Cohomology rings of almost-direct products of free groups

Abstract: An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg-Mac Lane space.


Resonance of basis-conjugating automorphism groups

Abstract: We determine the structure of the first resonance variety of the cohomology ring of the group of automorphisms of a finitely generated free group which act by conjugation on a given basis.


Schreier rewriting beyond the classical setting

Abstract: Using actions of free monoids and free associative algebras, we establish some Schreier-type formulas involving the ranks of actions and the ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish the generalization of the Schreier formula to the case of subgroups of infinite index. We also study and apply large modules over free associative algebras in the spirit of the paper Olshanskii, A. Yu.; Osin, D.V., Large groups and their periodic quotients, Proc. Amer. Math. Soc., 136 (2008), 753 - 759.


On localization properties of Fourier transforms of hyperfunctions

Abstract: In [Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space $\mathcal U(R^k)$ which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on $R^k$. It was shown that all Gelfand--Shilov spaces $S^{\prime 0}_\alpha(R^k)$ ($\alpha>1$) of analytic functionals are canonically embedded in $\mathcal U(R^k)$. While the usual definition of support of a generalized function is inapplicable to elements of $S^{\prime 0}_\alpha(R^k)$ and $\mathcal U(R^k)$, their localization properties can be consistently described using the concept of {\it carrier cone} introduced by Soloviev [Lett. Math. Phys. 33 (1995) 49-59; Comm. Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier cones of elements of $S^{\prime 0}_\alpha(R^k)$ and $\mathcal U(R^k)$ is studied. It is proved that an analytic functional $u\in S^{\prime 0}_\alpha(R^k)$ is carried by a cone $K\subset R^k$ if and only if its canonical image in $\mathcal U(R^k)$ is carried by $K$.


A Uniform Estimate for Fourier Restriction to Simple Curves

Abstract: We prove a uniform Fourier extension-restriction estimate for a certain class of curves in d-dimensional Euclidean space.


General Hormander and Mikhlin conditions for multipliers of Besov spaces

Abstract: Here a new condition for the geometry of Banach spaces is introduced and the operator--valued Fourier multiplier theorems in weighted Besov spaces are obtained. Particularly, connections between the geometry of Banach spaces and Hormander-Mikhlin conditions are established. As an application of main results the regularity properties of degenerate elliptic differential operator equations are investigated.


The orbit structure of the Gelfand-Zeitlin group on n x n matrices

Abstract: In recent work (\cite{KW1},\cite{KW2}), Kostant and Wallach construct an action of a simply connected Lie group $A\simeq \mathbb{C}^{{n\choose 2}}$ on $gl(n)$ using a completely integrable system derived from the Poisson analogue of the Gelfand-Zeitlin subalgebra of the enveloping algebra. In \cite{KW1}, the authors show that $A$-orbits of dimension ${n\choose 2}$ form Lagrangian submanifolds of regular adjoint orbits in $gl(n)$. They describe the orbit structure of $A$ on a certain Zariski open subset of regular semisimple elements. In this paper, we describe all $A$-orbits of dimension ${n\choose 2}$ and thus all polarizations of regular adjoint orbits obtained using Gelfand-Zeitlin theory.


Holomorphic Engel Structures

Abstract: Recently there has been renewed interest among differential geometers in the theory of Engel structures. We introduce holomorphic analogues of these structures, and pose the problem of classifying projective manifolds admitting them. Besides providing the basic properties of these varieties, we present two series of examples and characterize them by certain positivity conditions on the Engel structure.


On complex positive definite functions on Z_n vanishing on squares

Abstract: We generalize the Sarkozy-Furstenberg theorem on squares in difference sets of integers, and show that, given any positive definite function f:Z_N->C with density at least r(N), where r(N)=O((\log N)^{-c}), there is a perfect square s<=N/2 such that f(s) is non-zero. We do not rely on the usual analysis of the dichotomy of randomness and periodicity of a set and iterative application of the Hardy-Littlewood method. Instead, we find a bound for the van der Corput property of the set of squares.


Analytic continuation of Dirichlet series with almost periodic coefficients

Abstract: We prove that an ordinary Dirichlet series with coefficients a(n)=g(n b) has an abscissa of convergence 0 if g is an odd 1-periodic, real-analytic function and b is Diophantine. We also show that if g is odd and has bounded variation and b is of bounded Diophantine type r>1, then the abscissa of convergence is smaller or equal than 1-1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and b is Diophantine, then the ordinary Dirichlet series has an analytic continuation to the entire complex plane.


Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring

Abstract: We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i. e. non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors.


Non-holonomic Ideals in the Plane and Absolute Factoring

Abstract: We study {\it non-holonomic} overideals of a left differential ideal $J\subset F[\partial_x, \partial_y]$ in two variables where $F$ is a differentially closed field of characteristic zero. The main result states that a principal ideal $J=< P>$ generated by an operator $P$ with a separable {\it symbol} $symb(P)$, which is a homogeneous polynomial in two variables, has a finite number of maximal non-holonomic overideals. This statement is extended to non-holonomic ideals $J$ with a separable symbol. As an application we show that in case of a second-order operator $P$ the ideal $<P>$ has an infinite number of maximal non-holonomic overideals iff $P$ is essentially ordinary. In case of a third-order operator $P$ we give few sufficient conditions on $<P>$ to have a finite number of maximal non-holonomic overideals.


The Rees-Suschkewitsch Theorem for simple topological semigroups

Abstract: We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product of a topological group over topological spaces with a continuous sandwich function. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $S\times S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or each power of $S$ is countably compact. The last item generalizes an old Wallace's result saying that each simple compact topological semigroup is a topological paragroup.


Construction of universal Thom-Whitney-a stratifications, their functoriality and Sard-type Theorem for singular varieties

Abstract: {\bf Construction.} For a dominating polynomial mapping {$F: K^n\to K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {\it bundle} $G_F \subset T^{*}K^n $. We restrict $ G_F $ over the critical points $Sing(F)$ of $ F$ in $ F^{-1}(0)$ and partition $Sing(F)$ into {\it 'quasistrata'} of points with the fibers of $G_F$ of constant dimension. It turns out that T-W-a (Thom and Whitney-a) stratifications 'near' $F^{-1}(0)$ exist iff the fibers of bundle $G_F$ are orthogonal to the tangent spaces at the smooth points of the quasistrata (e. g. when $ l=1$). Also, the latter are the orthogonal complements over an irreducible component $ S $ of a quasistratum only if $S $ is {\bf universal} for the class of {T-W-a} stratifications, meaning that for any $\{S_j'\}_j $ in the class, $ \Sing (F) = \cup_j S'_j $, there is a component $S' $ of an $ S_j' $ with $S\cap S'$ being open and dense in both $S $ and $ S' $. {\bf Results.} We prove that T-W-a stratifications with only universal strata exist iff all fibers of $G_F$ are the orthogonal complements to the respective tangent spaces to the quasistrata, and then the partition of $\Sing(F)$ by the latter yields the coarsest {\it universal T-W-a stratification}. The key ingredient is our version of {\bf Sard-type Theorem for singular spaces} in which a singular point is considered to be noncritical iff nonsingular points nearby are 'uniformly noncritical' (e. g. for a dominating map $ F: X \to Z $ meaning that the sum of the absolute values of the $l\times l$ minors of the Jacobian matrix of $ F $, where $ l = \dim (Z) $, not only does not vanish but, moreover, is separated from zero by a positive constant).


Localized linear polynomial operators and quadrature formulas on the sphere

Abstract: The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere $\SS^q$ ($q\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.


Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension

Abstract: In this paper we explore two constructions of the same family of metric measure spaces. The first construction was introduced by Laakso in 2000 where he used it as an example that Poincar\'e inequalities can hold on spaces of arbitrary Hausdorff dimension. This was proved using minimal generalized upper gradients. Following Cheeger's work these upper gradients can be used to define a Sobolev space. We show that this leads to a Dirichlet form. The second construction was introduced by Barlow and Evans in 2004 as a way of producing exotic spaces along with Markov processes from simpler spaces and processes. We show that for the correct base process in the Barlow Evans construction that this Markov process corresponds to the Dirichlet form derived from the minimal generalized upper gradients.


Asymptotic behaviour of variation of pure polarized TERP structures

Abstract: The purpose of this paper is twofold. One is to give a survey of our study on the reductions of harmonic bundles, and the other is to explain a simple application in the study of TERP structure. In particular, we investigate the asymptotic behaviour of the "new supersymmetric index" for variation of pure polarized TERP structures.


The Figure Eight Knot Group is Conjugacy Separable

Abstract: We prove that torsion free subgroups of PGL(2,C) (abstractly) commensurable with the Euclidean Bianchi groups are conjugacy separable. As a consequence we deduce the result stated in the title.


On elements of order p^s in the plane Cremona group over a field of characteristic p

Abstract: We show that the plane Cremona group over a field of characteristic p > 0 does not contain elements of order of power of p larger than 2 and it does not contain elements of order p^2 unless p =2. Also we describe conjugacy classes of elements of order 4.


Some examples of lifting problems from quotient algebras

Abstract: We consider three lifting questions: Given a $C\sp{*}$-algebra $I$, if there is a unital $C\sp{*}$-algebra $A$ contains $I$ as an ideal, is every unitary from $A/I$ lifted to a unitary in $A$? is every unitary from $A/I$ lifted to an extremal partial isometry? is every extremal partial isometry from $A/I$ lifted to an extremal partial isometry? We show several constructions of $I$ which serve as working examples or counter-examples for above questions.


Polynomial Representation of E6 and Its Combinatorial and PDE Implications

Abstract: In this paper, we use partial differential equations to find the decomposition of the polynomial algebra over the basic irreducible module of E6 into a sum of irreducible submodules. It turns out that the cubic polynomial invariant corresponding to the Dicksons' invariant trilinear form is the unique fundamental invariant. Moreover, we obtain a combinatorial identity saying that the dimensions of certain irreducible modules of E6 are correlated by the binomial coefficients of twenty-six. Furthermore, we find all the polynomial solutions for the invariant differential operator corresponding to the Dickson trilinear form in terms of the irreducible submodules.


Intrinsically triple-linked graphs in RP^3

Abstract: Flapan--Naimi--Pommersheim showed that every spatial embedding of $K_{10}$, the complete graph on ten vertices, contains a non-split three-component link; that is, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}^3$. The work of Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known intrinsically triple-linked graphs in $\mathbb{R}^3$ to include several other families of graphs. In this paper, we will show that while some of these graphs can be embedded 3-linklessly in $\mathbb{R}P^3$, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}P^3$.


Conjectures about distinction and Asai $L$-functions of generic representations of general linear groups over local fields

Abstract: Let $K/F$ be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky's classification asserts that generic representations are parabolically induced from quasi-square-integrable representations. We show, following a method developed by Cogdell and Piatetski-Shapiro, that the equality of the Rankin-Selberg type Asai $L$-function of generic representations of $GL(n,K)$ and of the Asai $L$-function of the Langlands parameter, is equivalent to the truth of a conjecture about classification of distinguished generic representations in terms of the inducing quasi-square-integrable representations. As the conjecture is true for principal series representations, this gives the expression of the Asai L-function of such representations.


Spherical designs via Brouwer fixed point theorem

Abstract: For each N>=c_d*n^{2d*(d+1)/(d+2)} we prove the existence of a spherical n-design on S^d consisting of N points, where c_d is a constant depending only on $d$.


The Witten genus and vertex algebras

Abstract: This article is the first report of an ongoing project aimed at finding a geometric interpretation of the Witten genus and other tmf classes. Section 2 reviews the sheaves of chiral differential operators (CDOs) over a complex manifold, including their construction, obstructions and relation with the Witten genus. In section 3, the structure of each sheaf of CDOs is reorganized in terms of modules over the sheaf of holomorphic functions. This invokes the notion of a differential graded vertex algebroid. The construction of sheaves of CDOs is due to Gorbounov, Malikov and Schechtman, and so is the notion of a vertex algebroid; the differential graded version is first introduced here. Section 4 contains the main result, namely the construction of a sheaf of differential graded conformal vertex algebras that provides a fine resolution of a sheaf of CDOs. This `infinite dimensional Dolbeault complex' plays a role for the Witten genus similar to that of the Dolbeault complex for the Todd genus.


Global Well-posedness for the fourth order nonlinear Schr\"{o}dinger equations with small rough data in high demension

Abstract: For $n\geq 2$, we establish the smooth effects for the solutions of the linear fourth order Shr\"{o}dinger equation in anisotropic Lebesgue spaces with $\Box_k$-decomposition. Using these estimates, we study the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equations with three order derivatives and obtain the global well posedness for this problem with small data in modulation space $M^{9/2}_{2,1}({\Real^{n}})$.


Sharp Decay Estimates and Vanishing Viscosity for Diffusive Hamilton-Jacobi Equations

Abstract: Sharp temporal decay estimates are established for the gradient and time derivative of solutions to a viscous Hamilton-Jacobi equation as well the associated Hamilton-Jacobi equation. Special care is given to the dependence of the estimates on the viscosity. The initial condition being only continuous and either bounded or non-negative. The main requirement on the Hamiltonians is that it grows superlinearly or sublinearly at infinity, including in particular H(r) = r^p for r non-negatif and p positif and different from 1.


Multiplier ideal sheaves and the K\"ahler-Ricci flow on toric Fano manifolds with large symmetry

Abstract: The purpose of this paper is to calculate the support of the multiplier ideal sheaves derived from the K\"ahler-Ricci flow on certain toric Fano manifolds with large symmetry. The early idea of this paper has already been in Appendix of \cite{futaki-sano0711}.


Galois Theory, discriminants and torsion subgroups of elliptic curves

Abstract: We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.


A fully nonlinear problem with free boundary in the plane

Abstract: We prove that bounded solutions to an overdetermined fully nonlinear free boundary problem in the plane are one dimensional. Our proof relies on maximum principle techniques and convexity arguments.


Horseshoes in multidimensional scaling and local kernel methods

Abstract: Classical multidimensional scaling (MDS) is a method for visualizing high-dimensional point clouds by mapping to low-dimensional Euclidean space. This mapping is defined in terms of eigenfunctions of a matrix of interpoint dissimilarities. In this paper we analyze in detail multidimensional scaling applied to a specific dataset: the 2005 United States House of Representatives roll call votes. Certain MDS and kernel projections output ``horseshoes'' that are characteristic of dimensionality reduction techniques. We show that, in general, a latent ordering of the data gives rise to these patterns when one only has local information. That is, when only the interpoint distances for nearby points are known accurately. Our results provide a rigorous set of results and insight into manifold learning in the special case where the manifold is a curve.


Exact Categories

Abstract: We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel-Quillen embedding theorem in an appendix.


C*-Algebras Associated with Iterated Function Systems

Abstract: We review Kajiwara and Watatani's construction of a C*-algebra from an iterated function system (IFS). If the IFS satisfies the finite branch condition or the open set condition, we build an injective homomorphism from Kajiwara-Watatani algebras to the Cuntz algebra, which can be thought as the algebra of the lifted system, and we give the description of its image. Finally, if the IFS admits a left inverse we show that the Kajiwara-Watatani algebra is isomorphic to an Exel's crossed product.


Compactification minimale et mauvaise reduction

Abstract: We construct the minimal compactification of some modular Siegel varieties at their bad reduction places. These varieties parametrize principally polarized abelian schemes endowed with a parahoric level structure at a prime number $p$, and with an auxiliary level structure ; such varieties have bad reduction at $p$. We also sketch the arithmetic theory of the associated Siegel modular forms. ----- Nous construisons la compactification minimale de certaines varietes modulaires de Siegel en leurs places de mauvaise reduction. Ces varietes parametrent des schemas abeliens principalement polarises munis d'une structure de niveau parahorique en un nombre premier $p$, et d'une structure de niveau auxilliaire ; elles ont mauvaise reduction en $p$. Nous esquissons egalement une theorie arithmetique des formes modulaires de Siegel associees a ces varietes.


Matrix valued Brownian motion and a paper by Polya

Abstract: We give a geometric description of the motion of eigenvalues of a Brownian motion with values in some matrix spaces. In the second part we consider a paper by Polya where he introduced a function close to the Riemann zeta function, which satisfies Riemann hypothesis. We show that each of these two functions can be related to Brownian motion on a symmetric space.


On the problem of detecting linear dependence for products of abelian varieties and tori

Abstract: Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a point in G(K) and let L be a finitely generated subgroup of G(K). Suppose that for all but finitely many primes p of K the point (R mod p) belongs to (L mod p). Does it follow that R belongs to L? We answer this question affirmatively in three cases: if L is cyclic; if L is a free left End_K G-submodule of G(K); if L has a set of generators (as a group) which is a basis of a free left End_K G-submodule of G(K). In general we prove that there exists an integer m (depending only on G, K and the rank of L) such that mR belongs to the left End_K G-submodule of G(K) generated by L.


