On the Spectrum of geometric operators on K\"ahler manifolds

Abstract: On a compact K\"ahler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin^c-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved K\"ahler manifolds of odd complex dimension.


Noncommutative geometry and dual coalgebras

Abstract: In arXiv:math/0606241v2 M. Kontsevich and Y. Soibelman argue that the category of noncommutative (thin) schemes is equivalent to the category of coalgebras. We propose that under this correspondence the affine scheme of a k-algebra A is the dual coalgebra A^o and draw some consequences. In particular, we describe the dual coalgebra A^o of A in terms of the A-infinity structure on the Yoneda-space of all the simple finite dimensional A-representations.


Probabilistic representation for solutions of an irregular porous media type equation

Abstract: We consider a porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients.


Integer program with bimodular matrix

Abstract: This paper has been accepted for publication in Discrete Optimization.


The antipode of a dual quasi-Hopf algebra with nonzero integrals is bijective

Abstract: For $A$ a Hopf algebra of arbitrary dimension over a field $K$, it is well-known that if $A$ has nonzero integrals, or, in other words, if the coalgebra $A$ is co-Frobenius, then the space of integrals is one-dimensional and the antipode of $A$ is bijective. Bulacu and Caenepeel recently showed that if $H$ is a dual quasi-Hopf algebra with nonzero integrals, then the space of integrals is one-dimensional, and the antipode is injective. In this short note we show that the antipode is bijective.


Local recognition of reflection graphs on Coxeter groups

Abstract: We provide local recognition results for the reflection graphs on spherical Coxeter groups. In particular, we study the case $F_4$ which is locally recognizable under additional constraints only. It is then demonstrated in the cases $A_n$ and $F_4$ how these graph theoretical recognition results can be used to characterize the corresponding Coxeter groups in terms of their reflection centralizers. Finally, we outline the connection to the local recognition of Chevalley groups which is particularly important in the classification of the finite simple groups.


On knots with infinite smooth concordance order

Abstract: We use the Heegaard Floer obstructions defined by Grigsby, Ruberman, and Strle to show that forty-six of the sixty-seven knots through eleven crossings whose concordance orders were previously unknown have infinite concordance order.


Eulerian quasisymmetric functions and poset topology

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


The birational type of the moduli space of even spin curves

Abstract: We determine the Kodaira dimension of the moduli space of even spin curves for all genera, with one possible exception: The scheme S_g has negative Kodaira dimension for g<8 and it is of general type for g>8. The Kodaira dimension of S_8 is non-negative.


Minimal triangulations for an infinite family of lens spaces

Abstract: The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its "minimal layered triangulation." This paper proves that for each integer n>1, the minimal layered triangulation of the lens space L(2n,1) is its unique minimal triangulation. More generally, the minimal triangulations (and hence the complexity) are determined for an infinite family of lens spaces containing the lens spaces L(2n,1).


Isometric Immersions and Compensated Compactness

Abstract: A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. T


Studies on the second member of the second Painlev\'e hierarchy

Abstract: In this paper, we study the second member of the second Painlev\'e hierarchy $P_{II}^{(2)}$. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four, and this Hamiltonian system can be considered as a 1-parameter family of coupled Painlev\'e systems. This Hamiltonian is new. We also show that this system admits extended affine Weyl group symmetry of type $A_1^{(1)}$, and can be recovered by its holomorphy conditions. We also study a fifth-order ordinary differential equation satisfied by this Hamiltonian. After we transform this equation into a system of the first-order ordinary differential equations of polynomial type in dimension five by birational transformations, we give its symmetry and holomorphy conditions.


An abstract characterization of unital operator spaces

Abstract: In this article, we give an abstract characterization of the ``identity'' of an operator space $V$ by looking at a quantity $n_{cb}(V,u)$ which is defined in analogue to a well-known quantity in Banach space theory. More precisely, we show that there exists a complete isometry from $V$ to some $\mathcal{L}(H)$ sending $u$ to ${\rm id}_H$ if and only if $n_{cb}(V,u) =1$. We will use it to give an abstract characterization of operator systems. Moreover, we will show that if $V$ is a unital operator space and $W$ is a proper complete $M$-ideal, then $V/W$ is also a unital operator space. As a consequece, the quotient of an operator system by a proper complete $M$-ideal is again an operator system. In the appendix, we will also give an abstract characterisation of ``non-unital operator systems'' using an idea arose from the definition of $n_{cb}(V,u)$.


A pattern mixture model for a paired $2\times2$ crossover design

Abstract: When conducting a paired $2\times2$ crossover design, each subject is paired with another subject with similar characteristics. The pair is then randomized to the same sequence of two treatments. That is, the two subjects receive the first experimental treatment, and then they cross over and receive the other experimental treatment(s). The paired $2\times2$ crossover design that was used in the Beta Adrenergic Response by GEnotype (BARGE) Study conducted by the National Heart, Lung and Blood Institute's Asthma Clinical Research Network (ACRN) has been described elsewhere. When the data arising from such a design are balanced and complete -- or if at least any missingness that occurs is at random -- general linear mixed-effects model methods can be used to analyze the data. In this paper, we present a method based on a pattern-mixture model for analyzing the data arising from a paired $2\times2$ crossover design when some of the data are missing in a non-ignorable fashion. Because of its inherent scientific interest, we focus our particular attention on the estimation of the treatment-by-type of subject interaction term. Finally, we illustrate the pattern-mixture model methods described in this paper on the data arising from the BARGE study.


Projected likelihood contrasts for testing homogeneity in finite mixture models with nuisance parameters

Abstract: This paper develops a test for homogeneity in finite mixture models where the mixing proportions are known a priori (taken to be 0.5) and a common nuisance parameter is present. Statistical tests based on the notion of Projected Likelihood Contrasts (PLC) are considered. The PLC is a slight modification of the usual likelihood ratio statistic or the Wilk's $\Lambda$ and is similar in spirit to the Rao's score test. Theoretical investigations have been carried out to understand the large sample statistical properties of these tests. Simulation studies have been carried out to understand the behavior of the null distribution of the PLC statistic in the case of Gaussian mixtures with unknown means (common variance as nuisance parameter) and unknown variances (common mean as nuisance parameter). The results are in conformity with the theoretical results obtained. Power functions of these tests have been evaluated based on simulations from Gaussian mixtures.


Hook lengths and 3-cores

Abstract: Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that $A000731(n)=0$ if and only if $A033687(n)=0$. The numbers $A000731(n)$ are given in terms of hook lengths of partitions, while $A033687(n)$ equals the number of 3-core partitions of $n$. Here we prove this conjecture.


Discovering hook length formulas by expansion technique

Abstract: We introduce the hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook length expansion. The paper can be seen as a collection of hook length formulas for partitons and plane trees. All examples are illustrated by HookExp and, for many easy cases, expained by well-known combinatorial arguments.


Schr\"oder Paths and Pattern Avoiding Partitions

Abstract: In this paper, we show that both 12312-avoiding partitions and 12321-avoiding partitions of the set $[n+1]$ are in one-to-one correspondence with Schr\"oder paths of semilength $n$ without peaks at even level. As a consequence, the refined enumeration of 12312-avoiding (resp. 12321-avoiding) partitions according to the number of blocks can be reduced to the enumeration of certain Schr\"oder paths according to the number of peaks. Furthermore, we get the enumeration of irreducible 12312-avoiding (resp. 12321-avoiding) partitions, which are closely related to skew Dyck paths.


The Liouville phenomenon in the deformation problem of coisotropics

Abstract: The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L-infinity algebra on the shifted foliation complex (\Omega^*[1](\fol), d_\fol), which allows a concise description of deformations in terms of a Maurer-Cartan equation. Infinitesimal deformations are given by d_\fol-closed forms, and the relation between infinitesimal deformations and full deformations can be studied in terms of obstruction classes lying in the foliation cohomology H^*_\fol. Closely related to the foliation cohomology is Haefliger's group \Omega^*_c(T/H), an under-appreciated model for the leaf space of a foliation. We make integral use of this group in showing solvability and unsolvability of the obstruction equations. We also show the L-infinity apparatus to be capable of detecting the Liouville/diophantine distinction of KAM theory, and argue for the greater significance of Haefliger's integration-over-leaves map in passing this fine structure to a geometric model for the leaf space.


Bootstrapping the Grenander estimator

Abstract: The goal of this paper is to study the bootstrap for the Grenander estimator. The first result is a proof of the inconsistency of the nonparametric bootstrap for the Grenander estimator at a given point. The second result is the development and verification of a bootstrap for the $L_1$ confidence band for the Grenander estimator. As part of this work, kernel estimators are studied as alternatives to the Grenander estimator. We show that when the second derivative of the true density is assumed to be uniformly bounded, there exist kernel estimators with faster convergence rates than the Grenander estimator. We study the implications of this in developing $L_1$ and uniform confidence bands and discuss some open questions.


