Stability of equilibria for the $\mathfrak{so}(4)$ free rigid body

Abstract: The stability for all generic equilibria of the Lie-Poisson dynamics of the $\mathfrak{so}(4)$ rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of $\mathfrak{so}(n)$ are equilibrium points for the rigid body dynamics. In the case of $\mathfrak{so}(4)$ there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit give three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in $\mathfrak{so}(3)$. In addition to these coordinate type Cartan equilibria there are others that come in curves.


Function Spaces Related to the Dirichlet Space

Abstract: We present results about spaces of holomorphic functions associated to the classical Dirichlet space. The spaces we consider have roles similar to the roles of $H^{1}$ and $BMO$ in the Hardy space theory and we emphasize those analogies.


A uniqueness result on ordinary differential equations with singular coefficients

Abstract: We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the results.


Quantum Exchangeable Sequences of Algebras

Abstract: We extend the notion of quantum exchangeability, introduced by K\"ostler and Speicher in arXiv:0807.0677, to sequences (\rho_1,\rho_2,...c) of homomorphisms from an algebra C into a noncommutative probability space (A,\phi), and prove a free de Finetti theorem: an infinite quantum exchangeable sequence (\rho_1,\rho_2,...c) is freely independent and identically distributed with respect to a conditional expectation. As a corollary we obtain a free analogue of the Hewitt Savage zero-one law. As in the classical case, the theorem fails for finite sequences. We give a characterization of finite quantum exchangeable sequences, which can be viewed as a noncommutative analogue of sampling without replacement. We then give an approximation to how far a finite quantum exchangeable sequence is from being freely independent with amalgamation.


The projectively resolving of some classes over a direct product of rings

Abstract: In this paper, we study the resolving of $\mathcal{SGP}(-)$ and $\mathcal{SGF}(-)$, the classes of all strongly Gorenstein projective and flat modules respectively, over a direct product of commutative rings.


SK1 for graded division algebras

Abstract: The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\SK$ for generic abelian crossed products.


Priority Arguments and Epsilon Substitutions

Abstract: Kreisel has observed that the termination proof for Hilbert's epsilon-substitution method bears a resemblance to the priority arguments used in recursion theory. We make this precise by proving the termination using a framework for priority arguments due to Lerman and Lempp.


On maximal Subgroups of the multiplicative group of a division algebra

Abstract: The question of existence of a maximal subgroup in the multiplicative group D* of a division algebra D finite dimensional over its center F is investigated. We prove that if D* has no maximal subgroup, then deg(D) is not a power of 2, F^{*2} is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F.


Generalized Moonshine I: Genus zero functions

Abstract: We introduce a notion of Hecke-monicity for functions on certain moduli spaces associated to torsors of finite groups over elliptic curves, and show that it implies strong invariance properties under linear fractional transformations. Specifically, if a weakly Hecke-monic function has algebraic integer coefficients and a pole at infinity, then it is either a holomorphic genus-zero function invariant under a congruence group or of a certain degenerate type. As a special case, we prove the same conclusion for replicable functions of finite order, which were introduced by Conway and Norton in the context of monstrous moonshine. As an application, we introduce a class of Lie algebras with group actions, and show that the characters derived from them are weakly Hecke-monic. When the Lie algebras come from chiral conformal field theory in a certain sense, then the characters form holomorphic genus-zero functions invariant under a congruence group.


SK1 of Azumaya algebras over hensel pairs

Abstract: Let A be an Azumaya algebra of constant rank n^2 over a Hensel pair (R,I) where R is a semilocal ring with n invertible in R. Then the reduced Whitehead group SK(A) coincides with its reduction SK(A/IA).


Kodaira-Iitaka Dimension on a Normal Prime Divisor

Abstract: This paper was inspired by work by T. Peternell, M. Schneider and A.J. Sommese on the Kodaira dimension of subvarieties. In it I find a relation between the Kodaira-Iitaka dimension of a divisor on a normal variety and that of related divisors on an irreducible normal subvariety of codimension one. The main result may be stated in a simplified form as: For $X$ a complete normal variety, $Y \sub X$ an irreducible complete normal divisor and $\sL$ an invertible sheaf on $X$, there exist integers $n_1 > 0, n_2 \geq 0$ for which $\kappa(X,\sL) - 1 \leq \kappa(Y,\sL^{n_1}(-n_2Y)|_Y)$, where, if $Y$ is not a fixed component of large tensor powers of $\sL$, we may take $n_1 >> n_2$. This has implications for Kodaira-Iitaka dimension on a subvariety of any codimension.


Fixed points of compositions of earthquakes

Abstract: Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichm\"uller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichm\"uller space. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic AdS manifold. The proof uses some estimates from the geometry of those AdS manifolds.


Iterated destabilizing modifications for vector bundles with connection

Abstract: Given a vector bundle with integrable connection $(V,\nabla)$ on a curve, if $V$ is not itself semistable as a vector bundle then we can iterate a construction involving modification by the destabilizing subobject to obtain a Hodge-like filtration $F^p$ which satisfies Griffiths transversality. The associated graded Higgs bundle is the limit of $(V,t\nabla)$ under the de Rham to Dolbeault degeneration. We get a stratification of the moduli space of connections, with as minimal stratum the space of opers. The strata have fibrations whose fibers are Lagrangian subspaces of the moduli space.


Coarse dynamics and fixed point property

Abstract: We investigate the fixed point property of the group actions on a coarse space and its Higson corona. We deduce the coarse version of Brouwer's fixed point theorem.


A rank-based selection with cardinal payoffs and a cost of choice

Abstract: A version of the secretary problem is considered. The ranks of items, whose values are independent, identically distributed random variables $X_1,X_2,...,X_n$ from a uniform distribution on $[0; 1]$, are observed sequentially by the grader. He has to select exactly one item, when it appears, and receives a payoff which is a function of the unobserved realization of random variable assigned to the item diminished by some cost. The methods of analysis are based on the existence of an embedded Markov chain and use the technique of backward induction. The result is a generalization of the selection model considered by Bearden(2006). The asymptotic behaviour of the solution is also investigated.


Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution

Abstract: Consider a random sample from a bivariate distribution function $F$ in the max-domain of attraction of an extreme-value distribution function $G$. This $G$ is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of $F$. A major issue in multivariate extreme-value theory is the estimation of the spectral measure $\Phi_p$ with respect to the $L_p$ norm. For every $p\in[1,\infty]$, a nonparametric maximum empirical likelihood estimator is proposed for $\Phi_p$. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case studies illustrate how to implement the methods in practice.


Krull dimension of types in a class of first-order theories

Abstract: We study a class of first-order theories whose complete quantifier-free types with one free variable either have a trivial positive part or are isolated by a positive quantifier-free formula--plus a few other technical requirements. The theory of vector spaces and the theory fields are examples. We prove the amalgamation property and the existence of a model-companion. We show that the model-companion is strongly minimal. We also prove that the length of any increasing sequence of prime types is bounded, so every formula has finite Krull dimension.


Linear independence over tropical semirings and beyond

Abstract: We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrization of the max-plus algebra, establishing properties of linear spaces, linear systems, and matrices over the symmetrized max-plus algebra. In parallel we develop some general technique to prove combinatorial and polynomial identities for matrices over semirings that we illustrate by a number of examples.


An Effective Lower Bound for Group Complexity of Finite Semigroups and Automata

Abstract: The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, \textit{Complexity of finite semigroups}, Annals of Mathematics (2) \textbf{88} (1968), 128--160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, \textit{Algebraic theory of machines, {I}: {P}rime decomposition theorem for finite semigroups and machines}, Transactions of the American Mathematical Society \textbf{116} (1965), 450--464. Here we provide an effective lower bound for group complexity.


A deconvolution approach to estimation of a common shape in a shifted curves model

Abstract: This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature.


Canonical connection on quasi-Kaehler manifolds with Norden metric

Abstract: We study the geometry of the canonical connection on a quasi-Kaehler manifold with Norden metric. We consider the cases when the canonical connection has Kaehler curvature tensor and parallel torsion, and derive conditions for an isotropic-Kaehler manifold. We give the relation between the canonical connection, the B-connection, and the connection with totally skew-symmetric torsion on quasi-Kaehler manifolds with Norden metric.


Quintic surfaces with maximum and other Picard numbers

Abstract: This paper investigates the Picard numbers of quintic surfaces. We give the first example of a complex quintic surface in IP^3 with maximum Picard number 45. We also investigate its arithmetic and determine the zeta function. Similar techniques are applied to produce quintic surfaces with several other Picard numbers that have not been achieved before.


