilpotents (elements whose square vanishes). It also contains divisors of zero (elements with vanishing norm). The idempotents and nilpotents are subsets of the divisors of zero. These facts have been reported in the literature, but remain obscure through not being gathered together using modern notation and terminology. Explicit formulae for finding all the idempotents, nilpotents and divisors of zero appear not to be available in the literature, and we rectify this with the present paper. Using several different representations for biquaternions, we present simple formulae for the idempotents, nilpotents and divisors of zero, and we show that the complex components of a biquaternion divisor of zero must have a sum of squares that vanishes, and that this condition is equivalent to two conditions on the inner product of the real and imaginary parts of the biquaternion, and the equality of the norms of the real and imaginary parts. We give numerical examples of nilpotents, idempotents and other divisors of zero. Finally, we conclude with a statement about the composition of the set of biquaternion divisors of zero, and its subsets, the idempotents and the nilpotents.


Normal forms for almost non-integrable CR structures

Abstract: We propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-CR embeddings that are in some sense "maximally CR". One of the main differences with the classical integrable case is the presence of the non-integrability tensor at the same order as the Levi form, making impossible a good quadric approximation - a key tool in the Chern-Moser theory. Partial normal forms are obtained for general almost CR structures of any CR codimension, in particular, for almost-complex structures. Applications are given to the equivalence problem and the Lie group structure of the group of all CR-diffeomorphisms.


A note on the separability index

Abstract: In discriminating between objects from different classes, the more separable these classes are the less computationally expensive and complex a classifier can be used. One thus seeks a measure that can quickly capture this separability concept between classes whilst having an intuitive interpretation on what it is quantifying. A previously proposed separability measure, the separability index (SI) has been shown to intuitively capture the class separability property very well. This short note highlights the limitations of this measure and proposes a slight variation to it by combining it with another form of separability measure that captures a quantity not covered by the Separability Index.


A proof of the rooted tree alternative conjecture

Abstract: Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new conjectures.


A Unified Approach of Parameter Estimation

Abstract: We introduce a new distance and we use it to parameter estimation purposes. We observe how it operates and we use in its place the usual methods of estimation which we call the methods of the new approach. We realize that we obtain a discretization of the continuous case. Moreover, when it is necessary to consider truncated data nothing is changed in computations.


Algebraic A-hypergeometric Functions

Abstract: We formulate and prove a combinatorial criterion to decide if an A-hypergeometric system of differential equations has a full set of algebraic solutions or not. This criterion generalises the so-called interlacing criterion in the case of hypergeometric functions of one variable.


Katz's middle convolution and Yokoyama's extending operation

Abstract: We give a concrete relation between Katz's middle convolution and Yokoyama's extension and show the equivalence of both algorithms using these operations for the reduction of Fuchsian systems.


A simplicial model for proper homotopy types

Abstract: The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of_controlled sets_, yielding _simplicial controlled sets_, such that one can functorially produce a singular simplicial controlled set CSing(MaxCtl(X)) from a locally compact X. We then argue that this CSing(MaxCtl(X)) captures the (weak)_proper_ homotopy type of X. Moreover, our techniques strictly generalize the classical simplicial situation: e.g., one obtains, in a unified way, singular homology with compact supports and (Borel-Moore) singular homology with locally finite supports, as well as the corresponding cohomologies.


On minimal actions of finite simple groups on homology spheres and Euclidean spaces

Abstract: We consider the following problem: for which classes of finite groups, and in particular finite simple groups, does the minimal dimension of a faithful, smooth action on a homology sphere coincide with the minimal dimension of a faithful, linear action on a sphere? We prove that the two minimal dimensions coincide for the linear fractional groups PSL(2,p) as well as for various classes of alternating and symmetric groups. We prove analogous results also for actions on Euclidean spaces.


Sobolev spaces on multiple cones

Abstract: The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $\RR^n$. The analysis interestingly combines use of Poincar\'e inequalities and of some Hardy type inequalities.


The elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ and its vertex operators

Abstract: We construct a realization of the elliptic quantum algebra $U_{q,p}(\hat{sl_N})$ for any given level $k$ in terms of free boson fields and their twisted partners. It can be considered as the elliptic deformation of the Wakimoto realization of the quantum affine algebra $U_{q}(\hat{sl_N})$. We also construct a family of screening currents, which commute with the currents of $U_{q,p}(\hat{sl_N})$ up to total q-differences. And we give explicit twisted expressions for the type $I$ and the type $II$ vertex operators of $U_{q,p}(\hat{sl_N})$ by twisting the known results of the type $I$ vertex operators of the quantum affine algebra $U_{q}(\hat{sl_N})$ and the new results of the type $II$ vertex operators of $U_{q}(\hat{sl_N})$ we obtained in this paper.


Smoothness in Relative Geometry

Abstract: In \cite{tva}, Bertrand Toen and Michel Vaquie defined a scheme theory for a closed monoidal category $(C,\otimes,1)$. In this article, we define a notion of smoothness in this relative (and not necesarilly additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists practically in changing homological finiteness conditions into homotopical ones using Dold-Kahn correspondance. To do this, we provide the category $sC$ of simplicial objects in a monoidal category and all the categories $sA-mod$, $sA-alg$ ($a\in sComm(C)$) with compatible model structures using the work of Rezk in \cite{r}. We give then a general notions of smoothness in $sComm(C)$. We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and provide some examples of smooth morphisms in $N-alg$, $Comm(Set)$ and Comm(C).