Ratio tests for change point detection

Abstract: We propose new tests to detect a change in the mean of a time series. Like many existing tests, the new ones are based on the CUSUM process. Existing CUSUM tests require an estimator of a scale parameter to make them asymptotically distribution free under the no change null hypothesis. Even if the observations are independent, the estimation of the scale parameter is not simple since the estimator for the scale parameter should be at least consistent under the null as well as under the alternative. The situation is much more complicated in case of dependent data, where the empirical spectral density at 0 is used to scale the CUSUM process. To circumvent these difficulties, new tests are proposed which are ratios of CUSUM functionals. We demonstrate the applicability of our method to detect a change in the mean when the errors are AR(1) and GARCH(1,1) sequences.


A comparison of the Benjamini-Hochberg procedure with some Bayesian rules for multiple testing

Abstract: In the spirit of modeling inference for microarrays as multiple testing for sparse mixtures, we present a similar approach to a simplified version of quantitative trait loci (QTL) mapping. Unlike in case of microarrays, where the number of tests usually reaches tens of thousands, the number of tests performed in scans for QTL usually does not exceed several hundreds. However, in typical cases, the sparsity $p$ of significant alternatives for QTL mapping is in the same range as for microarrays. For methodological interest, as well as some related applications, we also consider non-sparse mixtures. Using simulations as well as theoretical observations we study false discovery rate (FDR), power and misclassification probability for the Benjamini-Hochberg (BH) procedure and its modifications, as well as for various parametric and nonparametric Bayes and Parametric Empirical Bayes procedures. Our results confirm the observation of Genovese and Wasserman (2002) that for small p the misclassification error of BH is close to optimal in the sense of attaining the Bayes oracle. This property is shared by some of the considered Bayes testing rules, which in general perform better than BH for large or moderate $p$'s.


The character table of a split extension of the Heisenberg group $H_1(q)$ by $Sp(2,q)$, $q$ odd

Abstract: In this paper we determine the full character table of a certain split extension $H_1(q)\rtimes Sp(2,q)$ of the Heisenberg group $H_1$ by the odd-characteristic symplectic group $Sp(2,q)$.


On estimating the change point in generalized linear models

Abstract: Statistical models incorporating change points are common in practice, especially in the area of biomedicine. This approach is appealing in that a specific parameter is introduced to account for the abrupt change in the response variable relating to a particular independent variable of interest. The statistical challenge one encounters is that the likelihood function is not differentiable with respect to this change point parameter. Consequently, the conventional asymptotic properties for the maximum likelihood estimators fail to hold in this situation. In this paper, we propose an estimating procedure for estimating the change point along with other regression coefficients under the generalized linear model framework. We show that the proposed estimators enjoy the conventional asymptotic properties including consistency and normality. Simulation work we conducted suggests that it performs well for the situations considered. We applied the proposed method to a case-control study aimed to examine the relationship between the risk of myocardial infarction and alcohol intake.


Using statistical smoothing to date medieval manuscripts

Abstract: We discuss the use of multivariate kernel smoothing methods to date manuscripts dating from the 11th to the 15th centuries, in the English county of Essex. The dataset consists of some 3300 dated and 5000 undated manuscripts, and the former are used as a training sample for imputing dates for the latter. It is assumed that two manuscripts that are ``close'', in a sense that may be defined by a vector of measures of distance for documents, will have close dates. Using this approach, statistical ideas are used to assess ``similarity'', by smoothing among distance measures, and thus to estimate dates for the 5000 undated manuscripts by reference to the dated ones.


Sequential nonparametrics and semiparametrics: Theory, implementation and applications to clinical trials

Abstract: One of Pranab K. Sen's major research areas is sequential nonparametrics and semiparametrics and their applications to clinical trials, to which he has made many important contributions. Herein we review a number of these contributions and related developments. We also describe some recent work on nonparametric and semiparametric inference and the associated computational methods in time-sequential clinical trials with survival endpoints.


Combinatorial cube packings in cube and torus

Abstract: We consider sequential random packing of cubes $z+[0,1]^n$ with $z\in \frac{1}{N}\ZZ^n$ into the cube $[0,2]^n$ and the torus $\QuotS{\RR^n}{2\ZZ^n}$ as $N\to\infty$. In the cube case $[0,2]^n$ as $N\to\infty$ the random cube packings thus obtained are reduced to a single cube with probability $1-O(\frac{1}{N})$. In the torus case the situation is different: for $n\leq 2$, sequential random cube packing yields cube tilings, but for $n\geq 3$ with strictly positive probability, one obtains non-extensible cube packings. So, we introduce the notion of combinatorial cube packing, which instead of depending on $N$ depend on some parameters. We use use them to derive an expansion of the packing density in powers of $\frac{1}{N}$. The explicit computation is done in the cube case. In the torus case, the situation is more complicate and we restrict ourselves to the case $N\to\infty$ of strictly positive probability. We prove the following results for torus combinatorial cube packings: We give a general Cartesian product construction. We prove that the number of parameters is at least $\frac{n(n+1)}{2}$ and we conjecture it to be at most $2^n-1$. We prove that cube packings with at least $2^n-3$ cubes are extensible. We find the minimal number of cubes in non-extensible cube packings for $n$ odd and $n\leq 6$.


Rectangle groups

Abstract: A class of groups is investigated, each of which has a fairly simple presentation . For example the group $R = (a, b, c, d | a^3 = b^3 = c^3 = d^3 = 1, ba^{-1} =dc^{-1}, ca^{-1} = db^{-1}) $ is in the class. Such a group does not have as a homomorphic image any group which is a 2-orbifold group or which is a group of isometries of the reals. However it does have incompatible splittings over subgroups which are not small. This contradicts some ideas I had about universal JSJ decompostions for finitely presented groups over finitely generated subgroups. Such a group also has an unstable action on an R-tree and a cocompact action on a CAT(0) cube complex with finite cyclic point stabilizers, and trivial edge stabilizers.


Estimating medical costs from a transition model

Abstract: Nonparametric estimators of the mean total cost have been proposed in a variety of settings. In clinical trials it is generally impractical to follow up patients until all have responded, and therefore censoring of patient outcomes and total cost will occur in practice. We describe a general longitudinal framework in which costs emanate from two streams, during sojourn in health states and in transition from one health state to another. We consider estimation of net present value for expenditures incurred over a finite time horizon from medical cost data that might be incompletely ascertained in some patients. Because patient specific demographic and clinical characteristics would influence total cost, we use a regression model to incorporate covariates. We discuss similarities and differences between our net present value estimator and other widely used estimators of total medical costs. Our model can accommodate heteroscedasticity, skewness and censoring in cost data and provides a flexible approach to analyses of health care cost.


Correcting for selection bias via cross-validation in the classification of microarray data

Abstract: There is increasing interest in the use of diagnostic rules based on microarray data. These rules are formed by considering the expression levels of thousands of genes in tissue samples taken on patients of known classification with respect to a number of classes, representing, say, disease status or treatment strategy. As the final versions of these rules are usually based on a small subset of the available genes, there is a selection bias that has to be corrected for in the estimation of the associated error rates. We consider the problem using cross-validation. In particular, we present explicit formulae that are useful in explaining the layers of validation that have to be performed in order to avoid improperly cross-validated estimates.


Some regular symmetric pairs

Abstract: In [AG2] we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, and in particular that any connected symmetric pair over C is a Gelfand pair. In this paper we show that the pairs $$(GL(V),O(V)), (GL(V),U(V)), (U(V),O(V)), (O(V \oplus W),O(V) \times O(W)), (U(V \oplus W),U(V) \times U(W))$$ are regular where V and W are quadratic or hermitian spaces over arbitrary local field of characteristic zero. We deduce from that that the pairs $(GL_n(\C),O_n(\C))$ and $(O_{n+m}(\C),O_n(\C) \times O_m(\C))$ are Gelfand pairs.


Parity-induced Selmer Growth For Symplectic, Ordinary Families

Abstract: Let $p$ be an odd prime, and let $K/K_0$ be a quadratic extension of number fields. Denote by $K_\pm$ the maximal $\mathbb{Z}_p$-power extensions of $K$ that are Galois over $K_0$, with $K_+$ abelian over $K_0$ and $K_-$ dihedral over $K_0$. In this paper we show that for a Galois representation over $K_0$ satisfying certain hypotheses, if it has odd Selmer rank over $K$ then for one of $K_\pm$ its Selmer rank over $L$ is bounded below by $[L:K]$ for $L$ ranging over the finite subextensions of $K$ in $K_\pm$. Our method or proof generalizes a method of Mazur--Rubin, building upon results of Nekov\'a\v{r}, and applies to abelian varieties of arbitrary dimension, (self-dual twists of) modular forms of even weight, and (twisted) Hida families.


Bimodule herds

Abstract: The notion of a bimodule herd is introduced and studied. A bimodule herd consists of a $B$-$A$ bimodule, its formal dual, called a pen, and a map, called a shepherd, which satisfies untiality and coassociativity conditions. It is shown that every bimodule herd gives rise to a pair of corings and coactions. If, in addition, a bimodule herd is tame i.e. it is faithfully flat and a progenerator, then these corings are associated to entwining structures; the bimodule herd is a Galois comodule of these corings. The notion of a bicomodule coherd is introduced as a formal dualisation of the definition of a bimodule herd. Every bicomodule coherd defines a pair of (non-unital) rings. It is shown that a tame $B$-$A$ bimodule herd defines a bicomodule coherd, and sufficient conditions for the derived rings to be isomorphic to $A$ and $B$ are discussed. The composition of bimodule herds via the tensor product is outlined. The notion of a bimodule herd is illustrated by the example of Galois co-objects of a commutative, faithfully flat Hopf algebra.


An asymptotically normal test for the selective neutrality hypothesis

Abstract: An important parameter in the study of population evolution is $\theta=4N\nu$, where $N$ is the effective population size and $\nu$ is the rate of mutation per locus per generation. Therefore, $\theta$ represents the mean number of mutations per site per generation. There are many estimators of $\theta$, one of them being the mean number of pairwise nucleotide differences, which we call $\mathcal{T}_2$. Other estimators are $\mathcal{T}_1$, based on the number of segregating sites and $\mathcal{T}_3$, based on the number of singletons. The concept of selective neutrality can be interpreted as a differentiated nucleotide distribution for mutant sites when compared to the overall nucleotide distribution. Tajima (1989) has proposed the so-called Tajima's test of selective neutrality based on $\mathcal{T}_2-\mathcal{T}_1$. Its complex empirical behavior (Kiihl, 2005) motivates us to propose a test statistic solely based on $\mathcal{T}_2$. We are thus able to prove asymptotic normality under different assumptions on the number of sequences and number of sites via $U$-statistics theory.


Complex structures on tangent and cotangent Lie algebras of dimension six

Abstract: This paper deals with complex structures on Lie algebras $\ct_{\pi} \hh=\hh \ltimes_{\pi} V$, where $\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\ct_{\pi} \hh$ for $\hh$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constrain $J\hh=V$. Whenever $\pi$ is the adjoint representation this kind of complex structures are associated to non singular derivations of $\hh$. This fact derives different kind of applications. Finally an approach to the pseudo K\"ahler geometry was done.


Model selection and sensitivity analysis for sequence pattern models

Abstract: In this article we propose a maximal a posteriori (MAP) criterion for model selection in the motif discovery problem and investigate conditions under which the MAP asymptotically gives a correct prediction of model size. We also investigate robustness of the MAP to prior specification and provide guidelines for choosing prior hyper-parameters for motif models based on sensitivity considerations.


Maximal Solutions of Semilinear Elliptic Equations with Locally Integrable Forcing Term

Abstract: We study the existence of a maximal solution of $-\Gd u+g(u)=f(x)$ in a domain $\Gw\subset \BBR^N$ with compact boundary, assuming that $f\in (L^1_{loc}(\Gw))_+$ and that $g$ is nondecreasing, $g(0)\geq 0$ and $g$ satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classical $C_{1,2}$ Wiener criterion then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition we discuss the question of uniqueness of large solutions.


Asymptotic behaviour for the gradient of large solutions to some nonlinear elliptic equations

Abstract: If $h$ is a nondecreasing real valued function and $0\leq q\leq 2$, we analyse the boundary behaviour of the gradient of any solution $u$ of $-\Delta u+h(u)+\abs {\nabla u}^q=f$ in a smooth N-dimensional domain $\Omega$ with the condition that $u$ tends to infinity when $x$ tends to $\partial\Omega$. We give precise expressions of the blow-up which, in particular, point out the fact that the phenomenon occurs essentially in the normal direction to $\partial\Omega$. Motivated by the blow--up argument in our proof, we also give in Appendix a symmetry result for some related problems in the half space.


The rationality of certain moduli spaces of curves of genus 3

Abstract: We show, for each algebraically closed field, the rationality of the following two moduli spaces: M(3,3) parametrizing pairs (C, \eta) where C has genus 3 and \eta is a 3-torsion divisor class, respectively of M(3,<3>) parametrizing pairs (C, <\eta>) as above and where <\eta> is the cyclic subgroup of order 3 in Pic_0(C) generated by \eta.


Symmetry of large solutions of nonlinear elliptic equations in a ball

Abstract: Let $g$ be a locally Lipschitz continuous real valued function which satisfies the Keller-Osserman condition and is convex at infinity, then any large solution of $-\Delta u+g(u)=0$ in a ball is radially symmetric.


A nonlinear inequality

Abstract: A quadratic inequality is formulated in the paper. An estimate on the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations.


The regular algebra of a poset

Abstract: Let $K$ be a field. We attach to each finite poset $\mathbb P$ a von Neumann regular $K$-algebra $Q_K(\mathbb P)$ in a functorial way. We show that the monoid of isomorphism classes of finitely generated projective $Q_K(\mathbb P)$-modules is the abelian monoid generated by $\mathbb P$ with the only relations given by $p=p+q$ whenever $q<p$ in $\mathbb P$. This extends the class of monoids for which there is a positive solution to the realization problem for von Neumann regular rings.


Ambiguity theory, old and new

Abstract: This is a introductory survey of some recent developments of "Galois ideas" in Arithmetic, Complex Analysis, Transcendental Number Theory and Quantum Field Theory, and of some of their interrelations.


Galois theory, motives and transcendental numbers

Abstract: From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn't one associate conjugates and a Galois group to transcendental numbers such as $\pi$? Beyond, can't one envision an appropriate Galois theory in the field of transcendental number theory? In which role? The aim of this text is to indicate what Grothendieck's theory of motives has to say, at least conjecturally, on these questions.


Triangulordinary Selmer Groups

Abstract: Let $p$ be a prime number, and let $K$ be a $p$-adic local field. We study a class of semistable $p$-adic Galois representations of $K$, which we call {\it triangulordinary} because it includes the ordinary ones yet allows non-\'etale behavior in the associated $(\phi,\Gamma_K)$-modules over the Robba ring. Our main result provides a description of the Bloch--Kato local condition of such representations. We also propose a program, using variational techniques, that would give a definition of the Selmer group along the eigencurve of Coleman--Mazur, including notably its nonordinary locus.


Nilpotent centralizers and Springer isomorphisms

Abstract: Let G be a semisimple algebraic group over a field K whose characteristic is very good for G, and let sigma be any G-equivariant isomorphism from the nilpotent variety to the unipotent variety; the map sigma is known as a Springer isomorphism. Let y in G(K), let Y in Lie(G)(K), and write C_y = C_G(y) and C_Y= C_G(Y) for the centralizers. We show that the center of C_y and the center of C_Y are smooth group schemes over K. The existence of a Springer isomorphism is used to treat the crucial cases where y is unipotent and where Y is nilpotent. Now suppose G to be quasisplit, and write C for the centralizer of a rational regular nilpotent element. We obtain a description of the normalizer N_G(C) of C, and we show that the automorphism of Lie(C) determined by the differential of sigma at zero is a scalar multiple of the identity; these results verify observations of J-P. Serre.


Lines on hypersurfaces

Abstract: This is a detailed study of the infinitesimal variation of the variety of lines through a point of a low degree hypersurface in pro jective space. The motion is governed by a system of partial differential equations which we describe explicitly.


The Yang-Mills stratification for surfaces revisited

Abstract: We revisit Atiyah and Bott's study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural map (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of the space of central Yang-Mills connections to the classifying space of the gauge group G(E). All of these results carry over to non-orientable surfaces via Ho and Liu's non-orientable Yang-Mills theory. A somewhat less detailed version of this paper (titled "On the Yang-Mills stratification for surfaces") will appear in the Proceedings of the AMS.


On Two Related Questions of Wilf Concerning Standard Young Tableaux

Abstract: We consider two questions of Wilf related to Standard Young Tableaux. We provide a partial answer to one question, and that will lead us to a more general answer to the other question. Our answers are purely combinatorial.


Overcrowding and hole probabilities for random zeros on complex manifolds

Abstract: We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $\delta>0$, the probability that this volume differs by more than $\delta N$ from its average value is less than $\exp(-C_{\delta,U}N^{m+1})$, for some constant $C_{\delta,U}>0$. As a consequence, the "hole probability" that a random section does not vanish in U has an upper bound of the form $\exp(-C_{U}N^{m+1})$.


On the homotopy theory of spectral categories

Abstract: This paper has been withdrawn by the author, due a critical mistake on page 3.


Separating algebras and finite reflection groups

Abstract: A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras.


A model category structure on the category of simplicial multicategories

Abstract: We establish a Quillen model category structure on the category of symmetric simplicial multicategories. This model structure extends the model structure on simplicial categories due to J. Bergner.


Convergence of nonlocal threshold dynamics approximations to front propagation

Abstract: In this note we prove that appropriately scaled threshold dynamics-type algorithms corresponding to the fractional Laplacian of order $\alpha \in (0,2)$ converge to moving fronts. When $\alpha \geqq 1$ the resulting interface moves by weighted mean curvature, while for $\alpha <1$ the normal velocity is nonlocal of ``fractional-type.'' The results easily extend to general nonlocal anisotropic threshold dynamics schemes.


(GL(2n,C),SP(2n,C)) is a Gelfand Pair

Abstract: We prove that (GL_{2n}(C),Sp_{2n}(C)) is a Gelfand pair. More precisely, we show that for an irreducible smooth admissible Frechet representation (\pi,E) of GL_{2n}(C) the space of continuous functionals Hom_{Sp_{2n}(\cc)}(E,C) is at most one dimensional. For this we show that any distribution on GL_{2n}(C) invariant with respect to the double action Sp_{2n}(C) \times Sp_{2n}(C) is transposition invariant. Such a result was previously proven for p-adic fields by M. Heumos and S. Rallis.


On the moduli space of negatively curved metrics of a hyperbolic manifold

Abstract: We study the moduli space of negatively curved metrics of a hyperbolic manifold.


On the $k$-free divisor problem

Abstract: Let $\Delta^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\geq 2$. In this paper we establish an asymptotic formula of the integral $\int_1^T|\Delta^{(k)}(x)|^2dx$ for each $k\geq 4.$


Regularity issues in the problem of fluid structure interaction

Abstract: We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier-Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have H\"older regularity $C^{1, \alpha}$, $0 < \alpha \le 1$. First, we show the existence and uniqueness of strong solutions up to collision. A key ingredient is a BMO bound on the velocity gradient, which substitutes to the standard $H^2$ estimate for smoother domains. Then, we study the asymptotic behaviour of one $C^{1, \alpha}$ body falling over a flat surface. We show that collision is possible in finite time if and only if $\alpha < 1/2$.


Universal Upper Bound for the Growth of Artin Monoids

Abstract: In this paper we study the growth rates of Artin monoids and we show that 4 is a universal upper bound. We also show that the generating functions of the associated right-angled Artin monoids are given by families of Chebyshev polynomials. Applications to Artin groups and positive braids are given.


Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group

Abstract: We prove pairwise disjointness of representations T_{z,w} of the infinite-dimensional unitary group. These representations provide a natural generalization of the regular representation for the case of "big" group U(\infty). They were introduced and studied by G.Olshanski and A.Borodin. Disjointness of the representations can be reduced to disjointness of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods.


On the solutions of Knizhnik-Zamolodchikov system

Abstract: We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of KZ system is rational too. We give the method of constructing the corresponding rational solution. We deduce the asymptotic formulas for the solution of KZ system when $\rho$ is integer.


Le probleme de Bogomolov effectif sur les varietes abeliennes

Abstract: We give a new lower bound for the essential minimum of subvarieties of abelian varieties with small codimension, under a conjecture about ordinary primes in abelian varieties. This lower bound is already known in the toric case since the work of Amoroso and David. It is the best expected, "up to an epsilon", in the degree of the subvariety.


Generalizations of Gronwall-Bihari Inequalities on Time Scales

Abstract: We establish some nonlinear integral inequalities for functions defined on a time scale. The results extend some previous Gronwall and Bihari type inequalities on time scales. Some examples of time scales for which our results can be applied are provided. An application to the qualitative analysis of a nonlinear dynamic equation is discussed.


Implicit higher derivatives, and a formula of Comtet and Fiolet

Abstract: Let F(x,y) be a function of two variables, and suppose y = f(x) satisfies F(x,y)=0 in some range. Then dy/dx = -Fx/Fy, where Fx and Fy denote the partial derivatives of F with respect to x and y. It is natural to seek a general expression for the higher derivatives d^ny/dx^n, in terms of partial derivatives of F, and such an expression was given in 1974 by L. Comtet and M. Fiolet. Their formula, however, contains some errors. In this note, we give a corrected expression. We give a derivation using Lagrange inversion and also an elementary proof by induction. We further correct a minor error in Comtet and Fiolet's expression for the number of terms in their formula.


Time-preserving structural stability of hyperbolic differential dynamics with noncompact phase spaces

Abstract: In this paper, we study the structural stability of hyperbolic differential systems on Euclidean spaces using Liao theory.


A Quasi Curtis-Tits-Phan theorem for the symplectic group

Abstract: We obtain the symplectic group $\SP(V)$ as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let $\SP(V)$ act flag-transitively on the geometry of maximal rank subspaces of $V$. We show that this geometry and its rank $\ge 3$ residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then obtained by applying Tits' lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups.


Analytic multiplicative cocycles over holomorphic dynamical systems

Abstract: We prove some properties of analytic multiplicative and sub-multiplicative cocycles. The results allow to construct natural invariant analytic sets associated to complex dynamical systems.


Whittaker Modules for the Virasoro Algebra

Abstract: Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.


Finite Rank Toeplitz Operators: Some Extensions of D.Luecking's Theorem

Abstract: The recent theorem by D.Luecking about finite rank Bergman-Toeplitz operators is extended to weights being distributions with compact support and to the spaces of harmonic functions.


Singular Solutions of Hessian Fully Nonlinear Elliptic Equations

Abstract: We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+\epsilon}, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<\alpha<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.


An improved estimate on sums of product sets

Abstract: In a recent paper \cite{Gl} A. Glibichuk proved that if $A,B$ are subsets of an arbitrary finite filed $\F_q$, such that $|A||B|>q$, then $16AB = \F_q$. We improve this to $10AB = \F_q.$


Recurrence Formulas for Fibonacci Sums

Abstract: In this article we present a new recurrence formula for a finite sum involving the Fibonacci sequence. Furthermore, we state an algorithm to compute the sum of a power series related to Fibonacci series, without the use of term-by-term differentiation theorem


Cops and robbers in random graphs

Abstract: We consider the pursuit and evasion game on finite, connected, undirected graphs known as cops and robbers. Meyniel conjectured that for every graph on n vertices a rootish number of cops can win the game. We prove that this holds up to a log(n) factor for random graphs G(n,p) if p is not very small, and this is close to be tight unless the graph is very dense. We analyze the area-defending strategy (used by Aigner in case of planar graphs) and show examples where it can not be too efficient.


Observable invariant measures

Abstract: For continuous maps on a compact manifold M, particularly for those that do not preserve the Lebesgue measure m, we define the observable invariant probability measures as a generalization of the physical measures. We prove that any continuous map has observable measures, and characterize those that are physical in terms of the observability. We prove that there exist physical measures whose basins cover Lebesgue a.e, if and only if the set of all observable measures is finite or infinite numerable. We define for any continuous map, its generalized attractors using the set of observable invariant measures where there is no physical measure, and prove that any continuous map defines a decomposition of the space in up to infinitely many generalized attractors whose basins cover Lebesgue a.e. We apply the results to the C1 expanding maps f in the circle, proving that the set of observable measures (even if f is not C1 plus Holder, is a subset of the equilibrium states.


Multi-bump Solutions for a Strongly Indefinite Semilinear Schr\"odinger Equation Without Symmetry or convexity Assumptions

Abstract: In this paper, we study the following semilinear Schr\"odinger equation with periodic coefficient: $$-\triangle u +V(x)u=f(x,u), u\in H^{1}(\mathbb{R}^{N}).$$ The functional corresponding to this equation possesses strongly indefinite structure. The nonlinear term $f(x,t)$ satisfies some superlinear growth conditions and need not be odd or increasing strictly in $t$. Using a new variational reduction method and a generalized Morse theory, we proved that this equation has infinitely many geometrically different solutions. Furthermore, if the solutions of this equation under some energy level are isolated, then we can show that this equation has infinitely many $m-$bump solutions for any positive integer $m\geq 2.$


Local solutions in Sobolev spaces with negative indices for the "good" Boussinesq equation

Abstract: We study the local well-posedness of the initial-value problem for the nonlinear "good" Boussinesq equation with data in Sobolev spaces \textit{$H^s$} for negative indices of $s$.


CW-groups associated with wrap groups

Abstract: This article is devoted to the investigation of wrap groups of connected fiber bundles. CW-groups associated with wrap groups are studied.


Intersections of base rings associated to transversal polymatroids

Abstract: The discrete polymatroids and their base rings are studied recently in many papers (see \cite{HH}, \cite{HHV}, \cite{V1}, \cite{V2}). It is important to give conditions when the base ring associated to a transversal polymatroid is Gorenstein (see \cite{HH}). In \cite{SA} we introduced a class of such base rings. In this paper we note that an intersection of such base rings (introduced in \cite{SA}) is Gorenstein and give necessary and sufficient conditions for the intersection of two base rings from \cite{SA} to be still a base ring of a transversal polymatroid. Also, we compute the $a$-invariant of those base rings. The results presented were discovered by extensive computer algebra experiments performed with {\it{Normaliz}} \cite{BK}.


On the continuity of separately continuous bihomomorphisms

Abstract: Separately continuous bihomomorphisms on a product of convergence or topological groups occur with great frequency. Of course, in general, these need not be jointly continuous. In this paper, we exhibit some results of Banach-Steinhaus type and use these to derive joint continuity from separate continuity. The setting of convergence groups offers two advantages. First, the continuous convergence structure is a powerful tool in many duality arguments. Second, local compactness and first countability, the usual requirements for joint continuity, are available in much greater abundance for convergence groups.


The local Langlands conjecture for Sp(4)

Abstract: We show that the local Langlands conjecture for $Sp(2n)$ follows from that for $GSp(2n)$. In particular, we prove the local Langlands conjecture for $Sp(4)$, based on our previous work on the local Langlands conjecture for $GSp(4)$. We also determine the possible sizes of $L$-packets for $Sp(4)$.


Connes' metric for states in group algebras

Abstract: In this article we follow the main idea of A. Connes for the construction of a metric in the state space of a C*-algebra. We focus in the reduced algebra of a discrete group $\Gamma$, and prove some equivalences and relations between two central objects of this category: the word-length growth (connected with the degree of the extension of $\Gamma$ when the group is an extension of Z by a finite group), and the topological equivalence between the w*-topology and the one introduced with this metric in the state space of $C_r*(\Gamma)$.


The quandle of the trefoil knot as the Dehn quandle of the torus

Abstract: We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space $\Z \oplus \Z$. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fundamental quandle of the long trefoil knot is isomorphic to the cord quandle on a 2-sphere with a hole and three punctures.


Symmetry in Data Mining and Analysis: A Unifying View based on Hierarchy

Abstract: Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. "Structure" has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Beginning with the role of number theory in expressing data, we show how we can naturally proceed to hierarchical structures. We show how this both encapsulates traditional paradigms in data analysis, and also opens up new perspectives towards issues that are on the order of the day, including data mining of massive, high dimensional, heterogeneous data sets. Linkages with other fields are also discussed including computational logic and symbolic dynamics. The structures in data surveyed here are based on hierarchy, represented as p-adic numbers or an ultrametric topology.


On the distribution of imaginary parts of zeros of the Riemann zeta function, II

Abstract: We continue our investigation of the distribution of the fractional parts of $a \gamma$, where $a$ is a fixed non-zero real number and $\gamma$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to pair correlation functions and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. This is a sequel to the paper math.NT/0405459.


A short proof of the Khukhro--Makarenko theorem on large characteristic subgroups with laws

Abstract: We give a short proof and some strengthening of the Khukhro--Makarenko theorem that each group virtually satisfying an outer commutator identity contains a finite-index characteristic subgroup satisfying this identity. An estimate for the index of this characteristic subgroup is obtained.


Universal Khovanov-Rozansky sl(2) cohomology

Abstract: We generalize the Khovanov-Rozansky cohomology for n=2 by means of a homogeneous potential that depends on two parameters, to obtain the universal Khovanov-Rozansky sl(2) link cohomology. This theory is equivalent to the universal foam sl(2) link cohomology, after tensoring both theories with appropriate rings.


The Remarkable Simplicity of Very High Dimensional Data: Application of Model-Based Clustering

Abstract: An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a hierarchical embedding. Such hierarchical structure can be global in the data set, or local. By quantifying extent or degree of ultrametricity in a data set, we show that ultrametricity becomes pervasive as dimensionality and/or spatial sparsity increases. This leads us to assert that very high dimensional data are of simple structure. We exemplify this finding through a range of simulated data cases. We discuss also application to very high frequency time series segmentation and modeling.


Invariant tori for commuting Hamiltonian PDEs

Abstract: We generalize to some PDEs a theorem by Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with $r$ integrals of motion and $n$ degrees of freedom, $r\leq n$. The result we get ensures the persistence of an $r$-parameter family of $r$-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunof-Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations. In the first case we construct 2 dimensional tori, while in the second case we construct 3 dimensional tori.


Correlation decay and recurrence estimates for some robust nonuniformly hyperbolic maps

Abstract: We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional non-uniformly hyperbolic transformations. Oliveira and Viana [15] proved that there is a unique equilibrium state for a large class of non- uniformly expanding transformations and Holder continuous potentials with small variation. For an open class of potentials with small variation, we prove quasi-compactness of the Ruelle-Perron-Frobenius operator in a space $V_\theta$ of functions with essential bounded variation that strictly contain Holder continuous observables. We deduce that the equilibrium states have exponential decay of correlations. Furthermore, we prove exponential asymptotic distribu- tion of hitting times and log-normal fluctuations of the return times around the average given by the metric entropy.


Weil-Petersson geometry of Teichmuller-Coxeter complex and its finite rank property

Abstract: We construct a Weil-Petersson geodesic completion of Teichmuller space through the formalism of Coxeter complex with the Teichmuller space as its non-linear non-homogeneous fundamental domain. We show that the metric and geodesic completions both satisfy a finite rank property, demonstrating a similarity with the non-compact symmetric spaces of semi-simple Lie groups.


Quantum D-modules, elliptic braid groups, and double affine Hecke algebras

Abstract: We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain geometric constructions of Calaque, Enriquez, and Etingof concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of U=U_t(sl_N), as t limits 1. In the latter case, we produce representations of the double affine Hecke algebra of type A_{n-1}, for each n.


A class of simple $C^*$-algebras arising from certain nonsofic subshifts

Abstract: We present a class of subshifts $Z_N, N = 1,2,...$ whose associated $C^*$-algebras ${\cal O}_{Z_N}$ are simple, purely infinite and not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The class of the subshifts is the first examples whose associated $C^*$-algebras are not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The subshifts $Z_N$ are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that KMS-state for gauge action on the associated $C^*$-algebra ${\cal O}_{Z_N}$ exists if and only if the logarithm of the inverse temperature is the topological entropy for the subshift $Z_N$, and the corresponding KMS-state is unique.


On the distribution of coefficients of residue polynomials

Abstract: Using the formalism of polynomials with positive coefficients, the fact that exactly half of all subsets of a finite set have even cardinality can be generalized asymptotically.


Integral representations for a generalized Hermite linear functional

Abstract: In this paper, we find new integral representations for the generalized Hermite linear functional in the real line and the complex plane. As an application, new integral representations for the Euler Gamma function are given.


Applications of Klee's Dehn-Sommerville relations

Abstract: We use Klee's Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai's conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify K\"uhnel's conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove K\"uhnel's conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee's Dehn-Sommerville relations and strengthen Kalai's result on the number of their edges.


Fraisse's construction from a topos-theoretic perspective

Abstract: We present a topos-theoretic interpretation of (a categorical generalization of) Fraisse's construction in model theory, with applications to countably categorical theories.


The srank Conjecture on Schur's $Q$-Functions

Abstract: We show that the shifted rank, or srank, of any partition $\lambda$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_{\lambda}$ function in terms of power sum symmetric functions. This gives an affirmative answer to a conjecture of Clifford. As pointed out by Clifford, the notion of the srank can be naturally extended to a skew partition $\lambda/\mu$ as the minimum number of bars among the corresponding skew bar tableaux. While the srank conjecture is not valid for skew partitions, we give an algorithm to compute the srank.


Logarithmically Improved Criteria for Navier-Stokes Equations

Abstract: In this paper we prove the logarithmically improved Serrin's criteria to the three-dimensional incompressible Navier-Stokes equations.


Ramanujan-type supercongruences

Abstract: We present several supercongruences that may be viewed as $p$-adic analogues of Ramanujan-type series for $1/\pi$ and $1/\pi^2$, and prove three of these examples.


The length of the shortest closed geodesics on a positively curved manifold

Abstract: We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.


Probability theory and its models

Abstract: This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark's concept of model based probabilities is examined and is used as a bridge between the formal theory and applications.


Long-time Asymptotics for the NLS equation via dbar methods

Abstract: We present a new method for obtaining sharp asymptotics of solutions of the defocussing nonlinear Schr\"odinger (NLS) equation, based on dbar methods and under essentially minimal regularity assumptions on initial data.


Dutch book in simple multivariate normal prediction: Another look

Abstract: In this expository paper we describe a relatively elementary method of establishing the existence of a Dutch book in a simple multivariate normal prediction setting. The method involves deriving a nonstandard predictive distribution that is motivated by invariance. This predictive distribution satisfies an interesting identity which in turn yields an elementary demonstration of the existence of a Dutch book for a variety of possible predictive distributions.


Continuous families of Hamiltonian torus actions

Abstract: We determine conditions under which two Hamiltonian torus actions on a symplectic manifold $M$ are homotopic by a family of Hamiltonian torus actions, when $M$ is a toric manifold and when $M$ is a coadjoint orbit.


Connectivity of the Uniform Random Intersection Graph

Abstract: A \emph{uniform random intersection graph} $G(n,m,k)$ is a random graph constructed as follows. Label each of $n$ nodes by a randomly chosen set of $k$ distinct colours taken from some finite set of possible colours of size $m$. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks. Such graphs arise in particular when modelling the network graph of the well known key predistribution technique due to Eschenauer and Gligor. The paper determines the threshold for connectivity of the graph $G(n,m,k)$ when $n\to \infty$ with $k$ a function of $n$ such that $k\geq 2$ and $m=\lfloor n^\alpha\rfloor$ for some fixed positive real number $\alpha$. In this situation, $G(n,m,k)$ is almost surely connected when \[ \liminf k^2n/m\log n>1, \] and $G(n,m,k)$ is almost surely disconnected when \[ \limsup k^2n/m\log n<1. \]


Connexions affines et projectives sur les surfaces complexes compactes

Abstract: We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection $\nabla$ on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if $\nabla$ is a generic connection on a principal elliptic bundle over a Riemann surface of genus $g \geq 2$, with odd first Betti number. In the last case, the local Killing Lie algebra is of dimension one, generated by the fundamental vector field of the principal fibration.


Local Unitary Cocycles of E_0-semigroups

Abstract: This paper concerns the structure of the group of local unitary cocycles, also called the gauge group, of an E_0-semigroup. The gauge group of a spatial E_0-semigroup has a natural action on the set of units by operator multiplication. Arveson has characterized completely the gauge group of E_0-semigroups of type I, and as a consequence it is known that in this case the gauge group action is transitive. In fact, if the semigroup has index k, then the gauge group action is transitive on the set of k+1-tuples of appropriately normalized independent units. An action of the gauge group having this property is called k+1-fold transitive. We construct examples of E_0-semigroups of type II and index 1 which are not 2-fold transitive. These new examples also illustrate that an E_0-semigroup of type II_k need not be a tensor product of an E_0-semigroup of type II_0 and another of type I_k.


R\'eseaux d'induction des repr\'esentations elliptiques de Lubin-Tate

Abstract: We study the reduction modulo $l$ of some elliptic representations; for each of these representations, we give a particular lattice naturally obtained by parabolic induction in giving the graph of extensions between its irreducible sub-quotient of its reduction modulo $l$. The principal motivation for this work, is that these lattices appear in the cohomology of Lubin-Tate towers.


Characteristics of hand and machine-assigned scores to college students' answers to open-ended tasks

Abstract: Assessment of learning in higher education is a critical concern to policy makers, educators, parents, and students. And, doing so appropriately is likely to require including constructed response tests in the assessment system. We examined whether scoring costs and other concerns with using open-end measures on a large scale (e.g., turnaround time and inter-reader consistency) could be addressed by machine grading the answers. Analyses with 1359 students from 14 colleges found that two human readers agreed highly with each other in the scores they assigned to the answers to three types of open-ended questions. These reader assigned scores also agreed highly with those assigned by a computer. The correlations of the machine-assigned scores with SAT scores, college grades, and other measures were comparable to the correlations of these variables with the hand-assigned scores. Machine scoring did not widen differences in mean scores between racial/ethnic or gender groups. Our findings demonstrated that machine scoring can facilitate the use of open-ended questions in large-scale testing programs by providing a fast, accurate, and economical way to grade responses.


Generating uniform random vectors in $\QTR{bf}{Z}_{p}^{k}$: the general case

Abstract: This paper is about the rate of convergence of the Markov chain $X_{n+1}=AX_{n}+B_{n}$ (mod $p$), where $A$ is an integer matrix with nonzero eigenvalues and ${B_{n}}_{n}$ is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of $Q^{k}$ invariant under $A$. If $|\lambda_{i}|\not=1$ for all eigenvalues $\lambda_{i}$ of $A$, then $n=O((\ln p)^{2}) $ steps are sufficient and $n=O(\ln p)$ steps are necessary to have $X_{n}$ sampling from a nearly uniform distribution. Conversely, if $A$ has the eigenvalues $\lambda_{i}$ that are roots of positive integer numbers, $|\lambda_{1}|=1$ and $|\lambda_{i}|>1$ for all $i\not=1$, then $O(p^{2}) $ steps are necessary and sufficient.


Plane curves as Pfaffians

Abstract: Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$. In this paper we parametrise all linear pfaffian representations of $F$ by an open subset in the moduli space $M_C(2,K_C)$. We construct an explicit correspondence between pfaffian representations of $C$ and rank 2 vector bundles on $C$ with canonical determinant and no sections.


Alternative formulas for synthetic dual system estimation in the 2000 census

Abstract: The U.S. Census Bureau provides an estimate of the true population as a supplement to the basic census numbers. This estimate is constructed from data in a post-censal survey. The overall procedure is referred to as dual system estimation. Dual system estimation is designed to produce revised estimates at all levels of geography, via a synthetic estimation procedure. We design three alternative formulas for dual system estimation and investigate the differences in area estimates produced as a result of using those formulas. The primary target of this exercise is to better understand the nature of the homogeneity assumptions involved in dual system estimation and their consequences when used for the enumeration data that occurs in an actual large scale application like the Census. (Assumptions of this nature are sometimes collectively referred to as the ``synthetic assumption'' for dual system estimation.) The specific focus of our study is the treatment of the category of census counts referred to as imputations in dual system estimation. Our results show the degree to which varying treatment of these imputation counts can result in differences in population estimates for local areas such as states or counties.


On the history and use of some standard statistical models

Abstract: This paper tries to tell the story of the general linear model, which saw the light of day 200 years ago, and the assumptions underlying it. We distinguish three principal stages (ignoring earlier more isolated instances). The model was first proposed in the context of astronomical and geodesic observations, where the main source of variation was observational error. This was the main use of the model during the 19th century. In the 1920's it was developed in a new direction by R.A. Fisher whose principal applications were in agriculture and biology. Finally, beginning in the 1930's and 40's it became an important tool for the social sciences. As new areas of applications were added, the assumptions underlying the model tended to become more questionable, and the resulting statistical techniques more prone to misuse.


Counting the homeless in Los Angeles County

Abstract: Over the past two decades, a variety of methods have been used to count the homeless in large metropolitan areas. In this paper, we report on an effort to count the homeless in Los Angeles County, one that employed the sampling of census tracts. A number of complications are discussed, including\^{E} the need to impute homeless counts to areas of \^{E}the County\^{E} not sampled. We conclude that, despite their imperfections, estimated counts provided useful and credible information to the stakeholders involved.


Statistical adjustment for a measure of healthy lifestyle doesn't yield the truth about hormone therapy

Abstract: The Women's Health Initiative randomized clinical trial of hormone therapy found no benefit of hormones in preventive cardiovascular disease, a finding in striking contrast with a large body of observational research. Understanding whether better methodology and/or statistical adjustment might have prevented the erroneous conclusions of observational research is important. This is a re-analysis of data from a case-control study examining the relationship of postmenopausal hormone therapy and the risks of myocardial infarction (MI) and ischemic stroke in which we reported no overall increase or decrease in the risk of either event. Variables measuring health behavior/lifestyle that are not likely to be causally with the risks of MI and stroke (e.g., sunscreen use) were included in multivariate analysis along with traditional confounders (age, hypertension, diabetes, smoking, body mass index, ethnicity, education, prior coronary heart disease for MI and prior stroke/TIA for stroke) to determine whether adjustment for the health behavior/lifestyle variables could reproduce or bring the results closer to the findings in a large and definitive randomized clinical trial of hormone therapy, the Women's Health Initiative. For both MI and stroke, measures of health behavior/lifestyle were associated with odds ratios (ORs) less than 1.0. Adjustment for traditional cardiovascular disease confounders did not alter the magnitude of the ORs for MI or stroke. Addition of a subset of these variables selected using stepwise regression to the final MI or stroke models along with the traditional cardiovascular disease confounders moved the ORs for estrogen and estrogen/progestin use closer to values observed in the Women Health Initiative clinical trial, but did not reliably reproduce the clinical trial results for these two endpoints.


Convergence of Bergman measures for high powers of a line bundle

Abstract: Let $L$ be a holomorphic line bundle on a compact complex manifold $X$ of dimension $n,$ and let $e^{-\phi}$ be a continuous metric on $L.$ Fixing a measure $d\mu$ on $X$ gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of $L.$ We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair $(K,\phi),$ where $K$ is the support of $d\mu,$ as long as $d\mu$ is stably Bernstein-Markov with respect to $(K,\phi).$ Here the Bergman measure denotes $d\mu$ times the restriction to the diagonal of the pointwise norm of the corresponding orthogonal projection operator. In particular, an extension to higher dimensions is obtained of results concerning random matrices and classical orthogonal polynomials.


On the Hochschild homology of elliptic Sklyanin algebras

Abstract: In this paper, we compute the Hochschild homology of elliptic Sklyanin algebras. These algebras are deformations of polynomial algebra with a Poisson bracket called the Sklyanin Poisson bracket.


Diagonal invariants and the refined multimahonian distribution

Abstract: Combinatorial aspects of multivariate diagonal invariants of the symmetric group are studied. As a consequence it is proved the existence of a multivariate extension of the classical Robinson-Schensted correspondence. Further byproduct are a pure combinatorial algorithm to describe the irreducible decomposition of the tensor product of two irreducible representations of the symmetric group, and new symmetry results on permutation enumeration with respect to descent sets.


Cohomology algebra of orbit spaces of free involutions on lens spaces

Abstract: Let $G$ be a group acting continuously on a space $X$ and let $X/G$ be its orbit space. Determining the topological or cohomological type of the orbit space $X/G$ is a classical problem in the theory of transformation groups. In this paper, we consider this problem for cohomology lens spaces. Let $X$ be a finitistic space having the mod 2 cohomology algebra of the lens space $L_p^{2m-1}(q_1,...,q_m)$. Then we classify completely the possible mod 2 cohomology algebra of orbit spaces of arbitrary free involutions on $X$. We also give examples of spaces realizing the possible cohomology algebras. In the end, we give an application of our results to non-existence of $\mathbb{Z}_2$-equivariant map $\mathbb{S}^n \to X$.


Diophantine approximation with arithmetic functions, I

Abstract: We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.


Natural Topologies on Colombeau Algebras

Abstract: We define natural topologies on the Colombeau algebras which are compatible with the algebraic structure. These topologies reduces do Scarpalezos sharp topologies when restricted. with this we take a positive step towards topological methods of solving P.D. Equations in Colombeau algebras. Applications will appear elsewhere.


On the classification of twisting maps between $K^n$ and $K^m$

Abstract: We define the notion of admissible pair for an algebra $A$, consisting on a couple $(\Gamma,R)$, where $\Gamma$ is a quiver and $R$ a unital, splitted and factorizable representation of $\Gamma$, and prove that the set of admissible pairs for $A$ is in one to one correspondence with the points of the variety of twisting maps $\mathcal{T}_A^n:=\mathcal{T}(K^n,A)$. We describe all these representations in the case $A=K^m$.


Multivariate data analysis: The French way

Abstract: This paper presents exploratory techniques for multivariate data, many of them well known to French statisticians and ecologists, but few well understood in North American culture. We present the general framework of duality diagrams which encompasses discriminant analysis, correspondence analysis and principal components, and we show how this framework can be generalized to the regression of graphs on covariates.


The future of census coverage surveys

Abstract: A quarter-century of statistical research has shown that census coverage surveys, valuable as they are in offering a report card on each decennial census, do not provide usable estimates of geographical differences in coverage. The determining reason is the large number of ``doubly missing'' people missing both from the census enumeration and from coverage survey estimates. Future coverage surveys should be designed to meet achievable goals, foregoing efforts at spatial specificity. One implication is a sample size no more than about $30,000$, setting free resources for controlling processing errors and investing in coverage improvement. Possible integration of coverage measurement with the American Community Survey would have many benefits and should be given careful consideration.


Hypergeometric functions over F_p and relations to elliptic curves and modular forms

Abstract: For primes p congruent to 1 mod 12, we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728/t and values of a particular 2F1-hypergeometric function over F_p. Additionally, we determine a formula for traces of Hecke operators T_k(p) on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's tau-function in terms of hypergeometric functions.


Quantization of pseudo-differential operators on the torus

Abstract: Pseudo-differential and Fourier series operators on the n-torus are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on $L^2$ under certain conditions on their phases and amplitudes.


Moderate deviations for stationary sequences of Hilbert valued bounded random variables

Abstract: In this paper, we derive the moderate deviation principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non adpated sequences of Hilbert-valued random variables. Applications to Cramer-Von Mises statistics, functions of linear processes and stable Markov chains are given.


Counter example to Strichartz estimates for the wave equation in domains

Abstract: Let U be a bounded, regular, strictly convex domain of R^2 and consider the wave equation on U with Dirichlet boundary condition. We prove that in such a domain the Strichartz estimates for the wave equation suffer losses when compared to the case U=R^2, at least for a subset of the usual range of indices.


Howe type duality for metaplectic group acting on symplectic spinor valued forms

Abstract: Let $\lambda: \tilde{G}\to G$ be the non-trivial double covering of the symplectic group $G=Sp(V,\omega)$ of the symplectic vector space $(V,\omega)$ by the metaplectic group $\tilde{G}=Mp(V,\omega).$ In this case, $\lambda$ is also a representation of $\tilde{G}$ on the vector space $V$ and thus, it gives rise to the representation of $\tilde{G}$ on the space of exterior forms $\bigwedge^{\bullet}V^*$ by taking wedge products. Let $S$ be the minimal globalization of the Harish-Chandra module of the complex Segal-Shale-Weil representation of the metaplectic group $\tilde{G}.$ We prove that the associative commutant algebra $\hbox{End}_{\tilde{G}}(\bigwedge^{\bullet}V^*\otimes S)$ of the metaplectic group $\tilde{G}$ acting on the $S$-valued exterior forms is generated by certain representation of the super ortho-symplectic Lie algebra $osp(1|2)$ and two distinguished operators. This establishes a Howe type duality between the metaplectic group and the super Lie algebra $\mathfrak{osp}(1|2).$ Also the space $\bigwedge^{\bullet}V^*\otimes S$ is decomposed wr. to the joint action of $Mp(V,\omega)$ and $osp(1|2).$


Vector partition function and generalized Dahmen-Micchelli spaces

Abstract: This is the first of two papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in a subsequent paper.


The line bundles on moduli stacks of principal bundles on a curve

Abstract: Let G be an affine reductive algebraic group over an algebraically closed field k. We determine the Picard group of the moduli stacks of principal G-bundles on any smooth projective curve over k.


Optimal reconstruction systems for erasures and for the q-potential

Abstract: We introduce the $q$-potential as an extension of the Benedetto-Fickus frame potential, defined on general reconstruction systems and we show that protocols are the minimizers of this potential under certain restrictions. We extend recent results of B.G. Bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst (normalized) reconstruction error is computed with respect to a compatible unitarily invariant norm. We finally describe necessary and sufficient (spectral) conditions, that we call $q$-fundamental inequalities, for the existence of protocols with prescribed properties by relating this problem to Klyachko's and Fulton's theory on sums of hermitian operators.


Asymptotically linear solutions in H^1 of the 2-d defocusing nonlinear Schroedinger and Hartree equations

Abstract: In the 2-d setting, given an $H^1$ solution $v(t)$ to the linear Schr\"odinger equation $i\partial_t v +\Delta v =0$, we prove the existence (but not uniqueness) of an $H^1$ solution $u(t)$ to the defocusing nonlinear Schr\"odinger (NLS) equation $i\partial_t u + \Delta u -|u|^{p-1}u=0$ for nonlinear powers $2<p<3$ and the existence of an $H^1$ solution $u(t)$ to the defocusing Hartree equation $i\partial_t u + \Delta u -(|x|^{-\gamma}\star|u|^{2})u=0$ for interaction powers $1<\gamma<2$, such that $\|u(t)-v(t)\|_{H^1} \to 0$ as $t\to +\infty$. This is a partial result toward the existence of well-defined continuous wave operators $H^1 \to H^1$ for these equations. For NLS in 2-d, such wave operators are known to exist for $p\geq 3$, while for $p\leq 2$ it is known that they cannot exist. The Hartree equation in 2-d only makes sense for $0<\gamma<2$, and it was previously known that wave operators cannot exist for $0<\gamma\leq 1$, while no result was previously known in the range $1<\gamma<2$. Our proof in the case of NLS applies a new estimate of Colliander-Grillakis-Tzirakis (2008) to a strategy devised by Nakanishi (2001). For the Hartree equation, we prove a new correlation estimate following the method of Colliander-Grillakis-Tzirakis (2008).


The Space of Symplectic Structures on Closed 4-Manifolds

Abstract: This is a survey paper on the space of symplectic structures on closed 4-manifolds, for the Proceedings ICCM 2004


Wintenberger's Functor for Abelian Extensions

Abstract: Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k((T))$ and $A$ is an abelian $p$-adic Lie subgroup of $\Aut_k(K)$. In this paper we extend this equivalence to allow $\Gal(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups.


Modified Schmidt games and Diophantine approximation with weights

Abstract: We show that the sets of weighted badly approximable vectors in $\Bbb R^n$ are winning sets of certain games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.


The Relative Symplectic Cone and T^2-Fibrations

Abstract: In this note we introduce the notion of the relative symplectic cone. As an application, we determine the symplectic cone of certain T^2-fibrations. In particular, for some elliptic surfaces we verify a conjecture on the symplectic cone of minimal Kaehler surfaces raised by the second author.


Projective C*-Algebras and Boundary Maps

Abstract: Both boundary maps in K-theory are expressed in terms of surjections from projective C*-algebras to semiprojective C*-algebras.


Adjoint functors and tree duality

Abstract: A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.


Generalized BSDE With 2-Reflecting Barriers and Stochastic Quadratic Growth

Abstract: We study the existence of a solution for a one-dimensional generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under assumptions on the input data which are weaker than that on the current literature. In particular, we construct a maximal solution for such a GRBSDE when the terminal condition \xi is only F_T-measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z without assuming any P-integrability conditions. The work is suggested by the interest the results might have in Dynkin game problem and American game option.


Cocycle Superrigidity for Profinite Actions of Property (T) Groups

Abstract: Consider a free ergodic measure preserving profinite action $\Gamma\curvearrowright X$ (i.e. an inverse limit of actions $\Gamma\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally of a group $\Gamma$ which admits an infinite normal subgroup $\Gamma_0$ such that the inclusion $\Gamma_0\subset\Gamma$ has relative property (T) and $\Gamma/\Gamma_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:\Gamma\times X\to \Lambda$ is a measurable cocycle with values in a countable group $\Lambda$, then $w$ is cohomologous to a cocycle $w'$ which factors through the map $\Gamma\times X\to \Gamma\times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $\Gamma\curvearrowright X$ with any free ergodic measure preserving action $\Lambda\curvearrowright Y$ comes from a (virtual) conjugacy of actions.


Multiple tests of association with biological annotation metadata

Abstract: We propose a general and formal statistical framework for multiple tests of association between known fixed features of a genome and unknown parameters of the distribution of variable features of this genome in a population of interest. The known gene-annotation profiles, corresponding to the fixed features of the genome, may concern Gene Ontology (GO) annotation, pathway membership, regulation by particular transcription factors, nucleotide sequences, or protein sequences. The unknown gene-parameter profiles, corresponding to the variable features of the genome, may be, for example, regression coefficients relating possibly censored biological and clinical outcomes to genome-wide transcript levels, DNA copy numbers, and other covariates. A generic question of great interest in current genomic research regards the detection of associations between biological annotation metadata and genome-wide expression measures. This biological question may be translated as the test of multiple hypotheses concerning association measures between gene-annotation profiles and gene-parameter profiles. A general and rigorous formulation of the statistical inference question allows us to apply the multiple hypothesis testing methodology developed in [Multiple Testing Procedures with Applications to Genomics (2008) Springer, New York] and related articles, to control a broad class of Type I error rates, defined as generalized tail probabilities and expected values for arbitrary functions of the numbers of Type I errors and rejected hypotheses. The resampling-based single-step and stepwise multiple testing procedures of [Multiple Testing Procedures with Applications to Genomics (2008) Springer, New York] take into account the joint distribution of the test statistics and provide Type I error control in testing problems involving general data generating distributions (with arbitrary dependence structures among variables), null hypotheses, and test statistics.


Three months journeying of a Hawaiian monk seal

Abstract: Hawaiian monk seals (Monachus schauinslandi) are endemic to the Hawaiian Islands and are the most endangered species of marine mammal that lives entirely within the jurisdiction of the United States. The species numbers around 1300 and has been declining owing, among other things, to poor juvenile survival which is evidently related to poor foraging success. Consequently, data have been collected recently on the foraging habitats, movements, and behaviors of monk seals throughout the Northwestern and main Hawaiian Islands. Our work here is directed to exploring a data set located in a relatively shallow offshore submerged bank (Penguin Bank) in our search of a model for a seal's journey. The work ends by fitting a stochastic differential equation (SDE) that mimics some aspects of the behavior of seals by working with location data collected for one seal. The SDE is found by developing a time varying potential function with two points of attraction. The times of location are irregularly spaced and not close together geographically, leading to some difficulties of interpretation. Synthetic plots generated using the model are employed to assess its reasonableness spatially and temporally. One aspect is that the animal stays mainly southwest of Molokai. The work led to the estimation of the lengths and locations of the seal's foraging trips.


Ces\`aro means of Jacobi expansions on the parabolic biangle

Abstract: We study Ces\`aro $(C,\delta)$ means for two-variable Jacobi polynomials on the parabolic biangle $B=\{(x_1,x_2)\in{\mathbb R}^2:0\leq x_1^2\leq x_2\leq 1\}$. Using the product formula derived by Koornwinder & Schwartz for this polynomial system, the Ces\`aro operator can be interpreted as a convolution operator. We then show that the Ces\`aro $(C,\delta)$ means of the orthogonal expansion on the biangle are uniformly bounded if $\delta>\alpha+\beta+1$, $\alpha-\frac 12\geq\beta\geq 0$. Furthermore, for $\delta\geq\alpha+2\beta+\frac 32$ the means define positive linear operators.


DNA Probabilities in People v. Prince: When are racial and ethnic statistics relevant?

Abstract: When a defendant's DNA matches a sample found at a crime scene, how compelling is the match? To answer this question, DNA analysts typically use relative frequencies, random-match probabilities or likelihood ratios. They compute these quantities for the major racial or ethnic groups in the United States, supplying prosecutors with such mind-boggling figures as ``one in nine hundred and fifty sextillion African Americans, one in one hundred and thirty septillion Caucasians, and one in nine hundred and thirty sextillion Hispanics." In People v. Prince, a California Court of Appeals rejected this practice on the theory that only the perpetrator's race is relevant to the crime; hence, it is impermissible to introduce statistics about other races. This paper critiques this reasoning. Relying on the concept of likelihood, it presents a logical justification for referring to a range of races and identifies some problems with the one-race-only rule. The paper also notes some ways to express the probative value of a DNA match quantitatively without referring to variations in DNA profile frequencies among races or ethnic groups.


A-infinity monads and completion

Abstract: Given an operad A of topological spaces, we consider A-monads in a topological category C . When A is an A-infinity-operad, any A-monad K : C -> C can be thought of as a monad up to coherent homotopies. We define the completion functor with respect to an A-infinity-monad and prove that it is an A-infinity-monad itself.


Testing earthquake predictions

Abstract: Statistical tests of earthquake predictions require a null hypothesis to model occasional chance successes. To define and quantify `chance success' is knotty. Some null hypotheses ascribe chance to the Earth: Seismicity is modeled as random. The null distribution of the number of successful predictions -- or any other test statistic -- is taken to be its distribution when the fixed set of predictions is applied to random seismicity. Such tests tacitly assume that the predictions do not depend on the observed seismicity. Conditioning on the predictions in this way sets a low hurdle for statistical significance. Consider this scheme: When an earthquake of magnitude 5.5 or greater occurs anywhere in the world, predict that an earthquake at least as large will occur within 21 days and within an epicentral distance of 50 km. We apply this rule to the Harvard centroid-moment-tensor (CMT) catalog for 2000--2004 to generate a set of predictions. The null hypothesis is that earthquake times are exchangeable conditional on their magnitudes and locations and on the predictions--a common ``nonparametric'' assumption in the literature. We generate random seismicity by permuting the times of events in the CMT catalog. We consider an event successfully predicted only if (i) it is predicted and (ii) there is no larger event within 50 km in the previous 21 days. The $P$-value for the observed success rate is $<0.001$: The method successfully predicts about 5% of earthquakes, far better than `chance,' because the predictor exploits the clustering of earthquakes -- occasional foreshocks -- which the null hypothesis lacks. Rather than condition on the predictions and use a stochastic model for seismicity, it is preferable to treat the observed seismicity as fixed, and to compare the success rate of the predictions to the success rate of simple-minded predictions like those just described. If the proffered predictions do no better than a simple scheme, they have little value.


Curse-of-dimensionality revisited: Collapse of the particle filter in very large scale systems

Abstract: It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate that the maximum of the weights associated with the sample ensemble converges to one as both the sample size and the system dimension tends to infinity. Specifically, under fairly weak assumptions, if the ensemble size grows sub-exponentially in the cube root of the system dimension, the convergence holds for a single update step in state-space models with independent and identically distributed kernels. Further, in an important special case, more refined arguments show (and our simulations suggest) that the convergence to unity occurs unless the ensemble grows super-exponentially in the system dimension. The weight singularity is also established in models with more general multivariate likelihoods, e.g. Gaussian and Cauchy. Although presented in the context of atmospheric data assimilation for numerical weather prediction, our results are generally valid for high-dimensional particle filters.


Semi-continuit\'e des cellules de Kazhdan-Lusztig

Abstract: Computations in small Coxeter groups and infinite dihedral groups suggest that Kazhdan-Lusztig cells for unequal parameters obey to some "semicontinuity" phenomenon (as the parameter vary). The aim of this paper is to provide a rigorous theoretical background that allows to state some precise conjectures.