Robust adaptive importance sampling for normal random vectors

Abstract: Adaptive Monte Carlo methods are very efficient techniques designed to tune simulation estimators on-line. In this work, we present an alternative to stochastic approximation to tune the optimal change of measure in the context of importance sampling for normal random vectors. Unlike stochastic approximation, which requires very fine tuning in practice, we propose to use sample average approximation and deterministic optimization techniques to devise a robust and fully automatic variance reduction methodology. The same samples are used in the sample optimization of the importance sampling parameter and in the Monte Carlo computation of the expectation of interest with the optimal measure computed in the previous step. We prove that this highly dependent Monte Carlo estimator is convergent and satisfies a central limit theorem with the optimal limiting variance. Numerical experiments confirm the performance of this estimator: in comparison with the crude Monte Carlo method, the computation time needed to achieve a given precision is divided by a factor between 3 and 15.


On the quadratic normality and the triple curve of three dimensional subvarieties of ${\mathbb P}^5$

Abstract: A well-known conjecture asserts that smooth threefolds $X\subset\{\mathbb P}^5$ are quadratically normal with the only exception of the Palatini scroll. As a corollary of a more general statement we obtain the following result, which is related to the previous conjecture: If $X\subset\{\mathbb P}^5$ is not quadratically normal, then its triple curve is reducible. Similar results are also given for higher dimensional varieties.


Generalized analytic functions on generalized domains

Abstract: We define the algebra of Colombeau generalized functions on a subset A of the space of d-dimensional generalized points. If the domain A is open, such generalized functions can be identified with pointwise maps from A into the ring of Colombeau generalized numbers. We study analyticity in these algebras, if the domain is an open subset of the complex generalized points. In particular, if the domain is an open ball for the sharp norm, we characterize analyticity and give a unicity theorem involving the values at generalized points.


Real Subpairs and Frobenius-Schur Indicators of Characters in 2-Blocks

Abstract: Let B be a real 2-block of a finite group G. Then B has a real defect class. Let g be an element of such a class. A defect couple of B is (D,E), where E is a Sylow 2-subgroup of the extended centralizer C^*(g) of g, and D is the intersection of E with the centralizer C(g). It is known that (D,E) is uniquely determined up to G-conjugacy. We show that (D,E) determines which B-subpairs are real. We also outline how (D,E) influences the vertices of components of the G-permutation module corresponding to the conjugation action of G on its involutions. We apply these methods to enumerate the Frobenius-Schur indicators of the irreducible characters in a real block that has a dihedral defect group. We also determine the vertices of the components of the involution module in such a block.


Voronoi cells of discrete point sets

Abstract: It is well known that all cells of the Voronoi diagram of a Delaunay set are polytopes. For a finite point set, all these cells are still polyhedra. So the question arises, if this observation holds for all discrete point sets: Are always all Voronoi cells of an arbitrary, infinite discrete point set polyhedral? In this paper, an answer to this question will be given. It will be shown that all Voronoi cells of a discrete point set are polytopes if and only if every point of the point set is an inner point. Furthermore, the term of a locally finitely generated discrete point set will be introduced and it will be shown that exactly these sets have the property of possessing only polyhedral Voronoi cells.


An Arakelov Inequality in Characteristic p and Upper Bound of p-Rank Zero Locus

Abstract: In this paper we show an Arakelov inequality for semi-stable families of algebraic curves of genus $g\geq 1$ over characteristic $p$ with nontrivial Kodaira-Spencer maps. We apply this inequality to obtain an upper bound of the number of algebraic curves of $p-$rank zero in a semi-stable family over characteristic $p$ with nontrivial Kodaira-Spencer map in terms of the genus of a general closed fiber, the genus of the base curve and the number of singular fibres. An extension of the above results to smooth families of Abelian varieties over $k$ with $W_2$-lifting assumption is also included.


Homotopy groups and twisted homology of arrangements

Abstract: Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generalizing both results.


A Hilbert C*-module admitting no frames

Abstract: We show that every infinite-dimensional commutative unital C*-algebra has a Hilbert C*-module admitting no frames. In particular, this shows that Kasparov's stabilization theorem for countably generated Hilbert C*-modules can not be extended to arbitrary Hilbert C*-modules.


Distributive Lattices, Polyhedra, and Generalized Flow

Abstract: A D-polyhedron is a polyhedron $P$ such that if $x,y$ are in $P$ then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every D-polyhedron corresponds to a directed graph with arc-parameters, such that every point in the polyhedron corresponds to a vertex potential on the graph. Alternatively, an edge-based description of the point set can be given. The objects in this model are dual to generalized flows, i.e., dual to flows with gains and losses. These models can be specialized to yield some cases of distributive lattices that have been studied previously. Particular specializations are: lattices of flows of planar digraphs (Khuller, Naor and Klein), of $\alpha$-orientations of planar graphs (Felsner), of c-orientations (Propp) and of $\Delta$-bonds of digraphs (Felsner and Knauer). As an additional application we exhibit a distributive lattice structure on generalized flow of breakeven planar digraphs.


Precise asymptotic of eigenvalues of resonant quasilinear systems

Abstract: In this work we study the sequence of variational eigenvalues of a system of resonant type involving $p-$ and $q-$laplacians on $\Omega \subset \R^N$, with a coupling term depending on two parameters $\alpha$ and $\beta$ satisfying $\alpha/p + \beta/q = 1$. We show that the order of growth of the $k^{th}$ eigenvalue depends on $\alpha+\beta$, $\lam_k = O(k^{\frac{\alpha+\beta}{N}})$.


On distribution of fractional parts of linear forms

Abstract: We prove a result on linear forms related to Peres-Schlag's theorem on badly approximable numbers with respect to lacunary sequences.


The space of subgroups of an abelian group

Abstract: We carry out the Cantor-Bendixson analysis of the space of all subgroups of any countable abelian group and we deduce a complete classification of such spaces up to homeomorphism.


Maximal Levi Subgroups Acting on the Building of GL_n(F)

Abstract: In this paper we give a complete invariant of the action of GL_n(F)\times GL_m(F) acting on the Euclidean building B[GL_n(F)], where F is a non-archimedian field. We then use this invariant to give a natural metric on the resulting quotient space. In the special case of the torus acting on B[GL_2(F)] this gives a method of calculating of any vertex to any fixed apartment.


On the zero set of G-equivariant maps

Abstract: Let $G$ be a finite group acting on vector spaces $V$ and $W$ and consider a smooth $G$-equivariant mapping $f:V\to W$. This paper addresses the question of the zero set near a zero $x$ of $f$ with isotropy subgroup $G$. It is known from results of Bierstone and Field on $G$-transversality theory that the zero set in a neighborhood of $x$ is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near $x$ using only information from the representations $V$ and $W$. We define an index $s(\Sigma)$ for isotropy subgroups $\Sigma$ of $G$ which is the difference of the dimension of the fixed point subspace of $\Sigma$ in $V$ and $W$. Our main result states that if $V$ contains a subspace $G$-isomorphic to $W$, then for every maximal isotropy subgroup $\Sigma$ satisfying $s(\Sigma)>s(G)$, the zero set of $f$ near $x$ contains a smooth manifold of zeros with isotropy subgroup $\Sigma$ of dimension $s(\Sigma)$. We also present a systematic method to study the zero sets for group representations $V$ and $W$ which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of $G$-reversible equivariant vector fields.


Invariant Hilbert schemes and Wonderful varieties

Abstract: The invariant Hilbert schemes considered in \cite{BC1} were proved to be affine spaces. The proof relied on the classification of strict wonderful varieties. We obtain in the present article a classification-free proof of the affinity of these invariant Hilbert scheme by means of deformation theoretical arguments. As a consequence we recover in a shorter way the classification of strict wonderful varieties. This provides an alternative and new approach to answer Luna's conjecture.


Kac's conjecture from Nakajima quiver varieties

Abstract: We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.


The quiver of an algebra associated to the Mantaci-Reutenauer descent algebra and the homology of regular semigroups

Abstract: We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola's results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao's algebra, which is related to the Mantaci-Reutenauer descent algebra.


The multivariate signed Bollobas-Riordan polynomial

Abstract: We generalise the signed Bollobas-Riordan polynomial of S. Chmutov and I. Pak [Moscow Math. J. 7 (2007), no. 3, 409-418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [J. Combinatorial Theory, Ser. B, 99 (3): 617-638, 2009] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.


Potential automorphy for certain Galois representations to GL_2n

Abstract: Building upon work of Clozel, Harris, Shepherd-Barron, and Taylor, this paper shows that certain Galois representations become automorphic after one makes a suitably large totally-real extension to the base field. The main innovation here is that the result applies to Galois representations to GL_{2n}, where previous work dealt with representations to GSp_n. The main technique is the consideration of the cohomology the Dwork hypersurface, and in particular, of pieces of this cohomology other than the invariants under the natural group action.


Meromorphic continuation for the zeta function of a Dwork hypersurface

Abstract: We consider the one-parameter family of hypersurfaces in $\Pj^5$ with projective equation (X_1^5+X_2^5+X_3^5+X_4^5+X_5^5) = 5\lambda X_1 X_2... X_5, (writing $\lambda$ for the parameter), proving that the Galois representations attached to their cohomologies are potentially automorphic, and hence that the zeta function of the family has meromorphic continuation throughout the complex plane.


A note on the Busemann-Petty problem for bodies of certain invariance

Abstract: The Busemann-Petty problem asks whether origin symmetric convex bodies in $\R^n$ with smaller hyperplane sections necessarily have smaller volume. The answer is affirmative if $n\leq 3$ and negative if $n\geq 4.$ We consider a class of convex bodies that have a certain invariance property with respect to their ordered k-tuples of coordinates in $\R^{kn}$ and prove the corresponding problem.


Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types

Abstract: For nonexceptional types, we prove a conjecture of Hatayama et al. about the prefectness of Kirillov-Reshetikhin crystals.


Markov switching negative binomial models: an application to vehicle accident frequencies

Abstract: In this paper, two-state Markov switching models are proposed to study accident frequencies. These models assume that there are two unobserved states of roadway safety, and that roadway entities (roadway segments) can switch between these states over time. The states are distinct, in the sense that in the different states accident frequencies are generated by separate counting processes (by separate Poisson or negative binomial processes). To demonstrate the applicability of the approach presented herein, two-state Markov switching negative binomial models are estimated using five-year accident frequencies on Indiana interstate highway segments. Bayesian inference methods and Markov Chain Monte Carlo (MCMC) simulations are used for model estimation. The estimated Markov switching models result in a superior statistical fit relative to the standard (single-state) negative binomial model. It is found that the more frequent state is safer and it is correlated with better weather conditions. The less frequent state is found to be less safe and to be correlated with adverse weather conditions.


On $k$-free-like groups

Abstract: A $k$-free like group is a $k$-generated group $G$ with a sequence of $k$-element generating sets $Z_n$ such that the girth of $G$ relative to $Z_n$ is unbounded and the Cheeger constant of $G$ relative to $Z_n$ is bounded away from 0. By a recent result of Benjamini-Nachmias-Peres, this implies that the critical bond percolation probability of the Cayley graph of $G$ relative to $Z_n$ tends to $1/(2k-1)$ as $n\to \infty$. Answering a question of Benjamini, we construct many non-free groups that are $k$-free like for all sufficiently large $k$.


A Priori Bounds for the Vorticity of Axis Symmetric Solutions to the Navier-Stokes Equations

Abstract: We obtain a pointwise, a priori bound for the vorticity of axis symmetric solutions to the 3 dimensional Navier-Stokes equations. The bound is in the form of a reciprocal of a power of the distance to the axis of symmetry. This seems to be the first general pointwise estimate established for the axis symmetric Navier-Stokes equations.


Stability of Localized Operators

Abstract: Let $\ell^p, 1\le p\le \infty$, be the space of all $p$-summable sequences and $C_a$ be the convolution operator associated with a summable sequence $a$. It is known that the $\ell^p$- stability of the convolution operator $C_a$ for different $1\le p\le \infty$ are equivalent to each other, i.e., if $C_a$ has $\ell^p$-stability for some $1\le p\le \infty$ then $C_a$ has $\ell^q$-stability for all $1\le q\le \infty$. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sj\"ostrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the $\ell^p$- stability (or $L^p$-stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.


Asymptotics for a free-boundary model in price formation

Abstract: We study the asymptotics for large time of solutions to a one dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria. The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.


For objective causal inference, design trumps analysis

Abstract: For obtaining causal inferences that are objective, and therefore have the best chance of revealing scientific truths, carefully designed and executed randomized experiments are generally considered to be the gold standard. Observational studies, in contrast, are generally fraught with problems that compromise any claim for objectivity of the resulting causal inferences. The thesis here is that observational studies have to be carefully designed to approximate randomized experiments, in particular, without examining any final outcome data. Often a candidate data set will have to be rejected as inadequate because of lack of data on key covariates, or because of lack of overlap in the distributions of key covariates between treatment and control groups, often revealed by careful propensity score analyses. Sometimes the template for the approximating randomized experiment will have to be altered, and the use of principal stratification can be helpful in doing this. These issues are discussed and illustrated using the framework of potential outcomes to define causal effects, which greatly clarifies critical issues.


Random survival forests

Abstract: We introduce random survival forests, a random forests method for the analysis of right-censored survival data. New survival splitting rules for growing survival trees are introduced, as is a new missing data algorithm for imputing missing data. A conservation-of-events principle for survival forests is introduced and used to define ensemble mortality, a simple interpretable measure of mortality that can be used as a predicted outcome. Several illustrative examples are given, including a case study of the prognostic implications of body mass for individuals with coronary artery disease. Computations for all examples were implemented using the freely available R-software package, randomSurvivalForest.


Geometric realizations of curvature models by manifolds with constant scalar curvature

Abstract: We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and *-scalar curvature.


Multiply generated dynamical systems and the duality of higher rank graph algebras

Abstract: We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=(T_{1},..., T_{r})$. We prove that this construction gives rise to amenable groupoids. The associated algebra is a Cuntz-like algebra. We use this construction for higher rank graph algebras in order to give a topological interpretation for the duality in $E$-theory between $C^{*}(\Lambda)$ and $C^{*}(\Lambda^{op})$.


Cohomology $C_{\infty}$-algebra and Rational Homotopy Type

Abstract: In the rational cohomology of a 1-connected space a structure of $C_{\infty}$-algebra is constructed and it is shown that this object determines the rational homotopy type


Open statistical issues in particle physics

Abstract: Many statistical issues arise in the analysis of Particle Physics experiments. We give a brief introduction to Particle Physics, before describing the techniques used by Particle Physicists for dealing with statistical problems, and also some of the open statistical questions.


Pseudodifferential operators on manifolds with linearization

Abstract: We present in this paper the construction of a pseudodifferential calculus on smooth non-compact manifolds associated to a globally defined and coordinate independant complete symbol calculus, that generalizes the standard pseudodifferential calculus on $\R^n$. We consider the case of manifolds $M$ with linearization in the sense of Bokobza-Haggiag, such that the associated (abstract) exponential map provides global diffeomorphisms of $M$ with $\R^n$ at any point. Cartan--Hadamard manifolds are special cases of such manifolds. The abstract exponential map encodes a notion of infinity on the manifold that allows, modulo some hypothesis of $S_\sigma$-bounded geometry, to define the Schwartz space of rapidly decaying functions, globally defined Fourier transformation and classes of symbols with uniform and decaying control over the $x$ variable. Given a linearization on the manifold with some properties of control at infinity, we construct symbol maps and $\la$-quantization, explicit Moyal star-product on the cotangent bundle, and classes of pseudodifferential operators. We show that these classes are stable under composition, and that the $\la$-quantization map gives an algebra isomorphism (which depends on the linearization) between symbols and pseudodifferential operators. We study, in our setting, $L^2$-continuity and give some examples. We show in particular that the hyperbolic 2-space $\HH$ has a $S_1$-bounded geometry, allowing the construction of a global symbol calculus of pseudodifferential operators on $\S(\HH)$.


An upper bound on the exceptional characteristics for Lusztig's character formula

Abstract: We develop and study a Lefschetz theory in a combinatorial category associated to a root system and derive an upper bound on the exceptional characteristics for Lusztig's formula for the simple rational characters of a reductive algebraic group. Our bound is huge compared to the Coxeter number. It is, however, given by an explicit formula.


On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

Abstract: To each once-punctured-torus bundle, $T_\phi$, over the circle with pseudo-Anosov monodromy $\phi$, there are associated two tessellations of the complex plane: one, $\Delta(\phi)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(\phi)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\infty$. In this paper, we study the relation between $\Delta(\phi)$ and $CW(\phi)$.


Predictive learning via rule ensembles

Abstract: General regression and classification models are constructed as linear combinations of simple rules derived from the data. Each rule consists of a conjunction of a small number of simple statements concerning the values of individual input variables. These rule ensembles are shown to produce predictive accuracy comparable to the best methods. However, their principal advantage lies in interpretation. Because of its simple form, each rule is easy to understand, as is its influence on individual predictions, selected subsets of predictions, or globally over the entire space of joint input variable values. Similarly, the degree of relevance of the respective input variables can be assessed globally, locally in different regions of the input space, or at individual prediction points. Techniques are presented for automatically identifying those variables that are involved in interactions with other variables, the strength and degree of those interactions, as well as the identities of the other variables with which they interact. Graphical representations are used to visualize both main and interaction effects.


Sequential category aggregation and partitioning approaches for multi-way contingency tables based on survey and census data

Abstract: Large contingency tables arise in many contexts but especially in the collection of survey and census data by government statistical agencies. Because the vast majority of the variables in this context have a large number of categories, agencies and users need a systematic way of constructing tables which are summaries of such contingency tables. We propose such an approach in this paper by finding members of a class of restricted log-linear models which maximize the likelihood of the data and use this to find a parsimonious means of representing the table. In contrast with more standard approaches for model search in hierarchical log-linear models (HLLM), our procedure systematically reduces the number of categories of the variables. Through a series of examples, we illustrate the extent to which it can preserve the interaction structure found with HLLMs and be used as a data simplification procedure prior to HLL modeling. A feature of the procedure is that it can easily be applied to many tables with millions of cells, providing a new way of summarizing large data sets in many disciplines. The focus is on information and description rather than statistical testing. The procedure may treat each variable in the table in different ways, preserving full detail, treating it as fully nominal, or preserving ordinality.


Energy dissipation and self-similar solutions for an unforced inviscid dyadic model

Abstract: A shell-type model of an inviscid fluid, previously considered in the literature, is investigated in absence of external force. Energy dissipation of positive solutions is proved and decay of energy like $t^{-2}$ is established. Self-similar decaying positive solutions are introduced and proved to exist and classified. Coalescence and blow-up are obtained as a consequence, in the class of arbitrary sign solutions.


Measure changes with extinction

Abstract: We consider a change of measure by a martingale $Z_t$ and clarify that in general $1/Z_t$ is only a supermartingale under the changed measure. We then give a necessary and sufficient condition for the event that the limit of the martingale is zero to coincide with the event that the martingale hits zero in finite time (up to a set of zero probability).


A Sharper discrepancy measure for post-election audits

Abstract: Post-election audits use the discrepancy between machine counts and a hand tally of votes in a random sample of precincts to infer whether error affected the electoral outcome. The maximum relative overstatement of pairwise margins (MRO) quantifies that discrepancy. The electoral outcome a full hand tally shows must agree with the apparent outcome if the MRO is less than 1. This condition is sharper than previous ones when there are more than two candidates or when voters may vote for more than one candidate. For the 2006 U.S. Senate race in Minnesota, a test using MRO gives a $P$-value of 4.05% for the hypothesis that a full hand tally would find a different winner, less than half the value Stark [Ann. Appl. Statist. 2 (2008) 550--581] finds.


Testing significance of features by lassoed principal components

Abstract: We consider the problem of testing the significance of features in high-dimensional settings. In particular, we test for differentially-expressed genes in a microarray experiment. We wish to identify genes that are associated with some type of outcome, such as survival time or cancer type. We propose a new procedure, called Lassoed Principal Components (LPC), that builds upon existing methods and can provide a sizable improvement. For instance, in the case of two-class data, a standard (albeit simple) approach might be to compute a two-sample $t$-statistic for each gene. The LPC method involves projecting these conventional gene scores onto the eigenvectors of the gene expression data covariance matrix and then applying an $L_1$ penalty in order to de-noise the resulting projections. We present a theoretical framework under which LPC is the logical choice for identifying significant genes, and we show that LPC can provide a marked reduction in false discovery rates over the conventional methods on both real and simulated data. Moreover, this flexible procedure can be applied to a variety of types of data and can be used to improve many existing methods for the identification of significant features.


Sufficient enlargements of minimal volume for finite dimensional normed linear spaces

Abstract: Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ such that $P(B_Y)\subset A$. The main results of the paper: {\bf (1)} Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. {\bf (2)} If a finite dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.


Skew-symmetric cluster algebras of finite mutation type

Abstract: In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices. Besides cluster algebras of rank 2 and cluster algebras associated with triangulations of surfaces there are exactly 11 exceptional skew-symmetric cluster algebras of finite mutation type. More precisely, 9 of them are associated with root systems $E_6$, $E_7$, $E_8$, $\widetilde E_6$, $\widetilde E_7$, $\widetilde E_8$, $E_6^{(1,1)}$, $E_7^{(1,1)}$, $E_8^{(1,1)}$; two remaining were recently found by Derksen and Owen. We also describe a criterion which determines if a skew-symmetric cluster algebra is of finite mutation type, and discuss growth rate of cluster algebras.


Branching Brownian motion: Almost sure growth along unscaled paths

Abstract: We give new results on the growth of the number of particles in a dyadic branching Brownian motion which follow within a fixed distance of a path $f:[0,\infty)\to \mathbb{R}$. We show that it is possible to count the number of particles without rescaling the paths. Our results reveal that the number of particles along certain paths can oscillate dramatically. The methods used are entirely probabilistic, taking advantage of the spine technique developed by, amongst others, Lyons et al, Kyprianou, and Hardy & Harris.


Inference using shape-restricted regression splines

Abstract: Regression splines are smooth, flexible, and parsimonious nonparametric function estimators. They are known to be sensitive to knot number and placement, but if assumptions such as monotonicity or convexity may be imposed on the regression function, the shape-restricted regression splines are robust to knot choices. Monotone regression splines were introduced by Ramsay [Statist. Sci. 3 (1998) 425--461], but were limited to quadratic and lower order. In this paper an algorithm for the cubic monotone case is proposed, and the method is extended to convex constraints and variants such as increasing-concave. The restricted versions have smaller squared error loss than the unrestricted splines, although they have the same convergence rates. The relatively small degrees of freedom of the model and the insensitivity of the fits to the knot choices allow for practical inference methods; the computational efficiency allows for back-fitting of additive models. Tests of constant versus increasing and linear versus convex regression function, when implemented with shape-restricted regression splines, have higher power than the standard version using ordinary shape-restricted regression.


Slowdown estimates for ballistic random walk in random environment

Abstract: For a random walk in an elliptic i.i.d. random environment in dimension greater than or equal to 4, satisfying the a ballisticity condition slightly weaker than condition (T'), We consider the probability of linear slowdown. We show an upper bound for this probability which is very close to the lower bound obtained by the "naive trap" analysis. As a tool for obtaining the main result, we show an almost local version of the quenched central limit theorem under the same ballisticity condition.


Bergman polynomials on an Archipelago: Estimates, Zeros and Shape Reconstruction

Abstract: Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains (called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago $|z^m-1|<r^m, 0<r<1,$ which consists of $m$ islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom of Szeg\H{o} orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators.


Hyperellipticity and Klein bottle companionship in systolic geometry

Abstract: Given a hyperelliptic Klein surface, we construct companion Klein bottles. Bavard's short loops on companion bottles are studied in relation to the surface to improve an inequality of Gromov's in systolic geometry.


Perturbation of essential spectra of exterior elliptic problems

Abstract: For a second-order strongly elliptic differential operator on an exterior domain in R^n it is known from works of Birman and Solomiak that a change of the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can augment, but not diminish, the essential spectrum by imposition of other Neumann-type non-elliptic boundary conditions. - The results are extended to 2m-order operators, where it is shown that for any selfadjoint realization defined by an elliptic normal boundary condition (other than the Dirichlet condition), one can augment the essential spectrum at will by adding a suitable operator to the mapping from free Diriclet data to Neumann data. We here also show an extension of the spectral asymptotics formula for the difference between inverses of elliptic problems. - The proofs rely on Krein-type formulas for differences between inverses, and cutoff techniques, combined with results on singular Green operators and their spectral asymptotics.


