Operad profiles of Nijenhuis structures

Abstract: Recently S. Merkulov established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on the Koszul dual of a certain quadratic operad. In this paper we prove, using the PBW-basis method of E. Hoffbeck, that the operad governing Nijenhuis structures is Koszul, thereby showing that Nijenhuis structures correspond to representations of the minimal resolution of this operad. We also construct an operad such that representations of its minimal resolution in a vector space V are in one-to-one correspondence with pairs of compatible Nijenhuis structures on the formal manifold associated to V.


Solvability of Rado systems in D-sets

Abstract: Rado's Theorem characterizes the systems of homogenous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions to those systems can in fact be found in every central set. (Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of $D$-sets. Moreover we will see that the conclusion of Furstenberg's Central Sets Theorem is true for all sets in this class.


New lower bounds for the number of blocks in balanced incomplete block designs

Abstract: Bose proved the inequality $b\geq v+r-1$ for resolvable balanced incomplete block designs (RBIBDs) and Kageyama improved it for RBIBDs which are not affine resolvable. In this note we prove a new lower bound on the number of blocks $b$ that holds for all BIBDs. We further prove that for a significantly large number of BIBDs our bound is tighter than the bounds given by the inequalities of Bose and Kageyama.


A Local Characterization of Combinatorial Multihedrality in Tilings

Abstract: A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in an earlier paper, which characterizes the case of combinatorial tile-transitivity.


High Degree Diophantine Equation c^q=a^p+b^p

Abstract: The main idea of this article is simply calculating integer functions in module. The algebraic in the integer modules is studied in completely new style. By a careful construction, a result is proven that two finite numbers is with unequal logarithms in a corresponding module, and is applied to solving a kind of high degree diophantine equation.


Asymptotic Unconditionality

Abstract: We show that a separable real Banach space embeds almost isometrically in a space $Y$ with a shrinking 1-unconditional basis if and only if $\lim_{n \to \infty} \|x^* + x_n^*\| = \lim_{n \to \infty} \|x^* - x_n^*\|$ whenever $x^* \in X^*$, $(x_n^*)$ is a weak$^*$-null sequence and both limits exist. If $X$ is reflexive then $Y$ can be assumed reflexive. These results provide the isometric counterparts of recent work of Johnson and Zheng.


A positively curved manifold homeomorphic to T$_1$S$^4$

Abstract: We construct a new 7-dimensional manifold with positive sectional curvature which is 2-connected with \pi_3=\Z_2 and admits an isometric group action with one dimensional quotient.


The Model Of Paths For Generalized Kac-Moody Algebras

Abstract: This submission has been withdrawn by arXiv administrators because of inappropriate authorship claims.


Virtual retractions, conjugacy separability and omnipotence

Abstract: We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infinite-order elements of Fuchsian groups (of the first type) are conjugacy distinguished, and a proof that surface groups are omnipotent.


Normal generation of line bundles on multiple coverings

Abstract: Any line bundle $\cl $ on a smooth curve $C$ of genus $g$ with $\deg \cl \ge 2g+1$ is normally generated, i.e., $\varphi_\cl (C)\subseteq \mathbb P H^0 (C,\cl)$ is projectively normal. However, it has known that more various line bundles of degree $d$ failing to be normally generated appear on multiple coverings of genus $g$ as $d$ becomes smaller than $2g+1$. Thus, investigating the normal generation of line bundles on multiple coverings can be an effective approach to the normal generation. In this paper, we obtain conditions for line bundles on multiple coverings being normally generated or not, respectively.


Maximal operator for pseudo-differential operators with homogeneous symbols

Abstract: The aim of the present paper is to obtain a Sj\"{o}lin-type maximal estimate for pseudo-differential operators with homogeneous symbols. The crux of the proof is to obtain a phase decomposition formula which does not involve the time traslation. The proof is somehow parallel to the paper by Pramanik and Terwilleger (P. Malabika and E. Terwilleger, A weak $L^2$ estimate for a maximal dyadic sum operator on $R^n$, Illinois J. Math, {\bf 47} (2003), no. 3, 775--813). In the present paper, we mainly concentrate on our new phase decomposition formula and the results in the Cotlar type estimate, which are different from the ones by Pramanik and Terwilleger.


Induced nilpotent orbits and birational geometry

Abstract: In general, a nilpotent orbit closure in a complex simple Lie algebra \g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when \g$ is classical. Here, the induced orbits play an important role instead of Richardson orbits.


Frame-type families of translates

Abstract: We construct a uniformly discrete, and even sparse, sequence of real numbers $\Lambda=\{\lambda_n\}$ and a function g in $L^2(R)$, such that for every q>2, every function f in $L^2(R)$ can be approximated with arbitrary small error by a linear combination $\sum c_n g(t-\lambda_n)$ with an $l_q$ estimate of the coefficients: \|\{c_n\}\|_{l_q}\leq C(q)\|f\|. This can not be done for q=2, according to [2].


The co-universal C*-algebra of a row-finite graph

Abstract: Let E be a row-finite directed graph. We prove that there exists a C*-algebra C*_{min}(E) with the following co-universal property: given any C*-algebra B generated by a Toeplitz-Cuntz-Krieger E-family in which all the vertex projections are nonzero, there is a canonical homomorphism from B onto C*_{min}(E). We also identify when a homomorphism from B to C*_{min}(E) obtained from the co-universal property is injective. When every loop in E has an entrance, C*_{min}(E) coincides with the graph C*-algebra C*(E), but in general, C*_{min}(E) is a quotient of C*(E). We investigate the properties of C*_{min}(E) with emphasis on the utility of co-universality as the defining property of the algebra.


Infinite paths and cliques in random graphs

Abstract: We study some percolation problems on the complete graph over $\mathbf N$. In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability, such as independency, is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory.


Two-dimensional Blaschke Products: Degree growth and ergodic consequences

Abstract: We study the dynamics of Blaschke products in two dimensions, particularly the rates of growth of the degrees of iterates and the corresponding implications for the ergodic properties of the map.


A constrained Nevanlinna-Pick interpolation problem for matrix-valued functions

Abstract: Recent results of Davidson-Paulsen-Raghupathi-Singh give necessary and sufficient conditions for the existence of a solution to the Nevanlinna-Pick interpolation problem on the unit disk with the additional restriction that the interpolant should have the value of its derivative at the origin equal to zero. This concrete mild generalization of the classical problem is prototypical of a number of other generalized Nevanlinna-Pick interpolation problems which have appeared in the literature (for example, on a finitely-connected planar domain or on the polydisk). We extend the results of Davidson-Paulsen-Raghupathi-Singh to the setting where the interpolant is allowed to be matrix-valued and elaborate further on the analogy with the theory of Nevanlinna-Pick interpolation on a finitely-connected planar domain.


Toward an inductive description of singularities of pairs

Abstract: Motivated by Shokurov's ACC Conjecture for log canonical thresholds, we propose an inductive point of view on singularities of pairs, in the case when the ambient variety is smooth. Our main result characterizes the log canonicity of a pair in dimension N+1 by the log canonicity of another pair (not effective, in general) in dimension N.


Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras

Abstract: In this note we prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional (nonassociative) algebra.


Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski space

Abstract: We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.


Measuring Singularity of Generalized Minimizers for Control-Affine Problems

Abstract: An open question contributed by Yu. Orlov to a recently published volume "Unsolved Problems in Mathematical Systems and Control Theory", V.D. Blondel, A. Megretski (eds), Princeton Univ. Press, 2004, concerns regularization of optimal control-affine problems. These noncoercive problems in general admit 'cheap (generalized) controls' as minimizers; it has been questioned whether and under what conditions infima of the regularized problems converge to the infimum of the original problem. Starting with a study of this question we show by simple functional-theoretic reasoning that it admits, in general, positive answer. This answer does not depend on commutativity/noncommtativity of controlled vector fields. It depends instead on presence or absence of a Lavrentiev gap. We set an alternative question of measuring "singularity" of minimizing sequences for control-affine optimal control problems by so-called degree of singularity. It is shown that, in the particular case of singular linear-quadratic problems, this degree is tightly related to the "order of singularity" of the problem. We formulate a similar question for nonlinear control-affine problem and establish partial results. Some conjectures and open questions are formulated.


Mapping stacks of topological stacks

Abstract: We prove that the mapping stack Map(Y,X) of topological stacks X and Y is again a topological stack if Y admits a compact groupoid presentation. If Y admits a locally compact groupoid presentation, we show that Map(Y,X) is a paratopological stack. In particular, it has a classifying space (hence, a natural weak homotopy type). We prove an invariance theorem which shows that the weak homotopy type of the mapping stack Map(Y,X) does not change if we replace X by its classifying space, provided that Y is paracompact topological space. As an example, we describe the loop stack of the classifying stack BG of a topological group G in terms of twisted loop groups of G.


Permutations with Kazhdan-Lusztig polynomial P_{id,w}(q) = 1 + q^h

Abstract: Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial P_{id,w}(q)=1+q^h for some h.


Testing Linear-Invariant Non-Linear Properties

Abstract: We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for "triangle freeness": a function $f:\cube^{n}\to\cube$ satisfies this property if $f(x),f(y),f(x+y)$ do not all equal 1, for any pair $x,y\in\cube^{n}$. Here we extend this test to a more systematic study of testing for linear-invariant non-linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by $k$ points $v_{1},...,v_{k}\in\cube^{k}$ and $f:\cube^{n}\to\cube$ satisfies the property that if for all linear maps $L:\cube^{k}\to\cube^{n}$ it is the case that $f(L(v_{1})),...,f(L(v_{k}))$ do not all equal 1. We show that this property is testable if the underlying matroid specified by $v_{1},...,v_{k}$ is a graphic matroid. This extends Green's result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of "1-complexity linear systems" of Green and Tao, and graphic matroids, to derive the results.


Homotopy Inner Products for Cyclic Operads

Abstract: We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\widehat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\widehat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\widehat{\mathcal O}$ in terms of derivations and module maps. As an application we construct a homotopy inner product over the commutative operad on the cochains of any Poincar\'e duality space.


Exotic automorphisms of the Schouten algebra of polyvector fields

Abstract: Using a new compactification of the (braid) configuration space of n points in the upper half plane we construct a family of exotic Lie-infinity automorphisms of the Schouten algebra of polyvector fields on an affine space depending on a Kontsevich type propagator.


Convergence of diagonal Pad\'e approximants for a class of definitizable functions

Abstract: Convergence of diagonal Pad\'e approximants is studied for a class of functions which admit the integral representation $ {\mathfrak F}(\lambda)=r_1(\lambda)\int_{-1}^1\frac{td\sigma(t)}{t-\lambda}+r_2(\lambda), $ where $\sigma$ is a finite nonnegative measure on $[-1,1]$, $r_1$, $r_2$ are real rational functions bounded at $\infty$, and $r_1$ is nonnegative for real $\lambda$. Sufficient conditions for the convergence of a subsequence of diagonal Pad\'e approximants of $ {\mathfrak F}$ on $\dR\setminus[-1,1]$ are found. Moreover, in the case when $r_1\equiv 1$, $r_2\equiv 0$ and $\sigma$ has a gap $(\alpha,\beta)$ containing 0, it turns out that this subsequence converges in the gap. The proofs are based on the operator representation of diagonal Pad\'e approximants of $ {\mathfrak F}$ in terms of the so-called generalized Jacobi matrix associated with the asymptotic expansion of $ {\mathfrak F}$ at infinity.


Number theoretic techniques in the theory of Lie groups and differential geometry

Abstract: The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense subgroups in a semi-simple Lie group. We have shown that for arithmetic subgroups, weak commensurability has surprisingly strong consequences. Our proofs make use of p-adic techniques and results from algebraic and transcendental number theory.


Estimation of a probability with optimum guaranteed confidence in inverse binomial sampling

Abstract: Sequential estimation of a probability $p$ by means of inverse binomial sampling is considered. For $\mu_1,\mu_2>1$ given, the accuracy of an estimator $\hat{p}$ is measured by the confidence level $P[p/\mu_2\leq\hat{p}\leq p\mu_1]$. The confidence levels $c_0$ that can be guaranteed for $p$ unknown, that is, such that $P[p/\mu_2\leq \hat{p}\leq p\mu_1]\geq c_0$ for all $p\in(0,1)$, are investigated. It is shown that within the general class of randomized or non-randomized estimators based on inverse binomial sampling, there is a maximum $c_0$ that can be guaranteed for arbitrary $p$. A non-randomized estimator is given that achieves this maximum guaranteed confidence under mild conditions on $\mu_1$, $\mu_2$.


A brief note on the spectrum of the basic Dirac operator

Abstract: In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation $(M,\mathcal{F})$ with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac operator on $M$. We discuss examples and also define a new version of the basic Laplacian whose spectrum does not depend on the choice of bundle-like metric.


Implementing Communication-Optimal Parallel and Sequential QR Factorizations

Abstract: We present parallel and sequential dense QR factorization algorithms for tall and skinny matrices and general rectangular matrices that both minimize communication, and are as stable as Householder QR. The sequential and parallel algorithms for tall and skinny matrices lead to significant speedups in practice over some of the existing algorithms, including LAPACK and ScaLAPACK, for example up to 6.7x over ScaLAPACK. The parallel algorithm for general rectangular matrices is estimated to show significant speedups over ScaLAPACK, up to 22x over ScaLAPACK.


Normal automorphisms of relatively hyperbolic groups

Abstract: An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no non-trivial finite normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with more than one end.


Shelling Coxeter-like Complexes and Sorting on Trees

Abstract: In their work on `Coxeter-like complexes', Babson and Reiner introduced a simplicial complex $\Delta_T$ associated to each tree $T$ on $n$ nodes, generalizing chessboard complexes and type A Coxeter complexes. They conjectured that $\Delta_T$ is $(n-b-1)$-connected when the tree has $b$ leaves. We provide a shelling for the $(n-b)$-skeleton of $\Delta_T$, thereby proving this conjecture. In the process, we introduce notions of weak order and inversion functions on the labellings of a tree $T$ which imply shellability of $\Delta_T$, and we construct such inversion functions for a large enough class of trees to deduce the aforementioned conjecture and also recover the shellability of chessboard complexes $M_{m,n}$ with $n \ge 2m-1$. We also prove that the existence or nonexistence of an inversion function for a fixed tree governs which networks with a tree structure admit greedy sorting algorithms by inversion elimination and provide an inversion function for trees where each vertex has capacity at least its degree minus one.


Chern classes of blow-ups

Abstract: We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the standard argument for the nonsingular case, involving Riemann-Roch without denominators. The new approach relies on the explicit computation of an ideal, and a mild generalization of the well-known formula for the normal bundle of a proper transform. We also discuss alternative, very short proofs of the standard formula in some cases: an approach relying on the theory of Chern-Schwartz-MacPherson classes (working in characteristic 0), and an argument reducing the formula to a straightforward computation of Chern classes for sheaves of differential 1-forms with logarithmic poles (when the center of the blow-up is a complete intersection).


On Coxeter Diagrams of complex reflection groups

Abstract: We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 \pi i/3}]$: there are only four such lattices, namely, the $\cE$-lattices whose real forms are $A_2$, $D_4$, $E_6$ and $E_8$. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Brou\'{e}, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group $G$, picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If $G$ is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally we observe that if $G$ is primitive and $G$ has a set of roots whose $\ZZ$--span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for $G$. The group $G$ has a presentation on these generators such that if we forget that the generators have finite order then we get a (Coxeter-like) presentation of the corresponding braid group. For some groups, such as $G_{33}$ and $G_{34}$, new diagrams are obtained. For $G_{34}$, our new diagram extends to an "affine diagram" with $\ZZ/7\ZZ$ symmetry.


Szasz Analytic Functions and Noncompact K\"{a}hler Toric Manifolds

Abstract: We show that the classical Szasz analytic function $S_N(f)(x)$ is obtained by applying the pseudo-differential operator $f(N^{-1}D_{\theta})$ to the Bergman kernels for the Bargmann-Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric \kahler manifold, defining the generalized Szasz analytic function $S_{h^N}(f)(x)$. About $S_{h^N}(f)(x)$, we prove that it admits complete asymptotics and there exists a universal scaling limit. % We also consider some dilation operator composed with $S_{h^N}(f)(x)$ and we give an estimate about this composition. As an example, we will further compute $S_{h^N}(f)(x)$ for the Bergman metric on the unit ball.


Wreath Product Symmetric Functions

Abstract: We systematically study wreath product Schur functions and give a combinatorial construction using colored partitions and tableaux. The Pieri rule and the Littlewood-Richardson rule are studied. We also discuss the connection with representations of generalized symmetric groups.


A priori convergence estimates for a rough Poisson-Dirichlet problem with natural vertical boundary conditions

Abstract: Stents are medical devices designed to modify blood flow in aneurysm sacs, in order to prevent their rupture. Some of them can be considered as a locally periodic rough boundary. In order to approximate blood flow in arteries and vessels of the cardio-vascular system containing stents, we use multi-scale techniques to construct boundary layers and wall laws. Simplifying the flow we turn to consider a 2-dimensional Poisson problem that conserves essential features related to the rough boundary. Then, we investigate convergence of boundary layer approximations and the corresponding wall laws in the case of Neumann type boundary conditions at the inlet and outlet parts of the domain. The difficulty comes from the fact that correctors, for the boundary layers near the rough surface, may introduce error terms on the other portions of the boundary. In order to correct these spurious oscillations, we introduce a vertical boundary layer. Trough a careful study of its behavior, we prove rigorously decay estimates. We then construct complete boundary layers that respect the macroscopic boundary conditions. We also derive error estimates in terms of the roughness size epsilon either for the full boundary layer approximation and for the corresponding averaged wall law.


Fractional diffusion limit for collisional kinetic equations

Abstract: This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.


Two new Probability inequalities and Concentration Results

Abstract: Concentration results and probabilistic analysis for combinatorial problems like the TSP, MWST, graph coloring have received much attention, but generally, for i.i.d. samples (i.i.d. points in the unit square for the TSP, for example). Here, we prove two probability inequalities which generalize and strengthen Martingale inequalities. The inequalities provide the tools to deal with more general heavy-tailed and inhomogeneous distributions for combinatorial problems. We prove a wide range of applications - in addition to the TSP, MWST, graph coloring, we also prove more general results than known previously for concentration in bin-packing, sub-graph counts, Johnson-Lindenstrauss random projection theorem. It is hoped that the strength of the inequalities will serve many more purposes.


Continuity of the radius of convergence of differential equations on $p$-adic analytic curves

Abstract: This paper deals with connections on $p$-adic analytic curves, in the sense of Berkovich. The curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them. For any choice of a semistable formal model of the curve, we define an intrinsic notion of normalized radius of convergence as a function on the curve, with values in $(0,1]$. For a sufficiently refined choice of the semistable model, we prove continuity and logarithmic concavity of that function. We characterize \emph{Robba connections}, that is connections whose sheaf of solutions is constant on any open disk contained in the curve.


The Polya-Tchebotarov problem

Abstract: We describe the solutions to the problem of identifying the continuum in the complex plane that minimizes the logarithmic capacity among all the continuum that contain a prefixed finite set of points. This description can be implemented numerically and this can be used to improve the estimates on the Bloch-Landau constant and other related problems as the maximal expected lifetime of the Brownian motion on domains of inner radius one or the principal eigenvalue for the Laplace operator on such domains.


Asymptotics of generalized Hadwiger numbers

Abstract: We give asymptotic estimates for the number of non-overlapping homothetic copies of some centrally symmetric oval $B$ which have a common point with a 2-dimensional domain $F$ having rectifiable boundary, extending previous work of the L.Fejes-Toth, K.Borockzy Jr., D.G.Larman, S.Sezgin, C.Zong and the authors. The asymptotics compute the length of the boundary $\partial F$ in the Minkowski metric determined by $B$. The core of the proof consists of a method for sliding convex beads along curves with positive reach in the Minkowski plane. We also prove that level sets are rectifiable subsets, extending a theorem of Erd\"os, Oleksiv and Pesin for the Euclidean space to the Minkowski space.


Hardness and Algorithms for Rainbow Connection

Abstract: An edge-colored graph $G$ is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connection} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In the first result of this paper we prove that computing $rc(G)$ is NP-Hard solving an open problem from \cite{Ca-Yu}. In fact, we prove that it is already NP-Complete to decide if $rc(G)=2$, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every $\epsilon >0$, a connected graph with minimum degree at least $\epsilon n$ has {\em bounded} rainbow connection, where the bound depends only on $\epsilon$, and a corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.


Coherence for Modalities

Abstract: Positive modalities in systems in the vicinity of S4 and S5 are investigated in terms of categorial proof theory. Coherence and maximality results are demonstrated, and connections with mixed distributive laws and Frobenius algebras are exhibited.


Ordinals in Frobenius Monads

Abstract: This paper provides geometrical descriptions of the Frobenius monad freely generated by a single object. These descriptions are related to results connecting Frobenius algebras and topological quantum field theories. In these descriptions, which are based on coherence results for self-adjunctions (adjunctions where an endofunctor is adjoint to itself), ordinals in epsilon zero play a prominent role. The paper ends by considering how the notion of Frobenius algebra induces the collapse of the hierarchy of ordinals in epsilon zero, and by raising the question of the exact categorial abstraction of the notion of Frobenius algebra.


Irrationality proof of a $q$-extension of $\zeta(2)$ using little $q$-Jacobi polynomials

Abstract: We show how one can use Hermite-Pad\'{e} approximation and little $q$-Jacobi polynomials to construct rational approximants for $\zeta_q(2)$. These numbers are $q$-analogues of the well known $\zeta(2)$. Here $q=\frac{1}{p}$, with $p$ an integer greater than one. These approximants are good enough to show the irrationality of $\zeta_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $\mu(\zeta_q(2))\leq 10\pi^2/(5\pi^2-24) \approx 3.8936$. This is sharper than the upper bound given by Zudilin (\textit{On the irrationality measure for a $q$-analogue of $\zeta(2)$}, Mat. Sb. \textbf{193} (2002), no. 8, 49--70).


Exponential functionals of Brownian motion and class-one Whittaker functions

Abstract: We consider exponential functionals of a multi-dimensional Brownian motion with drift, defined via a collection of linear functionals. We give a characterization of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrodinger-type partial differential equation. We derive a similar equation for the probability density. We then characterize all diffusion processes which can be interpreted as having the law of the Brownian motion with drift conditioned on the law of its exponential functionals. In the case where the family of linear functionals is a set of simple roots, the Laplace transform of the joint law of the corresponding exponential functionals can be expressed in terms of a (class-one) Whittaker function associated with the corresponding root system. In this setting, we establish some basic properties of the corresponding diffusion processes.


Integral and isocapacitary inequalities

Abstract: It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which imply Faber-Krahn property of the fundamental Dirichlet-Laplace eigenvalue are obtained. Necessary and sufficient conditions ensuring integral inequalities between a difference seminorm and the $L_p$-norm of the gradient are found.


Boundedness of the gradient of a solution to the Neumann-Laplace problem in a convex domain

Abstract: It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex $n$-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given.


A sequence to compute the Brauer group of certain quasi-triangular Hopf algebras

Abstract: A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra $B$ in a braided monoidal category $\C$, and under certain assumptions on the braiding (fulfilled if $\C$ is symmetric), we construct a sequence for the Brauer group $\BM(\C;B)$ of $B$-module algebras, generalizing Beattie's one. It allows one to prove that $\BM(\C;B) \cong \Br(\C) \times \Gal(\C;B),$ where $\Br(\C)$ is the Brauer group of $\C$ and $\Gal(\C;B)$ the group of $B$-Galois objects. We also show that $\BM(\C;B)$ contains a subgroup isomorphic to $\Br(\C) \times \Hc(\C;B,I),$ where $\Hc(\C;B,I)$ is the second Sweedler cohomology group of $B$ with values in the unit object $I$ of $\C$. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct $B \times H$, where $H$ is a usual Hopf algebra over a field $K$, the Hopf subalgebra generated by the quasi-triangular structure $\R$ is contained in $H$ and $B$ is a Hopf algebra in the category ${}_H\M$ of left $H$-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that $\BM(K,H,\R) \times \Hc({}_H\M;B,K)$ is a subgroup of the Brauer group $\BM(K,B \times H,\R),$ confirming the suspicion that a certain cohomology group of $B \times H$ (second lazy cohomology group was conjectured) embeds into $\BM(K,B \times H,\R).$ New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.


Quartic double solids with ordinary singularities

Abstract: We study the mixed Hodge structure on the third homology group of a threefold which is the double cover of projective three-space ramified over a quartic surface with a double conic. We deal with the Torelli problem for such threefolds.


Growth conditions and uniqueness of the Cauchy problem for the evolutionary infinity Laplacian

Abstract: We study the Cauchy problem for the parabolic infinity Laplace equation. We prove a new comparison principle and obtain uniqueness of viscosity solutions in the class of functions with a polinomial growth at infinity, improving previous results obtained assuming a linear growth.


Newton Binomial Formulas in Schubert Calculus

Abstract: We prove Newton's binomial formulas for Schubert Calculus to determine numbers of base point free linear series on the projective line with prescribed ramification divisor supported at given distinct points.


Locally semisimple and maximal subalgebras of the finitary Lie algebras $gl(\infty)$, $sl(\infty)$, $so(\infty)$, and $sp(\infty)$

Abstract: We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras $\gl(\infty), \sl(\infty), \so(\infty)$, and $\sp(\infty)$. For simple finite--dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin. Key words (2000 MSC): 17B05 and 17B65.


Spanning Trees in Grid Graphs

Abstract: Building on work by Desjarlais, Molina, Faase, and others, a general method is obtained for counting the number of spanning trees of graphs that are a product of an arbitrary graph and either a path or a cycle, of which grid graphs are a subclass. Results are obtained pertaining to recurrences obtained in this manner, and numerous new integer sequences are found.


On the mean curvature of Nash isometric embeddings

Abstract: J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{\mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent, imposing restrictions on the mean curvature vector of the embedding.


Uniqueness and Non-uniqueness in inverse radiative transfer

Abstract: We characterize the non-uniqueness in the inverse problem for the stationary transport model, in which the absorption "a" and the scattering coefficient "k" are to be recovered from the albedo operator. We show that "gauge equivalent" pairs (a,k) yield the same albedo operator, and that we can recover uniquely the class of gauge equivalent pairs. We apply this result to show unique determination of the media when the absorption "a" depends on the line of travel through each point while scattering "k" obeys a symmetry property. Previously known results concerned directional independent absorption.


Exercises in the birational geometry of algebraic varieties

Abstract: This a collection of about 100 exercises. It could be used as a supplement to the book Koll\'ar--Mori: Birational geometry of algebraic varieties.


Stability of noncharacteristic boundary layers in the standing shock limit

Abstract: We investigate one- and multi-dimensional stability of noncharacteristic boundary layers in the limit approaching a standing planar shock wave $\bar U(x_1)$, $x_1>0$, obtaining necessary conditions of (i) weak stability of the limiting shock, (ii) weak stability of the constant layer $u\equiv U_-:=\lim_{z\to -\infty} \bar U(z)$, and (iii) nonnegativity of a modified Lopatinski determinant similar to that of the inviscid shock case. For Lax 1-shocks, we obtain equally simple sufficient conditions; for $p$-shocks, $p>1$, the situation appears to be more complicated. Using these results, we determine stability of certain isentropic and full gas dynamical boundary-layers, generalizing earlier work of Serre--Zumbrun and Costanzino--Humphreys--Nguyen--Zumbrun.


On cluster algebras arising from unpunctured surfaces II

Abstract: We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincar\'e characteristic of quiver Grassmannians in Dynkin type $A$ and affine Dynkin type $\tilde A$.


Proximal Point Method for a Special Class of Nonconvex Functions on Hadamard Manifolds

Abstract: In this paper we present the proximal point method for a special class of nonconvex function on a Hadamard manifold. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each accumulation point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained.


Chow groups of K3 surfaces and spherical objects

Abstract: We show that for a K3 surface X the finitely generated subring R(X) of the Chow ring introduced by Beauville and Voisin is preserved under derived equivalences. This is proved by analyzing Chern characters of spherical bundles. As for a K3 surface X defined over a number field all spherical bundles on the associated complex K3 surface are defined over $\bar\QQ$, this is compatible with the Bloch-Beilinson conjecture. Besides the work of Beauville and Voisin, Lazarfeld's result on Brill-Noether theory for curves in K3 surfaces and the deformation theory developed with Macri and Stellari in arXiv:0710.1645 are central for the discussion.


Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

Abstract: We prove a monotonicity result at specific points for the Horizontal Perimeter for a class of surfaces in the Heisenberg group.


The Radio Number of Gear Graphs

Abstract: Let $d(u,v)$ denote the distance between two distinct vertices of a connected graph $G$, and $\diam(G)$ be the diameter of $G$. A radio labeling $c$ of $G$ is an assignment of positive integers to the vertices of $G$ satisfying $d(u,v)+|c(u)-c(v)|\geq \diam(G) + 1.$ The maximum integer in the range of the labeling is its span. The radio number of $G$, $rn(G)$, is the minimum possible span. The family of gear graphs of order $n$, $G_n$, consists of planar graphs with $2n+1$ vertices and $3n$ edges. We prove that the radio number of the $n$-gear is $4n+2$.


Quantifying the cost of simultaneous non-parametric approximation of several samples

Abstract: We consider the standard non-parametric regression model with Gaussian errors but where the data consist of different samples. The question to be answered is whether the samples can be adequately represented by the same regression function. To do this we define for each sample a universal, honest and non-asymptotic confidence region for the regression function. Any subset of the samples can be represented by the same function if and only if the intersection of the corresponding confidence regions is non-empty. If the empirical supports of the samples are disjoint then the intersection of the confidence regions is always non--empty and a negative answer can only be obtained by placing shape or quantitative smoothness conditions on the joint approximation. Alternatively a simplest joint approximation function can be calculated which gives a measure of the cost of the joint approximation, for example, the number of extra peaks required.


The Symmetry Preserving Removal Lemma

Abstract: In this note we observe that in the hyper-graph removal lemma the edge removal can be done in a way that the symmetries of the original hyper-graph remain preserved. As an application we prove the following generalization of Szemer\'edi's Theorem on arithmetic progressions. If in an Abelian group $A$ there are sets $S_1,S_2...,S_t$ such that the number of arithmetic progressions $x_1,x_2,...,x_t$ with $x_i\in S_i$ is $o(|A|^2)$ then we can shrink each $S_i$ by $o(|A|)$ elements such that the new sets don't have such a diagonal arithmetic progression.


Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence

Abstract: We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulae for the computation of the E_3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.


On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\ell_1$ Minimization

Abstract: We propose novel necessary and sufficient conditions for a sensing matrix to be "$s$-good" - to allow for exact $\ell_1$-recovery of sparse signals with $s$ nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect $\ell_1$-recovery (nonzero measurement noise, nearly $s$-sparse signal, near-optimal solution of the optimization problem yielding the $\ell_1$-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse $\ell_1$-recovery and to efficiently computable upper bounds on those $s$ for which a given sensing matrix is $s$-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties.


Logarithmic Sobolev inequalities: regularizing effect of L\'evy operators and asymptotic convergence in the L\'evy-Fokker-Planck equation

Abstract: In this paper we study some applications of the L\'evy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i. e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the L\'evy-Ornstein-Uhlenbeck process.


Balance laws with integrable unbounded sources

Abstract: We consider the Cauchy problem for a $n\times n$ strictly hyperbolic system of balance laws $$ \{{array}{c} u_t+f(u)_x=g(x,u), x \in \mathbb{R}, t>0 u(0,.)=u_o \in L^1 \cap BV(\mathbb{R}; \mathbb{R}^n), | \lambda_i(u)| \geq c > 0 {for all} i\in \{1,...,n\}, \|g(x,\cdot)\|_{\mathbf{C}^2}\leq \tilde M(x) \in L1, {array}. $$ each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the $\mathbf{L}^1$ norm of $\|g(x,\cdot)\|_{\mathbf{C}^1}$ and $\|u_o\|_{BV(\reali)}$ are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [1] to unbounded (in $L^\infty$) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.


Geometric aspects of transversal Killing spinors on Riemannian flows

Abstract: We study a Killing spinor type equation on spin Riemannian flows. We prove integrability conditions and partially classify those Riemannian flows $M$ carrying non-trivial solutions to that equation in case $M$ is a local Riemannian product, a Sasakian manifold or 3-dimensional.


Proportional fairness and its relationship with multi-class queueing networks

Abstract: We consider multi-class single-server queueing networks that have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. The stochastic flow level model found is insensitive. A large deviation principle for the stationary distribution of these multi-class queueing networks is also found. Its rate function has a dual form that coincides with proportional fairness. We then give the first rigorous proof that the stationary throughput of a multi-class single-server queueing network converges to a proportionally fair allocation. This work combines classical queueing networks with more recent work on stochastic flow level models and proportional fairness. One could view these seemingly different models as the same system described at different levels of granularity: a microscopic, queueing level description; a macroscopic, flow level description and a teleological, optimization description.


Representation of operators in the time-frequency domain and generalized Gabor multipliers

Abstract: Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator's best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator's best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.