Universal convex coverings

Abstract: In every dimension $d\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\mathcal U_d$ of $\mathbb R^d$ such that the following holds: $\mathcal C+\mathcal U_d=\mathbb R^d$ for every convex set $\mathcal C\subset \mathbb R^d$ of volume at least $v_d$ and $\mathcal U_d$ contains at most $\log(r)^{d-1}r^d$ points at distance at most $r$ from the origin, for every large $r$.


\'Equidistribution et diff\'erentiabilit\'e

Abstract: We propose a criterion of equidistribution by the differentiability of certain arithmetic invariants. Combined with the slope method and the asymptotic measures, this criterion gives a new "conceptual" proof to equidistribution results originally obtained via the variation principle.


On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

Abstract: We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure cental limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.


Efficient covariance estimation for asynchronous noisy high-frequency data

Abstract: We focus on estimating the integrated covariance of log-price processes in the presence of market microstructure noise. We construct an efficient unbiased estimator for the quadratic covariation of two It\^{o} processes in the case where high-frequency asynchronous discrete returns under market microstructure noise are observed. This estimator is based on synchronization and multi-scale methods and attains the optimal rate of convergence. A Monte Carlo study analyzes the finite sample size characteristics of our estimator.


Long-time behavior in scalar conservation laws

Abstract: We consider the long-time behavior of the entropy solution of a first-order scalar conservation law on a Riemannian manifold. In the case of the Torus, we show that, under a weak property of genuine non-linearity of the flux, the solution converges to its average value in $L^{p}$, $1\leq p<+\infty$. We give a partial result in the general case.


Geometry of the analytic loop group

Abstract: We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity with non trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of the quantum affine algebra (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters, are parameterized by certain G-bundles on an elliptic curve.


Collective Property of Numbers and Its Mathematical Refutation

Abstract: A number has the "collective" property if the number is the greatest lower bound of a bounded, strictly decreasing sequence on the real line. We prove that numbers with the collective property constitute an empty set.


Harmonics for Deformed Steenrod Operators

Abstract: We explore in this paper the spaces of common zeros of several deformations of Steenrod operators.


Simultaneous linearization of holomorphic germs in presence of resonances

Abstract: Let $f_1, ..., f_m$ be $m\ge 2$ germs of biholomorphisms of $\C^n$, fixing the origin, with $(\d f_1)_O$ diagonalizable and such that $f_1$ commutes with $f_h$ for any $h=2,..., m$. We prove that, under certain arithmetic conditions on the eigenvalues of $(\d f_1)_O$ and some restrictions on their resonances, $f_1, ..., f_m$ are simultaneously holomorphically linearizable if and only if there exists a particular complex manifold invariant under $f_1,..., f_m$.


On the transition to the normal phase for superconductors surrounded by normal conductors

Abstract: For a cylindrical superconductor surrounded by a normal material, we discuss transition to the normal phase of stable, locally stable and critical configurations. Associated with those phase transitions, we define critical magnetic fields and we provide a sufficient condition for which those critical fields coincide. In particular, when the conductivity ratio of the superconducting and the normal material is large, we show that the aforementioned critical magnetic fields coincide, thereby proving that the transition to the normal phase is sharp. One key-ingredient in the paper is the analysis of an elliptic boundary value problem involving `transmission' boundary conditions. Another key-ingredient involves a monotonicity result (with respect to the magnetic field strength) of the first eigenvalue of a magnetic Schroedinger operator with discontinuous coefficients.


Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions

Abstract: We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier--Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic "galloping", or "pulsating", solutions, in agreement with physical and numerical observation. In the process, we determine readily numerically verifiable stability and bifurcation conditions in terms of an associated Evans function, and obtain the first complete nonlinear stability result for strong detonations of the reacting Navier--Stokes equations, in the limit as amplitude (hence also heat release) goes to zero. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works.


A Remark on Gelfand Duality for Spectral Triples

Abstract: We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as morphisms and a suitable "metric" category of spectral triples over commutative pre-C*-algebras. We also construct an embedding of a "quotient" of the category of spectral triples introduced in arXiv:math/0502583v1 into the latter metric category. Finally we discuss a further related duality in the case of orientation and spin-preserving maps between manifolds of fixed dimension.


Semi log resolution

Abstract: The aim of this note is to discuss resolution theorems that are useful in the study of semi log canonical varieties.


A Spectral Theorem for Imprimitivity C*-bimodules

Abstract: After recalling in detail some basic definitions on Hilbert C*-bimodules, Morita equivalence and imprimitivity, we discuss a spectral reconstruction theorem for imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras and consider some of its applications in the theory of commutative full C*-categories.


A Horizontal Categorification of Gelfand Duality

Abstract: In the setting of C*-categories, we provide a definition of "spectrum" of a commutative full C*-category as a one-dimensional unital saturated Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gelfand duality theorem generalizing the usual Gelfand duality between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Although many of the individual ingredients that appear along the way are well-known, the somehow unconventional way we "glue" them together seems to shed some new light on the subject.


Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation

Abstract: We study the time dependent Hartree equation in the continuum, the semidiscrete, and the fully discrete setting. We prove existence-uniqueness, regularity, and approximation properties for the respective schemes, and set the stage for a controlled numerical computation of delicate nonlinear and nonlocal features of the Hartree dynamics in various physical applications.


Quotients by finite equivalence relations

Abstract: This note studies the existence of quotients by finite set theoretic equivalence relations. May 18: Substantial revisions with a new appendix by C. Raicu


Bias correction in a multivariate normal regression model with general parameterization

Abstract: This paper develops a bias correction scheme for a multivariate normal model under a general parameterization. In the model, the mean vector and the covariance matrix share the same parameters. It includes many important regression models available in the literature as special cases, such as (non)linear regression, errors-in-variables models, and so forth. Moreover, heteroscedastic situations may also be studied within our framework. We derive a general expression for the second-order biases of maximum likelihood estimates of the model parameters and show that it is always possible to obtain the second order bias by means of ordinary weighted lest-squares regressions. We enlighten such general expression with an errors-in-variables model and also conduct some simulations in order to verify the performance of the corrected estimates. The simulation results show that the bias correction scheme yields nearly unbiased estimators. We also present an empirical ilustration.


Stochastic Vs Worst-case Condition Numbers

Abstract: We compare Stochastic and Worst-case condition numbers and loss of precision for general computational problems. We show an upper bound for the ratio of Worst-case condition number to the Stochastic condition number of order O(sqrt m). We show an upper bound for the difference between the Worst-case loss of precision and the Stochastic loss of precision of order O(ln m). The results hold if the perturbations are measured norm-wise or componentwise.


On the annealed large deviation rate function for a multi-dimensional random walk in random environment

Abstract: We derive properties of the rate function in Varadhan's (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations.


A remark on a conjecture of Hain and Looijenga

Abstract: After recalling the various tautological algebras of the moduli space of curves and some of its partial compactifications and stating several well-known results and conjectures concerning these algebras, we prove that the natural extension to the case of pointed curves of a 1996 conjecture of Hain and Looijenga is true if and only if two of the stated conjectures are true.


Exponential decay for solutions to semilinear damped wave equation

Abstract: This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyaponuv function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in an article of Gazzola and Squassina.


Time management in a Poisson fishing model

Abstract: The aim of the paper is to extend the model of "fishing problem". The simple formulation is following. The angler goes to fishing. He buys fishing ticket for a fixed time. There are two places for fishing at the lake. The fishes are caught according to renewal processes which are different at both places. The fishes' weights and the inter-arrival times are given by the sequences of i.i.d. random variables with known distribution functions. These distributions are different for the first and second fishing place. The angler's satisfaction measure is given by difference between the utility function dependent on size of the caught fishes and the cost function connected with time. On each place the angler has another utility functions and another cost functions. In this way, the angler's relative opinion about these two places is modeled. For example, on the one place better sort of fish can be caught with bigger probability or one of the places is more comfortable. Obviously our angler wants to have as much satisfaction as possible and additionally he have to leave the lake before the fixed moment. Therefore his goal is to find two optimal stopping times in order to maximize his satisfaction. The first time corresponds to the moment, when he eventually should change the place and the second time, when he should stop fishing. These stopping times have to be less than the fixed time of fishing. The value of the problem and the optimal stopping times are derived.


Grid Diagrams, Braids, and Contact Geometry

Abstract: We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.


Ultraspherical type generating functions for orthogonal polynomials

Abstract: We characterize, up to a conjecture, probability distributions of all order finite moments having ultraspherical type generating functions for orthogonal polynomials.


On the period map for prime Fano threefolds of degree 10

Abstract: We prove that the deformations of a smooth complex Fano threefold X with Picard number 1, index 1, and degree 10, are unobstructed. The differential of the period map has two-dimensional kernel. We construct two two-dimensional components of the fiber of the period map through X: one is isomorphic to the variety of conics in X, modulo an involution, another is birationally isomorphic to a moduli space of semistable rank-2 torsion-free sheaves on X, modulo an involution. The threefolds corresponding to points of these components are obtained from X via conic and line (birational) transformations.


Regularized Multivariate Regression for Identifying Master Predictors with Application to Integrative Genomics Study of Breast Cancer

Abstract: In this paper, we propose a new method remMap -- REgularized Multivariate regression for identifying MAster Predictors -- for fitting multivariate response regression models under the high-dimension-low-sample-size setting. remMap is motivated by investigating the regulatory relationships among different biological molecules based on multiple types of high dimensional genomic data. Particularly, we are interested in studying the influence of DNA copy number alterations on RNA transcript levels. For this purpose, we model the dependence of the RNA expression levels on DNA copy numbers through multivariate linear regressions and utilize proper regularizations to deal with the high dimensionality as well as to incorporate desired network structures. Criteria for selecting the tuning parameters are also discussed. The performance of the proposed method is illustrated through extensive simulation studies. Finally, remMap is applied to a breast cancer study, in which genome wide RNA transcript levels and DNA copy numbers were measured for 172 tumor samples. We identify a tran-hub region in cytoband 17q12-q21, whose amplification influences the RNA expression levels of more than 30 unlinked genes. These findings may lead to a better understanding of breast cancer pathology.


Monge-Amp\`ere equations in big cohomology classes

Abstract: We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of a closed positive current in given big cohomology class. The solution is shown to have minimal singularities in the sense of Demailly if the measure is regular enough. These results are combined with a fixed point argument to construct singular Kaehler-Einstein volume forms with minimal singularities on varieties of general type.


Intersections on tropical moduli spaces

Abstract: This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties $\mathbb{P}^1$, $\mathbb{P}^2$, $\mathbb{P}^1 \times \mathbb{P}^1$ and with Psi-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\mathbb{P}^2$ in Markwig-Rau-2008). Our approach uses tropical intersection theory and can unify and simplify some parts of the existing tropical enumerative geometry (for rational curves).


Predictability in Spatially Extended Systems with Model Uncertainty

Abstract: Macroscopic models for spatially extended systems under random influences are often described by stochastic partial differential equations (SPDEs). Some techniques for understanding solutions of such equations, such as estimating correlations, Liapunov exponents and impact of noises, are discussed. They are relevant for understanding predictability in spatially extended systems with model uncertainty, for example, in physics, geophysics and biological sciences. The presentation is for a wide audience.


Symplectic forms and cohomology decomposition of almost complex 4-manifolds

Abstract: For any compact almost complex manifold $(M,J)$, the last two authors defined two subgroups $H_J^+(M)$, $H_J^-(M)$ of the degree 2 real de Rham cohomology group $H^2(M, \mathbb{R})$ in arXiv:0708.2520. These are the sets of cohomology classes which can be represented by $J$-invariant, respectively, $J$-anti-invariant real $2-$forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of $H^2(M, \mathbb{R})$. This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.


On multivariate Newton-like inequalities

Abstract: We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\bf Strongly Log-Concave} entire functions, generalizing {\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$ variables is called {\bf Strongly Log-Concave} if the function $(\partial x_1)^{c_1}...(\partial x_m)^{c_m} f$ is either zero or $\log((\partial x_1)^{c_1}...(\partial x_m)^{c_m} f)$ is concave on $R_{+}^{m}$. We start with yet another point of view (via {\it propagation}) on the standard univarite (or homogeneous bivariate) {\bf Newton Inequlities}. We prove analogues of (univariate) {\bf Newton Inequlities} in the (multivariate) {\bf Strongly Log-Concave} case. One of the corollaries of our new Newton(like) inequalities is the fact that the support $supp(f)$ of a {\bf Strongly Log-Concave} entire function $f$ is discretely convex ($D$-convex in our notation). The proofs are based on a natural convex relaxation of the derivatives $Der_{f}(r_1,...,r_m)$ of $f$ at zero and on the lower bounds on $Der_{f}(r_1,...,r_m)$, which generalize the {\bf Van Der Waerden-Falikman-Egorychev} inequality for the permanent of doubly-stochastic matrices. A few open questions are posed in the final section.


A New Family of Covariate-Adjusted Response Adaptive Designs and their Asymptotic Properties

Abstract: It is often important to incorporating covariate information in the design of clinical trials. In literature, there are many designs of using stratification and covariate-adaptive randomization to balance on certain known covariate. Recently Zhang, Hu, Cheung and Chan (2007) have proposed a family of covariate-adjusted response-adaptive (CARA) designs and studied their asymptotic properties. However, these CARA designs often have high variabilities. In this paper, we propose a new family of covariate-adjusted response-adaptive (CARA) designs. We show that the new designs have smaller variabilities and therefore more efficient.


The Penrose Transform in the Split Signature

Abstract: A version of the Penrose transform is introduced in the split signature. It relates the cohomological data with supports on the open subsets of the complex 3-projective space and kernel of differential operators on the (real) Grassmannian of 2-planes in the Euclidean 4-space. As an example we derive a cohomological interpretation of the so-called X-ray transform. Furthermore, a cohomological realization of the so-called "minimal" representation of SL(4,R) is given. We also present the split Penrose transform in split instanton backgrounds.


The Gaussian approximation for multi-color generalized Friedman's urn model

Abstract: The Friedman's urn model is a popular urn model which is widely used in many disciplines. In particular, it is extensively used in treatment allocation schemes in clinical trials. In this paper, we prove that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a multi-color generalized Friedman's urn model with non-homogeneous generating matrices. The Gaussian process is a solution of a stochastic differential equation. This Gaussian approximation together with the properties of the Gaussian process is important for the understanding of the behavior of the urn process and is also useful for statistical inferences. As an application, we obtain the asymptotic properties including the asymptotic normality and the law of the iterated logarithm for a multi-color generalized Friedman's urn model as well as the randomized-play-the-winner rule as a special case.


Immigrated urn models - asymptotic properties and applications

Abstract: Urn models have been widely studied and applied in both scientific and social science disciplines. In clinical studies, the adoption of urn models in treatment allocation schemes has been proved to be beneficial to both researchers, by providing more efficient clinical trials, and patients, by increasing the likelihood of receiving the better treatment. In this paper, we propose a new and general class of immigrated urn (IMU) models that incorporates the immigration mechanism into the urn process. Theoretical properties are developed and the advantages of the IMU models are discussed. In general, the IMU models have smaller variabilities than the classical urn models, yielding more powerful statistical inferences in applications. Illustrative examples are presented to demonstrate the wide applicability of the IMU models. The proposed IMU framework, including many popular classical urn models, not only offers a unify perspective for us to comprehend the urn process, but also enables us to generate several novel urn models with desirable properties.


Multi-color Randomly Reinforced Urn for Adaptive Designs

Abstract: This paper is withdrawn


Finite planar emulators for K_{4,5} - 4K_2 and K_{1,2,2,2} and Fellows' Conjecture

Abstract: In 1988 Fellows conjectured that if a finite, connected graph admits a finite planar emulator, then it admits a finite planar cover. We construct a finite planar emulator for K_{4,5} - 4K_2. Archdeacon showed that K_{4,5} - 4K_2 does not admit a finite planar cover; thus K_{4,5} - 4K_2 provides a counterexample to Fellows' Conjecture. It is known that Negami's Planar Cover Conjecture is true if and only if K_{1,2,2,2} admits no finite planar cover. We construct a finite planar emulator for K_{1,2,2,2}. The existence of a finite planar cover for K_{1,2,2,2} is still open.


Minimal Euclidean representations of graphs

Abstract: A simple graph G is said to be representable in a real vector space of dimension m if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values a or b, with distance a if the vertices are adjacent and distance b otherwise. The Euclidean representation number of G is the smallest dimension in which G is representable. In this note, we bound the Euclidean representation number of a graph using multiplicities of the eigenvalues of the adjacency matrix. We also give an exact formula for the Euclidean representation number using the main angles of the graph.


Curvature estimates for submanifolds with prescribed Gauss image and mean curvature

Abstract: We study that the $n-$graphs defining by smooth map $f:\Om\subset \ir{n}\to \ir{m}, m\ge 2,$ in $\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le\f{C}{R^2}$$ under the dimension limitations and the Gauss image restrictions. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le C R^{-a}\sup_{D_{2R}(x)}(2-\De_f)^{-(\f{3}{2}+\f{1}{s})}, \qquad s=\min(m, n)$$ with $a<1$ under the condition $$\De_f=[\text{det}(\de_{ij}+\sum_\a\pd{f^\a}{x^i}\pd{f^\a}{x^j})]^{\f{1}{2}}<2.$$ If the image under the Gauss map is contained in a geodesic ball of the radius $\f{\sqrt{2}}{4}\pi$ in $\grs{n}{m}$ we also derive corresponding estimates.


A Lieb-Thirring inequality for singular values

Abstract: Let $A$ and $B$ be positive semidefinite matrices. We investigate the conditions under which the Lieb-Thirring inequality can be extended to singular values. That is, for which values of $p$ does the majorisation $\sigma(B^p A^p) \prec_w \sigma((BA)^p)$ hold, and for which values its reversed inequality $\sigma(B^p A^p) \succ_w \sigma((BA)^p)$.


Loewner chains on the universal covering space of a Riemann surface

Abstract: Let R be a hyperbolic Riemann surface with boundary $\partial R$ and suppose that $\gamma:[0,T]\to R\cup\partial R$ is a simple curve growing from the boundary of R. By lifting $R_{t}=R\setminus \gamma(0,t]$ to the universal covering space of R (which we assume is the upper half-plane $\mathbb{H}=\{z\in\mathbb{C}:Im[z]>0\}$) via the covering map $\pi:\mathbb{H}\to R$, we can define a family of simply-connected domains $D_{t}=\pi^{-1}(R_{t})$. For each $t\in[0,T]$, suppose that f_{t} is a conformal map of \mathbb{H} onto D_{t} such that f(z,t)=f_{t}(z) is differentiable almost everywhere in (0,T) with respect to t. In this paper, we will derive a differential equation that describes how f(z,t) evolves in time t. This should be viewed as an extension of the Loewner differential equation to curves on Riemann surfaces with boundary. The motivation of this paper is the desire to extend Schramm's stochastic Loewner evolution (SLE) to multiply-connected domains and Riemann surfaces.


Optimal three-ball inequalities and quantitative uniqueness for the Stokes system

Abstract: In this paper we study the local behavior of a solution to the Stokes system with singular coefficients. One of the main results is the bound on the vanishing order of a nontrivial solution to the Stokes system, which is a quantitative version of the strong unique continuation property. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial \emph{optimal} three-ball inequalities. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution to the Stokes system from those three-ball inequalities.


Common fixed point theorems for occasionally weakly compatible maps

Abstract: In this paper, we establish a common fixed point theorem for two pairs of occasionally weakly compatible single and set-valued maps satisfying a strict contractive condition in a metric space. Our result extends many results existing in the literature as those of Aliouche and Popa [15-20]. Also we establish another common fixed point theorem for four owc single and set-valued maps of Gregu% \v{s} type which generalizes the results of Djoudi and Nisse, Pathak, Cho, Kang and Madharia and we end our work by giving a third theorem which extends the results given by Elamrani & Mehdaoui and Mbarki.


The Imprimitive Faithful Complex Characters of the Schur Covers of the Symmetric and Alternating Groups

Abstract: Using combinatorics and character theory, we determine the imprimitive faithful complex characters, i.e., the irreducible faithful complex characters which are induced from proper subgroups, of the Schur covers of the symmetric and alternating groups. Furthermore, for every imprimitive character we establish all its minimal block stabilizers. As a corollary, we also determine the monomial faithful characters of the Schur covers.


Semiring Properties of Heyting Algebras

Abstract: The relationship between Heyting algebras (HA) and semirings is explored. A new class of HAs called Symmetric Heyting algebras (SHAs) is proposed, and a necessary condition on SHAs to be consider semirings is given. We define a new mathematical family called Heyting structures, which are similar to semirings, but with Heyting-algebra operators in place of the usual arithmetic operators usually seen in semirings. The impact of the zero-sum free property of semirings on Heyting structures is shown as also the condition under which it is possible to extend one Heyting structure to another. It is also shown that the union of two or more sets forming Heyting structures is again a Heyting structure, if the operators on the new structure are suitably derived from those of the component structures. The analysis also provides a sufficient condition such that the larger Heyting structure satisfying a monotony law implies that the ones forming the union do so as well.


On Kummer 3-folds

Abstract: We investigate a generalization of Kummer construction, as introduced in a recent paper by M. Andreatta and J.A. Wisniewski. The aim of this work is to classify 3-dimensional Kummer varieties by computing their Poincare polynomials.


The duration problem with multiple exchanges

Abstract: We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the $m$--choice duration problem, there exists a sequence (s1,s2,...,sm) of critical numbers such that, whenever there remain k choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit an equivalence between the duration problem and the classical best-choice secretary problem. A simple recursive formula is given for calculating the critical numbers when the number of objects tends to infinity. Extensions are made to models involving an acquisition or replacement cost.


Quasi-complete homogeneous contact manifold associated to a cubic form

Abstract: Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic form is the determinant of a simple cubic Jordan algebra.


Overdetermined 2D Systems Invariant in One Direction and Their Transfer Functions

Abstract: In this work we develop a theory of Vessels. This object arises in the study of overdetermined 2D systems invariant in one of the variables, which are usually called time invariant. To each overdetermined time invariant 2D systems there is associated a vessel, which is a collection of system operators satisfying certain relations and vise versa. Such an invariance forces the theory of vessels to resemble a constant (classical) 1D case and as a result many notions are naturally redefined and most theorems are reproved in this setting. The notion of transfer function and its connection to the overdetermined 2D time invariant system (and the corresponding vessel) is one of the topics of this work. It is well known that multiplicative structure of a transfer function of a 1D system is closely connected to the decomposition of the state space into invariant subspaces of the state operator and we generalize this result to a wider class of functions. This class (denoted by $\boldsymbol {\mathcal I}$) arises as a class of transfer functions, which intertwine solutions of ODEs with spectral parameters. At the end we present solution of factorization problems for finite dimensional case.


Frameworks, Symmetry and Rigidity

Abstract: Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R^d. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear in CAD, body-pin frameworks, hybrid systems of distance constrained objects and infinite bar-joint frameworks. This leads to generalised forms of the Fowler-Guest character formula together with counting rules in terms of counts of symmetry-fixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partition-derived frameworks.


A non-abelian Stickelberger theorem

Abstract: Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring Z_(p)[G] that annihilates the p-part of the class group of L.


Beyond Totally Reflexive Modules and Back

Abstract: Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein rings in terms of total acyclicity of complexes.


Sur les automorphismes reguliers de C^k

Abstract: We show the uniqueness for the measure of maximal entropy for regular automorphisms of C^k.


Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel

Abstract: This paper is devoted to numerical simulation of a charged particle beam submitted to a strong oscillating electric field. For that, we consider a two-scale numerical approach as follows: we first recall the two-scale model which is obtained by using two-scale convergence techniques; then, we numerically solve this limit model by using a backward semi-lagrangian method and we propose a new mesh of the phase space which allows us to simplify the solution of the Poisson's equation. Finally, we present some numerical results which have been obtained by the new method, and we validate its efficiency through long time simulations.


Examples of stable embedded minimal spheres without area bounds

Abstract: The author proves that there is an open non empty set of metrics on any 3-manifold such that there exists a family of stably embedded minimal 2-spheres whose area is unbounded. This generalizes the work of T. Colding and W. Minicozzi who have shown an analogous result for the torus and B. Dean who showed the positive genus case.


Convergent finite element methods for compressible barotropic Stokes systems

Abstract: We propose finite element methods for compressible barotropic Stokes systems. We state convergence results for these methods and outline their proofs. The principal tools of the proofs are higher integrability estimates for the discrete density, equations for the discrete effective viscous flux, and renormalized formulations of the numerical method for the density equation.


The number of orientable small covers over cubes

Abstract: We count orientable small covers over cubes. We also get estimates for $O_n/R_n$, where $O_n$ is the number of orientable small covers and $R_n$ is the number of all small covers over an $n$-cube up to the Davis-Januszkiewicz equivalence.


Birational geometry of Fano double spaces of index two

Abstract: We study birational geometry of Fano varieties, realized as double covers $\sigma\colon V\to {\mathbb P}^M$, $M\geq 5$, branched over generic hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fiber space on $V$ are the pencils-subsystems of the free linear system $|-\frac12 K_V|$. The groups of birational and biregular self-maps of the variety $V$ coincide.


Depth Zero Representations of Nonlinear Covers of $p$-adic Groups

Abstract: We generalize the methods of Moy-Prasad, in order to define and study the genuine depth zero representations of some nonlinear covers of reductive groups over $p$-adic local fields. In particular, we construct all depth zero supercuspidal representations of the metaplectic group $Mp_{2n}$ over a $p$-adic field of odd residue characteristic.


A Probablistic Origin for a New Class of Bivariate Polynomials

Abstract: We present here a probabilistic approach to the generation of new polynomials in two discrete variables. This extends our earlier work on the 'classical' orthogonal polynomials in a previously unexplored direction, resulting in the discovery of an exactly soluble eigenvalue problem corresponding to a bivariate Markov chain with a transition kernel formed by a convolution of simple binomial and trinomial distributions. The solution of the relevant eigenfunction problem, giving the spectral resolution of the kernel, leads to what we believe to be a new class of orthogonal polynomials in two discrete variables. Possibilities for the extension of this approach are discussed.


Scaling limits for symmetric Ito-Levy processes in random medium

Abstract: We are concerned with scaling limits of the solutions to stochastic differential equations with stationary coefficients driven by Poisson random measures and Brownian motions. We state an annealed convergence theorem, in which the limit exhibits a diffusive or superdiffusive behavior, depending on the integrability properties of the Poisson random measure


Convergent Interpolation to Cauchy Integrals over Analytic Arcs

Abstract: We consider multipoint Pad\'e approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-smooth non-vanishing density, then the diagonal multipoint Pad\'e approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Pad\'e approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parametrization of the arc.


An introduction to motivic zeta functions of motives

Abstract: It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually multiple zeta values (Broadhurst, Kreimer). In order to determine, at least heuristically, whether this is the case in concrete instances, the philosophy of motives - more specifically, the theory of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting points of algebraic varieties related to $I$ modulo sufficiently many primes $p$ and checking that the number of points varies polynomially in $p$. On the other hand, Kapranov has introduced a new "zeta function", the role of which is precisely to "interpolate" between zeta functions of reductions modulo different primes $p$. In this survey, we outline this circle of ideas and some of their recent developments.


Slope filtrations

Abstract: Many slope filtrations occur in algebraic geometry, asymptotic analysis, ramification theory, p-adic theories, geometry of numbers... These functorial filtrations, which are indexed by rational (or sometimes real) numbers, have a lot of common properties. We propose a unified abstract treatment of slope filtrations, and survey how new ties between different domains have been woven by dint of deep correspondences between different concrete slope filtrations.


Optimal stopping of a risk process when claims are covered immediately

Abstract: The optimal stopping problem for the risk process with interests rates and when claims are covered immediately is considered. An insurance company receives premiums and pays out claims which have occured according to a renewal process and which have been recognized by them. The capital of the company is invested at interest rate $\alpha\in\Re^{+}$, the size of claims increase at rate $\beta\in\Re^{+}$ according to inflation process. The immediate payment of claims decreases the company investment by rate $\alpha_1$. The aim is to find the stopping time which maximizes the capital of the company. The improvement to the known models by taking into account different scheme of claims payment and the possibility of rejection of the request by the insurance company is made. It leads to essentially new risk process and the solution of optimal stopping problem is different.


Experimental designs for multiple-level responses, with application to a large-scale educational intervention

Abstract: Educational research often studies subjects that are in naturally clustered groups of classrooms or schools. When designing a randomized experiment to evaluate an intervention directed at teachers, but with effects on teachers and their students, the power or anticipated variance for the treatment effect needs to be examined at both levels. If the treatment is applied to clusters, power is usually reduced. At the same time, a cluster design decreases the probability of contamination, and contamination can also reduce power to detect a treatment effect. Designs that are optimal at one level may be inefficient for estimating the treatment effect at another level. In this paper we study the efficiency of three designs and their ability to detect a treatment effect: randomize schools to treatment, randomize teachers within schools to treatment, and completely randomize teachers to treatment. The three designs are compared for both the teacher and student level within the mixed model framework, and a simulation study is conducted to compare expected treatment variances for the three designs with various levels of correlation within and between clusters. We present a computer program that study designers can use to explore the anticipated variances of treatment effects under proposed experimental designs and settings.


Foliations on hypersurfaces in holomorphic symplectic manifolds

Abstract: Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction $\sigma|_Y$ of the holomorphic symplectic form induces a rank one foliation on Y. We investigate situations where this foliation has compact leaves; in such cases we obtain a space of leaves Y/F which has dimension 2n-2 and admits a holomorphic symplectic form.


On Galois Correspondence and Non-Commutative Martingales

Abstract: The subject of this thesis is Galois correspondence for von Neumann algebras and its interplay with non-commutative probability theory. After a brief introduction to representation theory for compact groups, in particular to Peter-Weyl theorem, and to operator algebras, including von Neumann algebras, automorphism groups, crossed products and decomposition theory, we formulate first steps of a non-commutative version of probability theory and introduce non-abelian analogues of stochastic processes and martingales. The central objects are a von Neumann algebra $\Ma$ and a compact group $\Gr$ acting on $\Ma$, for which we give in three consecutive steps, i.e. for inner, spatial and general automorphism groups one-to-one correspondences between subgroups of $\Gr$ and von Neumann subalgebras of $\Ma$. Furthermore, we identify non-abelian martingales in our approach and prove for them a convergence theorem.


Sectorial forms and degenerate differential operators

Abstract: If $a$ is a densely defined sectorial form in a Hilbert space which is possibly not closable, then we associate in a natural way a holomorphic semigroup generator with $a$. This allows us to remove in several theorems of semigroup theory the assumption that the form is closed or symmetric. Many examples are provided, ranging from complex sectorial differential operators, to Dirichlet-to-Neumann operators and operators with Robin or Wentzell boundary conditions.


Spaces of algebraic maps from real projective spaces into complex projective spaces

Abstract: We study the homotopy types of spaces of algebraic (rational) maps from real projective spaces into complex projective spaces. In a previous paper we have shown that the inclusion of the first space into the second one is a homotopy equivalence. In this paper we prove that the homotopy types of the terms of the natural "degree" filtration approximate closer and closer the homotopy type of the space of continuous maps and obtain bounds that describe the closeness of the approximation in terms of the degree. Moreover, we compute the homotopy groups of the spaces in low dimensions.


Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere

Abstract: Global quantization of pseudo-differential operators on compact Lie groups is introduced relying on the representation theory of the group rather than on expressions in local coordinates. Operators on the 3-dimensional sphere and on group SU(2) are analysed in detail. A new class of globally defined symbols is introduced giving rise to the usual Hormander's classes of operators $\Psi^m(G)$, $\Psi^m(S^3)$ and $\Psi^m(SU(2))$. Properties of the new class and symbolic calculus are analysed. Properties of symbols as well as $L^2$-boundedness and Sobolev $L^2$--boundedness of operators in this global quantization are established on general compact Lie groups.


The Siegel modular forms of genus 2 with the simplest divisor

Abstract: We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference "Black holes, Black Rings and Modular Forms" (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998.


Reflected Backward SDEs with General Jumps

Abstract: In the first part of this paper we give a solution for the one-dimensional reflected backward stochastic differential equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. The reflecting process is right continuous with left limits (rcll for short) whose jumps are arbitrary. We first prove existence and uniqueness of the solution for a specific coefficient in using a method based on a combination of penalization and the Snell envelope theory. To show the result in the general framework we use a fixed point argument in an appropriate space. The second part of the paper is related to BSDEs with two reflecting barriers. Once more we prove the existence and uniqueness of the solution of the BSDE.


Overdetermined conservative 2D Systems, Invariant in One Direction and a Generalization of Potapov's theorem

Abstract: This work is a direct continuation of the authors work arXiv:0812.3779v1. A special case of conservative overdetermined time invariant 2D systems is developed and studied. Defining transfer function of such a systems we obtain a class CI of inner functions $S(\lambda,t_2)$, which are identity for $\lambda=\infty$, satisfy certain regularity assumptions and intertwines solutions of ODEs with a spectral parameter $\lambda$. Using translation model, we prove that every function in the class CI can be realized as a transfer function of a certain vessel. The highlight of this theory is a generalization Potapov's theorem, which gives a very special formula for such a function in the form of multiplication of Blacke-Potapov products, corresponding to the discrete spectrum of certain system operator $A_1(t_2)$ and of multiplicative integral, corresponding to the continuous spectrum of $A_1(t_2)$. This theorem is proved under a slightly more restrictive assumptions, then the development of the whole theory. Namely, we suppose that the derivative of the transfer function is a continuous function of $t_2$ for almost all $\lambda$. At the last part zero/pole interpolation problem for matrix functions in CI is considered and a realization theorem of such functions appeared in arXiv:0812.3779v1 (theorem 8.1) is reproved. Hermitian case is also analyzed and the corresponding realization theorem is proved.


Revisiting R\'ev\'esz's stochastic approximation method for the estimation of a regression function

Abstract: In a pioneer work, R\'ev\'esz (1973) introduces the stochastic approximation method to build up a recursive kernel estimator of the regression function $x\mapsto E(Y|X=x)$. However, according to R\'ev\'esz (1977), his estimator has two main drawbacks: on the one hand, its convergence rate is smaller than that of the nonrecursive Nadaraya-Watson's kernel regression estimator, and, on the other hand, the required assumptions on the density of the random variable $X$ are stronger than those usually needed in the framework of regression estimation. We first come back on the study of the convergence rate of R\'ev\'esz's estimator. An approach in the proofs completely different from that used in R\'ev\'esz (1977) allows us to show that R\'ev\'esz's recursive estimator may reach the same optimal convergence rate as Nadaraya-Watson's estimator, but the required assumptions on the density of $X$ remain stronger than the usual ones, and this is inherent to the definition of R\'ev\'esz's estimator. To overcome this drawback, we introduce the averaging principle of stochastic approximation algorithms to construct the averaged R\'ev\'esz's regression estimator, and give its asymptotic behaviour. Our assumptions on the density of $X$ are then usual in the framework of regression estimation. We prove that the averaged R\'ev\'esz's regression estimator may reach the same optimal convergence rate as Nadaraya-Watson's estimator. Moreover, we show that, according to the estimation by confidence intervals point of view, it is better to use the averaged R\'ev\'esz's estimator rather than Nadaraya-Watson's estimator.


Braids, Shuffles and Symmetrizers

Abstract: Multiplicative analogues of the shuffle elements of the braid group rings are introduced; in local representations they give rise to certain graded associative algebras (b-shuffle algebras). For the Hecke and BMW algebras, the (anti)-symmetrizers have simple expressions in terms of the multiplicative shuffles. The (anti)-symmetrizers can be expressed in terms of the highest multiplicative 1-shuffles (for the Hecke and BMW algebras) and in terms of the highest additive 1-shuffles (for the Hecke algebras). The spectra and multiplicities of eigenvalues of the operators of the multiplication by the multiplicative and additive 1-shuffles are examined.


On the algebraic index for riemannian \'etale groupoids

Abstract: In this paper we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a riemannian \'etale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for riemannian \'etale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dim torus.


Holomorphic Cartan geometries and Calabi--Yau manifolds

Abstract: We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous d\'emontrons que les seules vari\'et\'es projectives de Calabi--Yau qui poss\`edent des g\'eom\'etrie holomorphes de Cartan sont les vari\'et\'es ab\'eliennes.)


On the validity of Chapman-Enskog expansions for shock waves with small strength

Abstract: We justify a Chapman-Enskog expansion for discontinuous solutions of hyperbolic conservation laws containing shock waves with small strength. Precisely, we establish pointwise uniform estimates for the difference between the traveling waves of a relaxation model and the traveling waves of the corresponding diffusive equations determined by a Chapman-Enskog expansion procedure to first- or second-order.


Geometric quantization for proper moment maps: the Vergne conjecture

Abstract: We establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a noncompact symplectic manifold with proper moment map.


A note on badly approximable affine forms and winning sets

Abstract: We give a sketch for an alternative proof of a recent result by J. Tseng.


Stability in the L1 norm via a linearization method for nonlinear hyperbolic systems

Abstract: We discuss the existence and uniqueness of discontinuous solutions to adjoint problems associated with nonlinear hyperbolic systems of conservation laws. By generalizing the Haar method for Glimm-type approximations to hyperbolic systems, we establish that entropy solutions depend continuously upon their initial data in the natural L1 norm.


Repetitions in beta-integers

Abstract: Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this paper, we study the maximal possible repetition of the same motif occurring in beta-integers -- one dimensional models of quasicrystals. We are interested in beta-integers realizing only a finite number of distinct distances between neighboring elements. In such a case, the problem may be reformulated in terms of combinatorics on words as a study of the index of infinite words coding beta-integers. We will solve a particular case for beta being a quadratic non-simple Parry number.


Diminishing functionals for nonclassical entropy solutions selected by kinetic relations

Abstract: We consider nonclassical entropy solutions to scalar conservation laws with concave-convex flux functions, whose set of left- and right-hand admissible states across undercompressive shocks is selected by a kinetic function \phi. We introduce a new definition for the (generalized) strength of classical and nonclassical shocks, allowing us to propose a generalized notion of total variation functional. Relying only upon the natural assumption that the composite function \phi o \phi is uniformly contracting, we prove that the generalized total variation of front-tracking approximations is non-increasing in time, and we conclude with the existence of nonclassical solutions to the initial-value problem. We also propose two definitions of generalized interaction potentials which are adapted to handle nonclassical entropy solutions and we investigate their monotonicity properties. In particular, we exhibit an interaction functional which is globally non-increasing along a splitting-merging interaction pattern.


Cordes characterization for pseudodifferential operators with symbols valued on a noncommutative C*-algebra

Abstract: Given a separable unital C*-algebra A, let E denote the Banach-space completion of the A-valued Schwartz space on Rn with norm induced by the A-valued inner product $<f,g>=\int f(x)^*g(x) dx$. The assignment of the pseudodifferential operator B=b(x,D) with A-valued symbol b(x,\xi) to each smooth function with bounded derivatives b defines an injective mapping O, from the set of all such symbols to the set of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E. It is known that O is surjective if A is commutative. In this paper, we show that, if O is surjective for A, then it is also surjective for the algebra of k-by-k matrices with entries in A.


Bounds on ternary cyclotomic coefficients

Abstract: We present a new bound on $A = \max_n |a_{pqr}(n)|$, where $a_{pqr}(n)$ are the coefficients of a ternary cyclotomic polynomial. We also prove that two consecutive coefficients of such a polynomial differ by at most one.


Stochastically stable globally coupled maps with bistable thermodynamic limit

Abstract: We study systems of globally coupled interval maps, where the identical individual maps have two expanding, fractional linear, onto branches, and where the coupling is introduced via a parameter - common to all individual maps - that depends in an analytic way on the mean field of the system. We show: 1) For the range of coupling parameters we consider, finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. 2) For the same range of parameters, the self-consistent Perron-Frobenius operator which captures essential aspects of the corresponding infinite-size system (arising as the limit of the above when the system size tends to infinity), undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium.


On powers of Stirling matrices

Abstract: The powers of matrices with Stirling number-coefficients are investigated. It is revealed that the elements of these matrices have a number of properties of the ordinary Stirling numbers. Moreover, "higher order" Bell, Fubini and Eulerian numbers can be defined. Hence we give a new interpretation for E. T. Bell's iterated exponential integers. In addition, it is worth to note that these numbers appear in combinatorial physics, in the problem of the normal ordering of quantum field theoretical operators.


Functionally recursive rings of matrices-Two examples

Abstract: We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 + log(2)/log(a) where a=1/2(1+sqrt(5)).


Generalization of n-ary Nambu algebras and beyond

Abstract: The aim of this paper is to introduce $n$-ary Hom-algebra structures generalizing the $n$-ary algebras of Lie type enclosing $n$-ary Nambu algebras, $n$-ary Nambu-Lie algebras, $n$-ary Lie algebras, and $n$-ary algebras of associative type enclosing $n$-ary totally associative and $n$-ary partially associative algebras. Also, we provide a way to construct examples starting from an $n$-ary algebra and an $n$-ary algebras endomorphism. Several examples could be derived using this process.


Separation theorems for compact Hausdorff foliations

Abstract: We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.


On the Supremum of Certain Families of Stochastic Processes

Abstract: We consider a family of stochastic processes $\{X_t^\epsilon, t \in T\}$ on a metric space $T$, with a parameter $\epsilon \downarrow 0$. We study the conditions under which \lim_{\e \to 0} \P \Big(\sup_{t \in T} |X_t^\e| < \delta \Big) =1 when one has the \textit{a priori} estimate on the modulus of continuity and the value at one point. We compare our problem to the celebrated Kolmogorov continuity criteria for stochastic processes, and finally give an application of our main result for stochastic intergrals with respect to compound Poisson random measures with infinite intensity measures.


Default times, non arbitrage conditions and change of probability measures

Abstract: In this paper we give a financial justification, based on non arbitrage conditions, of the $(H)$ hypothesis in default time modelling. We also show how the $(H)$ hypothesis is affected by an equivalent change of probability measure. The main technique used here is the theory of progressive enlargements of filtrations.


Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices

Abstract: Natural join of $di-bigraphs$ that is directed biparted graphs and their corresponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their $Hasse$ digraphs called $KoDAGs$. $KoDAGs$ are special orderable directed acyclic graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. $KoDAGs$ appear to be distinguished family of $Ferrers$ digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called $di-bicliques$. These digraphs serve to represent faithfully corresponding relations of arbitrary arity so that all relations of arbitrary arity are their subrelations. Being this $chain -way$ complete if compared with kompletne $Kuratowski$ bipartite graphs their $DAG$ denotation is accompanied with the letter $K$ in front of descriptive abbreviation $oDAG$. The way to join bipartite digraphs of binary into $multi-ary$ relations is the natural join operation either on relations or their digraph representatives. This natural join operation is denoted here by $\os$ symbol deliberately referring to the direct sum $\oplus$ of adjacency matrices as it becomes the case for disjoint $di-bigraphs$.


Gr\"obner bases for operads

Abstract: We define a new monoidal category on collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gr\"obner bases for operads, and present operadic versions of Bergman's Diamond Lemma and Buchberger's algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck's PBW criterion of Koszulness for (symmetric) quadratic operads.


Fast finite-energy planes in symplectizations and applications

Abstract: We define the notion of fast finite-energy planes in the symplectization of a closed 3-dimensional energy level $M$ of contact type. We use them to construct special open book decompositions of $M$ when the contact structure is tight and induced by a (non-degenerate) dynamically convex contact form. The obtained open books have disk-like pages that are global surfaces of section for the Hamiltonian dynamics. Let $S \subset \R^4$ be the boundary of a smooth, strictly convex, non-degenerate and bounded domain. We show that a necessary and sufficient condition for a closed Hamiltonian orbit $P\subset S$ to be the boundary of a disk-like global surface of section for the Hamiltonian dynamics is that $P$ is unknotted and has self-linking number -1.


Non-Hausdorff groupoids

Abstract: We present examples of non-Hausdorff, etale, essentially principal groupoids for which three results, known to hold in the Hausdorff case, fail. These results are: (A) the subalgebra of continuous functions on the unit space is maximal abelian within the reduced groupoid C*-algebra, (B) every nonzero ideal of the reduced groupoid C*-algebra has a nonzero intersection with the subalgebra of continuous functions on the unit space, and (C) the open support of a normalizer is a bissection.


Bijections on two variations of noncrossing partitions

Abstract: We find bijections on 2-distant noncrossing partitions, 12312-avoiding partitions, 3-Motzkin paths, UH-free Schr{\"o}der paths and Schr{\"o}der paths without peaks at even height. We also give a direct bijection between 2-distant noncrossing partitions and 12312-avoiding partitions.


K-stability of constant scalar curvature polarization

Abstract: In this paper, we shall show that a polarized algebraic manifold is K-stable if the polarization class admits a Kaehler metric of constant scalar curvature. This generalizes the results of Chen-Tian, Donaldson and Stoppa. (Parts of the arguments are based on a forthcoming paper "A stronger concept of K-stability." )


Fluctuations of the empirical quantiles of independent Brownian motions

Abstract: We consider $n$ independent, identically distributed one-dimensional Brownian motions, $B_j(t)$, where $B_j(0)$ has a rapidly decreasing, smooth density function $f$. The empirical quantiles, or pointwise order statistics, are denoted by $B_{j:n}(t)$, and we are interested in a sequence of quantiles $Q_n(t) = B_{j(n):n}(t)$, where $j(n)/n \to \alpha \in (0,1)$. This sequence converges in probability in $C[0,\infty)$ to $q(t)$, the $\alpha$-quantile of the law of $B_j(t)$. Our main result establishes the convergence in law in $C[0,\infty)$ of the fluctuation processes $F_n = n^{1/2}(Q_n - q)$. The limit process $F$ is a centered Gaussian process and we derive an explicit formula for its covariance function. We also show that $F$ has many of the same local properties as $B^{1/4}$, the fractional Brownian motion with Hurst parameter $H = 1/4$. For example, it is a quartic variation process, it has H\"older continuous paths with any exponent $\gamma < 1/4$, and (at least locally) it has increments whose correlation is negative and of the same order of magnitude as those of $B^{1/4}$.


Asymptotics of the Norm of Elliptical Random Vectors

Abstract: In this paper we consider elliptical random vectors X in R^d,d>1 with stochastic representation A R U where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A is a given matrix. The main result of this paper is an asymptotic expansion of the tail probability of the norm of X derived under the assumption that R has distribution function is in the Gumbel or the Weibull max-domain of attraction.


Regularity properties of the distance function to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and application in Riemannian geometry

Abstract: Given a continuous viscosity solution of a Dirichlet-type Hamilton-Jacobi equation, we show that the distance function to the conjugate locus which is associated to this problem is locally semiconcave on its domain. It allows us to provide a simple proof of the fact that the distance function to the cut locus associated to this problem is locally Lipschitz on its domain. This result, which was already an improvement of a previous one by Itoh and Tanaka, is due to Li and Nirenberg. Finally, we give applications of our results in Riemannian geometry. Namely, we show that the distance function to the conjugate locus on a Riemannian manifold is locally semiconcave. Then, we show that if a Riemannian manifold is a deformation of the round sphere, then all its tangent nonfocal domains are strictly uniformly convex.


Invariably Suboptimal - An attempt to improve the voting rules of Treaties of Nice and Lisbon

Abstract: We investigate the voting rules in the Council of the European Union. It is known that the current system, according to the Treaty of Nice, and the voting system proposed in the Lisbon treaty both strongly deviate from the square root law by Penrose. This is known to be the ideal voting rule under certain assumptions. In 2004 Slomczynski and Zyczkowski designed a voting system, now known as the Jagiellonian Compromise. It satisfies the square root law with very high accuracy. Each member state in this system obtains a voting weight proportional to the square root of the population. Additionally the quota is fixed in such a way that the voting power of each country is also proportional to the square root of the population. In this paper we investigate to which extent a change of the quota in the Treaty of Nice and the Treaty of Lisbon may bring the voting power closer to the ideal square root distribution. Our computations show that even with optimal quota both systems are way off the ideal power distribution.


Semisimplicity criteria for irreducible Hopf algebras in positive characteristic

Abstract: We prove that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple, if and only if it is commutative and semisimple, if and only if the restricted Lie algebra $P(H)$ of the primitives is a torus. This generalizes Hochschild's theorem on restricted Lie algebras, and also generalizes Demazure and Gabriel's and Sweedler's results on group schemes, in the special but essential situation with finiteness assumption added.


A class of nonlinear elliptic boundary value problems

Abstract: In this paper second order elliptic boundary value problems on bounded domains $\Omega\subset\dR^n$ with boundary conditions on $\partial\Omega$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework. For a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure. In the special case of rational Nevanlinna or Riesz-Herglotz functions on the boundary the solution operator is obtained in an explicit form in the product Hilbert space $L^2(\Omega)\oplus (L^2(\partial\Omega))^m$, which is a natural generalization of known results on $\lambda$-linear elliptic boundary value problems and $\lambda$-rational boundary value problems for ordinary second order differential equations.


Koszul duality for stratified algebras II. Standardly stratified algebras

Abstract: We give a complete picture of the interaction between Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras imposing the condition that standard filtrations of projective modules are finite, and develop the tilting theory for such algebras. Under the assumption of existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both Ringel and Koszul duals belong to the class, and that these two dualities on this class commute.


Interpolation Theorems for Self-adjoint Operators

Abstract: We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator $L$, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where $L$ is a uniformly elliptic operator or a Schr\"odinger operator with electro-magnetic potential.


On the 0-dimensional cusps of the Kahler moduli of a K3 surface

Abstract: Let S be a projective K3 surface. It is proved that the 0-dimensional cusps of the Kahler moduli of S are in one-to-one correspondence with the twisted Fourier-Mukai partners of S. This leads to a counting formula for the 0-dimensional cusps of the Kahler moduli. Applications to rational maps between K3 surfaces with large Picard numbers are given. When the Picard number of S is 1, the bijective correspondence is calculated explicitly.


The Enumeration of Vertex Induced Subgraphs with respect to the Number of Components

Abstract: Inspired by the study of community structure in connection networks, we introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of $Q(G;x,y)$ using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate $Q(G;x,y)$ to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimination polynomial introduced by I. Averbouch, B. Godlin and J.A. Makowsky (2008). We show that $Q(G;x,y)$ is vertex reconstructible in the sense of Kelly and Ulam, discuss its use in computing residual connectedness reliability. Finally we show that the computation of $Q(G;x,y)$ is $\sharp \mathbf{P}$-hard, but Fixed Parameter Tractable for graphs of bounded tree-width and clique-width.


Kan replacement of simplicial manifolds

Abstract: We establish a functor $Kan$ from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.


Backward errors and linearizations for palindromic matrix polynomials

Abstract: We derive computable expressions of structured backward errors of approximate eigenelements of *-palindromic and *-anti-palindromic matrix polynomials. We also characterize minimal structured perturbations such that approximate eigenelements are exact eigenelements of the perturbed polynomials. We detect structure preserving linearizations which have almost no adverse effect on the structured backward errors of approximate eigenelements of the *-palindromic and *-anti-palindromic polynomials.


On Borel complexity of the isomorphism problems for graph-related classes of Lie algebras and finite p-groups

Abstract: We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the isomorphism problem for graphs is harder than the two latter isomorphism problems in the sense of Borel reducibility. A computable analogue of Borel reducibility was introduced by S. Coskey, J.D. Hamkins, and R. Miller. A relation of the isomorphism problem for undirected graphs to the well-known problem of classifying pairs of matrices over a field (up to similarity) is also studied.


Yet Another Poincare's Polyhedron Theorem

Abstract: Poincar\'e's Polyhedron Theorem is a widely known valuable tool in constructing manifolds endowed with a prescribed geometric structure. It is one of the few criteria providing discreteness of groups of isometries. This work contains a version of Poincar\'e's Polyhedron Theorem that is applicable to constructing fibre bundles over surfaces and also suits geometries of nonconstant curvature. Most conditions of the theorem, being as local as possible, are easy to verify in practice.


Some convergence results on quadratic forms for random fields and application to empirical covariances

Abstract: Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. We obtain a non central limit theorem under a minimal integrability condition, which allows isotropic and anisotropic models. We apply our limit theorems and those of Ginovian (99) to obtain the asymptotic behavior of the empirical covariances of Gaussian fields, which is a particular example of quadratic forms. We show that it is possible to obtain a Gaussian limit when the spectral density is not in $L^2$. Therefore the dichotomy observed in dimension $d=1$ between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in $d>1$.


Schanuel's conjecture and the exceptional set of $\gamma$-th arithmetic zeta functions

Abstract: In this work, we study the arithmetic nature of the numbers of the form $n^{\g}$, for $n \in \N$ and $\g\in \C$. We also consider a related conjecture and we show that it is a consequence of the unipresent Schanuel's conjecture.


Consistency conditions for brane tilings

Abstract: Given a brane tiling on a torus, we provide a new way to prove and generalise the recent results of Szendroi, Mozgovoy and Reineke regarding the Donaldson-Thomas theory of the moduli space of framed cyclic representations of the associated algebra. Using only a natural cancellation-type consistency condition, we show that the algebras are 3-Calabi-Yau, and calculate Donaldson-Thomas type invariants of the moduli spaces. Two new ingredients to our proofs are a grading of the algebra by the path category of the associated quiver modulo relations, and a way of assigning winding numbers to pairs of paths in the lift of the brane tiling to R^2. These ideas allow us to generalise the above results to all consistent brane tilings on K(pi,1) surfaces. We also prove a converse: no consistent brane tiling on a sphere gives rise to a 3-Calabi-Yau algebra.


Embedding into manifolds with torsion

Abstract: We introduce a class of special geometries associated to the choice of a differential graded algebra contained in \Lambda R^n. We generalize some known embedding results, that effectively characterize the real analytic Riemannian manifolds that can be realized as submanifolds of a Riemannian manifold with special holonomy, to this more general context. In particular, we consider the case of hypersurfaces inside nearly-Kaehler and alpha-Einstein-Sasaki manifolds, proving that the corresponding evolution equations always admit a solution in the real analytic case.


Arithmetic McKay correspondence

Abstract: We propose an arithmetic McKay correspondence which relates suitably defined zeta functions of some Deligne-Mumford stacks to the zeta functions of their crepant resolutions. Some examples are discussed.


Hardy type spaces on certain noncompact manifolds and applications

Abstract: In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X^1(M), X^2(M), ... of new Hardy spaces on M, the sequence Y^1(M/, Y^2(M), ... of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for purely imaginary powers of the Laplace-Beltrami operator and for more general spectral multipliers associated to the Laplace--Beltrami operator L on M. Under the additional condition that the volume of the geodesic balls of radius r is controlled by C r^a e^{2\sqrt{b} r} for some real number a and for all large r, we prove also an endpoint result for first order Riesz transforms D L^{-1/2}. In particular, these results apply to Riemannian symmetric spaces of the noncompact type.


KMS States, Entropy and a Variational Principle for Pressure

Abstract: We want to relate the concepts of entropy and pressure to that of KMS states for $C^*$-Algebras. Several different definitions of entropy are known in our days. The one we describe here is quite natural and extends the usual one for Dynamical Systems in Thermodynamic Formalism Theory. It has the advantage of been very easy to be introduced. It is basically obtained from transfer operators (also called Ruelle operators). Later we introduce the concept of pressure as a min-max principle. Finally, we consider the concept of a KMS state as an equilibrium state for a potential (in the context of $C^*$-Algebras) and we show that there is a relation between KMS states for certain algebras coming from a continuous transformation and equilibrium measures.


Structure of the curvature tensor on symplectic spinors

Abstract: We study symplectic manifolds $(M^{2l},\omega)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\nabla$ and admitting a metaplectic structure. Let $\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative $\nabla^S$ associated to the Fedosov connection $\nabla.$ It is known that the space of symplectic spinor valued exterior differential 2-forms, $\Gamma(M,\bigwedge^2T^*M\otimes {\mathcal{S}}),$ decomposes into three invariant spaces with respect to the structure group, which is the metaplectic group $Mp(2l,\mathbb{R})$ in this case. For a symplectic spinor field $\phi \in \Gamma(M,\mathcal{S}),$ we compute explicitly the projections of $R^S\phi \in \Gamma(M,\bigwedge^2T^*M \otimes \mathcal{S})$ onto the three mentioned invariant spaces in terms of the symplectic Ricci and symplectic Weyl curvature tensor fields of the connection $\nabla.$ Using this decomposition, we derive a complex of first order differential operators provided the Weyl tensor of the Fedosov connection is trivial.


A Sliding Blocks Estimator for the Extremal Index

Abstract: In extreme value statistics for stationary sequences, blocks estimators are usually constructed by using disjoint blocks because exceedances over high thresholds of different blocks can be assumed asymptotically independent. In this paper we focus on the estimation of the extremal index which measures the degree of clustering of extremes. We consider disjoint and sliding blocks estimators and compare their asymptotic properties. In particular we show that the sliding blocks estimator is more efficient than the disjoint version and has a smaller asymptotic bias. Moreover we propose a method to reduce its bias when considering sufficiently large block sizes.


Critical points of solutions of degenerate elliptic equations in the plane

Abstract: We study the minimizer u of a convex functional in the plane which is not G\^ateaux-differentiable. Namely, we show that the set of critical points of any C^1-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler-Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense.


Polars of real singular curves

Abstract: Polar varieties have in recent years been used by Bank, Giusti, Heintz, Mbakop, and Pardo, and by Safey El Din and Schost, to find efficient procedures for determining points on all real components of a given non-singular algebraic variety. In this note we review the classical notion of polars and polar varieties, as well as the construction of what we here call reciprocal polar varieties. In particular we consider the case of real affine plane curves, and we give conditions for when the polar varieties of singular curves contain points on all real components.


Generalized Lagrangian mean curvature flow in K\"ahler manifolds that are almost Einstein

Abstract: We introduce the notion of K\"ahler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a K\"ahler manifold that is almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle.


Proving finitely presented groups are large by computer

Abstract: We present a theoretical algorithm which, given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We then implement a practical version of this algorithm using Magma and apply it to a range of presentations. Our main focus is on 2-generator 1-relator presentations where we have a complete picture of largeness if the relator has exponent sum zero in one generator and word length at most 12, as well as when the relator is in the commutator subgroup and has word length at most 18. Indeed all but a tiny number of presentations define large groups. Finally we look at fundamental groups of closed hyperbolic 3-manifolds, where the algorithm readily determines that a quarter of the groups in the Snappea closed census are large.


Weak convergence of the periodic multiplicative Selmer algorithm

Abstract: In order to prove weak convergence of the periodic multiplicative Selmer algorithm we ensure that the periodicity matrix is positive and establish a relation between its entries and eigenvalues. Since we can imply that the limit of these relations exist, we arrive at the desired result.