On the structure of left and right F-, SM- and E-quasigroups

Abstract: It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM- and E-quasigroups. Information on simple quasigroups from these quasigroup classes is given, for example, finite simple F-quasigroup is a simple group or a simple medial quasigroup. It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isomorphic the direct product of a group and a left S-loop (this is an answer to Belousov "1a", problem). Any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.


Moments, cumulants and diagram formulae for non-linear functionals of random measures

Abstract: This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and M\"obius functions. Gaussian and Poisson measures are treated in great detail. We also present several combinatorial interpretations of some recent CLTs involving sequences of random variables belonging to a fixed Wiener chaos.


Batch means and spectral variance estimators in Markov chain Monte Carlo

Abstract: Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean-square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality. Finally, we examine the empirical finite-sample properties of spectral variance and batch means estimators and provide recommendations for practitioners.


The Beilinson-Drinfeld Grassmannian and symplectic knot homology

Abstract: Seidel-Smith and Manolescu constructed knot homology theories using symplectic fibrations whose total spaces were certain varieties of matrices. These knot homology theories were associated to $SL(n) $ and tensor products of the standard and dual representations. In this paper, we place their geometric setups in a natural, general framework. For any complex reductive group and any sequence of minuscule dominant weights, we construct a fibration of affine varieties over a configuration space. The middle cohomology of these varieties is isomorphic to the space of invariants in the corresponding tensor product of representations. Our construction uses the Beilinson-Drinfeld Grassmannian and the geometric Satake correspondence.


The Euler adic dynamical system and path counts in the Euler graph

Abstract: We give a formula for generalized Eulerian numbers, prove monotonicity of sequences of certain ratios of the Eulerian numbers, and apply these results to obtain a new proof that the natural symmetric measure for the Bratteli-Vershik dynamical system based on the Euler graph is the unique fully supported invariant ergodic Borel probability measure. Key ingredients of the proof are a two-dimensional induction argument and a one-to-one correspondence between most paths from two vertices at the same level to another vertex.


How to compute the length of a geodesic on a Riemannian manifold with small error in arbitrary Sobolev norms

Abstract: We compute the length of geodesics on a Riemannian manifold by regular polynomial interpolation of the global solution of the eikonal equation related to the line element $ds^2=g_{ij}dx^idx^j$ of the manifold. Our algorithm approximates the length functional in arbitrarily strong Sobolev norms. Error estimates are obtained where the geometric information is used. It is pointed out how the algorithm can be used to get accurate approximation of solutions of parabolic partial differential equations leading obvious applications to finance and physics.


On free associative algebras linearly graded by finite groups

Abstract: As an instance of a linear action of a Hopf algebra on a free associative algebra, we consider finite group gradings of a free algebra induced by gradings on the space spanned by the free generators. The homogeneous component corresponding to the identity of the group is a free subalgebra which is graded by the usual degree. We look into its Hilbert series and prove that it is a rational function by giving an explicit formula. As an application, we show that, under suitable conditions, this subalgebra is finitely generated if and only if the grading on the base vector space is trivial.


Banach $\widetilde{\mathbb C}$-algebras

Abstract: We study Banach $\widetilde{\mathbb C}$-algebras, i.e., complete ultra-pseudo-normed algebras over the ring $\widetilde{\mathbb C}$ of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicit examples that important parts of classical Banach algebra theory do not hold for general Banach $\widetilde{\mathbb C}$-algebras and indicate a particular class of Banach $\widetilde{\mathbb C}$-algebras that overcomes these limitations to a large extent. We also investigate C*-algebras over $\widetilde{\mathbb C}$.


On a Structure of the Set of Differential Games Values

Abstract: In this paper the set of value functions of all-possible zero-sum differential games with terminal payoff is characterized. The necessary and sufficient condition for a given function to be a value of some differential game with terminal payoff is obtained.


Survival of branching random walks in random environment

Abstract: We study survival of nearest-neighbour branching random walks in random environment (BRWRE) on ${\mathbb Z}$. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. $2\times 2$ random matrices.


Orthogonal bundles over curves in characteristic two

Abstract: Let X be a smooth projective curve of genus g \geq 2 defined over a field of characteristic two. We give examples of stable orthogonal bundles with unstable underlying vector bundles and use them to give counterexamples to Behrend's conjecture on the canonical reduction of principal G-bundles for G= SO(n) with n \geq 7.


Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

Abstract: We show that the open unit ball of the space of operators from a finite dimensional Hilbert space into a separable Hilbert space (we call it "operator ball") has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In the appendix we present results of Itai Shafrir about hyperbolic metrics on the operator ball.


Compositions of projections in Banach spaces and relations between approximation properties

Abstract: A necessary and sufficient condition for existence of a Banach space with a finite dimensional decomposition but without the $\pi$-property in terms of norms of compositions of projections is found.


The representation category of any compact group is the bimodule category of a II_1 factor

Abstract: We prove that given any compact group G, there exists a minimal action of G on a II_1 factor M such that the bimodule category of the fixed-point II_1 factor M^G is naturally equivalent with the representation category of G. In particular, all subfactors of M^G with finite Jones index can be described explicitly.


Combinatorics of Tripartite Boundary Connections for Trees and Dimers

Abstract: A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection probabilities for the nodes in a random grove. In particular, for "tripartite" pairings of the nodes, the probability can be computed as a Pfaffian in the entries of the Dirichlet-to-Neumann matrix (discrete Hilbert transform) of the graph. These formulas generalize the determinant formulas given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings. These Pfaffian formulas are used to give exact expressions for reconstruction: reconstructing the conductances of a planar graph from boundary measurements. We prove similar theorems for the double-dimer model on bipartite planar graphs.


On locally extremal functions on connected spaces

Abstract: We construct an example of a real-valued continuous non-constant function $f$ defined on a connected complete metric space $X$ such that every point of $X$ is a point of local minimum or local maximum for $f$. The space $X$ is connected but fails to be separably connected.


On normal stability for nonlinear parabolic equations

Abstract: We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally stable.


Periodic elements of the free idempotent generated semigroup on a biordered set

Abstract: We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.


The large sieve and random walks on left cosets of arithmetic groups

Abstract: Applying E. Kowalski's recent generalization of the large sieve we prove that certain properties expected to be typical (irreducibility of the characteristic polynomial, absence of squares among the matrix coefficients...) are indeed verified by most (in a very explicit sense) of the elements of GL(n,A) with fixed determinant (where A is an intermediate ring between Z and Q that we specify) or by (special) orthogonal matrices with integral entries and fixed spinor norm.


A Lax Formalism for the Elliptic Difference Painlev\'e Equation

Abstract: A Lax formalism for the elliptic Painlev\'e equation is presented. The construction is based on the geometry of the curves on ${\mathbb P}^1\times{\mathbb P}^1$ and described in terms of the point configurations.


Bootstrapped Morawetz Estimates And Resonant Decomposition For Low Regularity Global Solutions Of Cubic NLS On R^{2}

Abstract: We prove global well-posedness for the L^{2}-critical cubic defocusing nonlinear Schr\"odinger equation on R^{2} with data u_{0} \in H^{s}(R^{2}) for s > {1/3}.


Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups

Abstract: We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.


Sur la lin\'earisation des tissus

Abstract: We give a simple analytic criterion which characterizes linearizable 1-codimensional webs. Then we give an invariant geometrical interpretation of it, in term of projective connection. We explain then how our approach allows to study linearization of more general objects than 1-codimensional webs. By way of illustration, we treat some explicit interesting examples.


A non-existence theorem for Morse type holomorphic foliations of codimension one transverse to spheres

Abstract: We prove that a Morse type codimension one holomorphic foliation is not transverse to a sphere in the complex affine space. Also we characterize the variety of contacts of a linear foliation with concentric spheres.


Surfaces of bounded mean curvature in Riemannian manifolds

Abstract: Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the inclusion maps converge in $C^0$ to a map from a surface of genus $g$ to $M$. We also show that, on passing to a further subsequence, the distance functions corresponding to pullback metrics converge to a pseudo-metric of fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface $F\subset M$ together with bounds on the geometry of $M$ give an upper bound on the diameter of $F$. Our proof is modelled on Gromov's compactness theorem for $J$-holomorphic curves.


Noncommutative Gelfand Duality and Applications I: The Existence of invariant subspaces

Abstract: Gelfand duality between unital commutative C*-algebras and Compact Hausdorff spaces is extended to all unital C*-algebras, where the dual objects are what we call compact Hausdorff quantum spaces. We apply this result to obtain, a characterization of unitary groups of C*-algebras, and, for arbitrary bounded Hilbert space operators, (i) A spectral theorem cum continuous functional calculus, and (ii) A proof of the general Invariant Subspace Theorem. Also described is a nonabelian generalization of Pontryagin duality of abelian locally compact groups.


An application of principal stratification to control for institutionalization at follow-up in studies of substance abuse treatment programs

Abstract: Participants in longitudinal studies on the effects of drug treatment and criminal justice system interventions are at high risk for institutionalization (e.g., spending time in an environment where their freedom to use drugs, commit crimes, or engage in risky behavior may be circumscribed). Methods used for estimating treatment effects in the presence of institutionalization during follow-up can be highly sensitive to assumptions that are unlikely to be met in applications and thus likely to yield misleading inferences. In this paper we consider the use of principal stratification to control for institutionalization at follow-up. Principal stratification has been suggested for similar problems where outcomes are unobservable for samples of study participants because of dropout, death or other forms of censoring. The method identifies principal strata within which causal effects are well defined and potentially estimable. We extend the method of principal stratification to model institutionalization at follow-up and estimate the effect of residential substance abuse treatment versus outpatient services in a large scale study of adolescent substance abuse treatment programs. Additionally, we discuss practical issues in applying the principal stratification model to data. We show via simulation studies that the model can only recover true effects provided the data meet strenuous demands and that there must be caution taken when implementing principal stratification as a technique to control for post-treatment confounders such as institutionalization.


A General formulation for standardization of rates as a method to control confounding by measured and unmeasured disease risk factors

Abstract: Standardization, a common approach for controlling confounding in population-studies or data from disease registries, is defined to be a weighted average of stratum specific rates. Typically, discussions on the construction of a particular standardized rate regard the strata as fixed, and focus on the considerations that affect the specification of weights. Each year the data from the SEER cancer registries are analyzed using a weighting procedure referred to as ``direct standardization for age.'' To evaluate the performance of direct standardization, we define a general class of standardization operators. We regard a particular standardized rate to be the output of an operator and a given data set. Based on the functional form of the operators, we define a subclass of standardization operators that controls for confounding by measured risk factors. Using the fundamental disease probability paradigm for inference, we establish the conclusions that can be drawn from year-to-year contrasts of standardized rates produced by these operators in the presence of unmeasured cancer risk factors. These conclusions take the form of falsifying specific assumptions about the conditional probabilities of disease given all the risk factors (both measured and unmeasured), and the conditional probabilities of the unmeasured risk factors given the measured risk factors. We show the one-to-one correspondence between these falsifications and the inferences made from the contrasts of directly standardized rates reported each year in the Annual Report to the Nation on the Status of Cancer.


Some Probabilistic and Statistical Properties of a Random Coefficient Autoregressive Model

Abstract: A statistical inference for random coefficient first-order autoregressive model $[RCAR(1)]$ was investigated by P.M. ROBINSON (1978) in which the coefficients varying over individuals. In this paper we attempt to generalize this result to random coefficient autoregressive model of order $p$ $[RCAR(p)]$. The stationarity condition will derived for this model.


Stochastic integrals and conditional full support

Abstract: We present conditions that imply the conditional full support (CFS) property, introduced by Guasoni, R\'asonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008), pp. 491--520], for processes Z := H + K \cdot W, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.


On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative L\'{e}vy processes

Abstract: We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries 2 (1957) 433--443]. Recently Avram, Palmowski and Pistorius [Ann. Appl. Probab. 17 (2007) 156--180] studied the case when the risk process is modeled by a general spectrally negative L\'{e}vy process. We draw upon their results and give sufficient conditions under which the optimal strategy is of barrier type, thereby helping to explain the fact that this particular strategy is not optimal in general. As a consequence, we are able to extend considerably the class of processes for which the barrier strategy proves to be optimal.


First Nonlinear Syzygies of Ideals Associated to Graphs

Abstract: Consider an ideal $I\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$ where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree $s+3$, and we compute the corresponding graded Betti number $\beta_{s,s+3}$. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.


Cyclic shifts of the van der Corput set

Abstract: In [13], K. Roth showed that the expected value of the $L^2$ discrepancy of the cyclic shifts of the $N$ point van der Corput set is bounded by a constant multiple of $\sqrt{\log N}$, thus guaranteeing the existence of a shift with asymptotically minimal $L^2$ discrepancy, [11]. In the present paper, we construct a specific example of such a shift.


Central Limit Theorem and the Bootstrap for U-Statistics of Strongly Mixing Data

Abstract: The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley to prove a new generalized covariance inequality similar to Yoshihara's. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions. The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see Bickel and Freedman). For mixing data, Politis and Romano proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean's distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results.


Aggregation of autoregressive processes and long memory

Abstract: We study the aggregation of AR processes and generalized Ornstein-Uhlenbeck (OU) processes. Mixture of spectral densities with random poles are the main tool. In this context, we apply our results for the aggregation of doubly stochastic interactives processes, see Dacunha-Castelle and Fermin (2006). Thus, we study the relationship between aggregation of autoregressive processes and long memory considering complex interaction structures. We precise a very interesting qualitative phenomena: how the long memory creation depends on the poles concentration near to the boundary of stability (measured in the Prokhorov sense). Our results extends the results given by Oppenheim and Viano (2004), and highlight the importance of the angular dispersion measure of poles in the appearance of the long memory.


A Hypergraph Dictatorship Test with Perfect Completeness

Abstract: A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based $\PCP$ construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are $o(1).$ Their test makes $q\geq3$ queries and has amortized query complexity $1+O(\frac{\log q}{q})$ but has an inherent loss of perfect completeness. In this paper we give an adaptive hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity $1+O(\frac{\log q}{q})$.


Enumeration of derangements with descents in prescribed positions

Abstract: We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated.


Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph

Abstract: We investigate the least common multiple of all subdeterminants, lcmd(A x B), of a Kronecker product of matrices, of which one is an integral matrix A with two columns and the other is the incidence matrix of a complete graph with n vertices. We prove that this quantity is the least common multiple of lcmd(A) to the power n-1 and certain binomial functions of the entries of A.


The \infty eigenvalue problem and a problem of optimal transportation

Abstract: The so-called eigenvalues and eigenfunctions of the infinite Laplacian $\Delta_\infty$ are defined through an asymptotic study of that of the usual $p$-Laplacian $\Delta_p$, this brings to a characterization via a non-linear eigenvalue problem for a PDE satisfied in the viscosity sense. In this paper, we obtain an other characterization of the first eigenvalue via a problem of optimal transportation, and recover properties of the first eigenvalue and corresponding positive eigenfunctions.


An Elementary Proof of Hawkes's Conjecture on Galton-Watson Trees

Abstract: In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees under a certain assumption on the tail at the infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as other other precise results on fractal properties of the boundary of Galton-Watson trees. The goal of this paper is to provide an elementary proof of Hawkes's conjecture under a less restrictive assumption than in T. Watanabe's paper, by use of size-biased Galton-Watson trees introduced by Lyons, Pemantle and Peres in 1995.


Optimum and equilibrium in a transport problem with queue effects

Abstract: Consider a distribution of citizens in an urban area in which some services (supermarkets, post offices...) are present. Each citizen, in order to use a service, spends an amount of time which is due both to the travel time to the service and to the queue time waiting in the service. The choice of the service to be used is made by every citizen in order to be served more quickly. Two types of problems can be considered: a global optimization of the total time spent by the citizens of the whole city (we define a global optimum and we study it with techniques from optimal mass transportation) and an individual optimization, in which each citizen chooses the service trying to minimize just his own time expense (we define the concept of equilibrium and we study it with techniques from game theory). In this framework we are also able to exhibit two time-dependent strategies (based on the notions of prudence and memory respectively) which converge to the equilibrium.


On right-angled Artin groups without surface subgroups

Abstract: We study the class N of graphs, the right-angled Artin groups defined on which do not contain surface subgroups. We prove that a presumably smaller class N' is closed under amalgamating along complete subgraphs, and also under adding bisimplicial edges. It follows that chordal graphs and chordal bipartite graphs belong to N'.


Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks

Abstract: We introduce a notion of Gieseker stability for coherent sheaves on tame Deligne-Mumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr \cite{MR2007396}. We prove that this stability condition is open, and pure dimensional semistable sheaves form a bounded family. We explicitly construct the moduli stack of semistable sheaves as a finite type global quotient, and study the moduli scheme of stable sheaves and its natural compactification in the same spirit as the seminal paper of Simpson \cite{MR1307297}. With this general machinery we are able to retrieve, as special cases, results of Lieblich \cite{MR2309155} and Yoshioka \cite{MR2306170} about moduli of twisted sheaves and parabolic stability introduced by Maruyama-Yokogawa in \cite{MR1162674}.


Fibrations and fundamental groups of Kaehler-Weyl manifolds

Abstract: We extend the Siu--Beauville theorem to a certain class of compact Kaehler--Weyl manifolds, proving that they fiber holomorphically over hyperbolic Riemannian surfaces whenever they satisfy the necessary topological hypotheses. As applications we obtain restrictions on the fundamental groups of such Kaehler--Weyl manifolds, and show that in certain cases they are in fact Kaehler.


Grothendieck Duality for Deligne-Mumford Stacks

Abstract: We prove the existence of the dualizing functor for a separated morphism of algebraic stacks with affine diagonal; then we explicitly develop duality for compact Deligne-Mumford stacks focusing in particular on the morphism from a stack to its coarse moduli space and on representable morphisms. We explicitly compute the dualizing complex for a smooth stack over an algebraically closed field and prove that Serre duality holds for smooth compact Deligne-Mumford stacks in its usual form. We prove also that a proper Cohen-Macaulay stack has a dualizing sheaf and it is an invertible sheaf when it is Gorenstein. As an application of this general machinery we compute the dualizing sheaf of a tame nodal curve.


A d-bar-theoretical proof of Hartogs' extension theorem on (n-1)-complete spaces

Abstract: Let X be a connected normal complex space of dimension n>=2 which is (n-1)-complete, and let p: M -> X be a resolution of singularities. By use of Takegoshi's generalization of the Grauert-Riemenschneider vanishing theorem, we deduce H^1_{cpt}(M,O)=0, which in turn implies Hartogs' extension theorem on X by the d-bar-technique of Ehrenpreis.


Quasi-convex sequences in the circle and the 3-adic integers

Abstract: In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J_3 of 3-adic integers. These sequences are determined by an increasing sequences of integers. For an increasing sequence \underline{a}=\{a_n\} of integers, put g_n=a_{n+1}-a_n. We prove that: (a) the set \{0\}\cup\{\pm 3^{-(a_n+1)} : n\in N\} is quasi-convex in T if and only if a_0>0 and g_n>1 for every n\in N; (b) the set \{0\}\cup\{\pm 3^{a_n} : n\in N\} is quasi-convex in the group J_3 of 3-adic integers if and only if g_n>1 for every n\in N. Moreover, we solve an open problem of Dikranjan and de Leo by providing a complete characterization of the sequences \underline{a} such that \{0\}\cup\{\pm 2^{-(a_n+1)} : n\in N\} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences \underline{a} such that the set \{0\}\cup\{\pm 2^{-(a_n+1)} : n\in N\} is quasi-convex in R.


The Index of Hypoelliptic Operators on Foliated Manifolds

Abstract: We present an index theorem for certain hypoelliptic differential operators on foliated manifolds. Our proof is a development of Alain Connes tangent groupoid proof of the Atiyah-Singer index theorem. The paper is largely self-contained.


Homomorphisms of infinitely generated analytic sheaves

Abstract: We prove that every homomorphism $\mathcal{O}^E_\zeta\to\mathcal{O}^F_\zeta$, with $E$ and $F$ Banach spaces and $\zeta\in\mathbb{C}^m$, is induced by a $\mathop{\mathrm{Hom}}(E,F)$-valued holomorphic germ, provided that $1\leq m<\infty$. A similar structure theorem is obtained for the homomorphisms of type $\mathcal{O}^E_\zeta\to\mathcal{S}_\zeta$, where $\mathcal{S}_\zeta$ is a stalk of a coherent sheaf of positive $\mathfrak{m}_\zeta$-depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the sense of Lempert-Patyi.


Differential posets and Smith normal forms

Abstract: We conjecture a strong property for the up and down maps U and D in an r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In particular, this would determine the integral structure of the maps U, D, UD, DU, including their ranks in any characteristic. As evidence, we prove the conjecture for the Young-Fibonacci lattice YF studied by Okada and its r-differential generalizations Z(r), as well as verifying many of its consequences for Young's lattice Y and the r-differential Cartesian products Y^r.


Representations of Multiloop Algebras

Abstract: We describe the finite-dimensional simple modules of all the (twisted and untwisted) multiloop algebras and classify them up to isomorphism.


Contractibility and the Hadwiger Conjecture

Abstract: Consider the following relaxation of the Hadwiger Conjecture: For each $t$ there exists $N_t$ such that every graph with no $K_t$-minor admits a vertex partition into $\ceil{\alpha t+\beta}$ parts, such that each component of the subgraph induced by each part has at most $N_t$ vertices. The Hadwiger Conjecture corresponds to the case $\alpha=1$, $\beta=-1$ and $N_t=1$. Kawarabayashi and Mohar [\emph{J. Combin. Theory Ser. B}, 2007] proved this relaxation with $\alpha={31/2}$ and $\beta=0$ (and $N_t$ a huge function of $t$). This paper proves this relaxation with $\alpha={7/2}$ and $\beta=-{3/2}$. The main ingredients in the proof are: (1) a list colouring argument due to Kawarabayashi and Mohar, (2) a recent result of Norine and Thomas that says that every sufficiently large $(t+1)$-connected graph contains a $K_t$-minor, and (3) a new sufficient condition for a graph to have a set of edges whose contraction increases the connectivity.


Crepant resolution conjecture in all genera for type A singularities

Abstract: We prove an all genera version of the Crepant Resolution Conjecture of Ruan and Bryan-Graber for type A surface singularities. We are based on a method that explicitly computes Hurwitz-Hodge integrals described in an earlier paper and some recent results by Liu-Xu for some intersection numbers on the Deligne-Mumford moduli spaces. We also generalize our results to some three-dimensional orbifolds.


The p-adic valuation of k-central binomial coefficients

Abstract: The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties.


Brownian moving averages have conditional full support

Abstract: We prove that any Brownian moving average \[X_t=\int_{-\infty}^t\bigl(f(s-t)-f(s)\bigr) dB_s,\qquad t\ge0,\] satisfies the conditional full support condition introduced by Guasoni, R\'{a}sonyi and Schachermayer [Ann. Appl. Probab. 18 (2008) 491--520].


Endpoint for the div-curl lemma in Hardy spaces

Abstract: We give a div-curl type lemma for the wedge product of closed differential forms on R^n when they have coefficients respectively in a Hardy space and L^infinity or BMO. In this last case, the wedge product belongs to an appropriate Hardy-Orlicz space.


On the laws of first hitting times of points for one-dimensional symmetric stable L\'evy processes

Abstract: Several aspects of the laws of first hitting times of points are investigated for one-dimensional symmetric stable L\'evy processes. It\^o's excursion theory plays a key role in this study.


A connection between extreme value theory and long time approximation of SDE's

Abstract: We consider a sequence $(\xi_n)_{n\ge1}$ of $i.i.d.$ random values living in the domain of attraction of an extreme value distribution. For such sequence, there exists $(a_n)$ and $(b_n)$, with $a_n>0$ and $b_n\in\ER$ for every $n\ge 1$, such that the sequence $(X_n)$ defined by $X_n=(\max(\xi_1,...,\xi_n)-b_n)/a_n$ converges in distribution to a non degenerated distribution. In this paper, we show that $(X_n)$ can be viewed as an Euler scheme with decreasing step of an ergodic Markov process solution to a SDE with jumps and we derive a functional limit theorem for the sequence $(X_n)$ from some methods used in the long time numerical approximation of ergodic SDE's.


The shifted plactic monoid

Abstract: We introduce a shifted analog of the plactic monoid of Lascoux and Sch\"utzenberger, the \emph{shifted plactic monoid}. It can be defined in two different ways: via the \emph{shifted Knuth relations}, or using Haiman's mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood-Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur $P$-function; a shifted counterpart of the Lascoux-Sch\"utzenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.