Set Linear Algebra and Set Fuzzy Linear Algebra

Abstract: In this book, the authors define the new notion of set vector spaces which is the most generalized form of vector spaces. Set vector spaces make use of the least number of algebraic operations, therefore, even a non-mathematician is comfortable working with it. It is with the passage of time, that we can think of set linear algebras as a paradigm shift from linear algebras. Here, the authors have also given the fuzzy parallels of these new classes of set linear algebras. This book is divided into seven chapters. The first chapter briefly recalls some of the basic concepts in order to make this book self-contained. Chapter two introduces the notion of set vector spaces which is the most generalized concept of vector spaces. Set vector spaces lends itself to define new classes of vector spaces like semigroup vector spaces and group vector spaces. These are also generalization of vector spaces. The fuzzy analogue of these concepts are given in Chapter three. In Chapter four, set vector spaces are generalized to biset bivector spaces and not set vector spaces. This is done taking into account the advanced information technology age in which we live. As mathematicians, we have to realize that our computer-dominated world needs special types of sets and algebraic structures. Set n-vector spaces and their generalizations are carried out in Chapter five. Fuzzy n-set vector spaces are introduced in the sixth chapter. The seventh chapter suggests more than three hundred problems.


A note on K\"ahler-Ricci soliton

Abstract: In this note we provide a proof of the following: Any compact KRS with positive bisectional curvature is biholomorphic to the complex projective space. As a corollary, we obtain an alternative proof of the Frankel conjecture by using the K\"ahler-Ricci flow.


A direct proof of Z-stability for AH algebras of bounded topological dimension

Abstract: We prove that a unital simple approximately homogeneous (AH) C*-algebra with no dimension growth absorbs the Jiang-Su algebra tensorially without appealing to the classification theory of these algebras. Our main result continues to hold under the slightly weaker hypothesis of exponentially slow dimension growth.


Generating sequences and Poincar\'e series for a finite set of plane divisorial valuations

Abstract: Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, $\gr_V R$. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ of divisorial valuations, and the relationship between the value semigroup of $C$ and the semigroups of the sets $V^{(k)}$, allow us to obtain the (finite) minimal generating sequences for $C$ as well as for $V$. We also analyze the structure of the homogeneous components of $\gr_V R$. The study of their dimensions allows us to relate the Poincar\'e series for $V$ and for a general curve $C$ of $V$. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincar\'e series of $V$. Moreover, the Poincar\'e series of $C$ could be seen as the limit of the series of $V^{(k)}$, $k\ge 0$.


Maximizing Sum Rates in Gaussian Interference-limited Channels

Abstract: We study the problem of maximizing sum rates in a Gaussian interference-limited channel that models multiuser communication in a CDMA wireless network or DSL cable binder. Using tools from nonnegative irreducible matrix theory, in particular the Perron-Frobenius Theorem and the Friedland-Karlin inequalities, we provide insights into the structural property of optimal power allocation strategies that maximize sum rates. Our approach is similar to the treatment of linear models in mathematical economies, where interference is viewed in the context of competition. We show that this maximum problem can be restated as a maximization problem of a convex function on a closed convex set. We suggest three algorithms to find the exact and approximate values of the optimal sum rates. In particular, our algorithms exploit the eigenspace of specially crafted nonnegative {\it interference matrices}, which, with the use of standard optimization tools, can provide useful upper bounds and feasible solutions to the nonconvex problem.


Rational convexity of non generic immersed lagrangian submanifolds

Abstract: We prove that an immersed lagrangian submanifold in $\C^n$ with quadratic self-tangencies is rationally convex. This generalizes former results for the embedded and the immersed transversal cases.


Thompson's Group F and Uniformly Finite Homology

Abstract: This paper demonstrates the uniformly finite homology developed by Block and Weinberger and its relationship to amenable spaces via applications to the Cayley graph of Thompson's Group F. In particular, a certain class of subgraph of F is shown to be non-amenable. This shows that if F is amenable, these subsets (which include every finitely generated submonoid of the positive monoid of F) must necessarily have measure zero.


Detecting automorphic orbits in free groups

Abstract: We present an effective algorithm for detecting automorphic orbits in free groups, as well as a number of algorithmic improvements of train tracks for free group automorphisms.


Statistical Behaviour of the Leaves of Riccati Foliations

Abstract: We introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension 1 and hyperbolic, corresponding to the unique complete metric of curvature -1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivisation of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation r:pi_1(S) -> GL(n,C). We give conditions under which the foliated geodesic flow has a generic repellor-attractor statistical dynamics. That is, there are measures m- and m+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to m- and for positive time to m+ (i.e. m+ is the unique SRB-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behavior of the leaves of the foliation.


Optimal designs for mixed models in experiments based on ordered units

Abstract: We consider experiments for comparing treatments using units that are ordered linearly over time or space within blocks. In addition to the block effect, we assume that a trend effect influences the response. The latter is modeled as a smooth component plus a random term that captures departures from the smooth trend. The model is flexible enough to cover a variety of situations; for instance, most of the effects may be either random or fixed. The information matrix for a design will be a function of several variance parameters. While data will shed light on the values of these parameters, at the design stage, they are unlikely to be known, so we suggest a maximin approach, in which a minimal information matrix is maximized. We derive maximin universally optimal designs and study their robustness. These designs are based on semibalanced arrays. Special cases correspond to results available in the literature.


Symplectic topology of SU(2)-representation varieties and link homology, I: Symplectic braid action and the first Chern class

Abstract: There are some similarities between cohomology of SU(2)-representation varieties of the fundamental group of some link complements and the Khovanov homology of the links. We start here a program to explain a possible source of these similarities. We introduce a symplectic manifold ${\mathscr M}$ with an action of the braid group $B_{2n}$ preserving the symplectic structure. The action allows to associate a Lagrangian submanifold of ${\mathscr M}$ to every braid. The representation variety of a link can then be described as the intersection of such Lagrangian submanifolds, given a braid presentation of the link. We expect this to go some way in explaining the similarities mentioned above.


Composite quantile regression and the oracle Model Selection Theory

Abstract: Coefficient estimation and variable selection in multiple linear regression is routinely done in the (penalized) least squares (LS) framework. The concept of model selection oracle introduced by Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] characterizes the optimal behavior of a model selection procedure. However, the least-squares oracle theory breaks down if the error variance is infinite. In the current paper we propose a new regression method called composite quantile regression (CQR). We show that the oracle model selection theory using the CQR oracle works beautifully even when the error variance is infinite. We develop a new oracular procedure to achieve the optimal properties of the CQR oracle. When the error variance is finite, CQR still enjoys great advantages in terms of estimation efficiency. We show that the relative efficiency of CQR compared to the least squares is greater than 70% regardless the error distribution. Moreover, CQR could be much more efficient and sometimes arbitrarily more efficient than the least squares. The same conclusions hold when comparing a CQR-oracular estimator with a LS-oracular estimator.


Lower order terms for the one-level densities of symmetric power $L$-functions in the level aspect

Abstract: In a previous paper, the authors determined, among other things, the main terms for the one-level densities for low-lying zeros of symmetric power L-functions in the level aspect. In this paper, the lower order terms of these one-level densities are found. The combinatorial difficulties, which should arise in such context, are drastically reduced thanks to Chebyshev polynomials, which are the characters of the irreducible representations of SU(2). %


Adaptive estimation of and oracle inequalities for probability densities and characteristic functions

Abstract: The theory of adaptive estimation and oracle inequalities for the case of Gaussian-shift--finite-interval experiments has made significant progress in recent years. In particular, sharp-minimax adaptive estimators and exact exponential-type oracle inequalities have been suggested for a vast set of functions including analytic and Sobolev with any positive index as well as for Efromovich--Pinsker and Stein blockwise-shrinkage estimators. Is it possible to obtain similar results for a more interesting applied problem of density estimation and/or the dual problem of characteristic function estimation? The answer is ``yes.'' In particular, the obtained results include exact exponential-type oracle inequalities which allow to consider, for the first time in the literature, a simultaneous sharp-minimax estimation of Sobolev densities with any positive index (not necessarily larger than 1/2), infinitely differentiable densities (including analytic, entire and stable), as well as of not absolutely integrable characteristic functions. The same adaptive estimator is also rate minimax over a familiar class of distributions with bounded spectrum where the density and the characteristic function can be estimated with the parametric rate.