The Weibull-Geometric distribution

Abstract: In this paper we introduce, for the first time, the Weibull-Geometric distribution which generalizes the exponential-geometric distribution proposed by Adamidis and Loukas (1998). The hazard function of the last distribution is monotone decreasing but the hazard function of the new distribution can take more general forms. Unlike the Weibull distribution, the proposed distribution is useful for modeling unimodal failure rates. We derive the cumulative distribution and hazard functions, the density of the order statistics and calculate expressions for its moments and for the moments of the order statistics. We give expressions for the R\'enyi and Shannon entropies. The maximum likelihood estimation procedure is discussed and an algorithm EM (Dempster et al., 1977; McLachlan and Krishnan, 1997) is provided for estimating the parameters. We obtain the information matrix and discuss inference. Applications to real data sets are given to show the flexibility and potentiality of the proposed distribution.


On the dimension of invariant measures of endomorphisms of $\mathbb{CP}^k$

Abstract: Let $f$ be an endomorphism of $\mathbb{CP}^k$ and $\nu$ be an $f$-invariant measure with positive Lyapunov exponents $(\lambda_1,\...,\lambda_k)$. We prove a lower bound for the pointwise dimension of $\nu$ in terms of the degree of $f$, the exponents of $\nu$ and the entropy of $\nu$. In particular our result can be applied for the maximal entropy measure $\mu$. When $k=2$, it implies that the Hausdorff dimension of $\mu$ is estimated by $\dim_{\cal H} \mu \geq {\log d \over \lambda_1} + {\log d \over \lambda_2}$, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the $\nu$-generic inverse branches of $f^n$ in $\mathbb{CP}^k$. Our tools are a volume growth estimate for the bounded holomorphic polydiscs in $\mathbb{CP}^k$ and a normalization theorem for the $\nu$-generic inverse branches of $f^n$.


On some new invariants for strong shift equivalence for shifts of finite type

Abstract: We introduce a new computable invariant for strong shift equivalence of shifts of finite type. The invariant is based on an invariant introduced by Trow, Boyle, and Marcus, but has the advantage of being readily computable. We summarize briefly a large-scale numerical experiment aimed at deciding strong shift equivalence for shifts of finite type given by irreducible $2\times 2$-matrices with entry sum less than 25, and give examples illustrating to power of the new invariant, i.e., examples where the new invariant can disprove strong shift equivalence whereas the other invariants that we use can not.


Quantum theta functions and Gabor frames for modulation spaces

Abstract: Representations of the celebrated Heisenberg commutation relations in quantum mechanics and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we try to bridge the two communities, represented by the two co--authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.


Criteria for strong and weak random attractors

Abstract: The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak attractors.


The Ricci flow does not preserve the set of Zoll metrics

Abstract: The question of whether or not the set of Zoll metrics on the 2-sphere is connected is still open. Here we show that a naive application of the Ricci flow is not sufficient to answer this problem.


Slicely Countably Determined Banach spaces

Abstract: We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $\ell_1$. We present many examples and several properties of this class. We give some applications to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $\ell_1$ and that operators which do not fix copies of $\ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.


Harmonic maps and Kaluza-Klein metrics on spheres

Abstract: This article studies the harmonicity of vector fields on Riemannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigidity conditions are obtained, especially for surfaces and vector fields of constant norm, and existence is proved on two-tori. Classifications are given for conformal, quadratic and Killing vector fields on spheres. Finally, the class of metric considered on the tangent bundle is enlarged, permitting new vector fields to become harmonic.


Remarques sur les spineurs de Killing transversaux

Abstract: We show necessary conditions for the existence of transversal Killing spinors on a spin manifold endowed with a Riemannian flow.


p(x)-Harmonic functions with unbounded exponent in a subdomain

Abstract: We study the Dirichlet problem $-\div(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $\Omega$. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as $n \to \infty$ of the solutions $u_n$ to the corresponding problem when $p_n(x) =p(x) \wedge n$, in particular, with $p_n = n$ in $D$. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, $\infty$-harmonic within $D$. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of $\Omega$.


Modified action and differential operators on the 3-D sub-Riemannian sphere

Abstract: Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approach, where the corresponding Hamitonian system is solved with mixed boundary conditions. A closed form of the modified action is given. It is a sub-Riemannian invariant and plays the role of a distance on $\mathbb S^3$.


McKay correspondence for the Poincar\'e series of Kleinian and Fuchsian singularities

Abstract: We give a conceptual proof that the Poincar\'e series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using triangulated categories and spherical twist functors.


Dynamics of vertex-reinforced random walks

Abstract: We generalize a result from Volkov [Ann. Probab. 29 (2001) 66--91] and prove that, on a large class of locally finite connected graphs of bounded degree $(G,\sim)$ and symmetric reinforcement matrices $a=(a_{i,j})_{i,j\in G}$, the vertex-reinforced random walk (VRRW) eventually localizes with positive probability on subsets which consist of a complete $d$-partite subgraph with possible loops plus its outer boundary. We first show that, in general, any stable equilibrium of a linear symmetric replicator dynamics with positive payoffs on a graph $G$ satisfies the property that its support is a complete $d$-partite subgraph of $G$ with possible loops, for some $d\ge1$. This result is used here for the study of VRRWs, but also applies to other contexts such as evolutionary models in population genetics and game theory. Next we generalize the result of Pemantle [Probab. Theory Related Fields 92 (1992) 117--136] and Bena\"{{\i}}m [Ann. Probab. 25 (1997) 361--392] relating the asymptotic behavior of the VRRW to replicator dynamics. This enables us to conclude that, given any neighborhood of a strictly stable equilibrium with support $S$, the following event occurs with positive probability: the walk localizes on $S\cup\partial S$ (where $\partial S$ is the outer boundary of $S$) and the density of occupation of the VRRW converges, with polynomial rate, to a strictly stable equilibrium in this neighborhood.


Ruan's conjecture and integral structures in quantum cohomology

Abstract: This is an expository article on the recent studies of Ruan's crepant resolution/flop conjecture and its possible relations to the K-theory integral structure in quantum cohomology.


$L^p$ continuity of wave operators in $\Bbb Z$

Abstract: We recover for discrete Schr\"odinger operators on the lattice Z, stronger analogues of the results by Weder, Galtabiar and Yajima and by D'Ancona and Fanelli on R.


Foliations with degenerate Gauss maps on $\mathbb P^4$

Abstract: We obtain a classification of codimension one holomorphic foliations on $\mathbb P^4$ with degenerate Gauss maps.


On the $q$-Bessel Fourier transform

Abstract: In this work, we are interested by the $q$-Bessel Fourier transform with a new approach. Many important results of this $q$-integral transform are proved with a new constructive demonstrations and we establish in particular the associated $q$-Fourier-Neumen expansion which involves the $q$-little Jacobi polynomials.


On Elkies subgroups of l-torsion points in elliptic curves defined over a finite field

Abstract: As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic p, there exists an algorithm that computes, for l an Elkies prime, l-torsion points in an extension of degree l-1 at cost O(l max(l, \log q)^2) bit operations in the favorable case where l < p/2. We combine in this work a fast algorithm for computing isogenies due to Bostan, Morain, Salvy and Schost with the p-adic approach followed by Joux and Lercier to get for the first time an algorithm valid without any limitation on l and p but of similar complexity.


Bounds on the frequency of 1 in the Kolakoski word

Abstract: We use a method of Goulden and Jackson to bound freq_1(K), the limiting frequency of 1 in the Kolakoski word K. We prove that |freq_1(K) - 1/2| <= 17/762, assuming the limit exists, and establish the semi-rigorous bound |freq_1(K) - 1/2| <= 1/46.


A topological approach to induction theorems in Springer theory

Abstract: We give a self-contained account of a construction due to Rossmann which lifts Springer's action of a Weyl group on the cohomology of a Springer fiber to an action on its homotopy type. We use this construction to produce a generalization of an "induction theorem" of Alvis and Lusztig, which relates the Springer representations attached to a reductive group to those attached to a Levi subgroup. Our generalization applies to more general centralizers and to representations of Weyl groups on mod p cohomology.


Smooth and weak synthesis of the anti-diagonal in Fourier algebras of Lie groups

Abstract: Let $G$ be a Lie group of dimension $n$, and let $A(G)$ be the Fourier algebra of $G$. We show that the anti-diagonal $\check{\Delta}_G=\{(g,g^{-1})\in G\times G \mid g\in G\}$ is both a set of local smooth synthesis and a set of local weak synthesis of degree at most $[\frac{n}{2}]+1$ for $A(G\times G)$. We achieve this by using the concept of the cone property in \cite{ludwig-turowska}. For compact $G$, we give an alternative approach to demonstrate the preceding results by applying the ideas developed in \cite{forrest-samei-spronk}. We also present similar results for sets of the form $HK$, where both $H$ and $K$ are subgroups of $G\times G\times G\times G$ of diagonal forms. Our results very much depend on both the geometric and the algebraic structure of these sets.


Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

Abstract: We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4.


Hodge realizations of 1-motives and the derived Albanese

Abstract: We prove that the embedding of the derived category of 1-motives into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor $LAlb$, commute with the Hodge realization.


Second cohomology groups for algebraic groups and their Frobenius kernels

Abstract: Let $G$ be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic $p > 0$. Let $T$ be a maximal split torus in $G$, $B \supset T$ be a Borel subgroup of $G$ and $U$ its unipotent radical. Let $F: G \rightarrow G$ be the Frobenius morphism. For $r \geq 1$ define the Frobenius kernel, $G_r$, to be the kernel of $F$ iterated with itself $r$ times. Define $U_r$ (respectively $B_r$) to be the kernel of the Frobenius map restricted to $U$ (respectively $B$). Let $X(T)$ be the integral weight lattice and $X(T)_+$ be the dominant integral weights. The computations of particular importance are $\h^2(U_1,k)$, $\h^2(B_r,\la)$ for $\la \in X(T)$, $\h^2(G_r,H^0(\la))$ for $\la \in X(T)_+$, and $\h^2(B,\la)$ for $\la \in X(T)$. The above cohomology groups for the case when the field has characteristic 2 one computed in this paper. These computations complete the picture started by Bendel, Nakano, and Pillen for $p \geq 3$ \cite{BNP2}.


The relative commutant of separable C*-algebras of real rank zero

Abstract: We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*-algebra in its ultrapower depend on the choice of the ultrafilter?


Free brace algebras are free prelie algebras

Abstract: Let g be a free brace algebra. This structure implies that g is also a prelie algebra and a Lie algebra. It is already known that g is a free Lie algebra. We prove here that g is also a free prelie algebra, using a description of g with the help of planar rooted trees, a permutative product, and anipulations on the Poincar\'e-Hilbert series of g.


On a computer-aided approach to the computation of Abelian integrals

Abstract: An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four.


A new approach to the family of singularities $Re(x+iy)^m$

Abstract: Assume that $m\ge 2$ and let $l$ be a nonnegative integer with $l\ge m-4$. We give an alternative proof of the fact that any smooth function defined locally around $(0,0)\in \mathbb{R}^2$ with the Taylor power series at $(0,0)$ beginning with $$Re(x+iy)^m+0+...+0$$ ($l$ zeros) is diffeomorphically equivalent to $Re(x+iy)^m$ at $(0,0)$. For $m\ge 5$ and $C\ne 0$ we show that the function $$Re(x+iy)^m+C(x^2+y^2)^{m-2}$$ is not diffeomorphically equivalent to $Re(x+iy)^m$ at $(0,0)$.


Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality

Abstract: We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow to draw some consequences about second order differential operators modeled on these nonsmooth Hormander's vector fields.


An analogue of the Szemeredi Regularity Lemma for bounded degree graphs

Abstract: We show that a sufficiently large graph of bounded degree can be decomposed into quasi-homogeneous pieces. The result can be viewed as a "finitarization" of the classical Farrell-Varadarajan Ergodic Decomposition Theorem.


La droite de Berkovich sur Z

Abstract: We study here the Berkovich line over the ring of integers of a number field. It is a natural object which contains complex and non-Archimedean analytic spaces associated to each place. We prove that this line satisfies good topological and algebraic properties and exhibit a few examples of Stein spaces that lie in it. We derive applications to the study of convergent arithmetic power series: choice of zeroes and poles, noetherianity of global rings and inverse Galois problem. Typical examples of such power series are given by analytic functions on the open complex unit disk whose Taylor development in 0 has integer coefficients.


The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

Abstract: The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of exponential decay of a damped membrane and an approximate controllability result for the bilinear Schr\"odinger equation.


The representation type of Ariki-Koike algebras and cyclotomic q-Schur algebras

Abstract: We give a necessary and sufficient condition on parameters for Ariki-Koike algebras (resp. cyclotomic q-Schur algebras) to be of finite representation type.


Transfer and Chern classes for extraspecial p-groups

Abstract: In the cohomology ring of an extraspecial p-group, the subring generated by Chern classes and transfers is studied. This subring is strictly larger than the Chern subring, but still not the whole cohomology ring, even modulo nilradical. A formula is obtained relating Chern classes to transfers.


Thom series of contact singularities

Abstract: Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterated residue identities are revealed.


On the notion of geometry over $\F_1$

Abstract: We refine the notion of variety over the "field with one element" developed by C. Soul\'e by introducing a grading in the associated functor to the category of sets, and show that this notion becomes compatible with the geometric viewpoint developed by J. Tits. We then solve an open question of C. Soul\'e by proving, using results of J. Tits and C. Chevalley, that Chevalley group schemes are examples of varieties over a quadratic extension of the above "field".


Stability Selection

Abstract: Estimation of structure, such as in variable selection, graphical modelling or cluster analysis is notoriously difficult, especially for high-dimensional data. We introduce stability selection. It is based on subsampling in combination with (high-dimensional) selection algorithms. As such, the method is extremely general and has a very wide range of applicability. Stability selection provides finite sample control for some error rates of false discoveries and hence a transparent principle to choose a proper amount of regularisation for structure estimation. Variable selection and structure estimation improve markedly for a range of selection methods if stability selection is applied. We prove for randomised Lasso that stability selection will be variable selection consistent even if the necessary conditions needed for consistency of the original Lasso method are violated. We demonstrate stability selection for variable selection and Gaussian graphical modelling, using real and simulated data.


Entropy and Poincar\'e recurrence from a geometrical viewpoint

Abstract: We study Poincar\'e recurrence from a purely geometrical viewpoint. We prove that the metric entropy is given by the exponential growth rate of return times to dynamical balls. This is the geometrical counterpart of Ornstein-Weiss theorem. Moreover, we show that minimal return times to dynamical balls grow linearly with respect to its length. Finally, some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures are given.


On $v$--domains and star operations

Abstract: Let $\ast$ be a star operation on an integral domain $D$. Let $\f(D)$ be the set of all nonzero finitely generated fractional ideals of $D$. Call $D$ a $\ast$--Pr\"ufer (respectively, $(\ast, v)$--Pr\"ufer) domain if $(FF^{-1})^{\ast}=D$ (respectively, $(F^vF^{-1})^{\ast}=D$) for all $F\in \f(D)$. We establish that $\ast$--Pr\"ufer domains (and $(\ast, v)$--Pr\"ufer domains) for various star operations $\ast $ span a major portion of the known generalizations of Pr\"{u}fer domains inside the class of $v$--domains. We also use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of Ideals, Academic Press, New York--London, 1971], which gives several equivalent conditions for an integral domain to be a Pr\"ufer domain, as a model, and we show which statements of that theorem on Pr\"ufer domains can be generalized in a natural way and proved for $\ast$--Pr\"ufer domains, and which cannot be. We also show that in a $\ast $--Pr\"ufer domain, each pair of $\ast $-invertible $\ast $-ideals admits a GCD in the set of $\ast $-invertible $\ast $-ideals, obtaining a remarkable generalization of a property holding for the "classical" class of Pr\"ufer $v$--multiplication domains. We also link $D$ being $\ast $--Pr\"ufer (or $(\ast, v)$--Pr\"ufer) with the group Inv$^{\ast}(D)$ of $\ast $-invertible $\ast $-ideals (under $\ast$-multiplication) being lattice-ordered.


Strong Law of Large Numbers for Fragmentation Processes

Abstract: In the spirit of a classical results for Crump-Mode-Jagers processes, we prove a strong law of large numbers for homogenous fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than $\eta$ for $1\geq \eta >0$.


New Discretization of Complex Analysis: The Euclidean and Hyperbolic Planes

Abstract: Few years ago we developed jointly with I.Dynnikov new discretization of complex analysis (DCA) based on the two-dimensional manifolds with colored black/white triangulation. Especially deep results were obtained for the Euclidean plane with equilateral triangle lattice. In the present work we develop a DCA theory for the analogs of equilateral triangle lattice in Hyperbolic plane. Some specific very interesting "dynamical phenomena" appear in this case solving most fundamental boundary problems. Mike Boyle from the University of Maryland helped to use here the methods of symbolic dynamics


Prime numbers in logarithmic intervals

Abstract: Let $X$ be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type $(p,p+h]$, where $p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove that for every $\lambda>1/2$ there exists a positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers $\lambda>1$ with the property that there is a positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains at least a prime number. The last application of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes $p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains no primes.


Poisson Limit for Associated Random Fields

Abstract: We prove that under an easily verifiable set of conditions a sequence of associated random fields converges under rescaling to the Poisson Point Process and give a couple of examples.


Thermodynamic Limit for Large Random Trees

Abstract: We consider Gibbs distributions on finite random plane trees with bounded branching. We show that as the order of the tree grows to infinity, the distribution of any finite neighborhood of the root of the tree converges to a limit. We compute the limiting distribution explicitly and study its properties. We introduce an infinite random tree consistent with these limiting distributions and show that it satisfies a certain form of the Markov property. We also study the growth of this tree and prove several limit theorems including a diffusion approximation.


Coloring Simple Hypergraphs

Abstract: Fix an integer $k \ge 3$. A $k$-uniform hypergraph is simple if every two edges share at most one vertex. We prove that there is a constant $c$ depending only on $k$ such that every simple $k$-uniform hypergraph $H$ with maximum degree $\D$ has chromatic number satisfying $$\chi(H) <c (\frac{\D}{\log \D})^{\frac{1}{k-1}}.$$ This implies a classical result of Ajtai-Koml\'os-Pintz-Spencer-Szemer\'edi and its strengthening due to Duke-Lefmann-R\"odl. The result is sharp apart from the constant $c$.


Presenting the cohomology of a Schubert variety

Abstract: We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform manner by introducing the essential set of a Coxeter group element, generalizing and giving a new characterization of [Fulton '92]'s definition for permutations. Further refinements are obtained in type A.


The Lawson-Yau Formula and its generalization

Abstract: The Euler characteristic of Chow varieties of algebraic cycles of a given degree in complex projective spaces was computed by Blaine Lawson and Stephen Yau by using holomorphic symmetries of cycles spaces. In this paper we compute this in a direct and elementary way and generalize this formula to the l-adic Euler-Poincare characteristic for Chow varieties over any algebraically closed field. Moreover, the Euler characteristic for Chow varieties with certain group action is calculated. In particular, we calculate the Euler characteristic of the space of right quaternionic cycles of a given dimension and degree in complex projective spaces.


Uniform estimates for order statistics and Orlicz functions

Abstract: We establish uniform estimates for order statistics of sequences of independent identically distributed random variables with log-concave distribution in terms of Orlicz norms associated with the distribution function of the random variables.


Derived categories of small toric Calabi-Yau 3-folds and counting invariants

Abstract: We first construct a derived equivalence between a small crepant resolution of an affine toric Calabi-Yau 3-fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wall-crossing formula for the generating function of the counting invariants of perverse coherent systems. As an application we provide certain equations on Donaldson-Thomas, Pandeharipande-Thomas and Szendroi's invariants. Finally, we show that moduli spaces associated with a quiver given by successive mutations are realized as the moduli spaces associated the original quiver by changing the stability conditions.


Multidimensional Dilation Operators, Boyd and Shimogaki Indices of Bilateral Weight Grand Lebesque Spaces

Abstract: In this paper we compute the norm of dilation operators, multidimensional Boyd`s and Shimogaki`s indices in the Bilateral Grand Lebesgue Spaces and consider some applications.


Partial duality and Bollobas and Riordan's ribbon graph polynomial

Abstract: Recently S. Chmutov introduced a generalization of the dual of a ribbon (or embedded) graph and proved a relation between Bollobas and Riordan's ribbon graph polynomial of a ribbon graph and its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation via the homfly polynomial of a knot.


Stable solutions of elliptic equations on Riemannian manifolds

Abstract: This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far as stable solutions are concerned, we derive a new weighted Poincar\'e inequality which allows to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.


Weakly group-theoretical and solvable fusion categories

Abstract: We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups - weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq^2, where p,q,r are distinct primes.


Simultaneous similarity and triangularization of sets of 2 by 2 matrices

Abstract: Let $\mathcal{A}=(A_{1},...,A_{n},...)$ be a finite or infinite sequence of $2\times2$ matrices with entries in an integral domain. We show that, except for a very special case, $\mathcal{A}$ is (simultaneously) triangularizable if and only if all pairs $(A_{j},A_{k})$ are triangularizable, for $1\leq j,k\leq\infty$. We also provide a simple numerical criterion for triangularization. Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic $\neq2$. We also describe canonical forms for sequences of $2\times2$ matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.


Partial Data for the Calderon Problem in Two Dimensions

Abstract: We show in two dimensions that measuring Dirichlet data for the conductivity equation on an open subset of the boundary and, roughly speaking, Neumann data in slightly larger set than the complement uniquely determines the conductivity on a simply connected domain. The proof is reduced to show a similar result for the Schr\"odinger equation. Using Carleman estimates with degenerate weights we construct appropriate complex geometrical optics solutions to prove the results.


Integrally Closed Ideals on Log Terminal Surfaces are Multiplier Ideals

Abstract: We show that all integrally closed ideals on log terminal surfaces are multiplier ideals by extending an existing proof for smooth surfaces.


Groups with the same cohomology as their pro-$p$ completions

Abstract: For any prime $p$ and group $G$, denote the pro-$p$ completion of $G$ by $\hat{G}^p$. Let $\mathcal{C}$ be the class of all groups $G$ such that, for each natural number $n$ and prime number $p$, $H^n(\hat{G^p},\mathbb Z/p)\cong H^n(G, \mathbb Z/p)$, where $\mathbb Z/p$ is viewed as a discrete, trivial $\hat{G}^p$-module. In this article we identify certain kinds of groups that lie in $\mathcal{C}$. In particular, we show that right-angled Artin groups are in $\mathcal{C}$ and that this class also contains some special types of free products with amalgamation.


Commutators on $\ell_1$

Abstract: The main result is that the commutators on $\ell_1$ are the operators not of the form $\lambda I + K$ with $\lambda\neq 0$ and $K$ compact. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 1513 - 1534) to obtain this result and use this generalization to obtain partial results about the commutators on spaces $\X$ which can be represented as $\displaystyle \X\simeq (\bigoplus_{i=0}^{\infty} \X)_{p}$ for some $1\leq p<\infty$ or $p=0$. In particular, it is shown that every compact operator on $L_1$ is a commutator. A characterization of the commutators on $\ell_{p_1}\oplus\ell_{p_2}\oplus...\oplus\ell_{p_n}$ is given. We also show that strictly singular operators on $\ell_{\infty}$ are commutators.


Numerical Solution of Multiple Nonlinear Volterra Integral Equations

Abstract: We analyze a discretization method for solving nonlinear integral equations that contain multiple integrals. These equations include integral equations with a Volterra series, instead of a single integral term, on one side of the equation. We prove existence and uniqueness of solutions, and convergence and estimates of the order of convergence for the numerical methods of solution.


Toeplitz-composition C*-algebras for certain finite Blaschke products

Abstract: Let R be a finite Blaschke product of degree at least two with R(0)=0. Then there exists a relation between the associated composition operator C_R on the Hardy space and the C*-algebra associated with the complex dynamical system on the Julia set of R. We study the C*-algebra generated by both the composition operator C_R and the Toeplitz operator T_z to show that the quotient algebra by the ideal of the compact operators is isomorphic to the C*-algebra associated with the complex dynamical system, which is simple and purely infinite.


On rigid Hirzebruch genera

Abstract: The classical multiplicative (Hirzebruch) genera of manifolds have the wonderful property which is called rigidity. Rigidity of a genus h means that if a compact connected Lie group G acts on a manifold X, then the equivariant genus h^G(X) is independent on G, i.e. h^G(X)=h(X). In this paper we are considering the rigidity problem for complex manifolds. In particular, we are proving that a genus is rigid if and only if it is a generalized Todd genus.


Some Notes on Standard Borel and Related Spaces

Abstract: These notes give an elementary approach to parts of the theory of standard Borel and analytic spaces.


Desingularization of some moduli scheme of stable sheaves on a surface

Abstract: Let $X$ be a nonsingular projective surface over an algebraically closed field with characteristic zero, and $H_-$ and $H_+$ ample line bundles on $X$ separated by only one wall of type $(c_1,c_2)$. Suppose the moduli scheme $M(H_-)$ of rank-two $H_-$-stable sheaves with Chern classes $(c_1,c_2)$ is non-singular. We shall construct a desingularization of $M(H_+)$ by using $M(H_-)$. As an application, we consider whether singularities of $M(H_+)$ are terminal or not when $X$ is ruled or elliptic.


Lifting Galois representations over arbitrary number fields

Abstract: It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero $p$-adic representation, if local lifting problems at places above $p$ are unobstructed.


Bandlet Image Estimation with Model Selection

Abstract: To estimate geometrically regular images in the white noise model and obtain an adaptive near asymptotic minimaxity result, we consider a model selection based bandlet estimator. This bandlet estimator combines the best basis selection behaviour of the model selection and the approximation properties of the bandlet dictionary. We derive its near asymptotic minimaxity for geometrically regular images as an example of model selection with general dictionary of orthogonal bases. This paper is thus also a self contained tutorial on model selection with orthogonal bases dictionary.


On the geometry of biharmonic submanifolds in Sasakian space forms

Abstract: We classify all proper-biharmonic Legendre curves in a Sasakian space form and point out some of their geometric properties. Then we provide a method for constructing anti-invariant proper-biharmonic submanifolds in Sasakian space forms. Finally, using the Boothby-Wang fibration, we determine all proper-biharmonic Hopf cylinders over homogeneous real hypersurfaces in complex projective spaces.


The vector-valued non-homogeneous Tb theorem

Abstract: The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces of functions with values in a Banach space with the unconditionality property of martingale differences (UMD). The new proof deals directly with all Lebesgue exponents p in the range 1<p<infinity, and relies on delicate estimates for the non-homogenous "Haar" functions, as well as McConnell's (1989) decoupling inequality for tangent martingale differences.


Degree-one maps, surgery and four-manifolds

Abstract: We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold $M$ to a closed, oriented 3-manifold $N$ if and only if $M$ can be obtained from $N$ by surgery about a link in $N$ each of whose components is an unknot. We use this to interpret the existence of degree-one maps between closed 3-manifolds in terms of smooth 4-manifolds. More precisely, we show that there is a degree-one map from $M$ to $N$ if and only if there is a smooth embedding of $M$ in $W=(N\times I)#_n \bar{\C P^2}#_m {\C P^2}$, for some $m\geq 0$, $n\geq 0$ which separates the boundary components of $W$. This is motivated by the relation to topological field theories, in particular the invariants of Ozsvath and Szabo.


How to Construct a Dirac Operator in Infinite Dimensions

Abstract: We define the notion of a co-Riemannian structure and show how it can be used to define the Dirac operator on an appropriate infinite dimensional manifold. In particular, this approach works for the smooth loop space of a so-called string manifold.


A Proof On Arnold-Chekanov Conjecture

Abstract: In this article, we give a proof on the Arnold-Chekanov Lagrangian intersection conjecture on the cotangent bundles and its generalizations.


T-entropy and Variational principle for the spectral radius of weighted shift operators

Abstract: In this paper we introduce a new functional invariant of discrete time dynamical systems -- the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on $L^1(X,m)$ generated by the dynamical system. This result is called the Variational principle and is similar to the classical variational principle for the topological pressure.


Incompressibility and Least-Area surfaces

Abstract: We show that if $F$ is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold $M$ such that for each Riemannian metric $g$ on $M$, $F$ is isotopic to a least-area surface $F(g)$, then $F$ is incompressible.


The co-Riemannian Structure of Smooth Loop Spaces

Abstract: We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.


Algebraic and Geometric intersection numbers for free groups

Abstract: We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of $S^2\times S^1$.


T-entropy and Variational Principle for the spectral radius of transfer and weighted shift operators

Abstract: The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or it is a topological Markov chain. As the main summands these principles contain the integrals over invariant measures and the Kolmogorov--Sinai entropy. In the article we derive the Variational Principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov--Sinai entropy but a new invariant of entropy type -- the t-entropy.


Steiner t-designs for large t

Abstract: One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.


An orthogonality relation for multivariable Bessel polynomials

Abstract: In a recent paper we introduced a multivariable generalisation of the Bessel polynomials, depending on one extra parameter, and related to the so-called hyperbolic Sutherland model with external Morse potential. In this paper we obtain a corresponding multivariable generalisation of a well-known orthogonality relation for the (one-variable) Bessel polynomials due to Krall and Frink.


On the norm of the Beurling-Ahlfors operator in several dimensions

Abstract: The Lp operator norm of the generalized Beurling-Ahlfors transformation in n variables is at most (n/2+1)(p-1) for p>2. This improves on earlier results in all dimensions n>2. The proof is based on the heat extension and relies at the bottom on Burkholder's sharp inequality for martingale transforms.


Anosov Automorphisms of Nilpotent Lie Algebras

Abstract: Each matrix A in GL_n(Z) naturally defines an automorphism f of the free r-step nilpotent Lie algebra on n generators. We study the relationship between the matrix A and the eigenvalues and rational invariant subspaces for f. We give applications to the study of Anosov automorphisms.


On Quasitoric Orbifolds

Abstract: Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invariants of quasitoric orbifolds. In particular, we discuss equivalent definitions and determine the orbifold fundamental group, rational homology groups and cohomology ring of a quasitoric orbifold. We determine whether any quasitoric orbifold can be the quotient of a smooth manifold by a finite group action or not. We prove existence of stable almost complex structure and describe the Chen-Ruan cohomology groups of an almost complex quasitoric orbifold.


Classifying smooth lattice polytopes via toric fibrations

Abstract: We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a recent paper Batyrev and Nill have suggested that there should be a bound, N(d), such that every lattice polytope of degree d and dimension at least N(d) decomposes as a Cayley sum. We give a sharp answer to this question for smooth Q-normal polytopes. We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a Cayley sum of strictly combinatorially equivalent polytopes if n is greater than or equal to 2d+1. The proof relies on the study of the nef value morphism associated to the corresponding toric embedding.


A time-variant norm constrained interpolation problem arising from relaxed commutant lifting

Abstract: A time-variant analogue of an interpolation problem equivalent to the relaxed commutant lifting problem is introduced and studied. In a somewhat less general form the problem already appears in the analysis of the set of all solutions to the three chain completion problem. The interpolants are upper triangular operator matrices of which the columns induce contractive operators. The set of all solutions of the problem is described explicitly. The results presented are time-variant analogues of the main theorems in [A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to the relaxed commutant lifting problem, Acta Sci. Math. (Szeged) 72 (2006), 299--318].


Milnor fibers over singular toric varieties and nearby cycle sheaves

Abstract: We propose a new sheaf-theoretical method for the calculation of the monodromy zeta functions of Milnor fibrations. As an application, classical formulas of Kushnirenko and Varchenko etc. concerning polynomials on $\CC^n$ will be generalized to polynomial functions on any toric variety.


Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves

Abstract: By using sheaf-theoretical methods such as constructible sheaves, we generalize the formula of Libgober-Sperber concerning the zeta functions of monodromy at infinity of polynomial maps into various directions. In particular, some formulas for the zeta functions of global monodromy along the fibers of bifurcation points of polynomial maps will be obtained.


Stochastic analysis of Bernoulli processes

Abstract: These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.


Economical toric spines via Cheeger's Inequality

Abstract: Let $G_{\infty}=(C_m^d)_{\infty}$ denote the graph whose set of vertices is $\{1,..., m\}^d$, where two distinct vertices are adjacent iff they are either equal or adjacent in $C_m$ in each coordinate. Let $G_{1}=(C_m^d)_1$ denote the graph on the same set of vertices in which two vertices are adjacent iff they are adjacent in one coordinate in $C_m$ and equal in all others. Both graphs can be viewed as graphs of the $d$-dimensional torus. We prove that one can delete $O(\sqrt d m^{d-1})$ vertices of $G_1$ so that no topologically nontrivial cycles remain. This improves an $O(d^{\log_2 (3/2)}m^{d-1})$ estimate of Bollob\'as, Kindler, Leader and O'Donnell. We also give a short proof of a result implicit in a recent paper of Raz: one can delete an $O(\sqrt d/m)$ fraction of the edges of $G_{\infty}$ so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O'Donnell, Rao and Wigderson; there is a subset of the continuous $d$-dimensional torus of surface area $O(\sqrt d)$ that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no- nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues.