Higher order influence functions and minimax estimation of nonlinear functionals

Abstract: We present a theory of point and interval estimation for nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions (Robins (2004), Section 9; Li et al. (2004), Tchetgen et al. (2006), Robins et al. (2007)). Higher order influence functions are higher order U-statistics. Our theory extends the first order semiparametric theory of Bickel et al. (1993) and van der Vaart (1991) by incorporating the theory of higher order scores considered by Pfanzagl (1990), Small and McLeish (1994) and Lindsay and Waterman (1996). The theory reproduces many previous results, produces new non-$\sqrt{n}$ results, and opens up the ability to perform optimal non-$\sqrt{n}$ inference in complex high dimensional models. We present novel rate-optimal point and interval estimators for various functionals of central importance to biostatistics in settings in which estimation at the expected $\sqrt{n}$ rate is not possible, owing to the curse of dimensionality. We also show that our higher order influence functions have a multi-robustness property that extends the double robustness property of first order influence functions described by Robins and Rotnitzky (2001) and van der Laan and Robins (2003).


Projection pursuit for discrete data

Abstract: This paper develops projection pursuit for discrete data using the discrete Radon transform. Discrete projection pursuit is presented as an exploratory method for finding informative low dimensional views of data such as binary vectors, rankings, phylogenetic trees or graphs. We show that for most data sets, most projections are close to uniform. Thus, informative summaries are ones deviating from uniformity. Syllabic data from several of Plato's great works is used to illustrate the methods. Along with some basic distribution theory, an automated procedure for computing informative projections is introduced.


Finitude homotopique et isotopique des structures de contact tendues

Abstract: Let V be a closed 3-manifold. In this paper we prove that the homotopy classes of plane fields on V that contain tight contact structures are in finite number and that, if V is atoroidal, the isotopy classes of tight contact structures are also in finite number.


Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions

Abstract: It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $ for $ j \in \mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $ j.$ The random reward (or scenery) $ \xi_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk $ S_n$ had independent increments limits.


Objective Bayesian analysis under sequential experimentation

Abstract: Objective priors for sequential experiments are considered. Common priors, such as the Jeffreys prior and the reference prior, will typically depend on the stopping rule used for the sequential experiment. New expressions for reference priors are obtained in various contexts, and computational issues involving such priors are considered.


J. K. Ghosh's contribution to statistics: A brief outline

Abstract: Professor Jayanta Kumar Ghosh has contributed massively to various areas of Statistics over the last five decades. Here, we survey some of his most important contributions. In roughly chronological order, we discuss his major results in the areas of sequential analysis, foundations, asymptotics, and Bayesian inference. It is seen that he progressed from thinking about data points, to thinking about data summarization, to the limiting cases of data summarization in as they relate to parameter estimation, and then to more general aspects of modeling including prior and model selection.


Sequential tests and estimates after overrunning based on $p$-value combination

Abstract: Often in sequential trials additional data become available after a stopping boundary has been reached. A method of incorporating such information from overrunning is developed, based on the ``adding weighted Zs'' method of combining $p$-values. This yields a combined $p$-value for the primary test and a median-unbiased estimate and confidence bounds for the parameter under test. When the amount of overrunning information is proportional to the amount available upon terminating the sequential test, exact inference methods are provided; otherwise, approximate methods are given and evaluated. The context is that of observing a Brownian motion with drift, with either linear stopping boundaries in continuous time or discrete-time group-sequential boundaries. The method is compared with other available methods and is exemplified with data from two sequential clinical trials.


Classification of complex naturally graded quasi-filiform Zinbiel algebras

Abstract: In this work the description up to isomorphism of complex naturally graded quasi-filiform Zinbiel algebras is obtained.


On predictive probability matching priors

Abstract: We revisit the question of priors that achieve approximate matching of Bayesian and frequentist predictive probabilities. Such priors may be thought of as providing frequentist calibration of Bayesian prediction or simply as devices for producing frequentist prediction regions. Here we analyse the $O(n^{-1})$ term in the expansion of the coverage probability of a Bayesian prediction region, as derived in [Ann. Statist. 28 (2000) 1414--1426]. Unlike the situation for parametric matching, asymptotic predictive matching priors may depend on the level $\alpha$. We investigate uniformly predictive matching priors (UPMPs); that is, priors for which this $O(n^{-1})$ term is zero for all $\alpha$. It was shown in [Ann. Statist. 28 (2000) 1414--1426] that, in the case of quantile matching and a scalar parameter, if such a prior exists then it must be Jeffreys' prior. In the present article we investigate UPMPs in the multiparameter case and present some general results about the form, and uniqueness or otherwise, of UPMPs for both quantile and highest predictive density matching.


A note on the ABC-PRC algorithm of Sissons et al. (2007)

Abstract: This note describes the results of some tests of the ABC-PRC algorithm of Sissons et al. (PNAS, 2007), and demonstrates with a toy example that the method does not converge on the true posterior distribution.


Galois actions on Neron models of Jacobians

Abstract: Let $X$ be a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, and let $K'/K$ be a tame extension. We study extensions of the $G = \Gal(K'/K)$-action on $ X_{K'} $ to certain regular models of $X_{K'}$ over $R'$, the integral closure of $R$ in $K'$. In particular, we consider the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model, and obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtration of the special fiber of the N\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $R$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber types for curves of genus 1 and 2.


Price dynamics in a strategic model of trade between two regions

Abstract: This paper develops a strategic model of trade between two regions in which, depending on the relation among output, financial resources and transportation costs, the adjustment of prices towards an equilibrium is studied. We derive conditions on the relations among output and financial resources which produce different types of Nash equilibria. The paths obtained in the process of converging toward a steady state for prices under discrete-time and continuous-time dynamics are derived and compared. It turns out that the results in the two cases differ substantially. Some of the effects of random disturbances on the price dynamics in continuous time are also studied.


Triply Periodic Minimal Surfaces Bounded by Vertical Symmetry Planes

Abstract: We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces.


Variational characterizations of the effective multiplication factor of a nuclear reactor core

Abstract: We derive some new inf-sup and sup-inf formulae for the so-called effective multiplication factor arising in the study of reactor analysis. We treat in a same formalism the transport equation and the energy-dependent diffusion equation.


The cone of lower semicontinuous traces on a C*-algebra

Abstract: The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely non-compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2-quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su algebra and have no non-zero simple subquotients, and simple C*-algebras that absorb the Jiang-Su algebra.


Derived deformations of Artin stacks

Abstract: We generalise the techniques of arXiv:0908.1963 to describe derived deformations in simplicial categories. This allows us to consider deformation problems with higher automorphisms, such as chain complexes (which have homotopies) and stacks (which have 2-automorphisms). We also give a general approach for studying deformations of diagrams.


Rigidity of Quasi-Einstein Metrics

Abstract: We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\"ahler quasi-Einstein metrics.


Asymptotic expansion of the heat kernel for orbifolds

Abstract: We study the relationship between the geometry and the Laplace spectrum of a Riemannian orbifold O via its heat kernel; as in the manifold case, the time-zero asymptotic expansion of the heat kernel furnishes geometric information about O. In the case of a good Riemannian orbifold (i.e., an orbifold arising as the orbit space of a manifold under the action of a discrete group of isometries), H. Donnelly proved the existence of the heat kernel and constructed the asymptotic expansion for the heat trace. We extend Donnelly's work to the case of general compact orbifolds. Moreover, in both the good case and the general case, we express the heat invariants in a form that clarifies the asymptotic contribution of each part of the singular set of the orbifold. We calculate several terms in the asymptotic expansion explicitly in the case of two-dimensional orbifolds; we use these terms to prove that the spectrum distinguishes elements within various classes of two-dimensional orbifolds.


Higher order derivative estimates for finite-difference schemes

Abstract: We give sufficient conditions under which solutions of finite-difference schemes in the space variable for second order possibly degenerate parabolic and elliptic equations admit estimates of spatial derivatives up to any given order independent of the mesh size.


An elementary proof of a series evaluation in terms of harmonic numbers

Abstract: An elementary proof of an identity by Lyons, Paule and Riese is given. It is simpler than all the 3 published proofs.


Smooth analysis of the condition number and the least singular value

Abstract: Let $\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $\a$ is gaussian. Our investigation reveals an interesting fact that the "core" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielman-Teng studies when $\a$ is gaussian. Consequently, our general estimate involves the norm $\|M\|$. In the special case when $\|M\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.


Hopf algebra extensions of group algebras and Tambara-Yamagami categories

Abstract: We determine the structure of Hopf algebra extensions of a group algebra by the cyclic group of order 2. We study the corepresentation theory of such Hopf algebras, which provide a generalization, at the Hopf algebra level, of the so called Tambara-Yamagami fusion categories. As a byproduct, we show that every semisimple Hopf algebra of dimension $<36$ is necessarily group-theoretical; thus 36 is the smallest possible dimension where a non group-theoretical example occurs.


Unknotting numbers of diagrams of a given nontrivial knot are unbounded

Abstract: We show that for any nontrivial knot $K$ and any natural number $n$ there is a diagram $D$ of $K$ such that the unknotting number of $D$ is greater than or equal to $n$. It is well known that twice the unknotting number of $K$ is less than or equal to the crossing number of $K$ minus one. We show that the equality holds only when $K$ is a $(2,p)$-torus knot.


Thue's Fundamentaltheorem, I: The General Case

Abstract: In this paper, Thue's Fundamentaltheorem is analysed. We show that it includes, and often strengthens, known effective irrationality measures obtained via the so-called hypergeometric method as well as showing that it can be applied to previously unconsidered families of algebraic numbers. Furthermore, we extend the method to also cover approximation by algebraic numbers in imaginary quadratic number fields.


A model of hydrodynamic interaction between swimming bacteria

Abstract: We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbell-type structures. We focus on alignment dynamics of a coplanar pair of elongated swimmers, which propel themselves either by pushing" or "pulling" both in three- and quasi-two-dimensional geometries of space. We derive asymptotic expressions for the dynamics of the pair, which, complemented by numerical experiments, indicate that the tendency of bacteria to swim in or swim off depends strongly on the position of the propulsion force. In particular, we observe that positioning of the effective propulsion force inside the dumbbell results in qualitative agreement with the dynamics observed in experiments, such as mutual alignment of converging bacteria.


Accurate Evaluation of Polynomials

Abstract: For a large class of polynomials, the standard method of polynomial evaluation, Horner's method, can be very inaccurate. The alternative method given here is on average 100 to 1000 times more accurate than Horner's Method. The number of floating point operations is twice that of Horner's method for a single evaluation. For repeated evaluations at nearby points, the number of floating point operations is only doubled for the first evaluation, and is the same as Horner's Method for all following evaluations. This new method is tested with random polynomials.


Hecke algebras from groups acting on trees and HNN extensions

Abstract: We study Hecke algebras of groups acting on trees with respect to geometrically defined subgroups. In particular, we consider Hecke algebras of groups of automorphisms of locally finite trees with respect to vertex and edge stabilizers and the stabilizer of an end relative to a vertex stabilizer, assuming that the actions are sufficiently transitive. We focus on identifying the structure of the resulting Hecke algebras, give explicit multiplication tables of the canonical generators and determine whether the Hecke algebra has a universal C*-completion. The paper unifies past algebraic and analytic approaches by focusing on the common geometric thread.The results have implications for the general theory of totally disconnected locally compact groups.


Data-dependent probability matching priors for empirical and related likelihoods

Abstract: We consider a general class of empirical-type likelihoods and develop higher order asymptotics with a view to characterizing members thereof that allow the existence of possibly data-dependent probability matching priors ensuring approximate frequentist validity of posterior quantiles. In particular, for the usual empirical likelihood, positive results are obtained. This is in contrast with what happens if only data-free priors are entertained.


Probability matching priors for some parameters of the bivariate normal distribution

Abstract: This paper develops some objective priors for certain parameters of the bivariate normal distribution. The parameters considered are the regression coefficient, the generalized variance, and the ratio of the conditional variance of one variable given the other to the marginal variance of the other variable. The criterion used is the asymptotic matching of coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. The paper uses various matching criteria, namely, quantile matching, matching of distribution functions, highest posterior density matching, and matching via inversion of test statistics. One particular prior is found which meets all the matching criteria individually for all the parameters of interest.


Fuzzy set representation of a prior distribution

Abstract: In the subjective Bayesian approach uncertainty is described by a prior distribution chosen by the statistician. Fuzzy set theory is another way of representing uncertainty. Here we give a decision theoretic approach which allows a Bayesian to convert their prior distribution into a fuzzy set membership function. This yields a formal relationship between these two different methods of expressing uncertainty.


Fuzzy sets in nonparametric Bayes regression

Abstract: A simple Bayesian approach to nonparametric regression is described using fuzzy sets and membership functions. Membership functions are interpreted as likelihood functions for the unknown regression function, so that with the help of a reference prior they can be transformed to prior density functions. The unknown regression function is decomposed into wavelets and a hierarchical Bayesian approach is employed for making inferences on the resulting wavelet coefficients.


Radical Parametrization of Algebraic Curves by Adjoint Curves

Abstract: We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less o equal to 4 and they are defined over an algebraically closed field of characteristic zero. In addition, we also present an algorithm for parametrizing by radicals any irreducible plane curve of degree $d$ having at least a point of multiplicity $d-r$, with $1\leq r \leq 4$ and, as a consequence, every irreducible plane curve of degree $d \leq 5$ and every irreducible singular plane curve of degree 6.


Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations

Abstract: S. Montgomery-Smith provided a one dimensional model for the three dimensional, incompressible Navier-Stokes equations, for which he proved the blow up of solutions associated to a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this note the model is adapted to the case of two and three space dimensions, with the additional feature that the divergence free condition is preserved. It is checked that the family of initial data constructed previously by J.-Y Chemin and I. Gallagher which is arbitrarily large but yet generates a global solution to the Navier-Stokes equations in three space dimensions, actually causes blow up for the toy model -- meaning that the precise structure of the nonlinear term is crucial to understand the dynamics of large solutions to the Navier-Stokes equations.


The initial value problem for a third-order dispersive flow into compact almost Hermitian manifolds

Abstract: We present a time-local existence theorem of the initial value problem for a third-order dispersive evolution equation for open curves on compact almost Hermitian manifolds arising in the geometric analysis of vortex filaments. This equation causes the so-called loss of one-derivative since the target manifold is not supposed to be a K\"ahler manifold. We overcome this difficulty by using a gauge transformation of a multiplier on the pull-back bundle to eliminate the bad first order terms essentially.


Objective Bayes testing of Poisson versus inflated Poisson models

Abstract: The Poisson distribution is often used as a standard model for count data. Quite often, however, such data sets are not well fit by a Poisson model because they have more zeros than are compatible with this model. For these situations, a zero-inflated Poisson (ZIP) distribution is often proposed. This article addresses testing a Poisson versus a ZIP model, using Bayesian methodology based on suitable objective priors. Specific choices of objective priors are justified and their properties investigated. The methodology is extended to include covariates in regression models. Several applications are given.


Consistent selection via the Lasso for high dimensional approximating regression models

Abstract: In this article we investigate consistency of selection in regression models via the popular Lasso method. Here we depart from the traditional linear regression assumption and consider approximations of the regression function $f$ with elements of a given dictionary of $M$ functions. The target for consistency is the index set of those functions from this dictionary that realize the most parsimonious approximation to $f$ among all linear combinations belonging to an $L_2$ ball centered at $f$ and of radius $r_{n,M}^2$. In this framework we show that a consistent estimate of this index set can be derived via $\ell_1$ penalized least squares, with a data dependent penalty and with tuning sequence $r_{n,M}>\sqrt{\log(Mn)/n}$, where $n$ is the sample size. Our results hold for any $1\leq M\leq n^{\gamma}$, for any $\gamma>0$.


Relative log convergent cohomology and relative rigid cohomology III

Abstract: In this paper, we prove the generic overconvergence of relative rigid cohomology with coefficient, by using the semistable reduction conjecture for overconvergent $F$-isocrystals (which is recently shown by Kedlaya).


Asymptotic optimality of a cross-validatory predictive approach to linear model selection

Abstract: In this article we study the asymptotic predictive optimality of a model selection criterion based on the cross-validatory predictive density, already available in the literature. For a dependent variable and associated explanatory variables, we consider a class of linear models as approximations to the true regression function. One selects a model among these using the criterion under study and predicts a future replicate of the dependent variable by an optimal predictor under the chosen model. We show that for squared error prediction loss, this scheme of prediction performs asymptotically as well as an oracle, where the oracle here refers to a model selection rule which minimizes this loss if the true regression were known.


Risk and resampling under model uncertainty

Abstract: In statistical exercises where there are several candidate models, the traditional approach is to select one model using some data driven criterion and use that model for estimation, testing and other purposes, ignoring the variability of the model selection process. We discuss some problems associated with this approach. An alternative scheme is to use a model-averaged estimator, that is, a weighted average of estimators obtained under different models, as an estimator of a parameter. We show that the risk associated with a Bayesian model-averaged estimator is bounded as a function of the sample size, when parameter values are fixed. We establish conditions which ensure that a model-averaged estimator's distribution can be consistently approximated using the bootstrap. A new, data-adaptive, model averaging scheme is proposed that balances efficiency of estimation without compromising applicability of the bootstrap. This paper illustrates that certain desirable risk and resampling properties of model-averaged estimators are obtainable when parameters are fixed but unknown; this complements several studies on minimaxity and other properties of post-model-selected and model-averaged estimators, where parameters are allowed to vary.


On a Monge-Amp\`{e}re type equation in the Cegrell class $\mathcal{E}_{\chi}$

Abstract: In this paper we prove an existence and uniqueness result for a Monge-Amp\`{e}re type equation in the Cegrell class $\mathcal{E}_{\chi}$.


Remarks on consistency of posterior distributions

Abstract: In recent years, the literature in the area of Bayesian asymptotics has been rapidly growing. It is increasingly important to understand the concept of posterior consistency and validate specific Bayesian methods, in terms of consistency of posterior distributions. In this paper, we build up some conceptual issues in consistency of posterior distributions, and discuss panoramic views of them by comparing various approaches to posterior consistency that have been investigated in the literature. In addition, we provide interesting results on posterior consistency that deal with non-exponential consistency, improper priors and non i.i.d. (independent but not identically distributed) observations. We describe a few examples for illustrative purposes.


Voting power and Qualified Majority Voting with a "no vote" option

Abstract: In recent years, enlargement of the European Union has led to increased interest in the allocation of voting weights to member states with hugely differing population numbers. While the eventually agreed voting scheme lacks any strict mathematical basis, the Polish government suggested a voting scheme based on the Penrose definition of voting power, leading to an allocation of voting weights proportional to the square root of the population (the "Jagiellonian Compromise"). The Penrose definition of voting power is derived from the citizens' freedom to vote either "yes" or "no". This paper defines a corresponding voting power based on "yes", "no" and "abstain" options, and it is found that this definition also leads to a square root law, and to the same optimal vote allocation as the Penrose scheme.


Reproducing kernel Hilbert spaces of Gaussian priors

Abstract: We review definitions and properties of reproducing kernel Hilbert spaces attached to Gaussian variables and processes, with a view to applications in nonparametric Bayesian statistics using Gaussian priors. The rate of contraction of posterior distributions based on Gaussian priors can be described through a concentration function that is expressed in the reproducing Hilbert space. Absolute continuity of Gaussian measures and concentration inequalities play an important role in understanding and deriving this result. Series expansions of Gaussian variables and transformations of their reproducing kernel Hilbert spaces under linear maps are useful tools to compute the concentration function.


Self-dual projective toric varieties

Abstract: Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T-module P(V). We determine when a projective toric subvariety X of P(V) is self-dual, in terms of the configuration of weights of V.


A Bayesian semi-parametric model for small area estimation

Abstract: In public health management there is a need to produce subnational estimates of health outcomes. Often, however, funds are not available to collect samples large enough to produce traditional survey sample estimates for each subnational area. Although parametric hierarchical methods have been successfully used to derive estimates from small samples, there is a concern that the geographic diversity of the U.S. population may be oversimplified in these models. In this paper, a semi-parametric model is used to describe the geographic variability component of the model. Specifically, we assume Dirichlet process mixtures of normals for county-specific random effects. Results are compared to a parametric model based on the base measure of the Dirichlet process, using binary health outcomes related to mammogram usage.


A Logical Calculus To Intuitively And Logically Denote Number Systems

Abstract: Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy sequences are common notations for number systems. In this note, first, it is proven that both simple continued fractions and base-b expansions fail to denote real numbers and thus lack logic; second, it is shown that Dedekind cuts and Cauchy sequences fail to join in algebraical operations and thus lack intuition; third, we construct a logical calculus and deduce numbers to intuitively and logically denote number systems.


On Perelman's Dilaton

Abstract: By means of a Kaluza-Klein type argument we show that the Perelman's F-functional is the Einstein-Hilbert action in a space with extra ``phantom'' dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his first famous paper. As a consequence the Ricci flow (modified by a diffeomorphism and a time-dependent factor) is the evolution of the ``real'' part of the metric under a constrained gradient flow of the Einstein-Hilbert gravitational action in higher dimension.


A hierarchical Bayesian approach for estimating the origin of a mixed population

Abstract: We propose a hierarchical Bayesian model to estimate the proportional contribution of source populations to a newly founded colony. Samples are derived from the first generation offspring in the colony, but mating may occur preferentially among migrants from the same source population. Genotypes of the newly founded colony and source populations are used to estimate the mixture proportions, and the mixture proportions are related to environmental and demographic factors that might affect the colonizing process. We estimate an assortative mating coefficient, mixture proportions, and regression relationships between environmental factors and the mixture proportions in a single hierarchical model. The first-stage likelihood for genotypes in the newly founded colony is a mixture multinomial distribution reflecting the colonizing process. The environmental and demographic data are incorporated into the model through a hierarchical prior structure. A simulation study is conducted to investigate the performance of the model by using different levels of population divergence and number of genetic markers included in the analysis. We use Markov chain Monte Carlo (MCMC) simulation to conduct inference for the posterior distributions of model parameters. We apply the model to a data set derived from grey seals in the Orkney Islands, Scotland. We compare our model with a similar model previously used to analyze these data. The results from both the simulation and application to real data indicate that our model provides better estimates for the covariate effects.


Finite element scheme for integro-partial differential equations

Abstract: We construct a finite element like scheme for fully non-linear integro-partial differential equations arising in optimal control of jump-processes. Special cases of these equations include optimal portfolio and option pricing equations in Finance. The schemes are monotone and robust. We prove that they converge in very general situations, including degenerate equations, multiple dimensions, relatively low regularity of the data, and for most (if not all) types of jump-models used in Finance. In all cases we provide (probably optimal) error bounds. These bounds apply when grids are unstructured and integral terms are very singular, two features that are new or highly unusual in this setting.


Kendall's tau in high-dimensional genomic parsimony

Abstract: High-dimensional data models, often with low sample size, abound in many interdisciplinary studies, genomics and large biological systems being most noteworthy. The conventional assumption of multinormality or linearity of regression may not be plausible for such models which are likely to be statistically complex due to a large number of parameters as well as various underlying restraints. As such, parametric approaches may not be very effective. Anything beyond parametrics, albeit, having increased scope and robustness perspectives, may generally be baffled by the low sample size and hence unable to give reasonable margins of errors. Kendall's tau statistic is exploited in this context with emphasis on dimensional rather than sample size asymptotics. The Chen--Stein theorem has been thoroughly appraised in this study. Applications of these findings in some microarray data models are illustrated.


A Note on Partial List Colorings

Abstract: Let $G$ be a simple graph with $n$ vertices and list chromatic number $\chi_\ell(G)=\chi_\ell$. Suppose that $0\leq t\leq \chi_\ell$ and each vertex of $G$ is assigned a list of $t$ colors. Albertson, Grossman and Haas [1] conjectured that at least $\frac{tn}{\chi_\ell}$ vertices of $G$ can be colored from these lists. In this paper we find some new results in partial list coloring which help us to show that the conjecture is true for at least half of the numbers of the set $\{1,2,...,\chi_\ell(G)-1\}$. In addition we introduce a new related conjecture and finally we present some results about this conjecture.


Orthogonalized smoothing for rescaled spike and slab models

Abstract: Rescaled spike and slab models are a new Bayesian variable selection method for linear regression models. In high dimensional orthogonal settings such models have been shown to possess optimal model selection properties. We review background theory and discuss applications of rescaled spike and slab models to prediction problems involving orthogonal polynomials. We first consider global smoothing and discuss potential weaknesses. Some of these deficiencies are remedied by using local regression. The local regression approach relies on an intimate connection between local weighted regression and weighted generalized ridge regression. An important implication is that one can trace the effective degrees of freedom of a curve as a way to visualize and classify curvature. Several motivating examples are presented.


Nonparametric statistics on manifolds with applications to shape spaces

Abstract: This article presents certain recent methodologies and some new results for the statistical analysis of probability distributions on manifolds. An important example considered in some detail here is the 2-D shape space of k-ads, comprising all configurations of $k$ planar landmarks ($k>2$)-modulo translation, scaling and rotation.


An ensemble approach to improved prediction from multitype data

Abstract: We have developed a strategy for the analysis of newly available binary data to improve outcome predictions based on existing data (binary or non-binary). Our strategy involves two modeling approaches for the newly available data, one combining binary covariate selection via LASSO with logistic regression and one based on logic trees. The results of these models are then compared to the results of a model based on existing data with the objective of combining model results to achieve the most accurate predictions. The combination of model predictions is aided by the use of support vector machines to identify subspaces of the covariate space in which specific models lead to successful predictions. We demonstrate our approach in the analysis of single nucleotide polymorphism (SNP) data and traditional clinical risk factors for the prediction of coronary heart disease.


Sharp failure rates for the bootstrap particle filter in high dimensions

Abstract: We prove that the maximum of the sample importance weights in a high-dimensional Gaussian particle filter converges to unity unless the ensemble size grows exponentially in the system dimension. Our work is motivated by and parallels the derivations of Bengtsson, Bickel and Li (2007); however, we weaken their assumptions on the eigenvalues of the covariance matrix of the prior distribution and establish rigorously their strong conjecture on when weight collapse occurs. Specifically, we remove the assumption that the nonzero eigenvalues are bounded away from zero, which, although the dimension of the involved vectors grow to infinity, essentially permits the effective system dimension to be bounded. Moreover, with some restrictions on the rate of growth of the maximum eigenvalue, we relax their assumption that the eigenvalues are bounded from above, allowing the system to be dominated by a single mode.


Handle moves in contact surgery diagrams

Abstract: We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots.


Hochschild and ordinary cohomology rings of small categories

Abstract: Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH*(kC) --> H*(|C|,k) and prove it is split surjective. This generalizes the well-known results for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, against a conjecture of Snashall and Solberg.


Proving the existence of the $n$th root by induction

Abstract: In this paper we prove by induction on $n$ that any positive real number has $n$th root.


On a certain generalization of the Balog-Szemeredi-Gowers Theorem

Abstract: In this note, we prove a certain hypergraph generalization of the Balog-Szemeredi-Gowers Theorem. Our result shares some features in common with a similar such generalizsation due to Sudakov, Szemeredi and Vu, though the conclusion of our result is somewhat different.


An Introduction to Smooth Infinitesimal Analysis

Abstract: An exposition of smooth infinitesimal analysis, which is a way to do calculus with nilsquare infinitesimals, is given.


Interior Cauchy-Schauder estimates for the heat flow in Carnot-Caratheodory spaces

Abstract: The purpose of this paper is to establish some basic interior estimates of Cauchy-Schauder type for the heat flow associated with a system of smooth vector fields satisfying Hormander's finite rank condition


Characterizing indecomposable plane continua from their complements

Abstract: We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit into the boundary of U_n, one can choose components of U_n minus A_n whose boundaries intersected with the continuum (which we call shadows) converge to the continuum.


Any counterexample to Makienko's conjecture is an indecomposable continuum

Abstract: Makienko's conjecture, a proposed addition to Sullivan's dictionary, can be stated as follows: The Julia set of a rational function R has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko's conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational functions whose Julia set is an indecomposable continuum.


Existence and stability of noncharacteristic boundary-layers for the compressible Navier-Stokes and viscous MHD equations

Abstract: For a general class of hyperbolic-parabolic systems including the compressible Navier-Stokes and compressible MHD equations, we prove existence and stability of noncharacteristic viscous boundary layers for a variety of boundary conditions including classical Navier-Stokes boundary conditions. Our first main result, using the abstract framework established by the authors in the companion work \cite{GMWZ6}, is to show that existence and stability of arbitrary amplitude exact boundary-layer solutions follow from a uniform spectral stability condition on layer profiles that is expressible in terms of an Evans function (uniform Evans stability). Whenever this condition holds we give a rigorous description of the small viscosity limit as the solution of a hyperbolic problem with "residual" boundary conditions. Our second is to show that uniform Evans stability for small-amplitude layers is equivalent to Evans stability of the limiting constant layer, which in turn can be checked by a linear-algebraic computation. Finally, for a class of symmetric-dissipative systems including the physical examples mentioned above, we carry out energy estimates showing that constant (and thus small-amplitude) layers always satisfy uniform Evans stability. This yields existence of small-amplitude multi-dimensional boundary layers for the compressible Navier-Stokes and MHD equations. For both equations these appear to be the first such results in the compressible case.


Natural Equivariant Dirac Operators

Abstract: We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-valued equivariant index coincide with those of an elliptic operator constructed from the original data.


Slices of motivic Landweber spectra

Abstract: We show that the Conjecture of Voevodsky concerning slices of the algebraic cobordism spectrum MGL implies a general statement about the slices of motivic Landweber spectra. In particular it confirms the possible approach suggested by Voevodsky for the computation of the slices of the homotopy algebraic K-theory spectrum KGL via a Conner-Floyd isomorphism complementing Levine's unconditional proof of these slices over perfect fields. A similar result, and Voevodsky's conjecture over fields of char. 0, are also announced by Hopkins-Morel.


Solubility of Fermat equations

Abstract: The arithmetic of ternary diagonal equation is considered for degree d >1, with the outcome that the set of coefficients for which the equation admits a non-zero integer solution is shown to have density zero.


On what Ontology Is and not-Is

Abstract: In this paper I study the connection between logic and metaphysics in Plato's participation theory, from the structural properties of the latter. Although Plato was the first ever to formulate the contradiction principle explicitly (in the Phaedo), the logic underlying his system appears to be paraconsistent. This confirms an earlier suggestion by G. Priest. Its technical characteristics and the textual evidence supporting this interpretation are both studied in detail.


An $L_\infty$ algebra structure on polyvector fields

Abstract: It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra $A=S(V^*)$ fails if the vector space $V$ is infinite-dimensional. In the present paper, we study the corresponding obstructions. We construct an $L_\infty$ structure on polyvector fields on $V$ having the even degree Taylor components, with the degree 2 component given by the Schouten-Nijenhuis bracket, but having as well higher non-vanishing Taylor components. We prove that this $L_\infty$ algebra is quasi-isomorphic to the corresponding Hochschild cochain complex. We prove that our $L_\infty$ algebra is $L_\infty$ quasi-isomorphic to the Lie algebra of polyvector fields on $V$ with the Schouten-Nijenhuis bracket, if $V$ is finite-dimensional.


On minimal extensions of rings

Abstract: Given two rings $R \subseteq S$, $S$ is said to be a minimal ring extension of $R$ if $R$ is a maximal subring of $S$. In this article, we study minimal extensions of an arbitrary ring $R$, with particular focus on those possessing nonzero ideals that intersect $R$ trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs & Shapiro, and Ferrand & Olivier on commutative minimal extensions.


Proper holomorphic mapppings between Reinhardt domains in $\mathbb C^2$

Abstract: We describe all possibilities of existence of non-elementary proper holomorphic maps between non-hyperbolic Reinhardt domains in $\mathbb C^2$ and the corresponding pairs of domains.


Bernstein measures on convex polytopes

Abstract: We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functions which is a generalization of the asymptotic expansion of the Bernstein polynomials on the standard $m$-simplex obtained by Abel-Ivan and H\"{o}rmander. These are different from the Bergman-Bernstein approximations over Delzant polytopes recently introduced by Zelditch. We discuss relations between Bernstein approximations defined in this paper and Zelditch's Bergman-Bernstein approximations.


Cross Curvature Flow on Locally Homogeneous Three-manifolds (II)

Abstract: In this paper, we study the positive cross curvature flow on locally homogeneous 3-manifolds. We describe the long time behavior of these flows. We combine this with earlier results concerning the asymptotic behavior of the negative cross curvature flow to describe the two sided behavior of maximal solutions of the cross curvature flow on locally homogeneous 3-manifolds. We show that, typically, the positive cross curvature flow on locally homogeneous 3-manifold produce an Heisenberg type sub-Riemannian geometry.


Differential Harnack Estimates for Backward Heat Equations with Potentials under the Ricci Flow

Abstract: In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.


A First Sight Towards Primitively Generated Connected Braided Bialgebras

Abstract: The main aim of this paper is to investigate the structure of primitively generated connected braided bialgebras $A$ with respect to the braided vector space $P$ consisting of their primitive elements. When the Nichols algebra of $P$ is obtained dividing out the tensor algebra $T(P) $ by the two-sided ideal generated by its primitive elements of degree at least two, we show that $A$ can be recovered as a sort of universal enveloping algebra of $P$. One of the main applications of our construction is the description, in terms of universal enveloping algebras, of connected braided bialgebras whose associated graded coalgebra is a quadratic algebra.


Estimation in models driven by fractional Brownian motion

Abstract: Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type \[X(t)=c+\int_0^t\sigma\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int _0^t\mu\bigl(X(u)\bigr)\mathrm {d}u.\] In different particular models where $\sigma(x)=\sigma$ or $\sigma(x)=\sigma x$ and $\mu(x)=\mu$ or $\mu(x)=\mu x$, we propose a central limit theorem for estimators of $H$ and of $\sigma$ based on regression methods. Then we give tests of the hypothesis on $\sigma$ for these models. We also consider functional estimation on $\sigma(\cdot)$ in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm.


On Asymptotic Stability of Solitary Waves in Discrete Schr\"odinger Equation Coupled to Nonlinear Oscillator

Abstract: The long-time asymptotics is analyzed for finite energy solutions of the 1D discrete Schr\"odinger equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group. For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Schr\"odinger equation. The proofs use the strategy of Buslaev-Perelman: the linerization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.


Cohomology of Courant algebroids with split base

Abstract: We study the (standard) cohomology $H^\bullet_{st}(E)$ of a Courant algebroid $E$. We prove that if $E$ is transitive, the standard cohomology coincides with the naive cohomology $H_{naive}^\bullet(E)$ as conjectured by Stienon and Xu. For a general Courant algebroid we define a spectral sequence converging to its standard cohomology. If $E$ is with split base, we prove that there exists a natural transgression homomorphism $T_3$ (with image in $H^3_{naive}(E)$) which, together with the naive cohomology, gives all $H^\bullet_{st}(E)$. For generalized exact Courant algebroids, we give an explicit formula for $T_3$ depending only on the \v{S}evera characteristic clas of $E$.


The least singular value of a random square matrix is O(n^{-1/2})

Abstract: Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value of A is of order n^{-1/2} with high probability. The lower estimate of this type was proved recently by the authors; in this note we establish the matching upper estimate.


A note on James spaces and superstrictly singular operators

Abstract: An elementary lemma is used in order to show that the natural inclusion $J_p\to J_q$ of James spaces is superstrictly singular for $p<q$. As a consequence, it is shown that an operator without nontrivial invariant subspaces constructed by Charles Read is superstrictly singular.


On Upper-Confidence Bound Policies for Non-Stationary Bandit Problems

Abstract: Multi-armed bandit problems are considered as a paradigm of the trade-off between exploring the environment to find profitable actions and exploiting what is already known. In the stationary case, the distributions of the rewards do not change in time, Upper-Confidence Bound (UCB) policies have been shown to be rate optimal. A challenging variant of the MABP is the non-stationary bandit problem where the gambler must decide which arm to play while facing the possibility of a changing environment. In this paper, we consider the situation where the distributions of rewards remain constant over epochs and change at unknown time instants. We analyze two algorithms: the discounted UCB and the sliding-window UCB. We establish for these two algorithms an upper-bound for the expected regret by upper-bounding the expectation of the number of times a suboptimal arm is played. For that purpose, we derive a Hoeffding type inequality for self normalized deviations with a random number of summands. We establish a lower-bound for the regret in presence of abrupt changes in the arms reward distributions. We show that the discounted UCB and the sliding-window UCB both match the lower-bound up to a logarithmic factor.


Vitesse de convergence dans le th\'{e}or\`{e}me limite central pour des cha\^{i}nes de Markov fortement ergodiques

Abstract: Let $Q$ be a transition probability on a measurable space $E$ which admits an invariant probability measure, let $(X_n)_n$ be a Markov chain associated to $Q$, and let $\xi$ be a real-valued measurable function on $E$, and $S_n=\sum _{k=1}^n\xi(X_k)$. Under functional hypotheses on the action of $Q$ and the Fourier kernels $Q(t)$, we investigate the rate of convergence in the central limit theorem for the sequence $(\frac{S_n}{\sqrt{n}})_n$. According to the hypotheses, we prove that the rate is, either $\mathrm{O}(n^{-{\tau}/{2}})$ for all $\tau<1$, or $\mathrm{O}(n^{-{1}/{2}})$. We apply the spectral Nagaev's method which is improved by using a perturbation theorem of Keller and Liverani, and a majoration of $|\mathbb{E}[\mathrm{e}^{\mat hrm{i}t{S_n}/{\sqrt{n}}}]-\mathrm{e}^{{-t^2}/{2}}|$ obtained by a method of martingale difference reduction. When $E$ is not compact or $\xi$ is not bounded, the conditions required here on $Q(t)$ (in substance, some moment conditions on $\xi$) are weaker than the ones usually imposed when the standard perturbation theorem is used in the spectral method. For example, in the case of $V$-geometric ergodic chains or Lipschitz iterative models, the rate of convergence in the c.l.t. is $\mathrm{O}(n^{-{1}/{2}})$ under a third moment condition on $\xi$.


Fesenko reciprocity map

Abstract: In recent papers, Fesenko has defined the non-abelian local reciprocity map for every totally-ramified arithmetically profinite ($APF$) Galois extension of a given local field $K$ by extending the works of Hazewinkel and Neukirch-Iwasawa. The theory of Fesenko extends the previous non-abelian generalizations of local class field theory given by Koch-de Shalit and by A. Gurevich. In this paper, which is research-expository in nature, we give a detailed account of Fesenko's work including all the skipped proofs.


Some results on the second Gaussian map for curves

Abstract: We study the second Gaussian map for a curve X of genus g, in relation with the second fundamental form of the period map. We exhibit a class of infinitely many curves with surjective second Gaussian map. We compute its rank on the hyperelliptic and trigonal locus and show global generation of its image for X not hyperelliptic nor trigonal.


Siegel metric and curvature of the moduli space of curves

Abstract: We study the curvature of the moduli space M_g of curves of genus g with the Siegel metric induced by the period map. We give an explicit formula for the holomorphic sectional curvature of M_g along a Schiffer variation at a point P on the curve X, in terms of the holomorphic sectional curvature of A_g and the second Gaussian map. Finally we extend the Kaehler form of the Siegel metric as a closed current on the Deligne-Mumford compatification of M_g and we determine its cohomology class as a multiple of the first Chern class of the Hodge bundle.


Generalized Fesenko reciprocity map

Abstract: In this paper, which is the natural continuation and generalization of Fesenko's non-abelian reciprocity map, we extend the theory of Fesenko to infinite $APF$-Galois extensions $L$ over a local field $K$, with finite residue-class field $\kappa_K$ of $q=p^f$ elements, satisfying $\pmb{\mu}_p(K^{sep})\subset K$ and $K\subset L\subset K_{\phi^d}$ where the residue-class degree $[\kappa_L:\kappa_K]=d$. More precisely, for such extensions $L/K$, fixing a Lubin-Tate splitting $\phi$ over $K$, we construct a 1-cocycle, \pmb{\Phi}_{L/K}^{(\phi)}:\text{Gal}(L/K)\to K^\times/N_{L_0/K}L_0^\times\times U_{\widetilde{\mathbb X}(L/K)}^\diamond /Y_{L/L_0}, where $L_0=L\cap K^{nr}$, and study its functorial and ramification-theoretic properties. The case $d=1$ recovers the theory of Fesenko.


L-R-smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules

Abstract: Let H be a bialgebra and D an H-bimodule algebra H-bicomodule coalgebra. We find sufficient conditions on D for the L-R-smash product algebra and coalgebra structures on D\otimes H to form a bialgebra (in this case we say that (H, D) is an L-R-admissible pair), called L-R-smash biproduct. The Radford biproduct is a particular case, and so is, up to isomorphism, a double biproduct with trivial pairing. We construct a prebraided monoidal category {\cal LR}(H), whose objects are H-bimodules H-bicomodules M endowed with left-left and right-right Yetter-Drinfeld module as well as left-right and right-left Long module structures over H, with the property that, if (H, D) is an L-R-admissible pair, then D is a bialgebra in {\cal LR}(H).


A Solution to the Monotonicity Problem for Unimodal Families

Abstract: In this note we consider a collection C of one parameter families of unimodal maps of [0,1]. Each family in the collection has the form uf where u is in [0,1]. Denoting the kneading sequence of uf by K(uf), we will prove that for each member of C, the map u->K(uf) is monotone. It then follows that for each member of C the map u -> h(uf) is monotone, where h(uf) is the topological entropy of uf. For interest, uf(x)=4ux(1-x) and uf(x)=usin(pi x) are shown to belong to C.


Quasi-elementary H-Azumaya algebras arising from generalized (anti) Yetter-Drinfeld modules

Abstract: Let H be a Hopf algebra with bijective antipode, let \alpha, \beta be two Hopf algebra automorphisms of H and M a finite dimensional (\alpha, \beta )-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k, H), the Brauer group of H.


On the Milnor fibers of cyclic quotient singularities

Abstract: The oriented link of the cyclic quotient singularity $\mathcal{X}_{p,q}$ is orientation-preserving diffeomorphic to the lens space $L(p,q)$ and carries the standard contact structure $\xi_{st}$. Lisca classified the Stein fillings of $(L(p,q), \xi_{st})$ up to diffeomorphisms and conjectured that they correspond bijectively through an {\it explicit} map to the Milnor fibers associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of $\mathcal{X}_{p,q}$. We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibers given by de Jong and van Straten, we also canonically identify these fibers with Lisca's fillings. Using these and a newly introduced additional structure - the order - associated with lens spaces, we prove that the above Milnor fibers are pairwise non-diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens.


Comparison between criteria leading to the weak invariance principle

Abstract: The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields 110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincar\'{e} Probab. Statist. 36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in every ergodic dynamical system with positive entropy, if we consider two of these criteria, we can find a function in $\mathbb{L}^2$ satisfying the first but not the second.


Synchronization in Networks of Identical Linear Systems

Abstract: The paper investigates the synchronization of a network of identical linear state-space models under a possibly time-varying and directed interconnection structure. The main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected. The result can be interpreted as a generalization of classical consensus algorithms. Stronger conditions are shown to be sufficient but to some extent, also necessary to ensure synchronization with the diffusive static output coupling often considered in the literature.


Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Abstract: Let $K$ be a one-variable function field over a field of constants of characteristic 0. Let $R$ be a holomorphy subring of $K$, not equal to $K$. We prove the following undecidability results for $R$: If $K$ is recursive, then Hilbert's Tenth Problem is undecidable in $R$. In general, there exist $x_1,...,x_n \in R$ such that there is no algorithm to tell whether a polynomial equation with coefficients in $\Q(x_1,...,x_n)$ has solutions in $R$.


Extended affine Lie algebras and other generalizations of affine Lie algebras - a survey

Abstract: This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras.


Leading coefficients of the Kazhdan-Lusztig polynomials for an Affine Weyl group of type $\widetilde{B_2}$

Abstract: In this paper we compute the leading coefficients $\mu (u,w)$ of the Kazhdan--Lusztig polynomials $P_{u,w}$ for an affine Weyl group of type $\tilde{B}_2$. By using the \textbf{a}-function of a Coxeter group defined by Lusztig (see [L1, \S2]), we compute most $\mu (u,w)$ explicitly. With part of these values $\mu (u,w)$, we show that a conjecture of Lusztig on distinguished involutions is true for an affine Weyl group of type $\tilde{B}_2$. We also show that the conjectural formula in [L3, (12)] needs a modification.


Singular value decomposition of large random matrices (for two-way classification of microarrays)

Abstract: Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to Wigner-noise is investigated.It is proved that such an m\times n matrix almost surely has a constant number of large singular values (of order \sqrt{mn}), while the rest of the singular values are of order \sqrt{m+n} as m,n\to\infty. Concentration results of Alon et al. for the eigenvalues of large symmetric random matrices are adapted to the rectangular case, and on this basis, almost sure results for the singular values as well as for the corresponding isotropic subspaces are proved. An algorithm, applicable to two-way classification of microarrays, is also given that finds the underlying block structure.


Locally Toroidal Polytopes and Modular Linear Groups

Abstract: When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d>1, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, we here develop methods to handle composite moduli and completely describe the corresponding modular polytopes when G is of spherical or Euclidean type. Using a modular variant of the quotient criterion, we then describe the locally toroidal polytopes provided by our construction, most of which are new.


Counting numerical sets with no small atoms

Abstract: A numerical set $S$ with Frobenius number $g$ is a set of integers with $\min(S) = 0$ and $\max(\Zbb - S)=g$, and its atom monoid is $A(S) = \setpres{n \in \Zbb}{$n+s \in S$ for all $s \in S$}$. Let $\gamma_g$ be the number of numerical sets $S$ having $A(S) = \set{0} \cup (g,\infty)$ divided by the total number of numerical sets with Frobenius number $g$. We show that the sequence $\set{\gamma_g}$ is decreasing and converges to a number $\gamma_\infty \approx .4844$ (with accuracy to within $.0050$). We also examine the singularities of the generating function for $\set{\gamma_g}$. Parallel results are obtained for the ratio $\gsymm{g}$ of the number of symmetric numerical sets $S$ with $A(S) = \set{0} \cup (g,\infty)$ by the number of symmetric numerical sets with Frobenius number $g$. These results yield information regarding the asymptotic behavior of the number of finite additive 2-bases.


Effective log Iitaka fibrations for surfaces and threefolds

Abstract: We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair $(X,\Delta)$ of Kodaira codimension one and dimension at most three such that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}$, there is a natural number $N$ that depends only on $\mathcal{A}$ for which the round down of $\N(K_X+\Delta)$ induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.


Riemannian manifolds with geometric structures

Abstract: Some geometric structures with associated Riemannian metrics have been considered in the book.


Extremely Non-symmetric, Non-multiplicative, Non-commutative Operator Spaces

Abstract: Motivated by importance of operator spaces contained in the set of all scalar multiples of isometries ($MI$-spaces) in a separable Hilbert space for $C^*$-algebras and E-semigroups we exhibit more properties of such spaces. For example, if an $MI$-space contains an isometry with shift part of finite multiplicity, then it is one-dimensional. We propose a simple model of a unilateral shift of arbitrary multiplicity and show that each separable subspace of a Hilbert space is the range of a shift. Also, we show that $MI$-spaces are non-symmetric, very unfriendly to multiplication, and prove a Commutator Identity which elucidates the extreme non-commutativity of these spaces.


Hyper-atoms and the critical pair Theory

Abstract: We introduce the notion of a hyper-atom. One of the main results of this paper is the $\frac{2|G|}3$--Theorem: Let $S$ be a finite generating subset of an abelian group $G$ of order $\ge 2$. Let $T$ be a finite subset of $G$ such that $2\le |S|\le |T|$, $S+T$ is aperiodic, $0\in S\cap T$ and $$ \frac{2|G|+2}3\ge |S+T|= |S|+|T|-1.$$ Let $H$ be a hyper-atom of $S$. Then $S$ and $T$ are $H$--quasi-periodic. Moreover $\phi(S)$ and $\phi(T)$ are arithmetic progressions with the same difference, where $\phi :G\mapsto G/H$ denotes the canonical morphism. This result implies easily the traditional critical pair Theory and its basic stone: Kemperman's Structure Theorem.


Deformation-obstruction theory for complexes via Atiyah and Kodaira--Spencer classes

Abstract: We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira--Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.


New cubature formulae and hyperinterpolation in three variables

Abstract: A new algebraic cubature formula of degree $2n+1$ for the product Chebyshev measure in the $d$-cube with $\approx n^d/2^{d-1}$ nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree $n$ in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube.


Poisson Geometry of Directed Networks in a Disk

Abstract: We investigate Poisson properties of Postnikov's map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grasmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriately chosen R-matrix Poisson-Lie structure. We also prove that Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.


Weighted Vogan diagrams associated to real nilpotent orbits

Abstract: We associate to each nilpotent orbit of a real semisimple Lie algebra $g_o$ a weighted Vogan diagram, that is a Dynkin diagram with an involution of the diagram, a subset of painted nodes and a weight for each node. Every nilpotent element of $g_o$ is noticed in some subalgebra of $g_o$. In this paper we characterize the weighted Vogan diagrams associated to orbits of noticed nilpotent elements.


Groups of volume-preserving diffeomorphisms of noncompact manifolds and mass flow toward ends

Abstract: Suppose M is a noncompact connected oriented C^infty n-manifold and omega is a positive volume form on M. Let D^+(M) denote the group of orientation preserving diffeomorphisms of M endowed with the compact-open C^infty topology and D(M; omega) denote the subgroup of omega-preserving diffeomorphisms of M. In this paper we propose a unified approach for realization of mass transfer toward ends by diffeomorphisms of M. This argument together with Moser's theorem enables us to deduce two selection theorems for the groups D^+(M) and D(M; omega). The first one is the extension of Moser's theorem to noncompact manifolds, that is, the existence of sections for the orbit maps under the action of D^+(M) on the space of volume forms. This implies that D(M; omega) is a strong deformation retract of the group D^+(M; E^omega_M) consisting of h in D^+(M) which preserves the set E^omega_M of omega-finite ends of M. The second one is related to the mass flow toward ends under volume-preserving diffeomorphisms of M. Let D_{E_M}(M; omega) denote the subgroup consisting of all h in D(M; omega) which fix the ends E_M of M. S.R.Alpern and V.S.Prasad introduced the topological vector space S(M; omega) of end charges of M and the end charge homomorphism c^omega : D_{E_M}(M; omega) to S(M; omega), which measures the mass flow toward ends induced by each h in D_{E_M}(M; omega). We show that the homomorphism c^omega has a continuous section. This induces the factorization D_{E_M}(M; omega) cong ker c^omega times S(M; omega) and it implies that ker c^omega is a strong deformation retract of D_{E_M}(M; omega).


Generalized improper integral definition for infinite limit

Abstract: A generalization of the definition of a one-dimensional improper integral with an infinite limit is presented. The new definition extends the range of valid integrals to include integrals which were previously considered to not be integrable. This definition is shown to preserve linearity and uniqueness. Integrals which are valid under the conventional definition have the same value under the new definition. Criteria for interchanging the order of integration and differentiation, and for interchanging the order of integration with a second integration, are determined. Examples are provided. A restriction on changing the variable of integration using integration by substitution with the new definition is demonstrated.


Homology rigidity of Grassmannians

Abstract: Applying the theory of Gr\"{o}bner basis to the Schubert presentation of the cohomology of Grassmannians, we extend the homology rigidity results known for the classical Grassmannians to the exceptional cases.


Averages of ratios of characteristic polynomials in circular beta-ensembles and super-Jack polynomials

Abstract: We study the averages of ratios of characteristic polynomials over circular $\beta$-ensembles, where $\beta$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as sums of super-Jack polynomials and another one is given by a hyperdeterminant. As applications, we give dual relations for ratio averages between $\beta$ and $4/\beta$.


Embedding property of $J$-holomorphic curves in Calabi-Yau manifolds for generic $J$

Abstract: In this paper, we prove that for a generic choice of tame (or compatible) almost complex structures $J$ on a symplectic manifold $(M^{2n},\omega)$ with $n \geq 3$ and with its first Chern class $c_1(M,\omega) = 0$, all somewhere injective $J$-holomorphic maps from any closed smooth Riemann surface into $M$ are \emph{embedded}. We derive this result as a consequence of the general optimal 1-jet evaluation transversality result of $J$-holomorphic maps in general symplectic manifolds that we also prove in this paper.


On adaptive Bayesian inference

Abstract: We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general in-probability theorems on the rate of convergence of the resulting posterior distributions. We extend their results to almost sure assertions. As an application we study log spline densities with a finite number of models and obtain that the Bayes procedure achieves the optimal minimax rate $n^{-\gamma/(2\gamma+1)}$ of convergence if the true density of the observations belongs to the H\"{o}lder space $C^{\gamma}[0,1]$. This strengthens a result in [1; 2]. We also study consistency of posterior distributions of the model index and give conditions ensuring that the posterior distributions concentrate their masses near the index of the best model.


Explicit error bounds for lazy reversible Markov Chain Monte Carlo

Abstract: We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte Carlo methods, such as the Metropolis algorithm. The problem is to compute the expectation (or integral) of f with respect to a measure which can be given by a density with respect to another measure. A straight simulation of the desired distribution by a random number generator is in general not possible. Thus it is reasonable to use Markov chain sampling with a burn-in. We study such an algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an explicit error bound.


Computationally Efficient Nonparametric Importance Sampling

Abstract: The variance reduction established by importance sampling strongly depends on the choice of the importance sampling distribution. A good choice is often hard to achieve especially for high-dimensional integration problems. Nonparametric estimation of the optimal importance sampling distribution (known as nonparametric importance sampling) is a reasonable alternative to parametric approaches.In this article nonparametric variants of both the self-normalized and the unnormalized importance sampling estimator are proposed and investigated. A common critique on nonparametric importance sampling is the increased computational burden compared to parametric methods. We solve this problem to a large degree by utilizing the linear blend frequency polygon estimator instead of a kernel estimator. Mean square error convergence properties are investigated leading to recommendations for the efficient application of nonparametric importance sampling. Particularly, we show that nonparametric importance sampling asymptotically attains optimal importance sampling variance. The efficiency of nonparametric importance sampling algorithms heavily relies on the computational efficiency of the employed nonparametric estimator. The linear blend frequency polygon outperforms kernel estimators in terms of certain criteria such as efficient sampling and evaluation. Furthermore, it is compatible with the inversion method for sample generation. This allows to combine our algorithms with other variance reduction techniques such as stratified sampling. Empirical evidence for the usefulness of the suggested algorithms is obtained by means of three benchmark integration problems. As an application we estimate the distribution of the queue length of a spam filter queueing system based on real data.


Gene profiling for determining pluripotent genes in a time course microarray experiment

Abstract: In microarray experiments, it is often of interest to identify genes which have a pre-specified gene expression profile with respect to time. Methods available in the literature are, however, typically not stringent enough in identifying such genes, particularly when the profile requires equivalence of gene expression levels at certain time points. In this paper, the authors introduce a new methodology, called gene profiling, that uses simultaneous differential and equivalent gene expression level testing to rank genes according to a pre-specified gene expression profile. Gene profiling treats the vector of true gene expression levels as a linear combination of appropriate vectors, i.e., vectors that give the required criteria for the profile. This gene-profile model is fitted to the data and the resultant parameter estimates are summarized in a single test statistic that is then used to rank the genes. The theoretical underpinnings of gene profiling (equivalence testing, intersection-union tests) are discussed in this paper, and the gene profiling methodology is applied to our motivating stem cell experiment.


Marginal Likelihood Integrals for Mixtures of Independence Models

Abstract: Inference in Bayesian statistics involves the evaluation of marginal likelihood integrals. We present algebraic algorithms for computing such integrals exactly for discrete data of small sample size. Our methods apply to both uniform priors and Dirichlet priors. The underlying statistical models are mixtures of independent distributions, or, in geometric language, secant varieties of Segre-Veronese varieties.


Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy

Abstract: We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition. We show that these solutions approach constant equilibrium state in the Lp-norm at a rate O(t^(-m/2(1-1/p))), as t tends to $\infty$, for p in [min (m,2),+ \infty]. Moreover, we can show that we can approximate, with a faster order of convergence, theconservative part of the solution in terms of the linearized hyperbolic operator for m >= 2, and by a parabolic equation in the spirit of Chapman-Enskog expansion. The main tool is given by a detailed analysis of the Green function for the linearized problem.


Simultaneous realization of function space structures in transitive Banach spaces

Abstract: This paper has been withdrawn, since it contains a corollary with impossible consequences and the source of an error is currently unknown.


Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

Abstract: Let C be a curve over a non-singular base variety S. We study algebraic cycles on the symmetric powers C^[n] and on the Jacobian J. The Chow homology of C^[*], the sum of all C^[n], is a ring using the Pontryagin product. We prove that this ring is isomorphic to CH(J)[t]<u>, the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over the Chow ring CH(J). We give two such isomorphisms that over a general base are different. Further we give some precise results on how CH(J) sits embedded in CH(C^[*]) and we give an explicit geometric description of how the derivations with regard to t and u act. Our results give rise to a new grading on the Chow ring of the Jacobian. After tensoring with Q the associated descending filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's. Finally we give a version of our main result for tautological classes, and we show how our methods give a very simple and geometric proof of some relations obtained by Herbaut and van der Geer-Kouvidakis.


Riemannian products which are conformally equivalent to Einstein metrics

Abstract: Necessary and sufficient conditions for a Riemannian product to be conformally equivalent to an Einstein manifold are given. Such spaces which are complete are characterized.


On Caccetta-Haggkvist Conjecture

Abstract: We show that we cannot avoid the existence of at least one directed circuit of length less than or equal to (n/r) in a digraph on n vertices with out-degree greater than or equal to r. This is well-known Caccetta-Haggkvist problem.


Spectral properties of general advection operators and weighted translation semigroups

Abstract: We investigate the spectral properties of a class of weighted shift semigroups associated to abstract transport equations with a Lipschitz--continuous vector field with no--reentry boundary conditions. We illustrate our results with various examples taken from collisionless kinetic theory.


Sym\'etrie des grandes solutions d'\'equations elliptiques semi lin\'eaire

Abstract: We prove that, if $g$ is a continuous asymptotically convex function, any solution $u$ of $-\Delta u+g(u)=0$ in a ball B which tends to infinity on $\partial B$ is radially symmetric.


Likelihood for generally coarsened observations from multi-state or counting process models

Abstract: We consider first the mixed discrete-continuous scheme of observation in multistate models; this is a classical pattern in epidemiology because very often clinical status is assessed at discrete visit times while times of death or other events are observed exactly. A heuristic likelihood can be written for such models, at least for Markov models; however, a formal proof is not easy and has not been given yet. We present a general class of possibly non-Markov multistate models which can be represented naturally as multivariate counting processes. We give a rigorous derivation of the likelihood based on applying Jacod's formula for the full likelihood and taking conditional expectation for the observed likelihood. A local description of the likelihood allows us to extend the result to a more general coarsening observation scheme proposed by Commenges & G\'egout-Petit. The approach is illustrated by considering models for dementia, institutionalization and death.


Diffusion versus absorption in semilinear parabolic equations

Abstract: We study the limit, when $k\to\infty$, of the solutions $u=u_{k}$ of (E) $\prt_{t}u-\Delta u+ h(t)u^q=0$ in $\BBR^N\ti (0,\infty)$, $u_{k}(.,0)=k\delta_{0}$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\gw(t)/t}$ where $\gw>0$ satisfies to $\int_{0}^1\sqrt{\gw(t)}t^{-1}dt<\infty$, the limit function $u_{\infty}$ is a solution of (E) with a single singularity at $(0,0)$, while if $\gw(t)\equiv 1$, $u_{\infty}$ is the maximal solution of (E). We examine similar questions for equations such as $\prt_{t}u-\Gd u^m+ h(t)u^q=0$ with $m>1$ and $\prt_{t}u-\Gd u+ h(t)e^{u}=0$.


Capacitary representations of positive solutions of semilinear parabolic equations

Abstract: We give a global bilateral estimate on the maximal solution $\bar u_F$ of $ \prt_tu-\Delta u+u^q=0$ in $\BBR^N\times (0,\infty)$, $q>1$, $N\geq 1$, which vanishes at $t=0 $ on the complement of a closed subset $F\subset \BBR^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q'}$. We deduce from this estimate that $\bar {u}_F$ is $\sigma$-moderate in Dynkin's sense.


Boundary singularities of solutions of N-harmonic equations with absorption

Abstract: We study the boundary behaviour of solutions $u$ of $-\Delta_{N}u+ |u|^{q-1}u=0$ in a bounded smooth domain $\Omega\subset\mathbb R^{N}$ subject to the boundary condition $u=0$ except at one point, in the range $q>N-1$. We prove that if $q\geq 2N-1$ such a $u$ is identically zero, while, if $N-1<q<2N-1$, $u$ inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.


A more intuitive definition of limit

Abstract: Limit can be defined by two axioms: 1. Strict inequality between limits implies, ultimately, strict inequality between functions. 2. For constant functions limit is trivial. How can basic results on convergence be derived from these axioms? In this paper we propose two answers: a) at the most elementary level- add two more axioms, b) at somewhat higher level, do it in three steps, and, in our forthcoming paper "Axiomatic definition of limit", a third answer- c) do it neater - in an abstract framework, where only order relations are present.


The singularities of the principal component of the Hilbert scheme of points

Abstract: We show that the principal component of the Hilbert scheme of 9 points in C^8 is not Cohen-Macaulay.


Morita Theory For Derived Categories: A Bicategorical Perspective

Abstract: We present a bicategorical perspective on derived Morita theory for rings, DG algebras, and spectra. This perspective draws a connection between Morita theory and the bicategorical Yoneda Lemma, yielding a conceptual unification of Morita theory in derived and bicategorical contexts. This is motivated by study of Rickard's theorem for derived equivalences of rings and of Morita theory for ring spectra, which we present in Sections 2 and 4. Along the way, we gain an understanding of the barriers to Morita theory for DG algebras and give a conceptual explanation for the counterexample of Dugger and Shipley.


Partial Crossed Product of a group G vs Crossed Product of S(G)

Abstract: In this work we present a new definition to the Partial Crossed Product by actions of inverse semigroups in a C^*-algebra, without using the covariant representations as Sieben did in [5]. Also we present an isomorphism between the partial crossed products by partial actions of a group G and the partial crossed product by actions of S(G), an inverse semigroup associated to G introduced by Exel in [2].


Gradient estimates for $u_t=\Delta F(u)$ on manifolds and some Liouville-type theorems

Abstract: In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $$u_t=\Delta F(u),$$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): $$u_t=\Delta (u^p),\qquad p>0,$$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of $p$ than that for Aronson-B\'enilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yaus celebrated Liouville theorem for positive harmonic functions.


In\'egalit\'es de Poincar\'e cin\'etiques

Abstract: In this note we prove Poincar\'e type inequalities for a family of kinetic equations. We apply this inequality to the variational solution of a linear kinetic model.


Amenability properties of the centres of group algebras

Abstract: Let G be a locally compact group, and ZL1(G) be the centre of its group algebra. We show that when $G$ is compact ZL1(G) is not amenable when G is either nonabelian and connected, or is a product of infinitely many finite nonabelian groups. We also, study, for some non-compact groups G, some conditions which imply amenability and hyper-Tauberian property, for ZL1(G).


The tropical analogue of polar cones

Abstract: We study the max-plus or tropical analogue of the notion of polar: the polar of a cone represents the set of linear inequalities satisfied by its elements. We establish an analogue of the bipolar theorem, which characterizes all the inequalities satisfied by the elements of a tropical convex cone. We derive this characterization from a new separation theorem. We also establish variants of these results concerning systems of linear equalities.


Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer

Abstract: We extend in several directions invariant theory results of Chevalley, Shephard and Todd, Mitchell and Springer. Their results compare the group algebra for a finite reflection group with its coinvariant algebra, and compare a group representation with its module of relative coinvariants. Our extensions apply to arbitrary finite groups in any characteristic.


A boundary value problem for minimal Lagrangian graphs

Abstract: Let \Omega and \tilde{\Omega} be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: \Omega \to \tilde{\Omega} such that the graph \Sigma = \{(x,f(x)): x \in \Omega\} is a minimal Lagrangian submanifold of \mathbb{R}^n \times \mathbb{R}^n.


Spectral Curves for Almost-Complex Tori in $ S ^6 $

Abstract: To each non-isotropic almost-complex immersion of a 2-torus into $ S ^ 6 $ we associate an algebraic curve, called the spectral curve, and a linear flow in the intersection of two Prym varieties on this spectral curve. We show that generically the spectral curve is smooth and compute the dimension of the moduli space of such curves and of the torus in which the eigenline bundles lie.


Embedded Associated Primes of Powers of Square-free Monomial Ideals

Abstract: An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all natural numbers t. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than beta_1, where beta_1 is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If in addition I fails to have the packing property, then embedded primes of I^t do occur when t=beta_1 +1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornu\'ejols.


Multiplicative functional for reflected Brownian motion via deterministic ODE

Abstract: We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.


Bijective Proofs of Identities from Colored Binary Trees

Abstract: This note provide bijective proofs of two combinatorial identities involving generalized Catalan number $C_{m,5}(n)={m\over 5n+m}{5n+m\choose n}$ recently proposed by Sun.


Pullback Attractors for the Non-autonomous FitzHugh-Nagumo System on Unbounded Domains

Abstract: The existence of a pullback attractor is established for the singularly perturbed FitzHugh-Nagumo system defined on the entire space $R^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero.


Constructing mean curvature 1 surfaces in $H^3$ with irregular ends

Abstract: With the developments of the last decade on complete constant mean curvature 1 (CMC 1) surfaces in the hyperbolic 3-space $H^3$, many examples of such surfaces are now known. However, most of the known examples have regular ends. (An end is irregular, resp. regular, if the hyperbolic Gauss map of the surface has an essential singularity, resp. at most a pole, there.) There are some known surfaces with irregular ends, but they are all either reducible or of infinite total curvature. (The surface is reducible if and only if the monodromy of the secondary Gauss map can be simultaneously diagonalized.) Up to now there have been no known complete irreducible CMC 1 surfaces in $H^3$ with finite total curvature and irregular ends. The purpose of this paper is to construct countably many 1-parameter families of genus zero CMC 1 surfaces with irregular ends and finite total curvature, which have either dihedral or Platonic symmetries. For all the examples we produce, we show that they have finite total curvature and irregular ends. For the examples with dihedral symmetry and the simplest example with tetrahedral symmetry, we show irreducibility. Moreover, we construct a genus one CMC 1 surface with four irregular ends, which is the first known example with positive genus whose ends are all irregular.


Birational geometry and localisation of categories

Abstract: This version corrects a wrong proof of Proposition 6.3.2 and simplifies the exposition in Section 6.


On q-deformed gl(l+1)-Whittaker function III

Abstract: We identify q-deformed gl(l+1)-Whittaker functions with a specialization of Macdonald polynomials. This provides a representation of q-deformed gl(l+1)-Whittaker functions in terms of Demazure characters of affine Lie algebra \hat{gl(l+1)}. We also define a system of dual Hamiltonians for q-deformed gl(l+1)-Toda chains and give a new integral representation for q-deformed gl(l+1)-Whittaker functions. Finally an expression of q-deformed gl(l+1)-Whittaker function as a matrix element of a quantum torus algebra is derived.


Contractions of low-dimensional nilpotent Jordan algebras

Abstract: In this paper we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among them. In particular, we prove that J2 and J3 are irreducible and that J4 is the union of the Zariski closures of two rigid Jordan algebras.


The Solvability and Subellipticity of Systems of Pseudodifferential Operators

Abstract: The paper studies the local solvability and subellipticity for square systems of principal type. These are the systems for which the principal symbol vanishes of first order on its kernel. For systems of principal type having constant characteristics, local solvability is equivalent to condition (PSI) on the eigenvalues, see arXiv:0801.4043. This is a condition on the sign changes of the imaginary part of the eigenvalue along the oriented bicharacteristics of the real part. In the generic case when the principal symbol does not have constant characteristics, condition (PSI) is not sufficient, not invariant and in general not well defined. Instead we study systems which are quasi-symmetrizable, these systems have natural invariance properties and are of principal type. We prove that quasi-symmetrizable systems are locally solvable. We also study the subellipticity of quasi-symmetrizable systems in the case when principal symbol vanishes of finite order along the bicharacteristics. In order to prove subellipticity, we assume that the principal symbol has the approximation property, which implies that there are no transversal bicharacteristics.


Period problems for mean curvature one surfaces in $H^3$ (with application to surfaces of low total curvature)

Abstract: We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute curvature, which is the total absolute curvature of the dual CMC 1 surface. Here we discuss results on both notions (proven in two other papers by the authors), and we introduce new results as well.


Some linear and nonlinear integral inequalities on time scales in two independent variables

Abstract: We establish some linear and nonlinear integral inequalities of Gronwall-Bellman-Bihari type for functions with two independent variables on general time scales. The results are illustrated with examples, obtained by fixing the time scales to concrete ones. An estimation result for the solution of a partial delta dynamic equation is given as an application.


Logarithmic bounds on Sobolev norms for time-dependent linear Schr\"odinger equations

Abstract: We prove that in 1-D the growth of Sobolev norms for time-dependent linear Schr\"odinger equations is at most logarithmic in time for any (fixed) potential which is analytic (or Gevrey). Recently it was proven in [N] that almost surely the Sobolev norms are unbounded, which indicates that the log is almost surely necessary. In [W], the author showed that the Sobolev norms remain bounded for an explicit time periodic potential. This is in the exceptional set in the sense of [N]. The present paper together with [N, W] give a rather complete picture of time dependent linear Schr\"odinger equations on the circle.


An Intrinsic Impulse Observability Criterion for Descriptor System

Abstract: Analyzing the order of unobservable impulse in descriptor system leads to a new testing criterion for impulse observability, both the statement and the proof of which use only the original system data.


On the generic and typical ranks of 3-tensors

Abstract: We study the generic and typical ranks of 3-tensors of dimension l x m x n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l,m,n not greater than 14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l=m, n=(m-1)^2+1, m=3,4,..., which have at least two typical ranks.