Vari\'et\'es rationnellement connexes sur un corps alg\'ebriquement clos

Abstract: These are lectures notes on rationally connected varieties, written for the "Etats de la Recherche" of the French Mathematical Society held in Strasbourg (May 2008). We focus on geometric aspects. These notes have been written in order that a wide audience can easily read them, except maybe the last section, a bit more technical, where we give the proof of Shokurov rational connectedness conjecture following Hacon and McKernan. ----- Ce sont les notes d'un mini-cours sur les vari\'et\'es rationnellement connexes, \'ecrit pour les Etats de la Recherche de la Soci\'et\'e Math\'ematique de France (Strasbourg, 2008). On met l'accent sur les aspects g\'eom\'etriques. Ce cours est r\'edig\'e dans l'espoir de s'adresser \`a un public large, \`a l'exception peut-\^etre du \S 7, o\`u nous donnons les grandes lignes de la preuve de la conjecture de connexit\'e rationnelle de Shokurov par Hacon et McKernan, plus technique et o\`u les pr\'erequis sont un peu plus importants.


Admissible predictive density estimation

Abstract: Let $X|\mu\sim N_p(\mu,v_xI)$ and $Y|\mu\sim N_p(\mu,v_yI)$ be independent $p$-dimensional multivariate normal vectors with common unknown mean $\mu$. Based on observing $X=x$, we consider the problem of estimating the true predictive density $p(y|\mu)$ of $Y$ under expected Kullback--Leibler loss. Our focus here is the characterization of admissible procedures for this problem. We show that the class of all generalized Bayes rules is a complete class, and that the easily interpretable conditions of Brown and Hwang [Statistical Decision Theory and Related Topics (1982) III 205--230] are sufficient for a formal Bayes rule to be admissible.


$L^2-$interpolation with error and size of spectra

Abstract: Given a compact set $S$ and a uniformly discrete sequence $\La$, we show that "approximate interpolation" of delta--functions on $\La$ by a bounded sequence of $L^2-$functions with spectra in $S$ implies an estimate on measure of $S$ through the density of $\La$.


Kostant's problem and parabolic subgroups

Abstract: Let $\frak g$ be a finite dimensional complex semi-simple Lie algebra with Weyl group $W$ and simple reflections $S$. For $I\subseteq S$ let $\frak g_I$ be the corresponding semi-simple subalgebra of $\frak g$. Denote by $W_I$ the Weyl group of $\frak g_I$ and let $w_o$ and $w^I_o$ be the longest elements of $W$ and $W_I$, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight $\frak g_I$-module $L_I(x)$ of highest weight $x\cdot 0$, $x\in W_I$, as the answer for the simple highest weight $\frak g$-module $L(x w^I_o w_o)$ of highest weight $(x w^I_o w_o)\cdot 0$. We also give a new description of the unique quasi-simple quotient of the Verma module $\Delta(e)$ with the same annihilator as $L(y)$, $y\in W$.


Wave equation and multiplier estimates on Damek-Ricci spaces

Abstract: Let S be a Damek-Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators associated with L. This generalizes previous results proved by D. Mueller and C. Thiele on ax+b-groups. We also prove pointwise estimates of the gradient of these convolution kernels. As a corollary we reprove basic multiplier estimates from previous papers of W. Hebisch and T. Steger and M. Vallarino by different methods. Finally we derive Sobolev estimates for the solution to the wave equation associated with L.


A Point is Normal for Almost All Maps $\beta x + \alpha \mod 1$ or Generalized $\beta$-Maps

Abstract: We consider the map $T_{\alpha,\beta}(x):= \beta x + \alpha \mod 1$, which admits a unique probability measure of maximal entropy $\mu_{\alpha,\beta}$. For $x \in [0,1]$, we show that the orbit of $x$ is $\mu_{\alpha,\beta}$-normal for almost all $(\alpha,\beta)\in[0,1)\times(1,\infty)$ (Lebesgue measure). Nevertheless we construct analytic curves in $[0,1)\times(1,\infty)$ along them the orbit of $x=0$ is at most at one point $\mu_{\alpha,\beta}$-normal. These curves are disjoint and they fill the set $[0,1)\times(1,\infty)$. We also study the generalized $\beta$-maps (in particular the tent map). We show that the critical orbit $x=1$ is normal with respect to the measure of maximal entropy for almost all $\beta$.


A Note on Dominant Contractions of Jordan Algebras

Abstract: In the paper we consider two positive contractions $T,S:L^{1}(A,\tau)\longrightarrow L^{1}(A,\tau)$ such that $T\leq S$, here $(A,\t)$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\in\mathbb{N}$ such that $\|S^{n_{0}}-T^{n_{0}}\|<1$. Then we prove that $\|S^{n}-T^{n}\|<1$ holds for every $n\geq n_{0}.$


Parametric bootstrap approximation to the distribution of EBLUP and related prediction intervals in linear mixed models

Abstract: Empirical best linear unbiased prediction (EBLUP) method uses a linear mixed model in combining information from different sources of information. This method is particularly useful in small area problems. The variability of an EBLUP is traditionally measured by the mean squared prediction error (MSPE), and interval estimates are generally constructed using estimates of the MSPE. Such methods have shortcomings like under-coverage or over-coverage, excessive length and lack of interpretability. We propose a parametric bootstrap approach to estimate the entire distribution of a suitably centered and scaled EBLUP. The bootstrap histogram is highly accurate, and differs from the true EBLUP distribution by only $O(d^3n^{-3/2})$, where $d$ is the number of parameters and $n$ the number of observations. This result is used to obtain highly accurate prediction intervals. Simulation results demonstrate the superiority of this method over existing techniques of constructing prediction intervals in linear mixed models.


On the ergodicity of the adaptive Metropolis algorithm on unbounded domains

Abstract: This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen [Bernoulli 7 (2001) 223--242] for target distributions with a noncompact support. The conditions ensuring a strong law of large numbers require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab. 16 (2006) 1462--1505].


Additivity of Heegaard genera of bounded surface sums

Abstract: Let $M$ be a surface sum of 3-manifolds $M_1$ and $M_2$ along a bounded connected surface $F$ and $\partial_i$ be the component of $\partial M_i$ containing $F$. If $M_i$ has a high distance Heegaard splitting, then any minimal Heegaard splitting of $M$ is the amalgamation of those of $M^1, M^2$ and $M^*$, where $M^i=M_i\setminus\partial_i\times I$, and $M^{*}=\partial_1\times I\cup_{F} \partial_2\times I$. Furthermore, once both $\partial_i\setminus F$ are connected, then $g(M) = Min\bigl\{g(M_1)+g(M_2), \alpha\bigr\}$, where $\alpha = g(M_1) + g(M_2) + 1/2(2\chi(F) + 2 - \chi(\partial_1) - \chi(\partial_2)) - Max\bigl\{g(\partial_1), g(\partial_2)\bigl\}$; in particular $g(M)=g(M_1)+g(M_2)$ if and only if $\chi(F)\geq 1/2Max\bigl\{\chi(\partial_1), \chi(\partial_2)\bigr\}.$ The proofs rely on Scharlemann-Tomova's theorem.


q,t-Fuss-Catalan numbers for complex reflection groups

Abstract: In type A, the q,t-Fuss -Catalan numbers can be defined as a bigraded Hilbert series of a module associated to the symmetric group S_n. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in q and t. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress.


Data-driven Sobolev tests of uniformity on compact Riemannian manifolds

Abstract: Data-driven versions of Sobolev tests of uniformity on compact Riemannian manifolds are proposed. These tests are invariant under isometries and are consistent against all alternatives. The large-sample asymptotic null distributions are given.


Construction of A Lattice on the completion space of an algebra and an isomorphism to its Caratheodory Extension

Abstract: In this paper, we will show how the Caratheodory Extension process is intimately related to the metric completion process. In particular, it will be shown how one is able to construct a lattice on the completion and to obtain an isomorphism to its Caratheodory Extension.


New Techniques for Empirical Process of Dependent Data

Abstract: We present a new technique for proving empirical process invariance principle for stationary processes $(X_n)_{n\geq 0}$. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions $(f(X_n))_{n\geq 0}$, not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques.


Decomposition rank and Z-stability

Abstract: We show that separable, simple, unital C*-algebras with finite decomposition rank absorb the Jiang-Su algebra Z tensorially. This has a number of consequences for Elliott's program to classify nuclear C*-algebras by their K-theory data. In particular, it completes the classification of C*-algebras associated to uniquely ergodic, smooth, minimal dynamical systems by their ordered K-groups.


Symmetry for solutions of two-phase semilinear elliptic equations on hyperbolic space

Abstract: Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $\Delta u = f(u)$ on hyperbolic space $\HH^n$ must reduce to functions of one variable provided they admit asymptotic boundary values on the infinite boundary of $\HH^n$ which are invariant under a cohomogeneity one subgroup of the group of isometries of $\HH^n$. We also prove existence of these one-dimensional solutions.


Branchfolds and rational conifolds

Abstract: We extend the concept of orbifold to that of branchfold, in order to allow any cone singularities with rational angles, and show why branchfolds naturally fit in the theory of branched coverings. Then, we obtain a geometric goodness theorem for branchfolds and apply it to prove that a conifold can be endowed with branchfold structure if and only if it has locally finite holonomy.


Remarks on an 1828 theorem of Clausen

Abstract: We list some explicit calculations related to a theorem of Clausen originally published in 1828, more commonly known as the result that describes the linear third order differential equation satisfied by the squares and the product of any two solutions of a linear second order differential equation in the real domain. The case of the cube of a solution dates to Appell, 1880. Although not commonly known and perhaps even new to some extent we show that the fourth and fifth powers of such solutions actually satisfy a linear differential equation of order five and six respectively provided the coefficients are sufficiently smooth. Indeed, it is the case that given any solution of a linear second order equation its m-th power satisfies an effectively computable linear differential equation of order m+1.


Optimal rank-based tests for homogeneity of scatter

Abstract: We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in $m$ elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavy-tailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normal-score version outperforms traditional Gaussian likelihood ratio tests and their pseudo-Gaussian robustifications under a very broad range of non-Gaussian densities including, for instance, all multivariate Student and power-exponential distributions.


Quasi-factors for infinite-measure preserving transformations

Abstract: This paper is a study of Glasner's definition of quasi-factors in the setting of infinite-measure preserving system. The existence of a system with zero Krengel entropy and a quasi-factor with positive entropy is obtained. On the other hand, relative zero-entropy for conservative systems implies relative zero-entropy of any quasi-factor with respect to its natural projection onto the factor. This extends (and is based upon) results of Glasner, Thouvenot and Weiss. Following and extending Glasner and Weiss, we also prove that any conservative measure preserving system with positive entropy in the sense of Danilenko and Rudolph admits any probability preserving system with positive entropy as a factor. Some applications and connections with Poisson-suspensions are presented.


Analytic pro-p groups of small dimensions

Abstract: According to Lazard, every p-adic Lie group contains an open pro-p subgroup which is saturable. This can be regarded as the starting point of p-adic Lie theory, as one can naturally associate to every saturable pro-p group G a Lie lattice L(G) over the p-adic integers. Essential features of saturable pro-p groups include that they are torsion-free and p-adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p-adic analytic pro-p group of dimension less than p is saturable. This leads to useful consequences and interesting questions. For instance, we give an effective classification of 3-dimensional soluble torsion-free p-adic analytic pro-p groups for p > 3. Our approach via Lie theory is comparable with the use of Lazard's correspondence in the classification of finite p-groups of small order.


Multivariate spacings based on data depth: I. Construction of nonparametric multivariate tolerance regions

Abstract: This paper introduces and studies multivariate spacings. The spacings are developed using the order statistics derived from data depth. Specifically, the spacing between two consecutive order statistics is the region which bridges the two order statistics, in the sense that the region contains all the points whose depth values fall between the depth values of the two consecutive order statistics. These multivariate spacings can be viewed as a data-driven realization of the so-called ``statistically equivalent blocks.'' These spacings assume a form of center-outward layers of ``shells'' (``rings'' in the two-dimensional case), where the shapes of the shells follow closely the underlying probabilistic geometry. The properties and applications of these spacings are studied. In particular, the spacings are used to construct tolerance regions. The construction of tolerance regions is nonparametric and completely data driven, and the resulting tolerance region reflects the true geometry of the underlying distribution. This is different from most existing approaches which require that the shape of the tolerance region be specified in advance. The proposed tolerance regions are shown to meet the prescribed specifications, in terms of $\beta$-content and $\beta$-expectation. They are also asymptotically minimal under elliptical distributions. Finally, a simulation and comparison study on the proposed tolerance regions is presented.


A general trimming approach to robust Cluster Analysis

Abstract: We introduce a new method for performing clustering with the aim of fitting clusters with different scatters and weights. It is designed by allowing to handle a proportion $\alpha$ of contaminating data to guarantee the robustness of the method. As a characteristic feature, restrictions on the ratio between the maximum and the minimum eigenvalues of the groups scatter matrices are introduced. This makes the problem to be well defined and guarantees the consistency of the sample solutions to the population ones. The method covers a wide range of clustering approaches depending on the strength of the chosen restrictions. Our proposal includes an algorithm for approximately solving the sample problem.


Energy Critical NLS in two space dimensions

Abstract: We investigate the initial value problem for a defocusing nonlinear Schr\"odingerequation with exponential nonlinearity. We identify subcritical, critical and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.


A fourth moment inequality for functionals of stationary processes

Abstract: In this paper, a fourth moment bound for partial sums of functional of strongly ergodic Markov chain is established. This type of inequality plays an important role in the study of empirical process invariance principle. This one is specially adapted to the technique of Dehling, Durieu and Voln\'y (2008). The same moment bound can be proved for dynamical system whose transfer operator has some spectral properties. Examples of applications are given.


Adaptive goodness-of-fit tests based on signed ranks

Abstract: Within the nonparametric regression model with unknown regression function $l$ and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis $l=0$ against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against H\"{o}lder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to 1 in case of homoscedastic Gaussian errors within a broad range of H\"{o}lder classes simultaneously.


Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets

Abstract: Let P be an hyperplane in R^N, and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an epsilon > 0, such that if K is a Reifenberg-flat set in B(0; 1), a ball in R^N, that contains the origin, with d_H(K; P) <epsilon, and if u is an energy minimizing function in B(0; 1)\K with restricted values on @B(0; 1)\K, then the normalized energy of u in B(0; r)\K is bounded by the normalized energy of u in B(0; 1)\K. We also prove the same result in R^3 when K is a epsilon-minimal set, that is a generalization of Reifenberg-flat sets with minimal cones of type Y and T. Moreover, the result is still true for a further generalization of sets called (eps; eps_0)-minimal. This article is a preliminary study for a forthcoming paper where a regularity result for the singular set of the Mumford-Shah functional close to minimal cones in R^3 is proved by the same author.


Regularity of the singular set for Mumford-Shah minimizers in R^3 near a minimal cone

Abstract: We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R^3, and if x, K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 degrees angles) or T (cone over a regular tetrahedron centered at the origin) in terms of Hausdorff distance in B(x; r), then K is C^1,alpha equivalent to the minimal cone in B(x; cr) where c < 1 is an universal constant.


Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves

Abstract: We describe the use of explicit isogenies to translate instances of the Discrete Logarithm Problem (DLP) from Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, where they are vulnerable to faster index calculus attacks. We provide explicit formulae for isogenies with kernel isomorphic to $(\ZZ/2\ZZ)^3$ (over an algebraic closure of the base field) for any hyperelliptic genus 3 curve over a field of characteristic not 2 or 3. These isogenies are rational for a positive fraction of all hyperelliptic genus 3 curves defined over a finite field of characteristic $p > 3$. Subject to reasonable assumptions, our constructions give an explicit and efficient reduction of instances of the DLP from hyperelliptic to non-hyperelliptic Jacobians for around 18.57% of all hyperelliptic genus 3 curves over a given finite field. We conclude with a discussion on extending these ideas to isogenies with more general kernels. A condensed version of this work appeared in the proceedings of the EUROCRYPT 2008 conference.


Invertible and nilpotent matrices over antirings

Abstract: In this paper we characterize invertible matrices over an arbitrary commutative antiring S and find the structure of GL_n (S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent $n \times n$ matrix over an entire antiring can be written as a sum of $\lceil \log_2 n \rceil$ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.


Abstract: Using Deodhar's decomposition of a double Schubert cell, we study the regular representations of finite groups of Lie type arising in the cohomology of Deligne-Lusztig varieties associated to tori. We deduce that the Deligne-Lusztig restriction of a Gelfand-Graev module is a shifted Gelfand-Graev module.


Stochastic control up to a hitting time: optimality and rolling-horizon implementation

Abstract: We present a dynamic programming-based solution to a stochastic optimal control problem up to a hitting time for a discrete-time Markov control process. Firstly, we determine an optimal control policy to steer the process toward a compact target set while simultaneously minimizing an expected discounted cost. We then provide a rolling-horizon strategy for approximating the optimal policy, together with quantitative characterization of its sub-optimality with respect to the optimal policy. Finally, we address related issues of asymptotic discount-optimality of the value-iteration policy. Both the state and action spaces are assumed to be Polish.


Asymptotic Uncorrelation for Mexican Needlets

Abstract: We recall Mexican needlets from [5]. We derive an estimate for certain types of Legendre series, which we apply to the statistical properties of Mexican needlets. More precisely, we shall show that, under isotropy and Gaussianity assumptions, the Mexican needlet coefficients of a random field on the sphere are asymptotically uncorrelated, as the frequency parameter goes to infinity. This property is important in the analysis of cosmic microwave background radiation.


Corks, Plugs and exotic structures

Abstract: We discuss corks, and introduce new objects which we call plugs. Though plugs are fundamentally different objects, they also detect exotic smooth structures in 4-manifolds like corks. We discuss relation between corks, plugs and rational blow-downs. We show how to detect corks and plugs inside of some exotic manifolds. Furthermore, we construct knotted corks and plugs.


Nonparametric estimation for an autoregressive model

Abstract: The paper deals with the nonparametric estimation problem at a given fixed point for an autoregressive model with unknown distributed noise. Kernel estimate modifications are proposed. Asymptotic minimax and efficiency properties for proposed estimators are shown.


Two-sided localizations of bimodules

Abstract: We extend to bimodules Schelter's localization of a ring with respect to Gabriel filters of left and right ideals. Our two-sided localization of bimodules provides an endofunctor on a convenient bicategory of rings with filters of ideals. This is used to study the Picard group of a ring relative to a filter of ideals.


Squaring rectangles for dumbbells

Abstract: The theorem on squaring a rectangle from a tiling of a quadrilateral (Schramm and Cannon-Floyd-Parry) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring as an approximation to the classical Riemamnn mapping.


Uniconvergence theorems for Sturm--Liouville operators with potentials from Sobolev space $W_2^{-1}[0,\pi]$

Abstract: We consider a Sturm--Liouville $Ly=-y''+q(x)y$ in space $L_2[0,\pi]$ with potential from Sobolev space $W_2^{-1}[0,\pi]$. Moreover, we assume, that $q=u'$, where $u\in L_2[0,\pi]$. We consider Direchlet boundary conditions $y(0)=y(\pi)=0$, although we can treat a boundary conditions of Sturm type. It is known, that operators of such class have a discrete spectr with only accumulation point $+\infty$ and the system $\{y_k\}_1^\infty$ of eigen and associated functions is a Riesz basis in $L_2[0,\pi]$. Moreover, this basis is a Hilbert--Schmidt perturbation of the basis $\{sin(kx)\}_1^\infty$. In this paper we prove the uniconvergence theorem: for any element $f\in L_2[0,\pi]$ the sequence $P_nf-S_nf\to0$ as $n\to\infty$ in $C[0,\pi]$ (here $P_n$ and $S_n$ are the Riesz projectors to $\{y_k\}_1^n$ and $\{\sin(kt)\}_1^n$ respectively).


Rates of contraction of posterior distributions based on Gaussian process priors

Abstract: We derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on Gaussian processes. The rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process and the small ball probabilities of the Gaussian process. We determine these quantities for a range of examples of Gaussian priors and in several statistical settings. For instance, we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a (fractionally integrated) Brownian motion. We also consider regression with Gaussian errors and smooth classification under a logistic or probit link function combined with various priors.


Codimension one generic homoclinic classes with interior

Abstract: We study generic diffeomorphisms with a homoclinc class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive.


A Filtration of (q,t)-Catalan numbers

Abstract: We define new generalizations of (q,t)-Catalan numbers applying nabla operator on k-Schur functions indexed by column partitions. In some special cases, we give a combinatorial interpretation of these numbers using configurations of Dyck paths. In some other special cases, we also interpret these new polynomial as Hilbert series of explicit sub-modules of the alternating diagonal harmonics built using differential operators.


Isoperimetry and Symmetrization for Sobolev spaces on metric spaces

Abstract: Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the P\'{o}% lya-Szeg\"{o} and Faber-Krahn principles.


Syzygies of the secant variety of a curve

Abstract: We show that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety.


Higher Order Birkhoff Averages

Abstract: There are well-known examples of dynamical systems for which the Birkhoff averages with respect to a given observable along some or all of the orbits do not converge. It has been suggested that such orbits could be classified using higher order averages. In the case of a bounded observable, we show that a classical result of G.H. Hardy implies that if the Birkhoff averages do not converge, then neither do the higher order averages. If the Birkhoff averages do not converge then we may denote by $[\alpha_k,\beta_k]$ the limit set of the $k$-th order averages. The sequence of intervals thus generated is nested: $[\alpha_{k+1},\beta_{k+1}] \subset [\alpha_k,\beta_k]$. We can thus make a distinction among nonconvergent Birkhoff averages; either: B1. $\cap_{k=1}^\infty [\alpha_k,\beta_k]$ is a point $B_\infty$, or, B2. $\cap_{k=1}^\infty [\alpha_k,\beta_k]$ is a non-trivial interval $[\alpha_\infty,\beta_\infty]$. We give characterizations of the types B1 and B2 in terms of how slowly they oscillate and we give examples that exhibit both behaviours B1 and B2 in the context of full shifts on finite symbols and "Bowen's example". For finite full shifts, we show that the set of orbits with type B2 behaviour has full topological entropy.


The algebraic concordance order of a knot

Abstract: Levine defined the rational algebraic knot concordance group and proved that each nontrivial element is of order two, of order four, or of infinite order. The determination of the order of an element depends on a p-adic analysis for all primes p. Here we develop effective means to determine the order of any element that is in the image of the integral algebraic concordance group by restricting the set of primes that need to be considered and by finding simple tests that often avoid p-adic considerations. The paper includes an outline of how the results apply to give the determination of the algebraic orders of all 2,977 prime knots of 12 or fewer crossings. The paper also includes a short expository account of the necessary background in p-adic numbers and Witt groups of bilinear forms.


Graphs of bounded degree and the $p$-harmonic boundary

Abstract: Let $p$ be a real number greater than one and let $G$ be a connected graph of bounded degree. In this paper we introduce the $p$-harmonic boundary of $G$. We use this boundary to characterize the graphs $G$ for which the constant functions are the only $p$-harmonic functions on $G$. It is shown that any continuous function on the $p$-harmonic boundary of $G$ can be extended to a function that is $p$-harmonic on $G$. Some properties of this boundary that are preserved under rough-isometries are also given. Now let $\Gamma$ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced $\ell^p$-cohomology of $\Gamma$ in terms of the cardinality of its $p$-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $\Gamma$, the $p$-harmonic boundary of $\Gamma$ with the first reduced $\ell^p$-cohomology of $\Gamma$.


A limit-method for solving period problems on minimal surfaces

Abstract: We introduce a new technique to solve period problems on minimal surfaces called limit-method. If a family of surfaces has Weierstrass-data converging to the data of a known example, and this presents a transversal solution of periods, then the original family contains a sub-family with closed periods.


Triply periodic minimal surfaces which converge to the Hoffman-Wohlgemuth example

Abstract: We get a continuous one-parameter new family of embedded minimal surfaces, of which the period problems are two-dimensional. Moreover, one proves that it has Scherk second surface and Hoffman-Wohlgemuth example as limit-members.


On a question of Erdos and Ulam

Abstract: Ulam asked in 1945 if there is an everywhere dense \emph{rational set}, i.e. a point set in the plane with all its pairwise distances rational. Erd\H os conjectured that if a set $S$ has a dense rational subset, then $S$ should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erd\H os's conjecture for algebraic curves, by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.


Elliptic and parabolic second-order PDEs with growing coefficients

Abstract: We consider a second-order parabolic equation in $\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally H\"older continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{\infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.


The Bass and Topological Stable Ranks of $H^\infty_\R(\D)$ and $A_\R(\D)$

Abstract: In this note we prove that the Bass stable rank of $H^\infty_\R(\D)$ is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of $H^\infty_\R(\D)$ is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of $A_\R(\D)$ is two and the topological stable rank of $A_\R(\D)$ is two, settling a conjecture by the second author. We also present a $\bar\partial$-free proof of the second author's characterization of the reducible pairs in $A_\R(\D)$.


Classification of singular Q-homology planes. I. Structure and singularities

Abstract: A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient singularity then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes.


Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal

Abstract: We give a necessary and sufficient condition for an MV polytope $P$ in a highest weight crystal to lie in an arbitrary fixed Demazure crystal (resp., opposite Demazure crystal), in terms of the lengths of edges along a path through the 1-skeleton of $P$ corresponding to a reduced word for the longest element of the Weyl group $W$. % Also, we give an explicit description as a pseudo-Weyl polytope for extremal MV polytopes in a highest weight crystal. % Finally, by combining the results above, we obtain a polytopal condition for an MV polytope $P$ to lie in an arbitrary fixed opposite Demazure crystal.


Classification and realizations of type III factor representations of Cuntz-Krieger algebras associated with quasi-free states

Abstract: We completely classify type III factor representations of Cuntz-Krieger algebras associated with quasi-free states up to unitary equivalence. Furthermore, we realize these representations on concrete Hilbert spaces without using GNS construction. Free groups and their type ${\rm II}_{1}$ factor representations are used in these realizations.


Laws of the iterated logarithm for a class of iterated processes

Abstract: Let $X=\{X(t), t\geq 0\}$ be a Brownian motion or a spectrally negative stable process of index $1<\a<2$. Let $E=\{E(t),t\geq 0\}$ be the hitting time of a stable subordinator of index $0<\beta<1$ independent of $X$. We use a connection between $X(E(t))$ and the stable subordinator of index $\beta/\a$ to derive information on the path behavior of $X(E_t)$. This is an extension of the connection of iterated Brownian motion and (1/4)-stable subordinator due to Bertoin \cite{bertoin}. Using this connection, we obtain various laws of the iterated logarithm for $X(E(t))$. In particular, we establish law of the iterated logarithm for local time Brownian motion, $X(L(t))$, where $X$ is a Brownian motion (the case $\a=2$) and $L(t)$ is the local time at zero of a stable process $Y$ of index $1<\gamma\leq 2$ independent of $X$. In this case $E(\rho t)=L(t)$ with $\beta=1-1/\gamma$ for some constant $\rho>0$. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper \cite{MNX}. We also obtain exact small ball probability for $X(E_t)$ using ideas from \cite{aurzada}.


On the uniqueness of promotion operators on tensor products of type A crystals

Abstract: The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$ crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type $A_n$ crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara's conjecture that all `good' affine crystals are tensor products of Kirillov-Reshetikhin crystals.


A comparison of formulations and solution methods for the Minimum-Envy Location Problem. Additional results

Abstract: We consider a discrete facility location problem with a new form of equity criterion. The model discussed in the paper analyzes the case where demand points only have strict preference order on the sites where the plants can be located. The goal is to find the location of the facilities minimizing the total envy felt by the entire set of demand points. We define this new total envy criterion and provide several integer linear programming formulations that reflect and model this approach. Extensive computational tests are reported, showing the potentials and limits of each formulation on several types of instances.


On De Giorgi Conjecture in Dimension $N \geq 9$

Abstract: A celebrated conjecture due to De Giorgi states that any bounded solution of the equation $\Delta u + (1-u^2) u = 0 \hbox{in} \R^N $ with $\pp_{y_N}u >0$ must be such that its level sets $\{u=\la\}$ are all hyperplanes, {\em \bf at least} for dimension $N\le 8$. A counterexample for $N\ge 9$ has long been believed to exist. Based on a minimal graph $\Gamma$ which is not a hyperplane, found by Bombieri, De Giorgi and Giusti in $\R^N$, $N\ge 9$, we prove that for any small $\alpha >0$ there is a bounded solution $u_\alpha(y)$ with $\pp_{y_N}u_\alpha >0$, which resembles $ \tanh (\frac t{\sqrt{2}}) $, where $t=t(y)$ denotes a choice of signed distance to the blown-up minimal graph $\Gamma_\alpha := \alpha^{-1}\Gamma$. This solution constitutes a counterexample to De Giorgi conjecture for $N\ge 9$.


Energy scattering for 2D critical wave equation

Abstract: We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schr\"odinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the Strichartz estimate together with Sobolev-type inequalities can not control the nonlinear term uniformly on each time interval, but with constants depending on how much the solution is concentrated. Thus we have to trace concentration of the energy along time, in order to set up favorable nonlinear estimates, and then to implement Bourgain's induction argument. We show the same result for the "subcritical" nonlinear Schr\"odinger equation.


Computation of a Feynman integral

Abstract: A Feymnan integral is computed exactly using LLL


Asymptotics of the number of partitions into p-cores and some trigonometric sums

Abstract: An asymptotic formula for the number of partitions into p-cores is derived. As a byproduct some integer valued trigonometric sums are found


Lower Bounds for Boxicity

Abstract: An axis-parallel $b$-dimensional box is a Cartesian product $R_1\times R_2\times...\times R_b$ where $R_i$ is a closed interval of the form $[a_i,b_i]$ on the real line. For a graph $G$, its \emph{boxicity} box(G) is the minimum dimension $b$, such that $G$ is representable as the intersection graph of boxes in $b$-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: (1) The boxicity of a graph on $n$ vertices with no universal vertices and minimum degree $\delta$ is at least $n/2(n-\delta-1)$. (2) Consider the $\mathcal{G}(n,p)$ model of random graphs. Let $ p \le 1- \frac{40 \log n}{n^2}$. Then, for $G \in \mathcal{G}(n,p)$, almost surely $box(G)=\Omega(np(1-p))$. On setting $p=1/2$ we immediately infer that almost all graphs have boxicity $\Omega(n)$. (3) Spectral lower bounds for the boxicity of $k$-regular graphs. (4) The boxicity of random $k$-regular graphs on $n$ vertices (where $k$ is fixed) is $\Omega(k/\log k)$. (5) There exists a positive constant$c$ such that almost all balanced bipartite graphs on $2n$ vertices with exactly $m$ edges have boxicity at least $c m/n$, for $ m\le c' n^2/3$ for any positive constant $c' < 1$.


Directionally Convex Ordering of Random Measures, Shot Noise Fields and Some Applications to Wireless Communications

Abstract: Directionally convex ($dcx$) ordering is a tool for comparison of dependence structure of random vectors that also takes into account the variability of the marginal distributions. When extended to random fields it concerns comparison of all finite dimensional distributions. Viewing locally finite measures as non-negative fields of measure-values indexed by the bounded Borel subsets of the space, in this paper we formulate and study the $dcx$ ordering of random measures on locally compact spaces. We show that the $dcx$ order is preserved under some of the natural operations considered on random measures and point processes, such as deterministic displacement of points, independent superposition and thinning as well as independent, identically distributed marking. Further operations such as position dependent marking and displacement of points though do not preserve the $dcx$ order on all point processes, are shown to preserve the order on Cox point processes. We also examine the impact of $dcx$ order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions, pair correlation functions as well as examples seem to indicate that point processes higher in $dcx$ order cluster more. As the main result, we show that non-negative integral shot-noise fields with respect to $dcx$ ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot-noise fields appear as key ingredients. We also mention a few pertinent open questions.


Existence of a critical point for the infinite divisibility of squares of Gaussian vectors in $R^{2}$ with non--zero mean

Abstract: Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\neq 0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ to be infinitely divisible for all $\alpha\in R^{1}$ is that \[ \Ga_{i,i}\geq \frac{c_{i}}{c_{j}}\Ga_{i,j}>0\qquad\forall 1\le i\ne j\le 2.\] In this paper we show that when this does not hold there exists an $0<\alpha_{0}<\ff $ such that $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ is infinitely divisible for all $|\alpha|\leq \alpha_{0}$ but not for any $|\al|>\al_{0}$.


Strongly Fillable Contact Manifolds and J-holomorphic Foliations

Abstract: We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of the 3-torus similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of the 3-torus are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on the cotangent bundle of the 2-torus is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result for contact manifolds with positive Giroux torsion.


Renewal series and square-root boundaries for Bessel processes

Abstract: We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor.


Holomorphic Functions of Exponential Type and Duality for Stein Groups with Algebraic Connected Component of Identity

Abstract: We suggest a generalization of Pontryagin duality from the category of commutative Stein groups to the category of (not necessarily commutative) Stein groups with algebraic connected component of identity. In contrast to the other similar generalizations, in our approach the enveloping category consists of Hopf algebras (in a proper symmetrical monoidal category).


Shephard-Todd-Chevalley Theorem for skew polynomial rings

Abstract: We prove the following generalization of the classical Shephard-Todd-Chevalley Theorem. Let $G$ be a finite group of graded algebra automorphisms of a skew polynomial ring $A:=k_{p_{ij}}[x_1,...,x_n]$. Then the fixed subring $A^G$ has finite global dimension if and only if $G$ is generated by quasi-reflections. In this case the fixed subring $A^G$ is isomorphic a skew polynomial ring with possibly different $p_{ij}$'s. A version of the theorem is proved also for abelian groups acting on general quantum polynomial rings.


Hydrodynamic limit of gradient exclusion processes with conductances

Abstract: Fix a strictly increasing right continuous with left limits function $W: \bb R \to \bb R$ and a smooth function $\Phi : [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0<b \le \Phi'\le b^{-1}$. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes, with conductances given by $W$, is described by the weak solutions of the non-linear differential equation $\partial_t \rho = (d/dx)(d/dW) \Phi(\rho)$. We derive some properties of the operator $(d/dx)(d/dW)$ and prove uniqueness of weak solutions of the previous non-linear differential equation.


Sparse Regularization with $l^q$ Penalty Term

Abstract: We consider the stable approximation of sparse solutions to non-linear operator equations by means of Tikhonov regularization with a subquadratic penalty term. Imposing certain assumptions, which for a linear operator are equivalent to the standard range condition, we derive the usual convergence rate $O(\sqrt{\delta})$ of the regularized solutions in dependence of the noise level $\delta$. Particular emphasis lies on the case, where the true solution is known to have a sparse representation in a given basis. In this case, if the differential of the operator satisfies a certain injectivity condition, we can show that the actual convergence rate improves up to $O(\delta)$.


Knot Group Epimorphisms, II

Abstract: We consider the relations $\ge$ and $\ge_p$ on the collection of all knots, where $k \ge k'$ (respectively, $k \ge_p k'$) if there exists an epimorphism $\pi k \to \pi k'$ of knot groups (respectively, preserving peripheral systems). When $k$ is a torus knot, the relations coincide and $k'$ must also be a torus knot; we determine the knots $k'$ that can occur. If $k$ is a 2-bridge knot and $k \ge_p k'$, then $k'$ is a 2-bridge knot with determinant a proper divisor of the determinant of $k$; only finitely many knots $k'$ are possible.


Linear precision for toric surface patches

Abstract: We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification also includes, as a proper subset, the classification of toric surface patches from geometric modeling which have linear precision. Besides the well-known tensor product patches and B\'ezier triangles, we identify a family of toric patches with trapezoidal shape, each of which has linear precision. B\'ezier triangles and tensor product patches are special cases of trapezoidal patches.


New Stability Conditions for Linear Differential Equations with Several Delays

Abstract: New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation $$ \dot{x}(t)+\sum_{k=1}^m a_k(t)x(h_k(t))=0 $$ with measurable delays and coefficients. These results are compared to known stability tests.


Non-perturbative approach to random walk in markovian environment

Abstract: We prove an averaged CLT for a random walk in a dynamical environment where the states of the environment at different sites are independent Markov chains.


Rational Witt classes of pretzel knots

Abstract: In his pioneering work from 1969, Jerry Levine introduced a complete set of invariants of algebraic concordance of knots. The evaluation of these invariants requires a factorization of the Alexander polynomial of the knot, and is therefore in practice often hard to realize. We thus propose the study of an alternative set of invariants of algebraic concordance - the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. We also obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference "Fifty years since Milnor and Fox" held at Brandeis University on June 2-5, 2008.


Maximum Likelihood Drift Estimation for Multiscale Diffusions

Abstract: We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarse-grained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarse-grained equation for the slow variable, using maximum likelihood. We show that, whereas the maximum likelihood estimator is asymptotically unbiased for the averaging problem, for the homogenization problem maximum likelihood fails unless we subsample the data at an appropriate rate. An explicit formula for the asymptotic error in the log likelihood function is presented. Our theory is applied to two simple examples from molecular dynamics.


Gale duality and Koszul duality

Abstract: Given an affine hyperplane arrangement with some additional structure, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.


The Bloch-Okounkov correlation functions, a classical half-integral case

Abstract: Bloch and Okounkov's correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of $\hgl_\infty$-modules of level one. Recent works have calculated these character functions for higher levels for $\hgl_\infty$ and its Lie subalgebras of classical type. Here we obtain these functions for the subalgebra of type $D$ of half-integral levels and as a byproduct, obtain $q$-dimension formulas for integral modules of type $D$ at half-integral level.


The cycle-convergence of restarted GMRES for normal matrices is sublinear

Abstract: We prove that the cycle-convergence of the restarted GMRES applied to a system of linear equations with a normal coefficient matrix is sublinear.


Envelope Algebras of Partial Actions as Groupoid C*-Algebras

Abstract: We describe the envelope C*-algebra associated to a partial action of a countable discrete group on a locally compact space as a groupoid C*-algebra (more precisely as a C*-algebra from an equivalence relation) and we use our approach to show that, for a large class of partial actions of Z on the Cantor set, the envelope C*-algebra is an AF-algebra. We also completely characterize partial actions of a countable discrete group on a compact space such that the envelope action acts in a Hausdorff space.


Sobre o papel dos Departamentos de Matem\'atica na vida e desenvolvimento da comunidade

Abstract: The objective of this article is to stimulate discussions in mathematical society about the role of mathematical departments in the life of the community. University community is the center of knowledge and promotes the intellectual development. However, this is questioned today because of its reduced participation on the global learning process as there are many other components.


Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology

Abstract: We prove a strengthened version of a theorem of Lionel Schwartz that says that certain modules over the Steenrod algebra cannot be the mod 2 cohomology of a space. What is most interesting is our method, which replaces his iterated use of the Eilenberg--Moore spectral sequence by a single use of the spectral sequence converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie tower for the suspension spectrum of Omega^nX. Much of the paper develops basic properties of this spectral sequence.


BART: Bayesian additive regression trees

Abstract: We develop a Bayesian "sum-of-trees" model where each tree is constrained by a regularization prior to be a weak learner, and fitting and inference are accomplished via an iterative Bayesian backfitting MCMC algorithm that generates samples from a posterior. Effectively, BART is a nonparametric Bayesian regression approach which uses dimensionally adaptive random basis elements. Motivated by ensemble methods in general, and boosting algorithms in particular, BART is defined by a statistical model: a prior and a likelihood. This approach enables full posterior inference including point and interval estimates of the unknown regression function as well as the marginal effects of potential predictors. By keeping track of predictor inclusion frequencies, BART can also be used for model-free variable selection. BART's many features are illustrated with a bake-off against competing methods on 42 different data sets, with a simulation experiment and on a drug discovery classification problem.


The Classification of Exceptional CDQL Webs on Compact Complex Surfaces

Abstract: Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the foliations defining the web reminiscent of Abel's addition theorem in classical algebraic geometry. The abelian relations of a given web form a finite dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number p(n,k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental problem in web geometry is the classification of exceptional webs, that is, webs of maximal rank not equivalent to the dual of a projective curve. Recently, J.-M. Trepreau proved that there are no exceptional k-webs for n>2 and k > 2n-1. In dimension two there are examples of exceptional k-webs for arbitrary k and the classification problem is wide open. In this paper, we classify the exceptional Completely Decomposable Quasi-Linear (CDQL) webs globally defined on compact complex surfaces. By definition, the CDQL (k+1)-webs are formed by the superposition of k linear foliations and one non-linear foliation. For instance, we show that up to projective transformations there are exactly four countable families and thirteen sporadic exceptional CDQL webs on the projective plane.


Jack polynomials and the coinvariant ring of $G(r,p,n)$

Abstract: We study the coinvariant ring of the complex reflection group $G(r,p,n)$ as a module for the corresponding rational Cherednik algebra $\HH$ and its generalized graded affine Hecke subalgebra $\mathcal{H}$. We construct a basis consisting of non-symmetric Jack polynomials, and using this basis decompose the coinvariant ring into irreducible modules for $\mathcal{H}$. The basis consists of certain non-symmetric Jack polynomials, whose leading terms are the ``descent monomials'' for $G(r,p,n)$ recently studied by Adin, Brenti, and Roichman and Bagno and Biagoli. The irreducible $\mathcal{H}$-submodules of the coinvariant ring are their ``colored descent representations''.


Mean representation number of integers as the sum of primes

Abstract: Assuming the Riemann Hypothesis we obtain asymptotic estimates for the mean value of the number of representations of an integer as a sum of two primes. By proving a corresponding Omega-term, we prove that our result is essentially the best possible.


F-injective singularities are Du Bois

Abstract: In this paper, we prove that singularities of $F$-injective type are Du Bois. This extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of Frobenius in positive characteristic.


Zariski-van Kampen method and transcendental lattices of certain singular K3 surfaces

Abstract: We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type $A_{10}+A_{9}$ that are defined over a real quadratic field and are conjugate to each other over the field of rational numbers.


The Volume of a Local Nodal Domain

Abstract: Let M either be a closed real analytic Riemannian manifold or a closed smooth Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball provided that this component enters the ball deeply enough.


Minimal Surfaces in the Four-Dimensional Euclidean Space

Abstract: We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for strongly regular minimal surfaces of general type in terms of their invariants. We introduce canonical parameters on strongly regular minimal surfaces of general type and prove that any such a surface is determined up to a motion by two invariant functions satisfying a system of two natural partial differential equations. On any minimal surface of the basic class of non strongly regular minimal surfaces we define canonical parameters and prove that any such a surface is determined up to a motion by two invariant functions of one variable satisfying a system of two natural ordinary differential equations. We find a geometric description of this class of non strongly regular minimal surfaces.


Cyclic orders on the quantum grassmannian

Abstract: The quantum grassmannian is known to be a graded quantum algebra with a straightening law when the poset of generating quantum minors is endowed with the standard partial ordering. In this paper it is shown that this result remains true when the ordering is subjected to cyclic shifts. The method involves proving that noncommutative dehomogenisation is possible at any consecutive quantum minor.


Fibr\'es vectoriels de rang deux sur $\P^2$ provenant d'un rev\^etement double

Abstract: Since Schwarzenberger and his celebrated paper called "Vector bundles on the projective plane" we know that any rank two vector bundle on $\P^2$ is a direct image of a line bundle on a double covering of the plane. This theorem suggests to study the rank two vector bundles according to the branch curve of the covering which they come from. Thus, in the first part we prove that, given a double covering ramified over an irreducible curve $C_{2r}$ with degree $2r$, the jumping lines of fixed order (order depending on $r$ and on the parity of the rank two vector bundle) of the direct images vector bundles are necessarely $r$-tangent to $C_{2r}$. In the second part we concentrate on the case $r=2$. Then we give a list of vector bundles for which the jumping lines are exactly the bitangent lines to the branch quartic.


Approximation of Holder continuous homeomorphisms by piecewise affine homeomorphisms

Abstract: This paper is concerned with the problem of approximating a homeomorphism by piecewise affine homeomorphisms. The main result is as follows: every homeomorphism from a planar domain with a polygonal boundary to R^2 that is globally Holder continuous of exponent \alpha, and whose inverse is also globally Holder continuous of exponent \alpha can be approximated in the Holder norm of exponent \beta by piecewise affine homeomorphisms, for some \beta that only depends on \alpha. The proof is constructive. We adapt the proof of simplicial approximation in the supremum norm, and measure the side lengths and angles of the triangulation over which the approximating homeomorphism is piecewise affine. The approximation in the supremum norm, and a control on the minimum angle and on the ratio between the maximum and minimum side lengths of the triangulation suffice to obtain approximation in the Holder norm.


Nonparametric adaptive estimation for pure jump L\'evy processes

Abstract: This paper is concerned with nonparametric estimation of the L\'evy density of a pure jump L\'evy process. The sample path is observed at $n$ discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the ${\mathbb L}^2$-risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator. The risk bound for the adaptive estimator is obtained under additional assumptions on the L\'evy density. Examples of models fitting in our framework are described and rates of convergence of the estimator are discussed.


Low frequency estimates for long range perturbations in divergence form

Abstract: We prove low frequency estimates for the boundary values of the resolvent of long range perturbations of the flat Laplacian in divergence form.


Well-posedness of the spatially homogeneous Landau equation for soft potentials

Abstract: We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderatley soft potentials, we assume additionnally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka.


Chow motives without projectivity

Abstract: In paper 0704.4003, Bondarko recently defined the notion of weight structure, and proved that the category $\DgM$ of geometrical motives over a perfect field k, as defined and studied by Voevodsky, Suslin and Friedlander, is canonically equipped with such a structure. Building on this result, and under a condition on the weights avoided by the boundary motive, we describe a method to construct intrinsically in $\DgM$ a motivic version of interior cohomology of smooth, but possibly non-projective schemes. In a sequel to this work, this method will be applied to Shimura varieties.


Log-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations

Abstract: Let $P_{n,k}$ be the number of permutations $\pi$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $\pi$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define $P_n(x)= \sum_k P_{n,k}x^k$ and $M_{2n}(x)=\sum_{k} M_{2n,k}x^k$. We propose some conjectures on the log-concavity and q-log-convexity of the polynomials $P_n(x)$ and $M_{2n}(x)$. We also introduce the notions of $\infty$-q-log-convexity and $\infty$-q-log-concavity, and the notion of higher order log-concavity with respect to $\infty$-q-log-convex or $\infty$-q-log-concavity. A conjecture on the $\infty$-q-log-convexity of the Boros-Moll polynomials is presented. It seems that $M_{2n}(x)$ are log-concave of any order with respect to $\infty$-q-log-convexity.


Calculating Effective Diffusivities in the Limit of Vanishing Molecular Diffusion

Abstract: In this paper we study the problem of the numerical calculation (by Monte Carlo Methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity filed, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume preserving property of the equations motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non trivial time-correlation structure and when inertial effects cannot be neglected. Modified equations are used to perform backward error analysis. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators.


Decay estimates for variable coefficient wave equations in exterior domains

Abstract: In this article we consider variable coefficient, time dependent wave equations in exterior domains. We prove localized energy estimates if the domain is star-shaped and global in time Strichartz estimates if the domain is strictly convex.


N-th root

Abstract: The quaternion equation X^n=A is solved for any integer number n > 1. A is a given quaternion with komplex numbers as its elements. We use the isomorphism between quaternions and (4,4)-matrices to solve this equation.


Overpseudoprimes, Mersenne Numbers and Wieferich primes

Abstract: We introduce a new class of pseudoprimes-so called "overpseudoprimes" which is a special subclass of super-Poulet pseudoprimes. Denoting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime iff value of h(n) is invariant of all divisors d>1 of n. In particular, we prove that all composite Mersenne numbers 2^p-1,where p is prime, and squares of Wieferich primes are overpseudoprimes. We give also a generalization of the results on arbitrary base a>1 and prove that every overpseudoprime is strong pseudoprime of the same base.


Norm Varieties and the Chain Lemma (after Markus Rost)

Abstract: The goal of this paper is to present proofs of two results of Markus Rost: the Chain Lemma and the Norm Principle. These are the final steps needed to complete the publishable verification of the Bloch-Kato conjecture, that the norm residue maps are isomorphisms between Milnor K-theory $K_n^M(k)/p$ and etale cohomology $H^n(k,\mu_p^n)$ for every prime p, every n and every field k containing 1/p. Our proofs of these two results are based on Rost's 1998 preprints, his web site and Rost's lectures at the Institute for Advanced Study in 1999-2000 and 2005.


M\'etodos de Multiresoluci\'on y su Aplicaci\'on a un Modelo de Ingenier\'ia

Abstract: The main objective of this dissertation is to present an adaptation of some finite volume methods used in the resolution of problems arising in sedimentation processes of flocculated suspensions (or sedimentation with compression). This adaptation is based on the utilization of multiresolution techniques, originally designed to reduce the computational cost incurred in solving using high resolution schemes in the numerical solution of hyperbolic systems of conservation laws.


Multiresolution Schemes and its Application to Sedimentation Models

Abstract: A numerical method is presented to obtain approximate solutions to problems arising from sedimentation models. These processes are widely utilized in minery for recovering water from suspensions coming out of flotation processes. The main idea is to apply a multiresolution method to the existing schemes developed by B\"urger et al. [2, 3, 4] and to observe the good performance of the multiresolution strategy when applied to these kind of problems. We obtain high rates of memory compression without affecting the quality of the solution.


Toeplitz Corona Theorems for the Polydisk and the Unit Ball

Abstract: The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in $\mathbb{C}^n$.


Minimal supporting subtrees for the free energy of polymers on disordered trees

Abstract: We consider a model of directed polymers on a regular tree with a disorder given by independent, identically distributed weights attached to the vertices. For suitable weight distributions this model undergoes a phase transition with respect to its localization behaviour. We show that, for high temperatures, the free energy is supported by a random tree of positive exponential growth rate, which is strictly smaller than that of the full tree. The growth rate of the minimal supporting subtree is decreasing to zero as the temperature decreases to the critical value. At the critical value and all lower temperatures, a single polymer suffices to support the free energy. Our proofs rely on elegant martingale methods adapted from the theory of branching random walks.


Boolean 2-designs and the embedding of a 2-design in a group

Abstract: We try to embed a t-design in a finite commutative group in such a way that the sum of the k points of a block is zero. We can compute the number of blocks of the boolean 2-design having all the non zero vectors of $(Z_2)^n$ as the set of points and the k-subsets of elements the sum of which is zero as blocks.


Stabilization of Three-Dimensional Collective Motion

Abstract: This paper proposes a methodology to stabilize relative equilibria in a model of identical, steered particles moving in three-dimensional Euclidean space. Exploiting the Lie group structure of the resulting dynamical system, the stabilization problem is reduced to a consensus problem on the Lie algebra. The resulting equilibria correspond to parallel, circular and helical formations. We first derive the stabilizing control laws in the presence of all-to-all communication. Providing each agent with a consensus estimator, we then extend the results to a general setting that allows for unidirectional and time-varying communication topologies.


Log minimal model program for the moduli space of stable curves: The first flip

Abstract: We give a geometric invariant theory (GIT) construction of the log canonical model $\bar M_g(\alpha)$ of the pairs $(\bar M_g, \alpha \delta)$ for $\alpha \in (7/10 - \epsilon, 7/10]$ for small $\epsilon \in \mathbb Q_+$. We show that $\bar M_g(7/10)$ is isomorphic to the GIT quotient of the Chow variety bicanonical curves; $\bar M_g(7/10-\epsilon)$ is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions, e.g., they do not contain elliptic bridges. We show that there is a small contraction $\Psi: \bar M_g(7/10+\epsilon) \to \bar M_g(7/10)$ that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction $\Psi^+ : \bar M_g(7/10-\epsilon) \to \bar M_g(7/10)$ that is the Mori flip of $\Psi$.


Shape derivative of potential energy and energy release rate in fracture mechanics

Abstract: We study general mathematical framework for variation of potential energy with respect to domain deformation. It enables rigorous derivation of the integral formulas for the energy release rate in crack problems. Applying a technique of the shape sensitivity analysis, we formulate the shape derivative of potential energy as a variational problem with a parameter. Key tools of our abstract theory are a new parameter variational principle and the classical implicit function theorem in Banach spaces.


On The Existence of Globally Solvable Vector Fields in Smooth Manifolds

Abstract: Let $(\mathrm{M}, \omega_{0})$ be a connected paracompact smooth oriented manifold. We establish a necessary and sufficient conditions on Lie subalgebra $\mathfrak{a}$ of $\mathrm{T M}$ such that its orbits becomes diffeomorphic to an open convex set of $\mathbb{R}^{n}$, where $n$ is the orbit dimension.


Conjugate Generators of Knot and Link Groups

Abstract: This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are peripheral.


$\zeta$-phenomenology

Abstract: It is well known that Euler experimentally discovered the functional equation of the Riemann zeta function. Indeed he detected the fundamental $s\mapsto 1-s$ invariance of $\zeta(s)$ by looking only at special values. In particular, via this functional equation, the permutation group on two letters, $S_2\simeq\Z/(2)$, is realized as a group of symmetries of $\zeta(s)$. In this paper, we use the theory of special-values of our characteristic $p$ zeta functions to experimentally detect a natural symmetry group $S_{(q)}$ for these functions of cardinality ${\mathfrak c}=2^{\aleph_0}$ (where $\mathfrak c$ is the cardinality of the continuum); $S_{(q)}$ is a realization of the permutation group on $\{0,1,2...\}$ as homeomorphisms of $\Zp$ stabilizing both the nonpositive and nonnegative integers. We present a number of distinct instances in which $S_{(q)}$ acts (or appears to act) as symmetries of our functions. In particular, we present a natural, but highly mysterious, action of $S_{(q)}$ on a large subset of the domain of our functions that appears to stabilize zeta-zeroes. As of this writing, we do not yet know an overarching formalism that unifies these examples; however, it would seem that this formalism will involve an interplay between the 1-unit group $U_1$ -- playing the role of a "gauge group" -- and $S_{(q)}$. Furthermore, we show that $S_{(q)}$ may be naturally realized as an automorphism group of the convolution algebras of characteristic $p$ valued measures.


The role of binomial type sequences in determination identities for Bell polynomials

Abstract: Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and binomial type sequences in first part, and, we generalize the results obtained in [4] in second part.


Rationality, irrationality, and Wilf equivalence in generalized factor order

Abstract: Let $P$ be a partially ordered set and consider the free monoid $P^*$ of all words over $P$. If $w,w'\in P^*$ then $w'$ is a factor of $w$ if there are words $u,v$ with $w=uw'v$. Define generalized factor order on $P^*$ by letting $u\le w$ if there is a factor $w'$ of $w$ having the same length as $u$ such that $u\le w'$, where the comparison of $u$ and $w'$ is done componentwise using the partial order in $P$. One obtains ordinary factor order by insisting that $u=w'$ or, equivalently, by taking $P$ to be an antichain. Given $u\in P^*$, we prove that the language $\cF(u)=\{w : w\ge u\}$ is accepted by a finite state automaton. If $P$ is finite then it follows that the generating function $F(u)=\sum_{w\ge u} w$ is rational. This is an analogue of a theorem of Bj\"orner and Sagan for generalized subword order. We also consider $P=\bbP$, the positive integers with the usual total order, so that $P^*$ is the set of compositions. In this case one obtains a weight generating function $F(u;t,x)$ by substituting $tx^n$ each time $n\in\bbP$ appears in $F(u)$. We show that this generating function is also rational by using the transfer-matrix method. Words $u,v$ are said to be Wilf equivalent if $F(u;t,x)=F(v;t,x)$ and we prove various Wilf equivalences combinatorially. Bj\"orner found a recursive formula for the M\"obius function of ordinary factor order on $P^*$. It follows that one always has $\mu(u,w)=0,\pm1$. Using the Pumping Lemma we show that the generating function $M(u)=\sum_{w\ge u} |\mu(u,w)| w$ can be irrational.


Highest weight categories arising from Khovanov's diagram algebra II: Koszulity

Abstract: This is the second of a series of four articles studying various generalisations of Khovanov's diagram algebra. In this article we develop the general theory of Khovanov's diagrammatically defined "projective functors" in our setting. As an application, we give a direct proof of the fact that the quasi-hereditary covers of generalised Khovanov algebras are Koszul.


The ancient Greeks present: Rational Trigonometry

Abstract: Pythagoras' theorem, the area of a triangle as one half the base times the height, and Heron's formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, using quadrance not distance. This leads to a simpler and more elegant trigonometry, in which angle is replaced by spread, and which extends to arbitrary fields and more general quadratic forms.


The twisted Floer homology of torus bundles

Abstract: Given a torus bundle $Y$ over the circle and a cohomology class $[\omega]\in H^2(Y;\mathbb{Z})$ which evaluates nontrivially on the fiber, we compute the Heegaard Floer homology of $Y$ with twisted coefficients in the universal Novikov ring.


Hedlund-Metrics and the Stable Norm

Abstract: The real homology of a compact Riemannian manifold $M$ is naturally endowed with the stable norm. The stable norm on $H_1(M,\mathbb{R})$ arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space $H_1(M,\mathbb{R})$ are stable norms of a Riemannian metric on $M$. If the dimension of $M$ is at least three, I. Babenko and F. Balacheff proved in \cite{baba} that every polyhedral norm ball in $H_1(M,\mathbb{R})$, whose vertices are rational with respect to the lattice of integer classes in $H_1(M,\mathbb{R})$, is the stable norm ball of a Riemannian metric on $M$. This metric can even be chosen to be conformally equivalent to any given metric. The proof in \cite{baba} uses singular Riemannian metrics on polyhedra which are finally smoothed. Here we present an alternative construction of such metrics which remains in the geometric framework of smooth Riemannian metrics.


White-noise-aided Control

Abstract: The issue of white-noise-aided control is considered and its availability is proved. And a noise-aiding way is developed to stabilize perturbed systems to be input-to-state stable (ISS) with respect to (w.r.t.) perturbations. To illustrate its effectiveness, the white-noise-aided control of a parameter perturbed chaotic Chen system is given as an example. And numerically, it shows that, comparing to the un-noise-aided case, noise-aided control can not only shorten the control's transient process but also save its cost. These are also demonstrated by various aiding noises such as common (symmetric) noise and non-common (independent or asymmetric) noise, where common noise is found to be the most efficient in enhancing the control.


Unbounded representations of $q$-deformation of Cuntz algebra

Abstract: We study a deformation of the Cuntz-Toeplitz $C^*$-algebra determined by the relations $a_i^*a_i=1+q a_ia_i^*, a_i^*a_j=0$. We define well-behaved unbounded *-representations of the *-algebra defined by relations above and classify all such irreducible representations up to unitary equivalence.


Riemannian Geometry of Lie Algebroids

Abstract: We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids.


Matrix valued polynomials generated by the scalar-type Rodrigues' formulas

Abstract: The properties of matrix valued polynomials generated by the scalar-type Rodrigues' formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are established, as well as, in the commutative case, the second order equations for which these polynomials are eigenfunctions and the corresponding eigenvalues, and ladder operators. The conjecture of Duran and Grunbaum that if the weights are self-adjoint and positive semidefinite then they are necessarily of scalar type is proved for Q(x)=x and Q(x)=x^2-1 in dimension two, and for any dimension under genericity assumptions. Commutative classes of quasi-orthogonal polynomials are found, which satisfy all the properties usually associated to orthogonal polynomials.


Cohomology of GKM Fiber Bundles

Abstract: The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.


Relaxed optimality conditions for mu-differentiable functions

Abstract: We prove some fundamental properties of mu-differentiable functions. A new notion of local minimizer and maximizer is introduced and several extremum conditions are formulated using the language of nonstandard analysis.


Abstract: Let $p(z,w)$ be a polynomial in two variables. We call the solution of the algebraic equation $p(z,w) = 0$ the algebraic correspondence. We regard it as the graph of the multivalued function $z \mapsto w$ defined implicitly by $p(z,w) = 0$. Algebraic correspondences on the Riemann sphere $\hat{\mathbb C}$ give a generalization of dynamical systems of Klein groups and rational functions. We introduce $C^*$-algebras associated with algebraic correspondences on the Riemann sphere. We show that if an algebraic correspondence is free and expansive on a closed $p$-invariant subset $J$ of $\hat{\mathbb C}$, then the associated $C^*$-algebra ${\mathcal O}_p(J)$ is simple and purely infinite.


Time-dependent Schr\"odinger perturbations of transition densities

Abstract: We construct the fundamental solution of $\partial_t-\Delta_y- q(t,y)$, for functions $q$ with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces. We also discuss specific applications to Schr\"odinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator.